probabilistic methods in fluids : proceedings of the swansea 2002 workshop : wales, uk, 14-19 april...

Probabilistic Methods in Fluids

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editors:

I M Davies

N Jacob

A Truman

Department of Mathematics University of Wales Swansea

UK

0 Hassan

K Morgan

N P Weatherill

School of Engineering University of Wales Swansea

Proceedings of the Swansea 2002 Workshop

Probabilistic

Methods in

Fluids Wales, UK 14 - 19 April 2002

ye World Scientific L NewJersey London Singapore Hong Kong

Published by

World Scientific Publishing Co. Pte. Ltd.

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PROBABILISTIC METHODS IN FLUIDS Proceedings of the Swansea 2002 Workshop

Copyright 0 2003 by World Scientific Publishing Co. Pte. Ltd.

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Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

IRIMA . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

Participants . . . . . . . . . . . . . . . . . . . . . . . . . xi

Sergio Albeverio and Yana Belopolskaya . . . . . . . . . . . . . 1

Probabilistic Approach to Hydrodynamic Equations

Hakima Bessaih and Franco Flandoli . . . . . . . . . . . . . . 22

A Mean Field Result for 3D Vortex Filaments

Bjorn Bottcher and Niels Jacob . . . . . . . . . . . . . . . . . 35

Remarks on Meixner-type Processes

Zdzistaw Brzeinaak . . . . . . . . . . . . . . . . . . . . . . 48

Some Remarks on It6 and Stratonovich Integration in 2-smooth

Banach Spaces

Tomas Caraballo . . . . . . . . . . . . . . . . . . . . . . . 70 The Long-time Behaviour of Stochastic 2D-Navier-Stokes Equations

Pao-Liu Chow . . . . . . . . . . . . . . . . . . . . . . . . 84

Semilinear Stochastic Wave Equations

Nigel J. Cutland . . . . . . . . . . . . . . . . . . . . . . . 97

Stochastic Navier-Stokes Equations: Loeb Space Techniques & Attractors

Arnaud Debussche . . . . . . . . . . . . . . . . . . . . . 115

The 2D-Navier-Stokes Equations Perturbed by a Delta Correlated Noise

Sergio Albeverio and Benedetta Ferrario . . . . . . . . . . . . 130 Invariant Measures of Lkvy-Khinchine Type for 2D Fluids

Franco Flandoli . . . . . . . . . . . . . . . . . . . . . . . 144

Some Remarks on a Statistical Theory of Turbulent Flows

Christophe Giraud . . . . . . . . . . . . . . . . . . . . . 161

Some Properties of Burgers Turbulence with White Noise

Initial Conditions

V

vi

Yuri E. Gliklikh . . . . . . . . . . . . . . . . . . . . . . . Deterministic Viscous Hydrodynamics via Stochastic Processes on Groups of Diffeomorphisms

Niels Jacob and Aubrey Truman Further Classes of Pseudo-differential Operators Applicable

to Modelling in Finance and Turbulence

Benjamin Jourdain and Tony Lel ihre . . . . . . . . . . . . . Mathematical Analysis of a Stochastic Differential Equation Arising

in the Micro-Macro Modelling of Polymeric Fluids

Hannelore Lisei and Michael Scheutzow . . . . . . . . . . . . On the Dispersion of Sets under the Action of an Isotropic

Brownian Flow

. . . . . . . . . . . . . . .

Aubrey Truman, Chris N. Reynolds and David Williams . . . . . Stochastic Burgers Equation in d-dimensions - A One-dimensional Analysis: Hot and Cool Caustics and Intermittence of Stochastic Turbulence

A m e n Shirikyan . . . . . . . . . . . . . . . . . . . . . . A Version of the Law of Large Numbers and Applications

Maricin SlodiEka . . . . . . . . . . . . . . . . . . . . . . Comprehensive Models for Wells

Enrique Thomann and Mina Ossiander Stochastic Cascades Applied to the Navier-Stokes Equations

Aubrey Truman and Jiang-Lun Wu . . . . . . . . . . . . . . Stochastic Burgers Equation with Lkvy Space-Time White Noise

TushengZhang . . . . . . . . . . . . . . . . . . . . . . . A Comparison Theorem for Solutions of Backward Stochastic

Differential Equations with Two Reflecting Barriers and

Its Applications

Aubrey Truman and Huaizhong Zhao Burgers Equation and the WKB-Langer Asymptotic L2 Approximation of Eigenfunctions and Their Derivatives

. . . . . . . . . . . .

. . . . . . . . . . . . .

179

191

205

224

239

263

272

287

298

324

332

Preface

This volume contains papers presented at the “Probabilistic Methods in

Fluids Workshop” which was hosted by the Department of Mathematics,

University of Wales Swansea between the 14th and l g t h of April 2002.

The aim of the meeting, the first IRIMA workshop, was to bring together

internationally reknowned researchers from the areas of Pure Mathemat-

ics, Applied Mathematics and Engineering to participate in a workshop,

on probabilistic methods for fluids, and through collaboration further the

mathematical understanding of the fundamental problems in this field.

This international workshop successfully allowed leading researchers to

present, reflect upon and discuss their recent work in the probabilistic

modelling of fluids. This field stretches across Pure Mathematics, Applied

Mathematics and Engineering and consequently is ideally placed to benefit

from regularly arranged workshops for collaborative purposes. The Work-

shop mainly concentrated on the understanding of turbulence in stochastic

fluid dynamics, a problem which has numerous applications in science and

engineering and has defied many attempts to success full^ model it. The

workshop bridged a gap between the recent year of activity at the Univer-

sity of Warwick and the year of emphasis at Princeton, which started in

Autumn 2002. As such the workshop ensured that the research momentum

in Britain, in this subject, was maintained.

In this volume probabilistic approaches to hydrodynamic equations are re-

viewed and deterministic viscous hydrodynamics is discussed in terms of

stochastic processes on groups of diffeomorphisms. At the Workshop sig-

nificant progress was made in understanding the intermittence of stochastic

turbulence for Burgers equation and the application of L6vy processes to

the Mathematics of Finance, both of which are represented in the pro-

ceedings. Other noteworthy developments concerned the Strong Law of

Large Numbers and ergodicity of the Gaussian invariant measures for 2-

dimensional Navier-Stokes equations with space-time white noise and pe-

riodic boundary conditions and mean field results for 3-dimensional vortex

filaments. Also, new results are presented on Loeb space techniques and

attractors for stochastic Navier-Stokes equations. The long time behaviour

of stochastic 2-dimensional Navier-Stokes equations is investigated as are

vii

... Vll l

perturbations by delta correlated noise. Burgers turbulence for white noise

initial conditions is discussed in detail and the Cauchy problem for stochas-

tic Burgers equation with L6vy space-time white noise is also examined. A complete mathematical analysis of stochastic differential equations arising

in micro-macro modelling of polymeric fluids is given.

Scientific Organising Committee S. Albeverio, Y.I. Belapolskaya, Z.

Brzezniak, A. Chorin, F. Flandoli, B. Rozovski, A. Truman

We are especially grateful to Zdzislaw Brzezniak for his contribution to

the organisation and success of the workshop, and to Roger Tribe for his

guidance.

finding The workshop was supported by EPSRC grant GR/96545/01 “Probabil-

isitic Methods for Fluids - IRIMA” and we are indebted to EPSRC for

their financial support and advice.

Local Organisation We wish to thank Jane Barham and Janice Lewis for their forebearance

before, during and after the workshop in providing secretarial and adminis-

trative support. We must thank also Bjorn Boettcher, Victoriya Knopova,

Scott Reasons and Chris Reynolds for their contribution towards the suc-

cessful running of the workshop.

Finally, we thank the referees for their important but anonymous contribu-

tion in helping us to finish this volume on time.

I M Davies 0 Hassan

N Jacob K Morgan

A Truman N P Weatherill

University of Wales Swansea, December 2002

International Research Institute in Mathematics and its

Applications

Patron: Sir Michael Atiyah OM, FRS

There is no such thing as Applied Science only the Applications of Science, Henri Poincark

We have established the International Research Institute in Mathematics

and Its Applications, IRIMA, (Sefydliad Ymchwil Rhyngwladol i Fathe-

mateg a’i Chymwysiadau, SYRIFAC) with the aim of conducting a series

of research programmes in Mathematics and its applications to Engineer-

ing and Science. In so doing we aim to accelerate the transfer of modern

Mathematics to Engineering and the Sciences. These programmes should

be seen to be interdisciplinary, with the express intention of providing a

forum for interaction between groups of mathematicians, engineers and sci-

entists, while at the same time preserving the integrity of the Mathematics

being utilised.

The Institute will be based in Swansea and will draw on existing strengths in

Stochastic Processes, Physical Mathematics, Finite Element Methods and

Theoretical Computer Science. Swansea (in the person of Oleg Zienkiewicz

FRS) pioneered the use of Finite Element Methods in Engineering. More

recently, his research group, which includes Profs. Nigel Weatherill, Ken

Morgan FREng and Roger Owen FREng, has done vitally important re-

search work in a number of different application areas, including work on

the European Airbus and Thrust SSC, the supersonic car. Swansea also

has international centres of research excellence in Probability Theory and

ix

X

Applications as represented by the presence of Profs. David Williams FRS,

Leonid Pastur and Aubrey Truman, in Theoretical Computer Science in

Professor John Tucker’s research group and in Theoretical Particle Physics

in the research team of Professor David Olive FRS.

Scientific Advisory Panel Prof. A. Truman (Chair), Prof. S. Albeverio (Bonn), Prof. C. Dafermos

(Brown, Providence RI), Prof. D. Elworthy (Warwick), Dr. N. Jacob,

Prof. R. Mackay FRS (Warwick), Prof. K. Morgan FREng, Prof. D. Olive

FRS, Dr. M. Overhaus (Deutsche Bank AG London), Dr. D.P. Rowse

(BAE Systems), Dr. D. Burridge (Meteorological Office), Prof. R. Owen

FREng., Prof. E. Rees (Edinburgh), Dr. C. Sparrow (IN1 Cambridge and

Warwick), Prof. J. Tucker, Prof. N. Weatherill, Prof. D. Williams FRS

and Prof. 0. Zienkiewicz FRS FREng.

Workshop Participants

Yana Belopolskaya, Department of Mathematics, St. Petersburg University

for Architecture and Civil Engineering

Hakima Bessaih, Dipartimento di Matematica Applicata, UniversitA di Pisa

Bjoern Boettcher, Department of Mathematics, University of Wales

Swansea

Zdzislaw Brzezniak, Department of Mathematics, The University of Hull

Tomiis Caraballo, Dpto.

Facultad de Matemiiticas, Sevilla

Pao-Liu (Paul) Chow, Dept. of Mathematics, Wayne State University

Nigel J. Cutland, Department of Mathematics, The University of Hull

Constantine Dafermos, Division of Applied Mathematics, Brown University

Ian M Davies, Department of Mathematics, University of Wales Swansea

Arnaud Debussche, ENS de Cachan, Bruz

Karl Doppel, Fachbereich Mathematik und Informatik, FU Berlin

Benedetta Ferrario, Institut fur Angewandte Mathematik, Bonn Univer-

sitat

Franco Flandoli, Dipartimento di Matematica, Universitg di Pisa

Mark Freidlin, Dept. of Mathematics, University of Maryland

Christophe Giraud, Laboratoire J.A. Dieudonne, Universite de Nice Sophia-

Anti polis

Yuri E. Gliklikh, Mathematics Faculty, Voronezh State University

Oleg Gulinskii, Moscow Institute of Information Transmission Problems,

Moscow

Oubay Hassan, School of Engineering, University of Wales Swansea

Niels Jacob, Department of Mathematics, University of Wales Swansea

Mark Kelbert, European Business Management School, University of Wales

Swansea

Viktoriya Knopova, Department of Mathematics, University of Wales

Swansea

Ecuaciones Diferenciales y Analisis Numerico,

xi

xii

Vassili Kolokoltsov, Department of Computing and Mathematics, Notting-

ham Trent University

Markus Kraft, Department of Chemical Engineering, University of Cam-

bridge

Sergei Kuksin, Department of Mathematics, Heriot-Watt University

Jose A. Langa-Rosado, Dpto.

merico, Facultad de Matemhticas, Sevilla

Tony Lelievre, CERMICS ENPC, Champs sur Marne

Nikolai Leonenko, School of Mathematics, Cardiff University

Yuhong Li, Department of Mathematics, The University of Hull

Hannelore Lisei, Institut fur Mathematik, Technische Universitat Berlin

Terry Lyons, The Mathematical Institute, University of Oxford

Salah Mohammed, Department of Mathematics, SIU-C Carbondale

Ken Morgan, School of Engineering, University of Wales Swansea

Szymon Peszat, Institute of Mathematics, Polish Academy of Sciences,

Krakow

Scott Reasons, Department of Mathematics, University of Wales Swansea

Chris Reynolds, Department of Mathematics, University of Wales Swansea

James Robinson, Mathematics Institute, University of Warwick

Francesco RUSSO, Institut Galilee, Mathematiques, Universite Paris 13

Michael Scheutzow, Institut fur Mathematik, TU Berlin

RenQ Schilling, Department of Mathematics, University of Sussex

Armen Shirikyan, Department of Mathematics, Heriot-Watt University

Maridn SlodiEka, Department of Mathematical Analysis, Faculty of Engi-

neering, Ghent University

Andrew Stuart, Mathematics Institute, University of Warwick

Enrique Thomann, Department of Mathematics, Oregon State University

Alexander Tokarev, Department of Mathematics, University of Wales

Swansea

Ecuaciones Diferenciales y Analisis Nu-

xiii

Michael Tretyakov, Department of Mathematics and Computer Science,

University of Leicester

Aubrey Truman, Department of Mathematics, University of Wales Swansea

Alexei Tyukov, School of Mathematical Sciences, University of Sussex

Nigel Weatherill, School of Engineering, University of Wales Swansea

David Williams, Department of Mathematics, University of Wales Swansea

Wojbor A. Woyczynski, Department of Statistics, Case Western Reserve

University

Jiang-Lun Wu, Department of Mathematics, University of Wales Swansea

Oleg Zaboronski, Mathematics Institute, University of Warwick

Tusheng Zhang, Department of Mathematics, University of Manchester

Huaizhong Zhao, Department of Mathematical Sciences, Loughborough

University

PROBABILISTIC APPROACH TO HYDRODYNAMIC EQUATIONS

S. ALBEVERIO

Institut fur Angewandte Mathematik, Universitat Bonn, Wegelerstr. 6, D-53115 Bonn,

Germany SFB 256,

Bonn, BiBoS, Bielefeld - Bonn, CERFIM, Locarno and USI (Switzerland)

YA. BELOPOLSKAYA

St. Petersburg State University for Architecture and Civil Engineering, Russia, 198005, St. Petersburg, 2-ja Krasnoarmejskaja 4

We construct diffusion processes associated with the Navier-Stokes system

in R3 and use them to prove the existence and uniquenes of local solution of the Cauchy problem for this system in some functional space.

AMS Subject classification: 60 H 15, 35 Q 30 Key words: Stochastic processes, Navier-Stokes system, probabilistic representa-

tions

1. Diffusion process and the Navier-Stokes equations

Among tremendous number of papers and books devoted to the investiga-

tion of the Navier-Stokes system there is relatively small number of works

with probabilistic background. The very idea to consider the stochastic

process with the drift velocity field subjected to the Navier-Stokes equa-

tion belongs to Nelson l and was presented in his functional analysis of the

finite energy Navier-Stokes flow.

To construct a diffusion process such that the Navier-Stokes equation

can be treated as the backward Kolmogorov equation (BKE) for this process

we consider the stochastic differential equation similar to one studied in '. The difference is in the relation used to define the drift coefficient.

We apply here the probabilistic approach to study the Cauchy problem

for nonlinear PDEs by reducing it to the investigation of stochastic differ-

ential equations with coefficients functionally depending on the distribution

of the SDE solution developed in papers by Dalecky and Belopolskaya 2-3.

1

Notice that among recent works the papers by Busnello and Flandoli,

Busnello, are mostly close to our approach. In these papers the authors

deal with the equation for the circulation of the velocity field and the Biot-

Savart law (instead of the original Navier-Stokes system) and study it by

probabilistic methods.

In the present paper we construct diffusion processes that can be used

for the probabilistic representation of the velocity field and the pressure

itself.

To this end we consider the Cauchy problem for the Navier-Stokes sys-

tem

divu = 0 (2)

where u(t, z) E R3, z E R3, t E [0, co), P is a positive constant and p ( t , x) E

R1 and change ( 2 ) for the Poisson equation

-Ap ( t , x) = y ( t , x), y = Tr[VuI2 (3)

connecting the velocity field and the pressure. The topic of main interest

for us in the present paper is the construction of diffusion processes in R3 associated with (1),(3). Let (0, F, P ) be a probability space and w( t ) E

R3, B( t ) E R3 be a couple of independent Wiener processes defined on it.

Denote by E the expectation with respect to P and by EE the conditional

expectation with respect to a stochastic process [ ( t ) . Consider the Cauchy

problem for the stochastic differential equation

d<(T) = -U ( t - 7 , <(T) )d7 + fJdw('T), <(o) = 5 (4)

and assume the unknown drift coefficient u to be determined by

Since p is an unknown function as well we close (4), (5) by the relation

div u(t, z) = 0. (6)

System (4)-(6) was considered previously in 6, 7.

In this paper we study a system consisting of (4), ( 5 ) and

instead of (6). It is easy to check using Ito's formula that if u is smooth

and together with p satisfy (1),(3) then u , p can be represented in the form

(5), (7).

2

3

The inverse statement is valid as well. Namely, given a solution to

(4), (5), (7) such that u,p are smooth enough we can state that (5), (7)

determine classical ( or respectively C') solution to (l), (3) and hence to

By general results of diffusion process theory and in particular the

we show that heuristic differentiation of (4),

(1) 0).

Bismut-Elworthy formula

(5), (7) leads to

and

where 6 i k is the Kronecker symbol and the usual convention of summation

over repeated indices is made.

In addition we notice that Bismut-Elworthy's formula allows to derive

from (7) the representation for Vp

(10) m l

V d t , .) = - -E [y ( t , + B(s))B(s)lds S

and use i t to eliminate the dependence on p from (5).

After that we obtain a nonlinear integral equation involving u, Vu, show

that it gives rise to a contractive mapping in a certain functional space

and find the fixed point of this mapping by a successive approximation

procedure. To realize this program we need some auxiliary results about the

behavior of the solution to the Poisson equation which can be proved using

standard techniques of integral estimates based on the Holder inequality.

Let us recall some results concerning the probabilistic representation of

the solution to the Poisson equation

where y ( t , .) : R3 -+ R3 is a measurable function depending on the param-

eter t E [0, m).

In the next two lemmas we recall the integral estimates for the solution

of the Poisson equation which we need below.

where 1 1 . 1 1 is a Euclidian norm in Rn. Let LY = L Y ( R ~ , ~ n ) = {j(.) E ~n : {sR3 ~lf(.)llydx}+ = Ilfll,

4

Lemma 1.1. Let y( t ) E Lq2 n L4(R3) for some 1 5 q 2 < 3 < q < 00

and arbitrary t from a compact [0, TI. Then for each x E R3 the integral

h" +[y(t, % + B(S))Bk (s)lds

converges. Moreover the function

belongs to Cb(R3) and

IPk(t, .)I100 5 Ilr(t, .)l142,4. (13)

If in addition q 2 > $, then Fk(t,x) = 2&p(t,z) .

[0, TI. Then for each r > 2 the functions x H Fk(t, x) belong to L' and Lemma 1.2. Let y( t ) E Lq2 nL4(R3) for some 1 < q 2 < 2 < 3 < q , t E

IIFk(t) llr I Kllr(t) 1142 , 4 (1 + Ilr(t) 1142 A). (14)

Let us construct the solution for (4), (5), (7) by a successive approxi-

To this end we consider

mation method.

uO(t,x) = uo(x), p ( t ) = x, (15) w

po(t, x) = ETr [Vu0l2(t, x + B(s))ds,

pk(t , x) = J ETr [Vukl2(t, x f B(s))ds 0

auk a u k where yk ( t , x ) = $$, ( I c = 1 , 2 , . . . and q , j = 1,2,3) and

t

uk+l(t, x) = E[UO(<"(t)) - / v p y t - 7,< ' " (7 ) )d7 ] . (18) 0

To prove the convergence of the successive approximations we have to add

and V u of the solutions to (4), (5), (7) and hence we need a probabilistic

representation for the gradient Vu. We recall the Bismut- Elworthy formula

for a diffusion process satisfying a stochastic differential equation.

Let v ( t , x ) , ( x E R3,t E [O,T]), be a differentiable vector field of

sublinear growth (in x) and let fo E C2(R3). Consider the Cauchy problem

for the stochastic differential equation

to (15)-(18) the successive approximations for the derivatives q(t) = at -&- (t)

d<z = -v(t, &(t))dt + adw, &(O) = z (19)

5

and put

u(t, .) = E[uo(Jz(t))]. (20)

Then Bismut- Elworthy’s formula states that if ‘u has bounded spatial

derivatives then u is a smooth function and Vu admits the representation

Here $(t) satisfies the linear equation

drlji = Vrn’ui(t, ,$Z(t))rlrnj(t)dt, V k i ( 0 ) = Sji (22)

and S j i is the Kronecker symbol. (We use the notation A = u @ g for

the matrix A = (aik) with matrix elements aik = u i g k and assume the

summation in the repeating indices. ) It results from (10) and Bismut-

Elworthy’s formula applied to f(t , z) = EVp(t, &(t ) ) that the heuristic

expressions for derivatives of (4),(5), (7) have the form

d V j i ( t ) = -Vmui(t - 7, J(T))Vrnj( t )dt , rlji(0) = S j i (23)

and

where

We set 0 = {O(t,z) = (ui ( t ,z) ,Vjui( t ,z))} , {e( t ,z) : Ile(t)IIL. < m} is denoted by 01 for < r < 2, and by 0 2 for r > 3. We then have

0 = 01 n 0 2 . Let M = C([O, TI, 0) be the linear space of continuous

functions defined on [O,T] and valued in 6 with the norm

l l ~ l I ~ , r x = S ~ P t G [ O , T ] [IP(t)llo + [V4t)lal.

We prove that (un, Vun) converges in the norm of the space M for some

fixed interval [0, TI. In fact we consider the space G = M U S where S is

the space of vector and matrix valued processes with the norm determined

by llE112 = sup&11,$(t)l12. In section 2 we determine the interval [O,T] and

prove the convergence of (15)-(18), (23)-(26) in 6. This leads us to the

following main results.

6

Theorem 1.1.Assume that (210, VUO) E 0. Then there exists a bounded interval [0, TI and a unique solution ( t ( t ) , u( t , z), p ( t , z), V ( t ) , Vu(t , z)) to

(4), (5), (7)-(9) in Theorem l.2.Assume that the conditions of theorem 1.1 hold. Then

there exists an interval [0, TI such that for all t E [0, TI there exists a unique solution t o (1), (2) in M .

The assertion of theorem 1.2 is a consequence of theorem 1.1 and the

results of the theory of diffusion processes according to which the function

u(t , z) given by (4),(5) satisfies the backward Kolmogorov equation of the

form (1). Notice that the solution constructed in this way is a generalized

solution to (1) since we can prove that u has a Holder continuous gradient

but the existence of the second spatial derivative for the function u is not

claimed.

for each r E [0, TI.

2. Convergence of successive approximations

To check the convergence of the successive approximations determined by

(15)-(18) we need some auxiliary results about the solution [,"(t) to the

equation

r t

with the smooth drift coefficient v ( t , x ) E R3, x E R3,t derivatives with respect to the initial data V"(t) given by

V v ( t - 7, &(.)) 0 VZ(T)dT,

(27)

E [0, co) and its

where I is the identity matrix and Vv o denotes the matrix product.

Lemma 2.1. Let v( t , z) E R3, ( t E [0, co), z E R3) satisfy the estimates

s~P"Ilv(t,z)1I2 I K,(t), s~PZIIVv(t,s)1I2 5 K,(t),

and

2 I I W 1 x ) - Vv(t,Y)1l2 I L:(t)Ilz - YII

where K, ( t ) , Kt ( t ) , LA ( t ) are positive time dependent scalar functions bounded on bounded interval [0, TI. Then

7

and

(31) 21 21 SOt [Kt ( t - -7 )+L t ( t - r ) ]d~

EIlr12(t) - rlY(t)1121 I 112 - YII e

hold for t E [O, TI. Proof. I t results from Ito's formula and Gronwall's lemma that

Elltdt) - t Y ( t ) l 1 2 1 t

5 1 l 2 - Y I l 2 ' + 2 1 1 El(4-7&(7)) - " ( t - 7 l t Y ( 7 ) ) , & ( 4 -Jy(7))1

- < 1 -+ 2 it E(Vv(t - 7, t(7)>17(.>1 rl(T))d7

21 21 lot K,'(t-r)dr Ilrz(.) - EY(7))/12(1-1)d7 i 115 - YII e

To prove (30) we notice that for 2 = 1 we have

Ellrl(t)l12

t

I 1 t 2 1 Kt ( t - 7)E11rl(7)/)2d7

where Ki( t ) = supZ((Vv(t, .)I/. Then for arbitrary I the required estimates

are derived in a standard way. Let us give some details on the proof of (31). To check (31) we consider the case 1 = 1 and derive the estimate for

EllrlZ(t) - 17Y( t ) l l 2 .

E11r15(t) - rlY(t)ll2 5

8

The solution &(t) of (27) gives rise to the stochastic flow x H &(t ) . In

addition

V d & ( T ) d,V&(T) = -V[v(t - T , < ~ ( T ) ] ~ T .

For given $( t ) = V&(t) we shall denote by J ( t , x) = det $( t ) the Jacobian

of the random map x -+ &(t). Lemma 2.2. The Jacobian J ( t , x) satisfies the equation

dt J ( t , x) = J ( t , x)div v(& ( t ) ) .

Proof. The determinant of a matrix is a mi

columns (or rows). Hence, fixing x, we have

dtVlE' V1t2 V1J3

d tV2 t1 V2t2 V2E3 d tV3 t1 V3t2 V3t3

V1C1 dtU1t2 V1J3

V2t' d tV2 t2 V2t3

V3J1 d tV3 t2 V3J3

tilinear function in the

By (27) we have d t V j [ l ( t ) = V j [ v i ( & ( t ) ) ] d t . Substituting this relation

along with vi [v j (<2(t)) ] = xi=, VkdVitk into the above expressions for

dt J we get

JV lv ' + JV2v2 + JV3v3 = (d iv U ) J. 0

Remark If the drift vector field v(t , x) in (27) possesses the property

Assume that v is a smooth divergence free vector field and consider

d i v v = 0 , we deduce d tJ = 0 and J ( t ) = J ( 0 ) = I .

functions p , and u given by

0

In what follows we denote by 1 1 . 1 1 either the Euclidian norm of a vector

or the norm of a matrix respectively. As a rule for matrices we choose

IlAll = rnaxjklajkl or the equivalent norm IJAIJ = TrA. Lemma 2.3. Let v(t),uo E C2 n L' and assume the estimates

II~oII, < tor, I I v w I I I ~ < c&, SUP, J J v ~ o ( z ) I J I K,, SUP, I l~o(x)I l I KO,

9

Il~(t)llT I CUT(t)I llVw(t)IIp < c:T(t)l SuPzIIW, .)I1 5 K,'(t)

hold. Then there exists an interval [0, TI ] (with TI depending on the functions U O , w )and functions ,B(t), y ( t ) bounded on this interval, such that the inequalities

supzIIV4t,z)ll 5 P(t) (34)

and

llvu(t)llT < y( t ) (35)

hold fo r the function u( t ,x ) given by (32), (33), 0 I t < TI and 9 < r < 2 or r > 3.

Proof. From the heuristic expressions (23)-(26) rewritten for the func-

tions determined by (32), (33) by Jensen and Holder inequalities we deduce

that

( (Vu( t -s ,x) ( l I m 1 ( t - s , x ) + m z ( t - s 1 s ) (36)

where

0

m ( t - 3 , ~ ) = 6" -$(EIIVp(t - ~,<z(Q)) l12)$( / Ellr1z(7)112d~)~]de. (38)

To derive the estimate for ml(t - s, x) we notice that by (30)

K: (t--7)dT mi( t - s , z ) I EllVuo(tS,,(t))/leLt

and derive the estimate for n1(t - s,z) = E / I V ~ O ( E , , ~ ( ~ ) ) / ( . Changing variables under the integral sign we deduce

Here Jl( t ) = det [~ ( t ) - l ] is the Jacobian of the random transformation

inverse to the transformation L H z e~ &(t ) determined by (27). By Lemma 2.2 we conclude that

To derive the estimate for mz we apply the Holder inequality to the right

hand side of (37). Taking into account (30) this yields

- ~,<z(e)) I Iz~ l I r1z(~)112)~de 5

10

To estimate \lm211r we apply the Holder inequality to the integral with

respect to e to get llm2(t - s)llr I eJ3t K t ( t - ~ ) d ~ v(t , s) where

for 1 + I = 1. Choose 43 < 2 and q4 > 2 to obtain 43 44

t where c6 =

that & 2 1 then we can apply Jensen’s inequality to get

do]& depends on t , s. If r > 4 and q4 are chosen so

where C7 depends on t , s. If & 5 1 a similar inequality can be derived by

the estimate a: < a valid for q , a > 1 assuming without loss of generality

that

1

[ (qvp(t - e,<Z(e))112q4)+d~ > 1.

Finally, we derive the required estimate for v in terms of the Lr- norm of

V p , using the properties of the Jacobian J proved in Lemma 2.2. In this

way we obtain

44 s ) I c7 I’ s,, IlVp(t - 0, z)llrEIJl(t - 8, z)lTdzdQ

t

I c7 L k3 IIVP(t - 8, z)IITdzdQ.

Recall that by (14) we have IIVp(t)ll, I K ~ ~ T r [ V ~ ] ~ ( t ) ~ l ~ ~ , ~ ~ [ l + l l T ~ [ V u ] ~ ( t ) / j ~ ~ , ~ ~ ] for r > 2 and < q1 < 2 and 3 < q 2 . Thus we ob-

tain

11

where Mt = tQCt and finally choosing t large enough we get

Using the notation Cur@) = IIVu(t)ll. we deduce from the above estimates

that

where K = Ka-,T 2Tl . Later we choose either 1 < r < 2 or r > 3. Let us derive next the estimate for Kt( t ) = s ~ p , I l V u ( t , x ) l 1 ~ . Using the

relation of the type (13) inspite of (14) and above considerations we obtain

Denote by P(t - s) and y(t - s) functions that satisfy the relations

t

dB1 (42) ? 2 ( t - e),J8' P ( ~ - - T ) ~ T P(t - s ) = K o e 1 J," ~ ~ ( t - 7 ~ 7 +

Finally, we notice that the functions y(t - s) and P(t - s) are governed by

the system of ODE

- d y = Py + KY2[1 + y2], y(0) = c;,, ds

= p2 + K y 2 , P(0) = KO. dP ds -

(44)

(45)

By the general theory of ODE systems we know that there exists a unique

bounded solution to this system over an interval [O,Tl] depending on

KJ,CJr. Finally we notice that if K,(t) 5 p(t) and C&(t) 5 y(t) then

IIVull, 5 y ( t ) and sup,IIVu(t,x)II 5 p(t) on the interval [O,Tl] as well. 0 Lemma 2.4 Under the conditions of Lemmas 2.1 - 2.3 there exist func-

tions M i @ ) , M,(t), Ku(t) and Z,(t) (bounded on [O,Tl) for above 7'1 ) such

12

that the vector fields u given by (33) and V p for p given by (32) obey the estimates

I I ~ ( t ) l l q l , r , qz 5 n ( t ) < 0 0 1 ll4t,x)ll 5 Pl(t),

supzIlVp(t.x)II 5 Z,(t)

l l ~ P ( t ) l l q l , T , q 2 -5

and

where 1 < r < $ or r = 4 and Proof. The proof of these estimates can be derived from (32), (33) using

the results of Lemma 1.2 and Lemmas 2.1 - 2.3. 0 Lemma 2.5.Assume that conditions of Lemma 2.1 - Lemma 2.3 hold.

Let in addition u0 E C2(R3) and llu0llcz 5 Kz. Then there exists an interval [0, TI ] with TI < T depending on KZ, KO and the function Lk(t - s ) bounded on this interval such that the estimate

< q1 < 2 , q 2 > 3.

I1Vu(t - s , x ) - Vu( t - s,y) l l2 < LL(t - s)11x - (46)

holds. Proof. It is easy to check that

$(t, x, Y) = 11v4t, .) - Vu(t , Y)1I2 I 2Cl( t , 5 , Y) + 2 C 2 ( t , Zl Y)

C l ( 4 Z,Y) = E / l [ ~ ~ o ( € z ( t ) ) 7 7 z ( t ) - ~~0(Jy(t))r ly( t )1I2,

where

By the assumptions about no and we derive form (23)-(26) that

Substituing (29)-(31) in (460 WE DEDUCE

13

To derive the estimate for ( 2 ( t ) we notice first that y(t, x) = Tr[V2w(t, x)] satisfies the estimate

and perform some computations based on Holder inequality and Fubini

theorem. For N l ( t ) we derive using (46)

where C is a positive constant depending on LA and K;. To estimate N2(t) we choose I c , I, m such that i+k+: = 1, use (45) and the Holder inequality

to derive

N2(t) 5 JT" :2K;(t)(EI(V.u(t,z + B(s)) - Vu(t, Y + B(s))Ilm)&

for q' = 5. Finally, choosing q < 2 and T < z q we prove that the integrals

in the latter expression converge that leads to the estimate

N( t ) L CIK,(t)L:(t)Jlz - Yll.

pUTTING

WE CAN WRITE n9T) IN THE FORM

14

Now we check that

where C is a positive constant depending on LA and KA. It remains to

apply the Holder inequality to derive the estimate

€or 2 + = 1 and a < 2. We use (30) to derive

$(t , 5 , Y) I IIx - YII’[C~ + u - ~ I C ~ ~ 0 st C4[KLv(t-T)+K]dT

with positive constants Cs, C4 depending only on t and /3 and by Lemma

We accomplish the proof of the Lemma by reasons similar to those used

in the proof of Lemma 2.3. Namely, let Lh(t) be the least scalar function

such that

2.4 K = Supo<t<TIK,(t) < 03 .

IIVu(k2) - V4t ,Y) / I2 5 L:(t)llx - YII 2 ‘

From the above estimates we get that there exist absolute positive constants

C3, C4 such that

L i ( t ) 5 [c3 + u-1]c4eJot C ~ [ K L : ( ~ - T ) + K I ~ T

Let us construct a majorizing function K ( t ) for Lh(t) as the function to be

governed by the equation

c4 [ K K ( f, - 7 ) +K]d7 K ( t - s) = MeL’

15

where A4 = C4[C3 + a-1]. Choose It - sI i TI to ensure that Ki(i-1) < K < 03. As a result we deduce that n(t - s) solves the Cauchy problem

dK(t - S ) = C ~ [ K + K ] K , ~ ( 0 ) = M

ds and can be explicitly represented in the form

M eC4 (t-s 1 K ( t - s, = c - eC4(t-s)

where C = 1 + c41c3+c-11. K Hence, if 0 < t < T3 where

T3 = min(Tl, T2) (53)

and

then K ( t - s) is bounded and Vu(t , z) possesses the required property. 0 Let the assumptions of Lemma 2.1 -2.4 hold. Then there

exists a positive function Lp( t ) bounded on the interval [0, T3] given by (53) such t h d the fu,nction. V p ( t , z) = E [ J r ~ - ~ T r [ V v ] ' ( t , z + B ( s ) ) B ( s ) d s ] satisfies the estimate

Lemma 2.6.

IlVP(tl.> - VP(t, Y>ll I LP(t)llZ - YII.

The assertion of this lemma can be deduced from the estimates derived

in section 1, Lemma 1.2 and Lemma 2.5. 0 Lemma 2.7. Under the conditions of Lemma 2.1 the estimates

Ell l2( t ) - E,""t)ll2 (54)

(55)

hold f o r the solutions [Zk ( t ) , ~ x ~ " k ( t ) to (27), (28) for 5 = I , 2. Proof. We deduce from (27) that

16

and Gronwall's lemma yields (54). We prove the second estimate applying

Gronwall's lemma to (28) that yields

q r l zv ' ( t ) - 77""2(t)1I2 I

Before coming back to successive approximation system (15)- (18) and

its derivatives in x variable

where

we need one important remark.

L' norm for t E [O,T3) and

Since the gradient Vuk was proved to be uniformly (in k) bounded in

m l vipyt, x) = .I -E[y'"(t, x + B(S))Bi(S)]dS

S

3 auk auf where y'"(t, x) = dF we deduce that V p k ( t , x) is uniformly (in

k) bounded in L' norm as well. Moreover given rk(t) E Lq n C1icy(R3), Q E

(0,1),1 5 q 5 4 we know by Schauder's theorem that IIV2p(t)Ilc; 5 KII YII Lqncg.

Let uk, Vuk be successive approximations of tensor fields u, Vu defined

by (15)-(18) and (55)-(58). Now we can prove our main result stated in

Theorem 1.1.

Proof of Theorem 1.1 Let us prove that uk( t ) , Vuk(t) given by (15)

- (18), (55) - (58) converge in L' norm. Set

17

We start our considerations with general remark that to estimate all above

functions we apply the Fatou-Fubini theorem to change the order of inte-

gration in t and z variables as well as Holder's and Jensen's inequalities.

We denote further by K,, i = 1 ' 2 , . . . , constants which depend only on

t and r and assume that s is choosen so that f + = 1. Many of our

computations use the Fubini theorem and the Holder inequality and the

properties of the Jacobian J that allows to change the variables under the

integral sign. Since we have used already this reasoning before in proving

previous Lemmas we do not give below the detailed description of them.

To estimate aL(t) we apply (34)' the Gronwall lemma and the Holder

inequality to deduce

t I K l [ e s," K Z P T ( t - T ) d T + 1 , q t - T ) d T ] ( 6 0 )

To derive the estimate for ,Li(t) by the Lipschitz property of uo(z) and

Holder inequality we deduce

By Lemma 2.4 and 2.5 we derive

In addition due to (7)

18

To derive the estimate for Z k r we rewrite it in the form

5

where

1;' ( t ) = El/ VUo(E,k-l ( t ) ) I/ ' 1 1 pk ( t ) - T p - ( t ) 1 1 ' dz L 3

and the last three terms are derived by Bismut-Elworthy's formula that

along with Fubini's theorem and the Holder inequality gives

To estimate Zkl, we use the Lipschitz property of Vuo(z) and estimates

from Lemma 2.7 to obtain

(64) I,&) 5 (L$aL(t)e-l;: /3 ( t - T ) d~

For 2zT(t) we derive from Lemma 2.7 estimates and properties of VUO that

19

In addition

Ak(t) 5

t

L K5[/ r;(t - T ) ( Q ~ ( T ) ) ~ ~ T ] .

By the Holder inequality we have

0

and by estimates from Lemma 2.1 and the Holder inequality applied in the

0 variable we deduce for ml = rm < 2

with positive constant K6 depending on t and ml. Finally taking into

account (67) we get

tO DERIVE THE ESTIMATE FOR iS(T) WE RECALL THAT WE HAVE

DUE TO THE ESTIMATES GIVEN IN SECTION 1. tHIS ALLOWS TO DEDUCE THAT

tO DERIVE THE ESTIMATE FOR (T0 WE USE THE ESTIMATES FROM lEMMA

20

By the properties of stochastic integrals we have

where

rl1 and - < I .

2

Let us combine the above estimates (58)- (70) to derive the following

inequality

t

IIGk(t)lIT I M ( t - ~)I IGk- l (~) I ITd~.

Here Gk(t, z) = ( a k ( t ) , Ak( t ) , Lk( t ) ,Zk( t ) ) and M ( t ) is the corresponding

positive scalar bounded function that can be read out of (58)-(70). By the

above arguments we deduce that

QnTn

n! vn = SuPO<t<TIIGn(t) - Gn-l(t)Il I - const

where Q is the fixed constant such that s u p ~ < ~ < r Z ( t ) - - 5 Q for t E [O,T) with T determined above. It results that

limn-mVn = 0.

To prove that the solution constructed in this way is unique in 0 1 ,

suppose on the contrary that there exist two solutions u( t ,z ) , <,"(t) and

v ( t , z), <,"(t) to (4), (5), (7) satisfying the same initial condition u(0, z) =

v(0, x) = ug(z). Using the estimates of Lemma 2.5 we derive

(71)

t

Ilu(t - T ) - w ( t - T)IITdT + 1 c1 Ilu(t - .) - v( t - T)II.dTde

where the positive constants C, C1 depend on the interval [0, T ) and es-

timates for functions Lh( t ) , Kk(t) derived in the above Lemmas. Finally

(71) yields that IJu(t) - v(t)lJ. = 0.

21

Acknowledgements

We are very grateful1 to Professor Aubrey Truman for the kind invitation

to an interesting and stimulating conference and to the University of Wales

for the hospitality. The financial support by DFG Project 436 RUS 113/593

and by Grant RFBR 02-01-00483 are gratefully acknowledged.

References

1. Nelson E. Les e‘coulements incompressibles d’energie f inie. Colloques intern. du CNRS” 117, 159, (1962).

2. Belopolskaya Ya., Dalecky Yu. Investigation of the Cauchy problem for quasi- linear parabolic systems with the help of Markov random processes. Izu, VUZ. Matematika, N 12, 5 (1978).

3. Belopolskaya Ya. I., Dalecky, Yu. L. Stochastic equations and differential geometry, Kluwer Acad. Publ., (1990).

4. Busnello B. A probabilistic approach to the two-dimensional Nauier-Stokes equations. The Annals of Prob. 27, N 4, 1750,(1999).

5. Busnello B., Flandoli F., Romito M. A probabilistic representation fo r the vorticity of a 3D viscous fluid and fo r general systems of parabolic equataons. Preprint (2002).

6. Belopolskaya Ya. Probabilistic representation of solutions to boundary-value problems for hydrodynamic equations Zap. nauchn.sem. POMI , 249, 77,

(1997). 7. Belopolskaya Ya. Burgers equation o n a Hilbert manifold and the mot ion of

incompressible fluid, Methods of Functional Analysis and Topology, 5 , N4, 15

8. Elworthy K.D., X-M.Li. Formulae for the derivatives of heat semigroup. JFA 125, 252, (1994).

(1999).

A MEAN F I E L D RESULT F O R 3D VORTEX F I L A M E N T S

H. BESSAIH AND F. FLANDOLI

Dipar t imento d i matemat ica applicata U. Din i , V i a B o n a n n o 25/B 56126 Pisa, IT

E-mai l : bessaihodma. unipi. it, j landoli@dma. unipi. it

A mean field result is proved for an abstract model, under a class of conditions on

the rescaling of the energy. Propagation of chaos, variational characterization of

the limit Gibbs density h and an equation for h are proved. The general results are

applied to a model of 3D vortex filaments described by stochastic processes, includ-

ing Brownian motion and Brownian Bridge, other semimartingales, and fractional

Brownian Motion.

1. In t roduct ion

The importance of thin vortex structures in 3D turbulence has been dis-

cussed intensively in the last ten years, see 4 , Some mathematical models

of vortex filaments, based on stochastic processes, have been proposed by

Chorin 4, Gallavotti Lions-Majda 14, Flandoli 5, Flandoli-Gubinelli ‘ and Flandoli-Minelli The importance of these models for the statistics

of turbulence or for the understanding of 3D Euler equations is under in-

vestigation.

The limit properties (mean field) of a collection of many interacting vor-

tices has been investigated by P. L. Lions and A. Majda l4 for a particular

model of “nearly parallel” vortices.

The aim of our work is to investigate a similar limit for the model intro-

duced in 5 , ‘. Here the expression for the kinetic energy is not approximated

and filaments may fold, so some features are more realistic. However, the

filament structures have a fractal cross section (as observed numerically) to

eliminate a divergence in the energy.

The structure of the paper is the following one. In the next section we

present the abstract frameworks and state a mean field result for them.

Section 3 is devoted to the proofs. Then in the final section we apply the

general result to some models of vortex filament.

22

23

2. Abstract mean field result

Let us define the abstract framework. Let ( R , d ) be a complete separable

metric space (it will be the space of vortex structures) and let B be its Bore1

cT-algebra. Let po be a probability measure on (R, B). Let H s and H I be

two random variables

H s : R -+ R, H I : R x R + R

with the meaning of self and interaction energy. Assume

N

i = l i#j

(here and below, the second summation is extended from 1 to N ) with the

understanding that for N = 1 it reduces to

H s ( w ) 2 0 po-a.s.

(5)

(hence also Sn2 HId& < a). Let us assume also the following conditions on H I :

H I ( W , w’)f(w)f(W’)&; 2 0, f E Lrn(R, Po), (6)

(7)

and that

2 HI(u,u’) is symmetric in w and w‘, po -a.s.

Let H N : ON 4 R, for any positive integer N , be the random variable

defined as

The variable H N has the meaning of a rescaled energy, where only the

interaction energy is reduced as N grows. A physical motivation for this

rescaling has to be found in each particular case.

Denote the product measure of N copies of po by p t . Let

hN :RN + R

be the probability density defined as

hN = (N)-’ e-pHN, Z ( N ) = lN e P P H N d p f

24

where ,8 > 0 is a given parameter, with the meaning of inverse temperature.

Denote by p N the measure (it is not a product measure)

dpN = hNdp,N.

Finally, denote by pN,k the k-marginals of p N on RN (by symmetry, the

choice of the k-components is irrelevant). We have

d p N h = h N , k k PO

where hNik : Rk ---f R are given by

hN’k(W1l.. . lwk) = hN(W1,...,WN)dpo(#k+1) . . .dPo(Wiy) s,r Under the assumptions (2), (5), (3), (6) and (7), we have the following

result.

Theorem 2.1. For each k 2 1, pNik converges weakly as N --+ 00 ( in the sense of probability measures on R k ) to a product measure @)zk=l p. I n addition

d p = hdpo

where h E L““ (R , PO) satisfies the equation

1 - P ( H s ( w ) + J , HI (w+J’)h(w’)&o ( w ’ ) ) h(w) = -e Z’

with

Moreover, h as the unique minimum of the following free energy functional

over the set { f L 0 pa - as . , f E Loo (0, PO) J, f (w)dpo = I}.

Remark 2.1. The theorem states that, in the limit N + 00, the filaments

behave independently (the so called propagation of chaos). Moreover, the

limit Gibbs density h of each filament is associated to an energy given

by the sum of the self-energy of the filament plus the interaction term

J, H I ( w , .)hdpo. The latter describes the interaction between the filament

and the mean field associated to h itself.

25

3. Proofs

We introduce new notations to shorten the formulas. We set

H ( q J l , . . . , UpJ) = HS(W,),

H ( " q U 1 , . . . , U N ) = H I ( W n , U m ) .

and

In the sequel, we simply write H(") and H("1") without their arguments.

3.1. Uniform bound on the marginals densities

Lemma 3.1. For every given k 2 1, there exists a positive constant C(k) such that

hN)k 5 C(k ) a.s. on Rk for all N 2 k . (16)

Proof. The proof of this lemma will be done in three steps. Let us define

In particular Z(N, N , 1) = ZN.

N 2 No(k)

Step 1 Given k 2 1, there exists a constant No(k) such that for all

2k N

Step 2 For every p > 0 and for every N 2 k 2 1,

hNYk 5 ( Z ~ J ) - ' Z ( N , N - k , 1 - -)

k Z", N, P ) 2 (CZ(IL)Y Z ( N , N - k, P + ;v)

where

0

Step 3 Let k 2 1 be given and let c k = 3k. Then, there exist constants C3(k) and Nl(Ic) such that

c k Z ( N , N , 1 - -) I C3(k)Z(N, N , l), ViV 2 Nl (k ) . N

To conclude the proof of the lemma, we collect the estimates of the three

steps and have (for sufficiently large N )

26

2k N

N N

hN'k 5 (ZN)- ' Z ( N , N - k , 1 - -)

c k = (ZN)- ' Z ( N , N - k , 1 - - $- -)

5 (ZN)- ' (CZ) -~ Z ( N , N , 1 - z) 5 (ZN)- ' c 3 ( k ) (CZ) -~ Z ( N , N , 1)

c k

= C3(k) (Cz)-"

The proof is complete.

3.2. Variational characterization of hN and known results

Let us introduce the following free energy functional

over the set

P = f 2 o p t - a s . , f E L ~ ( o N ; ~ : ) , { Lemma 3.2. The density hN is the unique minimum of F N .

Proof. The proof is classical, see 17. 0

3.3. Weak limit of hNyk

From the uniform bound (16), by a diagonal procedure, we can extract a

subsequence Nj of N , independent of k , still denoted by N in the sequel,

such that for all given k and for N + DC,

hNIk - h",k, weakly * in LO",

hNVk - hmik, weakly in Lp for all p 2 1

for some hoo%k E L" ( O k , p.,"). We easily have h"ik 2 0, Soh h">'dp.," = 1,

and h">k is symmetric (from the analogous properties of hN)k) . From the symmetry and Hewitt-Savage theorem lo we deduce that there

exists a measure if on P such that

27

3.4. Convergence of the variational problems

Let us denote by II the set of all probability measures 7r on P. We define

the following functional on II,

3.5. Properties of the limit variational problem

Up to now we have proved that the Gibbs densities hNik have a subsequence

converging to some density hmikl with symmetry properties] and such that

the associated measure i f minimizes the functional F ,

F(4 = s, F ( f ) d T ( f ) ,

where F is the functional given in theorem 2. Let us prove the following basic fact:

Lemma 3.4. That there exists h E P such that

?i = 6h (28)

and h is the unique minimum of F over P .

Due to (28), this implies that hm3k factorizes] i.e. the associated mea-

sure is a product measure. This proves the first claim of Theorem 2. More-

over, i t proves that the functional F has a unique minimum. So, let us

prove the lemma.

Proof. By definition] we have F(T) = S , F ( f ) d ~ ( f ) for all 7r E II. Let

us show that the set S of minimum points of F is non-empty and f is

28

concentrated on this set S. Since F is strictly convex, S reduces to a single

point h, and therefore the claims of the lemma are proved.

Let F be the infimum of F on P. For every 7r E II we have

F(%) = s, F ( f ) % ( d f ) 2 s, r;’?i(df) = F

and on the other side, if h, is a minimizing sequence for F ,

F(%) 5 F ( S h h ) = F( f )Sh , (d f ) = F(h,) -+ F J’, so

F(%) = F .

It follows that

Since F ( f ) - F 2 0, this implies F ( f ) - F = 0 %-as. This proves at the

same time that 5’ is non-empty and % is concentrated on 5’. The proof is

complete. Kl

4. Application to vortex filaments

First we describe in detail the application of the mean field theory to vor-

tex filaments modelled by Brownian trajectories. At the end of the section

we shortly describe a generalization to certain semimartingales (including

Brownian bridge and models of vortices a t a solid boundary) and to pro-

cesses with finite p variation, p < 2 (including fractional Brownian motion

with Hurst parameter H > i, and related nongaussian models).

4.1. Brownian vortex filaments

4.1.1. Introduction

We consider a fluid in EX3. The kinetic energy

of a velocity field u(x) can be written in terms of the vorticity field [(z) =

curlu(s) as

29

In the case of a vorticity field ideally concentrated along a curve y(t), t E

[0,1], the vorticity field is formally defined as

1

E(.) = r 1 b(x - y(t))?(t)dt,

where the parameter r is the circulation and the energy takes the form

In the case of N curves y' , ..., y N , we have

For regular curves, as well as for many examples of curves given by paths of

stochastic processes, this expression (with suitable interpretation for pro-

cesses) is divergent. Physical vort,ex structures, although very thin, have

a cross section. Re-introducing the cross section increases the degrees of

freedom and makes the model less intrinsic, but helps to eliminate the di-

vergence of the energy. To keep a closer relation with the vortex structures

observed in fluids, it is better to consider fractal cross sections instead of

simply a tubular mollification. In the previous papers ', vorticity fields

of such kind with finite energy have been constructed. They are formally

expressed as

where p is a probability measure describing the cross section and subject to

the assumptions given below, and (Wt)tE[o,ll is a Brownian motion in EX3. The corresponding energy takes the form

and in i t is proved to be meaningful and finite, with probability one. In

the case of N copies (W/)tE[o, l l , k = 1, ..., N , of 3-D Brownian motions,

and N probability measures p', . . . , pN on EX3, the energy takes the form

In this sum the terms of the form

30

represent the self energies of the single filaments] while the terms

with n # m give us the interaction energy between the filaments n and m. By easy manipulations with Fourier transform (see 6, we rewrite the self

energy in the form

and similarly we rewrite the interaction energy in the form

(44)

The presentation until now have been rather informal] but we give below

rigorous definitions.

4.1.2. Space of configurations

Following the previous description, a single vortex filament is defined to be

an element of the product space

R = c x M-1,

where C = C( [0, TI; R3) and M-1 is the space of probability measures p in

R3 defined below. The interpretation is that the filament has a core and

a cross-section. The core is a 3-D curve, i.e. an element of C([O, TI; R3).

The cross-section is a probability measure p, the support of the measure

represents the geometric cross-section, while the measure weights the inten-

sities of the different lines of vorticity. Thus R is the space of configurations

of a fluid when the vorticity field is made of a single vortex filament with

cross-section.

The space of configurations of a collection of N vortex filaments is ONl

the product of N copies of R.

4.1.3. Cross-section and its random selection

The cross-section of the vortex structures considered here will be described

by the probability measures p of the following form.

For any probability measure p on the Bore1 sets of R3, let us set

31

where b ( k ) = Jw3 ei"'"p(dz) and let us denote by M-1 the set of all p such

that 1 1 p llT1< 00. We recall from classical potential theory, see l2 that a

probability measure p E M-1 is called a measure with f in i te energy. Given

a set A in R3, there exists a probability measure p supported by A with

finite energy if and only if the capacity of A is strictly positive. Finally, by

Theorem 3.13 of 12, every compact set with Hausdorff dimension d > 1 has

positive capacity. Therefore, it supports a probability measure p satisfying

On the space M of all probability measures p on (R3, 23 (R')) there is a

metric d such that the convergence with respect to d is the weak convergence

of probability measures:

(46).

for all bounded continuous functions f . We endow the subset M-1 with

the metric d . Notice that ( M , d ) is complete, while (M-1,d) is not, but

this fact has no importance in the sequel. Let us denote by BM the Borel

c-algebra of (M-1, d ) . Let PM be a probability measure on (M-1, BM) . Our vortex structures

will have a cross-section measure p choosen at random with probability law

PM. In the sequel, we denote by B the product a-algebra Bc @ BM on R

a = & @ B M

where BC is the Borel a-algebra on C = C([0,T] ;R3) . Moreover, if PC denotes the Wiener measure on C, we set

PO = PC @ P M .

4.1.4. Reference measure

The statistics of vortex filaments are given by probability laws on the config-

uration space defined by a Gibbs weigth with respect to a reference measure.

In the case of a single vortex filament, we choose as reference measure the

product measure po on (R, B). In the case of N copies of vortex filaments

the reference measure is the product measure p r on ( O N , B N ) , product of

N copies of the measure po. Therefore we also have p r = P? @ P c , This

32

choice of the reference measure is very natural from a probabilistic point of

view, but rather arbitrary from the fluid dynamic viewpoint. However, we

do not have at the moment more accurate physical prescriptions.

4.1.5. The energy

With the motivations given above, we define the self-energy HS : 0 -+ R of

a vortex structure as the random variable

where (Wt)t,[o,ll is the canonical process on the Wiener space C. In the

case when p E M-1 is given, it is proved in that the random variable HS is well defined and it has finite expectation. As a joint function also of p, we prove a similar result. For the measurability of Hs, notice that it is

defined in terms of integrals in Ic of the product of measurable functions of

( p , k), namely ,6 (k) (see a previous subsection) and measurable functions

of k and the Wiener path.

In particular if we assume that

then, we have

4.2. Other models

4.2.1. Brownian semimartingales

This section is based on the paper ‘. The results described above for the

Brownian motion extend with the same proofs to the case when the self

energy is defined as

where (Xt)tE[o,ll is a Brownian semimartingale, i.e. a process of the form

X t = Wt + 1 b,ds t

33

with (Wt)t,[o,ll a 3D Brownian motion and (bt)tEjo,l l a progressively mea-

surable process. We need the condition

to have that H s is integrable, and therefore to apply the abstract result.

This model with a Brownian semimartingale is quite flexible. It covers

the Brownian bridge (hence closed filaments) and non Gaussian examples

(based on Bessel processes) like processes living in a half space, with end-

points on the boundary of the halfspace (modelling filaments on a solid

boundary). See for more details.

Without any change, the same fact holds true for the model based on

where p k is the projection on the plane orthogonal to k . For open filaments

there is an argument in showing that this expression is preferable from

the fluid dynamic point of view.

4.2.2. Processes with finite p variation

In

with finite a variation, for a E (1,2), and p fulfills the stronger condition

the self energy (53) has been defined in the case when X is a process

The assumption on the process (Xt ) tc [o , l l is that for every p 2 1 there

exists a constant C, > 0 such that

IE [IXt - X,(P] I C,lt - q ' a , v s , t E [O, 11.

In this way we cover the fractional Brownian motion with Hurst param-

eter H E (i, 1) and non Gaussian variants of it, like solutions to nonlinear

stochastic equations driven by the fractional Brownian motion, or modi-

fications of the fractional Brownian motion conditioned to live in a half

space. Restricted to the gaussian case, the same problem has been solved

by different techniques by l6 under less restrictive conditions on p.

Under the condition

the inequality (52) holds.

34

References

1. P. Blanchard., E. Briining. (1992). Variational Methods in Mathematical Physics. A unified Approach. Springer-Verlag.

2. E. Caglioti., P. L. Lions., C. Marchioro., M. Pulvirenti. (1992). A Special

Class of Stationary Flows for Two-Dimensional Euler Equations: A Statistical Mechanics Description. Comm. Math. Phys 143, no 3, 501-525.

3. E. Caglioti., P. L. Lions., C. Marchioro., M. Pulvirenti. (1995). A Special Class of Stationary Flows for Two-Dimensional Euler Equations: A Statistical Mechanics Description 11. Comm. Math. Phys 174, no 2, 229-260.

4. A. Chorin. (1994). Vorticity and Turbulence. Springer-Verlag, New York.

5. F. Flandoli. A probabilistic description of small scale structures in 3D fluids. To appear on Annales Inst. Henri Poincark, Probab. & Stat.

6. F. Flandoli, M. Gubinelli. Gibbs ensembles of Vortex filaments. To appear on Prob. Theory and Related Fields.

7. F. Flandoli, I. Minelli. Probabilistics models of vortex filaments. To appear on Czechoslovak Mathematical Journal.

8. U . Frisch. (1998). Turbulence, Cambridge Univ. Press, Cambridge. 9. G. Gallavotti. (1996). Meccanica dei jluidi. Quaderni CNR- GNAFA n. 52,

Roma. 10. E. Hewitt, L. J. Savage. (1955). Symmetric measures on Cartesian products.

Trans. Amer. Math. SOC 40, pp470-501.

11. H. Kunita. (1984). Stochastic Differential Equations and Stochastic Flows of Diffeomorphisms, Ecole d’6t6 de Saint-Flour XII, 1982, LNM 1097, P.L. Hennequin Ed., Springer-Verlag, Berlin.

12. N. S. Landkof. (1972) Foundations of Modern Potential Theory, Springer- Verlag, New York.

13. P.L. Lions. (1997). On Euler Equations and Statistical Physics, Scuola Nor- male Superiore.

14. P.L. Lions, A. Majda. (2000). Equilibrium Statistical Theory for Nearly Par- allel Vortex Filaments. C. P. A. M, Vol. LIII, pp 0076-0142.

15. C. Marchioro, M. Pulvirenti. (1994) Mathematical Theory of Incompressible Noviscous Fluids, Springer- Verlag, Berlin.

16. D. Nualart, C. Rovira, S. Tindel. Probabilstic models for vortex filaments based on fractional Brownian motion. In preparation

17. D. Ruelle. (1969). Statistical mechanics: rigorous results, W. A. Benjamin, New York- Amsterdam.

REMARKS ON MEIXNER-TYPE PROCESSES

BJORN BOTTCHER AND NIELS JACOB

Department of Mathematics, University of Wales Swansea, Singleton Park, Swansea, SA2 8PP, Wales, UK

E-mail: mabb @swan. ac.uk, n.jaco b@swansea. ac. uk

We construct Feller processes called Meixner-type processes by making the param-

eters of the characteristic exponent of a Meixner process state space dependent. Our main tool is the theory of pseudo-differential operators. A further aim of this

paper is to popularize these methods among probabilists. Key words: Meixner process, Meixner-type process, pseudo-differential operators, L6vy-type processes.

MSC-classification: 60J75, 60J35, 35899

1. Introduction

L6vy processes are becoming more and more important in modeling, com-

pare the collection of surveys recently edited by 0. Barndorff-Nielsen, Th.

Mikosch and S. I. Resnick '. Several applications are related to the mathe-

matics of finance, see in particular the contributions of 0. Barndorff-Nielsen

and N. Shephard 4, and E. Eberlein 7 , in '. By definition a L6vy process has stationary and independent increments.

This is reflected by the fact that its transition probabilities form a con-

volution semigroup and that the generator A of the corresponding Feller

semigroup is translation invariant. On smooth functions it is given by

where $ is the characteristic exponent of the LQvy process under consider-

ation and 6, is the Fourier transform of u. Often $ depends on parameters,

i.e.

(6). $(c> = +a,b,c,...

When we model with a specific LQvy process having the characteristic ex-

ponent $ u ~ b ~ c ~ ~ . . ( < ) it may happen that at certain threshold values the pa-

rameters a , b, c, . . . change. Thus for fixed zo in the state space we have

a Levy process with characteristic exponent $,a(.o)ib(.o),c(.o),... (<), but in

general z ++ $~(.)~'(.)~'(.)~...(~) is not constant, i.e. the process modeling

35

36

such a situation will not any longer be a Levy process, but a Lkvy-type

process. To our knowledge it had been 0. Barndorff-Nielsen and S. Z. Lev-

endorski; who first handled such a situation in case of normal inverse

Gaussian (type) processes. Their work had been stimulated by problems of

modeling in finance. Their approach via pseudo-differential operators had

partly been influenced by the survey 13.

In this paper we take up the ideas from of treating L&y(-type) pro-

cesses with state space dependent parameters by using the theory of pseudo-

differential operators. The family under consideration consists of Meixner-

type processes, i.e. we start with the characteristic exponent (1) of a

Meixner process and make the parameters state space dependent. The

choice of these processes is due to more recent work of W. Schoutens 21

and W. Schoutens and J. Teugels 22 on modelling in finance with Meixner

processes.

In a first section we recall basic facts of Meixner processes and in the fol-

lowing section we introduce symbols of Meixner-type, i.e. symbols q(z, <) where for fixed 20 the function < H q ( q , <) is a characteristic exponent of

a Meixner process. It is proved that under reasonable restrictions on the

2-dependence of the parameter functions these symbols are elliptic sym-

bols in the classical class S1(R). Finally in Section 3 we prove that to every

Meixner-type process corresponds a unique Feller process, and some short

time asymptotics of the corresponding transition functions is discussed. An

asymptotic of the transition function with respect to the state space vari-

able will go along the lines of the considerations in and is not discussed

here.

A remark to the style of the paper: We aim to give more, and in a certain

sense new tools into the hands of those who are modeling with LBvy pro-

cesses or more general jump processes. More details (on a technical level)

of the impact of pseudo-differential operators in the theory of Markov pro-

cesses are given in 13, the more recent survey '' and in the monographs l4

and 15.

Acknowledgement: The second author would like to thank Ole

Barndorff-Nielsen and Sergei Levendorski for stimulating discussions about

their paper. The first named author acknowledges financial support from

the EPSRC-Doctorial Training Grant of the Mathematics Department of

the University of Wales Swansea.

37

2. Meixner Processes

In this section we summarize various results on Meixner processes, i.e. real-

valued LBvy processes whose characteristic exponent is a continuous nega-

tive definite function of type

a< - ib 2

$m,6 ,a ,b(<) := -Zm< + 26 where m E R, 6 > 0, a > 0 and -7r < b < 7r.

For the corresponding process (Xy'6'"'b)t20 we find

According to W. the transition density

'$'m,6,a,b is given by

Schoutens and J. Teugels 22 and B. Grigelionis lo

for the Meixner process with characteristic exponent

For the L6vy characteristic of $m,b,a,b (or (Xr769a9b) t>o) we find the drift

and the LBvy measure

where is the one-dimensional Lebesgue measure. Clearly, there is

and its variance by

i.e.

The expectation of Xy ' * l a lb is . given by

we have $m,6 ,a ,b(<) = ir< + JR,(,,) 1 - eixt +

A straightforward calculation yields

38

and

Im ~ L + 5 , ~ , b ( < ) = -m< + 26tan-1

Moreover we find

a6 Re ‘$m,6,a,b(<) 2 yl<1

for all < E R, 161 2 y , as well as with some co > 0

(9)

IIm ‘$m,fi,a,b(()I 5 CO(1 + Re ‘$m,6,a,b(<)).

I‘$m,fi,a,b(<)I 5 cl(l + 151)

(11)

(12)

In addition it follows that

holds for all < E R.

ample 4.7.32, and those in Z.-M. Ma and M. Rockner l9 we derive

Proposition 2.1. The Meixner process ( X r ’ 6 ’ a ’ b ) t 2 0 is associated with the non-symmetric Dirichlet form given on S(R) by

From the considerations of Chr. Berg and G. Forst 5 , see also 14, Ex-

E(u, s, ‘&n,6,a,b(<)a(<).;(E) d<. (13)

Its domain D(E) is the classical Sobolev space H i ( R ) , hence ( I , D ( € ) ) is regular, and on H1(R) we have b y (4) and (5)

E(u, w ) = y u’(z)v(x) d x s,

Since for the study of symmetric Dirichlet forms much more (analytic)

techniques are available, compare the monograph of M. F’ukushima, Y. Os-

hima and M. Takeda let us have a short look at the symmetric part

of the Meixner process ( X r 1 6 1 a 1 b ) t 2 0 , i.e. the Lkvy process ( q s 1 a 3 b ) t 2 0

with characteristic exponent being the continuous negative definite func-

tion Re ‘$m,fi,a,b. w e find now

as well as

39

and

For b = 0 (3) yields the transition density of (yt6'a9b)t20

For the general case a longer calculation, see B. Bottcher 6 , leads to the

following series representation of pf 'a 'b :

where

To proceed further we need some estimates for the derivative of $m,6,a,b.

Details of these calculations are given in B. Bottcher '.

Theorem 2.2. For all a E No there exists c, > 0 such that

(20) ( a )

l$m,6,a,b(()l 5 + 1c12)9 holds for all ( E R.

This theorem tells us that in the sense of Definition (3.1) the continuous

negative definite function ?/),$,a$ is a symbol in the class s1 (&I). Note that

also Re $,,6,a,b belongs to S1(R).

3. Symbols of Meixner-type

As mentioned in the introduction we want to make the parameters m, 6,

a and b in ( 1 ) state space, i.e. x-dependent and then identify (under some

conditions) this function of x and ( as a symbol of a pseudo-differential

operator generating a Markov process. In the following we denote by S(IR)

the Schwartz space of rapidly decreasing functions.

Definition 3.1. A. An arbitrary often differentiable function q : IR x R 4

C is said to belong to the symbol class Sk(IR), k E IR, if for all a,P E No there are constants cap 2 0 such that

la,"a,Pq(T,r)l I cap(1 + IEI2)+ (21)

40

holds for all x E R and 6 E R.

B. A pseudo-differential operator with symbol q E Sk(R) is any exten-

sion of an operator of the form

4 ( z , w 4 x ) = (2T) S, eiZcq(x, c)ii(t) d t , u E s(R). (22)

The class of all pseudo-differential operators with a symbol in Sk(IR) is

denoted by Xb (R) .

Let us introduce a class of symbols which we would like to call (smooth)

Meixner symbols.

Definition 3.2. A function qm,&,a,b : R x &! -+ @. is called a (smooth)

Meixner symbol if it has the representation

with m, 6, a, b E C-(R) satisfying for all k E No and x E R

0 < a; 2 a(k)(x) I a; < 00 -7r < b , I b(’)(z) 5 b; < 7r

0 < 6, 5 S(”)(x) 5 6; < 00

(24)

(25)

(26)

Im(”(x)I 5 m k (27)

where a t , b t , 62 and m k are real constants.

The class of all Meixner symbols is denoted by MS(R).

Remark 3.3. By definition every Meixner symbol q E MS(R) is a negative

definite symbol in the sense that for all x E R the function < ++ q(x, <) is a

continuous negative definite function.

Theorem 3.4. The class MS(R) is a subset of S’(R). Moreover q E

MS(R) is elliptic in the sense that

Re d x , €) 2 Yo(1 + l€ I2)+ (28)

holds for all x E R and < E R, 151 large.

Proof. Our proof follows the dissertation of B. Bottcher where more

details are given. Instead of using a formula for higher order derivatives

for composite functions, see L.E. F’raenkel 8, we use the special structure of

symbols in MS(R) and some elementary results:

(29) 7 r 7 r

, x E R and y E (-- -), &J

1 - sin2 y 2 ’ 2 I tanh(x + iy) I I

41

1 sech2(z + iy)l 5 cy(l + Jz12)-f, z E R,y E (--, 7 r T -) and r 2 0, (30) 2 2

and

(sech’ z)’ = -(sech2 z ) tanh z. (31)

Recall that sech z = A. Note that for IyI 5 C < 5 (where C is a constant)

we get the right hand side of (30) independent of y. Now, for q = qm,s,a,b E MS(R) we find the estimates

by observing the structure of the derivatives. Since

+ 26’(5) tan( -)b’(x) b(x) 2

) a(.)( - ib(z) a’(z)< - ib’(z)

)( 2 + 26’(5) tanh(

2

b(z) b”(z) b(z) b’’(z) + %(z) tan(-)-- + 6(z) sec2(-)-

+ 2S(z) tanh(

2 2 2 2

1 2 2 (33)

a(..)[ - ib(x) a’(.)[ - ib‘(x) + 2S(z) sech2( 2 ) ( 2 1

we may use (31) to reduce the estimates for @q(z, e) , p 2 3, to the estimate

for a,”q(z, [) and the estimates for z H (sech’ z )zk , and then (29) and (30)

give the result. Next observe that

) a(z)< - ib(z)

2 L$q(x, <) = -im(z) + 6(z)a(z) tank( (34)

which implies the desired estimates (21) for Q = 1, ,B = 0. Further we have

1 a(.)< - ib(z)

a:a;q(z, <) = -zm’(z) + 6’(z)a(z) tanh( 2

1 a(.)< - ib(z)

2 + 6(z)a’(z) tanh(

1 a(z)< - ib(z) a’(z)E - ib’(z)

2 )( 2 + S(z)a(z) sech’(

and again (29) and (30) yields the estimate (21). For o = 2 we find

) 1 a(.)< - ib(z)

2 a,”q(x, [) = -a’(z)S(z) sech2(

2 (35)

42

which again by (29) and (30) leads to (21) for a = 2, /3 = 0. But now (32)-

(35) together with (31) as well as (29)-(30) imply q E S'(R). The ellipticity

condition (28) follows from the restrictions for the parameters and (10).

Corollary 3.5. Let q E MS(R). Then Re q E S1(R) and Re q is elliptic in the sense of (28).

Thus we may apply the theory of "classical" elliptic pseudo differential

operators to pseudo-differential operators with symbols in M S ( R ) . This

will be done in the next section.

4. Meixner-type Processes

The aim of this section is to show that every pseudo-differential operator

-q(x, D ) with q being a Meixner Symbol has an extension, in fact a unique

extension, generating a Feller semigroup, hence gives rise to a Feller process,

or equivalently, for every q E M S ( R ) there is a stochastic process (Xt) t>o with state space R such that

and (Tt)t?o, where

Ttu(z) = E"(u(Xt ) ) , (37)

is a Feller semigroup on Cm(R), compare l3 and R. Schilling 20.

Since in principle the desired result is by now easily quotable, compare

W. Hoh l1 and 12, or the more comprehensive treatment in 15, Chap-

ter 2, we just outline the arguments and ideas to obtain the result, but

we do not repeat longer calculations leading to the estimates needed.

The construction of the Feller semigroup is based on the following vari-

ant of the Hille-Yosida-Ray theorem: If a linear operator (A,D(A)) on

Cm(R) is densely defined and satisfies the positive maximum principle, i.e.

Vu E D(A) s.t. u(z) = supyEau(y) 2 0 implies (Av) (z ) 5 0, and if for

some X > 0 the range of X - A is dense in Cm(R), then (A,D(A)) is clos-

able and its closure generates a Feller semigroup.

(For a proof we refer to l4 and the references given there on the origin of

this result.)

Since q is a negative definite symbol, it is clear that ( - q ( z , D ) , C ~ ( R ) ) satisfies the positive maximum principle and if we extend -q(z, D ) to some

Sobolev space H t ( R ) such that q(z, D ) ( H t ( R ) ) is continuously embedded

into C,(R), then also (-q(z, D ) , Ht(IR)) satisfies the positive maximum

43

principle, compare Theorem 2.6.1 in 15. The serious problem is the solv-

ability of the equation Xu + q ( x , D)u = f . This problem is overcome in

several steps:

1. Show that for every f E L2(R) and X 2 0 sufficiently large there is a

weak solution u E H i , i.e. u satisfies

~ x ( u , $1 := ~ ( u , 4 ) + ~ ( u , 410 = (f, $10 for all 4 E H ~ ( R ) ,

where B(., .) is the continuous extension of (u, u) I+ ( q ( x , D)u, u)o from

H y R ) to H i ( R ) .

2. Show that for f E H S ( R ) , s 2 0, every weak solution belongs to

HS+’(R).

3. Finally, starting with ( - q ( x , D ) , H 3 ( R ) ) , note that

4x1 D ) ( H 3 ( R ) ) c H 2 ( R ) - cco(R)l

and apply the Hille-Yosida-Ray theorem.

Note that the fact M S ( R ) c S’(R) allows in fact an application of

classical pseudo-differential operator theory as discussed for example by H.

Kumano-go in 18, whereas in W. Hoh or in l5 larger classes of negative

definite symbols which are not classical symbols are treated.

Thus we arrive a t

Theorem 4.1. Let qm,6ia)b E MS(R) be given by

a(.)< - i b ( x ) 2 I-

where for m, S, a , b the restrictions (24)-(27) do apply. T h e n ( -qm~6~a~b(x , D ) , H 3 ( R ) ) extends uniquely t o a generator of a Feller semigroup (T,(oo))t20 = (Tp’6’a’b)t20 on Cco(R), and in addition (36) and (37) do hold.

Corollary 4.2. If q = Re q m , 6 , a , b l q m ~ 6 ~ a ~ b E M S ( R ) i s as in Theorem (4. l ) , t hen - q ( x , D ) extends t o a generator of a Feller semigroup too.

The proof of Theorem (4.1), more precisely working out step 1-3, yields

more, namely that there is A0 > 0 such that for X 2 A0 the operator

-qm,6,a,b(x, D ) - Aid extends also to a generator of an L2-sub-Markovian

semigroup which we denote by (T,(2)’x)t20. Clearly on Cco(R) n L2(R) we

have

e - ~ t ~ , ( c o ) u = T , ( ~ ) J U a,e. (39)

44

~ q ( z , +(2)1= (2n)-+ I J , e i x ~ z , <)&(<) d<

Moreover, one can prove the estimate

(40) 2 l141H$ 5 cBx(u1 u)1

llu112* 5 a ( u , .) (41)

(42)

which implies, by Sobolev's embedding theorem (borderline case, compare

D. Adams and L. Hedberg I , Theorem 1.2.4.(b), p. 14), that

for all finite p 2 2. Using Theorem 8.7 in W. Hoh l1 we find now

llTt (2) I I L m - L z I c't-2

for any K = s , p > 2.

We may ask the natural question whether for t > 0 the operator Tt = TiDc)) has a representation as pseudo-differential operator and if so, how we can

calculate or approximate the symbol a(Tt)(z,<) of Tt. In case where all

parameters are constant the answer is easy:

( E ) Q ( < ) d< (43) 1 s, i xc - t q m J A b

Ttu(z) = ( 2 ~ ) - 7 e e

which holds (at least) for all u E S(R). Thus we should long for

(zL)t + r ( t , 2, <), (44) - q m , 6 , a , b

O t ) ( 3 : , < ) = e

where r ( t , z, E ) satisfies certain smallness conditions.

To proceed further we need some preparations. The class Sk (R) as defined

by (21) is a Frkchet space if topologized with the serninorms

5 cm(u)plc,o(q) (46)

and from (46) we even get a uniform bound with respect

Now we arew in a position to solve our problem by a straIGHTFORWARD APPLI-

CATION OF tHEOREM 4.1, cHAPTER 7.IN h. kUMANO-GO

A CONSTINUOUS LINEAR FUNCTIONAL IS GIVEN ON s (r) BY

WE HAVE COMPARE h. kUMANO-GO

45

Theorem 4.3. Let q = qm,6,a,b E MS(E%) be as in Theorem (4.1). Then the operator Tt has on S(R) the representation

Ttu(x) = (27r)-3 eizEa(Tt)(z, J)G(J) dJ (47)

t-0 l ima(Tt)(z ,J) = 1 weakly in S'(IW), (48)

where a(Tt)(x, J) satisfies

a(Tt)(z, J) = e-q(z>E)t + ro(t, z,[) (49)

(50)

where ro(t, ., .) E S-l(R) and

t-io lim ro(t, z, <) = o weakly in s-'(R.),

and {?ro(t,x,J); 0 < t 5 T } is for each T > 0 a bounded set in So@).

I t follows from Theorem (4.3) that

and

l i i eizEro(t, z, [ ) G ( [ ) dJ = 0.

Finally, let us consider the result in a heuristic way. Using the semigroup

property of (Tt),>0 - we arrive for small t > 0 at

~,+,u(x) M (27r1-4 S, eizEe-q(z>E)t (TsuT(5) dJ1

and assuming that for Ix - zol and t > 0 small

46

In particular, if s is small and therefore p,(y,A) can be substituted by

~ ~ p ~ ~ y ~ ' " y ~ ~ u ~ y ~ ~ b ~ y ~ ( ~ ) dz we find now for Iz - zoI as well as s and t small

t ha t

should be an approximation for p t f s ( z , A). For some further considerations

in this direction we refere t o 16.

References

1.

2.

3.

4.

5.

6. 7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

Adams, D. and L. I. Hedberg, Function spaces and potential theory. Vol. 314 of Grundlehren der math. Wissenschaften. Springer Verlag, Berlin 1996. Barndorff-Nielsen, 0. and S. Z. Levendorski:, Feller processes of normal inverse Gaussian type. Quantitative Finance 1 (2001), pp. 318-331. Barndorff-Nielsen, O., T. Mikosch, and S. I. Resnick (eds.), Le'vy Processes - Theory and Applications. Birkhauser Verlag, Boston 2001. Barndorff-Nielsen, 0. and N. Shephard, Modelling by Le'vy processes for financial econometrics. In 3, pp. 283-318. Berg, C. and G. Forst, Non-symmetric translation invariant Dirichlet forms. Inventiones Math. 21 (1973), pp. 199-212. Bottcher, B., PhD-thesis, University of Wales Swansea. (In preparation). Eberlein, E., Application of generalized hyperbolic Le'vy motions to finance. In ', pp. 319-336. Fraenkel, L. E., Formulae for,higher derivatives of composite functions. Math. Proc. Cambridge Phil. SOC. 83 (1978), pp. 159-165. Fukushima, M., Y . Oshima, and M. Takeda, Dirichlet forms and symmetric Markov processes, Vol. 19 of de Gruyter Studies in Mathematics. Walter de

Gruyter Verlag, Berlin 1994. Grigelionis, B., Processes of Meixner type. Lithuanian Math. J . 39 (1999),

Hoh, W., Pseudo differential operators generating Markov processes. Habili-

tationsschrift. Universitat Bielefeld, Bielefeld, 1998. Hoh, W., A symbolic calculus for pseudo differential operators generating Feller semigroups. Osaka 3. Math. 35 (1998), pp. 789-820. Jacob, N., Pseudo-differential operators and Markov processes. Vol. 94 of

Mathematical Research. Akademie Verlag, Berlin 1996. Jacob, N., Pseudo-Differentia1 Operators and Markov Processes, Vol. I: Fourier Analysis and Semigroups. Imperial College Press, London 2001. Jacob, N., Pseudo-Differential Operators and Markov Processes, Vol. 11: Gen- erators and Their Potential Theory. Imperial College Press, London 2002. Jacob, N. and R. L. Schilling, Estimates for Feller semigroups generated by pseudo differential operators. In: Rakosnik, J . (ed.) , Function Spaces, Dif-

ferential Operators and Nonlinear Analysis. Prometheus Publishing House, Praha 1996, pp. 27-49.

Jacob, N. and R. L. Schilling, Le'vy-type processes and pseudo differential operators. In 3, pp. 139-168.

pp. 33-41.

47

18. Kumano-go, H., Pseudo-differential operators. MIT Press, Cambridge MA

1974.

19. Ma, Z.-M. and M. Rockner, A n introduction to the theory of (non-symmetirc) Dirichlet forms. Universitext. Springer Verlag, Berlin 1992.

20. Schilling, R. L., Conservativeness and extensions of Feller semigroups. Posi-

tivity 2 (1998), pp. 239-256. 21. Schoutens, W., The Meixner process: Theory and applications in finance.

Preprint 2002.

22. Schoutens, W. and J. L. Teugels, Le'wy processes, polynomials and mart in- gales. Commun. Statist.-Stochastic Models 14 (1,2) (1998), pp. 335-349.

SOME REMARKS ON IT0 AND STRATONOVICH

INTEGRATION IN 2-SMOOTH BANACH SPACES

ZDZISlAW BRZEZNIAK

Department of Mathematics

University of Hull Hull HU6 7RX, U.K.

E-mail: z. brzezniakQmaths.hu11.ac.uk

In this paper we study It8 integral in 2-smooth Banach spaces. Burkholder inequal-

ity is proved using It6 formula in certain subclass of such spaces. Relationship with

an integral introduced recently by Mikulevicius and Rozovskii is discussed. Finally,

Wong-Zakai type approximation for such integrals is proved.

1. Introduction

This paper has its origin in the author’s attempt to understand an impor-

tant work by Mikulevicius and Rozovskii 28. In order to study stochastic

Navier-Stokes equations in Rd for d = 2,3 in Sobolev space HS9P, the au-

thors introduce a new type of It6 integral for some Banach space valued

processes. One of the aims of the current presentation is to show that the

Mikulevicius-Rozovskii integral is a special case of an integral in 2-smooth

Banach spaces first introduced by Neidhardt in 31 and then extensively

studied and used by the present author and his collaborators. The main

object however is to present a concise and detailed exposition of the sub-

ject. The paper is organised as follows. In the section 2 we recall the basic

definitions, i.e. of 2-smooth Banach space and of It6 integral with valued

in 2-smooth Banach space. Section 3 is devoted to statement and proof of

the Burkholder inequality for It6 integrals taking values in certain class of

Banach spaces. Let us note here that Burkholder inequality is valid in 2-

smooth Banach spaces, see l6 and 32. The class of Banach spaces considered

in this section is big enough as it contains important examples of L P , p 2 2 and Besov and Sobolev-Slobodetski spaces. In section 4 we show how the

theory on It6 integration in 2-smooth Banach spaces can be used to solved

certain nonlocal stochastic differential equations. In the section 5 we inves-

tigate the relationship of the integral introduced by Mikulevicius-Rozovskii

with the one in 2-smooth Banach spaces. We show that the former is a

48

49

special case of the latter. For this we use a result of the author and Peszat

on identification of y-radonifying operators with values in LP-spaces with

certain class of integral operators.

We conlude the paper with a discussion of dependence of the It6 integral

on the Wiener process. We prove that the Stratovich integral is equal to

limits of the Riemann sums with the mid-point approximations is repalced

by interval averages. Our result should be seen as in conjunction with

the authour's paper with A Carroll on Wong-Zakai approximation for

stochastic differential equations in 2-smooth Banach spaces.

2. It6 integral in 2-smooth Banach spaces

In what follows X will be a real Banach space with norm I . I. A modulus

of smoothness of (X, I . I) is defined by

1 px ( t ) := sup - (12 + tyl + 12 - tyl) - 1.

Ix1=lyl=l 2

A Banach space (X, I . I) is called 2-smooth iff there exists a norm I . I on

X , equivalent to I + 1 and k > 0 such that the modulus of smoothness px of

(X, I . I ) satisfies

P X ( t ) 5 kt2, t E (0,1].

The notion of a 2-smooth Banach space was introduced by Pisier in 34.

Pisier proved there that X is a 2-smooth Banach space iff one of the fol-

lowing two conditions is satisfied

(i) There exists a constant A > 0 such that

12 + yI2 + 15 - y12 I 2(212 + AIyI2, 5, y E X. (1)

(ii) there exists a constant C = C z ( X ) > 0 such that for any X-valued finite

martingale { Mk} the following holds

In fact, the implication X is 2-smooth -----r. (i) had been earlier proven by

Figiel & Pisier in 19, see also l4 p. 144. The proof of converse implication,

only alluded to in 34, is rather straightforward.

A Banach space X satisfying property (2) is usually called an M-

type 2 Banach space. Although an It6 type integral for 2-smooth Banach

spaces was first introduced by Hoffmann-Jorgensen and Pisier l8 only for

1-dimensional square integrable martingales, a complete construction was

50

carried out by Neidhardt in 31, see also Belopolskaya and Daletskii 2, Det-

tweiler 16, Brzeiniak and references therein. In order to introduce this

integral we need one more new notion, i.e. of a y-radonifying operator. If

H and X are separable real Hilbert and resp. Banach spaces, a bounded

linear operator L : H -+ X is called y-radonifying iff L ( ~ H ) is a-additive,

where Y H is the canonical Gaussian distribution on H . If this is the case,

L ( ~ H ) has a unique extension to a a-additive Bore1 probability measure VL

on X. One can then also show that VL is a centered Gaussian measure on

X with Reproducing Kernel Hilbert Space (RKHS) (i.e. the Cameron Mar-

tin space) equal to H . In particular, in the spirit of L Gross 17, the triple

( H , X , VL) is a Abstract Wiener Space (AWS). The set of all y-radonifying

operators from H to X we will denote by R ( H , X ) . Note that in 31 and

earlier papers this set is often denoted by R ( H , X ) . For L E R ( H , X ) one

puts

Neidhardt in 31 proved that 1 1 . 1 1 is a norm on R ( H , X ) , that R ( H , X ) with that norm is a separable Banach space and that the set Cfi,(H,X) of bounded linear operators L : H + X with finite dimensional range, is

a dense subspace of R ( H , X). It follows from Baxendale that R(H, X ) is an operator ideal, i.e. if L E R(H, X ) , A E C(G, H ) and B E C ( X , Y ) (where G and Y is another separable Hilbert, resp. Banach space) then

also BLA E R(G,Y) and JJBLAIIR(G,Y) I CIBIqx,r) IILIIR(H,x)IAILc(G,H) for some constant C independent of A, B and L.

Let us fix an orthonormal basis (ONB) {ek}k of H and let us de-

note by I In the projection onto the space spanned by e l , . . . ,en. Let us

choose and fix an i.i.d. sequence of standard centered real valued Gaus-

sian random variables ,&, k E N. It follows from the Itb-Nisio Theorem,

see e.g. 23 then L E R ( H , X ) iff (IE)C,,&Lek1$)'/2 < 00. Moreover,

llLll = (IE I C , , P ~ L Q ~ $ ) ~ / ~ . One can also show that the exponent 2 above

can be replaced by any p E (1, m). Denote, for n E N, by CCfin,(H, X ) be space of L E C ( H , X ) such that L = LHn. Note that U,LCfin,(H,X) is dense in R ( H , X ) . We fix a filtered and complete probability space

'u = (n, F, (Ft)tc[O,~l,P). We have, see 26,

Definition 2.1. An (3t)-adapted canonical cylindrical Wiener process o n H is a family W(t ) , t 2 0 of bounded linear operators from H into

L2(R, F, P) such that:

and 13, the following

(i) for all t L 0, and $, cp E H , E W(t)+W(t)cp = t($, c p ) ~ ,

51

(ii) for each $J E H , W(t)+, t 2 0 is a real valued (Ft)-adapted Wiener

process.

One can show that if W(t) , t 2 0, canonical cylindrical Wiener process

on H iff there exists an orthonormal basis { e k } k of H and a sequence ,&(t), t 2 0, k E N of standard real valued (3t)-adapted Wiener processes such

that W(t)+ = C k P k ( t ) ( q b , e k ) , for all + E H and all t 2 0. If W(t ) , t 2 0

canonical cylindrical Wiener process on H then by @(t) we will denote the

series Ck P k ( t ) e k .

If S is a normed vector space endowed with some a-algebra p, then for

0 5 a < b I 00, N(a, b; S ) denotes the set of all progressively measurable

S-valued processes 7 : [a, b) x R + S. If p E [l, m), then we set

M P ( a , b; S ) := {e E N(a, b; S ) : IE (e(t)l$ d t < CO}, (4) I” Then, we define MP(a, b; S ) to be the space of all equivalence classes of

elements of &tP(a, b; S ) with respect to a natural equivalence relation, E N q iff IE Jab Ic(t)-q(t)lP dt = 0. Note that MP(a, b; S ) is complete if S is. Denote

finally by MfteP(a, b; &,(H, X ) ) the class of all < E MP(a, 6 ; R(H, X ) ) such that there exists rn E N and a partition 7r = {a = t o < t l < . . . < t, = b} of the interval (a,b) such that c(t) = [ ( t k ) & , t E [ t k , t k + l ) ,

k = 1,. . . , n - 1. One can show, similarly to 3 1 , that the latter space is

dense in MP(a, b; R(H, X ) ) . Now we will define a linear map I : MZtep(a, b; Lfi,(H, X ) ) + L2(R; X )

by the standard way. Thus, if c E Mstep(a, b; .&,(H, X)) with partition

7r = { a = t o < tl < . . . < t, = b} then we put, with F@(t) = C j , B j ( t ) e j ,

Since for L E Lf i , (H,X)) , Ll/ir(t) E L’(R,X), I is a well defined

linear map. Denoting, M k = x:Ii ( ( t j ) ( @ ( t j + l ) - @ ( t j ) then the

sequence ( M k ) k is an X-valued martingale (with respect to filtration

( 3 t k ) k . Indeed, if an FS measurable L : R --+ R ( H , X ) is such that

L = LII, and E : R + P,(H) is 3 measurable (and both are square

integrable) then IE (L<lFS) = LIE (<IF8). Therefore, IE(I(<)(’ = ElM,I2 5 c2(x) I E l ( ( t k ) ( @ ( t k + l ) - @ ( t k ) ) On the other hand, note that if

L E &,(H,X)) then L = LII, for some n E N and so E(L@(t)12 =

E (LII,@(t)( = IE 1 Cj”=, / ? j ( t ) L e j ( ’ = t l (L((2. Therefore, we have proved

)

52

that

One should mention here that in order to prove (6) both Neidhardt 31

and Dettweiler l5 used the property (i), while the author in and above

has used the M-type 2 property (ii). The last inequality (6) shows that

I is a bounded (obviously) linear map from I : Mstep(a,b;&,n(H,X)) to

L2(R; X ) . Since the former is dense in M2(a, b; R(H, X)), I has a unique

extension to a bounded linear map from the whole of M 2 ( a , b; R ( H , X ) ) with values in L2(R,IF,X). Moreover, this extension, also denoted by I satisfies

b

W(t)I2 I C 2 ( X ) E / Ilt(t)ll&H,X) dt. (7)

Let us recall, see e.g. 22 that a stopping time T is called accessible

iff there exists an increasing sequence of stopping times r, such that a.s.

T, < T and limn--tmrn = T . For a stopping time T we set Rt(7) = {w E

L? : t < ~ ( w ) } , [ o , ~ ) x R = { ( t , w ) E [0,00) x R : 0 5 t < ~ ( w ) } . For an

admissible processa Q : [ O , T ) x R --t X we define

I ' S ( s ) d W ( s ) = w [ o , T ) r ) ?

where I = la$. One can prove that for 0 I r 5 t !E s," c(s) dW(s)lFT) =

Ji t ( s ) dW(s). We also have, see 4 , the following

Proposition 2.1. If X is 2-smooth Banach space and 5 E ML,(O, a; R ( H , X)), then

t (1) The process x ( t ) := so [ ( s ) d W ( s ) , t 2 0 is an X-valued martingale,

(2) for any T 2 0, with almost all paths continuous; moreover x E MZ,(O, 00; X),

T

ESUP t<T I S t C ( S ) o W s ) 1 2 I c2(x)q 0 Ilt(4II;(H,X) ds.

In particular, x E L2(R, C(0, T ; X ) ) .

ai.e. (i)qlnt : Rt + X is Ft measurable, for any t 2 0; (ii) for almost all w E R, the function [O, .(LO)) 3 t H q(t , w ) E X is continuous.

53

Proof. According to the statement preceding the Proposition, the process

z(t) , t 2 0 is an X -valued martingale. To prove the remaining two claims,

we first assume that : 0 =

t o < ... < t, = T , c(t) = c(ti) E L2(R,.Ftti,P;X) for t E [ti,ti+l). Then

z(t) = ~ , , < , [ ( t i ) (@(ti+l A t ) - @(ti)), and so z E C(0 ,T ;X ) a.e. Since

also 1x1 is non-negative submartingale, applying the real version of Doob

inequality, see 21 or Theorem IV.8.2 in 26, we infer that

is an adapted step function with

Therefore the operator 1 : M&,(O, T ; &,(H, X)) + L2(R, C(0, T ; X)) defined above for simple functions can be uniquely extended to the whole

space M2(0 , T ; R(H, X ) ) . We use the fact that the space of progressively processes in

L2(R; C(0, T , X)) is closed therein.

We conclude this section with a statement of an It6 formula, see 31 and

8. But first we define an important concept of a trace of a bilinear map. If X, Y, Z are Banach spaces and A : X x Y -+ Z is a bounded bilinear map

and A E R(H, X) , B E R(H, Y ) , then, see 8, we put

j

The series is absolutely convergent and its sum is independent of the choice

of the ONB {e j } . If x = Y and A = B we write trAA instead of trA,AA.

Theorem 2.1. (It6 Formula) Assume that X and Y are %smooth Banach spaces. Let 0 5 c < d 5 00. Assume that a function f : [c, d ) x X 4

Y is of C172 class, i.e. f is Fre'chet differentiable, the Fre'chet derivative f ' :

[c, d ) x X + C(R x X , Y ) is continuous and differentiable in the X-direction with the resulting derivative being continuousb. Let, f o r a E JV&(C, d; X)

and < E n/12c(c, d; R(H, X)),

z(t) = ~ ( c ) + I " L a(.) ds + ((s) d W ( s ) , t E [c, 4. (8)

Then for all t E [c, d ) , a.s,

~ ~ ~~

bSimply, g, appropriate space.

and 3 exist and are continuous on [ c ,d ) x X with values in the

54

3. Burkholder inequality

In this section we assume that our

( H ) X is a real separable Banach space such that there exists p E [2 ,co) for which the function ' pp : X 3 x H l x l p E IR is of C2 class and there

are k 1 , kz > 0 such that for every z E X, lp'(x)I 5 klIz(p-' and ('p''(x)( 5 2k2 121p-2 .

Note that the Sobolev Hsip-spaces with p E [2, co) and s E IR satisfy the

condition (H). Moreover, a Banach space X satisfying (H) is 2-smooth7

see l4

If q 2 p , the following is a special case of Theorem 1.1 from 12.

Theorem 3.1. Assume that X is a Banach space satisfying the condition (HI. Assume that 5 E M ~ , ( O , c o , R ( H , X ) . Let x( t ) = J,"C(s)dW(s), t 2 0. Suppose q E ( 1 , a). Then there exists a constant Kq > 0 depending only o n q, H , X , and the constants kl, kz appearing in (H), such that for

every T > 0,

Remark 3.1. With a slight modification of the proof below one can show

that in fact the Burkholder inequality above is also valid for any accessible

(and hence any bounded) stopping time.

Theorem 2.1. was proved in 31 in the case a

is a bounded bilinear map, then

Let us state an important

55

Proof. The first step is to prove this result for q = p . We follow the above

mentioned paper l2 where a more general result is studied. Suppose first

that A is a bounded dissipative linear operator. Since ‘p(x) = 1xIP is of C2 class we can use It6’s formula of Neidhardt, see Theorem 2.1 above, and

obtain

Consider a process y(t) defined by

t

Y ( 4 := 1 9’ ( 4 s ) ) ( E ( 4 ) dW(s) ,

4 ( t ) = 1 (4 ( 4 s ) ) a s ) ) O (4s ) ) CW* ds-

t 2 0.

Obviously, y is an R-valued martingale with quadratic variation

t

From the inequality (25) in 31, i.e. for L E L ( X , W), B E R ( H , X),

2 l(LB) 0 (W*I I lFl:(x,R)l%(H,E) I: I~ l~(X,,) l I~ l l~(H,E,.

Therefore, by (H), we infer that

Applying next the Davis inequality, see also 33, we arrive at

Now, we shall deal with the second term on the RHS of (13). Since for L from R ( H , X ) and a bilinear mapping A : X x X 4 R, Itr A o (L , L ) ( I ]A1 . JJL))2, we have, again by ( H p ) ,

Combining this with the previous estimates we obtain

56

We shall study each term on the RHS of (13) separately. Let E > 0. First

we have

where we have used Holder and Young inequalities. Similarly for the second

term we have

Choosing now E > 0 such that

p - 1 p-2 1 kg & = - (ski- P + - I P 2

we obtain, for some generic K > 0,

which proves (12) for q = p. The proof in the case q > p follows the same

lines. It is enough to observe that if the Banach space satisfies the condition

(H) with p 2 2 then it also satisfies this condition with q 2 2. In order to

complete the proof we need to consider the case q < p. The proof in this

case is motivated by the proof of the Burkholder inequality given by Revuz

and Yor in 36. It is based on the following (see Proposition IV.4.7 therein)

57

Proposition 3.1. Suppose that a positive, adapted right-continuous process Z is dominated by an increasing process A, with An, i e . for every bounded stopping time 7, IEZ, 5 EAT. Then for any k E (0, l),

2-lc IE sup zk < - IEAL.

O l t < o o - 1 - k

Let now q E (1,p). We apply Proposition (3.1) to processes zt = )xtlPl{tlT) P/2

and At = (s," I IE (s ) l l " ) ) lttST} with k = q / p . We have just proven

that Z is dominated by A. Since Z is continuous (by Proposition 2.1),

This proves completely Theorem 3.1. 0

4. An Example

This Example is motivated by a question raised by Terry Lyons. Let S1 be the unit circle (with normalized Haar measure) and let H = H19"(S1)

and B = L2(S1). Let E = H"ip(S1) with < a < 3. Let us recall that

H"ip(S1) = [LP(S1); HIJ'(S1)la, the complex interpolation space. Then it

is well known that E is a Banach algebra. Note also that E is 2-smooth

Banach space.

Consider three maps

A : E 3 u H { H 3 H u . y E E } E R(H, E ) ,

A : E 3 u H { B 3 y H U * Y E E } E R(R,E) , B : E 3 u H { E 3 y H u.y E E } E L(E,E) .

(15)

(16)

(17)

Since E is a Banach algebra, B is a (well defined) bounded linear operator.

Since A(u) = B(u) o i , where i : H E is the natural embedding, the

map A : is well defined and bounded as well. Here we use the fact that

Since for f E L'(S1) the map A, : u H f * u is bounded from LP

into L P (by the Young inequality) and from HIJ'(S1) into H1+'(S1) (by the

former fact and equality D(f * u) = f * (Du)) we infer, by means of the

interpolation theory, that As is a bounded linear map from E = H"J'(SI)

into itself. Therefore, is a bounded linear map from E into L(L1,E) , hence into C ( f i , E ) . To prove that is a bonded linear map from E into

R(I?,E) we argue as follows. Let u E E. Then, Dau E LP(S'), where D"

i E R(H,E) .

58

is the fractional power of the derivative operator D . We will show first that

if u E El then

L2(S1) 3 y H D"(u * y) = (D"u) * y E Lp(S1)

is y-radonifying. Obviously, it's enough to show that for

linear operator

E LP(S1) the

K : L2 3 y - v * y E Lp

is y-radonifying. Note that K is an integral operator with a kernel k ( z , y ) :=

v(x - y ) (here we treat S1 as a group). Since

is finite, as v E LP C L2, the result follows, see l1 and Theorem 5.1 in the

next section. In fact we have proven that the map A : E -+ M ( H , E ) is

well defined, linear and bounded. Hence the following result follows directly

from 31 and 8 .

Theorem 4.1. Suppose in addition that W(t) , t 2 0 and l%'(t), t 2 0 are two independent cylindrical Wiener processes with respect to Halbert spaces H and H respectively. Then for every uo E E there exists a unique continuous E-valued process that i s a solution to

du ( t ) = u(t) d W ( t ) + u(t) * d W ( t ) u ( 0 ) =uo.

Since H L--) H we also have the following

Theorem 4.2. Suppose in addition that W(t) , t 2 0 is an H - cylindrical Wiener processes. Then for every uo E E there exists a unique continuous E-valued process that is a solution to

(19) du( t ) = u(t) d W ( t ) + u(t) * d W ( t ) u ( 0 ) = uo.

Remark 4.1. The reason we used the H"1p spaces and not the Sobolev-

Slobodetski W"tP was that for u E Wa,p, D"u may not be an element of

L2. Thus Theorem 4.1 may be not true in this case. However, in Theorem

4.2 we can use E = WaJ'(S1).

59

5. Relationship with the approach of Mikulevicius and Rozovskii

Suppose Y is a Hilbert space and W(t ) be a canonical Y-cylindrical Wiener

process. Consider an LP(0, Y)-valued process g ( r ) , r 2 0. Consider a map

* from LP(0, Y ) to R(Y, LP(0)) defined by the formula, with g E LP(0, Y)

where g j ( z ) := ( g ( z ) , e j ) , x E 0. Let us recall the following result, first

stated in l1 (see also lo, where a complete proof is given).

Theorem 5.1. Suppose Y is a separable real Hilbert space and let p E (1, a) be fixed. Let (0, F, u) be a a-finite measure space. For a bounded linear operator K : Y 4 LP(0) the following assertions are equivalent:

(1) K is y-radonifying; (2) There exists a u-measurable function K. : 0 --+ Y with

such that f o r all u-almost all x E 0 we have

( K ( Y ) ) ( X ) = ( K ( X ) , Y ) , Y E Y.

Moreover, there exists a constant C > 0 such that for all IC E LP(0, Y),

In fact, the above Theorem means that the map LP(0,Y) 3 K. H K E

M(Y,Lp(O)) is an isomorphism of Banach spaces. Since by the Parseval

formula, for y E Y and x E 0

& A x ) ( Y , e j ) = C ( 9 ( 4 , e j ) ( Y , e j ) = (S(X) ,Y) j j

we infer that the map A:g H 4 is nothing else but the isomorphism K. H K from Theorem 5.1. Therefore, for a process g E Mi,(O, a; LP(0, Y)) we

can define an LP(O)-valued integral s,” g( r ) d W ( r ) simply by putting

l g ( r ) d W ( r ) = i j ( r ) d W ( r ) , t 2 0. I’ This integral, being just a special case of the integral introduced earlier in

section 2 satisfies all its properties. In particular, it satisfies the Burkholder

60

inequality (12), a special case of which in the present situation takes the

following form. If p 2 2, then

Let us now show its another property whose a byproduct is that it coin-

cides with the It6 type integral of Mikulevicius-Rozovskii, see 28 (subsection

5.1 in the Appendix) and 27.

Proposition 5.1. Under the above assumptions, af cp E LP(0) with

9j(., 5, w ) = ( g ( r , 2, w ) , e j ) , then

Proof. This result is in fact a special case of the It6 formula, see Theo-

rem 2.1. Indeed, cp can be identified with a bounded linear map on LP(0). Since then for e E LP(0), = cp and cp”(J) = 0, we get that a.s.

E

L(Y,R) = Y* E Y and observing that L(Y,R) S R(Y,R) the integral

S,”c(r)dW(r) is again a special case of the It6 integral from section 2.

Hence, S,”S(r) dW(r) = C,”=, t j (r)dWj(r) = C&(gj(r),cp)dWj(r) what

concludes the proof of the Theorem. 0

(s,” g(r) dW(r) , 9) = J;(g(~)cp)dW(+ Denoting by E(r) = M r M

Remark 5.1. The above can be generalised to any Banach space X which

is isomorphic with the space LP(O), in particular for the Bessel spaces

He>p(Rd). Indeed, the isomorphism between the latter space and LP(Rd) is

given by f H (1 - A)e/2f.

6. Approximation of the Stratonovich integral

We conclude this paper with a brief discussion of the relationship between

the It6 and Stratonovich integrals in the framework of 2 smooth Banach

spaces. One should mention here a recent paper by Ledoux, Lyons and Qian

24, where a very novel approach this question for solutions of stochastic

difhential equations in Banach spaces via rough path theory of T Lyons is

discussed.

61

6.1. The result

w e fix a filtered and complete probability space % = (a, F, (Ft)t@J-], IF'), a

separable Hilbert space H and an (Ft)-adapted canonical cylindrical Wiener process on H , see Definition 2.1. Let us fix an ONB { e k } of H. @(t) :=

CF1(W(t ) , e j )e j . We suppose that the Wiener process @(t), t 2 0, lives

on a some Banach space E 2 H. With certain abuse of notation we will

denote the latter simply by W(t ) , t >_ 0. Recall that for A E L(E, E ; X ) ,

trA = C A ( e j , e j ) j

and the sum is independent of the ONB {e j } .

(23) t - t;

Wn(t) = W ( t l ) + ty+l - t; (w(t:+A - w w ) 7

where 0 = tt < t? < . . . < t"Nn, 5 T < t",n)+l < 00 is a partition of the

interval [O,T] . Recall that L(E, X ) is the space of bounded linear maps from E to X

and that the imbedding C ( E , X ) 3 A H Ao E Z R ( H , X ) is bounded. Here

i : H 4 E is the canonical imbedding. Our main result is the following

theorem.

Theorem 6.1. Suppose that the progressively measurable stochastic processes a(.) and b(s), respectively X and L(E, X)-valued, have almost all trajectories continuous, and for some fixed T < 00,

Assume that F : [O,T] x X -+ C ( E , X ) is of class C1 in t and C2 in x, with the second derivative bounded on bounded sets. Finally, let c(t) be an X-valued process such that

Then

62

Remark 6.1. We state and prove this theorem for the second moment.

A generalization to any p 2 2 is possible and will be discussed in ‘. For

simplicity and clarity of exposition we only give a proof in a special case of

one-dimensional Wiener process, i.e. H = E = R and (identifying C(W, X) with X ) F( t ,x ) = x , t E [O,T], x E X . We will also assume that a(t) = 0

for t E [O,T]. The proof in the general case will also be discussed in ‘.

Corollary 6.1. Suppose E = X and [ ( t ) = W ( t ) , t 2 0. Consider the following approximation sums

From Theorem 6.1 we infer their convergence to the Stratonovich in-

tegral of F ( W ( s ) ) (see for the Definition of the Stratonovich integral).

Therefore, the Stratonovich integral appears not only by the choice of mid-

point values of the integrand but also by taking its integral averages. See

also Mackevicius 2 5 .

6.2. The proof

We show that if [ ( t ) = s,” b(s) dW(s) and (24) then

where W, is defined by (23). Proof of (27) Let us denote

m,(t) = Sup{k : tk 5 t }

In what follows we shall try to drop the sub-(super-)script n whenever

we are not facing ambiguity. Moreover for simplicity we assume for the

time being that t; = . Thus we have

k m(t ) = m,(t) = sup{k : tk 5 t } = SUp{k : ; 5 t }

m(t) + 1 ) - WC"")) n

= I2(t) + I;@) + I;@)

Lemma 6.1. Under the above assumptions and notations we have

Proof. From the uniform continuity (on interval [0, TI) of paths of both

processes W(t ) and <(t) we have

sup II;(t)l2 -+ o a.e. o<t<T

Moreover ,

From (24) and the Doob inequality, see Theorem IV.8.2 in 26 and Proposi-

tion 2.1 in this paper, we infer that

We conclude the proof of Lemma 6.1 by applying Lebesgue dominated

convergence theorem. 0

Lemma 6.2.

64

Proof. E o m Proposition 2.1 we have

r t

where in(.) = [(k) for 5 s < % I m(T) and &(s) = 0 for M ( T ) 5 s 5 T and C is a generic constant (which value can change from line to

line).

We conclude the proof by observing that from Lebesgue dominated conver-

0 T

gence theorem EJo l[(s) - tn(s) l2ds + 0 as n -+ m.

The main point in the proof lies in the following

Lemma 6.3.

Proof. From the integration by parts formula we have

Therefore

The last equality can be written in the following way t

I z ( t ) - 1 b(s) d s = I i1( t ) + Iz2(t),

s)b(s) ds . k + l m(t)-1

k=O n

First we shall show that

65

(36)

* Since n Sk (w - s ) d s = n so& s ds = & we obtain - n

n

k + l c n l * ( D - s ) b ( s ) d s =

c nJ**(* n - s) (6(s) - b ( k ) ) ds

m(t)-1

k=O n

m(t)-1

k=O n

k n n

m(t)-1 + c n L * ( - - - r ) b ( - ) d s k + 1

k=O K

1 1 k m(t)-1 m(t1-1

k=O n ( ' k-0 ,!I(--) l c t l S) b ( s ) - b ( - ) d s + ?

and thus in order to prove (37) it is enough to prove

However (38) and (39) easily follow from continuity of paths of the

process b(s) and assumption (24) by applying the Lebesgue Dominated

Convergence Theorem. 0

66

The crux of the matter is to prove the following

Lemma 6.4.

Proof. Since 122(t) is constant on each time interval (k, y] we have

where

is a

, where c k = F w . For

m(T)-1 Now we are going to show that for fixed n, the sequence (Y?,)k=O martingale with respect to a filtration ( C k ) + O

this it is enough to show

m(T)-1

I E ( X p - 1 ) = 0 (43)

This follows from the following

In the last equality we used Corollary 2.1.

Therefore, from (41) by the M type 2 property of X , see (2), we obtain

m(T)-l m(T)-1

I IEIY,"(,)12 =El c X1l2 s C 2 ( X ) c IEIXTI2(44) i = O i=O

67

Let us observe that each term on the right hand side of (44) can be estimated

in the same way. Thus we may take i = 0 and get

By the Proposition 2.1 we have

where as usual, C > 0 is a generic constant. Similarly we have

From the last twoinequalities, (44), (45) and the fact that m(T) = m,(T) 5 0 cn we get (40). This concludes the proof.

Acknowledgments

The authour would like to thank Marek Capihski, David Elworthy, Terry

Lyons, Jan van Neerve, Martin Ondrejat, Szymon Peszat and Boris Ro-

zovskii for their helpful discussion on various topics related to this paper.

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12. Brzeiniak, Z. and Peszat, S., Maximal inequalities and exponential tail estimates for Stochastic Convolutions in Banach Spaces, pp. 55-64 in STOCHASTIC PROCESSES, PHYSICS AND GEOMETRY: NEW INTERPLAYS. I A Volume in Honour of Sergio Albeverio, CMS Conference Proceedings,

v. 28, Providence, Rhode Island (2000). 13. Brzeiniak, Z. and Peszat, S., Stochastic two dimensional Euler Equations,

Annals of Probability 29, 1796-1832 (2001) 14. R. Deville, G. Godefroy and V. Zizler, SMOOTHNESS AND RENORMING IN

BANACH SPACES, Pitman Monographs and Surveys in Pure and Applied Mathematics 64, Longman Scientific and Technical, 1993.

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DIFFUSION PROCESSES, North-Holland, Amsterdam - Oxford - New York

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INTEGRALS: SINGLE AND MULTIPLE, Probability and Its Applications,

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stochastic integral, Stochastics 13, 167-175 (1984) 26. Metivier, M. and Pellaumail, J., STOCHASTIC INTEGRATION, Academic

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McGraw-Hill, New York 1981.

(2001).

THE LONG-TIME BEHAVIOUR OF STOCHASTIC

2D-NAVIER-STOKES EQUATIONS

T. CARABALLO

Dpto. Ecuaciones Diferenciales y Andisis Nume'rico, Universidad de Sevilla,

Apdo. de Correos 11 60, 41 080-SE VILLA, Spain,

E-mail: caraball@us. es

Some results on the pathwise asymptotic behaviour of the weak solutions to a

stochastic 2D-Navier-Stokes equation are established. In fact we prove some results

concerning the asymptotic behaviour with general decay rate (exponential, sub and super-exponential).

1. Introduction

The long-time behaviour of flows is a very interesting and important prob-

lem in the theory of fluid dynamics, as the vast literature shows (see

Temam 26, Hale 18, Ladyzhenskaya 19, among others, and the references

therein), and has been receiving very much attention over the last three

decades.

One of the most studied models is the Navier-Stokes one (and its vari-

ants) since it provides a suitable model which covers several important

fluids (see Temam 24,26 and the references inside these).

On the other hand, another interesting question is to analyze the effects

produced on a deterministic system by some stochastic or random distur-

bances appeared in the problem. These facts motivated the analysis done

in Caraballo et al. l 1 and the one in the present work. Therefore, our main

objective is to show some aspects of the effects produced in the long-time

behaviour of the solution to a two dimensional Navier-Stokes equation un-

der the presence of stochastic perturbations, since it is very interesting to

investigate if a fluid subjected to random influences is asymptotically more

or less stable than the deterministic unperturbed one.

There exists a controversy concerning the different interpretations which

can be given to the stochastic terms used to model our problem. Two for-

mulations are the most commonly used for the noise in the literature: It6's

formulation and Stratonovich's one. Each interpretation gives a different

solution of the stochastic equation, so they provide different answers to the

70

same problem. There exist several reasons which make reasonable both

possibilities and there exists a rule which permits us to pass from one kind

of equation to the other (see Arnold Oksendal 21, Kunita 20, among

others). However, when one is analyzing the long-time behaviour of the

solutions, special care should be paid to the choice of the model since the

solutions of both stochastic equations can have totally different behaviour.

We will comment again about this in the final section.

In this work, we will first recall some results from Caraballo et al. l 1

concerning the exponential behaviour of the solutions to our stochastic 2D-

Navier-Stokes model. Then we will improve those results by giving some

information concerning the general decay rate of solutions. To this end, we

will consider the following stochastic 2D-Navier-Stokes equation:

dX = [vAX - ( X , 0) X + f (X) + Vp ld t + g ( t , X ) d W ( t ) divX = 0 in [O, co) x D, X = 0 on [O, co) x r, i X ( 0 , z ) = Xo(z ) , J: E D,

where D is a regular open bounded domain of R2 with boundary I?, u is

the velocity field of the fluid, p the pressure, v > 0 the kinematic viscosity,

Xo the initial velocity field, f the external force field and g(t,z)dW(t) the random field where W(t ) is an infinite dimensional Wiener process,

i.e., if (Q, P, 3) is a probability space on which an increasing and right

continuous family {3t}ZE~0,00) of complete sub-o-algebra of 3 is defined,

and &(t) (n = 1 ,2 ,3 , . ..) is a sequence of real valued one-dimensional

standard Brownian motions mutually independent on (0, P, S), then

M

n=l

where A; 2 0 (n = 1 , 2 , 3 . . . ) are nonnegative real numbers such that

C,"==, A; < +co, and {en} (n = 1 ,2 ,3 , . ..) is a complete orthonormal

basis in the real and separable Hilbert space K . Let Q E L ( K , K ) be the

operator defined by Qe, = xien. The above K-valued stochastic process

W(t ) is called a Q-Wiener process.

Our problem can be set in the usual abstract framework by considering

the following Hilbert spaces:

H = the closure of the set {u E C p ( D , R2) : divu = 0} in L2(D, R2) with the norm (u( = (u, u) ; , where for u, u E L2(D, R2),

71

72

V = the closure of the set {u E C r ( D , R 2 ) : divu = 0} in Hi(D,R2)

with the norm llull = ( ( u , v ) ) i , where for u,v E HA(D,R2),

Then, it follows that H and V are separable Hilbert spaces with associated

inner products (., .) and ((., .)) and the following is safisfied:

V c H = H' c V',

where injections are dense, continuous and compact. Now, we can set

A = -PA where P denotes here the orthogonal projector from L2(D, R2)

onto H, and define the trilinear form b by

As we shall need some properties on this trilinear form b, we list here the

ones we will use later on (see Temam 26):

Ib(u,v,w)l I c1 I u l i l l 4 + 11v11 I w l i IlWll+ , v u , v , w E v, b(u, 21, ?I) = 0, vu, v E v, (1) ~ ( u , u , w - U ) - ~ ( w , w , w - u ) = -b(v - U , U , W - u),'~u,v E V,

where c1 > 0 is an appropriate constant which depends on the regular open

domain D (see Constantin and Foias 13). Furthermore, we can define the

operator B : V x V + V' by

(B(u,v) , W) = b(u, w,w),Vu,v, w E V,

where (., .) denotes the duality (V', V ) . We also set

B(u) = B(u,u), vu E v. Thus the stochastic 2D-Navier-Stokes equation can be rewritten as fol-

lows in the abstract mathematical setting:

d X ( t ) = [ -vAX( t ) - B ( X ( t ) ) + f ( X ( t ) ) l d t + S ( t , X ( t ) ) d W ( t ) , (2)

where f : V + V', g : [ O , c o ) x V + L(K, H) are continuous functions

satisfying some additional assumptions (see conditions below). Also we

consider the deterministic version of this equation, namely,

d X ( t ) = [ -vAX( t ) - B ( X ( t ) ) + f ( X ( t ) ) ] dt . (3)

First, we give the definition of the weak solutions to stochastic 2D-Navier-

Stokes equation (2) .

73

Definition 1.1. A stochastic process X ( t ) , t 2 0, is said to be a weak

solution of (2) if

( la ) X ( t ) is Qt-adapted,

( lb ) X ( t ) E L”(0, T ; H ) n L2(0, T ; V ) almost surely for all T > 0,

(lc) the following identity in V’ holds almost surely, for t E [0, a)

X ( t ) = X ( 0 ) + s,” [ - -YAX(S) - B ( X ( s ) ) + f(X(s))] ds

+ s , ”ds , X(s ) )dW(s ) .

As we are mainly interested in the analysis of the asymptotic behaviour of

the weak solutions to the problem (2), we will assume the existence of such

weak solutions (see, for instance, Bensoussan

for some results on the existence and uniqueness of solutions).

We also recall some definitions from Caraballo et al. l 1

or Capinski and Gatarek

Definition 1.2. A weak solution X ( t ) to (2) is said to converge to z, E H exponentially in mean square if there exist a > 0 and Mo = Mo(X(0) ) > 0

(which may depend on X ( 0 ) ) such that

E ~x(t) - z,12 I MOePat, t 2 0,

In particular, if z, is a solution to (a ) , then i t is said that z, is expo-

nentially stable in mean square provided that every weak solution to (2)

converges to z, exponentially in mean square with the same exponential

order a > 0.

Definition 1.3. A weak solution X ( t ) to (2) is said to converge to z, E H almost surely exponentially if there exists y > 0 such that

1 lim - log IX ( t ) - t-m t

z,1 5 -7, almost surely.

In particular, if z, is a solution to (a ) , then i t is said that z, is almost

surely exponentially stable provided that every weak solution to (2) con-

verges to IC, almost surely exponentially with the same constant y.

2. The exponential stability of solutions

In this section we will deal with the moment exponential stability and

almost sure exponential stability of weak solutions to stochastic NSE (2).

Let A1 > 0 be the first eigenvalue of A. We remark that 11u112 2 A 1 luI2, Vu E

V. We also denote by

Ildt, u)l12; = t r ( d t , u)Qdt , u>*>.

74

Throughout this section we will use the following condition:

Assumption A. There exists p > 0 such that

I I f (u) - f (v) Il"& P I t 'u. - 2) I I , u, E v. We first recall a result ensuring existence of stationary solutions, i.e.,

solutions to the next equation

v ~ u + ~ ( u ) = f(u) (equality in v'). (4)

Indeed, we have the following lemma (see Caraballo et al. ' I)

Lemma 2.1. Suppose that Assumption A is satisfied and the function f satisfies that f(v,) converges to f(v) weakly in V' whenever {v,} c V converges to v E V weakly in V and strongly in H . Then,

(a ) i f v > p, there exists a stationary solution u, E V to (4); (b ) furthermore, if v > c l I ' f ( o ) I ' f i ( y - p " , ' + p, then the stationary solution to

(4) is unique.

Now, in order to study the long-time behaviour of weak solutions X ( t ) to

the stochastic Navier-Stokes equation (2) under some conditions including

that the kinematic viscosity v is sufficiently large, we will assume that there

exists a unique stationary solution u, E V to (4). Also, we will need the

following hypothesis.

where 5 > 0 is a constant and y ( t ) , 6( t ) are nonnegative integrable functions

such that there exist real numbers 0 > 0, M y , M6 2 1 with

2 Assumption B. I lg(t,U)ll;q I + (5 + W ) ) l'1L - u,I ,

b

y ( t ) 5 Mye-et, b( t ) 1. Mge-et, t 2 0.

Theorem 2.1. Let u, E V be the unique stationary solution to (4) and assume that 2v > A,'( + 2p + 3 1 1 u, 1 1 . Suppose that assumptions A and B are satisfied. Then, any weak solution X ( t ) to (2) converges to the stationary solution u, to (4) exponentially in mean square. That is, there ex& real numbers a E (0, e) , MO = Mo(X(0) ) > 0 such that

E IX( t ) - u,I2 1. MOe-at, t 2 0.

Proof (sketch). Since 2v > A;'5+2p+3 llu,ll, we can take a positive

real number a E (0, 0) such that 2v > A;'(< +a) + 2p + 2 llu,ll . Then,

by applying the It6 formula to the function eat IX(t) - u,I , taking into

account assumptions A and B and Gronwall's lemma, we can prove the

statement (see Caraballo et al. ll).

2

75

Now using the energy equality, Burkholder-Davis-Gundy's lemma,

Borel-Cantelli's lemma and the previous result, it can also be proved in

a standard way the following result.

Theorem 2.2. Suppose that all the conditions in Theorem 2.1 are satisfied. Then, any weak solution X ( t ) to (2) converges to the stationary solution u, of (4) almost surely exponentially.

In the particular case in which the stationary solution to (4) is also

solution to the stochastic equations, it holds the following result.

Theorem 2.3. Let u, E V be the unique stationary solution to (4). As- sume that condition A and the following ones hold:

( a ) g ( t , urn) -- 0, t 2 0,

(b) II S ( t , .) - d t , .) 11,; 5 cg I I 21 - 21 I I , cg > 0, u, v E v. If 2v > 2p + c i + & 1 1 u, 1 1 , then any weak solution X ( t ) to (2)

converges to u, exponentially an mean square and so u, is exponentially stable in mean square. That is, there exists a real number y > 0 such that

Furthermore, pathwise exponential stability with probability one of u, also holds.

3. Exponential stabilizability and stabilization

In the previous sections, the exponential pathwise stability has been proved

as a by product of the mean square stability. However, i t may happen that

a solution of a stochastic equation can be pathwise exponentially stable and

not exponentially stable in mean square.

Indeed, let us consider the following scalar ordinary differential equation

to illustrate this fact,

d z ( t ) = az ( t )d t + b z ( t ) d W ( t ) ,

where a , b are real numbers and W is a one dimensional Wiener process.

The solution is then given by

z( t ) = z(0)exp { ( a - T) t + bW(t)}

76

Thus, the zero solution is pathwise exponentially stable with probability

one if and only if a - $ < 0. Also, we have that

E Jx(t)I2 = E Jz(0)I2exp { (2a + b2) t } ,

and therefore, the zero solution is exponentially stable in mean square if

and only if a + < 0. So, we observe that there exist many possibilities

of being the zero solution pathwise exponentially stable and, at the same

time, exponentially unstable in mean square.

In Caraballo et al. l1 it is proved a result ensuring pathwise exponential

stability without using the previous mean square analysis but under more

restrictive assumptions on the terms appearing in the model.

To this end let us firstly state the following assumption

Assumpt ion C . f : H -i H , and satisfies

I f (u) - f (.)I 5 c 121 - 211 I c > 0, u, 21 6 H ,

g ( t , .) : H + L(K, H ) , and satisfies

Ildt, u ) - g( t ,2 " ) l lL (K ,H) I c, Iu - 4 , B E P I m), Vu, v E H.

Observe that if vX1 > c and f (0) = 0, then the zero solution to (3) is

exponentially stable (see Temam 25). But when vX1 5 c and f (0) = 0 we do

not know, in general, if the zero solution is exponentially stable or not. The

following theorem is going to state that , under some particular conditions,

any weak solution of the stochastic Navier-Stokes equation converges to zero

almost surely exponentially stable. So, in a sense, we can interpret that a

kind of stabilization could have taken place in the system, i.e., the stochastic

perturbation implies that the model exhibits better stability properties than

it had.

Theorem 3.1. In addition to Assumption C, assume that f(0) = 0 and g ( t , 0 ) = 0 for all t 2 0 , and that there exists po > 0 such that

a$(s, x) := tr [($,(x) @ $z(x))(ds, z)Qds, XI*)] 2 d lb14 ,

x, h E H ) . Then, there exists 00 c 0, P(R0) = 0 , such that for w @ 00 there exists T ( w ) > 0 such that any weak solution X ( t ) to (2) satisfies

where $(.) = 1xI2 (recall that ($z(.) @ $Z(X))(h) = &(.) (&(x), h) I for

cz 2 where y := ~ ( X I U - c - + + 9). In particular, exponential stability of

sample paths with probability one holds i f y > 0.

We omit the proof since this result is a particular case of Theorem 4.1.

77

4. Pathwise stability and stabilizability with general decay

It may happen in some occasions that some systems are asymptotically

stable but not exponentially, so it is very interesting to determine what is

the actual decay rate of solutions. Now we will prove some results in this

way concerning our Navier-Stokes model by adapting the techniques used

in the papers Caraballo et al. ‘1’ to this case.

First we will prove a general theorem which extends Theorem 3.1 and

then we will comment about its consequences.

Theorem 4.1. Assume that f : H + H,g(t , .) : H -+ L(K, H ) are such that f(0) = 0 and g(t, 0 ) = 0 for all t 2 0 , and that

rate

If(.) - f (u) l 5 c I u - 211 , h u E HI 2 < S ( t ) 1 2 ~ - ‘ ~ 1 , V U , TJ E HI t 2 0 , 2

lldtl.) - g(tluU)IIL(K,H) -

tr [(u 8 u)(g(t , u)Qg(t, u)*)l 2 ~ ( t ) I2l4 , vu E H , t 2 0,

where c > 0 and S ( . ) , p ( . ) are integrable nonnegative functions such that there exist 60 2 0, po > 0 satisfying

where A(.) is a nonnegative continuous function such that A ( t ) +co as t goes to +co. Then, if Alv - c > 0 , it follows for any weak solution X ( t ) to (2) defined for every t 2 0 and such that IX(t)I > 0 for all t 2 0 and P-as., that

Let us apply Ito's formula for our soution X(t) satisfying the

assumptions mentioned in the theorem. Then, it follows

78

and, applying once again ItG’s formula to the function log lX(t)I2 , and

taking into account the hypotheses, it follows

t t +q Ix;s)lz (X(S)I g(s1 X ( s ) ) d W ( s ) ) - 2 l p(s)ds.

Now, observe that M ( t ) = so !x!s,l ( ~ ( s ) , g(s, ~ ( s ) ) d ~ ( s ) ) is a real con-

tinuous local martingale and it is not difficult to prove, by means of the

law of iterated logarithm,

lim * = 0, P - almost surely. t++m log X ( t )

Indeed, if we denote by ( M ( t ) ) the quadratic variation process associated

to M ( t ) , we deduce from the assumptions that

and, as po > 0, it follows that limt++m ( M ( t ) ) = +m, what implies, by

means of the strong law of the large numbers, that limt+foo - 0

and, consequently ( M ( t ) ) -

Dividing now in both sides of (5) by log X ( t ) we obtain

79

and the proof is finished by taking limits when t goes to +m.

Remark. Observe that when Xlv - c > 0 the null solution to the de-

terministic Navier-Stokes equation is exponentially stable, i.e. , every weak

solution approaches zero with exponential decay rate. Then, by means of

Theorem 2.1, we have that when the perturbation term tends to zero ex-

ponentially fast, the weak solutions to the stochastic Navier-Stokes model

also approach the null solution with the same decay rate. But, what in

principle can be much more surprising is that when the perturbation is

large enough (in a suitable way), we also have asymptotic behaviour with

a decay rate which is similar to the growing of this perturbation term. To

illustrate this idea assume for simplicity that this term is linear and is given

by g ( t , z ) = a(t)z and W(t ) is a standard Wiener process. Now we can

easily check that 6 ( t ) = p( t ) = a2(t). If there exists X(t) such that

a2(s)ds lim = a0 > 0,

t+m logX(t)

then, Theorem 4.1 implies

Consequently, if for example we take a(t) = (for some positive ao) we

can take X ( t ) = t and it holds exponential stability of the null solution. If

a(t) has exponential decay to zero, then Theorem 2.1 ensures exponential

stability for the zero solution with probability one, and if a( t ) grows to

infinity with certain rate, say a(t) = t1/2, then choosing X ( t ) = expt2 it

follows that a0 = 1 and therefore the weak solutions to (2) converge to zero

with superexponential decay rate. So, we deduce from the previous analysis

that certain stochastic perturbation may improve the stability properties

of stable solutions to the deterministic equation.

Remark. However, even much more can be proved in the case that we do

not know what happens with the null solution t o the deterministic problem,

i.e, when Xlu - c < 0, we do not know whether the stationary solution of

the deterministic problem is exponentially stable or not, but if the growing

rate of the perturbation is super-exponential, then we can obtain super-

exponential decay asymptotic behaviour for the solutions to the perturbed

problem. See the next corollary.

Corollary 4.1. Assume that f : H 4 HI g( t , .) : H ---f L(K, H ) are such that f(0) = 0 and g ( t , 0 ) = 0 for all t 2 0 , and that

If(.) - f(v)l I c Ju - 211 , vu, v E HI

80

where A(.) is a nonnegative continuous function such that X ( t ) T +m as t goes to +m. Then, if Xlu - c 5 0 , and

t lim ~ - t-02 logX(t) - O1

it holds that

Proof. Proceeding as in Theorem 4.1 we get

1% lX(t)I2 1% IX(0)l2 + 2s: [ - u X ~ + c + "i"3 ds

1% X(t> logX(t) 1% X(t>

log X ( t ) log X ( t )

logX(t) 1% X(t)

M ( t ) 2 J," +-- log IX(0)l2 + 2 (-vX1 + c) t

sot (S(s) - 2 P ( S ) ) ds M ( t ) +- 1% Vt> log X ( t > I

+ and, taking into account the super-exponential growth of X(t) , the result

follows immediately.

5. Conclusions, comments and open problems

What we have first tried to point out in this work is that the theory of

stability for linear and semilinear stochastic differential equations is so gen-

eral that can be applied to the stochastic Navier-Stokes ones. Also, we

have proved that the stochastic versions of Navier-Stokes equations satisfy

similar stability properties to the deterministic unperturbed models.

On the other hand, we also have pointed out that , when the noise is

appropriately chosen, the perturbed stochastic model may exhibit better

81

stability properties than its deterministic counterpart. However, one obvi-

ous question is the following. The interpretation we have given to the noisy

term has been in the sense of Itd, so is it possible that happens the same if

we consider it in the sense of Stratonovich? This is of course an interesting

and challenging problem for which we can only give some partial results and

comments. In the finite dimensional case, there exits a wide literature on

this topic (see Arnold and the references therein) which proves that some

kind of multiplicative noise may produce a stabilization effect on determin-

istic unstable systems. However, for the infinite dimensional case, a similar

result has not been proved yet, mainly due to the fact that the technique

developed in the finite dimensional framework cannot be extended to this

case or, a t least, it is not known how to do that. The main result proved in

Arnold l ensures that an unstable linear differential system in Rn, namely

k( t ) = Az(t) with trace A < 0, can be stabilized by adding a multiplica-

tive noise in the Stratonovich sense containing a suitable skew-symmetric

matrix. One interesting remark is that when the stochastic multiplicative

perturbation is considered in the It6 sense, this uses to imply a general

stabilization effect on the system. In a limit sense, the It6 equations with

multiplicative noise correspond to deterministic equations with a mean-zero

fluctuating control plus a stabilizing systematic control. This would mean

that only the stabilization produced by Stratonovich terms could be con-

sidered as proper stabilization produced by random noise. However, in the

infinite-dimensional case we have been able to prove in the linear framework

that, if some kind of commutativeness holds, the deterministic systems and

their stochastic perturbed versions have the same behaviour when the noise

is considered in the sense of Stratonovich, while if the noise is considered

in It6’s sense, persistence of stability, stabilization and even destabiliza-

tion may happen (see also Caraballo and Langa lo for an analysis on these

topics).

Finally, we would like to mention that from a global point of view, the

analysis of the effects produced by random perturbations in determinis-

tic systems is being investigated right now by many authors within the

framework of the theory of random attractors recently introduced, among

others, by Crauel and Flandoli 14. On the one hand, existence of random

attractors is only known for specific random terms (see, for instance, Crauel

and Flandoli 14, Capinski and Cutland 7 , Flandoli and Lisei 17) . On the

other hand, almost nothing is known on the structure of these random sets,

so that many challenging open problems, as those related to stability and

instability, are still open.

82

Acknowledgments

I would like t o thank Aubrey Truman and Ian Davies for the kind invita-

tion to take part in this Conference on Probabilistic Methods in Fluids. I

finished this work during my stay in the Mathematics Institute (University

of Warwick, June-August 2002). I would like to thank the Royal Society of

London for their generosity, and especially, to James Robinson and Tania

Styles for the hospitality and friendship they offered me, what made me

feel as if I would have been at home.

This paper has been partially supported by Secretaria de Estado de

Universidades e Investigacibn (Spain).

References

1. L. Arnold, Stabilization by noise revisited, Z. angew. Math. Mech. 70 (1990),

2. L. Arnold, Stochastic Differential Equations: Theory and Applications, J. Wiley and Sons, New York, (1974).

3. A. Bensoussan, Stochastic Navier-Stokes equations, Acta Applicandae Math.,

4. Z. Brzezniak, M. Capinski and F. Flandoli, Pathwise global attractors for stationary random dynamical systems, Prob. Th. Rel. Fields, 95 (1993), 87- 102.

5. M. Capinski and D. Gatarek, Stochastic equations in Hilbert spaces with application to Navier-Stokes equations in any dimension, J. of Funct. Anal.

6. M. Capinski and N. Cutland, Measure attractors for stochastic Navier-

Stockes Equations, Electronic J. of Prob., 3(1998), 1-15. 7. M. Capinski and N.J. Cutland, Existence of global stochastic flow and at-

tractors for Navier-Stokes equations, Prob. Th. and Rel. Fields 115( 1999),

8. T. Caraballo, M. J. Garrido-Atienza and J. Real, Asymptotic stability of nonlinear stochastic evolution equations, Stoch. Anal. Appl., to appear.

9. T. Caraballo, M.J. Garrido-Atienza and J. Real, Stochastic stabilization of

differential systems with general decay rate, Systems and Control Letters, to appear.

10. T. Caraballo and J.A. Langa, Comparison of the long-time behaviour of linear It6 and Stratonovich partial differential equations, Stoch. Anal. Appl.

11. T. Caraballo, J.A. Langa and T. Taniguchi, The exponential behaviour and stabilizability of stochastic 2D-Navier-Stokes equations, J . Diff. Eqns.

12. T. Caraballo and K. Liu, On exponential stability criteriaof stochastic partial differential equations, Stochastic Processes and their Applications 83 (1999),

13. P. Constantin and C. Foias, "Navier-Stokes Equations", The University of

235-246.

38 (1995), 267-304.

126(1994), 26-35.

121-151.

19(2001), 183-195.

179(2002), 714-737.

289-301.

83

Chicago Press, Chicago and London, 1988.

14. H. Crauel and F. Flandoli, Attractors for random dynamical systems, Prob.

Th. Rel. Fields, 100(1994), 365-393.

15. G. Da Prato and J. Zabczyk," Stochastic Equations in Infinite Dimension;,

Cambridge, 1992.

16. F. Flandoli and D. Gatarek, Martingale and stationary solution for stochastic Navier-Stokes equations, Prob. Th. Rel. Fields, 102( 1995), 367-391.

17. F. Flandoli and H. Lisei, Stationary conjugation of flows for parabolilc SPDEs with multiplicative noise and some applications, preprint (2002).

18. J. Hale, Asymptotic behaviour of dissipative systems, Math. Surveys and Monographs 25, (1988).

19. 0. Ladyzhenskaya, Attractors for semigroups and evolution equations, Cam-

bridge University Press, (1991).

20. H. Kunita, Stochastic Partial Differential Equations Connected with Nonlin-

ear Filtering, in Lecture Notes in Mathematics, S.K. Mitter adn A. Moro,

Eds. Springer Berlin 1982, Vol 972, 100-169.

21. B. Oksendal, Stochastic Differential Equations, Springer-Verlag, Berlin

(1992). 22. E. Pardoux, Equations aux dhrivhes partielles stochastiques non linhaires

monotones. Etude de solutions fortes de type It8, These,1975 23. T . Taniguchi, Asymptotic stability theorems of semilinear stochastic evo-

lution equations in Hilbert spaces, Stochastics and Stochastics Reports,

53(1995), 41-52.

24. R. Temam, Navier-Stokes Equations, North-Holland, 1979.

25. R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, sec-

ond edition, SIAM, 1995.

26. R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, 1988.

SEMILINEAR STOCHASTIC WAVE EQUATIONS

P.L. CHOW

Wayne State University Detroit, Michigan 48202, USA

E-mail: plchow@math. Wayne. edu

The existence and uniqueness of solutions to a class of semilinear stochastic hy-

perbolic equations in Ed are considered. First an energy inequality for a linear

stochastic hyperbolic equation is established. Then it is proven that there ex-

ists a unique continuous local solution for the associated nonlinear equation in the

Sobolev space H I (Rd) when the nonlinear terms are locally bounded and Lipschitz-

continuous. Under an additionalcondition on the energy bound, the solution exists

for all time. The results are shown to be applicable to stochastic wave equations

with polynomial nonlinearities of degree m with m 5 3 for d = 3 , and for any

m 2 1 for d = 1 or 2.

1. Introduction

Consider the stochastic wave equation:

8,". = v 2 u + o(u)atW(x, t ) (1)

where at denotes the partial derivative in t , V2 the Laplacian; W(., t ) is a

Wiener random field. For d = 1 or 2, Mueller proved that the equation

Eq. (1) has a unique long-time global solution pointwise in (5, t ) E Rd x [0, m), provided that o(u) grows no faster then lul(logIu1)' with T E (0 , a ) .

In the case d > 1, the Weiner field W ( x , t ) must be smooth in z, because

nonlinear equations such as Eq. (1) and Eq. (2) below are not well defined

if atW(x, t ) is a space-time white noise (see Walsh '). In view of Mueller's

result, the following question arises naturally. That is, if o(u) grows like

u' for a sufficiently large T > 1, whether a solution to Eq. (1) may blow

up in a finite time. This question is still open. Here we consider a related

problem as follows:

(2) a,zu = v 2 u + f ( u ) + .(u)atw(x, t ) , z E Rd, t > 0 { u(x, 0) = g(z), dtu(z, 0) = h(x) ,

where nonlinear terms f ( u ) and a(u) are assumed to grow like polyno-

mials in u, and the initial data g and h are given functions. In general

84

85

such an equation admits only a local solution, if it exists. For example,

when f(u) is cubically nonlinear, we showed (see Chow that, under some

conditions on the data, the solution can explode in finite time. We also

proved the existence and uniqueness theorems for local and global solu-

tions in the case when f(u) is a polynomial of degree m under suitable

conditions, where m depends on the space dimension d 5 3. In this paper

we will consider the case where the Laplacian is replaced a second-order

strongly elliptic operator and the nonlinear terms are locally Lipschitz con-

tinuous in a Sobolev space. In particuIar we will show that such results are

applicable to equations with polynomial type of nonlinearities mentioned

above. To be specific, we shall first derive the basic energy inequality for a

linear stochastic hyperbolic equation in section 2. Then, for a class of non-

linear hyperbolic It6 equations in section 3, the existence and uniqueness

of a continuous local solution will be presented in Theorem 3.1 under the

assumptions that the nonlinear terms are locally bounded and Lipschitz

continuous in the Sobolev space H1(Rd). As stated in Theorem 3.2, if an

additional energy inequality can be established, the solution will become

global. In section 4, the theorems are applied to some polynomially nonlin-

ear stochastic wave equations to yield the existence and uniqueness results

obtained in the paper by Chow

2. Linear Stochastic Hyperbolic Equation

Let H := L 2 ( R d ) with the inner product and norm denoted by (., .) and

1 1 . 1 1 respectively. Let H1 = H1(Rd) be the L2-Sobolev space of order one

with norm ((.111. Let (0, F, P ) be a complete probability space for which a

filtration .Ft is given. Let M ( z , t ) , t 2 0 , x E Rd, be a continuous martingale

with a spatial parameter s E Rd and M ( z , 0) = 0 in the sense of Kunita ‘. Let its covariation function q(s, y, t ) be defined as in

< M ( z , .), M(y, .) >t= q(s, y, s)ds, x, y E Rd, t E [0, TI a.s. ( 3 )

Regarding Mt = M ( . , t ) as a continuous H-valued martingale with covari-

ation operator Qt defined by

I’

Let W(x , t ) be a continuous Wiener random field with mean zero and

covariance function ~ ( z , 9) defined by

EW(x, t>W(Yl, s> = ( t A S ) T ( Z , !/)I x, y E Rd,

86

where (t A s) = min(t, s) for 0 5 t , s 5 T . Let a(x,t) = o(x , t ,w) for

t 2 0,s E Rd and w E R, be a continuous predictable random field such

that

a2(x, t ) d t < oa, for each x E Rd a.s..

As a special case, let M be the stochastic integral

M ( x , t ) = lt o(x, s)W(x, ds), t > 0,x E E d ,

which is a continuous Wiener martingale with spatial parameter x and

covariation function given by

dx, Y, t ) = T ( 2 , y)a(x, t k ( Y , t )

for x , y E Rd,t E [O,T].

Now we consider the Cauchy problem for the linear hyperbolic equation

with a random perturbation:

(4) [a; - A(x, D)]u(x , t ) = f (z, t ) + & M ( x , t ) , 0 < t < T , i u(z, 0) = uo(x), &u(x, 0 ) = vo(2), 2 E Rd,

where A(x, D ) is a strongly elliptic operator of second order of the form:

d

A(& D)cp(x) = c ~ z , [ ~ i j ( ~ ) ~ z , c p ( 4 1 - b(z)cp(x), (5) i , j = l

where the coefficients aij = u jz and b are smooth functions such that

d

ao(1 + m2) 5 c U%)&<j I + ls1)2), t , x E Rd, i , j = l

for some constants a1 2 a0 2 0. This condition implies that ( -A) is a

self-adjoint, strictly positive linear operator on H = L2(Rd) with domain

D(A) = H z ( R d ) and its square root B = a is also a self-adjoint, strictly

positive operator with domain D ( B ) , which is a Hilbert space under the

inner product (9, h ) ~ := (Bg,Bh). Since the norms 1 1 . I I B and 1 1 . 111 are

equivalent, we have D ( B ) 2 HI.

Let ut = u(. , t ) , ut = &u(.,t) and rewrite the equation Eq. (2) as a

system:

r t

or equivalently,

r t

87

(7)

where we set

and

with I being the identity operator on H . Introduce the Hilbert space ‘FI =

( H I x H ) . As a linear operator in ‘FI, A generates a strongly continuous

semigroup etA on ‘H. Now regarding Eq. (7) as a stochastic evolution

equation in ‘FI in a distributional sense, we have the following lemma:

Lemma 2.1. For $0 = ( U O , UO) E ‘H, let f t be a continuous predictable process in H , and let Mt be a continuous H-valued martingale with covariance operator Qt such that

Then the equation Eq. (7), or Eq. (6) has a unique (mild) solution $t =

(ut, ut) which is a continuous predictable ‘FI-valued process. Moreover at satisfies the energy equation:

r t r t

for t E [0, TI. Moreover, if in addition to Eq. (8),

where the constants C1, C2 depend on p , T and the initial conditions.

Notice that, due to the lack of required smoothness of solutions, the general

It6 formula does not hold here. The energy equation Eq. (9 ) can be proved

by a smoothing technique, such as the Yosida approximation (Yosida 5 , as

done in (Chap. 5, Da Prato and Zabczyk 6 ) , and then taking a proper limit.

88

The energy inequality Eq. (11) can be shown to hold by applying the It6

formula to the energy equation and by invoking Burkholder's submartingale

inequality. The proof is similar to the special case given in Chow and will

be omitted.

3. Semilinear Stochastic Hyperbolic Equations

Let us consider the Cauchy problem for the following hyperbolic equation:

(12) (8," - A)ut = f t (Jut ) + &Mt(Ju) , t > O i uo = 9, &uo = h.

In the above equation, we assume g E H I and h E H , and set J u =

(21, &,u, ..., aZdu, at.), f(x, J u ( ~ ) , t ) = f t ( J u ) ( x ) and M ( z , Ju (x ) , t ) = Mt ( J u ) (x) defined by

r t

where, for x E Rd, E E Rd+', f(x, [, t ) and o(x, 6 , t ) are continuous pre-

dictable random fields, and Wt = W(., t ) is a continuous Wiener random

field with covariance operator R of kernel ~ ( x , y), for x, y E Rd. Let Ct(Ju) be defined by

[ C t ( J ~ ) h ] ( z ) = U ( Z , J u ( z ) , t )h (z ) h E H .

For brevity, let Ft (Ju) be a stochastic integral defined by

Again we rewrite the equation Eq. (12) as a stochastic system in the Hilbert

space IH: t

(15) ut = uo + J, usds

V t = YO + s,' Ausds + Ft(Ju) ,

which, similar to Eq. (7), yields the simple form:

t

$t = $0 + 1 A$& + 3t(4),

where dt and A are defined as before and

89

Let (p = ( u , v ) E ‘FI and set J u = (u0,u1,...,ud+l) E ( H ) d f 2 , with the

convention: uo = u, uj = dZju, j = 1, ..., d , and ud+’ = &u . Introduce the

energy function e ( 4 ) defined by

j = O

Theorem 3.1. Suppose the following conditions hold:

(1) Let f t ( J . ) : H1 4 H such that

llft(Ju)l12 I Cl(1 + 11~113 and

IlfdJu) - ft(Ju’)JI _< C2llu - u’JJ1 a s . ,

f o r all u, u’ E H I , t E [0, TI, and f o r some constants C1, C2 > 0 . (2) For any u E H I , the map C . ( J u ) : [O,T] ---f L ( H ) is continuous

as . , where L ( H ) denotes the space of bounded linear operators on H . There exist positive constants C3 and C4 such that

T r [ C t ( J u ) R C ; ( J 7 4 I C3(1 + llull:), and

T r { [ C t ( J u ) - Ct (Ju ’ ) ]R [C t (Ju ) - Ct(Ju’)]*

I C4llu - U’III: a s . ,

f o r any u, u‘ E H I , t E [0, TI, where * denotes the adjoint. (3) W, is a H-valued process with covariance operator R such that

Then, f o r g E H I , h E H , the system Eq. (15) or Eq. (16) has a unique (mild) solution ut on [0, T ] with u. E C( [0, TI, H I ) and u. E C( [0, TI, H ) . Moreover the following energy equation holds

e(u t , vt) = e(uo, uo) + 2

+2 I” (w,, C,(Ju,)dW,) + Tr [C, (Ju , )RC; (Ju , ) ]ds .

(‘us, f ( J u s ) ) d s I” (19) I”

90

Under the above conditions (1)-(3), it is easy to check that the coeffi-

cients of the evolution equation (3.5) in IFI satisfies the usual global Lip-

schitz continuity and linear growth conditions. Theorem 3.1 follows from

a standard existence theorem (Theorem 7.4, Da Prato and Zabczyk 6, for

stochastic evolution equations in a Hilbert space.

To be able to apply the theorem to equations with coefficients of a poly-

nomial growth, we relax the global conditions (1) and (2) to the lo-

cal ones. To this end, replace the constants by functions of the form

bl(s) , bz(s, t ) , b3(s), b4(s, t ) , for s, t E R such that they are positive, locally

bounded and monotonically increasing in each variable. Then the following

theorem holds.

Let conditions (Nl)-(N4) be given as follows:

(Nl) Let f t ( J . ) : H I + H such that

and

lIft(J.1 - ft(Ju')ll i bz(lluIl1, 11~1111)11~ - ullll a.s*>

for all u, u' E HI , t E [O, TI.

such that

(N2) For any u E H I , the map C . ( J u ) : [O,T] + C ( H ) is continuous a s .

and

Tr{ [C,(Ju) - Ct (Ju I ) ]R [C t (Ju ) - Ct(Ju I ) ] * )

for any u, u' E H I , t E [O,T]. (N3) Wt is a H-valued process with covariance operator R such that

and

sup T ( 2 , X ) < 00. X E R d

91

(N4) Suppose that, for any u. E C( [0, TI, HI) n C'( [0, TI, H) with &u =

v, there exist constants c l , c2 > 0 and IE < - such that, for any 1

2 t E [O, TI,

t

- < c1+ c2 J, e(us, v,)ds + IEe(ut, vt) a s . .

Theorem 3.2. If the conditions ( N l ) - (N3) hold, then, foruo E H1,vo E

H, the Cauchy problem Eq. (12) has a unique continuous local solution u( . , t ) E H I with &u(., t) E H . If, in addition, condition (N4) is satisfied, the solution u(., t ) exists on (0, T] for any T > 0.

The main idea of the proof, similar to that in Chow 3 , is to show that, by

a smooth HI-truncation, the conditions (Nl)-(N3) reduce to the conditions

(1)-(3) in Theorem 3.1. Therefore the truncated problem has a continuous

solution uN(., t ) E HI for t < (TN A T ) , where ( s A t ) = rnin.{s, t } , and TN

is a stopping time defined by

TN = inf{t > 0 : 11ur111 > N } ,

with N being a cut-off number. Hence, for t < (TN A T), u(., t) = uN(., t ) is the solution of Eq. (12) with &u = vN. Noting that TN increases with

N , let T = lim TN. Define u(. , t ) for t < (TN A T ) by u(. , t ) = uN(. , t )

if t < TN 5 T . Then u(. , t ) thus defined is the unique local solution. To

obtain a global solution, it is necessary to have an energy bound. This

can be established by imposing condition (N4). Then it can be shown that

Prob (r < m) = 0. Therefore the solution exists on any finite time interval

[0, TI as claimed.

Remark: In the above theorem, for simplicity, we assumed that the Wiener

random field W ( x , t ) is scalar or, in the integral Eq. (13), Wt is a H-valued

process. Theorem 3.2 still holds true when W ( x , t ) and ~ ( x , 5, t ) are both

random vector-fields such that the product in the integrand of Eq. (13) is

regarded as a scalar product.

t+m

4. Applications

Let us consider the following initial-value problem in R3:

(a; - c2v2 + y2)ut = ft(.t) + at(J..t)atwt, t > 0 ,

210 = g, dtuo = h,

92

where c and y are positive constants, while ft and ut are nonlinear (deter-

ministic) functions of polynomial type. In comparison with Eq. (12), we

have A = (c2V2 - y2), f t ( J u ) = ft(u) and Mt(Ju ) is defined by Eq. (13). In particular, we assume that the following conditions hold:

ft(s)(x) = f(x, s, t ) , x E Rd, s E Rd, t > 0, is a polynomial of the

form:

j =O

where aj(x, t ) is bounded and continuous on Rd x [0, T] for each

j = O , l , ..., m.

The function .t(E)(x) = (~(x, 6 , . . . ,&+I, t ) , for x E Rd, [ E Rd+2 and t > 0 , is continuous. There exist positive constants C,,C2 such that, with k 5 2m,

j = 1

and

for x E Rd; ,$, q E Rd+' and t E [0, TI. Let W ( x , t ) be a continuous Wiener random field as given before

with covariance function r(x, y) such that

Tr R = r(x, x)dx < 00 s and

To = sup ?-(x,x) < m. z € R d

To apply the previous theorem to Eq. ( la ) , i t is necessary to show that

the nonlinear terms are dominated by a HI-norm. For polynomial nonlin-

earities, one appeals to the Sobolev imbedding : H1(R3) c LP(Rd) (p.112,

Adams '). In particular we recall the following useful lemma (see, e.g. p.21,

Reed s). Denote the LP(Rd)-norm by I . I p and let C r stand for the set of

Cm-functions on Rd with a compact support.

93

Lemma 4.1. For u, v E C,W and 1 5 k 5 m, there exist positive constants ~ 1 , ~2 such that

and

where m = 3 ford = 3, and m 2 1 ford = 1 or 2.

With the aid of this lemma and conditions (P1)-(P3), we can apply The-

orem 3.2 to give a local existence theorem for Eq. (20) with polynomial

nonlinearities.

Theorem 4.1. Suppose that conditions (Pl)-(P3) given above hold true. Then, for g E H I and h E H , the Cauchy problem Eq. (20) in Rd, for d _< 3, has a unique continuous local solution ut E H I with &u, E H , provided that m 5 3 for d = 3, and m 2 1 for d = 1 or 2.

The proof of this theorem under the stated assumptions is to verify the

conditions (Nl)-(N3) in Theorem 3.2 are satisfied for d 5 3. We will sketch

the proof in steps:

Step 1) : In view of condition (Pl) and Lemma 4.1, we have,

m m

1 j=1

where a0 = max sup \a j (x , t ) l . Hence, for u E H1 and t E

[O, TI, we have l l j l m t E [ O , T ] , Z E R d

Ilft(u)112 5 b l ( l l ~ l l ) l l ~ l l f l (21)

where

m

bl(r) = ( ~ O C ~ ) ~ ( C ~ j - ' ) ~ .

j=1

Step 2) : Similar to Step 1, it can be shown that, for u, v E HI and t E [0, TI,

Ilft(.) - ft(v)l12 I b 2 ( l l ~ l l 1 , l l ~ l l 1 ) l l ~ - Vll?, (22)

94

where b 2 ( ~ , s) is a polynomial of degree 2(m - 1) in T , s E R with

positive coefficients. For instance, consider the case d = 3 and

m = 3. By condition ( P l ) , we have

3

IIft(.) - ft(41I2 I c1 c llJ - 412,

)lu2 - v2112 I c211u + vll?ll. - ull:

I 2C2(ll.llf + Il4lf)ll. - vll4,

(23) j=1

for some constant C1 > 0. By invoking Lemma 4.1,

(24)

and

1 1 ~ 3 - q = II(.~ + uw + G)(. - .)II~ I 8(IIu2(~ - v)(I2 + IIu2(u - u)l12) L C3(ll4I! + 1 1 ~ 1 1 ~ ) 1 1 ~ - 414.

(25)

In view of Eq. (23)-Eq. (25), condition (Nl) holds for d = 3. For

d < 3, it can be verified in a similar fashion.

Step 3) : By making use of conditions (P2) and (P3) together with

Lemma 4.1, we get, for u E H I and t E [O,T],

TT[o~(Ju)R~~(Ju)] = J T ( X , z)o2(z, Ju , t )dz

(26) I c4 J.(z, z)(l + IU12k + c;:: lujl2)dz I Cq(T7-R + r o l ~ l $ k + ~ 0 1 1 ~ 1 1 ~ ) I K ~ ( I + IIullf(k-l))(l + ~ ~ u ~ ~ ~ ) , for k I m.

The inequality Eq. (26) implies that

[.t ( J U ) Rat (J.)1 I b3 ( /I 11 II 1) (1 + II u I I 3, (27)

where b 3 ( ~ ) = K1(1+ T ~ ( ~ - ~ ) ) for some constant K1 > 0.

Similar to Step 3, by means of conditions (P2), (P3) and Step 4) :

Lemma 4.1, we deduce that

Tr{ [g t (Ju) - at (Jv) ]R[n(Ju) - .t(J.>]*}

= J?-(z,.>[cJt(Ju) - at(Jv)]2dz

- < KZ[l + (Iu112(k-1) + ~ ~ w ~ / 2 ( ~ - 1 ) ] ~ ~ u - .u11?,

TT{[gt(JU)-gt(JV)IR[gt(JU)-gt(J21)1*} F b4(11UII1, 11~111)11~-~11~,(29)

I c2 J.(z,z)([l+ J?q(k--1) + 1w12(k--1)]Iu - 2112 + C;f: IUj12)dz (28)

for some constant K2 > 0. I t follows from Eq. (28) that

with b 4 ( ~ , s) = Kz[l + T ~ ( ~ - ~ ) + s ~ ( ~ - ' ) ] . In view of Eq. (27) and

Eq. (29), the condition (N2) is valid.

95

Clearly condition (P3) is similar to (N3). Therefore, by Theorem 3.2, the

Cauchy problem Eq. (20) has a unique continuous local solution as stated.

Remark: As pointed out in the remark following Theorem 3.2, in Eq.(4.1),

the noise term may assume a more general form, such as

where Wj , j = 1, ..., n, are n independent Wiener processes in H with

covariance functions rj. Then Theorem 4.1 will hold if, for each j , the

conditions (P2) and (P3) are satisfied, (see the example given below).

To obtain a global solution, it is necessary to impose further conditions

on the functions f and CT so that the condition (N4) will be met. For

convenience, introduce the function G defined by

“ 2 f ( x , r, t )dr = - -aj (x, t)uj+l

j=1 3 + 1

Then the next theorem holds true, and its proof can be found in Chow 3 .

Theorem 4.2. Suppose that all conditions in Theorem 4.1 are fulfilled. In addition to (P l ) and (P2), assume that

( la) Given m = (2n + 1) for a positive integer n, there exist constants a 2 0 and ,L? 2 0 such that

G ( x , r , t ) 2 (a + PrZn)r2

for each x E Rd, r E R and t E [0, TI. (2a) Condition (P2) holds with k = (n + 1).

Then the solution obtained in Theorem 4.1 exists in any finite time interval

(0, TI.

As an example, consider the cubically nonlinear wave equation in R3 under

a random perturbation:

(31) (8,” - c 2 v 2 + y2)u = xu3 + atMt(Ju)

u(., 0) = g, a,.(., 0) = h, x E 723,

t > 0 ,

96

where c, y and X are real parameters, the functions g, h are given as before,

and Mt(Ju ) is assumed to be of the bilinear form:

3 t

M t ( J u ) ( z ) := M ( z , J u , t ) = c / [aZju(z, s)]Wj(z, ds). j=1 0

Here W j , j = 1 , 2,3, are independent Wiener random fields with covariance

functions r j (z , y) such that

~ j ( z , z ) ] d z + sup r j (x ,z ) < m. .€R3 j=1

Clearly Eq. (31) is special case of Eq. (20) with f t ( J u ) = Xu3, d = m = 3

and a more general noise term. In view of Eq. (32) and the remark following

Theorem 4.1 , i t is easy to check that, for any X I conditions (Pl)-(P3)

are met so that there is a unique continuous local solution as stated. By

definition given in Eq. (30), we have

-A 2

G(z, T , t ) = (-)r4,

so that, if X 5 0, condition (la) in Theorem 4.2 holds with a = 0 and

p = ($). Condition (2a) is obviously true. Therefore, for X 5 0, the

Cauchy problem Eq. (31) has unique continuous solution u E C( [O, TI, H l ) n C1 ( [0, TI , H ) on any finite interval [0, TI.

Acknowledgments

This work was supported in part by the NSF Grant DMS-9971608.

References

1. C. Mueller, Ann. Probab. 25, 133 (1997). 2. J.B. Walsh, Lect. Notes in Math., Springer-Verlag, Berlin, Heidelberg, New

York, 1180, 265 (1984). 3. P.L. Chow, Ann. Appl. Probab. 12, 361 (2002).

4. H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge Univ. Press, Cambridge, England, 1990.

5 . K. Yosida, Functional Analysis, Springer-Verlag, New York, 1968. 6. G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions,

Cambridge Univ. Press, Cambridge, England, 1992. 7. R.A. Adams, Sobelev Spaces, Academic Press, New York, 1975. 8. M. Reed, Abstract Non-linear Wave Equations, Springer-Verlag, Berlin, 1976.

STOCHASTIC NAVIER-STOKES EQUATIONS: LOEB SPACE TECHNIQUES & ATTRACTORS

NIGEL J. CUTLAND

Department of Mathematics, University of Hull

Hull, HU6 7RX, UK E-mail: n.j. [email protected]

We survey the use of Loeb space methods in stochastic fluid mechanics, with

particular emphasis on recent results concerning the existence of attractors for the

stochastic Navier-Stokes equations.

1. Introduction

A general version of the stochastic Navier-Stokes (sNS) equations in a bounded domain D c EXd takes form:

(1) du = [VAU - (u, V)U + f ( t , u) - V p ] d t + g ( t , u ) d ~ t { divu = 0

Here u( t ,x ,u ) is the (random) velocity of the fluid at the location x E D at time t , so that we have

u : [O, m) x D x R ---f Rd

where R is the domain of an underlying probability space. The initial

condition u(0) = u g is prescribed (and may be random); the boundary

condition is either u(t, x) = 0 for x E 8D or, occasionally, when d = 2 we

assume periodic boundary conditions.

These equations have been the subject of considerable study since they

were first solved in [6], for d 5 4, using Loeb space methods. Some time

after the publication of [6] a number of alternative proofs of existence ap-

peared (see below) so that now there is considerable interest in more delicate

issues such as the existence of a stochastic flow and attractors for the sNS

equations.

Loeb space methods have continued to prove powerful in this field, in

combination with the well-developed techniques of LLc1assica177 infinite di-

mensional stochastic analysis. The purpose of this paper is to survey what

97

98

has been achieved, with particular emphasis on recent work on attractors

for the sNS equations in d = 2,3.

2. Existence for the stochastic Navier-Stokes equations

The equations (1) with additive noise - that is, with g independent of -

were first discussed by Bensoussan & Temam in [3] - where they were solved

in 3 dimensions with 20 a 1-dimensional Wiener process and g = Identity.

Later contributions to the additive noise case were made by Viot [29] and

Vishik & F’ursikov [30].

For multiplicative noise the general equations (1) in dimensions d 5 4 (with only natural growth conditions on f , g ) were first solved in 1991

by the author and Marek Capinski [6] using the Loeb space techniques

to be elaborated upon below. Just prior to this (though published later)

Brzezniak, Capinski & Flandoli [2] obtained solutions for d = 2 with a

special form of multiplicative noise, and only for small initial conditions;

around the same time Bensoussan [4] established general existence ford = 2.

Some three years later, alternative proofs of existence for the general

equations in higher dimensions began to appear, beginning with the papers

of Capinski & Gatqrek [15] and followed by Bensoussan [5]. The latter

paper, curiously, has exactly the same title and appears in the same journal

as [6], which is not acknowledged even though the author of [5] was an editor

of the journal at the time.

Around the same time Flandoli & Gatqrek [24] proved existence of so-

lutions to a number of different formulations of the general sNS equations

(1) for d 5 4, as well as stationary solutions in each case.

2.1. Mathematical formulation

To proceed it is first necessary to note the precise mathematical formulation

of the equations (1). We adopt the usual Hilbert space setting as follows.

Denote by H the closure of the set {u E C r ( D , Rd): div u = 0} in the

L2 norm 1ul = ( ~ , u ) l / ~ , where

The space V is the closure of {u E C r ( D , Wd): div u = 0) in the stronger

norm IuI + llull where /lull = ((u, u ) ) ~ / ~ and

99

H and V are Hilbert spaces with scalar products (., .) and ((., .)) respectively,

and 1 . I 5 cII . 1 1 for some constant c.

By A we denote the self adjoint extension of the projection of -A in

H; A has an orthonormal basis {ek} of eigenfunctions with corresponding

eigenvalues X k r X k > 0 , X k m. For u E H we write U k = (u ,ek ) , and write Pr, for the projection of H on the subspace H, spanned by

{ e l , . . . , em}. Since each ek E V then H, C V. The trilinear form b defined by

d avi

(whenever the integrals make sense) has the well-known and crucial prop-

erty b(u, w , w) = --b(u, w, w) so that b(u, w, w) = 0.

In this framework, the stochastic Navier-Stokes equations (1) may be

formulated as a stochastic differential equation in H as follows:

du = l-vA.1~ - B(u) + f ( t , u) ]d t + g( t , u)dwt (2)

where B(u) = b(u, u, .). This is initially regarded as an equation in V’ (the

dual of V) although it turns out that the solution lives in H (and in fact

in V for almost all times). Compared to (l), note that the pressure has

disappeared, because V p = 0 in V’ (using divu = 0 in V and an integration

by parts).

The equation (2) is really an integral equation, with the first integral

being the Bochner integral and the second an extension of the It6 integral

to Hilbert spaces, due to Ichikawa [25]. The noise is given by a Wiener

process w : [0, m) x R t H with trace class covariance, and so the noise

coefficient g belongs to L(H, H). It is assumed that

g : [ O , o o ) x V + L(H,H)

while

f : [O,m) x v -+ V’.

(The restriction to V in the domains is sufficient because we will have the

solution in V for almost all times.)

2.1.1. Definition of solutions to the stochastic Navier-Stokes equations

The following makes precise what is meant by a solution to the stochastic

Navier-Stokes equations as formulated above. In fact there is a range of

100

solution concepts of varying strength, each of which is appropriate in certain

circumstances.

Definition 2.1. Suppose that u g E H and f, g as above are given, together

with a probability space R carrying an H-valued Wiener process w. A weak solution of the stochastic Navier-Stokes equations is a stochastic process

u : [0, co) x R 4 H such that for 8.8. w

(i) u E L2(0, T ; V) n L"(0, T ; H) n C(0, T ; Hweak) for all T < co , (ii) for all t 2 0

r t r t

A strong solution" has in addition that for a.a. w

for all T

The notion of solution for the deterministic case is given by taking g = 0

and removing the random parameter w throughout, so a weak solution is a

single function u E L2(0, T ; V) n Loo(O, T ; H) n C(0, T ; Hweak) for all T . The classical approach to the solution of the Navier-Stokes equations

(deterministic or stochastic) is to begin with an approximate version in the

finite dimensional space H, for each n, the so called Galerkin approxima-

tion, which can be solved easily using standard techniques from ODES (or

SDEs in the stochastic case) to give Galerkin approximate solutions u"(t). The hard part is then to find some way to pass to the limit to obtain a so-

lution to the Navier-Stokes equations. First, some specialized compactness

theorems are required to show that there is a subsequence of (u"(t)),cn that converges in an appropriate sense to a limit ~ ( t ) say. Second it is nec-

essary to show that this limit u(t) actually is a solution. The difficulties are

compounded in the case of the stochastic equations especially in dimension

2 3 because it seems necessary to work with a probability space that is

bigger than the Wiener space.

2.2. Loeb space methods

The methods used in [6] and subsequent papers are expounded in full detail

in the book Capiriski & Cutland [ll] and also in the monograph [16], so

"Some authors require a strong solution to have the stronger property

( s u p t 5 ~ llu(t)112 + JT 1Au(t)l2dt) < ca for all T ; we prefer to call this strictly strong.

101

here we only convey the key ideas.

A Loeb space is essentially an ultraproductb of probability spaces; the

particular Loeb space used in [6,11] is an ultraproduct of finite dimensional

Wiener spaces. This is simply a rather rich conventional probability space

with filtration] that happens to be constructed as an ultraproduct.

The power of Loeb spaces comes from the combination of their richness

and the fact that they are tractable: their richness can be easily exploited

using the ideas of Robinson’s nonstandard analysis. The heart of this is a

transfer principle that means that properties of the original spaces in the

ultraproduct are inherited in a precisely defined way by the Loeb space. In

the appendix we put a little more flesh on this idea, and point the reader

to sources where a full exposition may be found.

The methods of [6,11], which apply to dimensions d 5 4, can be infor-

mally described now as follows. Take R, to be the canonical Wiener space

of dimension n and let R be the Loeb space that is the ultraproduct of the

spaces (R , )nE~ . Let u, be a solution on R, to the n-dimensional Galerkin

approximation of the sNS equations (2). Then the ultraproduct U of the

solutions ( u , ) ~ € N is a “nonstandard” approximate solution to (2) that lives

on R. Formally] U has values in the ultraproduct of the spaces (Hn)nE~ ,

which is denoted HN, where N is an infinite natural number. That isc

U : R x R + HN

Most importantly] U inherits] via the transfer principle, the properties of

the Galerkin approximations (u,),€~, especially the usual energy estimates.

This enables the definition of a process u : R x R -+ H by

u(tl w ) = “U( t , w )

using a mapping O : HN -+ H called the standard part mapping, that is de-

fined for certain nearstandard members of HN. The energy estimates inher-

ited by U are crucial in showing that U(tl w ) is nearstandard for a.a. w E 0.

Once u is defined it is fairly routine to check that it is a solution to (2).

In 2-dimensions, if the noise g(tl u) has a special form - essentially that

it is orthogonal to the solution process u - it was shown in [8] how the

above method provides a construction of a global stochastic flow for (2).

The techniques developed in [6,11] for the stochastic equations origi-

nated in the paper [7] where the idea of Galerkin approximations of di-

mension N , with N an infinite natural number, were used to give a very

bThat is, a quotient of a product by an equivalence relation that is given by an ultrafilter. =In fact U : *R x R -+ HN where *R is the hyperreals, the extension of R given by the

ultraproduct of countably many copies of R, but in particular U is defined on all of R.

102

simple proof of existence for the deterministic Navier-Stokes equations in

dimensions d 5 4. An almost trivial extension of the method gave exis-

tence of statistical solutionsd in dimension d 5 4 with an arbitrary initial

measure. The idea applies also to the sNS equations to provide Foias and

Hopf statistical solutions of the stochastic Navier-Stokes equation - see [9].

The framework sketched above for solving the sNS equations allows a

more radical approach to solving the Foias equations. At the penultimate

stage of the construction, the “nonstandard” solution U(7 , w) lives in HN, which is isomorphic to RN and carries a nonstandard version of Lebesgue

measure. Thus the Foias equation for evolving measures may be recast as

an equation for an evolving density against Lebesgue measure on HN. In

the stochastic case this is a second order (nonstandard) PDE whose solution

readily gives a solution to the Foias equation using a simple Loeb measure

construction. Details may be found in (lo] or [ll] . A further extension of the basic existence theory and techniques devel-

oped in [6] gave one of the first solutions to the stochastic Euler equations

(that is, equation (2) with v = 0 ) in dimension d = 2 with periodic bound-

ary conditions [13]. It is also shown that the laws of solutions to (2) for

0 < v 5 1 are relatively compact and that for any convergent sequence of

laws for solutions with v, 4 0 there is a solution of the stochastic Euler

equations with the limiting law.

An alternative (but more or less equivalent) approach to solving the

sNS equations using Loeb space methods is to apply Keisler’s theory of

neocompact sets and rich adapted probability spaces [22,23]. A rich adapted

probability space is one that has those features of a Loeb space that are at

the heart of existence proofs such as in [6]. The theory captures these key

features as intrinsic properties of the space itself, rather than properties

that are derived from its construction. A typical existence result in this

theory is proved using a property called neocompactness - weaker than

classical compactness - to show for example that an intersection of sets

of approximate solutions to a stochastic equation (for example Galerkin

approximations) is non-empty, and contains a solution.

The paper [17] shows how to recast the basic existence proof of [6] in

the setting of a rich adapted spaces, and moreover proves the existence of

a wide range of optimal solutions to (2) for d 5 4. For example, there is a

dA statistical solution is a time-evolving family of probability measures that solves the

so-called Foias equation. This is derived heuristically from the Navier-Stokes equations,

and describes the evolution of the probability distribution of a solution to the equations

on the assumption of a random initial condition and uniqueness of solutions.

103

solution that minimizes the expected energy integral

E(u) = $lE J: lu(t)12dt

and there is a solution that minimizes the expected enstrophy integral

This may well have a bearing on the uniqueness question.

3. Att rac tors for stochastic Navier-Stokes equations

There are several ways to formulate the idea of an attractor for a system of

stochastic differential equations - for example by considering measure at-

tractors (see [12,27]), or by working with the notion of stochastic attractor

developed by Crauel & Flandoli [19]. A third approach is to extend the ap-

proach of Sell [28] that was used for deterministic Navier-Stokes equations

to overcome the problem of nonuniqueness.

In each case, to avoid unnecessary additional complications, the drift

and noise coefficients f , g in (2) are taken to be time-independent, so the

equations considered are

3.1. Measure attractors

This approach is currently applicable only to d = 2 since it is necessary

that the equation (4) has a unique solution. Thus it is assumed that f , g

satisfy an appropriate Lipschitz condition, to ensure that for each initial

condition u E H there is a unique solution u(t) = v(t, u) with u(0) = u (so w(0,u) = u). A semigroup St is now defined on Ml(H), the set of Bore1

probability measures on H, by putting Stp = pt where

s, d(u)dCLt(u) = s, IE 29(v(t1 U))dP(UZL)

for all bounded weakly continuous functions 0 : H -+ R. An attractor for the dynamical system (Ml(H) , St) is called a measure

attractor. The existence of measure attractors for the sNS equations was

first investigated by Schmallfufi in [27] for example. The paper [12] with

Capiriski establishes existence of a measure attractor for (4) under quite

general conditions:

104

Theorem 3.1. Suppose that f l g are Lipshitz and satisfy an appropriate growth conditione. Then there is a measure attractor A c Ml(H) for the stochastic Navier-Stokes equations (4). That is

(a) A is weakly compact; (b) StA = A for all t ; (c) for each open set 0 2 A, and for each r > 0

StBr 0

for all suficiently large t , where Br = { p E X : IuI2dp(u) 5 r }

The methods in [12] do not make essential use of Loeb spaces although at

some points they can be employed to assist the construction.

3.2. Stochastic attractors

For a stochastic system such as (4) the idea of a stochastic attractor devel-

oped by Crauel & Flandoli [19] takes into account the fact that a t all times

new noise is introduced into the evolution of each path of any solution to

(4). A stochastic attractor is defined to be a random set A ( w ) that, a t time

0, attracts trajectories “starting at -m” (compared to the usual idea of

an attractor being a set “at time m” that attracts trajectories starting at

time 0).

This idea is spelled out below, and involves the introduction of a one

parameter group Bt : R + R of measure preserving maps, which should be

thought of as a shift of the noise to the left by t . In proving the existence of

a stochastic attractor for the system (4) the nonstandard framework makes

it particularly easy to consider -co. Making this precise, suppose that cp is a stochastic flow of solutions to

(4). That is, cp is a measurable function

c p : [ O , c o ) x H x R + H

such that cp(.,.,w) is continuous for 8.8. w, and for each fixed initial

condition uo the process u ( t , w ) = c p ( t , u 0 , W ) is a solution to (4) with

The notion of a semigroup in the usual definition of a deterministic

attractor, along with the notion of an attractor itself, is now replaced by

the following.

u(0, w) = uo.

eFor example, a sufficient condition is that l j (u)&I 5 c + 61IIuII and Ig(u)IH,H 5 c + 621bll for Some &,& > 0 with 261 + 6;.trQ < 2u, where Q is the covariance of the H-valued Wiener process w.

105

Definition 3.1.

such that for all w E R, (i) The flow cp is a crude cocycle if for each s E R+ there is a full set R,

4 s + t , z, w ) = cp(t, cp(% 2 , w ) , Q3w)

holds for each z E H and t E R+.

(ii) A cocycle is perfect if R, does not depend on s.

(iii) Given a perfect cocycle c p , a global stochastic attractor is a random

compact subset A ( w ) of H such that for almost all w

cp(4 A ( w ) , w ) = A(Qtw), t L 0,

lim dist(cp(t, B, QPtw) , A ( w ) ) = 0 t+m

for each bounded set B c H.

Note that the existence of a perfect cocycle is necessary for the pos-

sibility of having a stochastic attractor. Constructing a perfect cocycle

is difficult for infinite dimensional systems, particularly for those that are

truly stochastic (as compared to random dynamical systems in which paths

may be treated individually).

3.2.1. Existence of a stochastic attractor for the Navier-Stokes equations

A stochastic attractor was constructed for the stochastic Navier-Stokes

equation with d = 2 by Crauel & Flandoli [19], but their version of (4)

reduced to a random equation that could be solved pathwise, giving es-

sentially a pathwise construction of the random attractor A(w) . The first

example of a stochastic attractor for a truly stochastic version of the Navier-

Stokes equations was constructed in [14] using Loeb space methods, seem-

ingly in an essential way. In the following, for simplicity the Wiener process

was taken to be one dimensional.

Theorem 3.2. (Capiliski & Cutland[l4]) (a) Suppose that (g (u ) -g(v), u- v) = 0 and (g(u), u ) = O.f With appropriate Lipschitz and growth condi- t ions o n f, g , there is an adapted Loeb space carrying a stochastic flow of solutions to the system (4) that is a perfect cocycle, and there is a stochastic attractor A ( w ) (compact in the strong topology of H) for this system.

(b) If g has the additional property that ((g(v),v)) = 0 for ZI E V the stochastic attractor is bounded and weakly compact in V.

fFor example g(u) = (h, 0 ) u for some h E H.

106

The proof of this result is quite long and complicated, and uses heav-

ily the fact that solutions to (4) may be obtained as standard parts of

Galerkin approximations of dimension N , infinite. A delicate extension of

the Kolmogorov continuity theorem as adapted to a nonstandard setting

by Lindstrom [l] is at the heart of the construction of the perfect cocycle.

An outline of the main steps and ideas of the proof is given in Chapter 2

of [16].

3.3. Process attractors

Sell’s radical approach [28] to the problem of attractors for the deterministic

Navier-Stokes equations for d = 3, bearing in mind the possible nonunique-

ness of solutions, was to replace the phase space H by a space W of entire

solutions to the Navier-Stokes equations. That is, each point in W is the

complete trajectory in H of a solution. The semigroup action St on W is

simply time translation. That is, if u = u(.) E W then Stu = w E W is

given by

w(s) = u(t + s).

Clearly this is well defined, and has the crucial semi-flow property

st, 0 st, = Stlft2

along with Sou = u. Using this idea, Sell was able to establish the existence of a global

attractor for the 3-dimensional (deterministic) Navier-Stokes gquations.

For the 3-dimensional stochastic case, Sell’s idea was used by Flan-

doli & Schmalfufi in the paper [20] for the Navier-Stokes equations with

a special form of multiplicative noise, using a mild solution concept. The

equation considered allowed essentially a pathwise solution, and then a ran-

dom attractor was obtained by combining Sell’s approach with the idea of

pulling back in time to -00, as developed by Crauel & Flandoli [19]. In a

later paper [21] Flandoli & Schmalfufi consider in the same framework the

Navier-Stokes equations with an irregular forcing term, but no feedback.

In the paper [18] with HJ Keisler we consider 3-d stochastic Navier-

Stokes equations with a general multiplicative noise g(u ) as in equation (4)

above. The idea is to use Sell’s approach at the level of processes rather than

paths. In this way the idea of an attractor is formulated in the conventional

sense, examining the long term behaviour of solutions as t 4 m. To do this,

it is necessary to have a single underlying probability space, rich enough to

carry a supply of solutions to the 3-d stochastic Navier-Stokes equations

107

that is sufficient for the concepts to make sense. For this an adapted Loeb

space is needed.

A precise formulation of the notion of a process attractor and the main

result of [18] is as follows.

On an arbitrary space R carrying a 1-dimensional Wiener process

(wt)t20 suppose that a class X of solutions to the sNS equations (4) is

defined. Suppose further that R is equipped with a family of measure

preserving maps Ot : R -+ R for t 2 0 with the following properties:

(el) B0 =identity and 8, 0 BS = 8t+s; (82) 8 t 3 s = Ft+s for all s , t 2 0, where (Ft ) is the filtration on R; (83) w ( t + s, &w) - w ( t , Otw) = W ( S , w) for all s 2 0.

Note that the property (83) tells us that for a fixed t the increments of

the process w ( t +s , 8,w) are the same as those of the process w(s ,w) . Thus

Ot can be thought of as a shift of the noise to the right by t . The family (8,) allows the following definition of a semiflow S, of

stochastic processes.

Definition 3.2. (Semiflow of Processes) Suppose that u = u( t ,w ) is

a stochastic process defined for t > 0. Then for any r 2 0 the process

u = S,u is defined by

v(t, w) = u ( r + t , 8,w)

It is clear that S, is a semigroup, and if u is adapted so is Stu. Suppose now that X is closed under St. Then a process attractor for the

class X can now be defined. In the following, if u is a stochastic process

then Law(u) is defined to be the probability law (on path space) of the

coupled process (u, w ) .

Definition 3.3. (a) A set of laws A c Law(X) is a Law-attractor if

(i) (Invariance) &A = A for all t 2 0, where St is the mapping of

(ii) (At t rac t ion) For any open set 0 2 A and bounded 2 c Law(X),

laws induced by the semigroup St.

$2 c 0

eventually (i.e. this holds for all t 2 t o ( O , 2 ) ) . (iii) (Compactness) A is compactg

108

(b) A (process) attractor for the semiflow St on X is a set of processes

A 5 X such that

(i) Law(A) is a Law-attractor (in particular Law(A) is compact and

(ii) (Invariance) StA = A for all t 2 0;

(iii) (Attraction) For any bounded set Z c X and compact set K

so A is bounded);

limt+,d(StZ, K ) 2 d(A, K )

(iv) A is closedh.

Remarks on Definition 3.3. (1) Since existence results for the stochastic

Navier-Stokes equations require a rather large probability space, it is to be

expected that any space carrying a whole class of solutions X as above

will be too big to allow an attractor A c X that is compact in the usual

sense. However, the attractor A of the following theorem is neo-compact, the key notion developed in [23] It is a consequence of neo-compactness

that Law(A) is compact.

(2) The attraction property 3.3(b)(iii) is equivalent to the following:

stz 5 0 (5)

eventually for any bounded Z and any open 0 3 A of the form 0 =

L2(R, M)\KsE, with K compact. Property 3.3(b)(i) means that in addition

(5) holds eventually for any open set 0 of the form 0 = LawP1(0’) where

0’ is an open set of laws with Law(A) 2 0’. The usual attraction property

for attractors, namely that StZ C 0 eventually for any bounded 2 and

any open 0 2 A is probably too much to expect. However, the attractor

in the following theorem has property (5) for a smaller class of open sets -

namely those that are neo-open, a further key notion of [23]. Sets 0 of the

form L2(R, M ) \ K s E or Law-l(O’) as above are neo-open.

We can now state the main theorem of [18].

Theorem 3.3. There is a Loeb space R (which carries solutions to the stochastic Navier-Stokes equations for all L2 30-measurable initial conditions) with a process attractor A for the class of solutions X described below.

where do is the Prohorov metric and pi ( i = 1,2) is the projection of X i onto the first

coordinate- that is, path space for the solutions of (4). hHere and in (iii) the topology is the L2 norm topology on processes in H given by

lu12 = IE som lu(t)12 exp(-t)dt.

109

The class of solutions in the following definition depends on the con-

stants k ~ , k ~ , k3, a, ,B, a, b. In the proof of Theorem 3.3 in [18] an explicit

choice of these is identified that ensures that X # 8. The condition (X5)

is the only one that needs explanation - see the remarks below.

Definition 3.4. (i) Denote by X the class of adapted stochastic processes

u : [0, 00) x R -+ H with the following properties.

(Xl) For a.a. w the path u(., w ) belongs to the following spaces:

L:~(O, 00; H) n L?~,[O, 00; H) n &(o, 00; V) n ~ ( 0 , 00; Hweak)

(X2) For all t l 2 t o > 0

t l

4 t l ) = u( to )+ lo [--vAu(t)-B(u(t))+f(u(t))ldt+ 9(u(t))dwt 1: (X3) For a.a. to > 0 and all tl 2 t o ,

IE(Iu(t1)12) I ~ ( I u ( t o ) I ~ ) exp(-kl(tl - t o ) ) + k2

(X4) For a.a. t o > 0 and all tl 2 t o ,

(X5) For a.a. to > 0 and all t~ 2 to, for all n 2 1

(X6) IE Ji Iu(t)12dt < co

(ii) Denote by xk the set of u E x with

(X6k) EJ i Iu(t)I2dt i k

Remarks 1. The above conditions tell us nothing about u ( t , w ) at t = 0

and there may be a singularity there. In this sense the class X is a class

of generalized weak solutions to the stochastic Navier-Stokes equations (cf.

[28] p.12).

2. It follows from (X6) that IE(Iu(t))I2) < 00 for 8.8. t E (0 , l ) . Thus,

from (X3) we see that IE(Iu(t))I2) is bounded on [A, m) for all n. 3. In condition (X5), the function cpn(u) is an explicit smooth approx-

imation to the function \~1~1{1..12,). The inequalities (X5) follow heuristi-

cally from the equation (4) as a particular instance of the Foias equation

corresponding to (4). The choice of the functions qn makes (X5) a kind

110

of uniform integrability condition for the random variables lu(t, u)lz for

t E Ito,m).

The proof of Theorem 3.3 proceeds as follows. First show that X # 0 by the construction outlined in Section 2.2. The heuristic argument for the

inequalities (X5) can be made precise for the approximate solution U living

in HN and it is this that gives (X5) for the solution u = " U . The other

properties in the definition of X follow naturally.

Next it is necessary to define an internal ("nonstandard") set of approx-

imate solutions X to (4) that is wider than the Galerkin approximations

on HN: X includes processes U where the equality (X2) is replaced by an

infinitesimal approximation. Then it is shown that X is precisely the set of

processes u such that u = "U for some process U E X that is nearstandard as a process: in symbols

X = " ( X n N S )

Finally, after defining a semigroup operation T, on X corresponding to S,, the set

c = 0 Tnxk n E N

is defined for a certain k (for which Xk is absorbing).

It is easily proved that T,C = C for finite times T and that C attracts

bounded sets in X - so C is a nonstandard attractor.

The key now is that C is non-empty (this follows from a kind of com-

pactness property of Loeb spaces) and also that C c NS. In consequence

the set A = "C is nonempty and in fact neocompact. The properties re-

quired for A to be an attractor follow from the corresponding properties of

the nonstandard attractor C. In the final part of [18] the class X of two-sided solutions to (4) is

discussed. It is shown that x # 0, and the attractor A is simply the

restriction of solutions in X to the nonnegative time interval [0, co[.

Appendix

Here we give a concise but mathematically complete construction of the

Loeb space used in the paper [18] and discussed in the previous section.

This is to take some of the mystery away from the notion of a Loeb space,

and to show that its construction is entirely algebraic. What we are not

able to do here is to expand on the properties of Loeb spaces that make

them so useful. For this see any of the introductions such as [1,11,16].

111

A . l . The hyperreals

To define the extension of the reals known as the hyperreals *R first fix

a nonprincapal ultrafilter U on N. That is, U is a collection of subsets of

N that is closed under intersections and supersets, does not contain any

finite sets, and is maximal with this property. This means that for every

set E C N either E E U or N \ E E U (but not both).

The hyperreals *R are defined by

*R = RN/U

meaning the quotient of RN by the equivalence relation

( ~ i ) -u (b i ) { i : ai = bi} E U

We say that ai = bi a.c.' Write [ (a i ) ] for the equivalence class (ai ) /U and

identify r E R with the constant sequence [ ( r ) ] , so that *R 2 R. Operations

of addition and multiplication are defined on *R pointwise, and it is easy

to check that this makes *R a field.

A hyperreal a = [(ai)] is said to be finite if there is n E N with (a1 5 n,

which means that Jail _< n a.c. The standard part " a E R of a finite

hyperreal is now defined byJ

"a = inf{r E R : ai 5 r a.c.}

It is easy to check that " ( a + b ) = ' a + " b and the same for products.

A.2. Construction of a Loeb space

Let W be two-sided Wiener measure on Co(R) = {x : R 4 R; x(0) = 0).

The set R is defined by

R = CO(R)"/IA

just like *R (so we could write R = *Co(R)).

An algebra 6 of subsets of R is given by sets of the form

A = r I i e~A i /U = [(Ai)]

where Ai C: Co(R). That is, for x = [(xi)] E R we define

x E A @ xi E Ai a.c

'A property Pi is said to hold a.c. if { i : Pi holds } 6 U. jIn the terminology of the subject, the standard part O a is the unique real number that

is infinitely close to a, written ' a N a

112

It is easily checked that 4 is indeed an algebra, and in fact the operations

n, U, \ are given pointwise.k

A finitely additive probability measure PO is now defined on Q by

Po(4 = "[(W(Ai)) l

for A = [(Ai)] E 4. Checking that PO is finitely additive is straightforward.'

The Loeb measure P on SZ is now the unique a-additive extension of

PO given by Loeb's fundamental result [26] which in this context is the

following.

Theorem A. l . (a) If (A,) is a sequence of sets from Q with nnEN A, = 0 then there i s a m E N with On<,, A, = 0.

(b) Hence, by Carathe'odory's Extension Theorem, there is a unique u-

additive extension P of Po to the a-algebra a(G).

Proof (a) Without loss of generality we may assume that the sets A, are

decreasing. Suppose for a contradiction that A, # 0 for each n. Then we

have for each n

0 # Anfl c A,

which means that if A, = [(A,,i)] we have 0 # A,+l,i C A,,i a.c. For

n = 1 ,2 ,3 , . . . in turn, systematically modify A,,i on a smallm set of indices

i , so that for each n

0 # A,+I,, 5 A,,% for all i E N

This does not alter the sets A, themselves. Now pick xi E Ai,i for each i and note that zi E A,,i for n 5 i. Consider the element x = [ (x i ) ] . Then

x E nnEN A, because for each n

{i : xi E A,,i} 2 {i : i 2 n} E ZA.

Thus A # 0, the required contradiction.

(b) Carathkodory's extension theorem shows that the finitely additive

probability PO on the algebra 4 extends uniquely to a a-additive probability

on a(G) provided that whenever nnENAn = 0 for a decreasing sequence

of sets from G then Po(A,) 4 0 with n. In our case this follows trivially

from (a).

The above construction gives a probability space

( 0 , 4 4 ) , P )

kFor example, [(Ai)] U [(Bi)] = [(Ai U Bi)]. ' In fact for disjoint A,B we have Po(A U B ) = Po([(Ai)] U [ (B i ) ] ) = Po([(Ai U Bi)]) = O[(W(Ai U Bi)) ] = " [ ( W ( A i ) + W(Bi ) ) ] = O[(W(Ai))] + O[W((Bi))] = Po(A) + Po(B). mThat is, a set not in the ultrafilter U.

113

The Loeb space resulting from the above construction is now the completion

of this space with respect to the measure P (that is, adding in the P-null

sets) which is still denoted P , giving the space (0, F, P ) say.

The a-algebra F is a Loeb algebra, and to indicate its origin i t is often

denoted F = L(G). Similarly we often write P = Q L , the Loeb measure

constructed from Q, where Q is the *R-valued function defined on 4 by

Q(A) = [(W(Ai))] , so that Po = "Q. The key to the use of Loeb spaces hinges on two main facts. The first

is due to Loeb [26] and shows that L(G) = G modulo null sets: for any

B E L ( 4 ) there is A E 6 with P(BAA) = 0.

The second is that the sets in 4 and their measures inherit (in a way

made precise by the Transfer Principle) the properties of the measurable

subsets of Co(R) and their Wiener measure. This makes the algebra 6 tractable, as expounded in any of the references cited above.

References

[l]. S.Albeverio, J.-E.Fenstad, R.H@egh-Krohn, and T.Lindstr@m, Nonstandard Methods in Stochastic Analysis and Mathematical Physics, Academic Press,

New York 1986.

[2]. Z.Brzeiniak, M.Capinski, and F.Flandoli, Stochastic Navier-Stokes equations with multiplicative noise, Stochastic Analysis and Applications 105

[3]. A.Bensoussan and R.Temam, Equations stochastiques du type Navier- Stokes, J. Functional Analysis 13 (1973), 195-222.

[4]. A. Bensoussan, A model of stochastic differential equation in Hilbert space applicable to Navier-Stokes equation in dimension 2, in: Stochastic Analy-

sis, Liber Amicorum for Moshe Zakai, eds. E.Mayer-Wolf, E.Merzbach, and A.Schwartz, Academic Press 1991, pp.51-73.

[5]. A. Bensoussan, Stochastic Navier-Stokes equations, Acta Applicanda Math- ematicae 38 (1995), 267-304.

[6]. M.Capiriski and N.J.Cutland, Stochastic Navier-Stokes equations, Acta Ap- plicanda Mathematicae 25 (1991), 59-85.

[7]. M.Capifiski and N.J.Cutland, A simple proof of existence of weak and statistical solutions of Navier-Stokes equations, Proceedings of the Royal Society, London, Ser.A, 436 (1992), 1-11.

181. M.Capiriski and N.J.Cutland, Navier-Stokes equations with multiplicative noise, Nonlinearity 6 (1993), 71-77.

[9]. M.Capifiski and N.J.Cutland, Foias and Hopf statistical solutions of Navier- Stokes equations, Stochastics and Stochastic Reports 52( 1995) , 193-205.

[lo]. M.Capihski and N.J.Cutland, Statistical solutions of stochastic Navier- Stokes equations, Indiana University Mathematics Journal 43( 1994), 927-

940. [ll]. M.Capiriski & N.J.Cutland, Nonstandard Methods for Stochastic Fluid Me-

chanics, World Scientific, Singapore, London, 1995.

(1992), 523-532.

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[la]. M. Capiriski & N.J. Cutland, Measure attractors for stochastic Navier- Stokes equations, Electronic J. Prob. 3(1998), Paper 8, 1-15.

[13]. M. Capinski & N.J. Cutland, Stochastic Euler equations on the torus, The Annals of Applied Probability, 9(1999), 688-705.

[14]. M. Capiriski & N.J. Cutland, Existence of global stochastic flow and

atractors for Navier-Stokes equations, Probability Theory & Related Fields

[15]. M.Capinski and D.Gqtarek, Stochastic equations in Hilbert space with application to Navier-Stokes equation in any dimension, Journal of Functional Analysis 126(1994), 26-35.

[16]. N.J. Cutland, Loeb Measures in Practice - Recent Advances, Springer Lec- ture Notes in Mathematics 1751(2000), Springer, Berlin.

[17]. N.J. Cutland & H.J. Keisler, Neocompact sets and stochastic Navier-Stokes equations, in Stochastic Partial Differential Equations, (Ed. A. Etheridge), LMS Lecture Notes Series 216, CUP, 1995, 31-54.

[18]. N.J. Cutland & H.J. Keisler, Global attractors for 3-dimensional stochastic Navier-Stokes equations, in preparation.

[19]. H.Craue1 and F.Flandoli, Attractors for random dynamical systems, Prob- ability Theory and Related Fields 100 (1994), 365-393.

[20]. F. Flandoli and B. Schmalfui.3, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative white noise, Stochastic & Stochas- tics Reports, 59(1996), 21-45.

[21]. F. Flandoli and B. Schmalfui.3, Weak solutions and attractors for three- dimensioanl Navier-Stokes equations with nonregular force, J . Dynamics and Differential Equations, 11( 1999), 355-398.

[22]. S. Fajardo and H.J. Keisler, Existence theorems in probability theory, Ad- vances in Mathematics, 120( 1996), 191-257.

[23]. S. Fajardo and H. J.Keisler, Neometric spaces, Advances in Mathematics,

[24]. F. Flandoli and D. Gatqrek, Martingale and stationary solutions for stochastic Navier-Stokes equations, Probability Theory 63 Related Fields 102( 1995),

[25]. AJchikawa, Stability of semilinear stochastic evolution equations, Journal of Mathematical Analysis and Applications 90 (1982), 12-44.

[26]. P.A.Loeb, Conversion from nonstandard to standard measure spaces and applications in probability theory, Trans. Amer. Math. SOC. 211( 1975), 113- 122.

[27]. B. Schmallfui.3, Measure attractors of the stochastic Navier-Stokes equation, Bremen Report No.258, 1991.

128). G.R. Sell, Global attractors for the three-dimensional Navier-Stokes equations, J. Dynamics and Diff. Equations 8( 1996), 1-33.

[29]. M.Viot, Solutions Faibles d 'Equations aux Derivees Partielles non Lin- eaires, Thesis, Universite Paris VI (1976).

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115(1999), 121-151.

118(1996), 134-175

367-391.

THE 2D-NAVIER-STOKES EQUATIONS PERTURBED BY A DELTA CORRELATED NOISE.

A. DEBUSSCHE

ENS de Cachan, an tenne de Bretayne Campus de K e r L a n n

351 70 Bruz cedex France

E-mail : arnaud. debussche@bretayne. ens-cachan. fr

We study the two-dimensional Navier-Stokes equations with periodic boundary

conditions perturbed by a space-time white noise. We prove that, for almost every

initial data with respect to a measure supported by spaces with negative regularity,

there exists a unique global solution in the strong probabilisticsense. The nonlinear

term is defined thanks to techniques borrowed from Wick renormalisation and

paraproducts in Besov spaces. Note however that no renormalisation is made here

and the nonlinear term is not modified. This result was given in g , here we give

simplified proofs. Then we prove ergodicity of the Gaussian invariant measure.

1. Introduction

We consider the two dimensional incompressible Navies-Stokes equations

in a periodic domain driven by a space time white noise: - du = (vAu - (u . V ) u - V p ) d t + dW, in [0, TI x 0, div u = 0, in [0, T ] x 0,

(1) 4 0 , t) = uo(t), 5' E 0, { u is periodic with period 27r,

where 0 = [0, 27rI2. The unknown are random processes: the velocity field

u(t, 6) = (ul(t , <), uz(t, 6)) and the pressure field p ( t , 6); these are defined

for (5'1, (2) E 0 and t 2 0. The kinematic viscosity v has no importance in

this work and we will take it equal to 1.

The equations are forced by a space time white noise. It is delta corre-

lated in time and in space, i .e. we formally have

It is the time derivative of a cylindrical Wiener process @ on ( L 2 ( 0 ) ) 2 associated to a stochastic basis (Q3, P, (.Tt)tzo) (see 12).

115

116

Stochastic Navier-Stokes equations have been investigated in many ar-

ticles 1 , 2 , 5 1 6 1 1 4 1 1 5 1 2 2 . In most cases, the noisy forcing term is white in time

and correlated in space. Recently much progress has been obtained in the

study of the associated invariant measures for noises which are very smooth

in space. Uniqueness and ergodicity properties have been proved 4 1 1 7 , 2 0 , 2 1 .

Also in 18, the singularities of the solutions in the three dimensional case

are studied.

In the work 16, a space-time white noise is also considered and equation

(1) has been studied through the associated Kolmogorov equation. They

prove directly the existence of a solution to this latter equation but are

unable to connect it to the original equation. The main difficulty is that,

as is well known, with such a rough noise i t is expected that a solution of

(1) is not regular.

In this work, we first observe that, using ideas borrowed from the theory

of the Wick product 1 1 , 2 2 1 2 4 , the nonlinear term can be defined for random

variables whose law is absolutely continuous with respect to a certain Gaus-

sian measure. Note however that we do not use renomalisation here. Then,

we split the unknown into a part whose law satisfies this property and a

smoother part. Using the bilinearity of the nonlinear term and using prod-

uct rules in Besov spaces, we show that the nonlinear term can be defined

for a sufficiently large class of random variables which contains a solution

of the equation.

Local existence is proved by a fixed point argument and an a priori

estimate is proved to get global existence. This a priori estimate is based on

the fact that we explictly know an invariant measure for equation (l), i t is

a Gaussian measure. Note that the idea to use an invariant measure is used

by J. Bourgain in the context of the deterministic nonlinear Schrodinger

equation. (See and the references therein).

Finally, we prove that this invariant measure is ergodic.

A space time white noise might not be relevant for the study of turbu-

lence where it is usually accepted that a spacially correlated noise should

be taken into account. However, in other circumstances, when a flow is

subjected to an external forcing with very small time and space correlation

length, a space-time white noise can be considered.

2. Notations

We introduce standard notations used for the Navier-Stokes equations (see

for instance 25). The subspace of ( L 2 ( 0 ) ) 2 consisting of periodic divergence

117

free functions with zero average is denoted by H :

Zl(O1 E2) = ZlP.rr , E2)r z2(51,0) = X2(E1,2.rr) 1 1 and P is the orthogonal projection onto H. The inner product of H is the

same as in ( L 2 ( 0 ) ) 2 and is denoted by (., .). It is convenient to use the complexification Hc of the space H . For

k = ( k l , k2) E Zi := Z2\{0, 0}, we write

2 112 kL = (k2 , lkl = (k? + k 2 ) 1

k' ik.E k . E = klEl + k2E2, e k ( E ) = - - e 7 E = ( E l , E2) E 0.

27T IN Then (ek)kcq (resp. (Re(ek))kEZ;) is a complete orthonormal system of

Hc (resp. H ) . We also use the space (R)"; := 7-t. We shall consider H as a subset of

x. The unbounded operator A is defined by

AX = P A X , z E D(A) = ( H ; ( O ) ) ~ n H .

Aek = -Ikl2ek, k E Z,. 2

We have

Here and in the following H&(O) is the subspace of the Sobolev space

H T ( 0 ) consisting of all periodic functions. For T E R, we use the fractional

power ( -A)T on the domain D(( -A)T) . It is classical that D( ( -A )T ) is the

closure in (H2 ' (0) )2 of the space spanned by (ek)&Z,2. Moreover . I is a norm on D(( -A)T) equivalent to the usual norm on (H2'(0))2. For

any T E R, P can be defined on ( H z ( 0 ) ) 2 and its image is D(( -A)T) . We set

W =PW. (2)

It is not difficult to see that W is a cylindrical Wiener process on H thus,

for any complete orthonormal system (ek)kcz; in H , we can write

W = Pkek

k€Z;

118

where (,&)kEZ; is a sequence of independent Brownian motions on the

stochastic basis (Q, F, PI (Ft)t?o). As is well known, thanks to the incompressibility condition, we can

rewrite the nonlinear term as

(u . 0 ) u = div (u @ u),

where

We will use this form which is better suited to the case of non smooth

velocities. Whenever it makes sense, we set

b(z, y) = P div (z 18 y), b(z) = b(z, z). (3)

When projecting equations (1) on HI we get

du = (Au + b(u))dt + dW,

(4) { u(0) = uo.

We wish to solve (4) and to find a solution which is a D((-A)') valued

process. Implicitly this means that we restrict our attention to zero average

initial data. This is no loss of generality since we can change the unknown

in (1) and consider only such initial data.

3. Preliminaries

It is not expected that (4) has a solution in D ( ( - A ) T ) for T 2 0. This is

not even true for the linear equation

d z = Azdt + dW,

4 0 ) = 201

(5)

whose solution is given by

t

~ ( t ) = etAzo + 1 e(t-s)AdW(s).

The second hrn in the right hand side is a continuous process with values

in D ( ( - A ) T ) for any T < 0 but does not take its values in D((-A)') for

any T 2 0. This follows from

119

for any r < a < 0 and

r t

for r 2 0, see 12. We have denoted by LHS (K1 , K2) the space of all Hilbert-

Schmidt operators from a Hilbert space K1 on a Hilbert space K2. It follows that we have to work with non smooth processes and this

creates difficulties when working with the nonlinear term. Here we pro-

ceed as is usual when dealing with parabolic equations in negative Sobolev

spaces. We use Littlewood-Paley decomposition and paraproduct to define

the nonlinear terms, see 7 1 8 1 2 3 .

However, working in the context of negative Sobolev spaces introduces

some technical difficulties and i t is convenient to consider Besov spaces.

We define, for N E N, PN as the orthogonal projector in Hc onto Span

(ek)lklSN, PN is also orthogonal in D((-A)T), r E R, and it can be easily

extended to 3-1. We also set, for q E N, 6, = P,, - P2,--1. Then 6,u i s

defined for all u E ‘H and contains the Fourier components of u between

2q-l and 24 :

For cr E R , p 2 1, p 2 1 we define

it is a Banach space with the norm

1 l P

Iulr3;,p = (~2~q~16qul;.;.))

The following result is crucial in our argument and is the main motivation

for working in Besov spaces, see 7,8.

Proposition 3.1. Let p , p 2 1, cy + p > 0 , a < 2 / p , p < 2 / p . Then zf u E B& and v E i3t,,p we have uv E Bz,P where y = a + P - p , and 2

1 4 B ; , p i C 1 4 B p q p l ~ l B g , p . (6)

Let us also recall that the nonlinear term verifies the following identities,

see 1i26

( ~ ( x ) , x ) = 0, ( b ( z ) , AX) = 0. (7)

120

These are true for any x such that the quantities on the left hand side make

sense.

Let us denote by p the product measure on 'H,

P = I-I "0, 1/(2142)). kEZ:

We write p = N(0, Q). Notice that p(D(( -A) ' ) ) = 1 if and only T < 0, so

the support of p is included in D((-A)'). This follows from the fact that

(-A)-l+" = Q(-A)z' is trace class if and only if T < 0.

Also, it is not difficult to prove that p(f?&) = 1 for any (T < 0, p , p 2 1.

This can be done using similar ideas as in lo.

Moreover, it is well known that in the case of periodic boundary con-

ditions considered here, thanks to (7), the measure 1-1 is formally invariant

for equation (4).

We use techniques borrowed from the theory of Wick renormalized prod-

uct to extend the definition of the nonlinear term. We shall denote by

H,, n = 0,1, ... the Hermite polynomials defined by the formula

It is convenient here to work on the space Hc as well as in the complex-

For x E 'H we write

ification of D((-A)') which for simplicity is still denoted by D((-A)').

1 2 X N = PNX = C (x, el)et, X N = ( z N , z N ) ,

I l l lN

and

bN(x) = b ( P N z ) .

We also define for x E If, and N E N :

where

and

121

As easily checked we have

: (XZ,), : ( I$) = ( X p ( I $ ) - p$, 2 = 1 , 2 , < E 0, N E N.

We will see that : X N @ X N : converges in some sense, the limit can be

defined as a renormalized tensor product. A key observation is that

N b (x) = b ( X N ) = P div ( X N @ X N ) = P div (: X N €9 X N :).

Thus b ( x N ) converges without any renormalization and the limit is a natural

definition of b(z). More precisely, we have

Lemma 3.1. For any o < 0 , p 2 1, p 2 1, k 2 1, the sequences (: (xL), :

) N E W , (: (x$), : ) N E W , ( X ~ X $ ) N E M are Cauchy in Lk(('FI, p; B&).

This result is proved in the context of Sobolev spaces in ', and for the clas-

sical Wick product in Besov spaces in lo. Using the techniques developed

there, it is not difficult to prove this lemma.

Using the continuity properties of P , we deduce

Proposition 3.2. For any 0 < 0 , p 2 1, p 2 1, k 2 1, the sequence ( b N ) N E n is convergent in L'"('FI, p; B:,;~).

Corollary 3.1. Let X be a random variable with a law vx which is absolutely continuous with respect to p and such that % E L'(('FI;p), 1 > 1,

then the sequence ( b N ( X ) ) N E N is convergent in Lk((R; BE,;') f o r any 0 < 0,

p 2 1, p 2 1, and k 2 1. W e denote by b ( X ) its limit.

Proof : It suffices to write

with + = 1. So that the result follows from Proposition 3.2. Let us now set

t

z ( t ) = e( t -S)AdW(s) , t E R, (8) .I, which is the stationary solution of

d z = Azd t + d W ( t )

with invariant law C(z( t ) ) = p.

Lemma 3.2. For any 0 < 0, p 2 p 2 2, we have

z E C(R; f?g,p), Pa.s.

122

Proof : It is not difficult to prove that for any S < 0 the trajectories of

(-A)&. are continuous with respect to t , II: on R x 0. This uses for instance

the Kolmogorov criterion of continuity, see 12.

Since for anyp 2 1, (C(a)>' c 13:,m, it follows that z has trajectories in

C(R; a;,,) for any 0 < 0. Moreover, it is easy to see that z has trajectories

in C([O,T]; D((-A)"/ ' ) ) = C([O,T];Bz,,), < 0, so that by interpolation

z E C([O,T]; BE,,), 0 < 0, p 2 p 2 2, P - as . . 0

By Corollary 3.1, b(a(t)) can be defined for each t E R so that we have

a well defined process (b(z( t ) ) tEw.

Lemma 3.3. For any T 2 0, lc 2 1, p, p 2 1 and 0 < 0 , we have

b(z ) E Lk(R x [0, TI;

Proof : We have by Fubini theorem, since L ( z ( t ) ) = p for any t E R,

and this is a finite quantity by Proposition 3.2. 0 We now want to extend the definition of b in a suitable way so that b(u)

makes sense for a solution of (4). The idea is that if we define v = u - z then v is expected to be smoother than both u and z . The following result

states that, if this is the case, b(u) can be defined in a nonambiguous way.

Proposition 3.3. Let X and 2 be random variables such that tke law of

Z is p and Y = X - Z E Lb(R; a:,,) where

2

P b > 2 , - > a > O ,

then the sequence of random variables (bN(X) )NEN converges in

If moreover, the law of X , ux, is absolutely continuous with respect to p and % E L'(7-i; p ) with 1 > 1 then the limit coincides with b ( X ) defined in Corollary 3.1 and

L ~ / ~ ( R ; B:,,) for any a < a - 1 - ;. 2

b ( X ) = b(Y) + 2b(X, Y ) + b ( 2 ) . (9)

Proof : Let a < a - 1 - 2 and set (T = a - a - 1 - 5. Clearly, 0 < 0

and 2 E Lb(R; a,",,). Thanks to Proposition 3.1, b(Y, 2) is well defined in

Lb12((R; a:,,) and

P

bN(y, Z) --+ b ( ~ , Y), in L ~ / ~ ( ( R ; B;,,).

123

Similarly, since a > u, B;,, c t3& and

P(Y) --t b ( ~ ) , in ~ ~ ' ~ ( ( 0 ; B;,,).

b y x ) = b y Y ) + 2 b N ( X , Y ) + b N ( 2 ) .

We have :

Using Corollary 3.1 the last term of the right hand side also converges. We

deduce that the left hand side converges. The last statement is clear. 0

Remark 3.1. It follows that b ( X ) can be defined whenever the hypotheses

of Corollary 3.1 or of Proposition 3.3 are satisfied. Moreover, if instead

of the assumptions of Proposition 3.3 we only have X - 2 E a;,, Pas.,

by a standard localization argument, we easily deduce that ( b N ( X ) ) ~ E ~ converges almost surely in a;,, .

4. Existence and uniqueness

The main result of this work is the following.

Theorem 4.1. Let CT < 0, p 2 p 2 2, ,O 2 1, and a > 0 such that

2 1 1 a l u -g < a < -, and - - - < - - - < -,

P P 2 2 P 2

then for any T 2 0 there exists a unique mild solution u to (4) such that

u - z E C([O, TI; B;,,) n La(O, T ; B;,,).

Moreover, for any 1 E N,

Remark 4.1. By Remark 3.1, the solution has the required properties to

ensure that b(u) is well defined. Indeed, u(t) - z ( t ) E B;,, P a s . w E 0 for

for almost every t E [O, TI.

Remark 4.2. Note the condition - $ < 5 implies that p > 2. This is the

reason for working in Besov spaces. Recall that Sobolev spaces correspond

to Besov spaces with p = 2.

Proof : We split the proof in two parts. We first prove local existence

on a random time interval depending on the initial data. Then an a priori

estimate enables us to get global solutions.

First step : Let uo E B;,,. We fix w and solve (4) pathwise, w is taken

in a set of probability one such that the various properties on z and b(z) proved above are true.

124

Let us fix and consider the mapping defined on

Thanks to Propossition 3.1 we know that

we deduce that

since, by assumption

since, by assumption

Similarly, we have

S9imilarly, we have

Furthermore,

125

so that, choosing k sufficiently large and using Lemma 3.3, we obtain

11 eA(t-s)b(z(s))dslETo 5 C ( Q 1 PI T)lb(z)Ilk(o,T;a~,,')'

Finally, it is clear that t H e"(u0 - z ( 0 ) ) is in E T ~ and

I."(.o - Z(0))lETo 1.4% (TIPI P,T)bo - "(0)lJ3,.,,.

This shows that 7 maps ET,, into itself.

It is standard to deduce from the above estimates that there

z(O)la;,,, I ~ ( Z ) I ~ ; , ~ I , IzIc(p,q;~;,,)) > 0 such that, for TO 5 T* , 7 is a strict

contraction on the ball of center 0 and radius R in E T ~ . We deduce that

there exists a unique solution on [O, T*]. Second step : It is clear that it is sufficient to obtain an a priori estimate

in f3& in order to have global existence. We will in fact prove that, if

u(t, uo) is a solution to (4)

exists T*(IUO - "(O)lB,.,,I Ib(z)la;,1, Izlc([o,T];a,.,,)) > 0 and " 0 -

r

This implies that for IJ. almost every uo we have

thus the local in time construction of the first step can be iterated leading

to a global solution. Thus Theorem 4.1 is proved.

Let us prove that (11) holds. We use a formal argument which can

be easily justified by a Galerkin approximation. Indeed, it is not difficult

to prove that the local solution constructed above is the limit of Galerkin

solutions.

We have

u(tl uo) = e"(u0 - z ( 0 ) ) + e(t-S)Ab(u(s, u0))ds + z ( t ) , 1" therefore

lu(k ~o)If?;,, 5 C ( P , PI ~)(IuOlB,.,, + Izola;,,)

126

We have, by Holder inequality in time and then in the expectation,

Then, integrating with respect to p, using again Holder inequality and the

invariance of p, we obtain

1/3 +C(P, p, a)W2 (Jx /b(uo)l;;,;l)~P(~O)) .

By Lemma 3.2 and Proposition 3.2, we know that the right hand side is

finite. This proves our claim (11).

The last statement (10) is proved in the same way.

5 . Ergodicity

As already mentionned, using a galerkin approximation, it is not difficult

to prove that the Gaussian measure p is invariant for (4). We now study

some of its properties.

Given a functional cp defined on 'Ti, we denote by (p its average with

respect to p r

We have the following result of exponential convergence which clearly

implies ergodicity.

127

Theorem 5.1. There exists a constant X > 0 such that

for any p E L2('FI, p) and t > 0.

Proof : Again the proof is formal and could be justified by approximation.

Replacing 'p by p - p, we can assume that (p = 0. Let U ( t , u g ) =

E ( p ( u ( t , ug))), then U is formally a solution to the Kolmogorov equation

(see 13)

d t - - 1TrD2U 2 + (Au + b(u), DU), { U(0, uo) = 4 . u . o ) .

Recall that thanks to the invariance of p, we have

Jx ( ;T rD2U(4 + (Au + b(u), W 4 ) ) U(U)dP(U)

= -; J, IDU(u)l2dp(u).

Therefore

I t is well known that the Gaussian measure satisfies the spectral gap in-

equality

for any $ E W1i2(7d, p) , with X > 0. I t follows

Hence, the result follows by integration. 0

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EQUATIONS IN HILBERT SPACES, London Mathematical Society, Lecture Note

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ANALYSIS, SIAM, Philadelphia.

INVARIANT MEASURES OF LEVY-KHINCHINE TYPE FOR 2D FLUIDS

S. ALBEVERIO

Institut f u r Ang. Mathematik, Universitat Bonn, Wegelerstr. 6, 0 -531 15 Bonn; SFB 61 1, Bonn; BiBoS, Bielefeld; CERFIM, Locarno; Accademia d i Architettura, USI, CH-6850 Mendrisio;

Dipartimento d i Matematica, Universitci di Trento, I-38050 Povo; E-mail: albeverio@uni-bonn. de

B. FERRARIO

Institut fur Ang. Mathematik, Universitat Bonn, Wegelerstr. 6, 0-531 15 Bonn; Dipartimento di Matematica, Universitci d i Pavia, via Ferrata 1, I-271 00 Pavia;

E-mail: [email protected]; [email protected]

A survey of results on invariant measures of the L6vy-Khinchine type for 2D Eu-

ler and stochastic Navier-Stokes equations is given. Uniqueness results of the

corresponding Liouville respectively Kolmogorov flows are discussed. Stochastic

dynamics associated with the invariant measures are also discussed (stochastic

Stokes equation for the vorticity in the Gaussian case, Doob's independent Brow-

nian motions process in the compound Poisson case).

1. Introduction

Let us begin considering the classical motion of an ideal incompressible

fluid, that is the Euler equations

with suitable boundary conditions: u . n = 0 on OD, where n is the exte-

rior normal to the boundary dD of the smooth domain ED or the periodic

boundary condition for w . n when ID is the torus. The unknowns are the

velocity vector u = w(t,z) and the pressure (scalar) p = p( t ,z) . Here

An equivalent formulation can be expressed in terms of the vorticity w :=

V A u. For space dimension d = 2, w is a scalar field w = 2 - 2 z V' . w

z = (21,. . .,zd), V = (%, a . . . , &) and w . w is the scalar product in I@.

130

131

and applying the V' = (-&, &-) operator to the first equation in (l),

we obtain { ~ + u . V w = 0 (t1x) E ( O I T I x JD (2)

w = v L . u

with the tangential boundary conditions for the velocity. In (2) there is no

pressure, but this has to be recovered from the velocity field u: applying V to the first equation (1) we get -Ap = V . [(u . V)u].

Since w evolves according to a transport equation, the solution is w ( t , x) =

w ( 0 , E t x ) where Et is the flow of material points in the fluid (&x ( t ) =

v( t ,x ( t ) ) ) . Since the vector field w is divergence free, any solution to sys-

tem (2) corresponds to a volume preserving flow Et (i.e. the Lebesgue

measure on ID is preserved in time). Moreover there are two other con-

served quantities

energy

enstrophy S = J, w ( z ) ' d z E = L J 2 , I +)I2da:

This has to be understood as follows: if an Euler flow (2) with finite energy

is defined, then the energy is indeed constant. The same holds for the

enstrophy in a two dimensional spatial domain. The computations showing

this invariance in time are easily checked in these cases, e.g.

dS

d t - - - J, w ( t , .) at4 t1 .) dx

= - J, ~ ( t , X) ~ ( t , 2) . V w ( t , X) dx

= J, V U ( ~ , X) . U ( t , X) w ( t , X) dx + J, ~ ( t , x)V . u( t , X) w ( t , X) dx

Since V . w ( t , x) = 0, then 2 = 0. Notice that all the quantities have been

assumed to be well defined, i.e. the solution w is regular enough.

By means of these conserved quantities, heuristic expressions of invariant

measures can be given. In the next section, we will deal with probability

measures m of Lkvy-Khinchine type. They are supported on distribution

spaces. Therefore the Euler dynamics with initial data in the support of

the measure rn (if it exists) is not a classical one. An overview on the

study of a deterministic dynamics having m as invariant measure will be

presented in section 3. According to the Koopman-von Neumann theory,

as soon as a (candidate) invariant measure m is known, any flow St, t E R, in the space of distributions S' a gives rise to a flow in C2(rn) , represented

aBy S' we denote the vector space of continuous linear functionals on C,"(ID) or, when

the spatial domain is the torus, CpMer(T).

132

by a unitary group: (Utf)(w) = f(Stw),t E R. And viceversa, under

some assumption on the group Ut, it is possible to construct a flow St, m as . . For this reason the infinitesimal generator B of the unitary group

(Ut = eitB) will be analyzed. On the other hand, in section 4 a stochastic

dynamics having m as invariant measure will be introduced, as the Markov

process associated to the classical Dirichlet form given by the measure m. In

contrast to the deterministic nonlinear case, this is an easy (linear) problem

to study. The corresponding flow in C 2 ( m ) is represented by a contraction

semigroup Tt,t E R+; an analysis of its infinitesimal generator Q will be

given (Tt = etQ). Finally, in section 5 we merge the deterministic and the

stochastic frame. Partial results about the stochastic nonlinear problems,

which arise in this case, will be given and open problems will be presented.

2. Invariant measures of LQvy-Khinchine type

One important feature of the evolution (2) is that the underlying flow Et in D preserves the Lebesgue measure. This allows to construct a family of

probability measures on the space of distributions S‘ which are invariant

for any given flow (2). In fact, let m be a probability measure on S’; this

corresponds to a family of random variables {x,},,~, realized canonically

as random variables on the measure space (9, B(S’))

Assume now that the random variables are identically distributed and

X, independent of X, ’dp, II, E S such that pII, = 0

This expresses, in a distributional sense, independence in distinct points.

Assuming some continuity (e.g. in the sense of X,,, --+ 0 in proba-

bility if pn -+ 0 in S), this is the definition of white noise (see, e.g.,

Gel’fand&Vilenkin’‘).

Then the law of the random variable (w( t , .), p) = (w(0, .), p o is inde-

pendent of time, where Et is the pointwise flow in D corresponding to (2). (Of course, this is rigorous if Et : D + D is “smooth enough”.) Because of

independence, the knowledge of the law of each (w ( t , .), determines any

joint distribution for the family { ( w ( t , .), p ) } r p E ~ . Hence the white noise m is a time invariant measure.

It is well known (see Gel’fand&Vilenkinl‘) that any Lkvy-Khinchine prob-

ability measure is a white noise in the sense specified above. The Lkvy-

Khinchine representation for infinitely divisible laws gives the characteristic

133

functional

where the characteristic exponent is

with a E R, b 2 0,

When a = 0, b > 0,O = 0, we have a centered Gaussian measure p. When

b = 0 and a = J,, vl{lvl<l) dO(v), we have a compound Poisson measure

IT (this holds in particular if a = 0 and B is a symmetric measure). We

briefly consider these two cases. For the results on the Gaussian case, we

refer to Albeverio et a1.2. For the result on the Poisson case we refer to

Albeverio&Ferrario4 and references therein.

Gaussian measure

Let us denote by A the Laplace operator in D with homogeneous Dirichlet

boundary condition and let R"(D) = D( ( -A )u /2 ) ( a > 0); for negative

index a < 0, the Hilbert space is defined by duality: 3-la(D) = ( ' W U ( D ) ) ' . Hence,

&,(l A v2)de(v) < 00 and RO = R \ (0 ) .

p( 'Hb(D)) = 1 V b < -1

that is, the support of the Gaussian measure p is given by nb<-lRb(IO). Similarly when D is the torus.

Compound Poisson measure

The support of the compound Poisson measure IT is the space r of config-

urations. More precisely, for any n = 1 , 2 , . . . l let

;i(", = { ( (m Z l ) , . . ., (vn, 4) E. (Rl x D)n : 51 # 21, for 1 # I C )

The space of n point configurations is defined as

n

r ( n ) = {w = C q s z l : vl E R O , z1 E D~ z1 + Xk for 1 # I C ) 1=1

where 6, is the Dirac measure concentrated in z. For each index nl there is a bijection

j ( n ) ;i(n)/s(n) + r(n)

where S(") denotes the permutation group over (1,. . . , n). Consider on the

Bore1 a-algebra of subsets of i ( n ) / S ( n ) the measure o @ ' ~ = (dO(v)dz)@'n, where for simplicity we assume the LBvy measure 6 to be finite. The image

134

measure on I '(n), under the bijection J(") , is denoted by on

Set = (8) and a 0 = Si0). The space of configurations

r = u;=,W

is defined as disjoint union of topological spaces, with the corresponding

Bore1 a-algebra B ( r ) . The compound Poisson measure II is defined by

Remark. Notice that the above measures are not the only invariant measures

known for the Euler equation (2). For instance, Albeverio et al.'l5 consid-

ered more general Gaussian white noises p ~ , ~ (y > 0, p > -y), expressed by

means of the enstrophy and of the renormalized energy. For other types of

invariant measures related to the Gaussian ones, see Capiriski&Cutland",

Ciprianoll. Anyway we consider here only white noise distributions for

the vorticity w , in order to have a unified approach ( p and II as particular

cases of a Lkvy-Khinchine measure). Infinitesimal invariance of measures of

Gaussian and Poisson type for the Euler equation has also been discussed

in Boldrighini&F'rigiog. 0

3. Deterministic dynamics

From now on, we choose the spatial domain D to be the torus T = [0, 27rI2; hence periodic boundary conditions are assumed. In this section this choice

is done for mathematical convenience; in the next one it will anyway appear

necessary for a right physical interpretation.

Let w be a periodic distribution; it can be developed in Fourier series

with respect to the complete orthonormal basis { & e i k ' z } k E Z z in the (com-

plex) L2(T). Let denote for short by P k the k-th element in this basis. Then

w k = W - k , because w is real. Adding a constant to the velocity, solving

(l), yields again a solution of (1). We select that one of zero mean value.

Hence also the mean value of the vorticity is assumed to vanish: wo = O.b

1 W = 2;; x k E Z 2 w k p k with w k := s ' ( W , p - k ) S . The Coefficients w k E and

bThe starting problem indeed is formulated in the real framework. Now the complex

structure arises in a somewhat artificial but practical way - via Fourier transform. Ac-

tually the relevant variables are {%~k,Swk}~.~~,~>~ or { w k } k E Z ~ , k > O , where k > 0

means either !q > 0 or kl = 0, Icz > 0. Anyway, whenever the whole sequence { ~ k } ~ ~ ~ z

appears, the condition wl, = w-k is assumed.

135

Albeverio et a1.2 show that equation (2) can be rewritten as a system of

infinite equations for the Fourier coefficients W k , for any k E Z2, k > 0

-- dwk( t ) - ChkWh(t)wk-h(t) dt h#k,h#O (4)

where the r.h.s. Bk is a quadratic expression of the Fourier components

with coefficients C h k =

This is obtained formally from equation (2). We point out that the “Euler

to

Here

as if Y f

dynamics” with state space the support of the measure p or II has to be

understood in the generalized sense, i.e. this is not a classical dynamics

with function valued solution w( t , .). When dealing with the compound

Poisson measure II, equation (2) is actually the equation of vortices (see

Marchioro&P~lvirenti~~). For any integer n 2 2, the vorticity w at time t is

concentrated in n distinct points ~ l ( t ) , . . . , xCn(t) of T with given intensity vj

of each vortex xj ( w( t , 11:) = C:=, ~ j S , ~ ( ~ ) ( 1 1 : ) ), the z j ( t ) evolving according

d dt

Y.- X j ( t ) = vL ” j

l # j , l , j = l

g is the Green’s function of -A on T: g(y) = -& Ckfo & eik’y, 0. For n = 1 ( w ( t , x ) = v1SX,(,~(z) ), the single vortex moves

there were two vortices a t the points x1 and - X I , with intensity

v1 and -v1 respectively, namely the vortex point moves according to

%11:1(t) = - v 1 m CkZo 6 eik.2xl(t) and the velocity field in any point

11: E T distinct from the vortex is w ( t , z) = Vig(11: - z l ( t ) ) .

d 1

The main properties of the B k are given in the following

Proposi t ion 3.1. If m = II, assume that the finite Le‘vy measure satisfies

Then for both cases m = p or m = II, we have that

B k E P ( m ) for any 1 5 p < 00

dBk - (w )=O m-a.e. w auk - Bk(u) = B - ~ ( w ) m - a.e. w

for any IC E Z2,k # 0.

136

For the proofs, we refer to Albeverio et al.2,334; the LP-summability comes

from Ciprianoll.

What is the meaning of the functions Bk? If equation (4) would give a

flow St (t E EX) in the support of the measure VI, then this would induce a

flow in the Hilbert space L2(m) by

( U t f ) ( W ) = f ( S t w ) , f E C2(m) (6 )

The strongly continuous unitary group Ut in C2(m) (unitarity is given

by the fact that the measure m is invariant) is characterized by its in-

finitesimal generator B , which is a self-adjoint operator with domain

D ( B ) = { f E L2(m) : 3 L2 - limt+o v}. Its expression when act-

ing on the dense subset 3 C r of smooth cylindrical functions ( 3 C r 3

f : f(w) = F ( w j l , . . . , wj,) for some integer N and F E C F ( ( C N ) ) is the

following

U t f - f 1 dF Bf = L 2 - l i m v = - ~ B k -

t-+O zt i k a u k (7)

This is a well defined expression in Cz(m), because the sum is finite and,

according to Proposition 3.1, each B k is square summable. Let us call

(B, 3Cp) the Liouville operator.

The Liouville operator is symmetric, i.e.

and has self-adjoint extensions (according to von Neumann theorem, since

B commutes with the conjugation J defined in C2(m) by J f ( w ) = T(-w) ). Actually, one of the self-adjoint extensions is B (when it exists, that is

when a flow St is given). The question of uniqueness of the self-adjoint

extensions of the Liouville operator was posed in Albeverio et a1.' (this is

formulated as essential self-adjointness of the Liouville operator). This is

interesting since any self-adjoint extension Be generates a strongly continu-

ous unitary group. Among these groups, the positivity preserving ones are

in one-to-one correspondence with a dynamics St, in the sense that there is

a one-to-one correspondence between positivity-preserving unit-preserving

(Utl = 1) unitary groups Ut in L2(m) and weakly measurable measure

preserving flows St in the support of the measure m (see Goodrich et al.17,

Albeverio&Ferrario4).

At this point, we have to distinguish which measure m is considered.

For m = p, it has been proven by Albeverio&Cruzeiro' that there exists a

flow (4) for p-a.e. initial data. Hence, for the Euler problem, the essential

self-adjointness of the Liouville operator is equivalent with the uniqueness

137

of this generalized Euler flow, having p as invariant measure. But so far,

the essential self-adjointness of the Liouville operator in L2(p) has not

been proven. On the other hand, for m = II, a unique flow St, t E R, to equation (4) exists for II-a.e. initial data (see Albeverio&Ferrario*).

II-a.e. is justified, indeed for some initial data the vortices can collapse.

DUrr&Pulvirentil4 prove that for any number of vortices n and for any

choice of the vortex intensities, there exists a unique (global in time) flow of

(5) for each initial data in the complementary set of a (Lebesgue-)negligible

subset of Ti". Therefore, keeping in mind the definition of the pre-image

measure II on each for any TI L 1, there exists a set N" E f?(r(")) with IT(N") = 0, such that there exists a unique (globaI in time) flow

of (4) for each initial data w(0) in the complementary set of N" in I'("). This holds also for negative time t . Hence we can define, II-a.s., a flow

St : w ( 0 ) H w ( t ) , r --t r for t E R. More precisely St : r(n) 4 I'("); for

each w(") the vortex intensities uj , j = 1,. . . , n, do not change in time,

only the points xj on which the vorticity is concentrated evolve in time.

This flow is volume preserving, since the point flow given by equations (5)

preserves the Lebesgue measure. Therefore, St of (4) gives a IT-measure

preserving flow on each component of r. This is expressed by

ITost=n, t E R (8)

Hence there exists a unique strongly continuous positivity preserving uni-

tary group U,, defined by (6). Let us call this a Markov uniqueness result,

adopting the same terminology of Markov uniqueness as used to denote a

second order (Kolmogorov) dissipative operator which has a unique exten-

sion generating a Markov strongly continuous semigroup in a Banach space

(see, e.g., Albeverio et aL8, Eberle15, Stannat'O). We have therefore

Proposition 3.2. The Liouville operator ( B , FCT) in L2( r , II) is Marlcov unique, that is there exists only one self-adjoint extension Be 2 B which generates a positivity preserving strongly continuous unitary group in P(r, n).

Remark. The measure II is invariant for the group Ut, i.e.

or, equivalently, the measure II is invariant for the infinitesimal generator

(B, D(B) )

138

(The equivalence of infinitesimal and full invariance is due to the fact that

Finally, let us notice that even if any Gaussian measure can be approx-

imated by a sequence of Poisson measures, the II-a.s. well posed dynamics

(in S') is not helpful to define in the limit a p a s . dynamics. Indeed,

p and II are singular measures: supp II c supp p and II(F) = 1,

p ( r ) = 0 (see the proof by Colella and Lanford, in the modified version

in Albeverio&Ferrario4).

1 E D ( B ) . )

4. Stochastic dynamics

One way to define a stochastic dynamics with a given invariant measure,

is by means of the theory of Dirichlet forms. We consider the two cases

separately.

Gaussian measure

Let & be the classical pre-Dirichlet form given by p:

where the measure p is the infinite product of centered Gaussian measures

p k on @; since w k = x k + z y k ( X k , lJk E R), each measure p k is in fact defined

as a measure on R x lR

Therefore the integration of a function f : {Wk}k#O .+ C with respect to the

measure p has to be understood as the integration of a complex function

of the real variables Z k , Y k , through W k = X k + i y k . In particular the pre-

Dirichlet form can be rewritten as

because -2- auk = l( 2 a p i & ) , for k > o . This form is closable, as easily seen by integration by parts, rewrit-

ing & as the positive symmetric sesquilinear form associated with a

densely defined positive symmetric operator (see, e.g., Albeverio&Rockner7,

Ma&Rockner" for this technique). The closure is a classical Dirichlet

form, quasi-regular and local; moreover, the minimal and the maximal ex-

tension coincide (we refer, e.g., to Ma&Rockner" for results of this type

in the Gaussian case). Its associated classical Dirichlet operator is the

139

Ornstein-Uhlenbeck operator in f? (p ) , which is the closure of

This Dirichlet operator generates a strongly continuous Markov semigroup

dolt E R+, in ,C2(p); the Markov process properly associated solves the

stochastic linear differential equation

where P,(t) = WL"'(t) + iW,"'(t), with {WL"), W, ("1 } k>O a sequence of

independent standard real-valued Brownian motions. The measure p is

invariant for this process. This is a stochastic Stokes equation

in which b / 2 represents the viscosity of the fluid ( b > 0). This corresponds

to the following equation for the velocity vector fields

b dv(t, x) = ,A v(tl x)dt + Vp(t , x)dt

v . v( t ,2 ) = 0

Therefore, the noise is defined by means of a Brownian motion, cylindrical

in Lz(T) for the velocity. Interpretation of (10) as an equation of motion of

a viscous fluid is possible only in the frame of periodic boundary conditions.

Indeed, the boundary conditions in a bounded domain ID are v . n 1 a D = 0

for an ideal fluid (viscosity b = 0) and ~ 1 8 ~ = 0 for a viscous fluid ( b > 0). Hence the torus is the only case in which the boundary conditions for the two different fluids coincide, and therefore the functional spaces

introduced for the Euler problem fit also for the Stokes problem (and in

the next section for the Navier-Stokes problem). It is worth at this point

to say that all the results of section 3 require the spatial domain ID to be

bounded. (For a formulation of the Euler problem as an infinite system of

nonlinear equations (4) when ID is a bounded domain in IR2 with piecewise

C1 boundary 8D1 see Albeverio&H@egh-Krohn5 .)

Compound Poisson measure

Similarly as before, we introduce the classical pre-Dirichlet form given by

We refer to Albeverio,Kondratiev&Rockner' for the definition of the intrin-

sic gradient Vr on r, of the tangent bundle Tu(l?), as well as for the basic

framework and results used in the following. We have

Any pre-Dirichlet form ( E n J , FCT) is closable, as seen "by integration

by parts". Thus the pre-Dirichlet form (& ,FCT) is closable, being the

sum of closable forms. Let us denote by ( z , D ( z ) ) the closure form. It is

easy to see that this is a classical Dirichlet form, local quasi-regular. The

corresponding classical Dirichlet operator is the closure of (Ar, FCF), the

Laplacian on the space of configurations r; there is no drift term, since

the reference measure on T is the (flat) Lebesgue measure. Therefore the

Markov process properly associated to r is a Brownian motion on the space

of configurations, i.e.

X € W ( O )

where { W:}zE~ are independent standard Brownian motions on the torus

and W$ = x (x E w(0) means that the sum runs over all z on which w(0) is concentrated). The measure II is invariant for this process (which has

been originally discussed, in other terms, by Doob, see Albeverio et al.')).

5 . Final remarks

Given the Liouville operator B and the diffusion operator Q, it is possible

to merge them together in the following sense. Since both the operators B and Q are well defined on the dense subset FCF of C2(m) , we can consider

the sum operator (Kolmogorov operator)

K = Q + iB, D ( K ) = 3 C r (12)

which corresponds to a non-symmetric sesquilinear form. The operator

(Q, FC?) is negative definite, the operator (B , FCT) is skew-symmetric;

140

the compound Posssion measure II

hence, the operator ( K , FCT) is dissipative in C 2 ( m ) , able. The measure m is infinitesimally invariant for K

J K f d m = 0 Vf E F C ~

141

and therefore clos-

since it is so separately for Q and for B. Our analysis is based on the follow-

ing: if the closure generates a strongly continuous Markov semigroup in

C z ( m ) , then there would exist a unique Markov process (a diffusion) solving

the associated stochastic nonlinear equation. This property is called C2(rn)- uniqueness (or strong uniqueness) of the Kolmogorov operator (K , FCF). For the Gaussian case, this Markov process would be the (unique) weak

solution to the stochastic Navier-Stokes equation

b lkl 2 Jz d W k ( t ) = [ - -Ik12wk(t) + B k ( ~ ( t ) ) ] d t + - d P k ( t ) , k E Z2, k > 0 (13)

having p as invariant measure.

Albeverio&Cruzeirol have proven that there exists a weak solution to equa-

tion (13); Da Prato&Debu~sche'~ (see also Debussche in these proceedings)

have proven the existence of a strong solution. (Here, weak and strong are

to be understood in the probabilistic sense). But there is still no proof of

uniqueness.

Results of Cp-uniqueness have been proven for some approximation op-

erators. Preliminarily, we remark that the operator ( K , FCT) can be con-

sidered as an operator in any space P ( p ) ( p < a). First, consider the

finite dimensional (Galerkin) operators K N , defined restricting the vari-

ables indices to vary in the subset I N = { j E 2' : 0 < Ijl 5 N} of Z2 (hence B[ (w) = x h , k - h , k g I N ch kwhwk-h)

For any N , the operator ( K N , FCF) is CP-unique. For 1 5 p < 2, a proof

can be given according to the following footnote (c). But for the finite di-

mensional case, it can be proven directly that there exists a unique solution

of the stochastic (Galerkin-) Navier-Stokes equation, having p as invariant

measure. (For a proof, see, e.g., Cruzeiro". In fact, the coefficients of the

equation are locally Lipschitz and therefore there exists a unique solution,

local in time. Conservation of the energy and It6 formula yield mean-square

a priori estimates, hence the solution does not explode in finite time. This

fails for the infinite dimensional problem, since the covariance of the noise

is not trace class.)

142

Moreover, CP-uniqueness (1 5 p < 2) holds for the following approximated

Kolmogorov operator

which is still an infinite dimensional operator, but only a finite number of components B k appear. This is proven by Albeverio&Ferrario3, based on

a result by Eberle15, since the Bk are smooth (quadratic expression of the

wj 's) and Lq(p)-integrable for any q < 00 '. Let us point out that, if the

C2(p)-norms of the components B k would decay fast enough so that

then this same technique would give C1-uniqueness for the Kolmogorov

operator K defined in (12). Unfortunately, the best estimates are (see

Albeverio&Ferrario4)

1 I B k I 2 d p N llc13+& for (any) E > 0, as ~kl-+ co

We remark that for a different "regularization" of B in (12) (as well as in

(7)), C1-uniqueness as been proven by Stannat21.

For the Poisson case, the dynamics obtained merging the motion of

vortices and Brownian motion is a stochastic inviscous equation of vortices;

for any n 2 2, the n points in which the vorticity is concentrated evolve

according to the following equation

v j d z j ( t ) = V$ 2 vj. lg(zj(t)-zl(t))ctt+d~~"'(t) , j = 1,. . . , n(14)

l#j,l,j=l

We write the Kolmogorov operator with respect to the scalar product given by the

symmetric part as

so that the k-th component of the first order perturbation operator is 2 2 3 Ikl '

Then, in our setting the LP-uniqueness result of Eberle15 (Th. 5 . 2 ) holds true if the

components a satisfy the integrability condition Ikl

for 1 5 p < 2.

143

L1-uniqueness of the corresponding diffusion operator would give a unique

weak solution of this problem, for II-a.e. initial data. Not even the existence

in known so far; an analysis of this problem is postponed t o future work.

Acknowledgments

We would like to thank the organizers of the Conference on Probabilistic

Methods in Fluids for the interesting meeting and for arranging a very pleas-

ant stay in Swansea. The second author gratefully acknowledges financial

support from the Alexander von Humboldt Stiftung.

References

1. S. Albeverio and A.B. Cruzeiro, Comm. Math. Phys. 129, 431 (1990).

2. S. Albeverio, M. Ribeiro de Faria and R. H0egh-Krohn, J . Statist. Phys. 20 No. 6, 585 (1979).

3. S. Albeverio and B. Ferrario, J . Funct. Anal. 193 No. 1, 77 (2002).

4. S. Albeverio and B. Ferrario, Infin. Dimens. Anal. Quantum Probab. Relat. Top. (2002) in press.

5 . S. Albeverio and R. H0egh-Krohn, Stochastic Process. Appl. 31, 1 (1989).

6. S. Albeverio, Yu.G. Kondratiev and M. Rockner, J . Funct. Anal. 154, 444

(1998) and 157, 242 (1998).

7. S. Albeverio and M. Rockner, J . Funct. Anal. 88, 395 (1990).

8. S. Albeverio, M. Rockner and T.S. Zhang, Markov uniqueness for a class of infinite dimensional Dirichlet operators, in Stochastic processes and optimal control Stochastics Monogr. 7 (eds. H.J. Engelbert, I. Karatzas and M. Rockner) Gordon and Breach, Montreux, pp. 1-26 (1993).

9. C. Boldrighini and S. F'rigio, Comm. Math. Phys. 72 , 55 (1980); Errata: ibid. 78, 303 (1980).

10. M. Capiriski and N.J. Cutland, Nonstandard methods for stochastic fluid mechanics, World Scientific Series on Advances in Mathematics for Applied

Sciences, Vol. 27 (1995).

11. F. Cipriano, Comm. Math. Phys. 201, 139 (1999).

12. A.B. Cruzeiro, Expo. Math. 7, 73 (1989).

13. G. Da Prato and A. Debussche, J. Funct. Anal. (2002). To appear. 14. D. Diirr and M. Pulvirenti, Comm. Math. Phys. 85, 265 (1982).

15. A. Eberle, Uniqueness and non-uniqueness of semigroups generated b y singular diffusion operators LNM 1718, Springer, Berlin (1999).

16. I.M. Gel'fand and N.Ya. Vilenkin, Generalized Functions Vol. 4, Academic Press (1964).

17. R. Goodrich, K. Gustafson and B. Misra, Physica 102A, 379 (1980). 18. Z.M. Ma and M. Rockner, Introduction to the theory of (non-symmetric)

Dirichlet forms, Springer, Berlin (1992).

19. C. Marchioro and M. Pulvirenti, Vortex methods in two-dimensional fluid mechanics, LNP 203, Springer (1984).

20. W. Stannat, Ann. Scuola Norm. Sup. Pisa C1. Sci. (4) 28, 99 (1999). 21. W. Stannat, Preprint Bielefeld (2002).

SOME REMARKS ON A STATISTICAL THEORY OF

TURBULENT FLOWS

FRANC0 FLANDOLI

Dipartimento d i Matematica Applicata, Universitci d i Pisa Via Bonanno 25b, 56126 Pisa E-mail: flandoli@dma. unipi. at

Some recent notions and results, like invariant memures for the Navier-Stokes

equations, random attractors, random invariant measures and vortex filaments are

reviewed. Some conjectures about their relation are expressed.

1. Introduction

The statistical theory of turbulent fluids contains a number of scaling laws

derived on the basis of phenomenological arguments and experimental re-

sults, like the Kolmogorov K41 scaling law for the energy spectrum which

asserts that E ( k ) behaves as k - 3 for wave numbers in the inertial range

(between the integral scale and the dissipation scale); here E ( k ) is the

mean value of J',,! 1C(k)l2 dk, where S ( k ) is the sphere of wave numbers k of modulus k and u(k) is the Fourier transform of the velocity of the fluid.

Moreover, in some cases the experiments and certain pieces of the theory

have some discrepancies, like the scaling of the pmoments of velocity in-

crements, $ ~ ~ ( r ) = (lu(z + r ) - u(z)Ip), that are not correctly described by

Kolmogorov theory and seem to require proper intermittency corrections.

The Kolmogorov theory would predict for the structure function 4p(r ) a

scaling of the form r c ( p ) with < ( p ) = f (for small r in a suitable range), but

the experiments clearly show different exponents q ( p ) for p > 2. A number

of models have been proposed to recover exponents close to the experimen-

tal ones but a final model is not known. See the review of F'rischZ0 for an

extensive discussion of these topics.

The most rational approach to the analysis of fluids is by means of

the Navier-Stokes equations, but the previous facts and theories on the

statistical properties of fluids have not been explained on such a basis.

Of course a number of attempts to fill the gap between the Navier-Stokes

equations and the statistical theory of turbulence have been performed, but

the present understanding of this subject is very incomplete.

144

145

In the last ten years there has been some interest in the concept of

statistics of vortex filaments. It is quite clear from numerical simulations

and experiments that the vorticity field of a turbulent fluid presents some

degree of geometrical organization and the concept of coherent structure

has been introduced. Particularly interesting seem to be structures having

the shape of filaments] therefore called vortex filaments. The importance of

these structures for the statistics of turbulent fluids is not clarified yet, but

the question whether a relation exists between them must be considered.

In addition, the 3D geometric concreteness of these objects with respect

to the vague concepts of eddies (K41 theory and many others) or fractal

sets of singularities (multifractal models) and others, usually advocated in

phenomenological studies of turbulence, may open the door to a more rig-

orous connection with the Navier-Stokes equations. In other words, there is

some hope that vortex filaments (and maybe other structures not identified

yet) may constitute the bridge between the Navier-Stokes equations and

the phenomenological laws of turbulence.

Whether the typical scalings of turbulence can be derived from statis-

tical models of vortex filaments is still an open problem, with some pre-

liminary indications in the works of Chorin4, and some work in progress.

See also She et a1 26 and Boyer et a1 We devote this note to the other

question] namely the possible connection between the Navier-Stokes equa-

tions and the ensembles of vortex filaments. We describe just a few rigorous

results that could build up such a bridge with the addition of.many other

still unclear ingredients.

In a sense, we meet in turbulence the same situation as in statistical

mechanics] as described by R. Feynman. The theory of statistical mechanics

is like a mountain: the ascent is the path from the Hamiltonian dynamics of

particles (or other miscroscopic models) to the Gibbs measures, the descent

goes from Gibbs measures to macroscopic predictions and laws. In fluid

mechanics we see the ascent from the Navier-Stokes equations to statistical

ensembles of vortex structures (filaments or others) and the descent from

the latter to the laws of turbulence.

This note is restricted to a few fragments of a possible path of the ascent.

The main tools will be SPDEs, random dynamical systems and stochastic

analysis.

2. Vortex filaments

We first describe the concept of vortex filaments following Flandoli et a1 12,15,161 which is a generalization to continuous processes of the ideas of

Chorin4. In the next sections we re-start from the Navier-Stokes equations]

146

as promised in the introduction.

Vortex filaments have been seen in numerical simulation of turbulent

fluids, see the references in Chorin4, Frisch2'. The regions of space where

the vorticity field is particularly intense seem to have the form of filament

(instead of blobs or other geometric shape that appear in other sectors

of Physics). Idealizing, we may think that the vorticity is concentrated

on lines, around which the fluid rotates. This seems to be the 3D analog

of the observed vortex points of 2D fluids (or fluids with reasonable 2D

symmetry). As point vortex statistics, following O n ~ a g e r ~ ~ and a lot of

subsequent work, proved to be interesting for 2D fluids, there is a similar

hope for the statistics of vortex filaments.

However, we want immediately to point out the transient aspect of this

picture, in contrast to other statistical models. Vortex filaments are not

stable objects. New vortex filaments continuously arise from various kind

of instabilities (the Kelvin-Helmoltz is most famous one, but also others

may be very important, see Pradeep et a1 24) . They persist for some time,

but they undergo a number of modifications that eventually destroy them,

producing for instance larger scale structures (see a mechanism described

by Bonn e t a1 '). It is more like in Biology than in Physics. We presum-

ably have a number of different structures, some of them more eddy-like,

others more sheet-like, others like filaments, and maybe others; they may

have different scales and different properties of scaling; and we observe a

continuous evolution where new structures arise, evolve, and disappear into

other structures. How this picture is correct we do not know exactly, but

it may be a first intuitive approximation.

From this viewpoint, a concept of statistical ensemble of vortex filaments

cannot represent the long time statistics of certain eternal structures, like

particles are in classical statistical mechanics. They do not have a long-time

existence. Therefore we see two directions. One is to consider statistical

ensembles like the grand-canonical, where the number of objects is not

given a priori (here in each realization of the ensemble we shall see certain

objects instead of others, depending on the realization). The other is that

the ensemble of vortex filaments represents some sort of quasi-stationary

measure, or another concept of measure having a meaning just for short or

transient times. We shall explain this second appealing possibility in the

next sections, with the help of dynamical systems.

2.1. Random 1 -currents

Following Flandoli et a1 16, we base the definition of vortex filament on the

one of current.

147

We denote by D1 the space of all infinitely differentiable and compactly

supported 1-forms on Rd. Such forms can be identified with vector fields

cp : Rd + Rd. A 1-dimensional current is a linear continuous functional on

V'. We denote by V1 the space of 1-currents. A common example is the

mapping T : V1 + R defined as T (cp) = Jt ( c p ( X t ) , X t ) dt.

Definition 2.1. Given a complete probability space (R, A, P) , a random

1-current is a continuous linear mapping from the space 23' to the space

Lo (0) of real valued random variables on (R, A, P) , endowed with the

convergence in probability.

Example 2.1. Given a continuous semimartingale (Xt) tGIo, l l in Rd, the

It6 and Stratonovich integrals

1 1

I (cp) = Jlo (cp ( X t ) , d X t ) I s (9) = Jlo ('p ( X t ) I OdXt )

are typical examples of random 1-currents.

Definition 2.2. We say that the random 1-current cp H S (cp) has a pathwise realization if there exists a measurable mapping

w I-+ S(w)

from (0, A, P ) to the space V1 of deterministic currents (endowed with the

natural topology of distributions) , such that

[S (p)] (w) = [S (w)] (cp) for P-a.e. w E R. (3)

for every p E V'.

A general theorem of Minlos in nuclear spaces implies that the usual It6

and Stratonovich integrals have a pathwise realization. A direct spectral

argument provides (presumably optimal) Sobolev regularity properties of

the pathwise realization, see Flandoli et a1 14. We state here only the result

for the 3D Brownian motion, to minimize the digression.

Theorem 2.1. Let (Wt) be a 3-dimensional Brownian motion. Then the random 1-current S(p) defined by the Stratonouich integral above has a pathwise realization S (w), with

S ( . ) E L2 (R, H-" (R3,R3)) .

for all s > $.

148

It will be clear below that we are interested mainly in H-l currents.

The Stratonovich integral (or the It6 one) does not have this property. Let

us use the expressive notation

(z) = 1' 6 (z - Wt) 0 dWt

for the random distribution such that 5' (cp) = s,'(cp(Wt) ,odWt ) =

[S (w) ] (9). If we want a random 1-current similar to 6' but with a path-

wise realization in H- ' , a natural idea is to mollify the 6 Dirac, just to

the needed extent. Geometrically it means that in place of a single curve,

namely a path of (Wt), we consider a sort of Brownian sausage, with a

cross section that is not necessarily a ball. In place of set-theoretic sausage

we prefer to work with a smoothing based on a measure p. Here is the

definition.

Given a probability measure p on R3, consider the random current

or in more rigorous terms, the mapping

defined over all cp E D1, with values in Lo (0). With the same arguments

that yield the previous theorem we have:

Theorem 2.2. Assume that the measure p has finite energy, in the following sense:

Then the random current cp H 5 (cp) just defined has a pathwise realization < ( w ) , with

< E L2 (0, H-1 (R3, R3)) .

Remark 2.1. If A is a compact set in R3 with Hausdorff dimension > 1,

then there exists at least one measure p supported on A (for instance the so

called equilibrium measure of potential theory) which satisfies the previous

condition. Therefore, if we want H-l samples, it is sufficient to mollify the

current <' just by means of a fractal cross section with Hausdorff dimension

> 1.

149

Remark 2.2. It is possible to show that the H-'-norm of < is given (up

to a multiplicative constant) by the following double stochastic integral in

the Stratonovich sense:

where the "interaction" energy Hzy is given by

1

In Flandoli12, this double integral is rigorously defined and analyzed. It is proved that s s H,,p ( d z ) p ( d y ) has finite expectation. This provides a

different proof of the last theorem above, not based on random currents.

With such approach the previous hypothesis on p turns out to be necessary

and sufficient. An interesting fact is that Hzy can be expressed as a double

It6 stochastic integral plus the self-intersection local time of the Brownian

motion (plus boundary terms). Another proof of the previous theorem can

be found in Flandoli e t a1 15.

2.2. Back to vortex f i laments i n 3D fluids

The previous set-up and results are motivated by probabilistic models of

vortex filament. We interpret the random distribution < as a vorticity field of a fluid, concentrated in a tubular region around the curve (Wt), a region

having a possibly fractal cross section p. The previous regularity property of < implies that it defines a velocity

field with f in i te kinetic energy. To explain this, consider a 3D fluid, in

the whole space R3, with velocity field u(z) (we do not consider the time

dependence here). The kinetic energy is

The vorticity field is defined as

< (z) = curl u(z).

The relation between the regularities of u and < is that u E L2 implies

< E H- ' , and given < E H-' one can reconstruct (by Biot-Savard law) a

velocity field u E L2. Therefore the requirement H(u) < 00 is equivalent

So, up to now we have defined random vorticity fields, concentrated

on narrow sets, such that the associated velocity fields have finite kinetic

energy. The law po of < on H-' is the image law of the Wiener measure

to < E H - l .

150

in R3. We may, as a first approximation, consider po itself as a possible

statistical ensemble of vortex structures. More natural is to introduce the

Gibbs measures

PO ( d t ) = z i le-OH(u)po (&) .

where u is recovered from E by the Biot-Savard law. In Flandoli et al l5 it is

proved that pp is well defined for all 0 greater than some 0 0 < 0, hence also

for some megative inverse temperature (and it is also proved that e-@H(u) is not po-exponentially integrable for sufficiently large negative 0). The

measures p~p are similar to those introduced by Chorin on the lattice. In

that case po was the law of the self-avoiding walk, but also here there is,

hidden in H(u), the presence of the self-intersection local time, see a remark

above.

We are not sure that the measures p~p are the best candidate to describe

the statistics of vortex structures in turbulent fluids. Variants of them

could be more interesting, as a work in progress indicate us, where many

different vortex structures are taken into account simultaneously. Therefore

the research on such measures is still a t the beginning. In spite of our

ignorance about them, we indicate in the sequel an hypothetical path to

relate them to the Navier-Stokes dynamics.

3. 3D stochastic Navier-Stokes equation: weak stationary solutions

As we have remarked above, the statistical description of turbulence is

based on certain relevant expected values, like the energy spectrum, the

structure function, or the mean dissipation rate. A sound mathematical

basis for them should be to take the expectation of certain observables

with respect to a suitable measure p on the configuration space of the fluid

(the space of all relevant velocity fields, or vorticity fields); for instance,

E ( k ) = (Jsc,, 10(k)12dk) . Let us restrict our attention to persistent

turbulence (in contrast to decaying turbulence), which is a stationary long

time phenomena. Under such a viewpoint, p should be an invariant measure for the Navier-Stokes dynamics.

The final aim is to have quantitative informations on mean quantities

related to turbulence, but for the time being let us comment on the pre-

liminary question of the rigorous facts known about existence, uniqueness

and ergodicity of invariant measures for the Navier-Stokes equations. The

picture is different depending on the space dimension d = 2 or 3 and on the

deterministic or stochastic nature of the equation.

P

151

1. For the deterministic 2D Navier-Stokes equations one can prove the

existence of an invariant measure p, supported by the compact global at-

tractor. The proof is a straightforward application of the existence Krylov-

Bogoliubov theorem for invariant measures of continuous flows on compact

metric spaces, along with the existence of the compact global attractor

(see for instance Constantin et a1 '). Uniqueness of p is certainly not a

general property and especially it is not expected at high Reynolds num-

bers. For instance, many flows with an unstable stationary solution are

known; in such a case the delta Dirac at the stationary solution is an in-

variant measure, but another invariant measure certainly exists. At high

Reynolds numbers one could even expect to have infinitely many invariant

measures, as suggested by the Ruelle-Sinai-Bowen (RSB) theory. The first

question is then how to identify the physical measure p which gives us the

mean values of interest for turbulence. Again the RSB theory provides

fundamental paradigms in this direction, but we have to remind that it is

applicable, a t present and in spite of great recent extensions to partially

hyperbolic systems, only to rather artificial dynamical systems quite far

from the Navier-Stokes equations.

2. For many classes of stochastic 2D Navier-Stokes equations one can

prove the existence of an invariant measure p, and under several different

assumptions on the noise also the uniqueness and ergodicity of p. This is

one of the most notable achievements of the recent probabilistic efforts in

fluid dynamics.

3. For both the deterministic and stochastic 3D Navier-Stokes equations

one can prove the existence of a shift-invariant measure ji on the path space

of solutions to the equations. In other words, there exists a stochastic

process (u (t)),Lo that is a solution of the 3D Navier-Stokes equations and

is also a stationary process, hence its law ji in the path space is shift-

invariant. We shall state in a moment a rigorous theorem of this kind.

The law p of u(t) is then independent of t and it can be considered as a

measure on the space of configuration that may represent the stationary

regime. Unfortunately, the lack of well-posedness of the 3D Navier-Stokes

equations does not allow us to prove uniqueness and ergodicity of p under

stochastic perturbations; but this seems to be a technical aspect, perhaps

transient in the history of this subject (see for instance the irreducibility

property proved in Flandoli"). On the contrary, the lack of uniqueness of p in the deterministic case is a true fact for many flows and has a fundamental

origin.

The results just quoted appeared in a long series of works by many

authors, like Foias, Prodi, Temam and many others in the deterministic

152

case, and several works in the stochastic case, among which we just quote * and references therein, Ferrariog, Flandoli et a1 17, Kuksin et a1 21 and subsequent works, Vishik et a1 27.

We complete this section with the precise statement of a rigorous result on point 3 above.

In a sufficiently regular domain D c R3, consider the SPDE of Navier- Stokes type

d u + [(u . V) u + Vp] dt = [VAU + f] dt + G (u) d W

divu = 0, U I ~ D = 0

where u, p , f , W are functions of space x E D, time t 2 0 (or sometimes t E R), and the random element w E 0, where (0, A, P ) is an underline probability space. The field p is scalar and has the meaning of pressure, u is a 3D vector field with the meaning of velocity, f is a given 3D force field, W is a cylindrical Wiener process in a suitable Hilbert space, so G (u) d W is a random perturbation of the classical incompressible, Newtonian, constant- density Navier-Stokes equations. Denote by H the function space

1 3 H = Q : D 3 R31Q E [L2 ( D ) ] , divQ = 0 , Q . nlaD = 0 { where n is the outer normal to i3D

V = {Q E [H' (D) I3 IdivQ = 0 , Q l a ~ = O}.

Let { e i } be a complete orthonormal system in H and let {p i } be a sequence of independent standard Brownian motions on (0, A, {Ft} , P ) , where F =

{F t } is a given filtration. Formally we set W ( t , x ) = Ci ei (.)pi ( t ) (this series converges only in suitable distributional spaces). Let G : H --f L2 (H) be a continuous mapping, where L2 (H) denotes the space of Hilbert- Schmidt operators on H. Even if W ( t , . ) is not an element of HI the stochastic integral G (u ( s ) ) d W (s) is well defined, for instance when u is an F-adapted process with paths in L" (0, T ; H ) . The following theorem of existence of martingale weak solutions has been proved by Flandoli et a1 l3 in a great generality (even in some cases when G is defined only on V , so it may depend on the space derivatives of u), but close results may be found also in works of Schamlfuss, Capinski and Cutland, among others.

Theorem 3.1. Assume that uo E H , f E V', and G : H + L2(H) is continuous with linear growth. Then there exists a stochastic basis (R, A, {Ft}, P) , a sequence of independent standard Brownian motions

153

{pi} o n it, and a weakly continuous adapted process u in H , satisfying the stochastic Navier-Stokes equations as a n ident i ty in V', with the property

f o r all T 2 0. If in addition 2

IIG ( m 2 ( H ) 5 A0 Il.11; + c f o r all x E V and f o r a suf ic ient ly small A0 2 0 , then there exists a stationary process u with the previous properties.

For the sequel of our discussion we take a stationary solution u given by the second part of the previous theorem. Then the measure

p = law of u( t )

is independent of time and therefore it is a candidate to describe the long time statistics of the fluid. In principle we do not know whether the stochastic Navier-Stokes equations define a Markov process, so we cannot speak of invariant measures in the usual sense, but p is clearly a substitute of such a concept (there are open paths to give a formal definition, like the coiicept of infinitesimally invariant measure, or the possibility to prove the existence of a Markov selection, that we believe to exist).

Remark 3.1. If u is a stationary measure then E IIu (t)11; is constant, so

An inequality of the form E S U ~ ~ ~ [ ~ , ~ I IIu (t)[lt] < 00 would imply the well posedness of the Navier-Stokes equations, but we do not have such a strong estimate. Anyway, the weaker estimate (22) is sufficient to prove interesting improvements of the theory of singularities, see Flandoli & Romito".

[

4. The viewpoint of random dynamical systems

Having in mind the search for quantitative properties of invariant measures of the Navier-Stokes equations, let us comment on the directions opened by the results of the previous section.

RSB theory. In the deterministic case, 2D for sake of rigor, p is concentrated on a compact set of configuration space, presumably a rather complicate geometrical object at high Reynolds numbers (there exists lower

154

bounds on the Hausdorff dimension of the attractor that show that the dimension diverges with the Reynolds number, see Liu22). Therefore we do not expect p to have a simple form, like a Gibbs measure. The paradigm arising from the RSB theory, however, open the door to a quantitative analysis, even if very difficult. The picture that emerges in the RSB theory is that a typical trajectory on the attractor crosses continuously local unstable manifolds Wp (such unstable manifolds are sets of points close to each other and approaching each other exponentially in the reversed motion). The measure p conditioned to Wz is a Gibbs measure with energy propor- tional to a certain logarithmic determinant of the flow, so a quantity that in principle one can try to analyze to get quantitative informations. Hence the statistics of plwp reflect into statistics of the flow. The latter sentences require careful analysis since one has to mix up in a rather complex way the measures p[wp for different manifolds Wp to get statistical properties of a trajectory. Rigorously speaking, the point is still unclear. However, localization on the attractor (which is related to conditioning to W:) seems to be compatible with scaling properties of p: one can presume that universal scaling properties do not depend so much on the local piece of attractor we observe, while more large scale properties (depending for instance on the particular geometry of the boundary) may vary in essential way over the attractor.

Fokker-Planck equation. In the stochastic case, again 2D, we think that the unique invariant measure p is a sort of diffused regularization of the invariant measure p d e t of the deterministic system. Again for the models rigorously covered by the RSB theory, the measures ps of suitable random perturbations of order E converge to p d e t as E -+ 0. The additional regularity of the measure p has the good consequence that i t satisfies certain elliptic infinite dimensional equations of Fokker-Planck type, so in principle there could be a way to obtain quantitative results from these equations. However, at present, really promising results in this direction are not known, especially as far as scaling properties of local quantities are concerned. This approach seems to be more promising to get large scale informations by suitable finite dimensional or large eddies approximations. An argument in favor of the SRB approach instead of the Fokker-Planck one is the following. In the section on vortex filaments we underlined the transient features of such vortex structures. The restrictions plw2 may capture features that change in time, while i t looks less reasonable to see them directly from p using global tools like the Fokker-Planck equation.

155

RSB theory for random dynamical systems. In view of the previous facts and the additional ergodic properties of stochastic systems, we think that the most promising direction at present is an approach based simultaneously on the RSB paradigm and the invariant measure of the stochastic system. We remarked above that in the deterministic case there could be very many invariant measures and the physical one with the good RSB properties has to be identified. The viewpoint of random dynamical system (see Arnold') comes to help us. In the stochastic case it is still possible to introduce concepts intimately related to the geometry of configuration space as in the deterministic case, by means of the theory of random dynamical systems. In such a framework there exists a concept of random attractor and of random invariant measure p,, whose expected values are the classical invariant measures p. At the level of p, it seems possible to develop the concepts of the RSB theory. The lack of uniqueness of invariant measures suffered by the deterministic models is met again here: even if p is unique, it is not clear that p, is unique. But there is a theorem asserting that under a condition of ergodicity of the 2-point motion, there is a constructive way to identify a unique p,, with some properties similar to those of the RSB theory.

4.1. Random attractors

Consider in this section the case of additive noise:

G(u) = G constant.

Extensions of the following facts to multiplicative noise are of great interest, but only a few results have been proved until now. When the noise is additive it is possible to study the stochastic equation path by path, as a deterministic equation with a distributional forcing term G F . See the details in Flandoli & Schmalfu~s'~. For P-a.e. w E R, considered as given, the following fact can be proved: for every uo E H there exists a weak solution u = u(w), namely a weakly continuous function from [0, co) to H , with

that satisfies the (deterministic) Navier-Stokes equation

au aW - + ( u . V ) U + V ~ = V A U + ~ + G - at at

in the distributional sense and the initial condition u(0) = U O . We do not know whether this solution is unique, as in the deterministic case.

156

Denote by P ( H ) the family of all subsets of H . Consider as R the two-sided Wiener space R = Co (R, R)N, with the product a-algebra and product Wiener measure (on a single component CO (R, R), the two-sided Wiener measure is the measure of a process ,Ll ( t ) , t E R, such that { p (t)}t>o

and {,Ll(-t)},,o are two independent Br0wnia.n motions). Consider on R the shift &, t E R, defined as

-

(&w) (s) = w (t + s) - iJ ( t ) .

The previous existence result defines a multivalued random dynamical system, namely a family of mappings

cp(t,w) : H + P ( H )

with t 2 0 and w E R, such that

cp ( t , w ) = cp (t - s, Qsw) 0 cp (s, w )

(as composition of multivalued maps).

Remark 4.1. One can also associate a random dynamical system by lifting the dynamics in the path space L2 (0,oo; H) . This approach, introduced by Sell in the deterministic case, has been developed also in the stochastic one, see Cutland8, Flandoli et a1 19.

Remark 4.2. The map cp ( t , w ) does not have good continuity properties (similarly to the lack of uniqueness). Just the following very weak form of continuity can be proved: b'x, E HI b'y, E cp ( t , w ) x,, 3 { n k } , z E H , y E cp ( t , w ) x such that x,, - x and y n k - y in H .

Remark 4.3. In 2-dimensions the map cp ( t , w ) is single valued and continuous.

The following definition has been given in Crauel et a1 '.

Definition 4.1. A random set A (w ) is a compact global attractor if 1) it is compact and non empty,

3) for every bounded set B C H we have d ( c p ( t , 1 9 - t ~ ) B, A ( w ) ) + 0 as 2) cp ( t , w ) A (w ) = A ( O N ) ,

t + +m.

The following theorem is proved in the series of papers and the part on the weakly compact attractor in H by standard analysis is still a work in progress. See Cutland', Flandoli et a1 l9 and references therein. One of the

157

claims require the following assumption, which represents one of the main open problems in the theory of the Navier-Stokes equations:

t'uo E V there exists a global solution with

u ( . , w ) E C([O,oo) ;V) ,P-a .s . (29)

Theorem 4.1. For the 3 0 stochastic Navier-Stokes equations (with additive noise) there exists both a weakly compact global attractor in H for the multivalued random dynamical system and compact global attractor for the shift in the path space. Under the assumption (29), the flow is single valued and there exists a compact global attractor in H .

In 2-dimensions the compact global attractor exists and (at least for certain noise) has finite Hausdorff dimension.

4.2. Random invariant measures

In this subsection we describe a few general facts for random dynamical systems, so it is not assumed that the dynamics come form the Navier- Stokes equations. The main facts are taken from works of Crauel'.

Let H be a Polish space and let cp ( t , w) be a random dynamical system on it (see Arnold'). Let Pr ( H ) be the set of all Bore1 probability measures on H and cb ( H ) be the space of all bounded continuous functions on H . A random measure w H p, from R to Pr ( H ) is called invariant for cp ( t , w) if

Recall on the other side that a probability measure p E Pr ( H ) is invariant for the Markov semigroup if

P (f) = (Ptf)

where Ptf (x) = E [f ('p ( t , .) x)]. Denote by F<o - the a-algebra generated by the mappings w H cp (t , &,w) x, with 0 5 t 5 s. It describes the past.

Theorem 4.2. If p (t , w) is continuous and has a compact global attractor A ( w ) then there exists a random invariant measure p,, with suppp, c A ( w ) .

Theorem 4.3. If in addition A (.) is F<o-measurable, - then there exists an F<o-measurable - invariant p,, with suppp, c A (w).

158

Theorem 4.4. For any .?'<o-measurable - random invariant measure p,, the measure

is invariant for the Marlcov semigroup.

These theorems give us a strategy to construct Markov invariant measures (but usually they may be constructed in easier ways). More than this, they provide a richer structure for the Markov invariant measures.

4.3. RSB properties of the random invariant measures

The following result is a version of known facts proved in a series of works by Kifer, Baxendale-Stroock, Ledrappier-Young, Le Jan and others, see Dolgopyat et a1 lo. Its says that under the ergodicity of the 2-point motion there is a random invariant measure with some RSB properties. We restrict ourselves to discrete times for sake of simplicity.

Let 'p (n, w), n E N, be a continuous random dynamical system on a Polish space H , having a compact global attractor A (w). Denote by On, as above, the underlying flow. Assume that cp (n , w ) generates a discrete time Markov process, with transition operator Pn. Assume in addition that 'p (n, w ) and 'p (k, O-kw) are independent, as it happens for systems generated by stochastic equations driven by white noise.

Denote by P?' the transition operator of the 2-point process:

for all g E c b ( H x H ) . Let C C Pr ( H ) be a set of measures closed by the action of p (n, w).

Proposition 4.1. Assume that P, has an invariant measure p E C and that the %point motion is exponentially mixing on C, in the sense that there is a constant X > 0 such that for every ul , u2 E C and every g E cb ( H x H ) one has

for some constant C > 0 depending only on g . Then there exists a (unique) random invariant measure p,, supported by A ( w ) , such that for every u E

C, every f E C b ( H ) , every A' < X and P-a.e. w E R one has

159

for some random variable C’ (w ) > 0. Moreover,

P-a.s., for all f E Cb(H) .

The application of this result to stochastic equations of Navier-Stokes type is still an open problem, but it is reasonable to expect positive results in the next future. General sufficient conditions on the coefficients of ordinary stochastic differential equations on compact manifolds to have the ergodicity of the two-point motion are known and they are generic in a suitable sense; see Dolgopyat et al lo. They are based on Hormander type conditions. The first step is to try to understand these conditions for finite dimensional approximations of the 3D Navier-Stokes equations, the investigation of which is now at a good stage, see R ~ m i t o ~ ~ . Extension to the infinite dimensional full 3D Navier-Stokes equations, or at least to the well-posed 2D case, is another more open step.

5 . Conclusions

We arised the question whether the physical invariant measure p of the Navier-Stokes equations, conditioned to the unstable manifolds W:, is ‘ related to the statistics of vortex structures, like the Gibbs measure of vortex filaments. We believe that a statistical analysis of some typical instability of fluid flows could throw some light.

Up to now, just a few objects of such a story are known, like invariant measures for the Navier-Stokes equations, random attractors, random invariant measures possibly with some RSB properties, and some ensembles of vortex filaments.

References

1. L. Arnold, Random Dynamical Systems, Springer, Berlin 1998. 2. D. Bonn, Y. Couder, P. H. J. van Damm, S. Douady, Phys. Rev. E. 47, 28

(1993). 3. D. Boyer, J. C. Elicer-Cortbs, J. Phys. A : Math. Gen. 33, 6859 (2000). 4. A. Chorin, Vorticity and Turbulence, Springer-Verlag, 1994. 5. P. Constantin, C. Foias, R. Temam, Attractors Representing Turbulent Flows,

Memoirs Amer. Math. SOC. 314, Providence 1985. 6. H. Crauel, Random Probability Measures on Polish Spaces, Habilitations-

schrift, Universitat Bremen, 1995. 7. H. Crauel, F. Flandoli, Prvbab. Theory Rel. Fields 100, 365 (1994). 8. N. J. Cutland, these proceedings.

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9. B. Ferrario, Stochastics and Stoch. Reports 60, 271 (1997). 10. D. Dolgopyat, V. Kaloshin, L. Koralov, Sample path properties of the

stochastic flows, preprint 2001. 11. F. Flandoli, J . Funct. Anal. 149, 160 (1997). 12. F. Flandoli, Ann. I. H. Poincari, P. Ei S. 38, 207 (2002). 13. F Flandoli, D. Gatarek, Probab. Theory and Rel. Fields 102, 367 (1995). 14. F. Flandoli, M. Giaquinta, M. Gubinelli, V. M. Tortorelli, On a relation

between stochastic integration and geometric measure theory, preprint 2002. 15. F. Flandoli, M. Gubinelli, Probab. Theory Rel. Fields 122, 317 (2002). 16. F. Flandoli, M. Gubinelli, Random currents and probabilistic models of vor-

tex filaments, preprint. 17. F. Flandoli, B. Maslowski, Comm. Math. Phyls. 171, 119 (1995). 18. F. Flandoli, M. Romito, Trans. Amer. Math. SOC. 354, 2207 (2002). 19. F. Flandoli, B. Schmalfuss, J . of Dynamics and Di f f . Eq. 11, 355 (1999). 20. U. Frisch, Turbulence, Cambridge Univ. Press, Cambridge 1998. 21. S. Kuksin, A. Shirikyan, Comm. Math. Phys. 213, 291 (2000). 22. V. X. Liu, Comm. Math. Phys. 147, 217 (1992). 23. L. Onsager, Nuovo Cimento 6, 279 (1949). 24. D. S. Pradeep, F. Hussain, J . Fluid Mech. 447, 247 (2001). 25. M. Romito, Ergodicity of the finite dimensional approximation of the 3D

Navier-Stokes equations forced by a degenerate noise, preprint 2002. 26. Z.-S. She, E. Leveque, Phys. Rev. Letters 72, 336 (1994). 27. M. I. Vishik, A. V. Fursikov, Mathematical Problems of Statistical Hydrome-

chanics, Kluwer, Dordrecht, 1980.

SOME PROPERTIES OF BURGERS TURBULENCE

WITH WHITE NOISE INITIAL CONDITIONS

CHRISTOPHE GIRAUD

Laboratoire J.A.Dieudonne UMR CNRS 6621 Universite de Nice Sophia-Antipolis

Parc Valrose 06108 Nice Cedex 2, FRANCE E-mail: [email protected]. fr

This paper intends to review the main properties of the solutions of Burgers equa-

tion with random initial conditionsof white noise type. These properties are closely

related to those of the convex hull of a Brownian motion with parabolic drift. A special attention is given to the latter.

1. Introduction

This text aims at surveying some key properties of the solutions of the

one-dimensional (inviscid) Burgers equation

&u+ud,u=O (1)

with initial condition of white noise” type. Burgers introduced this equation

around 1940 in its multidimensional form, &u + u . Du = 0, as a toy model

for hydrodynamic turbulence. It is known nowadays this far from accurate;

see Kraichnan’l for a discussion on the similarities and the differences with

Navier-Stokes equation. Yet, Burgers equation appears in many fields of

mathematical physics, such as the formation of the large scale structures of

the universe, or the dynamics of growing surfaces, see e.g. Woyc~ynski’~.

The study of the solution of Burgers equation (1) with white noise initial

condition takes place in the field of the analysis of solutions of PDE’s with

random initial data. If we think to the phenomenon of turbulence, i t seems

interesting to exhibit the statistical properties of the solutions of some

PDE of fluid mechanics, with random and chaotic initial conditions. Such

studies also appear in astrophysics, when one considers the formation of

the structures of the universe. Solutions of Burgers equation with random

aA white noise is the derivative, in the sense of distribution, of a Brownian motion.

161

162

Gaussian initial data seem to be in this case of particular interest] see

Vergassolla et a1 26 for an up-to-date survey. Roughly, the analysis of

Burgers turbulence may be viewed as a first step for depicting the solutions

of more sophisticated PDE’s with random initial data.

The choice of white noise as initial condition stems from the fact that it

appears as a natural model for chaos. Some others initial conditions have

yet also been considered. We refer to Bertoin for the analysis of the Brow-

nian case5 and a survey on the stable noise case6, and to Leonenko” and

WoyczynskiZ7 for other cases. The white noise initial data also arise natu-

raIly in statistical physics. Consider a time t = 0 particles of mass 1 spread

on a regular lattice, say Z, with random initial velocities independent and

identically distributed (i.i.d.) with centered law of finite variance. Next,

let the system evolve according to the dynamics of free sticky particles: be-

tween collisions particles move at constant speed, and when some of them

meet, they merge into a single particle] whose mass and momentum are

given by the sum of the masses and momenta of the particles involved into

the collision. Then, the velocity field of the hydrodynamic limit of such a

system of ballistic aggregation is a solution to Burgers equation with white

noise initial condition; see l2 and also next section for further explanations.

Investigating solutions of Burgers equation with random initial data can

lead to interesting problems in probability theory. Indeed, according to the

celebrated Hopf-Cole formula, the solution u(. , t ) of (1) at time t can be

expressed in terms of the convex hull of the path

1

2t 2 H LZ u(5,O) dz + -2’.

In the case of a white noise initial condition u(., 0), the analysis of u thus

requires a deep analysis of the convex hull of a Brownian motion with

parabolic drift, which is mainly based on the work of Groeneboomlg. See

Section 3 for a sketch of this analysis. There are also some interesting

connections with the phenomenon of coalescence and fragmentation] see

Bertoin5.

The rest of the paper intends to review the main properties of the so-

lutions of Burgers equation (1) with initial condition of white noise type.

Section 2 recalls necessary background on Burgers equation (with deter-

ministic initial condition). In Section 3, various results on the convex hull

of a Brownian motion with parabolic drift are collected. Even if at first

sight they seem to have little to do with Burgers turbulence] they are the

key for the understanding of the proofs of the next sections. In Section 4

the main properties of the solution of (1) with white noise initial condition

are depicted. A special attention is given to its time-evolution. Some other

163

types of white noise initial condition are presented in Section 5. Section 6

concludes with few open problems.

2. Some background on Burgers equation

The purpose of this section is to present some standard features on solutions

of Burgers equation (1). We refer to 11j12,20 for proofs.

Even for very smooth initial conditions, solutions may develop shocks

(discontinuities) at finite time. We then lose the existence of strong solu-

tions, as well as the uniqueness of weak solutions. We will focus henceforth

on a special (weak) solution of (l), the so-called entropy solution, since it

is the physically meaningful1 solution of (l), see g. This special solution

can be obtained in adding a vanishing viscosity term to equation (1). More

precisely, the viscid equation

d t u + u d d 3 3 u = & d ~ Z U

has a unique solution u, which converges as E + 0, except maybe on a set

of Lebesgue measure 0, to the entropy solution u of (1).

u(z, 0) dz fulfills

the condition

Provided that the so-called initial potential W ( z ) :=

W ( z ) = o ( 2 ) as 121 + 00, (2)

it is remarkable that the (entropy) solution u(., t ) at time t can be expressed

in terms of the convex hull 7-It of

1

2t z H W ( z ) + -2.

Indeed, write a ( z , t) for the right-most location of the minimum of

1 2 z I--+ W ( z ) + - (z - x) . 2t

Then, on the one hand a(x, t) coincides with the right-continuous inverse of

t times the derivative of the convex hull ' l i t . On the other hand, a versionb

of the entropy solution u of (1) is given by the Hopf-Cole formula

x - a ( z , t)

t ' u(2, t ) =

see 11$20, Notice already that the discontinuities of x ++ u(x, t ) come from

the discontinuities of z H u(z , t ) . Since z H a(z,t) is right-continuous and

bA weak solution is only defined up t o a set of Lebesgue measure 0, we can thus only

speak of a version of it.

164

increasing, they are of the first kind and always negative (this is precisely

the entropy condition).

As mentioned before, we can interpret the entropy solution in terms

of a system of ballistic aggregation. Consider a t time t = 0, infinitesimal

particles spread on the real line according to the uniform density p(dz, 0) =

dz, with velocities given by the velocity field u,(., 0). Then, let the system

evolve according to the dynamics of free sticky particles described in the

introduction. At time t , the velocity field of the system fits with (a version

of) the entropy solution u(., t ) with initial condition u(., 0). Moreover, the

function a ( z , t ) defined above represents the right-most initial location of

the particles lying in ] - 00, z] at time t. In other words, the density of

mass in the system is given at time t by the Stieltjes measure

d l z , Yl, t ) = 4 Y , t ) - 4x1 t ) .

Therefore, the jumps of z H a(z, t ) , which correspond to the shocks of z H

u(z, t ) , also correspond to the macroscopic clusters of particles (clusters of

positive mass) present in the system at time t . Actually, a jump of a(., t ) at

z corresponds exactly to a macroscopic cluster located in z, whose mass is

given by a ( z , t ) -a(z- , t ) , where the notation a(z- , t ) refers to the left limit

of a(., t ) at z. The velocity V of this cluster is enforced by the conservation

of momentum a ( x , t ) 2z - a ( z , t ) - a(z- , t )

u ( z , 0) dz = V = 451 t ) - I S 4 z - > t ) a(z-, t ) 2t

In the special case where z H a(z, t ) is a step function, we say that the

shock structure is discrete at time t . The path z H u(z , t ) is then shaped

as a toothpath made of pieces of line of slope l / t separated by negative

jumps (shocks). In terms of ballistic aggregation, a discrete shock structure

corresponds to a state of the system where all particles have clumped into

macroscopic clusters, whose locations form a discrete sequence on the real

line. From a geometrical point of view, the shock structure is discrete if

and only if the convex hull ?it of z H W ( z ) + &z2 is piecewise linear. It

is convenient in this case to introduce the so-called &-parabolic hull Pt of

the initial potential W , defined by

When the convex hull ?it is piecewise linear, the parabolic hull Pt is made

of pieces of parabola. Indeed, to a linear piece of X t with slope X / t , say

( z H 5 X z + k ; a I z 5 b ) , corresponds a piece of parabola of Pt

165

with leading coefficient -$ and vertex of abscissa X . A moment of thought

then shows that there is a one-to-one correspondence between the (pieces

of) parabolas of 'Pt and the macroscopic clusters present in the system of

ballistic aggregation at time t. Indeed, to a parabola of ?t corresponds

a cluster whose location X is given by the abscissa of the vertex of the

parabola. Consider the two extremal contact points between this parabola

and the initial potential W . Then, the distance between the abscissae of

these contact points gives the mass of the cluster, whereas the slope of the

segment linking these two points coincides with its velocity, see Figure 1.

The state of the system is thus completely determined by Pt.

Figure 1. Geometrical interpretation of a shock

Finally, we emphasize that the above analysis still makes sense when the

initial condition u(., 0) is not a real function, but is only the derivative in

the sense of Schwartz of an initial potential W fulfilling condition (2). The

solution u(., t ) is then a real function at any time t > 0 and when t -+ 0+,

it converges in the sense of Schwartz to u(., 0), which is still said to be the

initial condition. The white noise initial condition is to be understood in

this sense.

3. Parabolic hull of a Brownian motion

According to the work of Groeneboom'' (see also Pitman23), the convex

hull of a Brownian motion W is 8.5. piecewise linear. A standard applica-

166

tion of Girsanov Theorem shows that this property still holds for the convex

hull of a Brownian motion with parabolic drift, see Groeneboomlg and also

Avellaneda & E4.

Theorem 3.1. The convex hull 7tt of a (two-sided) Brownian motion with parabolic drift (Wz + &t2; t E R) is piecewise linear with probability one.

Recall that when the convex hull 7 t t is piecewise linear, the &-parabolic

hull of W is made of pieces of parabola. We can index these pieces of

parabola by Z, with indices increasing from left to right and the convention

that parabola number 1 is the first parabola whose vertex is located at the th right of 0. We write X, for the abscissa of the vertex of the piece of n

parabola and also Mn-l and M, for the abscissae of its end-points; see

Figure 1. One may notice that, in the notation of the previous section,

The parabolic hull Pt is fully determined by the sequence (X,, M n ) n E ~ . A characterization of the distribution of this sequence can be easily derived

from the work of Groeneboomlg on Brownian motions with parabolic drift.

It involves the Laplace transform C(X) of the integral of a Brownian excur-

sion e of duration 1. According to Groeneboom's formula (see l9 Lemma

4.2.(iii))

M, = .(X,,t).

n=l

= IE (exp (-A Jiu' e, d s ) )

for X > 0, where 0 > -w1 > -w2 > . . . denotes the zeros of the Airy

function A i (see on p 446). We also introduce, following Groeneboom's

notations, the function g : R + Rf defined by its Fourier transform

Theorem 3.2.

The sequences ((0, Mo), ( X n , Mn)n>l} and ((0, Mo), (X-n+l, M-n)n>l} are two Marlcov chains, independent conditionally on Mo , with transitions given by

167

3 ') dz,dm,. (an-i - z,) - (an-l - zn-i)

6t2

Moreover, the law of MO is given by

1 1

P(Mo E da) = -9 2 5 / 3 t 2 / 3 ( - (2 t ) -2 /3a) g ( (2t)-2/3a da.

(5)

This result has been recently recovered by F'rachebourg and Martin14.

It is known that the "excursions" of the Brownian motion above its con-

vex hull are distributed, conditionally on its convex hull, as independent

Brownian excursions, see Groeneboom" and Pitman23. The next theorem

states a similar path decomposition of the Brownian motion conditionally

on its parabolic hull, see l5 for proof. We write elrn] for a Brownian excur-

sion of duration m and

a(m) = min { h e i r n I ; z E 10, m/}

~ ( m ) = right-most location of this minimum.

Theorem 3.3. The "excursions" of the Brownian motion above i ts parabolic hull Pt

€("I = (W(Mn-l + z) - Pt(M,-1 + z); 0 <_ z <_ Mn - Mn- I )

are independent conditionally on Pt, with as conditional law, the law

v(mn,t) of

where m, = Mn - M,-1.

Remark: A straightforward application of Girsanov Theorem shows that

the law v(m, t ) is absolutely continuous with respect to the law P[rn] of dml. Actually,

) 1

2 H eirnnl - - 2tz(mn - X) I a(mn) 2 I/t (

exp (- $ ST eiml d z )

IE (exp (-$ Sr eiml d z ) ) d v ( m , t ) = dPIrn]

The law of the variables u(m) and q ( m ) plays a key role in the analysis of

Burgers turbulence with white noise initial data. It is specified in the next

theorem, in terms of the function C defined above. See l6 for proof.

168

Theorem 3.4. The scaling property of Brownian excursions entails the identity in law

(a(m), rl(m)) kW ( m - 3 / 2 a ( 1 ) , .

For any a > 0 and 0 < x < 1, the probability density function of ( a ( l ) , ~ ( l ) ) is given by

e-a2/24

P(a(1) E da, ~ ( 1 ) E dx) = C (ax3/') C (a (1 - x ) ~ / ~ ) dadx. diG$Tj

4. Burgers turbulence with white noise initial velocity

In this section, we turn our attention to the solutions of Burgers equation

( 1 ) with initial condition u(.,O) distributed as a white noise. In other

words, we consider an initial potential (W,; 5 E R) distributed as a two-

sided Brownian motion. We first describe the solution at a fixed time t > 0,

and then focus on its time-evolution.

4.1. State at a f ixed t i m e t > 0

According to Theorem 3.1, when W is distributed as a Brownian motion,

the convex hull of the path x H W, + $x2 is piecewise linear with prob-

ability one. As a consequence (see Section 2), when u(.,O) is a white

noise, the shock structure is discrete a.s. We recall that in this case,

the solution x H u(x, t ) is a toothpath, fully determined by the sequence

( ( X n , M n ) ; n E Z) described in Theorem 3.2. Indeed, X , gives the lo-

cation of the nth shock at the right of the origin, and (M , - M,-I)/t the strength of this shock. In terms of ballistic aggregation, the state of

the system is the following. All particles have a.s. clumped into macro-

scopic clusters located at (X,; n E Z), with masses and velocities given by

(m, = M, - M,-l; n E Z) and

2Xn - M, - Mn-l ( vn = 2t

Besides, it has to be mentioned that the scaling property of the white noise

propagates to the turbulence and induces the identity in law (see e.g. 4) ,

(u(x , t ) ; x E R) 'EW (t-'/3u xt-2/3 1 x E R . ( 1 ) ; )

169

4.2. Time evolution of the turbulence

The previous section gives a complete description of the state of the tur-

bulence at a fixed time t > 0. The natural question is now to understand

its time evolution. I t will be convenient in this view to use the ballistic

interpretation of the turbulence.

As time runs, the clusters present in the system aggregate according to

the dynamics of sticky particles. This clustering is deterministic, because

so are the dynamics. Clearly it induces a loss of information in the sense

that we cannot recover the state of the system at a time tl from the state

of the system at a time t 2 > t l . Suppose now that time runs backwards. Then, clusters dislocate and due to the loss of information, dislocations

occur randomly. If we do understand how a cluster breaks into pieces in

backwards times, then we will understand how it did aggregate in forwards

times. Roughly, in this subsection we will answer the question: what does

the genealogical tree of a given cluster look like?

Figure 2. Genealogical tree of a Cluster

Henceforth, we focus on the fragmentation of the clusters in backwards

times. The next theorem specifies the parameters on which the fragmenta-

tion of a cluster depends.

Theorem 4.1. Conditionally o n the state of the system at t ime t , each cluster present at t ime t breaks into pieces independently of the others, and according t o a conditional law only depending on i ts mass and t ime t .

Physically, the independence of the fragmentation of a cluster from its lo-

cation and velocity may be viewed as a consequence of the invariance of

the system under translation and Galilean transformations. The fact that

i t does not depend of the other clusters may be understood as follow. Con-

sider at time 0 two (infinitesimal) particles, which belong at time t to two

170

different clusters. These two particles cannot interact up to time t , else

they would stick and belong to the same cluster. Therefore, the particles

which made up a cluster a t time t cannot interact before time t with the

other particles. Since, in addition, the initial velocities of the particles are

uncorrelated, the aggregation processes of the clusters are expected to be

independent.

Proof: We only sketch the proof of Theorem 4.1, and refer to l5 for details.

The main point is to translate the fragmentation of the clusters in terms

of the parabolic hull of the initial potential W . Recall there is a one-to-

one correspondence between the clusters present a t time t in the system

and the (pieces of) parabolas of the &parabolic hull of the initial poten-

tial. Consider a given cluster a t time t and its corresponding parabola with

leading coefficient -A. At time s < t , its corresponding parabola of the

&-parabolic hull of W is stretched in the vertical direction, since its lead-

ing coefficient -& is larger. Let time s decrease from t to 0. The parabola

corresponding to the cluster gets more and more stretched, up to a time

t* < t where it enters into contact with the initial potential W . This time

t* corresponds to the time at which the cluster splits into two clusters. Let

time s decrease further. We now have two parabolas corresponding to the

two clusters. They are stretched in the vertical direction, up to the moment

where one of them touches W at a new point, and also splits into two new

parabolas, giving at all three parabolas/clusters. And so on.

2->-

Figure 3. Time t' of splitting.

A moment of thought thus shows that the fragmentation of a given cluster

a t time t only depends on the "excursion" E of the initial potential W above the parabola corresponding to the cluster. When W is distributed as

a Brownian motion, it follows from Theorem 3.3 that conditionally on the

state of the system at time t , each cluster breaks into pieces independently of

171

the others. Moreover, since the conditional law of & given Pt only depends

on time t and the mass m of the cluster, the fragmentation of the cluster

According to the previous theorem, we can focus on a single cluster of

only depends on m and t , and not on its velocity or location.

mass m at time t. We now turn our attention to its first splitting.

Theorem 4.2. With probability one a cluster splits into exactly two clusters at its first splitting. The law of the time t* of the splitting of a cluster of mass m at time t and of the mass m* of the left-most cluster arising from this splitting is given by

P(t* E ds,m* E d m l )

for ( s , mi) E]O, t [x ]O,m[ , with the notation m2 = m - ml and C defined

bY (3). Moreover, we have for 0 < s < t

We refer to l 5 for numerical illustrations of these laws.

Proof: We write as before & for the "excursion" of the initial potential W above the parabola corresponding to the cluster at time t . Recall from the

proof of the previous theorem that the time t* corresponds to the time at

which the parabola enters into contact with the initial potential a t a new

point. When the initial potential is distributed as a Brownian motion, the

cluster splits a s into two clusters, because the parabola enters as. into

contact with the Brownian motion at a single new point, see l5 for proof.

The location of this contact point gives the distribution of mass between the

two new clusters. Indeed, it should be plain from the mechanism described

above that l / t* and m* correspond to the maximum and the location of

the maximum of

When W is distributed as a Brownian motion, the conditional law of & given Pt is u(m, t ) . Therefore, l / t * and m* are distributed as the variables

a(m) and v(m) conditioned by {a(m) 2 l / t } . Formulaes ( 6 ) and (7) follow

thus from Theorem 3.3. I

172

The previous result depicts the first splitting. Combined with a Markov

property at the times of fragmentation (see 1 5 ) , i t yields a complete descrip-

tion of the fragmentation of a cluster. This description can be formulated

as follows. We write ml, . . . , m k for the masses of the clusters resulting at

time s = t - r of the fragmentation of a cluster of mass m at time t . The

mass ml refers to the mass of the left-most cluster, the mass mk to the one

of the right-most cluster. We write also

Theorem 4.3. The process ( r H M("it)(r); 0 < r < t ) is a pure-jump (inhomogeneous) strong Markov process, with rate of jump at time r

M(m) t ) ( r + h) = (ml, . . . , mi,l, mi,2, . . . , m k ) 1 M(mit)(r) =

(ml, . . .,mi,. . . , m k )

with the function C defined b y (3) and A2 = mi - X I .

We refer to l5 for the proof of the Markov property and l6 for the compu-

We end this section with a remark about the dynamics of fragmen-

tation. The property stated in Theorem 4.1 bears the same flavor as

the so-called fragmentation property considered by Aldous', PitmanZ4 and

Bertoin7. Nevertheless, the fragmentation process r ++ M("lt)(r) we study

here is not homogeneous in time and therefore differs from those considered

by Aldous et al. Besides, a cluster of mass m at time t statistically breaks

into pieces in the same way as a cluster of mass mt-'l3 at time 1. This per-

mits us to associate a time homogeneous Markov process to r H M(m,t)(r) . Indeed, the process

tation of the rate of jump.

fi(Wt)(') := t-2/3e2"/3M(m,t)(te-s), s E Rf

is a time homogeneous strong Markov process, whose dynamic can be de-

picted as follows. Each cluster making up M("st) grows deterministically as

s H e2s/3 and also splits randomly, independently of the others, according

to the fragmentation rate

A3/2

J87rA1 (A - A,)

c (A;y) c ((A - X 1 ) 3 / 2 )

F(A1, x - A,) = X C ( A 3 9

173

5. Burgers turbulence with some other initial velocities of white noise type

In this section, we consider other initial conditions of white noise type for

equation (1). We outline in Section 5.1 the main properties of the solution

of Burgers equation (1) with as initial condition u(., 0), a white noise on Rf and 0 on R-. In Section 5.2, we depict the case where u(., 0) is a periodic

white noise. We omit the proofs.

5.1. The one-sided white noise case

In this subsection, we deal with the initial condition

on ] - m,O] white noise on 10, m[ .

U ( . , O ) =

In terms of ballistic aggregation, such an initial condition arises a t the

hydrodynamic limit of the following system. At time t = 0 the sticky

particles are spread uniformly on Z; those on the right of the origin receive

random i.i.d. velocities (with finite variance), whereas those on the left of

the origin stay at rest.

The phenomenon of main interest here is the propagation to the left

of the chaos initially located on the right of 0. The solution z ++ u(x, t ) has a shock front, which travels to the left as time t runs. At the left of

this shock front u(. , t ) equals 0, whereas a t its right z H u(x,t) is a s .

a toothpath, made of pieces of line of slope l / t separated by a discrete

sequence of shocks, see Figure 4. The location X , and the strength Mn/t of the nth shock at the right of the shock front form a Markov chain, with

transitions given by (5). We write henceforth xt and Mt for the location

and t times the strength of the shock front.

Figure 4. Shape of x H u(x , t ) .

It is convenient to use the ballistic description of u( . , t ) . There ex-

ists a so-called front cluster, travelling to the left, on the left of which

174

there are infinitesimal particles a t rest. On its right, all particles have

clumped into macroscopic clusters, whose locations and masses are given

by (Xnl M n ) n E ~ . The location and the mass of the front cluster correspond

to xt and Mt.

Figure 5 . Shape of the system of sticky particles.

The first property to mention about the shock front is the time-scaling

identity in law

(xt, Mt) ’aw (t2l3x1, t2l3M1) .

This property originates from the scaling property of the white noise and

permits to focus on time t = 1. The second property to be noticed, is that

the shock front is completely described at time t = 1 by the variables z1

and M I . Indeed, according to the conservation of mass and momentum the

velocity & of the shock front is given by V1 = - ;MI . This equality can

be extended at any time t > 0 by

It is an easy task to derive from the work of Groeneboomlg the law of

( X I , M I ) , in terms of the function g defined by (4) and the function h(m, .) : R+ 4 R+ defined by the series

O0 A i (2’I3m - wn) h(m, x) = 2 l l3 exp ( -21/”zWn)

Ai’ (-tun) n=l

where, as before, 0 > -w1 > -w2 > . . . represent the zeros of the Airy

function A i ranked in decreasing order. See l7 for proof and also the law

of x1 alone.

Theorem 5.1. In the above notation, the law of (x1 ,Ml ) is given by

for M , x > 0.

175

We now turn our attention to the time-evolution of the shock front. It is

conspicuous from the ballistic description of the system, that the dynamics

of the shock front are governed by two phenomena. First its movement to

the left is continuously slowed down by the infinitesimal particles a t rest on

its left. Second, macroscopic clusters on its right sometimes catch it and

then increase sharply its velocity. We are mainly interested by the evolution

of the location xt of the shock front. The identity (8) suggest that xt behave

roughly as t H -t2l3. But we stress that the identity (8) is only true for a

fixed time t > 0 and therefore does not give the time-evolution of t H xi. The identity (9) implies the equality

so that the evolution of the shock front can be fully expressed in terms of

the process t H Mt, which is characterized in the following theorem.

Theorem 5.2. T h e process t H Gt := t-'l2 Mt i s a pure- jump inhomogeneous and increasing Markov process, wi th rate of j u m p

kt+h - Mf E d m I kt = M>

for any M , m, t > 0.

We can also give the asymptotic behaviour o f t H xt for small and large

time t

Proposition 5.1. W h e n t ime t tends t o 0 or 03, we have wi th probability one the asymptotics

Some other aspects of the solution u(., t ) have also been investigated. The

main contributions are perhaps the description of the flux of particles cross-

ing a given point and the study of the different scaling regimes of the solu-

tion by Frachebourg, Jacquemet and Martin13, see also '. Besides, it can be

noticed that the genealogy of a macroscopic cluster present in the system,

is statistically the same as the genealogy considered in Section 4. Finally,

we mention the work of Tribe & ZaboronskiZ5 and also of Frachebourg et

176

a l l 3 in the case where the initial condition is given by a white noise on a

finite interval, and 0 elsewhere.

5.2. The periodic white noise case

We focus henceforth on the solution of Burgers equation (1) with initial

condition u(.,O) distributed as a periodic white noise. In other words,

we consider the case where the initial potential W is 1-periodic and is

distributed on [0,1] as a Brownian bridge of duration 1. Since the solution

x H u(x, t ) is also 1-periodic at any time t > 0, we can focus on a period.

It is convenient for investigating such a solution to use the ballistic de-

scription of J: w u(z, t ) . The system of sticky particles associated to u(., t ) is 1-periodic and can therefore be thought of as a circular system, corre-

sponding to the hydrodynamic limit of the following system. Consider at

time t = 0, N particles uniformly spread on the unit circle, with random

angular velocities ( W ~ ) I , N i.i.d., of finite variance and fulfilling wi = 0. Then, let the system evolve according to the next dynamic. Between colli-

sions the particles evolve on the circle with constant angular velocities and

when some particles meet, they merge into a new particle with conservation

of mass and momenta.

As before, the shock structure of u(., t ) is discrete a s . at any time t > 0.

From a circular point of view it means that all particles have clumped into

a finite number of macroscopic clusters. Moreover, it can be shown that

when time t tends to 03 there remains a s . a single cluster of mass 1 and

velocity 0. Its location follows the uniform law on the circle. The genealogy

of this final cluster is distributed according to the law of the genealogy of

a cluster of mass 1 at time t in Section 4, in the limit t -+ 03. This permits

to compute the probability density of a given state in terms of the function

C defined by (3). Indeed, the probability density to have at time t exactly

N clusters of mass ml, . . . , m N (fulfilling ml + . . . + m N = 1) located at

N

81 < ‘ < 8~ equals

where 4 is a completely determined ” polynomial-like” function of

(mi, & ) l , N , see l6 Section 4 Proposition 1. Since the formula of 4 is some-

what complicated, we refer to l6 for its very definition.

177

6. Some open problems

To conclude we mention some open problems. Many questions on the one-

dimensional Burgers turbulence remain open. For example, concerning the

periodic case, it would be interesting to obtain a simple formula for the

law of the number N of clusters present at time t . For more general initial

conditions, we may wonder whether it is possible to extend some of the

above results (see for a discussion in the stable noise case)? Yet, going in

higher dimensions appears now as the most challenging problem in Burgers

turbulence, see Vergassola et al.26 for motivations and simulations.

Besides, for a better understanding of the phenomenon of turbulence,

it would be intersting to exhibit some statistical properties of the solution

of PDE’s of fluid mechanics (especially of Navier-Stokes equation), with

random initial conditions.

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Kluwer, 1999

on Stochastic Processes (1982), Birkhauser, Boston.

4, pp 1870-1902

DETERMINISTIC VISCOUS HYDRODYNAMICS VIA STOCHASTIC PROCESSES ON GROUPS OF

DIFFEOMORPHISMS

Y. E. GLIKLIKH

Mathematics Faculty Voronezh State University

Universitetskaya p l . , 1 394006 Voronezh, Russia E-mail: [email protected]

The flow of viscous incompressible fluid on an n-dimensional flat torus is presented

a s the expectation of a certain stochastic process on the group of diffeomorphisms

of the torus. The above-mentioned process is governed by a stochastic analogue

of the second Newton’s law subjected to the mechanical constraint that garantees

incompressibility. The diffusion term of the process is connected with viscosity

coefficient of the fluid. The constraint is given in invariant geometric terms, the

Newton’s law is formulated in terms of Nelson’s mean backward derivatives. The

Navier-Stokes equation is derived as an Euler type equation in “algebra” of the

group. The construction is translated into the finite-dimensional language of pro-

cesses on the torus (as far as it is possible). Relations with some other stochastic

approaches to viscous hydrodynamics is discussed.

1. Preliminaries and Introduction

The paper is devoted to the approach to hydrodynamics in terms of geom-

etry of groups of diffeomorphisms, suggested for perfect fluids by Arnold

and Ebin and Marsden ‘. In previous papers by the author it was found

that the adequate description of viscous fluids in this language requires in-

volving stochastic processes (see, e.g., and s). In particular, the second

Newton’s law on the groups of diffeomorphisms, used in the case of perfect

fluids, is replaced by its special stochastic analogue in terms of Nelson’s

mean derivatives. Here we engage some additional geometric machinery

that provides clear finite-dimensional interpretation of the construction.

Consider a stochastic process [ ( t ) in Rn, t E [O,Z], given on a certain

probability space (0, F, P) and such that [ ( t ) is an L1-random variable for

all t. The ”present” (”now”) for [ ( t ) is the least complete cT-subalgebra Nf of 3 that includes preimages of Bore1 set of R” under the map [ ( t ) : R +

179

180

R". We denote by Ei the conditional expectation with rcspcct to Nf. Below we shall most often deal with the diffusion processes of the form

in R" and flat torus In as well as natural analogues of such processes on

groups (infinite-dimensional manifulds) of diffeornorphisms. In (1) w ( t ) is

a Wiener process, adapted to ( ( t ) , a( t , x) is a vector field and 0 > 0 is a

real constant.

Following Nelson (see, e.g., - 11) we give the next:

Definition 1.1. (i) The forward mean derivative D l ( t ) of process e( t ) at

t is the L1-random variable of the form

where the limit is supposed to exist in L1(R, F, P ) and At 4 f O means

that A t ---f 0 and A t > 0.

(ii) The backward mean derivative D,<(t) of ( ( t ) at t is L 1 - I a ~ ~ d o ~ ~ ~

variable

where (as well as in (i)) the limit is supposed to exist in L1(R, F, P ) and

At ---f +O means the same as in (i).

Notice that generally speaking D[( t ) # D*(( t ) (but, if [ ( t ) a.s. has

smooth sample trajectories, those derivatives evidently coincide). jFrom

the properties of conditional expectation it follows that D(( t ) and D*( ( t ) can be represented as compositions of l ( t ) and Bore1 measurable vector

fields

on R" (following Parthasarathy we call them the regressions): D(( t ) =

Yo@, l ( t ) ) and D*r( t ) = Y,O(t, l ( t ) ) .

4t l x).

Lemma 1.1. For a process of type (1) D[( t ) = a( t , ( ( t ) ) and so Yo(t , x) =

181

See details of proof, e.g. in

Mean derivatives of Definition 1.1 are particular cases of the notions

determined as follows. Let x ( t ) and y ( t ) be L1-stochastic processes in F defined on ( R , 3 , P). Introduce y-forward derivative of x ( t ) by the formula

and '.

1 x ( t + At) - x ( t )

DYx( t ) = lim E,Y( At-+O At

and y-backward derivative of x ( t ) by the formula

1 x ( t ) - x( t - At)

at D,Yx(t) = lim E,Y( At++O

(5)

where, of course, the limits are assumed to exist in L1(R, 3, P). Let Z ( t , x) be C2-smooth vector field on R".

Definition 1.2. L1-limits of the form

are called forward and backward, respectively, mean derivatives of Z along

I(.) at time instant t .

Certainly D Z ( t , J ( t ) ) and D*Z( t , [ ( t ) ) can be represented in terms of

corresponding regressions, defined analogously to (4). If it does not yield a

confusion, we shall denote those regressions by DZ and D, Z .

Lemma 1.2. For process (1) an R" the following formulae take place:

(9) a o2

at 2 DZ = -2 + (YO. 0)Z + -v2z,

a o2

at 2 D*Z = -Z+ (Y*". V ) 2 - - P Z ,

where V = (A, ..., &), V2 i s the Laplacian, the dot denotes the scalar product in Rn and the vector f ields Y o ( t , x) an,d Yf( t , x) are introduced in

(4).

The main idea of description of viscous hydrodynamics in the language

of mean derivatives is as follows.

For the sake of convenience we deal with fluids moving in a flat n- dimensional torus I". It is the quotient space of €2" with respect the

integral lattice where the Riemannian metric is inherited from Rn. Consider

182

the vector space Vect(s) of all Sobolev HS-vector fields (s > Introduce the L2-scalar product in Vect(s) by the formula

+ 1) on I".

(X, Y ) = 1 < X(X), Y ( x ) > CL(dx) (11) I n

where < ., . > is the Riemannian metric on 7" and p is the form of Rie-

mannian volume (here it is the ordinary Lebesgue measure on 7"). Denote by p the subspace of Vecds) consisting of all divergence-free

vector fields. Then consider the projector

P : vecds) -+ p (12)

orthogonal with respect to (11). Notice that from Hodge decomposition it

follows that the kernel of P is the subspace consisting of all gradients. Thus

for any Y E Vect(s) we have

P ( Y ) = Y - gradp (13)

where p is a certain HS+l function on I" that is unique to within the

constants for given Y . Let a random flow [ ( t ) be given on a flat n-dimensional torus 7". Sup-

pose that it is a general solution of a stochastic differential equation of the

type

dJ( t ) = a(s, J(s ) )ds + udw(t) (14)

where u > 0 is a real constant. Let o,c ( t ) = u( t ,E( t ) ) , where u(t ,z) is

a divergence-free vector field on I", C1-smooth in t and C2-smooth in

m E 7". Suppose that [ ( t , x) satisfies the relation

PD*D*J(t) = F( t l < ( t ) ) , (15)

where F ( t , x) is a divergence-free vector field on 7". Taking into account

formulae (10) and (13), we obtain

d U2

at 2 PD*D*[ ( t , x) = P(-u + (21,V)u - -0%)

a U 2

at 2 = -u + (ul V)u - -V2u - gradp.

Thus (15) means that the divergence-free vector field u ( t l x ) satisfies the

relation

a U 2 -u + (u, 0 ) u - -V2u - gradp = F, at 2

that is the Navier-Stokes equation with viscosity $ and external force

F ( t , XI.

183

We interpret (15) as a stochastic analogue of the second Newton's law

on the group of Sobolev diffeomorphisms D'(7") of the torus, subjected to

a certain mechanical constraint expressed in geometrically invariant form.

In spite of the fact that the constraint is holonomic (i.e., integrable), we

do not restrict the consideration to its integral manifolds. This allows us

to apply both finite and infinite-dimensional language to the investigation

more easily. Involving constraints is a new point of our presentation.

2. Basic notion from the geometry of groups of diffeomorphisms

Consider a flat n-dimensional torus I" as in 51. The tangent bundle to

I" is trivial: T I " = 7" x R" and so any tangent space to T I " admits

the decomposition T(,,x)TIn = R" x R" where the first multiplier, called

horizontal (denote it by H(,,x)) , is tangent to I" and the second one,

called vertical (denote it by V,,,,)), is tangent to R". The family of sub-

spaces H(,,x) in all tangent space T(,,x)TI" is a flat connection on the

torus. Introduce the Riemannian metric < ., . > on I" such that given

X , Y E TmIn the value < XI Y > is their ordinary scalar product in R". This metric is called flat and I" with this metric is called the flat torus.

Everywhere below we deal with the flat torus.

Notice that both 'H(z,x) and V(,,X) are isomorphic to Tm7" (here all

three spaces are canonically isomorphic to R", see above). Thus we can

send any vector X E Tm7" into 'H(,,x) and into V(, ,X) . The former is

called the horizontal lift of X and denoted by XT while the latter is called

the vertical lift of X and denoted by X 1 . The same notations will be in use

for the groups of diffeomorphisms below.

Thus there is a natural map K : TTI " 4 T I " that sends the vector

Y E T(,,x)TIn into the second factor in T(,,x)TIn = R" x R", i.e.,

K : T(,,x)TI" = 'H(,,x) x V(,,x) + V(,,X) = R" = TmIn. This map is

called the connector. The connection 'H is its kernel.

At any point (5 , X ) E T I " consider the vector 2( , ,~ ) that belongs

to ?t(,,x) and satisfies the relation T7r2(,,x) = X , E TmIn where 7r : TI " 4 I" is the natural projection and TT : TTI " -+ TI " is its tangent

map. For the flat torus, taking into account the above decomposition of the

second tangent space, the vector 2(,,x) is described as 2(,,x) = (X ,O) E

7i(,,x) x V(,,X). The vector field 2 on TI " is called the geodesic spray of

the connection.

Consider the set D"(7") of all diffeomorphisms of 7" belonging to the

Sobolev space HS, s > $n + 1. Recall that for s > $n + 1 the maps from

184

H" are C1-smooth.

There is a structure of a smooth (and separable) Hilbert manifold on

D"(7") as well as the natural group structures with the composition in-

volved as multiplication. A detailed description of the structures and their

interconnections can be found in 6. Note that at the unit element e = id the tangent space TeDs(7" ) = Vecd") (see above). As above denote by ,Ll its subspace consisting of all divergent-free vector fields on 7" belonging

to H". The space TfD"(( I " ) , f E D S ( I n ) , consists of the maps Y : I" + T M

such that .rrY(z) = f(z). Obviously for any Y E T f D 5 ( I n ) there exists

unique X E TeD"('Tn) such that Y = X o f . In any T f D " ( I " ) we can

define the L2-scalar product in analogy with (11) by the formula

(X, Y ) f = 1 < X ( z ) , Y(,) >f(.) l l (dz) . (18) I"

The family of these scalar products form the weak Riemannian metric on

D s ( I n ) (it generates the topology, weaker than H') . The right-hand translation Rf : D"(I") + D " ( I " ) , R f o 0 = 0 o f ,

8, f E V s ( I n ) , is Cw-smooth and thus one may consider right-invariant

vector fields on D s ( I n ) . Note that the tangent to right translation takes

the form: T R f X = X o f for X E T D " ( I n ) . A right-invariant vector field X on D i ( 7 " ) generated by a vector X E

T e D " ( l n ) is C'-smooth iff the vector field X on 7" is Hs+'--smooth This

fact is a consequence of the so-called w-lemma (see 6 , and it is valid also

for more complicated fields. For example, if a tensor (or any other) field

on 7" is Cw-smooth, the corresponding right-invariant field on D s ( 7 " ) is

Cw-smooth as well.

One can easily check that the second tangent bundle TTD' (7" ) con-

sists of H" maps from 7" to TT7" with additional properties that they

are projected into maps from D S ( 7 " ) . Thus we can apply the connector

K : TT7" -+ TI " of introduced above to obtain the connector on

TTV"(7") by the formula

K : TTD" (7 " ) 4 TD" (7 " ) . (19)

The family of its kernels in second tangent spaces form the connection on

D"( I " ) , denoted by 7-1. The geodesic spray 2 of is described as follows:

2 ( X ) = 2 o X (20)

for X E T V S ( I n ) , where 2 is the geodesic spray of the connection 'Ft on

I" (see above). One can easily obtain from (20) the following statement:

185

2 is V s (In)-right-invariant and C"-smooth on TDs (7"). Introduce the subspace p f c TfDs('Tn) as TRfP. Thus we obtain the

smooth subbundle p of T P ( 7 " ) that will play the role of constraint below.

Consider the map P : TDS(7") + 6 determined for each f E D S ( 7 " ) by

the formula

Pf = T R f o P o TRf ' .

where P = P, : Vecd") = T e D S ( I n ) --f /3 = Be is the projection intro-

duced in (12). It is obvious that P is D;(I")-right-invariant. There is an

important and rather complicated result (see 6 , that P is C"-smooth.

Construct the vector field S on the manifold p by the formula

S ( X ) = T B ( 2 o X ) , X E p. (21)

Since P and 2 are P(P)-r ight- invariant and C"-smooth on TDS(7" ) , it evidently follows from (21) that so is S.

Introduce the operators:

B : T I " --f R",

the projection onto the second factor in 7" x R";

A(z) : R" 4 Tm7", (22)

the converse to B linear isomorphism from Rn onto the tangent space to

I" at m E I", and

Q g ( z ) = A ( g ( z ) ) 0 B (23)

where g E D s ( I " ) , m E 7". For a vector Y E T f D s ( I " ) we get QgY = A(g(z) ) 0 B(Y(z) ) E

T,DS(7") for any f E Ds(In). In particular, Q,Y E Vec t ( " ) . Notice

that for Y E pf the vector QeY may not belong to Be. The operation Q , is a formalization for D s ( I " ) of the usual finite-dimensional operation that

allows one to consider the composition X o f of a vector X E Vecds ) and

diffeomorphism f E D s ( l n ) as a vector in Vec t (s ) . It denotes the shift of

a vector, applied at the point f(z), to the point z with respect to global

parallelism of the tangent bundle to torus.

The map A has the following property. For the natural orthonormal

frame b in R" we have an orthonormal frame A,(b) in T,P, the field of

frames A(b) on T7" consists of frames inherited from the constant frame

b. Thus for a fixed vector X E R" the vector field A ( X ) on In is constant

(i.e., it is obtained from the constant vector field X on R" and has constant

coordinates with respect to A(b)) and in particular A ( X ) is Coo-smooth and

186

divergent-free since such is the constant vector field X on R". So, A may

be considered as a map A : R" -+ p = 0, c T e D s ( l n ) . Consider the map A : D'(7") x R" -+ T D S ( I n ) such that A, : Rn 4

T,DS(7") is equal to A, and for every g E D'(7") the map A, : R" -+

T g D S ( 7 " ) is obtained from A, by means of the right-translation:

A, (X) = TR,A,(X) = ( A o g ) ( X ) .

Since A is Cm-srnooth, it follows from w-lemma that A is Cm-smooth

jointly in X E R" and g E D'(7") .

3. Description of viscous hydrodynamics

For the sake of simplicity of presentation, in this section we suppose s >

We shall deal with It6 type equations on D'(7") . We refer the reader to

3 , and for global geometric-invariant constructions of such equations on

manifolds suggested by Belopolskaya and Daletsky in terms of exponential

maps of connections (in particular, in and equations on Ds(ln) are

considered). Local presentation in charts of those equations are known as

the Baxendale form of It6 equations. In, e.g., and it is shown that

Lemma 1.1 is true for It8 equations in Belopol'skaya-Daletsky form and so

this is an adequate machinery for working with mean derivatives.

Since the connection on D'(7") is generated by the flat connection on

the torus, the corresponding exponential map is like that on a linear space.

So, without loss of generality we use the notations, usual for It6 equations

in linear spaces. Below we consider a certain equation on the manifold

in general form with respect to the exponential map of some special

connection.

Let a( t , z) be a divergence-free H' vector field on 7". Denote by a(t, f ) the corresponding right-invariant vector field on DS('Tn). The flow on I", generated by equation (14), is a solution of the equation

+ 2. This means that H S vector fields on 7" are a t least C2.

dl( t ) = q t , l ( t ) )dt + cA(l(t))dw(t) (25)

on Ds (7").

Definition 3.1. If<(t) satisfies an equation of (25) type with some (maybe

random) initial condition, we say that it is a process with diffusion term

aA.

Suppose that a process ( ( t ) with diffusion term aA is well-posed for

t E [O,T] for some T > 0. Recall the well-known fact that the process

187

v(t) = [(T - t ) with inverse time direction has the same diffusion term aA but, generally speaking, different drift.

The definition of mean derivatives for processes on V s (7") is analogous

to that on R" and on 7". In order to distinguish the derivatives on Ds('Tn) and on Tn we denote the former by D and D* while D and D, remain valid

for I". The mechanical meaning of the subbundle f l is a constraint. According

to the ideology of geometric description of constraints suggested by Vershik

and Faddeev, we give the following

Definition 3.2. A stochastic process ( ( t ) is called forward admissible to

the constraint f l if D[( t ) E &( t ) a.s. for all t. A stochastic process [ ( t ) is called backward admissible to the constraint

f l if D,[(t) E f l ~ ( ~ ) a s . for all t . A vector field X is called admissible, if X f E bf at any f E D'(7") .

Notice that for a solution [ ( t ) of (25) we have D[ ( t ) = a( t , [ ( t ) ) (see

Following general ideas of mechanics with constraints we can introduce

Lemma 1.1). Thus this [ ( t ) is forward admissible.

the notions of covariant mean derivatives with respect to a constraint.

Definition 3.3. For an admissible vector field X and forward admissible

process [ ( t ) the expression P D X ( t , [ ( t ) ) is called covariant forward mean

derivative with respect to the constraint /3. For an admissible vector field X and backward admissible process [ ( t )

the expression PD,X( t , [(t)) is called covariant backward mean derivative

with respect to the constraint p.

Let v(t) be a backward admissible process. Then, according to Def-

inition 3.3, we can consider the covariant backward mean derivative

PB,,D,[(t). Let F( t , x) be a divergence-free Hs-vector field on In, i.e., it

can be considered as a time-dependent vector F ( t ) E fie. Denote by p( t , f )

the right-invariant vector field on Vs(7") generated by F ( t ) .

Theorem 3.1. Let a process [ ( t ) on D'(7") has the diffusion term aA

and let D* [ ( t ) = u(t , [ ( t ) ) where G ( t l f ) is a right-invariant vector field on VS(7"), generated by a divergence-free HS-uector f ield u ( t , x ) on 7". If [ ( t ) satisfies the constraint Newton's law

u(t, x) on 7" satisfies Navier-Stokes equation (1 7).

188

The proof of Theorem 3.1 is reduced to the finite-dimensional arguments

The divergence-free vector field u(t , x) on I" from Theorem 3.1, i.e., a

time-dependent vector in ,Be c T e D S ( I n ) , can be obtained by right trans-

lation of backward velocity o*[(t) at e , and so the Navier-Stokes equation

(17) plays the role of Euler equation in the "algebra" T,DS( In ) according

to general approach to Euler equations. The flow of u(t, x) on I", that is

a curve on D s ( I n ) describing the motion of viscous incompressible fluid,

may be considered as the expectation of the process [ ( t ) . So, we need to construct a backward admissible process on D s ( l n )

with diffusion term aA satisfying (26). It is a complicated problem to find

a process with given backward mean derivatives. That is why we shall try

to construct [ ( t ) by solving first a certain equation of (25) type and then

changing the time direction in its solution.

Let a process q( t ) on DE(I" ) be a solution of stochastic differential

equation of (25) type with initial condition q(0) = e and let it exist for t from a certain non-random time interval [0, TI. Consider the process with

inverse time direction [ ( t ) = q(T - t ) . Our aim now is to construct an

equation for q such that (26) is fulfilled for [ ( t ) , and D,[(t) = G ( t , E ( t ) )

where u(t, f ) is an admissible right-invariant vector field with initial condi-

tion u(0, e ) = uo E ,Be where uo = uo(x) is a divergence-free HS-vector field

on I". Since the backward mean derivative for [ ( t ) is equal to the forward

mean derivative for q(T - t ) with minus, we have Dq(t) = -D,[(T - t ) =

-u(T - t , q( t ) ) . Hence, taking into account Lemma 1.1 and the fact that

T r S ( X ) = X and TTF' = 0, we can derive that ( ( t ) will satisfy (26) if

q(t) satisfies the equality

of 31.

d q ( t ) = -G(T - t , q( t ) )d t + oA(q( t ) )dw( t ) . (27)

and the process u(T - t , q( t ) ) in satisfies the equality

DVG(T - t , q ( t ) ) = -S(G(T - t , ~ ( t ) ) ) - F'(T - t , G(T - t , ~ ( t ) ) ) (28)

where F'(T - t , G(T - t , q( t ) ) ) is the vertical lift of F(T - t , G(T - t , q( t ) ) ) . Denote by AT the horizontal lift of the field A onto TD' (7" ) . On

p there is a natural connection such that the projections of its geodesics

onto D'(7") are geodesics of the connection 7? (see, e.g., '). Denote the

exponential map of this connection by expT.

Theorem 3.2. If the process u(T - t , q( t ) ) on p satisfies the It; equation in Belopols~aya-Daletskii f o r m

T du(T-t , q ( t ) ) = exp,(T-t,l)(t))(-S(iZ(T-t, q( t ) ) )d t -F ' ( t , q( t ) ) )d t

189

(29) -T - +aA - t , r l ( t ) ) )dw(t)) ,

the process q( t ) and the right-invariant admissible vector field ii on V s (I") satisfy (27) and (28) and so ( ( t ) = q(T- t ) satisfies (26) and the divergence- free vector f ield u(t, x) on I" is a solution of (17).

Theorem 3.2 follows from a statement of Lemma 1.1 type for equations

in Belopolskaya-Daletskii form (see, e.g., 7, 8 ) .

The next finite-dimensional interpretation makes the construction more

clear. Notice that the process q( t ) with initial condition q(0) = e on

V8(In), that satisfies (27), is a random flow on I". Denote this flow

by q( t , x ) with q(0,x) = x. It is the general solution of It6 stochastic

differential equation on I"

dq(t, X) = -u(T - t , q( t , x ) )d t -t adw(t) (30)

with divu(t, x) = 0, the finite-dimensional version of (27). By direct calcu-

lation of forward mean derivatives for the finite dimensional process q( t , x) we show that

Drl(t1.) = -4T - t177(4 x)),

PDDq(t , z) =

--v(T - t , q(t, .)) f (G - t , 77(t1 z)), V)U(T - t , d t , x))- a at

U2 -V2u(T - t , q( t , x)) - gradp. 2

The latter equality is turned into (16) under the change of variables q( t , x) =

[ (T - t ) . Thus equation (29) guarantees that for the process q( t ) , satisfying

(30), the relation PDDq( t ,x ) = F ( t , x ) holds. The same relation can be

achieved also by another way.

For a stochastic differential equation with respect to a process ( ( t ) on

Vs(In) denote by & ( s ) its solution with initial condition &(t) = e. Con-

sider the following system on V s (I"):

dq(t) = -G(T - t , q( t ) )dt + aA(q( t ) )dw(t ) P t

where Qe is introduced in (23) and u0 = u(0) E Pe is the initial value for

u(t), introduced above. Notice that the first equation of (31) is (27).

Theorem 3.3. If the process q( t ) and the vector u( t ) satisfy (31), then u ( t ) , considered as a divergence-free vector field on I", satisfies (17).

190

Indeed, taking into account the routine stochastic presentation of solu-

tions of PDE’s one can easily derive from the second equation of (3.3) t ha t

It should be pointed out that system (31) is similar t o that considered

by Ya. Belopolskaya (see also 4). Equation (30) as a part of another

system of stochastic differential equations, connected with Navier-Stokes

equation, was considered also by B. Busnello (the problem was set up by

M. F’reidlin).

PDDq(t, x) = F ( t , x).

Acknowledgments

The research is supported in part by Grant 99-00559 from INTAS, Grant

UR.04.01.008 of the program Universities of Russia and by U.S. CRDF -

RF Ministry of Education Award VZ-010-0.

References

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2. Belopolskaya Ya.1. Probabilistic presentation for solutions of boundary-value problems for hydrodynamical equations. Trudy POMI, V. 249, 71-102 (1997).

3. Belopolskaya Ya.1. and Dalecky, Yu.L. Stochastic processes and differential

geometry. Kluwer Academic Publishers, Dordrecht 1989 4. Belopolskaya Ya.I., Gliklikh Yu.E. Diffusion processes on groups of diffeomor-

phisms and hydrodynamics of viscous incompressible fluid. Transactions of RANS, ser. MMMIC, V. 3, N. 2 , 27-35 (1999).

5. Busnello B. A probabilistic approach to the two-dimensional Navier-Stokes equation. The Annals of Probability, V. 27, No. 4, 1750-1780 (1999).

6. Ebin D.G. and Marsden J. Groups of diffeomorphisms and the motion of an incompressible fluid Annals of Math.,V.92, N 1, 102-163 (1970).

7. Gliklikh Yu.E. Ordinary and Stochastic Differential Geometry as a Tool for Mathematical Physics.- Dordrecht: Kluwer, 1996.

8. Gliklikh Yu.E. Global Analysis in Mathematical Physics. Geometric and

Stochastic Methods.- N .Y. : Springer-Verlag , 1997.

9. Nelson, E. Derivation of the Schrodinger equation from Newtonian mechanics. Phys. Reviews, 150 (4), 1079-1085 (1966)

10. Nelson, E. Dynamical theory of Brownian motion.-Princeton: Princeton Uni- versity Press, 1967

11. Nelson E. Quantum Fluctuations.-Princeton: Princeton University Press, 1985.

FURTHER CLASSES OF PSEUDO-DIFFERENTIAL OPERATORS APPLICABLE TO MODELLING IN FINANCE

AND TURBULENCE

NIELS JACOB AND AUBREY TRUMAN

University of Wales, Swansea Department of Mathematics

Singleton Park Swansea SA2 8 P P

E-mail: N . [email protected] A . [email protected]

0. Barndorff-Nielsen and S. Levendorskii used some classical (SE6 -) pseudo-

differential operators to construct Markov processes in order to model some situ-

ations in finance (and turbulence). In our note we describe various symbol classes

consisting of non-classical but smooth symbols which lead to Markov processes

and obey a symbolic calculus. In particular it is pointed out that in many cases

it is possible to make parameters of the characteristic exponent of a LBvy pro-

cess state-space dependent to get a corresponding Markov process generated by a

psuedo-differential operator.

1. Introduction

Since the pioneering work of E. EberleinlO on comparing solutions for fi-

nance market models obtained from models driven by diffusions with real-

world data it is clear that jump processes yield much better models. The

most widely used class of jump processes for modelling are L&y processes.

We refer to the surveys of E. Eberleing and 0. Barndorff-Nielsen and N.

Shephard4, respectively, and the references given therein. The fact that

L6vy processes have stationary and independent increments implies a cer-

tain “translation-invariance” of the distribution corresponding to the under-

lying process. This fact excludes for example that a change of parameters

occurs when a certain threshold (for prices for example) is reached.

In the very original paper, 0. Barndorff-Nielsen and S. Levendorskii3

therefore started to model finance markets by using distributions in-

volving parameters depending on the price, or by an abuse of the lan-

guage of physics: price-homogeneous distributions were replaced by price-

inhomogeneous distributions. Tracking back to the generator of the un-

191

192

derlying Markov process, this change necessitates the switch from con-

stant coefficient operators to variable coefficient operators. In their paper,

Barndorff-Nielsen and Levendorskii used the representation of the generator

as a pseudo-differential operator following l6 where it was emphasised that

pseudo-differential operators are canonical tools in the theory of Markov

processes. They chose to work with classical pseudo-differential operators,

i.e. with symbols q(x,<) in the class Sz6 with the additional assumption

that for all z E R", q(z, .) : Rn -+ C is a continuous negative definite func-

tion to ensure that the generated semigroup is a Feller semigroup. Their

model symbol is:

which gives for frozen coefficients, i.e. z = ZO, just a normal inverse Gaus-

sian distribution with parameters ~ ( z o ) , ~ ( z o ) , ~ ( z o ) and P(s0). The fact

that modelling with normal inverse Gaussian distribution is rather suc-

cessful and that Eq. (1) belongs to the Hormander class allowed them to

emphasise the need for having smooth symbols in modelling finance mar-

kets.

The purpose of this note is to show that there are large and rather

general classes of smooth but non-classical symbols leading to pseudo-

differential operators generating Feller semigroups. After we have discussed

some basic facts on pseudo-differential operators and Markov processes, we

will first discuss W. Hoh's l 1> l2 symbol class and then the Weyl calculus

approach due to F. Baldus '. Further we will have a short look a t subor-

dination in the sense of Bochner as well as to pseudo-differential operators

of variable order of differentiation.

Many results in finance have a counterpart in turbulence, compare for

example Barndorff-Nielsen2 and Barndorff-Nielsen and Shephard4. In par-

ticular experimental observations show that the time derivative of a fluid's

velocity field is not log normally distributed. It has instead a hyperbolic or

normally inverse Gaussian type of distribution such as those arising for the

above jump processes. Thus, the classes of pseudo-differential operators

introduced here may also be helpful in modelling turbulence problems, e.g.

by considering classical models of fluid dynamics in a random environment

where the driving noise is a jump process like those described in what fol-

lows. Extensive results are already known for models involving Burgers

equation when the driving noise is white noise. Here we have exact solu-

tions for the Burgers velocity field in terms of a stochastic mechanics with

additive white noise. The challenge is to replace this noise with a jump

193

process and Burgers dynamics with Navier-Stokes dynamics so as to match

the experimental observations (Barndorff-Nielsen2, Barndorff-Nielsen and

Shephard4, Davies, Truman and Zhao', Truman and Z h a ~ ~ ~ ) .

As explained in Barndorff-Nielsen and Levendorskii3 , or more precisely

in Bogarchenko and L e ~ e n d o r s k i i ~ > ~ > ~ , the application in finance follows by

solving a generalised Black-Scholes (backward Kolmogorov) equation

&u(t, .) - (A + q(z, O x ) ) 4 6 .) = 0 (2)

with u(t, z) being the price of a contingent claim. But the calculi introduced

by W. Hoh and F. Baldus allow to attack this equation for non-classical

symbols in a straight forward way analogously to the classical case.

Our report is intended to inform those who do modelling in finance and

turbulence of the mathematical tools available. In particular i t should be

emphasised that i t is often possible to pass from LBvy processes to jump

processes with a pseudo-differential operator as generator by making the

parameters "location dependent".

The authors are very grateful to 0. Barndorff-Nielsen and S. Leven-

dorskii for discussions on their work.

2. Some basic facts on Feller processes

We restrict ourselves to Feller processes (Xt) t>O , PX) R". The fundamental quantity characterising the process is its symbol

with state space ( - XERn

which reduces in the case of L6vy processes to the character is t ic expo-

nent $([) of the LBvy process, for details see Jacob" or the survey Jacob

and Schillinglg. From Eq. (3) i t is clear that for z E Rn fixed < H q(z,<)

must be a characteristic exponent, i.e. we have the LBvy-Khinchin repre-

sentation

n

q(z, C) = 4.) + ib(z) . E + c a l ( z ) E k E l l , k = l

and E H q ( z , [ ) is therefore a cont inuous negat ive definite funct ion,

i.e. has a LBvy-Khinchin representation. We call such symbols by a small

abuse of language a negat ive definite symbol.

194

The basic observation is that (for reasonably nice processes) the genera-

tor of the semigroup Ttu(z) = Ex (u ( X t ) ) is given by the pseudo-differential

operator

-q(x, D)u(x) = - (27r-" eiz'cq(z, <)ii(<)d< , (5) " S Wn

ii denoting the Fourier transform of u. For translation invariant operators,

i.e. operators with constant coefficients (i.e. generators of LBvy processes)

we find the well-known formulae

and

$I being the characteristic exponent of the L6vy process. Thus modelling a

phenomenon with "varying parameters" simply means to pass from the gen-

erator -$I(D) (LBvy process case) with symbol -$(<) (characteristic expo-

nent) to the generator -q (x , D ) with symbol -q(x, <) where q(xo, <) N $(<) for all 20. Thus the fundamental problem is twofold, construct the process

starting with q(x, <) and study the process (if constructed) by using q(x, <), in particular try to prove that close to zo it behaves like a LeGy process

with characteristic exponent q(z0, <). Note that so far no smoothness as-

sumptions on (5, <) H q(x, <) were imposed.

3. Hoh's symbolic calculus

The philosophy of the theory of pseudo-differential operators is to have a

symbolic calculus which allows one to reduce operations on the level of

operators to operations on the level of their symbols. Such a calculus needs

some smoothness for (z, <) H q(z, I). Hormander's calculus requires C"- smoothness and in addition some type of homogeneity of (the principal

symbol of) q(z,<). W. Hoh11y'2 had developed a symbolic calculus for

negative definite symbols without assuming homogeneity properties. We

just describe his approach.

Given a continuous negative definite function $I : Rn -+ R with LBvy-

Khinchin representation

$(<) = 1 (1 -cosY+(dY) . (8)

W n \ { o )

195

The L6vy measure v has to integrate y H 1 A IyI2, and the integrability

properties of v determine the smoothness of $. To see this just differentiate

in Eq. (8) formally under the integral sign to find for a multi-index Q

aa$(E) = 1 a; (1 - cosy. E ) v (dy)

a-\{o)

= ca 1 ya cos) (Y . E ) v (dy) 7

Wn\to)

ca being fl .

Thus, if s wn\{o}

lyal v (dy) < 00, then a"$ exists by the dominated con-

vergence theorem. In fact, we find more. For Q = ~ j , 1 I j I n,

or with Mz := IyI2 v (dy) it follows R n \ t o )

P""(E)I I (2-'M2)+ ($ ( 0 1 4 and for (a( 2 2 we always find

la"$ (511 I Mlal

with Mlal = s ly l la ' v (dy). an\{0)

Finally we proved the following result due to W. Hoh":

Lemma 3.1. If $ has the representat ion Eq. (8) and if Mla1 exists for 2 5 JQJ 5 m. t h e n E C" (Rn) and

(11) Z - P ( l u l ) ppm1 I CIaI (1 + + (I))' > IQI I m 1

holds where p ( k ) = k A 2, k E No.

196

Note that the continuous negative definite function < H 1 -cos a.<, a E R", shows that Eq. (11) is optimal. Given any continuous negative definite

function with representation Eq. (8) we may always construct a continuous

negative definite function $R E C" (R") satisfying Eq. (11) for all m E No. We just have to consider

$ R ( E ) = J (1 - C O S Y . <) v (dy) = J (1 - cosy . <I xaR(o) (y) v (dy) .

BR(O)\{O} an\{o}

Now the way is open to construct a symbolic calculus related to a fixed

continuous negative definite function $I. Let $I : Rn 4 R have the repre-

sent at ion

= J (1 - cosy. <I v (dy) (13) BR(O)\{O}

with some L&y measure v on Rn\{O} and define

A(<) = (1 + $ ( E l ) + . (14)

We consider Hoh's symbol class SF,' consisting of all p : R" x R" + @, p E C" (R" x R"), satisfying

IaE"a:P(x, 01 I ca,&m"-P(IaI) . (15)

As worked out by W. Hoh11,12, it is possible to develop a complete symbolic

calculus including the parametrix construction for "elliptic" elements for

pseudo-differential operators p(x, D) with symbol p E S~~'. In particular,

if in addition $ satisfies a growth condition from below,

NE) 2 co IEI? , r > 0 and IEI large, (16)

and if p(z , .) : R" -+ R is a continuous negative definite function for each

x E R", then we have

Theorem 3.1. (W. Hoh) Assume Eq. (16), p E 5':)' such that p(x, .) : R" R i s a negative definite funct ion, and

P ( Z , 0 2 f ix2 ( E l (17)

for large 111 and some 6 > 0. C, (R") and the closure generates a Feller semigroup.

T h e n (-p(z, D ) , C r ( E X n ) ) is closable in

Corollary 3.1. In case of Theorem 3.1 we find that p(x, 5) i s the symbol of a Feller process.

197

Of course we still have the restriction that p ( . , .) must be a Cm-function

with respect to x and < (as observed in 3 ) . But there is a way to over-

come this restriction too. Suppose that p : R" x R" + R is a contin-

uous function such that p(z, . ) is negative definite with uniform bound

p ( z , <) 5 c (1 + /(I2) and L6vy-Khinchin representation

P(Z,O = .i' (1 -COSY . 5 )N(GdY) .

P ( X , C ) = .i' (1-cosY.<)N(x ,dY)+ (1-cos(Y.t))"z,dY) s

(18)

an\{o)

When we make a uniform decomposition

BR(O)\{O) ak(o)

= PR(x , t) + $R(x, 6) >

it is often possible to identify ~ R ( z , () as a symbol in some class sat-

isfying Eq. (17). Further, P R ( z , D ) is a bounded operator which is an ad-

missible perturbation of p ~ ( x , D ) in the sense that if - p ~ ( z , D ) generates

a Feller semigroup, so will -p (z , D ) = - ~ R ( z , 0) - I j ~ ( z , 0). For details

we refer to W. Hoh l1 and 12.

In conclusion: Hoh's symbolic calculus works almost as Hormander's

SE6-calculus and allows to construct Feller semigroups leading to Feller

processes with C"-symbols not belonging to the Hormander class.

4. Baldus' Weyl calculus approach

In this section we will briefly discuss results due to F. Baldus' who used

the Weyl calculus to construct Feller semigroups. Unfortunately we need

quite a lot of special notions to state the result. For a detailed discussion

we refer to F. Baldus' and also to the original paper and monograph by L.

HOrmander'sl4> 15.

Denote by 0 the standard symplectic form on Rn x R", i.e.

* ((z, 0 7 (Y, 7 ) ) = Y . < - z ' 7 7

and for a positive definite quadratic form y on R" x Iw" we set

In the following a metric on Rm simply means a family y = (yz)zEWm of

positive definite quadratic forms on W" which we may interpret as Rieman-

198

nian metric and denote it sometimes by y(dz ,dz) . Given a metric y on

R" x R", we say that it splits if we have for each (y, 17) E R" x R"

Definition 4.1. A. A metric y on R" is called a slowly varying metric if there exists a constant cy such that for z,y E R" satisfying

yz (z - y, z - y) 5 1 it follows that c-7

holds for all z E R". B. Let y be a slowly varying metric on R". A function M : R" 4 R+ is called y-slowly varying if there is a constant

CM such that for all z, y E R" with y,(z - y, z - y) 5 & we have

Next we have to introduce the notion of a Hormander metric and that of

(sub-) admissible weight functions.

Definition 4.2. A. A slowly varying metric y on R" x R" is called a

Hormander metric if there exist constants cy > 0 and Ny E N such that

for all (z, [) E R" x R" we have

B. Let y be a slowly varying metric on R" x R". We call M : Rn x R" + R+

a y-admissible weight function if M is y-slowly varying and satisfies

with C M > 0 and NM E N

for all (z, <), (y, 7) E R" x R".

Denoting for a metric y on R" x R" the function h, by

199

we have:

Definition 4.3. Given a Hormander metric y on Rn x R". A function

M : R" x R" ---f R+ is called a sub-y-admissible weight function if

there exists a y-admissible weight function MO such that M 5 MO and for

some m E N and c > 0 it follows that

hTM0 5 CM . (28)

If M and 6 are both sub-y-admissible we call M an invertible sub-y- admissible weight function.

For ( y , ~ ) E Rn x R" and ZL : Rn x R" -+ @, we set

d(,,,)+ E ) = ( ( I l l 77) I V2nU (2, 0) I where Van is the gradient in R" x R" and (., .) is the scalar product. Further

we set for a metric y on R" x R" and Ic E No

J~l:"'q') (z, 5) (30)

Definition 4.4. Given a metric y on R" x R" and a weight function M :

Iw" x R" 4 R+. The symbol class S(M, y) consists of all functions q E

C" (R" x R") which satisfy for all Ic E NO

Now we can introduce the operators associated with S(M, y).

Definition 4.5. Given q E S ( M , y ) . pseudo-differential operator qw(x1 D ) : S(Rn) -+ S'(Rn) by

We define the associated Weyl-

qw (Z] D) U ( Z ) = (27r)-" J' ei(z-Y).Eq (F, <) u(y)dyd[ . (32) P" W n

The set of all operators qw(zl D ) with symbol q E S(Ml y) is denoted by

Q ( M , 7).

Example 4.1. For 0 5 6 5 p 5 1, 6 < 1, a Hormander metric is given by

200

m - Taking in addition the weight function M ( z , S ) = (1 + we find

S ( M , y ) = SE6 . (34)

Note that the Weyl-pseudo-differential operators qw(z, 0) we are interested

in can always be transformed into the "usual" form

q(z, D)u(z ) = (27r)-? ] eiz.5q(zl [)ii(J)d< . ( 3 5 ) IWn

Let us denote by B (L2 (R"))-' the set of all bounded linear operators

A : L2(Rn) -+ L2(Rn) which have a bounded inverse, and denote by

9 ( M , y)-l the set of all qw(z, D) E 9 (MI y) with inverse in 9 ( M , y). Now we can state the result of F. Baldus ':

Theorem 4.1. Let y be a Hormander metric o n R" x Rn which splits and assume

Q ( I , ~ ) ~ B ( L ~ ( w ~ ) ) - ' = s ( i , y ) . (36)

Further let M be an invertible sub-y-admissible weight function and m 5 1 an arbitrary sub-y-admissible weight function satisfying with some k E N and C M > 0

where h, is given by Eq. (27). If q E S ( M , y) satisfies

for all k E No, as well as

IX + 4(z, E l + cql 1 54 (A + M ( z , <)I (39)

for all (z,c) E R" x R", X 2 A, 2 0 and cq,Cq 2 0 , and

E H q(z, I ) is a negative definite function, (40)

then the operator -q(z, D ) : C r (Rn) 4 C , (R") is a densely defined operator on C, (R") which extends to a generator of a Feller semigroup, hence q(x, <) i s the symbol of a Feller process.

Example 4.2. A. Elliptic elements p E Sz6 such that for all z E R" the function 5 H p(x, <) is negative definite are included in Theorem 4.1.

20 1

B. The class Sr,' considered by W. Hoh, see Section 2 is included when

working with the metric

Note however that certain extensions of Hoh's results, i.e. the perturba-

tion theory, is not covered. C . Symbols of mixed homogeneity are partly

included.

5. Relations to subordination in the sense of Bochner and operators of variable order of differentiability

Subordination in the sense of Bochner is a procedure to construct a new

stochastic process out of a given one by a random time change. Most

importantly, i t has a nice analytic counterpart. A non-technical outline is

given in Jacob", Chapter 5, and in Jacob and Schillinglg, Section 4. In this

section we sketch only very briefly how to get using subordination in the

sense of Bochner further pseudo-differential operators generating (Feller)

processes.

By definition a Bernstein function f : ( 0 , ~ ) + R is a function with

represent ation

f ( z ) = a + bz + (1 - e-zt) p (dt) (42) i: where a, b 2 0 and p integrates t H 1 A t , t > 0. To every Bernstein function

f there is associated a one-sided L6vy process (St)t>O called a subordinator

the paths of which are almost surely monotone increasing. If (X t ) t20 is any

Markov process and (St)t20 is an independent subordinator, then rt := Xs , is a new Markov process called the subordinated (in the sense of Bochner)

process. For the case where (Xt ) t>o is a Lkvy process with characteristic

exponent + then the characteristic exponent of ( X S , ) ~ > ~ is f o $. It is a

fact that f o $ is always a continuous negative definite function for f being

a Bernstein function and $ being a continuous negative definite function.

Now suppose that q : Rn x R" + R is a symbol of a generator of a

Feller process. In particular q(z,.) : R" 4 R is a continuous negative

definite function. It follows that for every Bernstein function f the function

f o q : Rn x RT2 + R, (z, I ) H f ( q (x ,E)) is negative definite too. Hence we

may try to construct a Feller process starting with the symbol f ( q (z, 0). Clearly, if q(z,J) is independent of z then we just get the subordinated

L6vy process with characteristic exponent f ($ (t)) = f ( q (z0,t)) for some,

hence all z o E R". However, if -q(z, D ) generates a Feller process (Xt)t,O -

202

and q(xl<) depends on x, then the subordinated process X,f := Xs, and

the process yt with symbol f (q (zl 6)) are clearly distinct! The symbolic

calculi introduced in Sections 2 and 3 may be used to relate X s , and Yt. We

refer to Jacob and Schilling18 for a first simple approach and to F. Baldus',

Section 6.5. What we may expect (and what holds true in many situations)

is that the generator -f (-4 (x, D)) of X [ and the operator - (f o q) (x, D) differ only by a "low order" term which will follow a reasonable asymptotics

of the transition function of ( X ! ) versa.

in terms of that of (K)t>O and vice - t2o

For modelling purposes maybe a different] but very related concept is

more important. Stable, especially symmetric stable processes, are very

often used for modelling] in finance and turbulence, but also in other

problems. We may interpret the symmetric stable process with index 2a, 0 < Q < 1, as the process obtained from Brownian motion by subordinat-

ing with the one-sided L6vy process associated with the Bernstein function

fa(s) = s". Indeed the characteristic exponent of the symmetric stable

process of index 2a is given by $a (I) = 1 < 1 2 " which is just fa ( 1 < 1 2 ) and

of course < H [ < I 2 is the characteristic exponent of Brownian motion. As indicated in the introduction, often in modelling it is useful to make pa-

rameters z - (state space) dependent. Thus we may consider the function

(x, c ) H I < 1 2 a ( z ) l or more generally] if q(x1 6) is the symbol of a Feller pro-

cess we may have a look to the symbol

This function has the property that if < H q(xl E ) is a continuous negative

definite function, so is < H q(zl<)"("). Hence, it may lead to a stochastic

process with generator

In case of q(xl <) = 1 < 1 2 (or more generally q(xl <) = ~ i , ~ = ~ ak,l(x)<k<l) we

enter the field of stable-like processes. For these processes a lot of results

(probabilistic as well as analytic) are known and we refer to the work of A. Negoro20)21222 and coworkers and the references given therein.

W. Hoh13 was able to make his symbolic calculus also work for the

"stable-like" situation, an extension of F. Baldus' approach seems to be

possible. It should be mentioned that some work in this direction had been

stimulated by the paper Jacob and Schilling18.

203

In conclusion : stable-like processes have generalisations to processes

generated by pseudo-differential operators of variable order of differentia-

tion, and these classes of processes (or operators) are at our disposal when

modelling. We feel that because of empirical merits such processes should

come into their own in the modelling of turbulence and financial markets.

References

1. Baldus, F., S ( M , g)-pseudo-differential calculus with spectral invariance on

Rn and manifolds for Banach function spaces. Dissertation Universitat Mainz

2000, Logos Verlag, Berlin 2001. 2. Barndorff-Nielsen, 0. E., Probability and statistics: Selfdecomposability fi-

nance and turbulence. In : Probability Towards 2000 (eds. L. Accardi and

C. C. Heyde), Springer Verlag, Berlin 1998, 47 - 57. 3. Barndorff-Nielsen, 0. E., and Levendorskii, S. Z., Feller processes of normal

inverse Gaussian type. Quantitative Finance (to appear). 4. Barndorff-Nielsen, 0. E., and Shephard, N., Modelling by LBvy processes for

financial econometrics. In: L6vy processes: Theory and applications (eds. 0. E. Barndorff-Nielsen, T. Mikosch, S. J . Resnick), Birkhauser Verlag, Boston

5. Bogarchenko, S. J., and Levendorskii, S. Z., Option pricing for truncated Le'vy processes. Intern. J. Theor. Appl. Finance (to appear).

6. Bogarchenko, S. J., and Levendorskii, S. Z., Perpetual American options under LBvy processes. Preprint 2000.

7. Bogarchenko, S. J., and Levendorskii, S. Z., Barrier options and touch-and-

out options under regular LBvy processes of exponential type. Preprint 2000. 8. Davies, I. M., Truman, A., and Zhao, H. Z., Stochastic heat andSurgers

equations and their singularities - geometrical and analytical properties (The fish and the butterfly, and why). Preprint 2001.

9. Eberlein, E., Application of generalised hyperbolic Le'vy motion to finance. In: LBvy processes: Theory and applications (eds. 0. E. Barndorff-Nielsen,

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10. Eberlein, E., and Keller, U., Hyperbolic distributions in finance. Bernoulli 1

11. Hoh, W., Pseudo differential operators generating Markov processes. Habili- tatsionschrift Universitait Bielefeld, 1998.

12. Hoh, W., A symbolic calculus for pseudo differential operators generating Feller semigroups. Osaka J. Math. 35 (1998), 789-820.

13. Hoh, W., Pseudo differential operators with negative definite symbols of variable order. Revista Mat. Iberoamericana 16 (2000), 219-241.

14. Hormander, L., The Weyl calculus of pseudo-differential operators. Comm. Pure Appl. Math. 32 (1979), 359 - 443.

15. Hormander, L., The analysis of linear partial differential operators, vol. 3, Springer Verlag, Berlin, 1985.

16. Jacob, N., Pseudo-differential operators and Markov processes. Akademie Verlag, Berlin, 1996,

17. Jacob, N., and Leopold, H. G., Pseudo-differential operators with variable

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(1995), 281 - 299.

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order of differentiation generating Feller semigroups. Integr. Equat. Oper. Th. 17 (1993), 544-553.

18. Jacob, N., and Schilling, R. L., Subordination in the sense of Bochner - An approach through pseudo differential operators. Math. Nachr. 178 (1996),

19. Jacob, N., and Schilling, R. L., Le'vy-type processes and pseudo differential operators. In: LBvy processes: Theory and applications (eds. 0. E. Barndorff- Nielsen, T. Mikosch, S. J. Resnick), Birkhauser Verlag, Boston 2001, 139 - 168.

20. Kikuchi, K. , and Negoro, A, , On Markov processes generated by pseudo differential operators of variable order. Osaka J. Math. 34 (1997), 319 - 335.

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199-231.

MATHEMATICAL ANALYSIS OF A STOCHASTIC DIFFERENTIAL EQUATION ARISING IN THE

MICRO-MACRO MODELLING OF POLYMERIC FLUIDS.

BENJAMIN JOURDAIN, TONY LELIEVRE

CERMICS, Ecole Nationale des Ponts et Chausse‘es, 6 & 8 Av. Blaise Pascal, 77455 Champs-sur-Marne, France.

E-mail: {jourdain, lelievre}@cermics. enpc. f r

We analyze the properties of a stochastic differential equation (SDE) arising in

the modeling of polymeric fluids. More precisely, we focus on the so-called FENE

(Finite Extensible Nonlinear Elastic) model, for which the drift term in the SDE is singular.

1. Introduction

The rheology of non-newtonian fluids is a very lively field of modern fluid

mechanics. The challenge is to find a good relation linking within the fluid

the stress tensor to the velocity field in order to reproduce the behavior of

the fluid in some classical situations (shear flow, elongational flow) and to

simulate it in some more complex cases. This relation may be complicated

since the stress generally depends on the whole history of the velocity field.

Many approaches consist in deriving this relation from the microscopic

structure of the fluid. In some cases, it is possible to directly attack the

full system coupling the evolution of these microscopic structures to the

macroscopic quantities (such as velocity or pressure) : this is the so-called

micro-macro approach.

We are here interested in the modeling of polymeric fluids. More pre-

cisely, we consider dilute solutions of polymers, so that the chains of poly-

mers (the ‘hicroscopic structures”) do not interact with each other. In

order to describe the microscopic structure of this fluid, one can model a

polymer by a chain of beads and rods (this is the Kramers model) or more

simply by some beads linked by springs (see Figure 1). We consider here

the simplest model consisting in two beads linked by one spring : this is

the dumbbell model. In this model, the evolution of the end-to-end vector

(which joins the two beads) is described by a SDE. We refer the interested

205

206

reader to Refs 10)1,2,6 for the general physical background of these mod-

els. This SDE is actually coupled to the Navier-Stokes equation through

the expression of the stress tensor as an expectation value built from the

end-to-end vector.

Figure 1.: A hierarchy of models : from Kramers chain (top) to dumbbell

(bottom).

The spring force can be linear (Hookean dumbbell model) or explosive

(Finite Extensible Nonlinear Elastic dumbbell model).

In the following, we consider the start-up of a Couette flow of a poly-

meric fluid (see Figure 2) : the fluid is initially at rest, and for t > 0, the

upper plate moves with a constant velocity. For a complete analysis (ex-

istence, uniqueness, convergence of a finite element method coupled with

a Monte Carlo method) of this model in the Hookean dumbbell case, we

refer to Ref. 8. This reference also contains a more detailed introduction to

these types of models and the way to discretize the corresponding system

of coupled PDE-SDE.

We here complement the mathematical analysis of the FENE model

presented in Ref. by focusing on the SDE modeling the evolution of the

conformation of the polymers in the FENE case. It is proven in Ref.

that a solution to the coupled micro-macro system uniquely exists under

natural assumptions. Our concern in the present paper is in particular

to investigate the role played by the finite extensibility coefficient b (see

formulas (2) and (3) below) in the existence and uniqueness of solution of

the SDE itself, the fluid velocity being considered known.

207

u=o

Figure 2.: Velocity profile in a shear flow of a dilute solution of polymers.

Let us now introduce the equations we deal with. They read, in a non-

dimensional form :

where the parameter b > 0 measures the finite extensibility of the poly-

mer. The space variable y varies in c? = (0 , l ) and t varies in the

whole of R+. The random variables are defined on a filtered proba-

bility space (R, .FIFt l lP) . The random process (&, Wt) is a (.Ft)-two-

dimensional Brownian motion. We take Dirichlet boundary conditions on

the velocity. The initial velocity is u(t = 0, .) = U O , and ( X O , YO) is a FO- measurable random variable. We will suppose that (Xo , Yo) is either such

that P ( X i + Y: > b) = 0 (Section 2) or such that P ( X i + Y; 2 b) = 0

(Sections 3 and 4).

We fix y in 0, set g ( t ) = a,u(y,t) and suppose throughout this paper

that we have at least the following regularity on g :

where R+ = [O,+w). We are then interested in solving for t 2 0 the

following SDE, which is a rewriting of the SDE (3) of the initial coupled

208

system : { dXf = [-i1- ( 5 )

dY,g = -1. 2 1- ( X ? P + ( Y , 9 ) 2 qg ) d t + d W t ,

Let us begin by recalling from Ref.

b

with initial condition ( X O , YO).

we give to (5).

Definition 1.1. Let Xo = (Xo, Yo) and Wt = (K, Wt). We shall say that

a (Ft)-adapted process Xi = (Xa, yt”) is a solution to (5) when : for lP-a.e.

the precise mathematical meaning

w , lft 2 0 ,

Remark 1.1. Because of the convention = +m if )zI2 = b, we 1 - ! 2

deduce that a solution to (6) is such that the subset of R+ (0 5 u < 00, IX:12 = b} has lP-as. zero Lebesgue measure.

The paper is organized as follows : in Section 2 , we prove the existence

and uniqueness of the solution to (6) with values in B, where

B = B(0, h) = { (x, y) , x2 + y2 < b} .

The existence of such a solution is derived from results concerning mul-

tivalued SDEs (see Refs 4 1 5 ) . We then focus on the probability for this

solution to reach the boundary of B (see Section 3). When b < 2 and

I P ( ( X O ( ~ < b) = 1, this probability is equal to one. This enables us

to construct (for g = 0) a solution to (6) that leaves a.s. B. Hence, if

b < 2 , uniqueness of solutions does not hold for solutions to (6) without

the additional requirement to take values in B. When b 2 2 and again

lP(IX0l2 < b) = 1, the probability to reach the boundary is equal to zero

and trajectorial uniqueness holds. We exhibit the unique invariant proba-

bility measure of the SDE (6) with g = 0 (see Section 4). All these results

on the SDE have an impact on the analysis and the understanding of the

coupled SDE-PDE system (for which we refer to Ref. They show that

the assumption b 2 2 adopted in Ref. to prove existence and uniqueness

of solution to the coupled system is in some sense “optimal”.

-

209

2. Existence and uniqueness

In this section, we suppose that ( X O , Yo) is such that IP (Xi + Y: > b) = 0.

Our aim is to prove the following :

Proposition 2.1. Under assumption (4), for any b > 0 and for any initial condition ( X O , YO) such that IP (Xi + Y$ > b) = 0 , there exists a unique solution to (6) with values in B.

We first prove the uniqueness statement (Section 2.1), then turn to the

existence first when g E L y (Section 2.2) and finally when g E L:,c (Section

2.3). In the following, the point is to notice that the singular term in the

drift derives from a convex potential II : R2 +] - 00, +m] :

(1 - 2 if x2 + y2 < b, W X , Y ) = "'1 '>

otherwise. (7)

We have : Vx E B, W I ( x ) = +&. Moreover, the function II is a

continuous convex function with domain B. b

2.1. Trajectorial uniqueness for solutions with values in B Let us begin with the uniqueness.

Proposition 2.2. Let us suppose we have two solutions Xf and Xi to (6) and such that IP-a.s., X: = X,". Then these two solutions are indistiguishable until one of the processes leaves B. In addition, if P (3 t 2 0IXfl2 = b ) = 0 , then X: and X: are indistiguishable.

Proof : Let us consider r = inf{t 2 0 , ( ( X f ( 2 V (X ; l2 ) > b} and

Z t = X: - X, . By ItG's formula, we have : ( - g,

dlZl: = 22t .dZt ,

= -2(VrI(X,s) - on(x;)).z, dt + 2g(t)(X,g - x , g ) ( q g - P)dt,

where x.y denotes the scalar product of x and y E lR2.

VrI(%)).(x - 2) 2 0, we obtain, for any t 2 0 :

Using the fact that, since II is convex, for a,ny x and 2 E B, (VrI(x) -

210

Using Gronwall Lemma and the fact that g E L~o,(R+), we have thus shown

that P-a.s, 'dt 2 0, XfAT = X t A T . Therefore, on {T < m}, IX$12 = b. We - 9

deduce that in case P(3t 2 0, IX:lz = b) = 0, T = 03 P-a.s. . 0

2.2. Existence in the case g E L r

In this section, we suppose :

9 E L" @+). (8)

In order to prove an existence result, we will use a multivalued stochastic

differential equation. In this section, we use the results of E. C6pa4 and

E. Ckpa and D. Lepingle5.

Since the function II is convex on the open set B, its subdifferential XI is

a simple-valued maximal monotone operator on R2 with domain B :

{VII(z)} if z E B, { 0 i f x f B . 8rI(X) =

Let us now consider the two-dimensional process X t solution of the follow-

ing multivalued SDE :

(9) d X i + BII(X:) d t 3 (g( t )Kg, 0) dt + d W t , Lo - - xo = (xo,yo)l

We first recall the precise meaning of a solution to (9).

Definition 2.1. We shall say that a continuous (.Ft)-adapted process X; = ( X a , Kg) with values in B is a solution to (9) if and only if X i = X O and

the process K: = Wt + ~ ~ ( g ( s ) Y ~ , O ) ds - ( X : - X i ) is a continuous

process with finite variation such that : for any continuous (.Ft)-adapted

process at with values in R2, for P-a.e. w , 'do 5 s 5 t < 03,

t t t l I I (X t ) du 5 l II(a,) du + l ( X t - a,).dKt. (10)

Remark 2.1. A condition equivalent to (10) is the following : for any

continuous (&)-adapted process at with values in B, the measure on R+ :

( X t - a,). (dKt - VrI(aU) du)

is P-a.s. nonnegative.

Since (8) ensures that x = ( 5 , ~ ) ++ (g(t)y,O) is (uniformly in time)

Lipschitz and with linear growth, according to E. C6pa4, we have :

Proposition 2.3. Under the assumption (8), for any b > 0 , the multivalued SDE (9) has a unique strong solution.

211

We are now going to recover a solution to (6) from the solution of (9).

More precisely, we follow the method of E. C6pa and D. L6pingle5 (see

Lemmas 3.3, 3.4 and 3.6) in order to identify the process Kf . We can thus show that for all 0 < t < co, we have :

IE ( l l d I I ( X t ) l du < 03, with convention ldII(z)I = +co if x @ B. ) As a consequence, for any 0 < t < co, IP-as.,

du < 00 with convention & = +co. (11)

Moreover, the process K i is IP-a.s. absolutely continuous on (0 5 u < co, X t E B} , with density VII (X: ) so that dKt has the following form :

dK: = VII(X:) du + dG:, (12)

where Gg is a continuous boundary process with finite variation IGgl :

Finally, one can identify this process Gf : for all t 2 0,

where, for any x E dB, n(x ) = 5 is the unitary outward normal to B at

the point x. Hence the process X : is solution of the following SDE with normal

reflexion at the boundary of B :

d X i = -V I I (X i ) dt + (g(t)Ytg,O) dt + d W t - l{x;Eas}n(X:)dlGglt.

It just remains to show that IGgl, = 0, for u 2 0, in order to recover (6).

Notice in particular that by ( l l ) , the property of integrability of the drift

term in (6) holds for the solution X : of the multivalued SDE (9).

Lemma 2.1. IGgl = 0.

Proof : We follow here again the ideas of E. Ckpa and D. L6pingle5 (see

Lemma 3.8 p. 438) to prove that IGgl = 0. Let us consider Rf = b - IXfI’. By It6’s formula,

dR: = -2X:.dXi - 2dt,

= -2VII(Xf).Xf dt - 2g(t)X/Kg dt - 2dt - 2Xf.dWt + 2 l lxg l2 dlGglt, 4

b2 = - dt - 2g(t)X/Kg dt - ( 2 + b) dt - 2Xf.dWt + 2&dlGglt, (14)

R:

212

the last equality using the fact that d J G g J t = l ( X ; E B B ) d J G g l t . We know that Rf is a continuous semimartingale with values in [0, b ] . We

want to prove that dRf = lR ;>OdRf . Using Tanah's formula (see" p. 213) ,

where, for any a E [0, b ] , Lg denotes the local time in a of Rg. Using now

the occupation times formula (see Ref. l 1 p. 215), we know (using (11) ) that, for any fixed t > 0 :

Since a + Lg is a.s. cadlag (see" p. 216), we deduce that for any t > 0,

IP-a.s., L: = 0. Using this in (15) , we obtain

dRf = 1Rf>O d R f .

Using this equality in (14) , we have : V t 2 0,

1 b2 - s" 1Rj=O (-z ds + 2g(s)X,SY,S d s + ( 2 + 13) d s + 2 X : . d W s 2& 0

Since, according to ( l l ) , IP-a.s., ( 0 5 t < m,Rf = 0 ) has zero Lebesgue

measure, the right hand side is null. We conclude by using dlGgl t = lR:=odlGglt. 0 We have thus shown the following properties on the process X: :

d u < co, t 1 h 1-w 0 for any 0 < t < co, IP-as-.,

0 d X f = -VII(X:) d t + (g(t)Kg,O) dt + d W t .

We have thus built a solution Xf = ( X a , q g ) to our initial problem ( 6 ) in case g 6 L"(R+). This result is not sufficient in our context since the

energy estimates on the coupled system (1-3) yields less regularity on g (see

Ref. 8 ) .

2.3. Existence in the case g E L$,(R+)

We now want to build a solution to ( 6 ) using the multivalued SDE (9), but

with a weaker assumption on g, namely (4). In this case, the general results

of existence on multivalued SDE do not apply immediately.

213

Therefore, we consider the following sequence of approximations of this

problem :

dX: " + aII(X:") d t 3 ( g n ( t ) q g " , 0) d t + d W t , x;" = xo,

where n E IN* and g n ( t ) = -nV ( n A g ( t ) ) . Since gn is bounded, the results

of the previous section apply and we obtain a unique solution Xi " of the

multivalued SDE (16). Moreover, these processes Xz" are such that :

d u < 00, I', 0 for any 0 < t < cc, P-a.s.,

t"

0 x;"=x0- L V I I ( X : " ) d s + l ( g " ( s ) Y : " , O ) d s + W t . (17)

We now want to let n go to cc in Definition 2.1 (notice that by (17),

dK: " = VII(X:") d t ) . In the following, we choose T > 0 and we work on

the time interval [O,T]. We know that for all n, supt>, - IXfnI 5 b. For

any n 2 m, we have, by ItB's formula,

d (X:" - X;" l 2 = - (VII(X;") - VII(xfm)) . (X;" - X:") d t

+ ( g n ( t ) q 9 " - gm( t )Yg" . ( t ) ) (X;" - X:") d t .

2

Using the fact that, since II is convex, for any x and y E B, (VII(x) -

VII(y)).(z - y) 2 0, we obtain : Vt E [O,T], t

(X:" - X:m 1' 5 1 ( g n ( s ) Y g n ( s ) - gm(s)Ygm(s)) (X:" - X.gm) d s

so that : V t E [O,T],

2 Ixin - x:" I t

5 1 lgn(s)Y;" - p ( s ) Y ; " I 1X.g" - x:" I d s

5 J' ( lgn(s ) l jY;" - Y:* 1 + lY;m 1 Ign(s) - g m ( s ) l ) 1X:" - X:" 1 d s

5 i 1 l g l (s ) IX:" - X;m I 2 d s + 2b

0

0 t

l gn (s ) - g"(s)l d s . I' Using Gronwall Lemma, we then obtain :

214

From this inequality and the fact that g E Li ([0, TI), we deduce that there

exists a continuous adapted process X : with values in B such that X:" --+ X : in L,"(L,oO([O, TI)).

One has the following estimate on the total variation of VII(XE") du on [O,T] :

By ItB's formula, we know that : W E [O,T],

Ix:"12 t t t

= IXOl2 - 1 Ix:" IXp ds + 2 g"(s) x;" Y,"" ds + 2t + 2 1 x:".dWs,

which yields : V t E [0 , TI,

It is obvious that s,' X;" . d W , + Ji X,.dW, in L:(Lp([O, TI))-norm. Up

to the extraction of a subsequence, we can suppose that this convergence

holds for almost every w . Using this property together with (18) and (19) , we deduce that for a.e. w, the measure VII(X:")dt on [O,T] is such that Jz IVlr(Xig")l d t < C(T,w) where C(T,w) is a constant only depending

on T and w . 'One can thus extract a weakly converging subsequence of

the other hand, taking the limit n --+ co in (17) ,

1 du uniformly converges on [0, TI to K: satisfying :

t

K: = 1 (g(u)Y,, 0) du + Wt - (Xf - XO).

By identification of the limit, we have VII(X:") dt 2 dK: weakly.

By Definition 2.1, the processes X:" are such that for any continuous

(.Ft)-adapted process a t with values in R', for P-a.e. w , VO 5 s 5 t < co, t t [ I I(X5") d u 5 rI(a,) du + 1 (Xt" - au) .V r I (X~" ) du. (20)

215

One can pass to the limit n -+ co in (20), using the fact that Il(Xt") + II(Xt) pointwise in u and that II(Xt") is uniformly integrable. Indeed,

for any A 2 $, if we set Mu =

is decreasing on [e, +co)) :

, we have (since x ++

T

so that 1 11n(xzn,12AII(Xtn) du + 0 uniformly in n when A + co. We

have thus obtained a continuous process Xf on [O,T] and a continuous

process with finite variation Kf = Ji(g(u)Y, lv, 0 ) d u + Wt - (Xi - XO) on

[0, TI such that for any continuous (Ft)-adapted process at with values in

lR2, for IP-a.e. w , VO 5 s 5 t < TI t t I t II(Xt) d u 5 1 II(a,) du + (Xt - a,).dKt.

This shows that we have built a solution to the rnultivalued SDE (9) on

the time interval [0, TI. Since T is arbitrary, using Proposition 2.2, we have

built a solution on lR+. Following again the arguments of the last section

it is easy to show that :

0 for any 0 < t < 00, IP-a.s., J" & d U < n , 0 1 -

dXf = -VII(Xi) dt + ( g ( t ) q g , 0 ) dt + d W t .

This shows that Xi is a solution to (6) and completes the proof of Propo-

sition 2.1.

3. Does the solution reach the boundary ?

In this section, we want to determine whether or not the process Xi we

have built in the previous section reaches the boundary of B. Should the

occasion arise, we deduce that uniqueness does not hold for (6), a t least

in the case g = 0. Throughout this section, we suppose that the initial

condition is such that IP(IXo12 < b) = 1.

3.1. Necessary and suflcient conditions

In this section, we want to analyze the event (3 > 0, IXf12 = b} . We are

going to prove :

216

Proposition 3.1. A s s u m e

9 E Jm+), (21)

and that P(IX0J2 < b) = 1. Let u s consider the process X : solution t o (6) built above. We have :

if b _> 2, t hen P (3 > 0 , IX:12 = b) = 0 , 0 if b < 2 , then P (3 > 0, IX:12 = b) = 1.

In view of Proposition 2.2, we deduce immediatly :

Corollary 3.1. If b 2 2 and P(IXOl2 < b) = 1, t hen trajectorial uniqueness holds for (6).

Proof. First, by Girsanov Lemma, one can suppose g = 0. Indeed, let us

consider the process X i we have built in last section. Under the probability

Pg defined by

the process (Gg, WE) = (& + s," g(s)Y,S d s , Wt) is a Brownian motion and

therefore ( X f , y,", Kg, WE, lPg)tER+ is a weak solution of the SDE ( 5 ) with

g = 0. Since this solution is with values in B, it is also a weak solution of

the multivalued SDE (9), with g = 0, for which uniqueness in law holds.

Since IPg and P are equivalent on F, we can then deduce the properties of

Proposition 3.1 in case g E L2(R+) from the properties of Proposition 3.1

in case g = 0.

In the following, we focus on the solution to (9) with g = .O, which we

denote by X t = (Xt,Y,). We fix x E B and the superscript x means that

we consider the solution to (9) with g = 0 such that Xo = x. Let us first suppose that 1x1 > 0. Let us consider the process R," =

b - lXT12. We know that :

b2 dR," = - dt - ( 2 + b) dt - 2X,".dWt.

R," Let us introduce the stopping time

Let fix t > 0. By Girsanov Lemma, one shows that P -as . ,

Indeed, by definition of r:,

P(lX~A,l = 0) = P(JXTI = 0 and t < r;).

217

Let IF': be defined by :

and IE," denote the corresponding expectation.

(B: = x + W , - s; VIl(X;,,,) du)

ing from x. Since on t 5 T:, XT = BrS

By Girsanov Theorem,

is a IP:--Brownian motion start- s<t

IP(lXr1 = 0 and t < T,") 5 IP(IBY1 = 0)

= 0.

One can therefore show that IXrl > 0 on [O,T"), where

T" = lim T," = inf { t 2 0, 1XF12 = b } = inf {t 2 0, R," = O}. n+o3

Thus, one can write, for t E [0, T") :

b2 dR: = - d t - (2 + b) d t + 2 d q d , & ,

R,"

(23)

where Pt is a Ft-adapted 1-dimensional Brownian motion.

Let us now introduce the stopping time

S" = inf {t 2 0, R," 4 (0, b ) )

We have, IP-a.s., S" 5 T". We refer here to I. Karatzas and S.E. Shreveg

(see Section 5.5 p. 342-351).

We introduce a scale function p such that :

(; - ( 2 + b) ) p ' ( r ) + 2(b - r )p"(r) = 0,

which leads to :

p'(r) = C(b - r ) - l ~ - - ~ / ~ ,

where C > 0. We have thereforep(b-) = +m and ( b < 2 p(O+) > -m). Using this property of the scale function and the results of I. Karatzas

and S.E. Shreve, one can conclude that :

0 if b 2 2, then IP (So = +m) = IP (T" = +m) = 1, (25)

if b < 2, then P ( lim IX:lz = b = 1. 1 t+S"

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In case b < 2, we can deduce from the second item that 5'" = T". We now

want to know whether S" = +m or not in this case. Let us introduce the

speed measure m on (0, b) defined by

I- b / 2 dr - -

2 dr m(dr) =

4 ( b - r ) p t ( r ) 2C '

and the function v such that, for any r E (O,b),

We have p(b-) = +m and therefore v(b-) = +m. In case b < 2, it is

easy to check that v(O+) < 00. Using again the results of I. Karatzas and

S.E. Shreve, we can deduce from this that in case b < 2, we have

P(S" < m) = P(T" < 0) = 1. (26)

In case 1x1 = 0, the former results (25) and (26) still hold. Indeed,

let us suppose that x = 0 and let us introduce the stopping time T = inf { t 2 0, IX:12 2 i}. Obvisouly, one has :

IP (3 > 0, IX:12 = b) = IP (3 > 0, IX:12 = b and T < m) .

In case b 2 2, using the strong Markov property of Xa (see E. C6pa4

p. 86), one has :

IP (3 > 0, IX,"12 = b) = IP (3 > 0, lX:12 = b and T < 00) , = (lT<COIP ( j t > O, IxyIz = b, l X = X , ) >

= 0.

In case b < 2, we use the fact that, due to the proof of (23), IP(IX~,,,I = 0) = 0. By the strong Markov property and

since IP-a.s., S U ~ ~ ~ [ ~ , ~ ~ ~ ~ IXt 0 2 I < b, we have IP (3 > 0, lX:12 = b) =

E (P (3 > 0, Ix:lz = b) lx=X1/J = 1.

In case of a non-deterministic initial condition Xo with law po, we can

deduce the properties of Proposition 3.1 from the fact that (by uniqueness

of the solution) :

IP (3 > 0, lXt12 = b) = IP (3 > 0, IX:12 = b) dpo(x). 0

Remark 3.1. In case g E L~o,(R+), what we can conclude is the following :

J 0 if b 2 2 , then IP (3 > 0, IX:lz = b) = 0,

0 if b < 2, then IP (3 > 0, IX:12 = b) > 0.

219

3.2. Non-uniqueness in case b < 2

In this section, we suppose b < 2 and P(IX0l2 < b) = 1. We restrict our

attention to the case g = 0. We are going to construct another process Xt

weak solution to (6) and such that lP(3t > 0, X t 4 B) = 1. In other words,

we will build a solution to (6) which, unlike Xt, goes out of the ball B. This will show that (6) admits at least two different solutions.

Let us consider the solution Xt to (6) we have built in Section 2. We

know that IP-as., the process Xt reaches the boundary of B in finite time

(see Proposition 3.1). Let us introduce the stopping time T = inf{t 2 0, IXtI2 2 b}. In polar coordinate, we write XT = (&, 6 0 ) : ( X T , YT) = (&os(60), &sin(eo)), where 60 E [0,27r) denotes the polar angle. We

now want to construct a solution to (6 ) , which takes ( X T , YT) as initial . .

value, and lives outside of the ball B. Let us introduce a standard Brownian motion (Pi, rt) independent of Wt.

representation (fi ,et) of the process we want to build.

solution rt to the following multivalued SDE :

drt + d f (rt) dt 3 (2 + b) dt + 2&dPt, { TO = b,

where f : R -+I - co, +MI is the convex function defined

-b21n(r - b) if T > b, otherwise.

two-dimensional

We use a polar

We consider the

by :

so that af is a simple-valued maximal monotone operator with domain

I = (b,co) (for all T > b, d f ( r ) = { V f ( r ) } = {&}). By E. C6pa4, there

exists a unique process rt solution to (27). Following exactly the arguments

of Lemma 2.1, one can show that this process rt is such that :

t 1 for any 0 < t < co, P-a.s., 1 151 du < M, with convention

1 - - +W,

drt = -& dt + (2 + b) dt + 2fidPt.

Let us now consider the process 6t defined by :

and the random process X t in R2 defined by :

x t = (fi cos(dt), fi sin(&))

220

b < 2. Existence.

By ItB's formula, we have :

1 x t

21-1x,12 d X t = -- d t + (-sin(&), cos(8t))dyt + (cos(&), sin(Ot))dpt.

b

Using Paul L6vy characterisation, one can show that

(- sin(&), cos(Ot))dyt + (cos(&), sin(Ot))d,Bt = d B t

where Bt is a two-dimensional Brownian motion, independent of W t .

b 2 2. Existence.

Let us now consider Xt defined by 2, = lo<t<TXt + l t > T x t - T and

the process w t defined by wt = WtAT + l t>FBt-T. It is obvious (for

example by Paul L6vy characterisation) that Wt is a Brownian motion.

In addition, the process X t is a solution to (6) with g = 0, such that

IP(3t > 0 , X t @ B) = 1. This shows that the problem (6) with g = 0 does

not admit a unique solution.

Remark 3.2. In case g E LEc(R+), using the solution (rt, 0,) of the mul-

tivalued SDE : (TO, 00) = (b , 190) and

d(r t ,e t ) + ah(rt,ot) dt 3

((2 + b) + rt sin(&)g(t), - sin2(Qg(t)) dt + (2&, &)d(Pt,yt),

where h : R2 +] - co, +co] is the convex function defined by h(r, 0) = f(r)

(see formula (28)), one can by the same arguments prove that there is

non-uniqueness in law for the solutions to (6).

IP (3 2 0, lXt12 = b) = 1.

IP(IXo12 = b) = 0.

We have summarized in Table 1 some of the results we have obtained

in the last two sections.

IP (3 2 0, lXt12 = b) = 0.

I Non-uniaueness. I Uniaueness

Existence. Existence.

Non-uniqueness. Non-uniqueness I IP(IXo12 = b) > 0.

Table 1.: Properties of solutions to (6) when g = 0. We suppose

IP(I_XO~~ 5 b) = 1. In any case, uniqueness holds for solutions with values

in B according to Proposition 2.2. The terminology uniqueness and non

uniqueness relates to a solution that is not enforced to take values in B.

221

4. Invariant probability measure in case g = 0 and b 2 2

In this section we are interested in invariant probability measures for the

SDE (6 ) with g = 0 in case b 2 2.

The motivation for this study is twofold. First, since we consider a

fluid which is initially a t rest, it is natural from a physical point of view to

choose an invariant probability for the SDE (6) with g = 0 as law for Xo. Second, in the analysis of the coupled system (1-3), we are interested in the

regularity of the stress 7 ( t , y ) = IE ( 1- (X ,YP+(Yt? lP xTT ) which, by Girsanov,

can also be written in the following form :

where X i = ( X i , & ) denotes (as in last section) the solution with values

in This expression of the stress yields the

following estimate (using Holder inequality) : for almost all y and t , to (6) with g = 0 (see Ref.

wherep= L. 9-1

It is thus important to estimate the quantities IE

which is simple if we identify and start under an invariant probability mea-

sure (see formula (31)).

The density po defined by :

(30) exp(-2II(x)) b + 2 ( b’2

- 1-- lI=IZ<b = Jexp(-2II(x)) dx 27rb

obviously solves div

ral candidate to be invariant. This is indeed the case as shown by :

Proposition 4.1. For b 2 2, po(x) dx is the unique invar iant probability measure on B f o r the SDE (6) with g = 0 .

This proposition is a consequence of the following lemma :

Lemma 4.1. Let b 2 2. For any x E B, t > 0 , the solution XT of the SDE (6) with g = 0 and XO = x has a density p ( t , x, y ) with respect t o the Lebesgue measure o n B. In addition, V t 2 0 ,

(-(V,II)po + $ ( V , p o ) ) = 0 and is therefore a natu-

(2) d x dY-a.e., exP(-2WX))P(t, 2 , Y) = eXP(-an(Y))P(t, Y , x),

222

(ai) Vx E B, dy-a.e., p(t, x, y) > 0.

Indeed, by (i), one easily checks that po(x) d z is invariant. By (ii), any invariant probability measure is equivalent to the Lebesgue measure on B which implies uniqueness (see Proposition 6.1.9 p. 188 of M. DuAo 7).

With Proposition 4.1, it is then straightforward to prove that, if X O has

the density po(x), then we have :

Let us now prove Lemma 4.1.

Proof. In order to prove (i), we regularize the potential II so that the

results of L.C.G. Rogers l 2 (see p. 161) apply. Let II, be defined by :

Un(x) = nn(Ix12), (32)

and T, is increasing and C2(R+,R+), so that VII, is bounded with con-

tinuous derivatives of first order. Let t > 0 and x E R2. According to

L.C.G. Rogers, the solution Xn7" of the SDE :

r t

has a density p,(t, x, y) with respect to the Lebesgue measure on R2 which

satisfies dx dy-a.e., exp(-211n(x))p,(t, x, y) = exp(-2IIn(y))p,(t, y , x). For x E B, let 7," = inf{t 2 0,1XT12 2 b ( l - k) } . Since

IP (X;'" # XF) 5 P(T," < t ) , according to Proposition 3.1,

n+cc Iim P (X: 'z # X r ) = 0. (35)

We deduce that for a fixed x E B, p n ( t , x , y ) converges in L$(lR2) to

p(t, x , y), which is the density of XF. As the non-negative potential II, converges pointwise to II in B, we

deduce that exp(-211n(x))p,(t, x, y) converges to exp(-2II(x))p(t, x, y) in Lk,,(B x B ) and conclude that (i) holds.

We are now going to check (ii) for a fixed x E B and t > 0. Let A be

a Bore1 subset of B such that 1~ dx > 0. We choose n E N' such that

223

1zI2 < b ( l - i) and S 1 A n dx > 0 where A, = A n B

Girsanov Theorem, under IP: defined by :

where 7,” is as above, (X:AT;)s5t is a Brownian motion starting

from z and stopped at the boundary of B

IP: (X:AT; E A,) > 0. Therefore, IP(X: E A) 2 IP (X&,, E A,) =

CI EE ( IA , , (xFAT;) &) > 0, which concludes the proof.

Acknowledgments

This work has partly been motivated by some remarks of Claude Le Bris.

Bibliography

1.

2.

3.

4.

5.

6.

7. 8.

9.

10. 11.

12.

R.B. Bird, R.C. Armstrong, and 0. Hassager. Dynamics of polymeric liquids, volume 1. Wiley Interscience, 1987.

R.B. Bird, C.F. Curtiss, R.C. Armstrong, and 0. Hassager. Dynamics of

polymeric liquids, volume 2. Wiley Interscience, 1987.

M. BOSSY, B. Jourdain, T. Leli$vre, C. Le Bris, and D. Talay. Existence of

solution for a micro-macro model of polymeric fluid : the FENE model. In preparation. E. CCpa. Equations diffkrentielles stochastiques multivoques. ThGse, Univer- sit6 d’orlkans, 1994. E. CCpa and D. Lepingle. Diffusing particles with electrostatic repulsion. Probab. Theory Relat. Fields, 107:429-449, 1997.

M. Doi and S.F. Edwards. The Theory of Polymer Dynamics. International Series of Monographs on Physics. Clarendon Press, 1988. M. Duflo. Random iterative models. Springer, 1997. B. Jourdain, T. LeliBvre, and C. Le Bris. Numerical analysis of micro-macro simulations of polymeric fluid flows : a simple case. to appear in Math.

Models and Methods in Applied Sciences. I. Karatzas and S.E. Shreve. Brownian mot ion and stochastic calculus. Springer-Verlag, 1988. H.C. Ottinger. Stochastic Processes in Polymeric Fluids. Springer, 1995.

D. Revuz and M. Yor. Continuous martingales and Brownian motion. Springer-Verlag, 1994. L.C.G. Rogers. Smooth transition densities for one-dimensional diffusions. Bull. London Math. SOC., 17:157-161, 1985.

ON THE DISPERSION OF SETS UNDER THE ACTION OF AN ISOTROPIC BROWNIAN FLOW*

H. LISEI

Faculty of Mathematics a n d Computer Science,

Babeg-Bolyai University,

Str. Koga"1niceanu Nr. 1, RO - 3400 Cluj-Napoca, Romania

E-mail: [email protected]

M. SCHEUTZOW

Institut fur Mathematik, MA 7-5, Technische Universitat Berlin,

Straj3e des 17. Juni 136, 10623 Berlin, Germany

E-mail: [email protected]. de

We give a survey on results about the growth of the diameter of the image of a

bounded subset X of Rd under the action of a stochastic flow. We provide a new proof of the fact that, under reasonable assumptions, the diameter of this image set

will almost surely grow at most linearly in time, and we establish an explicit upper

bound for the linear growth rate which is both simpler and better than previous

bounds. Our main tool is the Garsia-Rodemich-Rumsey Lemma.

1. Introduction

Imagine that at time t = 0 an oil slick on the surface of the ocean covers

the set X and that each oil particle moves randomly according to a random

differential equation or a stochastic differential equation. Let &(z) be the

location of the particle at time t 2 0 which started at z E X at time 0. It is

of considerable practical importance to predict some characteristics of the

random set &(X) := { &(z), z E X}. We regard the particles as passive

tracers, which means that we assume they are being carried by the fluid

without interacting with the fluid or with other particles. This assumption

is rather unrealistic for oil particles but is in good agreement with reality

*This work is supported by the DFG-Schwerpunktprogramm Interugierende stochasti-

sche Systeme won hoher Komplexitat.

224

225

for light pollutants like dust. It has been conjectured by R. Carmona and

Y. Sinai3 that under reasonable assumptions, the diameter of the set &(X) will grow linearly in t . Proving the conjecture consists in showing that

the set will grow at most linearly, i. e. in giving an upper bound for the

linear growth rate, and that it grows at least linearly, i. e. that it has a non

trivial linear lower bound. A linear upper bound was proved for a certain

class of stochastic flows by Cranston, Scheutzow and Steinsaltz' and by

the authors" using somewhat different methods. In section 3 we will use

yet another method - namely the Garsia-Rodemich-Rumsey Lemma (in

short: GRR) - to prove an upper linear bound which in fact happens to be

better than the previous ones. In addition, our proof seems to be shorter

and more transparent. We state the GRR-Lemma in the appendix. Lower

linear bounds have been proved under various assumptions by Cranston,

Scheutzow and Steinsaltz5, Scheutzow and Steinsaltz12 and Cranston and

Scheutzow4. We state a corresponding result for isotropic flows in section

4 but only provide an idea of the proof. The reader is referred to the

references for more general results and detailed proofs. Finally we state

some open problems.

time 0 time T

Figure 1. dispersion of an oil spot

226

2. Isotropic Brownian Flows

We will first define the concept of an isotropic covariance tensor (or matrix)

b, then we will introduce isotropic Brownian fields and finally isotropic

Brownian flows (driven by an isotropic Brownian field).

Definition 2.1. Let b = (b i j (z ) ) i , j= l , , , , ,d be a positive semidefinite real

matrix for each x E Rd. We say that b is an isotropic covariance tensor or

matrix if

(i) z H b(z) is four times continuously differentiable.

(ii) b(0) = Ed (the identity matrix)

(iii) z H b(z) is not constant.

(iv) b(z) = G*b(Gz)G for all z E Rd, G E O(d) .

(i) is a convenient and not too restrictive smoothness assumption, (ii) a

normalization condition, (iii) is assumed to avoid rigid motions later and

(iv) ensures that b is invariant under orthogonal transformations -justifying

the term isotropic. Following Baxendale and Harris2, we define the longitudinal and trans-

verse correlation functions Br, and BN by

BL(r) = bii(rei),

BN(r) = bi i ( re j ) ,

r 2 0

r 2 0, j # i,

where e k , Ic = 1,. . . , d denotes the standard basis of Rd. Due to isotropy

the functions BL and BN do not depend on the choice of i and j . For later

reference, we introduce the strictly positive parameters

PL := -Bg(O),

PN := -BZ(O).

If U ( z ) , z E Rd is a zero mean, Rd-valued Gaussian vector field with

covariance cov(V(y + x ) , U(y)) = b(z), then it is easy to check that U has

a continuously differentiable modification and we have

for any i # j .

227

Definition 2.2. Let b = ( b i j ( ~ ) ) ~ , ~ = ~ , , , , , d be an isotropic covariance ten-

sor. An Rd-valued random field M ( t , x), t 4 0 , x € Rd defined on some

probability space (R, F, P) is called an isotropic Brownian f ield, if

( t , x) H M ( t , x) is a zero-mean Gaussian process.

( t , x ) H M ( t , x ) is continuous for almost all w E R. COV(M(S, x), M ( t , y)) = (S A t ) b(x - y).

From this definition it is easy to obtain the following properties of M .

Corollary 2.1. Let M be an Rd-valued isotropic Brownian f ield. the following holds:

Then

t H M ( t , x) is a d-dimensional standard Brownian motion for each

<'M(. , x), M ( . , y) >t= b(x - y) t for each x, y E Rd. x E R d .

Next, we consider the Kunita-type stochastic differential equation (sde)

d X ( t ) = M(dt , X ( t ) ) , (1)

where M is an isotropic Brownian field. It wits shown by Kunitag, Theorem

4.5.1, that this equation does not only have a unique solution for every

initial condition X ( 0 ) = x E Rd but that it even generates a stochastic

flow of homeomorphisms, i. e. that there exists a family ( @ s l ) ~ j s , t < o o of

random homeomorphisms of Rd such that

$szL = $tu. 0 $st for all 0 5 s, t , u < 00 and all w E R.

0 $ss = IdlRd for all s 2 0 and all w E 0. 0 For each s 2 0, z E Rd ($st(x))t>s - solves (1) for t 2 s with initial

0 The map ( s , t , x ) H $,t(x) is continuous for all w E R. condition X ( s ) = x.

We will call any such stochastic flow of homeomorphisms (based on a

Kunita-type sde driven by an isotropic Brownian field M ) an isotropic Brownian flow. It is easy to see that for each x E Rd, $ ~ t ( x ) , t 2 0 is

a standard d-dimensional Brownian motion starting in x. We point out

however, that for x # y the RZd-valued process ($ot(z),$ot(y))t20 is not Gaussian. In the following we will write $t instead of $ot.

We will need the following facts concerning isotropic Brownian flows

(see Baxendale and Harris2):

0 For each z # y, t H ll$t(x) - $t(y)ll =: pt is a diffusion on (0, m)

228

with generator

1 - B N ( z ) A g ( z ) = (1 - B L ( z ) ) g ” ( z ) + (d - 1) ( ) S’(z),

where g E C;. Therefore pt satisfies the sde

where w is a suitable standard Brownian motion

0 For each x E Rd, v E Rd\{O}

1 1 t’cc t 2

X is called top Lyapunov exponent of the flow.

X := lim -logII(D&)(x)vII = - ( ( d - l),Ojv - P L ) a. s. (3)

3. The Upper Bound

We will formulate and prove an upper bound under the following condition.

Condition (C) : (Cl ) ( t , x) H &(x) is a continuous random field on [ O , o o ) x Rd such that

there exist A 2 0, u > 0 and b > 0 such that for each x, y E Rd there exists

a one dimensional standard Brownian motion W such that

ll$t(x) - dt(Y)II L: 1 1 % - YII +aV) ,

0 I t I inf{s 2 0 : llds(x) - @S(y)ll = b } ,

where W: := S U ~ ~ < ~ ~ ~ Ws. (C2) There exist A > 0 ,B 2 0 such that for each x E Rd and each Ic 2 0

we have

where r+ = r V 0 denotes the positive part of r E R. We recall the concept of upper entropy dimension (see e.g. Hoffmann-

J@rgensen8). Let X be a bounded subset of Rd and let N ( X , r ) be the

minimal number of subsets of diameter a t most r which cover X. Then the

upper entropy dimension A of X is defined as

log N ( X , r ) 7-10 log f .

A := lim sup

Remark 3.1. In Cranston, Scheutzow and Steinsaltz6 and Lisei and

Scheutzow” an upper linear bound was established under the assumption

that the so called local characteristics of the flow are bounded and Lipschitz,

229

which implies Condition (C) , see Cranston, Scheutzow and Steinsaltz',

Lemma 5.1. for (Cl) and Lisei and Scheutzow", equation (9) for (C2).

Isotropic Brownian flows possess bounded and Lipschitz characteristics and

therefore satisfy (C). In fact we can infer from (a), using It6's formula ap-

plied to logpt, that for an isotropic Brownian flow and E > 0 there exists

some b > 0 such that condition (Cl ) holds with A = ( A + € ) + and 0 = a. Since the one-point motion of an isotropic Brownian flow is a standard

d-dimensional Brownian motion it follows that (C2) holds with B = 0 and

A = l .

Theorem 3.1. Assume that q5 satisfies condition (C) and that X c Rd i s a compact subset with upper entropy dimension A > 0. Then we have

where

2c2d d- A where A0 = ) .

For an isotropic Brownian flow with top Lyapunov exponent X 2 0 we get the result above with

Proof. Choose E > 0 and ro > 0 such that logN(X, r ) 5 (A + E ) log 5 for

all 0 < r _< ro. Further, let y, T > 0 satisfy e-yT _< rg. Then N ( X , e - T T ) 5 exp{yT(A+e)}. Let Xi, i = 1, . . . , N ( X , e-TT) be compact sets of diameter

at most e-YT which cover X and choose arbitrary points xi E Xi. Define - x := {X i , i = 1,. . . , N ( X , C ' T ) } .

For K > 0 we have

P{ sup ll$t(x) - zll 2 KT + b for some x E X } 5 S1 + S 2 , O<t<T

where

S1 := exp{yT(A + E ) } maxP{ sup ll&(x) - 511 2 KT - eCYT} s E X W t < T

230

and

S 2 := exp{yT(A + E ) } maxP{ sup diam($t(Xi)) 2 b} . O<t<T

Using (C2) we get

Our aim is to identify the infimum k over all r; for which there exists some

y > 0 and E > 0 such that the upper bounds of both S1 and S, above decay

to zero exponentially fast as T --+ 00. A simple Borel-Cantelli argument

will then show that k is indeed an upper bound for the linear growth

rate. Observing (6) we get k = B + A m , where YO is the infimum

of all y > 0 for which there exists some E > 0 such that 5’2 decays to 0

exponentially fast as T .+ 00. Rather than identifying 70, we will instead

provide some yo 2 70. Then

K : = B + A ~

will turn out to be an upper bound for the linear growth rate.

We will estimate SZ using the Lemma of Garsia-Rodemich-Rumsey (see

Lemma 5.1).

Define

0

We will use the abbreviation

c := -T(1 U2 + 6) . 2

We have

and we will use the following estimates

23 1

and

Therefore

We fix T > 0,y > 0 and i E (1 , . . ., N ( X , e - Y T ) } and define

We choose ,B = ecT with < 2 -A. Using (7) and ( C l ) we get

By the GRR Lemma 5.1 applied with the metric

d ( f , 9) = SUP Ilf(t) - g(t)II A b OSt lT

and by (8) it follows that

where

I := "Texp { / s } d t . 0

232

We have

Define

y - 2 f i J 7 2 ’ ( C G J if y 2 a2ci(1 + 6) -a%( 1 + 6) otherwise.

U = U ( y , 6 ) :=

Then

1 5 2 exp{-UT) (1 + 6 ~ 2 0 2 4 1 + 6)T)

Assuming betT 2 121 we have

P(ZT 2 b) 5 P(exp{ {-} 7T Cd 2 g) < P ( v > -

Using Chebyshev’s inequality, we obtain

P ( ~ T 2 b) < E V 8 p e x p { Cd - ‘(i.g(E))2}. 4 C 1 2 1

Using we have

1 (t + N2 - (5 + w2 2a2(1 + 6) <

lim sup - log S2 5 y(A + E ) - 2dy + 2026 T-oo

for some E > 0 provided that

(i) t + U > 0;

(ii) 5 + A 2 0;

(iii) Ay - 2dy + (E + A>2 - (5 + w2 2026 2+(1 + 6) < O.

If

2a2d(d - A) A ’

A 2

then it is easy to check that

70 := A + a2A + Jo4A2 + 2A02A,

60 := (70 - A)2d

Eo := -U(yo, 60)

- 1’ a27oA2

233

satisfy (i)-(iii) above, provided “>” and ‘(<” are replaced by “,” and “5” in (i) and (iii) respectively, and that 60 > 0. Further, it is easy to see that

yo is greater or equal than the infimum of all y’s for which there exist 6 > 0

and ‘$ such that (i)-(iii) are satisfied. If ~ on the other hand -

2 a 2 d ( d - A) A ’ A <

then i t is again easy to check that

A

a 2 d so := - + 1,

‘$0 := -U(Yo, 60)

satisfy (i)-(iii) above, provided “>” and “<” are replaced by “2” and “5” in (i) and (iii) respectively. Further, it is easy to see that yo is greater or

equal than the infimum of all y’s for which there exist 6 > 0 and E such

that (i)-(iii) are satisfied.

Therefore, for each €0 > 0 we have

Using the Borel-Cantelli Lemma and letting €0 go to 0 we obtain

limsup - sup sup Ilqbt(x)II 5 K := B + A m a. s. 1

T-ca T x E X O<t<T

which proves (4) of Theorem 3.1.

B = 0, a =

Formula (5) for the isotropic case follows from (4) by inserting A = 1,

[7 and A = A, and using formula ( 3 ) and Remark 3.1.

Remark 3.2. If & is a homeomorphism of Rd for every t 2 0 and if the

upper entropy dimension A of the set X is greater than d - 1, then (4)

and (5) remain true when replacing A in the definition of K by the smaller

number d - 1. To see this, one can take a closed ball B which contains X and apply Theorem 3.1 with X replaced by aB, which has (upper entropy)

dimension d - 1. Due to the homeomorphic property of 4t, the upper linear

growth rate of & ( X ) is bounded by that of &(as).

4. The Lower Bound

In the following we will call a subset of Rd nontrivial if it contains a t least

two points.

234

Theorem 4.1. Let (4t)t>O - be an isotropic Brownian jlow on Rd, d 2 2. There exists a number c* > 0 such that for any nontrivial, connected, compact subset X c Rd we have

1 P liminf -diam(q5t (X)) 2 c* diam(q5t (X)) = 0} = 1 (10) { t+m t

and the first of the two probabilities is strictly positive.

Since the two events in (10) are disjoint Theorem 4.1 says that for any

subset X as above one of the following two cases will occur almost surely:

either the diameter will grow to infinity with at least linear speed c* or

the diameter will shrink to zero. Even if the top Lyapunov exponent X is

negative, linear growth will occur with strictly positive probability.

Remark 4.1. It is easy to see that Theorem 4.1 will no longer hold if

we either allow the set X to be finite or if d = 1. In the first case the

diameter of 4 t ( X ) equals the maximum of the distance of a finite number

of (correlated) Brownian motions in Rd which grows at most like a constant

times (t log log t) ' /2. In the second case the compact set X c R is contained

in a compact interval [a, b] , and hence diam (& (2)) 5 &(b) - &(a) , which

again grows at most like a constant times (t loglogt)1/2.

* *

St St 1 coordinate t=O 1 coordinate t=l

Figure 2. linear expansion in the first coordinate direction

235

Idea of the proof of Theorem 4.1. We sketch the competition and

selection procedure to show that as long as the diameter of the set $ t ( X ) does not become too small, supzEx 4t(x) will grow at least linearly in t (the upper index 1 stands for the first coordinate). Using isotropy of the

flow, this implies that &(X) will grow at least linearly in every direction,

which is actually more than what we claim in the theorem. A complete

proof (even for more general stochastic flows) can be found in Scheutzow

and Steinsaltz12. Consider two points x and y in X such that 112 - yII = 1

and z1 2 y1 (assume that X has diameter at least 1). Since t H $;(x) is a

martingale, we have

E (+:(~)lFo) = x1 = x1 V yl.

Further it is plausible (and true) that there exists some p > 0 (not depend-

ing on the particular choice of z and y) such that

p (&Y) 2 4 ; ( 4 + 1 l F o ) 1 p a. s.

Observe that a t this point we need the assumption d 2 2: for d = 1 it is

impossible for a trajectory to pass another trajectory. Therefore

E ( 4 m v 4 Z Y ) I F o ) = E ( 4 x 4 + (4XY) - 4 : ( 4 ) + I F o )

2 x1 v y' + p a. s.

Now we iterate the procedure by selecting x or y depending on whether

$;(x) or $:(y) is larger. Assume that x is the winner. Then we pick a new

competitor z E X for which I l$:(z) - $i(x)II = 1 and so on. Therefore in

each unit time step the right frontier of the set 4 t ( X ) moves to the right by

an average at least p. Now a suitable version of the law of large numbers

0 for martingales (essentially) finishes the proof.

Remark 4.2. Under weaker conditions than in Theorem 4.1, Scheutzow

and Steinsaltz12 proved much stronger results than 4.1, namely so-called bull chasing properties. We formulate one result for isotropic Brownian flows in

dimension 2 or greater with a nonnegative Lyapunov exponent: there exist

numbers c1 > 0 and c2 2 0 such that for any process II, : [ O , o o ) -+ Rd which is adapted to the filtration of 4 and which is Lipschitz continuous

with constant c1, and for any nontrivial connected subset X, there exists

almost surely some x E X for which

5 c2. lim sup IlM.) - +tll

t-cc log t

236

5. Open Problems

In this section we assume that (q&)t?o is an isotropic Brownian flow which

has a nonnegative top Lyapunov exponent A. We list some open problems.

Is i t true that for any (reasonable) nontrivial compact subset X C Rd the limit

1

T-cc T lim - diam((bT(X))

exists almost surely? If so, is i t deterministic?

If the answer to both questions is yes, does this limit depend on the

set X (e.g. on its dimension A)? Since our upper bound depends

on A we conjecture that the linear growth rate will depend on A in

general.

Let X be a curve in Rd of finite length LO > 0, and let LT be the

length of the curve ~ T ( X ) . How does LT grow as T + oo? I t seems

reasonable to conjecture that l i m $ l o g L ~ = X almost surely but

we conjecture that LT will grow faster, namely that

1 PL

T-oo T 2 lirn - log LT = X + - almost surely.

It has been shown by G. Dimitrofl, using martingale arguments,

that

1 1 PL X 5 lim inf - log LT 5 lim sup - log LT 5 X + - a. s. T-too T T-m T 2

Let X be a compact subset of positive d-dimensional Lebesgue mea-

sure and let V, be the d-dimensional Lebesgue measure of 4 t ( X ) . It is not hard to see that (V,)t?o is a (nonnegative) martingale (see

Baxendale and Harris2). By the martingale convergence theorem

V, converges almost surely to a (finite) random variable V,. We

conjecture that V, > 0 almost surely.

Is it true that q5t(X) becomes dense in Rd as t -+ 03 in case X is

a nontrivial, connected and compact subset of Rd? More precisely

we can ask if

lim P {w : &(u, X) n B # 0} = 1 t-,

holds for any nonempty open subset B C Rd. Let X be a compact subset of Rd with nonempty interior and denote

by Ad the d-dimensional Lebesgue measure. Does there exist a

function I : [0, oo) --$ [0, m) such that

1

T-+m T lim -logXd {x E X : I I ~ T ( w , z ) I I L y T } = -I(y) a. s.?

237

If such a function I exists, then i t will take the value +00 for suf-

ficiently large values of y by Theorem 3.1. The following simple

observation shows that if such a function I exists, then I ( y ) 2 y2/2

for all y 2 0: using Chebychev's inequality and Fubini's theorem,

we get for any E > 0

Since

we get I(?) 2 $. If the flow is volume-preserving, equation (11)

provides an upper bound for the probability that the amount of oil

(say) which is found outside a ball of radius yT at time T exceeds

the value exp - - E T (since i t is easy to find an explicit

upper bound for P { \ I ~ T ( x ) ~ ( 2 yT}). This bound does not use any

information about the correlation of several tracers and it is likely

that it can be improved considerably by using such information.

H 7 2 * 1 1

Appendix

We state the following lemma which is originally due to Garsia, Rodemich

and Rumsey and which we briefly refer to as the GRR-Lemma. A proof

(of a more general version) can be found in Arnold and Imkeller'.

Lemma 5.1. Let B be a compact subset of Rd, (E ,d ) a metric space, Q : [0,00) -+ [0,00) a right-continuous and strictly increasing function satisfying Q(0) = 0 and assume that f : B -+ E is continuous. If

then we have

where Cd denotes the square of the volume of a ball of radius 1 in Rd.

238

References

1. L. Arnold and P. Imkeller, Stratonovich calculus with spatial parameters and

anticipative problems in multiplicative ergodic theory, Stoch. Proc. Appl. 62, 19-54 (1996).

2. P. Baxendale and T. Harris, Isotropic stochastic flows, Ann. Probab. 14, 1155-

1179 (1986). 3. R. Carmona and F. Cerou, Transport b y incompressible random velocityfields:

simulations and mathematical conjectures, in: Stochastic partial differential

equations: six perspectives, eds. R. Carmona and B. Rozovskii, AMS, 1999. 4. M. Cranston and M. Scheutzow, Dispersion rates under finite mode Kol-

mogorov flows, Ann. Appl. Probab., 12, 511-532 (2002). 5. M. Cranston, M. Scheutzow and D. Steinsaltz, Linear expansion of isotropic

Brownian flows, Electron. Commun. Probab. 4, 91-101 (1999).

6. M. Cranston, M. Scheutzow and D. Steinsaltz, Linear bounds for stochastic

dispersion, Ann. Probab. 28, 1852-1869 (2000).

7. G. Dimitroff, forthcoming Ph. D. thesis, Technische Universitiit Berlin.

8. J. Hoffmann-J~rgensen, Probability with a view toward statistics, Vol. II, Chapman & Hall, 1994.

9. H. Kunita, Stochastic flows and stochastic differential equations, Cambridge

University Press, 1990.

10. M. Ledoux and M. Talagrand, Probability in Banach spaces, Springer, 1991.

11. H. Lisei and M. Scheutzow, Linear bounds and Gaussian tails in a stochastic

12. M. Scheutzow and D. Steinsaltz, Chasing balls through martingale fields,

dispersion model, Stochastics and Dynamics 1, 389-403 (2001).

Ann. Probab. 30, 2046-2080 (2002).

STOCHASTIC BURGERS EQUATION IN D-DIMENSIONS - A ONE-DIMENSIONAL ANALYSIS: HOT AND COOL CAUSTICS AND INTERMITTENCE OF STOCHASTIC

TURBULENCE

A. TRUMAN*, C. N. REYNOLDS AND D. WILLIAMS

Department of Mathematics, University of Wales Swansea, Singleton Park, Swansea, SA2 8PP, Wales, UK

We give a one dimensional analysis of the solution vp of the stochastic Burgers

equation in d dimensions, with viscosity p2 N 0, as obtained by Davies, Truman

and Zhao. Our analysis shows how the graph of a simple action functional in one

space variable can be used to decompose the caustics into hot and cool parts. The

inviscid limiting Burgers velocity field has a jump discontinuity across a cool part

but is continuous as you cross the hot part. Our analysis also enables us to get a

hold on the intermittence of stochastic turbulence in terms of the recurrence of a

one dimensional stochastic process C simply related to the reduced action. Some

detailed examples are discussed.

1. In t roduct ion

Burgers equation has been used to model the large scale structure of space-

time (Shandarin and Zeldovich and in a noisy environment in studies of

turbulence ( E, Khanin, Maze1 and Sinai Here we develop some related

results.

We begin by giving a brief account of the results of Davies, Truman and

Z h a ~ ~ > ~ . Let Wt be a B M ( R ) process on the probability space (a, 3, P ) with

lE[WtW,] = (s A t ) .

Consider the stochastic viscous Burgers equation for up = up(z, t ) , IC E Rd,

t > 0.

* [email protected]

239

240

V P ( Z , 0 ) = VSO(Z) ,

Wt being white noise, pz coefficient of viscosity. We are interested in

Burgulence, that is the advent of discontinuities in

v’(z, t) = lim v ~ ( z , t) . P V

The corresponding Stratonovich heat equation is

UYX, 0) = exp (-so(x)/p2)To(Z),

the convergence factor TO being related to the initial Burgers fluid density.

Here the connection is the Hopf-Cole transformation

V P ( Z , t) = - p 2 v lnuP(z, t) .

Following Donsker, Freidlin et a1 l1 we expect as p \ 0

X ( 0 ) -pz lnuP(z, t) 4 inf [So(X(O)) + A(X(O), z, t ) ] = S(x , t ) ,

where

A(X(O), Z, t ) = inf A [ X ] , X ( S )

X ( t ) = x

t t t

A [ X ] = 2-1 X 2 ( s ) ds - 1 c ( X ( s ) ) ds - E 1 k , (X(s ) ) dW,

This gives the minimal entropy solution of the Burgers equation 12. Set

d [ X ] := A [ X ] + So(X(0) ) .

Then necessary conditions for X to be an extremiser of A are :

0 = dX(s) + V c ( X ( s ) ) ds + &VlCs(X(s)) dW, , X ( 0 ) = VSo(X(0)) .

Minimising A [ X ] over X ( 0 ) gives S(z,t) which satisfies the Hamilton-

Jacobi equation

dSt + (2-1 (VStj2 + c(z)) dt + &(x) dWt = 0 ,

St=o(.) = So(.) .

Definition 1.1. We define the stochastic wavefront Wt in x by {x :

S ( Z , t) = O}.

24 1

For small p , the heat equation solution u p switches continuously from being

exponentially large to small as we cross the wavefront. up can also switch

discontinuously.

Define the classical flow m a p Q S : Rd 4 Rd by

d&s + VC(@,) ds + E V ~ ~ ( @ , ) dWs = 0 I

with = id, 60 = 0 5 ' 0 . So, since by definition X ( t ) = x,

X ( s ) = @ , q 1 X ]

where we accept that xo(x, t ) = @,lz is not necessarily unique.

caustic time T(d) such that for s < T(w), Moreover, for t < T ( w ) ,

Given some regularity, the global inverse function theorem gives the

is a random diffeomorphism.

U O ( 5 , t ) = d&@;lz )

is a classical C1 solution of Burgers equation.

From the method of characteristics, we expect non-uniqueness of xo(z, t ) to be associated with discontinuities in vo(x, t ) . The simplest way €or this

to arise is if a positive, infinitesimal volume of points is focused into zero

volume by @t.

Definition 1.2.

Det- ax(t) = 0 8x0

Pre-caustic in zo, QT'C~,

Detax(t) 1 = 0 Caustic in x, Ct. 8x0 so=zo(z,t)

When @,'{z} = { ~ ~ ( z , t ) , zz(x, t ) , . . . , xE(x, t ) } , for a non-degenerate

critical point,

n

u p ( z , t ) N C ~i exp (-~:(z, t ) / p 2 , i=l

where for i = 1 , 2 , . . . , n

SA("Cl t ) = So(x9(z1 t ) ) + A(sA(z, t ) , z, t ) I

and Oi is an asymptotic series in p2.

vp(x, t ) is given for each integer m 2 0 by

When zo(x,t) is unique, t < T ( w ) , the asymptotic series for up =

242

m t

w p (x, t ) = c p2jvj (x, t ) - p2V lnE{ exp ( - V . um(y;, t - s) ds j =O

Here

U j ( X , t ) = VSj (2 , t ) , Sj satisfying

for j = 0 , 1 , 2 , . . .with the convention 2-lAS-l = -c-&lctWt, can be found

explicitly5.

Moreover, the Nelson diffusion process

j =O

yo” = x .

Here, we see as p - 0, the leading term

V P ( Z , t ) - VSO(Z, t ) + O ( 2 ) , where SO is the solution of the Hamilton-Jacobi equation, which minimises

the action A. When @T’{x} = {xA(x, t ) , xg(x, t ) , . . . , $(x, t ) } , there is a similar

asymptotic series 64 for the ith term in the series. Since

S(x,t) = min Sh(x,t) , z=1,2, ..., n

we define the zero level surface by

H,“ = {x : ~ h ( r c , t ) = o , for some i} , where H: includes the wavefront. The dominant term for wo(x,t) comes

from the minimising 20(x,t) (assumed unique) and we obtain the corre-

sponding Burgers velocity field

VO(X, t ) = & q ’ z = &tS,(x,t)

Two x;(x,t)’s can coalesce a.nd disappear as we cross the caustic. When

this corresponds to the minimiser jumping, up=o has a jump discontinuity

and we say that this part of the caustic is cool.

243

Example 1.1. (Cusp and Tricorn). In two dimensions c = 0, kt z 0, &(z, y) = 9. The multiplicity of 20’s changes as we cross the caustic.

cusp

Ht” : z(z0,t) =

Y(X0,t) =

Figure 1.

3t 2 1 -zo - - . 2 t

s(ii

Multiplicity of

Evidently, n, the multiplicity of zo(z,t), depends on z and t . This

multiplicity changes by multiples of 2 as we cross the caustic surface. This

is associated with level surfaces of Hamilton’s principal function having

cusps on the caustic caused by 2 different zo(z,t)’s coalescing. This is

illustrated in 1-dimension by considering

G(zo)eiF(xO , a ? t ) / P z dzo , G E C,-(R) 7 .I I(x, t) =

R

where i = G.

crosses the caustic. (D,F” > 0 in neighbourhood of (l).)

Consider the graph of the phase function F,,t(so) = F(zo,z, t ) as z

244

\ \ , , ,

Cusped Side of Caustic On cool part of Caustic Beyond Caustic

, , I

I

2 coalesce at the

Begin moving x in direction n. Zo(x,t) is the global

point of inflection. New ZO(Z, t ) here.

minimiser of Fz,t (.) .

i). Zo(x, t ) jumps from position (1) to position (2). This causes u p and

ii). This only happens when a point of inflexion is the global minimiser

iii). Some parts of the caustic (the cool parts) will be jump discontinu-

ities in wo and uo. Coalescing 2x0’s is associated with level surfaces of

Hamilton’s function having cusps on the caustic. I t is therefore im-

portant to know when Ht has a cusp on the caustic. We investigate

this in the next section.

up to change discontinuously as we cross the caustic.

of &(.I.

1.1. W h e n does Ht have cusps?

I I 4

Figure 2: Pre-Caustic and

Pre-Level Surface

Figure 3: Cusp, Tricorn and

Line Pair

An important insight about where Ht has cusps is that the cusped part

245

where @FICt is the pre-caustic and @TIHt the pre-level surface, determined

algebraically. If you want to find cusps on the level surfaces of Hamilton's

principal function you look for images of intersections of the corresponding

pre-level surfaces with the pre-caustic.

Figure 4: Pre-Level Surface and

Pre-Caustic Caustic

As we shall see, if you want nc(t), the number of cusped curves in

(Ct n H,) to change, the simplest way is for the pre-surfaces @FICt and

@FIHt to touch, or for the Burgers velocity field to be zero on the caus-

tic, or orthogonal to the caustic. The turbulent times t are when nc(t)

changes. For the stochastic Burgers equation, such times t are the zeros of

a stochastic process C, i.e. times t satisfying c(t) = 0.

Typically these zeros form a perfect set - an infinite set containing no

isolated points. At such times the geometry of the surface of discontinuity

of vo can change infinitely rapidly - reflecting turbulent behaviour of the

fluid. If C is recurrent to 0, the scale of random fluctuations varies,in

a random periodic way. This will be seen as intermittence of stochastic

turbulence, when the cusp is on the minimising part of the level surface

of the Hamilton Jacobi function i.e. on the cool part of the caustic. We

shall see this can be investigated by the one-dimensional graph above. Our

analysis also shows which part of the caustic corresponds to discontinuities

in vo. in both deterministic and stochastic cases.

Figure 5: Level Surface and

246

2. Some Geometrical Results ( E = 1)

We investigate the geometrical relationship between curves on level surfaces

of the Hamilton Jacobi function and caustics for Burgers equation. In 2 dimensions the curves are the level surfaces themselves. In 3 dimensions

we think of them as arising by taking planar cross sections.

Definition 2.1. A curve x = x(y), y E N(yo,6) is said to have a gen-

eralised cusp at y = yo, y being an intrinsic variable such as arc-length,

if

Consider first the deterministic case E = 0. Here

where

The corresponding Euler-Lagrange equations read

X ( s ) = -Vc(X(s)) ,

and X ( t ) = 2, X(0) = ZO. The free case corresponds to c = 0,

Consider the level surface H t obtained by eliminating 20 between

i3A -(xo, x, t ) = 0 , A(Q, x,t) = 0 and cli = 1 , 2 , . . .d . ax,.

Eliminating 5 alternatively gives the pre-level surface @T1H:.

XO) between

Similarly the pre-caustic (and caustic) are obtained by eliminating 2 (or

D e t ( e ( x o , x , t ) ) = 0 and - ( x o , ~ , t ) = O , i3A a=1,2 , . . . d . ax0 ax;

247

We denote these by @T'Ct and Ct.

(In passing, we point out that the processes of determining @FICt and

@L'Ht are algebraic. So @;'Ct and @T'Ht are algebraic inverse images

not the topological inverse images @F ' (Ct) and @F1 (Ht ) .)

In the free case the equation for the zero pre-level surface is the eikonal equation

t 2 - /VS0(zo)l2 + So(x0) = 0 ,

D@t(zo) = ( I + tV2So(zo)) .

and the derivative map DDt(xo) : T,, + T, is given by

The following elementary identity is the key to the free case

Vx, { 5 IPS0(zo)l2 + So(x0) = ( I + tV2So(xo)) VSo(z0) 1 The next lemma and proposition illustrate the scope of our results in 2

dimensions.

Lemma 2.1. Assume the pre-level surface meets the pre-caustic at xo where [ ( I + tV2So(zo))VSO(xo)( # 0 and dim (Ker (I+tV2So(zo))) =

1. Then the tangent plane to the pre-level surface T,, i s spanned by

Ker( ( I+ tV2So(xo))).

Proof. At the point of intersection, the normal to the pre-level surface is

a linear combination of the eigenvectors of ( I + tV2So(zo)) corresponding

to non zero eigenvalues. Let eo be the eigenvector corresponding to the

eigenvalue zero. This normal is orthogonal to eo, so T,, =< eo >.

Proposi t ion 2.1. Assume that [(I+tV2So(xo))VSo(xo)I # 0 , so that xo i s not a singular point of @;'Ht. T h e n @t(xo) can only be a generalised cusp, i f @t(xo) E Ct, the caustic. Moreover, i f z = @txo E @t(@;lCt n @;'Ht), x will indeed be a generalised cusp of the level surface.

Proof. We have normal n(z0) # 0 and % (7) I # 0 and from above Y=YO

For this to be zero it is necessary that Det ( I + tV2So(xo)) = 0, so zo E BTlCt. Trivially from Lemma (2.1) l Y = Y o = 0, since &( ) 1 1 eo. dy

248

It is very easy to generalise the above to d dimensions and to include

noise. Let the stochastic action be defined by

where X , = X ( s ) = X ( s , xo,po) E Rd and

dX(s) = -VC(X(S) ) ds - E V ~ ( X ( S ) , S) dW, , s E [O, tl 1

with X ( 0 ) = xo, X ( 0 ) = PO; z o 1 p o E Rd. We assume X, is F,-measurable

and unique. If du, dX, = 0, we have from ItB’s formula

In particular this is true when us =

Kunita6, mild regularity gives with above Equation (1).

for any a! = 1 , 2 , . . . , d. Using

Q = 1 , 2 , . . ., d ,

almost surely. This gives:

Lemma 2.2. Assume SO, c E C2 and k E C2io, V c , Vk Lipschitz, with Hessians V 2 c , V 2 k and all second derivatives with respect to space variables of c and k are bounded. Then, for po possibly xo dependent, we have

a! = 1 , 2 , . . ., d -(xo,po,t) d A = X(t) .% - X,(O) 7

ax,. ax,.

Now let

4x01 5 1 t ) = A ( X o 1 Po, t)lpo=po(zo,z,t) 1

where PO = po(xo,z , t ) is the (random) minimiser (assumed unique)

of A(zo,po,t) with X(t ,xo,po) = z. (Here we need the map po H

X ( t , x0,po) E Rd to be onto for all 20. Methods of Kolokoltsov et a18>’

guarantee this for small t.)

Theorem 2.1. The classical stochastic flow map @t is defined by

a! = 1 , 2 , . . .d , d - [SO(XO) + A(zo, 2, t)l = 0 7 ax,.

so that x = Qtxo.

249

Assume now that A(x0, x , t ) is C4 in space variables and Det (a) # 0.

Then we can show that:

Lemma 2.3. The random classical flow map has Frechet derivative a s .

Proposition 2.2. The random pre-level surface at a point xo is obtained by

Then the normal to the pre-level surface at the point xo is eliminating x between A(xo, x , t ) = c and ~ ( x o , d d x , t),O, (Y = 1 , 2 , . . . , d .

dx,

We content ourselves here by quoting a result in 3 dimensions.

Theorem 2.2. Let x E cusp(^^) = { x E at (@;lCt n @ ; ' H ~ ) , x = atxO, n(xo) # o } . Then in 3 dimensions in the stochastic case, T,, the tangent space to the level surface at x , is at most one-dimensional.

Proof. On the caustic at Qt(xo), Det (s) = 0, so there exists eo E

Ker (@(xo, 2, t ) ) , eo # 0. From the above eo . n = 0, so eo E T,,, the

tangent plane to the pre-level surface. Similarly (n A eo) E Txo. From

the explicit form of D@t(xo) we see that D@t(xo)eo = 0. Therefore, T, is

spanned by DQt (n A eo).

The above explains the geometry of level surfaces of the Hamilton Jacobi

function. We know that u p changes dramatically as we cross Cusp(Ct n H t ) in the cool region. What about discontinuities in up as p N O? Let us now

see how a simple one-dimensional analysis reveals all.

Definition 2.2. The classical flow map is globally reducible if

d 1 2 d Y = %Yo , Y = (Y' , Y 2 , . . . , Y 1 1 Yo = (Yo , Vo, . . . > Yo 1 1 2 y& = y k ( y , y o , y o , . . . , yyo'-',t) , r = d , d - l , d - 2 , . . . , 2 .

Given some differentiability and non-vanishing of derivatives this will be

true locally. We want a global result.

We want C2 functions yo", yod-', . . . , such that

Yy," = Y~(Y ,Y~ ,Y~ , . . . ,Yod- l , t )

250

where y,”( ) = y$(y, yo, 1 2 yo, . . . yt-’, t ) . No root is repeated so second

derivatives of A do not vanish. (Evidently we are assuming a favoured

ordering of coordinates and a corresponding decomposition of @t, so that

non-uniqueness is reduced to the level of the y: coordinate.)

Proposition 2.3. Assume the @t map is globally reducible. Define the reduced action

f (YA, 9, t ) = 4 Y k Y 3 Y > YIL t ) , V03(Yj YIL YE(Y, Y h , t ) ) , 1 Y, t )

Then

, a f a) . T(yA, y, t ) = 0 and Equations (2) * y = a t y o 8Yo

i i). Equations (2) and 7 ( y A 1 y1 t ) = - 2(YILYlt) = 0 * a f a2f dY0 (aY; 1

y = a t y o is such that the number of solutions yo of this equation changes.

Lemma 2.4.

where the last term i s f”(yh, y, t ) and the first ( d - 1) terms are non-zero as above.

25 1

The above results follow by applying the principle of stationary phase to

For instance by stationary phase, if we assume % (yh, y, t ) = 0 and y is

such that q ( y h , y, t ) # 0, then the first equation will have n roots yh =

a:(y , t ) , ai(y, t ) , . . . , aA(y, t ) . If we vary y now so that %(yi, 9, t ) = 0,

typically two of the above critical points will coalesce - a local maximum and

a local minimum forming a point of inflection. Then, if D, azf , (yh , y, t ) # 0, D, directional derivative, we have the picture shown below.

ayo

(ago)

( a d )

( a d )

Here the picture de-

forms as we move in di-

rection n.

Here 2 a:'s coalesce, say Here the point of inflec-

U A - ~ and uA. tion at (1) has disap-

a: (y , t ) , a repeated

root.

anPl(y, t ) 1 = uA(g,t) = peared.

Because the value f (a:(y, t ) , y, t ) < mini=1,2, ...,,-2 f (ai(y, t ) , y, t ) the

disappearing root So the minimiser

jumps from (1) to (2) . Hence wo is discontinuous and uo is exponentially

discontinuous. Hence the function fy,t(yi) = f (yi , y, t ) gives a complete

analysis of the discontinuities. A similar analysis may be given if you only

have local reducibility. This explains how to analyse hot and cool parts of

the caustic.

= a; is the minimising one.

3. Intermittence of Stochastic Turbulence

Here we illustrate how turbulent times and turbulent processes C can be

determined when at is globally reducible. For simplicity we work in two

dimensions.

252

Proposition 3.1. Assume @t i s globally reducible. Let f(x,t,(xA), the reduced action, be defined as above, so that

f(x,t , (4) = 4:, xo2(x, x:, t ) , 2, t ) ,

where x = ( z : ) , xo = (xi) and 5 0

x = @txo s f{x,t)(xA) = o and xg = xi(., xo, 1 t ) .

W h e n x E Ct, the random caustic, let f[x,t,(xh) = 0 have the repeated root

xi = xg(x,t), Let X H xt(X) be a parameterisation of Ct, X E R, such that X = A0 corresponds to a cusp o n the caustic, or a point o n the caustic where the Burgers velocity field is zero or orthogonal to the caustic. T h e n the < processes f o r stochastic turbulence at xt(X0) are given by

< C ( t ) = f(z,(x,,,t)(.;;(.t(XO),t)) - c,

for c E R.

Remark 3.1. The (0 processes are just the stochastic action evaluated at

the relevant points on the caustic and their inverse images. Similar results

hold in &dimensions and for more general noise.

Proof. Firstly, X H x2;(xt(A), t ) (the equal root vector (xh, xi(x, xA, t ) , . . .) evaluated at x; = zE(x, t ) , x = xt(X)) is a parameterisation of the precaus-

tic @FICt. Hence, the number of cusps on the level surface 5' = c is given

by

# { A E R : f(xc,(x,,t,(.2;(xt(X),t)) = c} .

< c ( t ) = 0.

Differentiating our last equation with respect to X gives X = Xo and

The above suggests the nomenclature for the three kinds of turbulence

- cusp, zero and orthogonal turbulence. We expect orthogonal turbulence

to be the most important. Similar results hold in higher dimensions.

4. Some Analytical Results (Small E )

Here we summarise some of the (small E ) analytical results of Davies, Tru-

man and Zhao3i4. Consider

dv + (u.V) dt = -VC(Z) dt - ~ V k ( 2 ) dWt ,

253

and the corresponding stochastic classical mechanics

dXi"(zo, S) = -VC(X'(ZO, s ) ) ds - E V ~ ( X ' ( Z O , s)) dW9 ,

with X'(z0,O) = zo and Xi"(xo,O) = VSo(zo), 0 < s < t . Let X o ( z , s) =

@:xo satisfy the deterministic ( E = 0) version of the above equation and let

4 be given by Bi, = {X," (u) ,X! (s) } 8(s - u), the product of the Poisson

bracket { } and the Heaviside function 8.

Lemma 4.1. G satisfies the matrix Jacobi equation

with boundary condition

Let

X ' ( ~ O , s ) = @;ZO - EL' SO, s,u)Vk(@;zo) dW, ,

for s E [O, t ] . This is the first term in, the perturbation expansion for X' .

Theorem 4.1. Given some mild conditions on continuity and boundedness of c and k and their derivatives, there exists a constant A4 > 0 such that for any 6 > 0 and suficiently small E > 0

and

In particular,

and

V X " ( x 0 , s) - VX'(z0, s ) = O(&+) ,

as E \ 0 in probability.

254

It is not difficult to prove from the above that the pre-caustic surface of

the stochastic mechanics converges to the pre-caustic surface of classical

mechanics as E \ 0 in probability. Caustic surfaces are stable in probability.

What about the stability of level surfaces of the Hamilton-Jacobi function?

We can prove:

2 Theorem 4.2. Let @, be the minimiser of 2-1 Ji l&sx~l ds+So(@txo) -

Jot c(@,xo) ds satisfying Qtxo = x, with corresponding minimum

So(x, t ) and let @: be the minimiser of 2-1 s," I&:xo/ ds + So(@zxo) -

s," c(@:xo) d s - E J i k(@zxo) dW,, satisfying @zxo = x for almost all w E R, with corresponding minimum SE(x , t ) . Then we have for almost all w E R

2

t

So(x, t ) - EL k(@;xo) dW, I SE(x, t ) 5 So(x, t ) - E

In particular, as E \ 0, SE(x , t ) + So(x, t ) a s .

Finally, if we assume there exists a unique z o for fixed t and x such that

Qtxo = x, then the first approximation is

r t

S'(Z, t ) = S0(z, t ) - E k ( @ . , ~ o ) dW, + O ( E ) , l o where So(x, t ) is Hamilton's principal function for the path Xo(xo , s). Sim-

ilar results hold for xh(z, t ) and corresponding S'.

5. Some Applications

We give two elementary results illustrating the kind of applications now

accessible.

5.1. Hot and Cool Parts of the Caustic

Recall that when the level surface of Hamilton's principal function (with a

cusp at the point of intersection with the caustic) is the minimiser of the

action at the point ( x , t ) of intersection we say the caustic is cool. The

corresponding solution of the heat equation will have a jump discontinuity

here because fio(z, t ) will jump.

Theorem 5.1. (Polynomial Swallowtail in 2 dimensions). Let c = 0 , kt(x,y) = x, So(xo,yo) = x; + xgyo. We have global reducibility and

255

yo(y, XO) = ?/ - txg. Then the graph in question is

2 t x; xzo t as, ~ ( z o ) = - 2t - -+ t - 2 ( - (xo ,Yo(Y,xo) ) ) dyo - 7 1 Ws ds

where

As expected

2 (;) = Qt (;:) - f’(X0) = 0 and yo = y - txo

Moreover, additionally

(5 , y ) E ct * f” (Z0) = 0 .

Analysis of the graph o f f ( . ) yields the hot and cool parts of swallowtail, as shown in Figure 6, where

t 1 t3 (9 - 4) 450

= ( - t5(3 + 8v@) - E L Wsds,--+ 18000 2t

and K = ( -&-EL t W s d r , - - ~ + ~ ) .

2t 50

Proof. I t may be shown7 that, for k t ( z , y ) 3 x, the effect of noise is to

bodily translate the whole picture in the direction ( - 1 , O ) . Hence we need

only consider the deterministic setting, in which

t 4 5 0 2 x2 f(xo) = Z; - -xi + -(1+ 2ty) - - + -

2 2t t 2t

We consider the roots of f’(x0) for the following two cases.

Case 1 : y < -2t or y > --% + &. Since (x, y ) E Ct we know f’(x0) = 0 has only one solution, namely the

repeated solution x2; ( ( z, y ) , t ) . Thus f (zo) has only one stationary point

which is a point of inflection and so one side of this part of CL is cool.

1 l 3

Case 2 : -% 1 5 y 5 -& + &. We adopt the labelling scheme for the caustic shown in Figure 6. On branch

(A), f’(zo) = 0 will have one solution which is repeated and as in Case 1

one side of this part of Ct is cool.

256

Figure 6. Hot and Cool Parts of the Polynomial Swallowtail

If (2, y) is a point on branch (D) then f ‘(zo) = 0 will have three solutions

xA((x, y), t), z6((x, y), t ) and zi((z, y), t ) where the middle one is repeated.

This implies f ( x 0 ) will have three stationary points occurring from left

to right as maximum, inflection and minimum. Hence f(z6((x,y),t)) > f ( x g ( ( z , y), t ) ) meaning that the coalescing cusped level surfaces do not

correspond to the minimiser and so one side of branch (D) is hot.

Full details of the analysis for branches (B) and (C) are omitted for the

sake of brevity. It may be shown that one side of branch (B) is hot, whilst

on branch (C) there exists a point X at which Ct will switch from cool to

hot. This is found by solving the four equations:-

in four unknowns z, y, zZ;((x, y), t) and zi((x, y), t ) . Solving these yields

450 = ( -t5(3 +8&) 1

18000 2t , -- +

257

A similar numerical study works in three dimensions. The effect of the

noise here is to bodily translate the whole picture in the direction (-l1 010)

by E s,” W, ds. (See Reynolds

5.2. Intermittence of Stochastic Turbulence - a simple example in two dimensions

Theorem 5.2. Let c = 0 , kt(x, y ) = x and So(x0, yo) = f(x0) + g(xo)yo, where f l g , f’ and g‘ are zero at xo = a , g”(a) # 0 . The turbulent t imes t at which nc(t), the number of cusps on the zero pre-level surface of the Hamilton- Jacobi function changes are the zeros of the stochastic turbulence process (0

{t : <o( t ) = 0 } is a perfect set and ( ( t ) is recurrent to 0. Cc(t) = <o( t ) - c has exactly the same properties, where zeros of &(t) are times at which the number of cusps on the c pre-level surface of the Hamilton- Jacobi function changes.

Proof. <O is the result of carrying out the above analysis of orthogonal

0 turbulence. The remainder follows from the argument below.

Lemma 5.1. Let W be a B M ( R ) process starting at 0 and c a real constant. Define

Then with probability one there exists a sequence of t imes (a,) with a, / 00 such that Y,,, = 0 for every n.

Proof. We begin by finding a sequence of times tending to infinity at which

yt 2 0. Define f ( r ) := T for 0 5 r 5 1 so that clearly f is absolutely con-

tinuous] f(0) = 0 and Jt f’(u)’du < 1. Thus f ( r ) is a Strassen function]

f E K . Hence by Strassen’s Law of the Iterated Logarithm we know that after

throwing away a null set of paths, we can path-wise find a sequence t, such

that if

h(t) := (2 t ln ln t ) i

258

then

h(tn)-lWrt, + f ( r ) 7

uniformly over r in [ O , 1 ] .

We show that for each w with t , = t n (w) we have h(t,)-2t;1yt, + $. Let us consider each of the terms that comprise the stochastic process y t (w) .

i) . a&h(tn)-2t,lWt, + 0 .

ii) .

iii).

Combining the above we see that for each w with t, = tn (w) we have

To conclude we must find a sequence of times tending to infinity at

which yt 5 0. If c > 0 then we simply choose times when Wt = 0. For

c I 0 we must choose a Strassen function such that

Taking

i t may be easily shown that f E K and f(1) J: f(u) du = -& < 0.

5.3. C process fo r small noise in 2 dimensions

We perturb an underlying deterministic classical mechanical system by

adding a small noise potential term ~IC~(z)r't., to see its effect on stochastic

turbulence at the displaced cusp zt (X0) of the deterministic caustic. We use

259

the above notation for the globally reducible deterministic map, wit,*

X: = @;zo, and with X: = z;(z, t ) , z = X: = zt(Xo), z; the repeated

root vector. When Xf = 0, there is a very simple result for the small noise

stochastic turbulence process. (There are numerous examples with X: = 0

in the free case.) Let the corresponding < processes be <: for stochastic tur-

bulence at the cusp on the deterministic caustic, zt(X0). These are simple

deterministic functions coming from the reduced classical action, ft’,t) (.A), x; = zE(z, t ) , z = zt(X0).

Proposition 5.1. I f X: = 0 , formal ly correct to first order in E , the stochastic turbulence processes < are given by

t

Mt) = e ( t ) - E l k s (Q;(.;;(.t(Xo), t ) ) ) d W s , c E R, 0

where z.~(zt(Xo), t ) is the repeated root vector evaluated at zt(X0) the cusp o n the deterministic caustic, (2 the reduced classical action at zt(Xo) and .;;(zt(Xo), t ) .

Remark 5.1. Observe that Cc(t) i s possibly not Markov if Ic(s,t) =

k, (@;(zg(~(X), t ) ) ) depends upon t.

Proof. A simple consequence of Theorem 4.2 and a calculation.

Question: What properties of the underlying system give rise to recur-

rence of < and the intermittence of stochastic turbulence? We include an

example here, very similar to the above, to show that the explanation of

intermittence of stochastic turbulence is sometimes very simple.

Example 5.1. (Harmonic Oscillator Potential). Let kt(z, y ) = z and

c = i(z, y)Q2(z, Y ) ~ , where R2 is a real symmetric 2 x 2 positive definite

matrix with

wi if i = j , 0 otherwise .

If we take So(z0,yo) = f(z0) + g(z0)yO where f , g , f’,g’, f”’ and g”’ are

zero at 20 = ai and g”(ai) # 0 for i = 1 , 2 , . . . , n, then the zeros t ( w ) of

the stochastic process

1 sin(2wzt) ( f ” (Qi ) + w1 Cot(Wlt))2

1 2 4 4 g/ ’ (a i )

[ t ( w ) := - -a iw l s in(2wl t ) - -wz

sin(wl(r - t ) ) 0 aW, -

260

sin(wl(r - t)) o aW,

will be turbulent times.

We show that there exists an increasing sequence {t,} with t, 7 00 such

that <tt,(w) = 0 almost surely. Observe that the stochastic process <t(w)

may be written as

w2 sin(2w2t) cosec2(wlt) {sin(cjit).Y(cr,) + w1 cos(w1t)I2 M W ) =

1 2 +ecosec(wlt)Rt(w) - -a ,wl sin(2wlt) - c , 4

where &(w) is a stochastic process well defined for all t.

we have cosec2(wlt) + m. Let { t k } denote an increasing

sequence a t which cosec2(wltk) = 00, then limt,t, [t = -m if 4&cSd > 0

but limt+tk [t = +m if s ’n(2wztk) < 0. However, we can find an infinite

increasing subsequence {tk,} such that It is continuous on (tk, , tk,+l) and

sgn (s in(2~Ztk~) = - sgn (~in(2w2tk,+~) ,

so that limt->t, <t successively switches between plus and minus infinity.

Hence, by continuity and the intermediate value theorem, there will exist

an increasing sequence {t3} with t, /” co a t which st, = 0 almost surely.

As t 4

d ’ ( 4

g” (at )

We remark that the above argument fails if

sgn (sin (F) ) ,

is the same for all k E Z+. This will only be the case if % = 2n7r, namely

w2 = nwl, for some n E Z.

We conclude with an elementary result in the direction of Proposition 5.1.

Assume that in Proposition 5.1, k , (@y(zL(zt(Xa), t))) = k x 0 ( s ) , is indepen-

dent of t . (See Reynolds for examples like this.) Then, for small noise,

for a BM(R) process B,

Sdt) = <,OM - &B(V(t)),

where v(t) = s,” k:,(s) ds. This gives:

Proposition 5.2. Assume that v(t) is bounded and that v(t) /” co as t /” 03. Then a suficient condition for Cc to be recurrent is that

<;(t)/(2v(t) loglogv(t))+ + o as t /” 00.

261

Remark 5.2. This means that t he stochastic turbulence at cusp zt(Xo) will

be intermittent as long as xL(xt(Xo), t ) is the minimising critical repeated

root.

Proof. A simple consequence of the Law of the Iterated Logarithm. [7

Needless to say most of the above results can be extended to d-dimensions

and to more general kinds of noise. However, we should add tha t the

physical interpretation of t he small noise process C is fraught with difficulty.

Acknowledgement

It is a pleasure for one of us (AT) to acknowledge helpful conversations with

Professor Costas Dafermos (Brown), Professor Mark Freidlin (Maryland)

and Professor Oleg Smolyanov (Moscow).

References

1. S. F. Shandarin and Ya. B Zeldovich, The large-scale structure of the uni-

verse: turbulence, intermittency, structures in a self gravitating medium, Rev. Mod. Phys. 6, 185-220 (1989).

2. W. E, K. Khanin, A. Maze1 and Ya Sinai, Invariant measures for Burgers equation with stochastic forcing, Ann. Math. 151, 877-960 (2000).

3. I.M. Davies, A. Truman and Huaizhong Zhao. Stochastic heat and Burgers equations and their singularities I - geometrical properties, J. Math. Phys.

4. I.M. Davies, A. Truman and Huaizhong Zhao. Stochastic heat and Burg- ers equations and their singularities - geometrical and analytical p r o p erties (the fish and the butterfly, and why.), UWS MRRS preprint, http://www.ma.utexas.edu/mp~arc-bin/mpa?yn=Ol-45, 2001.

5 . A. Truman and H.Z. Zhao. Stochastic Burgers’ equations and their semi classical expansions, Comm. Math Phys. 194, 231-248 (1998).

6. H. Kunita. “Stochastic Differential Equations and Stochastic Flows of

Homeomorphisms” in Stochastic Analysis and Applications, edited by M. A. Pinsky, Advances in Probability and Related Topics (Marcel Dekker,

New York, 1984), Vol. 7, pp. 269 - 291. 7. C. Reynolds. On the polynomial swallowtail and cusp singularities of

stochastic Burgers equation, PhD thesis, University of Wales, Swansea, 2002. 8. V. N. Kolokoltsov, R. L. Schilling, A. E. Tyukov. Estimates for multiple

stochastic integrals and stochastic Hamilton-Jacobi equations, to appear in Revista Matematica Iberoamericana.

9. V. N. Kolokoltsov, A. E. Tyukov. Small time and semiclassical asymptotics for stochastic heat equation driven by LBvy noise, Stoch. Stoch. Rep. 75,

10. K.D. Elworthy, A. Truman and H.Z. Zhao. Stochastic elementary formulae on caustics I: One dimensional linear heat equations, UWS MRRS preprint.

43, 3293-3328 (2002).

1-38 (2003).

262

11. M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical systems, (Springer-Verlag, New York, 1998).

12. C. Dafermos, Hyperbolic conservation laws in cont inuum physics, Grundlehren der Mathematischen Wissenschaften 325, (Springer-Verlag, Berlin, 2000).

A VERSION OF THE LAW OF LARGE NUMBERS AND

APPLICATIONS

ARMEN SHIRIKYAN

Laboratoire de Mathe'matiques Universite' de Paris-Sud X I , Bhtiment 425

91405 Orsay Cedex, France E-mail: [email protected]

We establish a version of the strong law of large numbers (SLLN) for mixing-type

Markov chains and apply it to a class of random dynamical systems with additive

noise. The result obtained implies the SLLN for solutions of the 2D Navier-Stokes

system and the complex Ginzburg-Landau equation perturbed by a non-degenerate

random force.

1. Introduction

We study the 2D Navier-Stokes (NS) system perturbed by an external

random force:

li - Au + (u, 0 ) u + Vp = q(t, z), div u = 0, z E D, (1)

u = 0, x E dD. (2)

Here D c R2 is a bounded domain with smooth boundary dD and 7 is a

random process of the form

k = l

where v k are i.i.d. random variables in L2(D,R2) and S ( t ) is the Dirac

measure concentrated at t = 0. It was established in 5 1 1 1 1 0 > 1 1 9 6 9 1 3 1 1 4 that,

if the distribution of T k is sufficiently non-degenerate, then the family of

Markov chains associated with the problem (l), (2) has a unique stationary

measure p and possesses an exponential mixing property. Namely, for a

large class of functionals f and any solution u(t) of (1) - (3), the average

of f ( u ( k ) ) converges exponentially, as k t 00, to the mean value o f f with

respect to p:

I ~ f ( u ( k ) ) - ( f ,p ) l I conSte+, k 2 1, (4)

263

264

where /? > 0 is a constant not depending on u(t). Moreover, as was shown

in 7 , the strong law of large numbers (SLLN) for stationary processes com-

bined with the coupling of solutions constructed in lo implies an SLLN for

solutions of the problem (1)-(3): for any solution u(t) , with probability 1 we have

k-I

We note that similar properties were established for perturbations of the

We refer the reader to 12,’ for a detailed discussion of the results obtained

in this direction.

The aim of this article is to derive the SLLN (5) from the mixing prop-

erty (4) without using the coupling of solutions and to estimate the rate of

convergence. To this end, we establish a simple version of SLLN for a class

of Markov chains (Section 2) and show that it applies to the problem in

question (Section 3) . We note that the result of this paper remains valid for

the 2D NS system perturbed by a random force white in time and smooth

in the space variables.

NS system by a random force smooth in x and white in t (see 5 , 3 , 4 , 2 1 1 2 , 7 >.

Notation

Let H be a real Hilbert space with norm 1) . 1 ) . We shall use the following

notation:

BH(R) is the ball in H of radius R > 0 centred at zero;

B(H) is the Bore1 a-algebra in H ; P ( H ) is the family of probability measures on ( H , B ( H ) ) ; C ( H ) is the space of continuous functions f : H -+ R; Cb(H) is the space of bounded functions f C ( H ) endowed with the norm

llflloo := SUP lf(.>I. u E H

C ( H ) is the space of Lipschitz-continuous functions f E Cb(H) with norm

If f : H 4 R is a B(H)-measurable function and p E P ( H ) , then we denote

by (f, p) the integral of f over H with respect to p.

265

2. Strong law of large numbers for mixing-type Markov chains

2.1. Formulation of the result

Let (R, F, P) be a probability space and let H be a real Hilbert space with

norm 1 1 . 1 1 . We consider a family of Markov chains ( U k , P U ) in H with

transition function pk(U,r) = P,{?lk E I?}, u E H , r E B ( H ) . Recall that

the corresponding Markov semi-groups are defined by the formulas

yk : Cb(H) Cb(H), q k f ( U ) = 1 p k ( % dv)f (v),

A measure p E P ( H ) is said to be stationary for the family ( U k , P u )

if Q l p = p.

Definition 2.1. We shall say that the family ('ilk, P,) is uniformly mixing if it has a unique stationary measure p E P ( H ) and there is a continuous

function p : R+ + R+ and a sequence { Y k } of positive numbers such that,

for any f E C ( H ) and u E H, we have

l ? k f ( U ) - ( f ,P) I 5 ’-Ykp(llull)llfllL, 2 0 . (6)

The following theorem shows that “sufficiently fast” mixing combined with

a dissipation property implies an SLLN.

Theorem 2.1. Let ( U k , P,) be a uniformly mixing family of Markov chains in H such that

k=O

Suppose there is a continuous function h : R+ -+ R+ such that

pkp(u) := &p(IIukll) 5 h(llU11) for all k 2 0 , (8)

where lE, is the expectation with respect to P,. Then there exists a constant D > 0 such that for any f € C ( H ) , u E H, and S > 0 the following statements hold:

(i) There is a P,-a.s. finite random integer K(w) 2 1 depending on f ,

u, and S such that

266

(ii) For 0 < r < 36, we have

E d T 5 1 + & IIsll~~(llull)~ (10)

We note that Theorem 2.1 remains valid (with trivial modifications) for

Markov processes with continuous time. Moreover, under some additional

assumptions, one can take in (9) functionals f with polynomial growth at

infinity.

We also note that inequality (9) immediately implies the following esti-

mate:

where M ( w ) = D + 2 K ( w ) g-'.

2.2. Proof of Theorem 2.1

Let us fix an arbitrary function f E L ( H ) and set

k-1

There is no loss of generality in assuming that I l f l l m I 1 and (f, p ) = 0.

Step 1. We first show that

k2, I c~~~ f l l L~ ( l l ~ l~ )~ - l l k 2 1. (11)

Here and henceforth, we denote by Ci positive constants that do not depend

on f , u, lc and 6. Let us note that

Hence using (8) and the inequality

By the Markoa-property,

267

Substitution of this inequality into (12) results in (11) with C1 = 2C,

where C is the constant in (7).

Step 2. We now prove (9). To this end, we fix 6 E (0, f ) and set

where [a] is the integer part of a 2 0. Let us consider the events

G, = {W E R : ISk,l > K' } , ?I 2 1.

Using (11) and the Chebyshev inequality, we derive

P(Gn) 5 n 2 & I s k , 1 2 5 CZllf11Lh(11.11) n-l-'. (13)

Hence, by the Borel-Cantelli lemma, there is a Pu-a.s. finite random integer

m ( w ) 2 1 such that

ISk , (W) l I n-l for n 2 rn(w). (14)

We shall assume that m ( w ) 2 1 is the smallest integer satisfying (14). In

particular, if m ( w ) 2 2, then

Isk,(w)I > n-l for n = m ( w ) - 1. (15)

(16)

To estimate ( S k i for kn-l < k < k,, we note that

k n - k n - 1 l s k - Sk, 1 5 ( i - k) lsk, I + s k -Sk,, l 5 2 r .

1 _- Since ~ k n - kn - 1 < C3n-' and n-l 5 kn ''' = k i B f 6 , it follows from (14)

and (16) that kn-1 -

k - k n - l lSkl 5 l s k - sk,l + ISk,l 5 2 "ic,, + n-' 5 (2c3 + 1)n-l

5 (2c3 + l ) k i 5 + 6 5 (2c3 + 1)k-i+6,

where n 2 m ( w ) and kn-l < k < k,. Thus, inequality (9) holds with

Step 3. It remains to establish (10). To this end, we first note that,

for 0 < q < 0,

K(w) = [m(w)3+P].

M M

1=1 1=2

268

where we used inequalities (13), (15) and the definition of m ( w ) and G,.

Since K = [m3+O], we see that, for 0 < T < 36,

E,KT 5 E,mT(3+O) 5 1 + a-r(3+p) c4 w4) IlfllL 5 1 + 3(& w-11) IlfllL.

The proof of Theorem 2.1 is complete.

3. Applications

3.1. Dissipative PDE’s perturbed by a bounded kick force

Let H be a real Hilbert space with norm 11 . 1 1 and orthonormal base {ej}.

We consider the random dynamical system (RDS)

u k = s ( u k - 1 ) + q k , (17)

where S : H + H is a continuous operator such that S(0) = 0 and { q k }

is a sequence of i.i.d. random variables. As was explained in 8 , 9 1 1 0 , RDS of

the form (17) naturally arise in the study of dissipative PDE’s perturbed

by the random force (3) , and in this case S is the time-one shift along

trajectories of the unperturbed equation. We assume that S satisfies the

following three conditions introduced in 8,10:

(A) For any R > T > 0 there are positive constants a = a ( R , r ) < 1

and C = C(R) and an integer no = no(R, r ) 2 1 such that

IIS(u1) - S(m)11 5 C ( R ) \ l U l - U Z I I for all ul, uz E BH(R) , for u E BH(R), n 2 no IISn(u)ll 5 max{aIIu.II,r}

(B) For any compact set K c H and any bounded set B c H there

is R > 0 such that the sets A k ( K , B ) defined recursively by the

formulas do(K,B) = B and d k ( K , B ) = S ( d k - l ( K , B ) ) + K are

contained in the ball BH(R) for all k 2 0.

(C) For any R > 0 there is an integer N 2 1 such that

l l Q N ( s ( ~ 1 ) - S(uz))ll I illui - uzll for all ~ 1 , 2 1 2 E B H ( R ) ,

where Q N is the orthogonal projection onto the subspace spanned

by { e j , j 2 N + 1).

We note that the above conditions are satisfied for the resolving operators of

the 2D Navier-Stokes system and the complex Ginzburg-Landau equation.

As for the i.i.d. random variables q k , we assume that they have the form

00

q k = bjtjkej, (18) j = 1

269

where bj 2 0 are some constants such that

00

j=1

and [jk are independent scalar random variables whose distributions rj

satisfy the following condition:

(D) For any j 2 1 there is a function of bounded variation p j ( r ) such

that r J ( d r ) = pj(r)dr, where dr is the Lebesgue measure on R. Moreover, suppr j c [-1,1] and J , ,<Ep . ( r )d r > 0 for all j 2 1

and E > 0.

Let (uk,pu) be the family of Markov chains that is associated with the

RDS (17) and is parametrized by the initial condition u E H . We de-

note by Pk(U, r) the corresponding transition function and by q k and pz the Markov operators generated by Pk. It was proved in 10,11i6 that, if

conditions (A)-(D) are fulfilled and

I - 3

b j # 0 for j = l , . . . , N , (20)

where N 2 1 is sufficiently large, then the RDS (17) has a unique stationary

measure p , and for any f E C ( H ) we have

JPkf(4 - (f&)) L P(l l~ l l ) l l f l lLe-pk, k 2 11 (21)

where p : R+ 4 R+ is a continuous increasing function and ,D > 0 is a

constant not depending on f and u. Thus, the family (uk, Pu) is uniformly

mixing, and condition (7) is satisfied. We claim that (8) also holds. Indeed,

let us define the compact set

where b j 2 0 are the constants in (18). It follows from condition (D)

that the support of the distribution of r lk is contained in K. Therefore, by

assumption (B), there is a continuous increasing function R = R(d), d 2 0 , such that

pu{IIukll 5 R ( d ) } = 1 for llull 5 d, k 2 0. Hence, since p is increasing, for ))u)) _< d we obtain

';PkP(.) = ~uP( l lUk l l ) i P ( W ) ) ,

which means that (8) holds with h(d) = p(R(d)) .

for the RDS (17).

Thus, Theorem 2.1 applies, and therefore inequalities (9) and (10) hold

270

3.2. The Navier-Stokes s y s t e m perturbed by an unbounded kick force

We now consider the problem (1)-(3). It is assumed that V k are i.i.d.

random variables of the form (18), where bj 2 0 are some constants for

which (19) holds, and [ j jk are independent scalar random variables satisfying

the following condition (cf. (D)):

(D') For any j 2 1 the distribution of cjjk possesses a density p j ( r ) (with

respect to the Lebesgue measure) that is a function of bounded

variation such that

A e p 2 p j ( r ) d r 5 Q , p3( r ) > o for all r E IR,

where Q > 0 is a constant not depending on j .

The problem (1)-(3) reduces to an RDS of the form (17). Namely, let us

introduce the Hilbert space (endowed with the L2-norm)

H = u E L2(D,R2) : divu = 0, (u, " ) I a D = 0}, { where v is the unit normal to 6'D (see l5 for further details on the space H).

Let S : H 4 H be the time-one shift along trajectories of the NS system (l),

(2) with 7 z 0. Setting Uk = u ( k , z), we obtain (17) (see ',lo for details).

Let (uk,pu) be the family of Markov chains associated with the

RDS (17). As is shown in '114, if the non-degeneracy condition (20) is

satisfied for N >> 1, then the family (uk,pu) has a unique stationary mea-

sure p , and (21) holds with p(d) = C,(l+ d ) , where C1 and p are positive

constants not depending on f , u, and k . Moreover, by Theorem 1.3 in ', we have

K d l l u k l l ) I c2(1 + llull) for all k 2 0.

Thus, the conditions of Theorem 2.1 are fulfilled, and we obtain the SLLN

for solutions of the NS system (1)-(3).

References

1. J. Bricmont, A. Kupiainen, and R. Lefevere, Ergodicity of the 2D Navier- Stokes equations with random forcing, Comm. Math. Phys. 224 (2001), 65-

81. 2. J. Bricmont, A. Kupiainen, and R. Lefevere, Exponential mixing for the

2D stochastic Navier-Stokes dynamics, Comm. Muth. Phys. 230 (2002), no. 1, 87-132.

271

3. W. E, J. C. Mattingly, and Ya. G. Sinai, Gibbsian dynamics and ergodicity for

the stochastically forced Navier-Stokes equation, Comm. Math. Phys. 224

4. J.-P. Eckmann and M. Hairer, Uniqueness of the invariant measure for a

stochastic PDE driven by degenerate noise, Comm. Math. Phys. 219 (2001),

5. F. Flandoli and B. Maslowski, Ergodicity of the 2-D Navier-Stokes equation

under random perturbations, Comm. Math. Phys. 172 (1995), 119-141.

6. S. Kuksin, On exponential convergence to a stationary measure for nonlin-

ear PDE’s, perturbed by random kick-forces, and the turbulence-limit, The M. I. Vishik Moscow PDE seminar, AMS Translations, 2002.

7. S. Kuksin, Ergodic theorems for 2D statistical hydrodynamics, Rev. Math. Physics 14 (2002), no. 6, 585-600.

8. S. Kuksin and A. Shirikyan, Stochastic dissipative PDE’s and Gibbs mea-

sures, Comm. Math. Phys. 213 (2000), 291-330. 9. S. Kuksin and A. Shirikyan, Ergodicity for the randomly forced 2D Navier-

Stokes equations, Math. Phys. Anal. Geom. 4 (2001), no. 2, 147-195. 10. S. Kuksin and A. Shirikyan, A coupling approach to randomly forced non-

linear PDE’s. I, Comm. Math. Phys. 221 (2001), no. 2, 351-366.

11. S. Kuksin, A. Piatnitski and A. Shirikyan, A coupling approach to randomly

forced nonlinear PDE’s. 11, Comm. Math. Phys. 230 (2002), no. 1, 81-85. 12. S. Kuksin and A. Shirikyan, Coupling approach to white-forced nonlinear

PDE’s, J. Math. Pures Appl. 81 (2002), 567-602. 13. N. Masmoudi and L.-S. Young, Ergodic theory of infinite dimensional systems

with applications to dissipative parabolic PDEs, Comm. Math. Phys. 227

14. A. Shirikyan, Exponential mixing for 2D Navier-Stokes equations perturbed by an unbounded noise, J . Math. Fluid Mechanics, to appear.

15. R. Temam, Nauier-Stokes Equations. Theory and Numerical Analysis, North-

Holland, Amsterdam-New York-Oxford, 1977.

(2001), 83-106.

523-565.

(2002), 461-481.

COMPREHENSIVE MODELS FOR WELLS

MARIAN SLODICKA

Department of Mathematical Analysis, Ghent University,

Galglaan 2, B-9000 Ghent, Belgium

E-mail: [email protected]. be, web page: http://cage.rug.ac. be/-ms

The aim of this paper is to present various mathematical models for wells. Special

attention is paid to a non-standard description using nonlocal boundary conditions (BCs). We also develop numerical algorithms t o handle nonlocal BCs. The choice of the appropriate model depends, of course, on concrete situation.

1. Introduction

Many ground-water hydrologists are interested in the determination of

water-table elevations resulting from inputs and outputs such as natural

replenishment, artificial recharges and pumping. Some of them are inter-

ested in the general flow pattern in the whole aquifer, other study the details

in a vicinity of a well. Here, wells represent inputs or outputs, which affect

flow in a soil matrix. These sinks/sources are concentrated, i.e., their di-

ameters are relatively small compared with the whole aquifer. This feature

makes the modeling more complicated. Of course, wells are not only used

in the ground-water hydrology, but also by oil extraction or soil venting,

which is used for soil remediation (for cleaning of unsaturated zone from

chlorinated hydrocarbons or other volatile organic compounds). The main

difference among all these applications are (a) the substance (water, oil,

gas) for which wells are used, (b) different geological conditions.

2. Point sources

Let us consider the steady-state case with a single extraction well with an

infinitely small diameter located at the origin. We suppose that our do-

main is infinite in all directions and we consider a homogeneous unconfined

aquifer with the conductivity KO. Then a fundamental solution of

-v . (KOVUO) = ss

272

273

(classical outside the origin) for a single point sink is given by

lnlxl in 2D

in 3D. 25KO (1)

uo(x) = rL 4.rrKo I 2 I This solution so far has not included any realistic BCs and it generates

drawdownsa everywhere. Further, the seepage face at the well is omitted

because of a negligible well radius. This is not realistic for a small vicinity

of the wellbore.

Method of images is a simple technique to create some basic BCs.

Adding imaginary wells to the real point sink at strategic locations allows

to generate infinitely long straight equipotentials or no-flow boundaries (cf.

' i 2 ) . For the analytical description of a single-phase flow caused by a single

extraction well for a perfectly layered subsurface we refer the reader to '. Bounded domains. We consider a bounded domain R E Co?l in IRN

( N = 2,3) with boundary r = I'D U r N , where J?D has a positive measure.

We study

-V . (KVu) = SS in R u=O o n r o

K V u . v = 0 on rN. Problem (2) is linear, but the right-hand side does not belong to the H-l(R)

(dual space to H1(R)), thus we cannot directly apply the theory of linear

elliptic equations. When the conductivity K is Holder continuous (with the

coefficient a, Q > 0 in 2D, a > in 3D) near the well, then one can use

the method of subtraction of singularities. Then ( 2 ) is rewritten in terms

of an new unknown function ii = u - U O , uo being defined by (1). The

reformulated problem will contain the right-hand side from Lz(R), due to

the Holder continuity of K near the origin.

The case when the conductivity is not Holder continuous is more diffi-

cult. Such a situation can appear, e.g., when a well is located at an interface

of two different layers, or there is a rock at the well tube. In such situa-

tion, we cannot suppose the regularity of K , thus the right-haad side of

the modified problem (after subtraction of singularities) will not belong to

the Lebesgue space. Nevertheless, one can overcome this using the so-called

very weak solution as it has been proposed in '. Here, the solution is defined

aPumping from a phreatic aquifer removes water from the void space leaving there a

certain quantity of water which is held against gravity. As a result, the watertable at each point is lowered with respect to its initial position by a vertical distance called

drawdown.

274

in terms of an adjoint problem. The author also describes the numerical

schemes based on finite elements.

3. Wells with a non-negligible radius

Method of images helps in some cases to model BCs. For more complicated

but also more realistic situations we have to use variational calculus, where

the differential equation can be equipped by various types of BCs. It always

depends on the concrete case which BC has to be chosenb. We briefly

discuss typical cases and later we focus our attention to nonlocal conditions.

Pressure Condition. Pressure is prescribed on the well, i.e., we

speak about a Dirichlet type condition. This is frequently used for

passive wells by soil venting. Here, clean air enters the contam-

inated domain. One can suppose that a constant atmospherical

pressure is given on passive wells.

Flux Condition. Flux through the well boundary is prescribed

pointwise, i.e., we consider a Neumann type condition. This case is

doubtful in many real cases, because the flux distribution is com-

pletely unknown. This cannot be used for inhomogeneous vicinity

of the well or in the case when the well is located near a boundary

(e.g., lake, river, . . . ). Signorini Condition. When a well diameter cannot be neglected,

then a storage capacity of the well tube has to be taken into account

(see 6,7,8). Then one part of the probe discharge comes from the soil

matrix and the other one from the well tube. By this situation the

waterhead inside and outside the extraction tube can be different,

i.e., the seepage face can appear (cf. Figure 1).

The length of the seepage face depends on the well diameter.

For a large well radius one can observe a very small seepage face.

This can be explained by a large storativity of the well tube.

This model can be mathematically described as (cf. 9) : Find p such that

bModels describing air-, water- or oil-pumping wells differ from each other.

275

R rN (impervious layer)

Figure 1. A vertical cross section through a well

with the initial and BCs

u(0) = d o in 52 q(t) . u = 0 on r N

u(t) = do on r D (4) p(t) 5 0, q(t) . v 2 0, p(t)q(t) . v = 0 for z 2 w(t)

p(t) = w(t) - z for z < w(t)

Continuity equation for water inside the well tube is

nR2&W(t> = 2nR q . u - Q , (5) I D where 0 denotes the saturation, K conductivity, p pressure, q the

mass flow, R the well radius, Q the discharge of the well. D is the

thickness of the aquifer. The Neumann, Dirichlet and Signorini

boundaries (see lo) are denoted by r N , r D , rs, respectively.

(D) Discharge Condition. It is assumed that a constant but un-

known pressure builds up on the well boundary such that the pre-

scribed discharge is obtained. Such a type of BC can be used for

active wells by soil venting. Here, the total discharge of the well is

given and one can assume that the pressure along the well tube is

constant.

We demonstrate this on the following study case:

v . (-KVU) = f in R

U = gD On r D

(6) -KVU, Y = gN on r N

u = unknown constant on r, G(u) = L, ( -KVu) . v = s E R

276

Here, apart from a standard Dirichlet BC on r D and Neumann

BC on r N , there is a nonlocal BC on F,, where the total flux

through r, (a well) is given along with a condition that the solution

(pressure) is constant but unknown on r,. One can define a suitable variational framework for (6) by in-

troducing the subspace V of H1(R)

v = {'p E H'(R); p = o on r D , 'p = const on rn}. (7)

Adopting standard assumptions on the data-functions appearing

in (6), one can show the well-posedness of a weak solution.

The choice (7) of the test space V is not standard. Therefore,

there could be problems by a space discretization, due to the non-

local BC on I?,. One can show (cf. 11) that the solution can be

obtained via a linear combination of solutions to the following two

BVPs with standard BCs:

V . ( -KVV) = f in R

( 8 ) 21 = g D on r D

v = o on r, - K V V . V = g N On r N

and

V . ( -KVz) = 0 in R

(9) z = o on r D

-KVz .V = 0 on r N

z = 1 on rn.

The problems (8) and (9) are well-posed. Now, applying the prin-

ciple of linear superposition, we see that u, = v+az for any Q E R

solves

V . (-KVU,) = f in R

(10) '& = g D on r D

u, = Q on rn. - K V U , ' U = g N On r N

The total flux through F n is a linear operator, thus G(u,) = G(v)+

aG(z). Setting G(u,) = s we get

s - G(v) a =

G(z ) '

Thus, taking this value of Q for u,, we see that u, solves (6).

277

(R) Robin Condition. This type of BC is developed for a well in a

confined aquifer. It also represents a kind of nonlocal BC, where the

pressure along the well-boundary is assumed to be constant (as by

the discharge condition), but the total discharge is also unknown.

Its dependence on the pressure inside the well is known. Next

section is devoted to the study of this case.

4. Robin type boundary condition

An aquifer that is sandwiched between two impermeable layers is called a

confined aquifer if it is totally saturated from top to bottom. If a recharge

area for the aquifer is located at a higher elevation that the top of the

aquifer, and a well is drilled into the aquifer, the water will rise above

the top of the well without additional forces. Such an aquifer is known as

artesian. Similar situation can appear by oil pumping. The oil is usually

stored in a large deepness under the soil surface. In fact, it is a mixture of

oil and gas.

By standard pumping one creates an under-pressure at a well and in this

way oil or water come out from the soil. This situation can be described in

various ways, e.g., by a Dirichlet or by a discharge condition. The question

is how to describe a flowing well, see Figure 2. Here, the liquid is flowing

t

Figure 2. Cross-section through a well

out without pumping. The pressure at the bottom of the well is unknown.

It varies in time depending on situation in the aquifer. One can measure

the total flux through the well tube, but we cannot expect that it will

remain constant. The total flux through the well clearly depends on the

pressure at the bottom of the well. This can be taken as a space constant

(along the well boundary at the bottom). In fact, this value can change in

time and it is a priori unknown. But the dependence of the total flux on

278

the well-pressure can be known. It is a nonnegative function, monotonically

increasing, zero up to a given point PO. Roughly speaking, po is the minimal

value of pressure which has to be achieved in order to push the fluid up to

the soil surface.

Therefore, the situation at the bottom of the well rn (suction area) can

be described as follows

p = unknown space constant r

where q is the flux vector and u is for the outer normal vector at rn. The derivation of a flow equation for water in a confined aquifer can be

found in 13 . We assume that the flow is governed by Darcy’s law. Thus,

the flow equation for a saturated flow reads as

where S is the storativity, K is the conductivity tensor, p stands for the

water density, f describes possible spatially distributed sources, and g de-

notes the gravitation vector. If the confined aquifer is located horizontally,

then (12) will be independent of the gravitation vector.

To avoid un-necessary technical details we study the following problem

8tP = AP + f in R

P = P D on r D

vp .v=o on rN

p = unknown space constant on r, (13)

in R,

where R c RN for N 2 2 is a bounded domain with a Lipschitz continuous

boundary I?. This is split into three mutually disjoint parts r D , rN and I?,,

which describe the Dirichlet, Neumann and nonlocal boundary part. We

assume that all three parts have a positive measure.

For the function g describing the total flux through the well we adopt

the following assumptions ( L is the Lipschitz constant of 9)

279

4.1. Variational formulation, well-posedness

We denote (w, z),,,, = sM wz, and the corresponding norm I I w ~ ~ ~ , ~ =

d m . Let the Hilbert space H l ( 0 ) be equipped with the norm

The symbol lrnl denotes the measure of the boundary part rn. Taking into

account (14) and (7), one can easily deduce the well-posedness of a weak

solution p to the IBVP 1.

Throughout the paper we tacitly assume that the data-functions ap-

pearing in the problem setting are sufficiently smooth.

In that follows C, E and C, denote generic positive constants depending

only on the data, where E is a small one and C, is a large one.

4.2. Numerical scheme

We divide the time interval [O,T] into n E N equidistant subintervals

( tz- l r tz ) for t, = ir, where T = E . Applying the discretization in time

(Backward Euler method) we get

1 (17) (Szz, 4 0 + P z z , VF), + - (dzz), P)r, = ( f z , d n

Irn I for i = 1,. . ., n, Sz, = - and the starting datum zo = pa. The

well-posedness of (17) is guaranteed by the theory of monotone operators.

One can use an iteration scheme by computations at each time step to

avoid the nonlinearity. There are more possibilities. Newton like iteration

Problem 1. Find a esuple (u,a) such that

be defined by(7), Now, we give the variational formulation of the

280

schemes need to start close to the exact solution. This implicitly means that

the time step T is small. One can use the following linearization scheme,

which is robust and converges for any initial datum pi,^ ( k E N, p E V )

We define p , , ~ = pi-]. This choice can diminish the number of relaxation

iterations. Similar relaxation schemes have also been used in 1 2 1 1 4 .

The problem (18) is linear and well-posed. This follows from the V- ellipticity of the left-hand side and from the Lax-Milgramm lemma.

For a given time-index i we perform relaxation iterations for k = 1,. . . , ki,maa: until the stopping criterion

IlPi,k - P i , k - l \ I o , r , 5 7' (19)

is achieved for some 77 > 0. Then we set p i = p i , k t ,maz and we switch to the

next time step.

Now, we introduce a sequence of auxiliary nonlinear elliptic BVPs,

which are defined in terms of pi, for i = 1, . . . , n in the following way

The existence and uniqueness of a weak solution ui E V for i = 1,. . . , n follows from the theory of monotone operators. For convenience we define

uo = Po. We show that relaxation iterations pi& converge towards ui as k 4 00

in appropriate function spaces. Let us note that ui differs from z i because

we stop the iteration process after a finite number of steps.

Lemma 4.1. There exist positive constants CO and TO such that for any r 1. TO and for all k E N the following estimates hold:

Proof. (2) We subtract (20) from (18) and set p = p i & - u i . We get

28 1

for P(s ) := g(s) - Ls. Using the Cauchy-Schwarz and Young's inequalities

to the right-hand side, and the Lipschitz continuity of the function p, we

deduce

Thus

2 lrnl 2 2 - IIpi,k - uiI10,n +2 IrnI I I V ( P ~ , ~ - ui)llo,n + L I I P ~ , ~ - uiIIi,r,

(21) 2 T

5 L l l p i , k - l - uillo,r,, *

The generalized Friedrichs inequality implies that the following relation is

valid for any w E H1(R) (due to the fact that > 0)

ll'Ulll0,n 5 c ll~'UlIl0,n~ (22)

(23)

(24)

The combination of the trace inequality and (22) yields for some Co > 0

2 2 CO I I P ~ , ~ - utIIo,r, 5 co I I P ~ , ~ - uiIIo,r i 2 IrnI I I V ( P ~ , ~ - ui)11;,n.

Now, we deduce from (21) and (23)

2 2 ( L + CO) i h , k - uiIlo,r, 5 L lIpi,k-l - ui))o,r, .

This iterative relation gives rise to the following estimate

(ii) The desired result is a consequence of the part (i) and (21).

We point out that the choice of the time step r is free. Next, the

relaxation iterations can start from any starting datum from H1(R) and

they converge in the H1(R)-norm to a function u, E V , which is defined

by (20). Please note that u, = p( t , ) . We stop the relaxation iterations if

l l p z , k t , m a z - Pz ,k t ,maz- i /lo,r, I 7'. Moreover, we know from (24) that

I h , k - uzllo,r, I 4 l l ~ z , k - l - uzllo,r,

for some 0 < q < 1 and for any k. Thus

\ \ p z , k t , m a z - 1 - uzl\o,r,, 5 I l p z , k z , m a z - l - ~ z , k ~ , m a z \\o,r, + I I P ~ , ~ ~ , ~ ~ ~ - uzllo,rn

5 7' + 4 Ilp, ,kz,maz-l - utllo,rn . The last inequality yields

282

This estimate together with (21) for k = ki,maz imply

llPd%maz - uillo,R 5 [ lPi,kt,maz-l - % l l o , r , I CTV,

IIv(Pi,k%,maz - ui)llo,n I c ~~P i , kz ,mas- l - UiII o,r, I CTV1

(25)

(26)

and

which are valid for any i = 1,. . . , n. Now, we derive suitable a priori estimates for ui.

Lemma 4.2. Le t q 2 1. We assume (19) for all i = 1,. . . , n. T h e n

i=l i=l

take place for all m = 1,. . . , n

Now, we set 'p = uir and sum the relation up for i = 1,. . . , m. W-e get

The lower bound for the left-hand side is (the last term is nonnegative)

) m m

1 ( ~ ~ ~ m l ~ ~ , n + c I I U ~ - ui-ll/i,n + c I I ~ ~ ~ I I : , ~ T . i= 1 i=l

Applying the Cauchy and Young inequalities and (25) to the right-hand

side we easily get the upper bound

2

283

The rest is a consequence of Gronwall's lemma.

for i = 1,. . . ,m we get

(ii) We start from the relation (27). Setting cp = 6uir and summing up

Now, we introduce the convex function Q g ( z ) = g(s) ds. According to

the properties of g we have

which holds for any z1, zz E R. Moreover, one can prove

Using these properties we can write

Thus, the lower bound for the left-hand side of (28) is

Applying the Cauchy and Young inequalities and (25) to the right-hand

side of (28) we easily get the upper bound

m m

C& (1 + 3q-1) ) + €C ll~uzll:,n 7 5 C& + ll~~ill02,n 7, i=l i=l

which is valid for any E > 0. Therefore, we have

m. m. m

i=l i=l i=l

Fixing a sufficiently small positive E, we conclude the proof. CI

284

4.3. Convergence of the scheme

Now, let us introduce the following piecewise linear in time function

and the step function En

- un(0) = uo, z,(t) = uil for t E (t i- l,t i ].

Exactly in the same way we also define the step functions ji, and7, as well

as the piecewise linear function p,. Using this notation we rewrite (27) into

Lemma 4.3. Let the assumptions of Lemma 4.2 be fulfilled. Moreover we assume that 7 2 g. Then there exists a positive number C such that

Proof. We subtract (16) from (29), set 'p = Ti, - p , integrate the equality

over the time interval (0, t ) and get

The last term on the left-hand side is nonnegative due to the monotonicity

of g. Further, Lemma 4.2(ii) implies

285

Using the Cauchy-Schwarz inequality and (25) we deduce

i”

The Cauchy-Schwarz inequality and &u,, &p E L2 ( (0 , T ) , L2(R)) imply

The last term of (30) containing the function f can be estimated analo-

gously taking into account the properties of f . Therefore, we can write

t t

l lUn ( t ) - P( t ) l l&2 + J’ llV(% - P)ll:,sl I c (T + IIun - PIl i$) 0

Applying the Gronwall argument we arrive at

from which we easily conclude the proof.

Now, we are in a position to derive the error estimates for p ,

Theorem 4.1. Le t the assumpt ions of L e m m a 4.3 be satisfied. T h e n there exists a posit ive number C such tha t

Proof. The desired result is a consequence of Lemma 4.3, (25) and (26)u

The error estimates from Theorem 4.1 fully correspond to the known

results for semilinear problems starting from po E H1(R). When starting

from more regular initial datum po E H2(R), assuming that po is compatible

with BCs and taking 7 2 2 , one can get the convergence rate 0 (r2).

286

References

1. H.M. Haitjema. Analytic Element Modeling of Groundwater Flow. Academic

Press, Inc, New York, 1995. 2. D.J. Wilson. Modeling of Insitu Techniques for Treatment of Contaminated

Soils: Soil Vapor Extraction, Sparging, and Bzoventing. Technomic Publish- ing Company, Inc., Lancester-Basel, 1995.

3. R. Nieuwenhuizen, W. Zijl, and Van Veldhuizen. Flow pattern analysis for a

well defined by point sinks. Bansport Porous Med., 21:209-223, 1995. 4. M. SlodiEka. Finite elements in modeling of flow in porous media: How to

describe wells. Acta Mathematica Universitatis Comenianae, LXVII( 1):197-

214, 1998. 5 . M. SlodiEka. Mathematical treatment of point sources in a flow through

porous media governed by Darcy's law. Tramport Porous Med., 28(1):51-

67, 1997. 6. I.S. Papadopulos and H.H. Cooper Jr. Drawdown in a well of large diameter.

Water Resources Research, 3(1):241-244, 1967.

7. S.P. Neumann and P.A. Witherspoon. Analysis of nonsteady flow with a free surface using the finite element method. Water Resources Research, 7(3):611-

623, 1971. 8. E.A. Sudicky, A.J.A. Unger, and S. Lacombe. A noniterative technique for the

direct implementation of well bore boundary conditions in three dimensional

heterogeneous formations. Water Resources Research, 31 (2) :411-415, 1995.

9. S. Schumacher, M. Sloditka, and U. Jaekel. Well modeling and estimation of hydraulic parameters. Computat. Geosci., 1 (3,4):317-331, 1997.

10. C. Baiocchi and A. Capelo. Variational and Quasivariational Inequalities. John Wiley and Sons, Chichester . New York, 1984.

11. M. SlodiEka and R. Van Keer. A nonlinear elliptic equation with a nonlocal boundary condition solved by linearization. International Journal of Applied Mathematics, 6(1):1-22, 2001.

12. M. SlodiEka. Error estimates of an efficient linearization scheme for a nonlinear elliptic problem with a nonlocal boundary condition. M 2 A N , Math. Model. Numer. Anal., 35(4):691-711, 2001.

13. J.W. Delleur (Editor-in Chief). The handbook of groundwater engineering. Springer, Heidelberg, 1998.

14. M. SlodiEka. A robust and efficient linearization scheme for doubly nonlinear and degenerate parabolic problems arising in flow in porous media. SIAM J . Sci. Comput., 23(5):1593-1614, 2002.

STOCHASTIC CASCADES APPLIED TO THE NAVIER-STOKES EQUATIONS

ENRIQUE THOMANN AND MINA OSSIANDER

Department of Mathematics Oregon State University

Cornallis, Or 97331 E-mail: thomann@math. orst. edu

In this paper a representation of the Fourier transform of solutions to the Navier-

Stokes Equations are obtained in terms of a stochastic recursion generated by

a branching random walk. The notion of majorizing kernel is introduced and

used to study regularity and existence of solutions of the Navier-Stokes equations.

Similar representation of solutions to other equations are also discussed and its

corresponding multiplicative recursion in the physical space are presented. This is

joint work with R. Bhattacharya, L. Chen, S. Dobson, R. Guenther, C. Orurn and E . Way mire.

1. Introduction

The study of properties of solutions of the Navier Stokes equations remains

one of the most notable problems in mathematics. While a substantial body

of literature is available on this subject, see e.g. Ternamlo and Galdi4, the

recent work of LeJan and Sznitman' has opened new opportunities for anal-

ysis and a novel application of branching processes. Indeed, in their work,

LeJan and Sznitman obtained a representation of the Fourier transform

of the solution of the Navier-Stokes equations as an expected value of a

multiplicative functional defined on a branching random walk. This repre-

sentation uses exponential random variables with means depending on wave

number in a way naturally related to the equation. The distribution of the

offsprings at each branching is on the other hand determined by a kernel

conveniently introduced to use the quadratic nonlinearity of the equation.

Three basic extensions of this approach are presented in this paper.

First, the notion of majorizing kernels is introduced in order to analyze and

control the regularity of solutions of the Navier-Stokes equations. While

the solutions determined in the work of LeJan and Sznitman have to be

understood in a weak sense, we show that it is possible to use an appropriate

majorizing kernel to maintain or improve the regularity of the solutions.

287

288

Second, we remove the restriction to three space dimensions present in

the work of LeJan and Sznitman. Using the example of the Kolmogorov

Petrovskii and Piskunov (KPP) equation holding in one space dimension, a

representation for the Fourier transform of its solution is obtained also as an

expected value of a multiplicative functional. From the work of McKeang

it is known that such a representation is available in physical space. In this

paper we establish a direct relation between both approaches.

Third, solutions to a linearly damped Burgers equations as an expected

value of a multiplicative functional defined on a branching process in physi-

cal space are obtained. An outstanding problem is to obtain representations

in physical space for the Navier-Stokes equations. While no solution for this

problem is suggested in this paper, recent work of E Frolova contains some

related ideas.

While not developed in this paper, it should be noted that a similar

approach applies to other evolution equations including linear parabolic

equations such as Schrodinger equation with a potential that is the Fourier

transform of a complex measure and to evolutions equations that involve a

fractional power of the Laplace operator. Further details and examples of

the methods presented in this paper can be found in Bhattacharya et a1.l

and Chen et aL3.

The organization of the paper follows the three extensions described

above. In the next section, the stochastic branching process and multi-

plicative functional corresponding to the Navier-Stokes equations are intro-

duced. Also in this section, the notion and basic properties of majorizing

kernels are developed. Finally, a correspondence with more standard Picard

iteration schemes is made. In section 3, the example of the KPP equations

is considered both in physical and Fourier space. The relations between the

corresponding representations is also described. Section 4 includes a treat-

ment of the multiplicative functional and corresponding branching process

for the damped Burgers equation as well as concluding remarks.

2. Applications to the Navier-Stokes Equations

Recall that the 3d incompressible Navier-Stokes equation can be expressed

in the Fourier domain as follows:

289

where for complex vectors w, z

(1) E w CQ z = - i (q . z ) I I c ~ w , ec = M I

v > 0 is the viscosity parameter, and IIELW is the projection of w orthog-

onal to < and ij is the Fourier transform of known exterior body forces.

For < # 0, LeJan and Sznitman' rescale the equation (FNS) to normal-

ize the integrating factor e-'lc12s to the exponential probability density

v)<)2e--vlf12s. The resulting equation is precisely the form for a branching

random walk recursion for G(E, t)/vlE12 for a transition probability kernel

naturally constrained by normalization requirements to dimensions d 2 3. To extend this approach introduce non-negative measurable functions

h such that

h * h(E) I Bl<lh(E), E # 0, B > 0. (2)

Refer to such a function h as a majorizing kern.el with, constant B or in the

case B = 1 as a standard majorizing kernel. Note that if h is a majorizing

kernel with constant B then 5 is a standard majorizing kernel. Also, if

h( [ ) is a majorizing kernel then so is ce".ch(<) for arbitrary fixed vector

a and positive scalar c. To avoid unnecessary technicalities regarding their

supports, attention is restricted in this paper to positive majorizing kernels

h( ( ) defined for E # 0. Such kernels are said to be fully supported. Examples

of majorizing kernels are given by the following proposition.

Proposition 2.1. For 0 # E E R3, 0 5 p 5 1, CY > 0,

defines a majorizing kernel.

Given a majorizing kernel we consider the Fourier transformed equation

(FNS) rescaled by factors of the form &, for < # 0. Namely, we consider

where

290

Notice that for each fixed < with h * h(<) # 0, the convolution h * h(<) simply normalizes the product h(ql)h(772) to be a probability kernel on

the set 71 + 772 = (. In particular, while a majorizing kernel need not be

integrable, i t is required that the convolution h * h(<) be finite for each

< E R3\{O}. It is then possible to show the existence of globally defined

solutions of the (FNS) equations, the regularity of which depends on the

particular majorizing kernel applied as follows.

Introduce the Banach space Fh,T as the completion of the set

in the given norm. In the case h = ho this is the Besov type space intro-

duced by Cannone and Planchon2.

Note that considering h = hz with p < 1, from Proposition 2.1, the

Banach space corresponding to such a majorizing kernel contains initial

data 210 which are infinitely differentiable functions of compact support.

One of the main results obtained using majorizing kernels is the follow-

ing theorem.

Theorem 2.1. Let h(J) be a standard majorizing kernel. Fix T > 0 and suppose that ICo(<)I 5 (~ ‘%)~ ;h ( ( ) , and I@([,t)l 5 (~%i)~(3~1<1~h(<), ( # 0,O 5 t 5 T. Then (FNS) has a unique solution in the ball of radius R = (&)’v/2 centered at 0 in the space Fh,T.

Theorem 2.1 illustrates how majorizing kernels can be used to maintain

regularity of the solutions. For example, if the majorizing kernel h = h g ) for p > 0 is being used, the solution remains infinitely differentiable. A further example is that it is possible to obtain spatial analyticity of the

solutions, for t > 0, provided the initial data satisfies for some majorizing

kernel h and appropriate constants A, C independent of v

ICo(<)l 5 Ch(<)ve-A’v.

In the case that h(( ) = 1/1[12 G ho((), this result was obtained by LemariQ-

Rieusset’. However, the following proposition shows that there are majoriz-

ing kernels that exhibit a stronger singularity at the origin and decay slower

at infinity than ho(<).

Proposition 2.2. For < E R3 such that C,”=, S t j , o < 2 } , let

291

T h e n H3 i s a major iz ing kernel, and H3 = G(</l(l)/l<12 wi th

lim G(w) = 00 w-iv

where v is an e lement of the standard basis of R3

Using the majorizing kernel obtained in Proposition 2.2 with Theorem

2.2 exhibits solutions of the Navier-Stokes equations with initial data whose

Fourier transform blows up at the origin at a faster rate than l/l<12. See

Bhattacharya et a1 for details.

The solution obtained in Theorem 2.1 can be obtained as an expected

value of a multiplicative functional defined on a branching process. The

following subsection provides the main points of this idea.

2.1. Stochastic Recursion

Denote by V the vertex set of a complete binary tree rooted at 8 coded as

v = u,o~,~{i , 2}j = {el < 1 >, < 2 >, < 11 >, . . .}, (6)

where {1,2}O = {O}. Also let aV = n ~ o { l , 2) = (1, 2}N.

A stochastic model consistent with (3) is obtained by consideration of

a multitype branching random walk of nonzero Fourier wavenumbers <, thought of as particle types, as follows: A particle of type < # 0 initially

at the root O holds for an exponentially distributed length of time So with

holding time parameter A(<) = V I J ~ ~ ; i.e. ESo = &. When this exponen-

tial clock rings, a coin KO is tossed and either with probability the event

[ K O = 01 occurs and the particle is terminated, or with probability f one

has [KO = 11 and the particle is replaced by two offspring particles of types

71, r/2 selected from the set 71 + r/z = < according to the probability kernel

This process is repeated independently for the particle types 71,772 rooted

at the vertices < 1 >, < 2 >, respectively.

Now, recalling (4), for given initial data and forcings xo(<) and p(<, t ) , ( # 0, t 2 0, define a functional X(O, t ) by the following stochastic recursion:

xo (Ee) , if so 2 t cp(t - S O , < ) ! if SO < t , KO = 0, { m((o)X(< 1 >,t - So) @,cs X(< 2 >, t - So) else

X(O,t) =

where 71 + 7 2 = are distributed according to K c s ( d q l , d 7 2 ) and

r<1>, 7<2> are the trees defined by re-rooting at the vertices < 1 >, < 2 > of

292

new types 71, 7 2 , respectively. Standard results on critical branching show

that this recursion will terminate in finite time with probability one. In

particular there can be no explosion of the branching random walk in finite

time. Thus X(0, t ) is a finite random variable for each time t and wavenum-

ber 5. Indeed, for the evaluation of the stochastic functional X(e,t), for a

given (0 = E , it is useful to identify a particular tree structure intrinsic to

the stochastic branching model. Let

where

Ivl-1

j =O

B~ = 0, B, = C s,,~, e # v E v. (9)

Then the stochastic functional X(0, t ) on a particular tree is obtained as a

product of m’s, X O ’ S , and cp’s appropriately evaluated at the nodes of this

tree.

Moreover, decomposing the functional X in terms of the events [SO 2 t ] , [So < t , no = 01 and [SO < t , no = 11, one may check the following

consequence of the strong Markov property.

Theorem 2.2. If EIX(Q, t)I < co, for each E # 0, then x(<, t ) = EX(0, t ) solves (3).

Theorem 2.1 is obtained from this by simply noting that if m(() 5 1,

Icp(E,t)l 5 1 and Ixo(J)I 5 1, then the finite number of factors appear-

ing in the product functional [I(@, t)l are bounded by 1, and consequently

Ix(<, t)l <_ 1 for all < and t . In this sense the notion of majorizing kernel as

described above simply exploits sufficient bounds on the stochastic times

functional X(e, t ) . However, the essential property of the majorizing kernel

is the finiteness of the convolution h*h(<) for normalization to a probability.

In particular, this suggests that significantly sharper results are possible by

so relaxing (2) and more detailed analysis of the stochastic structure of the

branching random walk.

2.2. Successive Iterations of a Contraction Map

It is possible to relate the stochastic theory presented so far with an iterative

method based on Picard iterations such as the one considered by Kato

For this, write (FNS) as

293

where

Now, consider the iteration

Gn+l(t,t) = Q[an;.clo,Gl(E,t), (12)

where iil(t, t ) = Q[u(O); GO, GI((, t ) , for u(')([, t ) = e-'IE12tii~(<). Note the

particular initialization of the iteration, which is the one utilized by Kato',

and it is the appropriate one to relate the iteration to the stochastic process

introduced in subsection 2.1

To establish this relation, define the replacement time of a vertex v as

IVI

k=O

and let

A,(@, t ) = [lvl 5 n Vv E ~ ( t ) ] n [R, > t Vv E {u E 7e( t ) : 1 ~ 1 = n}],

with l [n ; 19, t] being the indicator of the event A,(O, t ) .

Prop 2.1. Let

u k ( t , t ) = h ( E ) X k ( t , t )

= h ( < ) E d l [ k E , tIX(I9, 4 ) and denote by &(t, t ) the Fourier transform of the kth iterate of the itera-

tion scheme defined in (12). Then W k ( < , t ) = file([, t ) .

A consequence of the proposition is that the convergence of the iteration

scheme (12) and the existence of the expected value in Theorem 2.2 are

essentially equivalent.

3. Application to KPP Equations

Recall that from the work of McKeang the solution to the initial value

problem

is given by

r Nt 1

294

where Bv(t) is the location of a branching Brownian motion defined re-

cursively as follows. Let Xe(t), denote the location at time t of a stan-

dard Brownian motion, and let Te be an exponential random variable

with parameter 1 independent of this Brownian motion. If To 2 t , set

Bs(t) = z + X,(t). Else, start two independent Brownian paths X<1> and

X<z> each with its own independent exponential time T<1> and T<z> respectively and iterate on this process. Let

Ivl-1 IVI

j = O j = O

ye(z, t ) = {V E V : Rv = C Tvlj < t I C Tvlj}

Then for v E ye(z, t ) ,

Ivl-1

j =O

Bv(t) = z + c XVlj(TVlj) + Xv(t - RV).

Finally, let M(ye(2 , t ) ) = max{ IvI : v E ye} and let 1[k; z, t] the indicator

of the event [M(yo(<, t ) ) 5 k]. Let

~ ( z , t ) = Ex [n [uo(Bv(t))l l[k : z, ti] . (15)

It follows that

u(z , t ) = lim uk(z, t ) k+cc

On the other hand, consideration of the Fourier transform of the KPP

equation leads after a simple integration to the integral equation

where

1 A(<) = 1 + ~ 1 < 1 2 .

Proceeding as done with the Navier Stokes equations, scale (16) by

l /h(<) to obtain

where

295

Define the recursive functional

where < 1 >, < 2 > are re-rooted trees a t vertices of types <<I>, Ec2> re-

spectively and the distribution of types is given on v~ + q 2 = & by

Note that the only difference with the recursive functional corresponding

to the Navier-Stokes equations is the node operation which for the KPP

equations is standard multiplication.

Using the strong Markov property it follows that the solution of (16) is

given by

fi(<, t ) h(t)EIX(TB(t, t ) ) ] .

provided the expected value is finite. The analogue of a majorizing kernel

for the KPP equation is given by

1 ( h * h)(E) I B(1+ 21<I2)h(<).

It is simple to check that Cauchy densities,

are majorizing kernels.

representation of the solutions

is furnished by the following proposition which is identical to proposition

2.1 Recall that A,(<, t ) defined above Proposition 2.1 denotes the event

that all vertex on a tree rooted at E are of length less than or equal to n and those vertex of exactly length n are replaced after time t. As in that

proposition, let l [ k ; E , t] denote the indicator of A k ( [ , t ) .

Finally, the relation with the McKean

denote the Fouriuer transform of the function

296

4. Damped Burgers Equation, Some Open Problems

An application of the Duhamel principle shows that the solution of the

damped Burger equation

(17) au 1 1 au2

at 2 a x 2 2 ax -- +----‘11, _ - -

with initial data u(x, 0) = U O ( X ) , satisfies the integral equation

where

It then follows, using integration by parts on the second integral, and

that

J

1 +I” / e-5 (E) g( t - s, 2 - y)-u2(y, 2 s)dyyds.

Using the same branching Brownian process used by McKean for the KPP

equation as done in section 3, it is possible to define a recursive multiplica-

tive functional such that the solution of (17) is obtained as an expected

value. Indeed, let

Then, using the strong Markov property it follows that

U ( X , t ) = EX(x, t ) .

It should be remarked that the introduction of the damping term in

the equation (17) is only done for simplicity of presentation. The main

point of this example is to illustrate that recursive multiplicative function-

als can be used to obtain solutions of nonlinear partial differential equa-

tions. As indicated in the introduction, a similar representation for the

Navier-Stokes equations is not presently available. The major difficulty to

be overcome appears to be the projection on divergence free vector fields

used to eliminate the pressure term. In the Fourier space, this projection

297

becomes part of the node operation as i t is a local operator and is given by

l$ = 1 - & @ & given in (1). By contrast, the same projection in the

physical space involves the Riesz transforms tha t are nonlocal operators.

Despite the results of Gundy and Silverstein5 , that provides a probabilistic

interpretation of the Riesz transform in terms of Brownian motions, the

representation of solutions of the Navier-Stokes equation as an expected

value of an appropriate stochastic functional remains an open problem.

Acknowledgments

The work presented here is joint work with R. Bhattacharya, L. Chen, S.

Dobson, R. Guenther, C. Orum and E. Waymire and i t is partially funded

by US NSF Grant 0073958.

References

1. R. N. Bhattacharya, L. Chen, S. Dobson, R. B. Guenther, C. Orum, M. Os- siander, E. Thomann, and E. C. Waymire, Majorizing Kernels & Stochastic Cascades With Applications To Incompressible Navier-Stokes Equations. To

appear in Transactions of the AMS. 2. Cannone, M. and F. P1anchon:On the regularity of the bilinear term for

solutions to the incompressible Navier-Stokes equations Revista Matema’tica Iberoamericana 16 1-16, (2000).

3. Larry Chen, Scott Dobson, Ronald Guenther, Chris Orum, Mina Ossiander, Enrique Thomann, Edward Waymire. On ItB’s Complex Measure Condition

For a Feynman-Kac Formula. To appear in IMS Lecture-Notes Monographs Series, Papers in Honor of Rabi Bhattacharya, eds. K. Athreya, M. Majumdar, M. Puri, E. Waymire.

4. G. Galdi, “An Introduction to the Mathematical Theory of the Navier-Stokes Equations” Vol 1 and 2. Springer Tracts in Natural Philosophy, Vol 38 and

39. Springer 1994. 5. Gundy, R and M.L. Silverstein: “On a probabilistic interpretation for the

Riesz Transforms” in Functional analysis in Markov processes, Lecture Notes

in Mathematics, 923, 199-203. Springer 1982. 6. LeJan, Y. and A.S. Sznitman: Stochastic cascades and 3-dimensional Navier-

Stokes equations, Prob. Theory and Rel. Fields 109 343-366, (1997).

7. LemariB-Rieusset, P.G. Une remarque sur l’analyticitB des soutions milds des Bquations de Navier-Stokes dans R3, C.R. Acad. Sci. Paris, t.330, SBrie 1,

8. Kato, T.: Strong Lp solutions of the Navier-Stokes equations in Rm with applications to weak solutions, Math. Z., 187 471-480, (1984).

9. H. McKean, Applications of Brownian motion to the equation of Kolmogorov,

Petrovskii and Piskunov. Comm. Pure and Applied Math. Vol 28, 323-331, (1975).

10. R. Temam, “Navier-Stokes equations and nonlinear functional analysis”. SIAM 1995.

183-186, (2000).

STOCHASTIC BURGERS EQUATION WITH LEVY SPACE-TIME WHITE NOISE

AUBREY TRUMAN AND JIANG-LUN W U

Department of Mathematics, Unversity of Wales Swansea

Singleton Park, Swansea SA2 8PP, UK E-mail: A . Duman@swansea. ac . uk, J . L. Wu@swansea. ac. uk

The purpose of this paper is t o investigate the Cauchy problem for the following

stochastic Burgers equation

with suitable initial condition (for all ( t , x) E [0,03) xW), where Ft,, is a L6vy space-

time white noise. The problem is interpreted as a stochastic integral equation of

jump type involving the heat kernel. We obtain existence of a unique local solution

in the L2 sense and show that it gives rise to a (local) stochastic flow (in time).

Mathematics Subject Classification (1991): 60H15, 35R60.

Key Words and Phrases: Stochastic Burgers equation, LBvy space-time white

noise, stochastic integral equations of jump type, local existence and uniqueness,

flow property.

1. Introduction

This paper is mainly concerned with the Cauchy problem for the following

stochastic Burgers equation

on the given domain [0, m)xR with L2 initial condition, where Ft,z is the so-

called Lkvy space-time white noise consisting of Gaussian space-time white

noise (i.e. a Brownian sheet on [0, m) x R) and Poisson space-time white

noise (see 52 for the definition). There has recently been increasing interest

in solving stochastic partial differential equations with non-Gaussian white

noise (see, e.g. Bertoin5i6, Giraud18, Winke143, Mueller31, M ~ t n i k ~ ~ and

Shlesinger et a136 and references therein).

In particular, Gaussian white noise driven parabolic SPDEs have been

intensively studied (see e.g. W a l ~ h ~ ~ and references therein). SPDEs driven

by Poisson white noise are less well known and were first investigated in

298

299

Albeverio et all. Let us also mention that Saint Loubert BiB 35 formulated a

parabolic SPDE driven by a Poisson random measure in a different way from

Albeverio et all and he obtained very.interesting results on the existence of

the unique solution. Moreover, parabolic SPDEs driven by L6vy space-time

white noise are studied in Applebaum and Wu' and besides the existence

of the unique solution, the flow property of the system obtained is also

discussed. While very interesting studies of heat equations driven by Q-

stable LBvy noise have been carried out by Mueller31 and by Mytnik3'.

On the other hand, as is well-known (see e.g. Burgers7), the Burgers

equation

au la(u2) d'u - +--=- at 2 ax 8x2

has been used extensively, under the name of Burgers turbulence, to model

a variety of physical phenomena where shock creation is an important in-

gredient. The solution to Burgers equation is then called Burgers turbu-

lent fluid flow. In recent years there appears to be a great interest to

investigate Burgers turbulence in the presence of random forces, that is,

to study stochastic Burgers equations with (Gaussian) white noise as ran-

dom forces and/or with random inital data, see e.g. Bertini et a13, Bertini

and Giacomin4, Bertoin5y6, Da Prato et als, Da Prato and Gatarekg, Da

Prato and Zabczykl', Davies et all', E et all', Giraud17>18, Gyongy and

Nualart2', Holden et a123,24, Kifer27, Le6n et a12', Sinai37>38, Tribe and

Z a b o r o n ~ k i ~ ~ , Truman and Zhao40,41, Winke143. Burgers equation has,also

been used to study efficient stock markets, see Hodges and Carverhill" and

references cited there.

One of the main investigations of Burgers equation is based on the

intriguing connection between the (nonlinear) Burgers equation and the

somehow simpler linear heat equaiton, via the celebrated Hopf-Cole trans-

formation. This technique can be still adapted to stochastic Burgers equa-

tion with additive Gaussian white noise (see e.g. Bertini et a13, Holden

et a123124), but it is no longer available in the case of stochastic Burg-

ers equations driven by more general Gaussian white noise (for instance,

multiplicative Gaussian space-time white noise). Another method is used

successfully, e.g. in Da Prato et a18, Da Prato and Gatarekg, Da Prato and

Zabczyk" and Gyongy and Nualart" (here we just mention a few refer-

ences), to study the mild solutions to stochastic Burgers equations driven

by Gaussian space-time white noise.

In this paper we introduce a stochastic Burgers equation driven by L6vy

space-time white noise which generalizes all stochastic Burgers equations

with white noises considered in the literature mentioned above. We will

300

prove existence of a unique, local, mild solution to the stochastic Burgers

equation we posed above.

The paper consists of three sections. In the next section, we set up what

we call Poisson white noise and the corresponding stochastic integrals. In

Section 3, in order to make the problem we are considering precise, we first

elucidate briefly what L6vy space-time white noise is and then interpret

the (heuristic) stochastic Burgers equation driven by Lkvy space-time white

noise (weakly) as a rigorous jump type stochastic integral equation which

involves evolution heat kernels. We present existence of a unique local L2- solution. Namely, for any initial function from L2(R), we obtain a local

solution with c&dl&g (i.e., right continuous with left hand limits in the time

variable t E [0, cm)) trajectories in L2(R)). Finally, we discuss the flow

property of the local solution.

Our approach is based on combining the methods for solving stochas-

tic Burgers equations driven by Gaussian space-time white noise in Da

Prato et a18, Da Prato and Gatarekg, Da Prato and ZabczyklO, GyOngylg,

Gyongy and Nualart20, Gyongy and Rovira2' with the techniques for solv-

ing parabolic SPDE's driven by L4vy white noise in Albeverio et all, Ap-

plebaum and Wu2, Mueller31, M ~ t n i k ~ ~ , Saint Loubert Bi635.

2. Poisson White Noise and Stochast ic Integrat ion

In this section, we set up some notations and recall some facts for our later

presentation. We start with a general account of Poisson white noise in an

abstract setting. Let ( R , 3 , P) be a given complete probability space and

(U, B ( U ) , v ) be an arbitrary a-finite measure space.

Definit ion 2.1. Let (E ,& ,p ) be a a-finite measure space. By a Poisson

white noise on ( E , E , p ) we mean an integer-valued random measure

N : ( E , &, p ) x (U, w>, .) x (0, F, P ) N u (01 u I..) with the following properties:

a Poisson distributed random variable with

(i) for A E & and B E B(U) , N(A, B, .) : (R, 3, P ) 4 N U (0) U {cm} is

e-f i(A)u(B) [p(A)v(B)]" n!

P{w E R : N ( A , B , w ) = n } =

for each n E N~{O}u{co}. Here we use the convention that N ( A , B, .) = 03,

P-as . whenever p ( A ) = 00 or v(B) = a; (ii) for any fixed B E B ( U ) and any n 2 2, if A l , . . . , A , E & are

all disjoint of one another, then N(A1, B, .), . . . , N(An, B, .) are mutually

301

independent random variables such that

Clearly, the mean measure of N is

E[N(A, B, .)] = p ( A ) u ( B ) , A E I , B E B ( U ) .

Moreover, N is nothing but a Poisson random measure on the Cartesian

product measure space ( E x U, & x B ( U ) , p @ v ) as formulated e.g. in Ikeda

and Watanabe25. Hence, by a similar argument to that of Theorem 1.8.1 of

Ikeda and Watanabe25, we have the following existence result for Poisson

white noise, namely, given any a-finite measure p on the measurable space

(E, &), there exists a Poisson white noise N on ( E , &, p) with mean measure

E[N(A, B, .)] = p(A)v(B) , A E E , B E B(U) . In fact, N can be constructed

as follows

7 n ( W )

N ( A , B , W ) := C C l ( A n E , , ) x ( B n l r n ) ( [ ~ ) ( ~ ) ) l ( w E R : a n ( W ) ~ i ~ ( ~ ) (2) ,EN j=1

w E R for A E & and B E B(U) , where

(a) { E n } n E ~ c & is a partition of E (i.e., En, n E N, are disjoint of one

another and U n E ~ E , = E ) with 0 < p(E,) < m , n E N, and {U,},,N c B ( U ) is a partition of U with 0 < v(Un) < a, n E N;

(b) V n , j E N, [:,) : s1 ---f E, x U, is .F/& x B(Un)-measurable with

where &, := & n E, and B(U,) := B(U) n U,;

variable with

(c) V n E N, 7, : R --+ N U {0} U {m} is a Poisson distributed random

e-p(En)V(Un) [ p ( ~ ~ ) v ( u,)] k!

P{w E R : vn(w) = k } = , k E N U {0}u {a};

(d) and qn are mutually independent for all n, j E N.

Thus, given any a-finite measure p on ( E , &) and any a-finite measure Y

on (U, B( U ) ) , there is always a Poisson random measure N on the product

measure space ( E x U, & x B ( U ) , p@v) which can be constructed in the above

manner. We call such a N canonical Poisson random measure associated

with p and v .

302

Now let us give an example of Poisson white noise. Take (E , &, p) =

( [ O , c o ) x Rd,B([O,co)) x B(Rd) ,d t 18 dx) ,d E N. Then the Poisson white

noise N can be well defined. Denote

Nt,,(B, w ) := N([O, tl x (-00,x], B , w ) , (B , w) E WJ) x d

for t E [O,m) and x = (xj)15j5d E Bd, where (-co,x] := n j = l ( - m , x j ] . We can define (formally) the Radon-Nikodym derivative

for (t , x) E [O, m) x Rd. We call Nt,z Poisson space-time white noise.

In the sequel in this section, we will take the measurable space (I?,&) in Definition 2.1 to be a product space ( [ O , m) x E , B([0, m)) x &) where

( E , & ) in the latter is a Lusin space. Let p be a o-finite measure

on (E ,& ) (note that from the next section onwards, ( E , & , p ) will be

taken to be (EX, B(B), d x ) ) , then there exists a Poisson white noise N on

([O, m) x E, B([0, m)) x E , dt 8 p ) with mean measure

E{N([O, t] x A, B, .)} = t p (A )v (B ) , ( t , A, B ) E [0,03) x & x B(U).

Let {3 t } tE [0 ,00 ) be a right continuous increasing family of sub a-algebras

of 3, each containing all P-null sets of 3, such that the Poisson white noise

N has the property that (i) N([O, t ] x A, B,.) : R + N U (0) U {co} is

Ft/P(N U (0) U {co})-measurable V(t, A, B ) E [0, m) x & x B(U) and (ii)

{N([O, t+ s] x A, .) - N([O, t ] x A, . ) } ~ > o , ( A , B ) E E ~ B ( u ) is independent of 3 t

for any t 2 0, where P(N U (0) U {m}) is the power set of N U (0) U {co}. (For instance, we may directly take

3 t := m ( { N ( [ O , t ] x A, B, .) : (A, B ) E E x B(U)} ) V N , t E [0, a)

where N denotes the totality of P-null sets of F.) In what follows, let us set up related stochastic integrals by following

the procedure of Section 11.3 of Ikeda and Watanabe25 (see also Applebaum

and Wu')). First of all, for those integrands f : [0,00) x E x U x R + R

which are {Ft}-predictable and satisfy

for some (A, B ) E & x B(U) , the stochastic integral

is well defined as the usual Lebesgue-Stieltjes integral.

303

Now we define the (compensating) martingale measure

M ( t , A, B, U ) := N([O, t], A, B,w) - tp(A)v(B) (3)

for any (t, A, B ) E [0, co) x E x B(U) with p(A)v(B) < co. Obviously,

E[M(t, A, B, , ) I = 0

and

E([M(t , A, B, . ) I2) = tP(A)V(B) . (4)

For any {Ft}-predictable integrand f : [0, co) x E x U x R --+ R which

satisfies

E I" s, If(s, X I Y, .)ldsp(dz)v(dy) < 00, a.s. vt > 0

for some (A, B ) E & x B(U) , we can define the stochastic integral

Moreover, stochastic integrals with respect to M are also well defined for

{&}-predictable integrands f satisfying

for some (A,B) E E x B(U) by a limit procedure (see the argu-

ment in Section 11.3 of Ikeda and Watanabe25) and t E [ O , o a ) H

304

Ji' S, S, f(s, x, y , . )M(ds, dx, d y , .) E R is a square integrable {.Ft}- martingale with the quadratic variation process

On the other hand, it is clear that A4 defined by (3) is a worthy, orthog-

onal, {&}-martingale measure in the context of Walsh4'. Thus stochas-

tic integrals of {Ft}-predictable integrands with respect to M can be also

well defined (alternatively) by the method in Section 11.3 of Ikeda and

WatanabeZ5. Furthermore, the following stochastic Fubini's theorem for

changing the order of integration in iterated stochastic integrals with re-

spect to M was presented in Applebaum and Wu2:

3. Burgers Equation Driven by LQvy Space-Time White Noise

Let ( R , 3 , P ) be a given complete probability space with a usual filtration

{3t}t~[0,~). In this section we will consider the Cauchy problem for the

following stochastic Burgers equation

(g - &) u(t, x, w ) + +g[u2( t1 x, w ) l = a(t, x, u(t, x, w))+

+qt , 2, u(t, 5 , w))Ft,,(w) I ( t , 5 , w ) E (0, 03) x R x R (9)

u(O,x, w ) = uo(x, w ) , (x, w ) E R x R

305

where a, b : [0, 00) x R x R ---f R are measurable and the initial condition uo is .&measurable, and F is an Lkvy space-time white noise, which includes

terms not only controlled by a Gaussian space-time white noise but also by

a Poisson space-time white noise, so that in fact the noise we shall consider

has a formal structure similar to that of a Lkvy process:

where c1, c2 : [O, 00) x R x U 4 R are measurable, Wt,, is a Gaussian

space-time white noise on [O, 00) x R used initially by W a l ~ h ~ ~ (formally,

Wt,, := *, where W(t , x) is a Brownian sheet on [0, co) x R), Mt,, and Nt,z are defined formally (in the previous section) as Radon-Nikodym

derivatives as follows

for ( t , x) E [ O , o o ) x R, while as given in Section 2, N is the Poisson white

noise on ([0, co) x R, B( [0, 00) x R)) with respect to a given o-finite measure

space (U, B ( U ) , v), and UO E B(U) with v(U\Uo) < 00, M is the associated

(compensating) martingale measure.

Following Walsh4’, let us introduce a notion of (weak) solution to Equa-

tion (9). An L2(R)-valued and {&}tEio,w)-adapted cAdlAg (in the variable

t E [ O , c o ) ) process u : [O,m) x R x R + R is a solution to (9) if for any

cp E S(R), the Schwartz space of rapidly decreasing Cw-functions on R,

306

holds for all t E [0, oo). Based on this notion, we can present a weak (but rigorous) formulation

of Equation (9). Let Gt be the Green’s function for the operator & - 6 in the domain [0, oo) x R, which is given by the following formula:

} (x - z)2

ezp{ - ___ 1

4t Gt(z, 2) z= - rn (13)

for t > 0; and Go(x, z ) = S(z - 2). We will need the following facts:

(i) J, Gt(z, z ) d z = 1, Jw[Gt(zl z) I2d t = (27rt)-? , (ii) Gt(z, t) = Gt(z, z) , (iii) J, Gt(x, z’)G,(z‘, z)dz’ = Gt+s(x, z ) , (iv) Vm, n E N U {0}, there exist some constants K , C > 0 such that

Vt E [0, oo),Vz E R ; t E [0, oo) , 2, t E R;

s, t E [0, co) , z, z E R ;

For the property (iv), cf. e.g., Friedman14, or Ladyzhenskaya et a128. Based

on (iv), we have the following particular estimates which will be used later

on

and

for all t E (0, oo), z, z E R.

The following heuristic discussion paves the way for us to give a rigorous

(weak) formulation of Equation (9). (Of course, our derivation can be made

rigorous in the sense of Schwartz distributions.) First of all, we notice that

the solution of (9) can be (formally) written in terms of the Green’s function

as follows

1 au2 u(t , 5, W ) = [G * (210 - - - + CL + bF)](t, Z, W ) .

2 at In other words, if u solves the following convolution equation

307

1 a u y , ., w ) az

~ ( t , Z, U ) = (G * UO( . , w) ) ( t , Z) + {G * [-z +a(*, ., U ( . , * , w ) ) -k b(., ., u(., . ,w))F.,.(U)]>(t, 7 (14)

then u satisfies (9). Furthermore, (14) is formally equivalent to the follow-

ing stochastic integral equation

since from (10) we have the following (heuristic) derivation for the second

term in the right hand side of (14)

308

P t P

xM(ds , dz, dy, w )

xN(ds , dz, dy, w) .

where we have used “integration by parts” for the first term. Moreover, by

observing that v(V \ Vo) < 00 and using formula (6), we see that equation

(15) can be written in the equivalent form

+ Gt-s(x:r z)b(s, z , ~ ( s - , z , w ) ) [ci (s, z ; y)lu0 (Y)

+CZ(S, z ; V)1U\Uo(Y)] M(ds, dz, dY, w ) . (16)

Based on the above discussion, without loss of generality, we shall con-

sider the equations in the following form

309

where f,g : [ O , o o ) x R x R --t R and h : [ O , o o ) x R x R x U + R are

measurable and the coefficient function q : [O,co) x R x R -+ R is measurable

and satisfies the following growth condition

Iq(t,z, .)I I Kl(Z) + K2(z)I4 + const.l.12 (18)

for all ( t , z , z ) E [ O , o o ) x R x R, for some nonnegative functions K1 E

L1(R) and KZ E L2(R)”. Clearly, the term containing the quadratic u2 in Equations (15) and (16) satisfies the above growth condition for q with

q ( t , z , z ) = z2. Moreover, the case that replacing u2 by a more general

form of \ulr for T E [1,2] also satisfies the conditions posed for q (with

q(t, 2, z ) = 1 . ~ 1 ~ ) . Therefore, the condition for the coefficient q we posed

above covers at least these two important and interesting cases. Also,

it is obvious that q( t , z, z ) = z is another special case under our growth

condition, which corresponds to the second term containing the linear u

instead of u2 on the right hand side of Equations (15) and (16).

Clearly, Equation (17) is a weak (but rigorous) formulation of the fol-

lowing (formal) equation

= f(t , 5, u(t, 2, w ) ) + g( t , 5, u(t, 2 , w ) ) W t , z ( w )

q t , z,u(t, 2, w ) ; Y)Mt,z(dY, w ) . +s, Let us now give a precise formulation of solutions for Equation (17). By

a (global) solution of (17) on the set-up (0, F, P; {Ft}tE[~,cro)), we mean an

{Ft}-adapted function u : [0, co) x R x R + R which is c&dl&g in the variable

t E [ O , o o ) for all z E R and for almost all w E R such that (17) holds.

Furthermore, we say that the solution is (pathwise) unique if whenever u(l )

a.e., V ( t , x) E [0, m) x R. Moreover, one can formulate a (global) solution

over a finite time interval [O,T] for any 0 < T < co in the same pattern.

Furthermore, an {.Ft}-adapted function u : [O,T] x IR x R + R which

is c&dl&g in t E [O.T] is called a local solution to Equation (17) if there

exists an increasing sequence { T ~ } ~ ~ N of stopping times such that W E

[0, TI and Vn E N, the stopped process u(t A T,, 2, w ) satisfies Equation

and u ( ~ ) are any two solutions of (17), then u(l)(t, z, .) = u ( ~ ) ( t ,z , . ) , p-

aHere and in the sequel, by “const.” we mean a generic positive constant whose value might vary from line to line.

310

(17) almost surely. Clearly, a local solution becomes a global solution if

T , , := supnEW~, = T . Moreover, a local solution to Equation (17) is

(pathwise) unique if for any other local solution fi : [O,T] x R x R R, u(t, x , w ) = fi(t, 2, w ) for all ( t , x, w) E [0, T, A ?,) x R x := { ( t , x , w ) E

[0, T] x R x R : 0 I t < T,,(w) A?,(w)}. We have the following main result:

Theorem 3.1. Let T > 0 be arbitrarily fixed. Assume that there exist (positive) functions L1, L2, L3 E L1(R) such that the following growth conditions

I f ( t , z, z)J2 I L ~ ( z ) + const.)zI2, (19)

(20) Idt, 2, .)I2 + / Ih(t, 2, z ; Y)l2Y(dY) I L2(x) + const.1zI2 U

and Lipschitz conditions

Iq(t, 5 , a) - 4( t , 2, z2)12 + If ( t , x, z1) - f (t , 5, .2)12

I [L3(x) + const.(lzl12 + Iz21')]1z1 - z2I2 (21)

Ig(t, x,z1) - g( t , x, z2)I2 + Ih(t, 2, z i ; Y ) - h(t ,x, z2; Y ) ~ ~ Y ( ~ Y )

(22)

2 L 5 const.jz1 - z21

hold f o r all ( t , x ) E [O,T] x R and z , z l , z 2 E R. Then f o r every Fo- measurable uo : R x R -+ R with EJw(luo(x,.)12)dz < co, there exists a unique local solution u to Equation (17) with the following property

for any t E [0, TI.

We need some preparation before the proof to Theorem 3.1. For any fixed n E N, let the mapping 7rn : L2(R) 4 B, := {u E L2(R) : 11ullL2 5 n } be defined via

Clearly, for any n E N, we have

I I7rn(u) I ILz 5 TL .

ll7rnllL2 := sup ll7rn+ 5 1

Moreover, it is clear that the norm

l l 4 l L Z l l

311

that is, 7rn : L2(R) -i L2(R) is a contraction.

Notice that if u is a solution to Equation (17), then u is L2(R)-valued,

{&}-progressive process. Thus, by Theorem 2.1.6 in Ethier and Kurtz13

(page 55), for any n E N,

defines a stopping time. It is clear that {T,},~N is an increasing sequence

of stopping times determined by u. Moreover, for any fixed n E N, the

stopped process u(t A 7,) satisfies the following equation

On the other hand, any solution to Equation (23) is a local solution to

Equation (17). Therefore, the existence of a unique local solution to Equa-

tion (17) is equivalent to the existence of a unique solution to Equation

(23). Hence, we will focus our attention on showing the existence of a unique solution to Equation (23).

The following proposition is a reformulation of some estimates obtained

in Gyongy and NualartZ0 in a way convenient for our setting here. One

can, alternatively, verify them by utilizing inqualities of Holder, Minkowski

and Young.

Proposition 3.2. (i) For u : [O,T] x lR 3 R, the following estimates hold

and

312

in particular,

(ii) For 0 5 tl 5 t 2 5 T , there exist Q E (0, i ) , ~ E (4, co) and ,8 E

(0 , l - :), such that the following estimates hold

Proof of Theorem 3.1 We will carry out the proof by the following three

steps.

Step 1 Suppose that u : [O,T] x R x f2 -+ R is an L2(R)-valued, {Ft}- adapted, c&dl&g process. For any fixed n E N, set

with

and

313

By (24) in Proposition 3.2, the condition (19) and Schwarz inequality, we

have

5 const.ti 5 const.T? < 00 .

Notice that here and after the constant “const.” depends also on n (of

course on T as well). By (25) in Proposition 3.2 and the condition (18), we

get

5 c0nst.t’ 5 const.T2 < 00 .

314

On the other hand, by our Proposition 2.2 (Fubini’s theorem) and It6’s

isometry property for stochastic integrals with respect to (both continuous

and c&dl&g) martingales, we have

and

Thus, by the condition (20), we get

Therefore, we obtain that

for any fixed t E [0, TI.

Step 2 Now let X > 0 be arbitarily fixed. For any L’(IR)-valued, {&}- adapted, c&dl&g process u : [O,T] x IR x R 4 IR with initial condition

u(0, z, w ) = UO(Z, w ) , we define

315

Clearly, 1 1 . I Ix is a norm. Let B denote the collection of all L2(W)-valued,

{Ft}-adapted, c&dl&g process u : [0, TI x R x R -+ W with initial condition

u(O,x, w ) = u ~ ( z , w ) , such that

Then (B , ( 1 . Ilx) is a Banach space. Now Vu E B, Ju is well defined and for

any fixed t E [O, TI

E(L/ ( Ju ) ( t , z , . ) l ' dz <const.(T4 +T2) < 03.

Thus

fi - 3 00

<_ const. ( t i + t')e-xtdt = const.[-A + 2 ~ - 3 1

I const.(A-$ + A - ~ ) < 00

2

that is, J u E B, which implies that J : B + B.

Step 3 Now Vu, v E B, by (24), (25), (26), (27) and (28) in Proposition 3.2 and the Lipschitz condition (21), together with utilizing Fubini's theorem,

the Young inequality and the Schwarz inquality, we get for any t E [0, TI

5 c0nst.E [ l ( t - s ) - i L (L3(z) + const.[(nnu)2(s, z, a ) )

317

and by It6’s isometry for both stochastic integrals with respect to W ( d s , dz ) and M(ds , dz, dy, w), we have

318

Now let us take X large enough so that

const. (5 + 5) < 1

which implies that J : B 4 B is a contraction. Therefore there must be a

unique fixed point in B for J and this fixed point is the unique solution for

our Equation (23). To see that this gives us a local solution to Equation

(17), let us denote by u, the unique solution of Equation (23) for each

n E N. For this u,, let us set the stopping time

T,(w) := inf{t E [0, T ] : u;(t, z, w)dz 2 n2} . s, Clearly by the contraction property of J , we have for all j 2 n and for

almost all w E R

Therefore we define

for any ( t , z, w ) E [0, ~ , ( w ) ) x IR x R and

Too (w) := sup 7, (w) . nEN

Then

{ 4 t 7 X , W ) : ( t , z ,w ) E [0,7,(W)) x JR x 0)

is a local solution to Equation (17).

Finally, for the uniqueness of the local solution to Equation (17), sup-

pose that there are two local solutions u and v to Equation (17). Then u and v must satisfy Equation (23) for any fixed n E N. On the other hand,

by the uniqueness of solution to Equation (23), we get

U ( t , X , W ) = W ( t , X , W ) , V ( t , X , W ) E [O, 7,(w)) x R x a.

319

Now let n -+ co, we deduce

u(t, 2, w) = v(t, 2, w), V(t , 2, w) E [O , 7,(w)) x IR x R .

Hence we obtain the uniqueness. Q.E.D.

Remark 3.3. In the case M -- 0, Equation (17) becomes a Burgers equa-

tion with Gaussian (space-time) white noise. Unique global L2 solutions

are obtained by Gyongy and Nualart2' in the whole space line and by Da

Prato, Debussche and Teman in Da Prato et al' (see also Da Prato and

Zabczyk'') in bounded space intervals. Their methods depended critically

on some uniform estimates which employed Burkholder's inequalities for

continuous martingales. We observe that we are unable to follow this route

herein as the corresponding inequalities for cadlag martingales (see e.g Ja-

cod and Shiryaev26) do not behave so nicely.

Finally let us consider the flow property of the local solution to Equation

(17) starting with an L2 function as the initial condition. We refer to the

references Fujiwara and Kunita15 and Fujiwara" for investigations of LBvy

flows associated with (ordinary) stochastic differential equations of jump

type. For (T, t , w) E { ( T , t , w) E [0, TI x [0, TI x R : T F t < T,(w) I TI w E

Q}, and 'p E L2(R), we define

for z E R. Then it follows clearly by Theorem 3.1 that almost surely

: L2(R) + L2(R)

for all 0 5 r _< t < rm(w) 5 T . Furthermore, we have

320

Proposition 3.4.

act = identi ty, V t E [0, t m ( w ) ) , P - a s . for w E R ; (30)

and

0 a:,., = a:', V0 I r F r' I t < T,(w), P - a s . f o r w E R. (31)

Therefore we conclude that { @ ~ t } o ~ r ~ t < T , ( w ) , w E o forms a (local) stochastic %ow.

since Go(z, z ) = 6(z - z ) and N ( { t } , {z}, dy, w ) = 0, P - a s . for w E R. Thus (30) comes directly from (29) and (32).

To verify (31), by (29), it suffices to show the following equality

x M ( d s , d z , d y , W ) , P - U.S.

(33) is obtained by a straightforward derivation using the (usual) Fubini's

theorem (see, e.g. Theorem 7.8 in R ~ d i n ~ ~ ) , integration by parts, Theorem

2.6 of W a l ~ h ~ ~ , our Proposition 2.2, property (iii) of the Green's function

G, and our equality (32). Q.E.D.

321

Acknowledgements

We thank Ian M. Davies for the great help in setting of the manuscript.

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A COMPARISON THEOREM FOR SOLUTIONS OF

BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS

WITH TWO REFLECTING BARRIERS AND ITS

APPLICATIONS

T.S.ZHANG

Department of Mathematics, University of Manchester, Oxford Road,

Manchester M13 9PL, England.

In this note, we prove a comparison result for the solutions of backward stochastic

differential equations with two reflecting barrier processes. The result is then applied to obtain an existence result for solutionsof a backward SDE with reflecting

barrier processes under weak assumptions on the coefficients.

AMS Subject Classifications: Primary 60H20. Secondary 60H10,60H30.

1. Introduction

The notion of backward stochastic differential equation was introduced by

Pardoux and Peng' in (1990), there they obtained existence and unique-

ness of adapted solutions under suitable conditions on the coefficients and

terminal random variables. Certain backward SDE were also independently

used by Duffie and Epstein in (1992) to study stochastic differential utili-

ties in economics models. This subject has attracted a lot of attention and

has developed rapidly in recent years, which is partly due to the applica-

tions found in the theory of partial differential equations and mathematical

finance, etc. See 2,5,8 and references therein.

We particularly mention the paper by J.Cvitanid and I.Karatzasl, which

motivates our work here. In this paper they studied backward stochastic

differential equations with two reflecting barrier processes and obtained

existence and uniqueness of solutions under various conditions. They also

proved that the solution of a backward SDE with two reflecting barrier

processes is the value function of certain Dynkin game. If the coefficients

depend also on the state variable, the Lipschitz condition is required.

The aim of this paper is to prove a comparison theorem for solutions of

backward stochastic differential equations with two reflecting barrier pro-

324

325

cesses. The theorem is applied to obtain a new existence result for solutions

of a backward SDE with reflecting barrier processes.

2. Backward SDE With Two Reflecting Barrier Processes

In this section we follow closely the notations in '. Let ( R , F , P ) be a

complete probability space. Bt , t 2 0 denotes a standard &dimensional

Brownian motion. Denote by F = (.F~}o<T - the argumentation of the nat-

ural filtration of the Brownian motion B. Let < be a given FT measurable

random variable in L2(R), and f : [O,T] x R x R -+ R be a P 8 B(R)-

measurable function, where P denotes the a-algebra of predictable sets in

[0, TI x R. Given two continuous F-progressive processes L, U such that

L( t ) 5 U ( t ) , YO 5 t 5 T and L(T) 5 5 5 U ( T ) a s (1)

These two processes will serve as two reflecting barriers.

backward SDE with two reflecting barrier processes.

Consider the

d X ( t ) = - f ( t , w , X( t ) )d t - dK1(t) + dK2(t) + Y'(t)dB(t) (2)

As in

Definition 2.1. We say that ( X , Y, K 1 , K 2 ) : X : [0, TI x R 4 R, Y : [0, T ] x R 4 Rd, and K1, K 2 : [0, T ] x R - R is a solution of the backward

SDE (2) with reflecting barriers U ( . ) , L( . ) and terminal condition < if the

following holds

we introduce the following

(i) X , Y, K 1 and K 2 are continuous and F-progressive,

(ii) K1(t), K2( t ) , t 2 0 are increasing with K1(0) = K2(0) = 0,

(iii)

T

X ( t ) = < + 1 f(S, w , X ( s ) ) d s + K1(T) - K1( t )

T

- (K2(T) - K2( t ) ) - / Y' (s )dB(s) , 0 I t I T, (3) 0

(iv) LJt) 5 X ( t ) I U( t ) ,

almost surely.

0 5 t 5 T, (v) J, ( X ( t ) - L( t ) )dKl ( t ) = J?(U(t) - X( t ) )dK2( t ) = 0,

Let fl(s, w , x) and f2(s , w , x) be two P@B(R)-measurable functions. Let

K:, K?) be a solution to equation (2) with f replaced by fi, terminal ( X i , condition <i and reflecting barrier processes L( t ) , U ( t ) , t 2 0, i = 1 , 2 .

Theorem 2.2. Assume one of fl and f2 , say f2 , satisfies a Lipschitz

condition in x uniformly w.r.t. (s, w ) , i.e.,

l f2 ( s ,w7x1) - f2(s ,w,x2)1 I c151 - 5 2 1

326

for some constant c. If 51 L E2 and f ~ ( s , w , x) L f i ( s , w , x ) almost surely,

then

Xl(t) 5 X2(t ) , Vt E [O,T], a.s. ( 4 )

Proof. Choose a sequence {&,n 2 1) of functions that satisfy & E

C2(R), &(x) = 0, for 5 5 0, 0 5 4L(x) I 1, &(x) 2 0 and &(x) ,/ x+. This is always possible, see ,for example, the proof of Theorem 3.2 in

Observe that

Applying It6’s formula, we have

This gives

327

Taking expectation we see that

where we have used the fact that

Letting n + 00, we obtain

5 &, and (iv), (v) in the definition 2.1.

Iterating the above inequality n times, we arrive at the following

(9) 1

n! E[(Xl(S) - XZ(S))+] I M--c"(T - t)"

where M = suptcT E[IXl(t) - Xz(t)(]. We complete the proof of the theo-

rem by sending n to +00.

328

3. Existence of Solutions of A Backward SDE with Two Reflecting Barriers

In this section, [ denotes a F+ measurable random variable with E[l[I2] < 00, which will be the terminal condition. L( t ) , U ( t ) , t 2 0 denote the re-

flecting barrier processes as in section 2. Let g( t , w) : [0, T ] x R + R be a

P-measurable process. Consider the backward SDE with reflecting barriers

L( t ) , U ( t ) , t 2 0 :

d X ( t ) = -g ( t , w)dt - dKl( t ) + dK2(t) + Y'( t )dB(t)

X ( T ) = 5. (10)

Condition 3.1. The backward SDE with reflecting barriers L(t) , U ( t ) , t 2 0 admits a unique solution ( X , Y, K 1 , K2) for every ?-measurable process

g with E[JT g( t , ~ ) ~ d t ] < 00.

Remark 3.2. Condition 3.1 is fullfilled if L( t ) < U ( t ) , 0 I t < T and

L W X { t < T } + [X{ t=T} I E[EIFtT,] I U(t )X{ t<T} + 5 X { t = T } (11)

See for details.

Let f : [0, T ] x R x R + R be a P 8 B(R)-measurable function described

as in section 2.

Theorem 3.3. Suppose that f is bounded and f(t,w,.) : R + R is uniformly continuous on bounded intervals uniformly with respect to

( t , w) E [0, T ] x R} . In addition, condition 3.1 holds. Then there exists a

solution to the backward SDE (2) with two reflecting barrers L, U , terminal

condition [ and the coefficient f .

Proof of the theorem. Choose a decreasing sequence fn : [0, T] x R x R +

R, n 2 1 of P 8 B(R)-measurable functions that satisfy

Ifn(t,w,.) - fn(t,w,Y)I B CnI.-Yyl

for some constant c, and that for any fixed ( t , w ) E [0, T ] x R, fn( t , w, .) converges to f ( t , w , x) uniformly on bounded intervals in R. It is easy to

see that such a sequence f,, n 2 1 exists under our assumptions on f . It

was proved in ' that when f is repaced by fn the equation (2) has a unique

solution. Let us denote it by (Xn ( t ) , Y,(t), KA(t), K:(t)). By Theorem 2.1,

we have

X , ( t ) 2 X2(t) 2 X3(t ) 2 . . . 2 X n ( t ) 2 . . . 2 X M ( t ) , a.s (12)

where X M stands for the solution to the backward SDE with reflecting bar-

riers corresponding to the coefficient given by the lower bound of f (t , w, .).

329

This shows that the sequence {Xn( t ) }n>l has a limit, which we will denote

by k(t). Observe that for fixed ( t , w ) , k n ( t ) lies in a compact interval of

R for all n 2 1. Hence,

fn(t, w, Xn( t ) ) - f ( t , w, X ( t ) )

= f n ( 4 w, Xn( t ) ) - f(t, w, X n ( t ) ) + f ( t , u, X n ( t ) ) - f(t, w , R t ) ) + O , as 7 2 4 0 0 (13)

which yields, by the dominated convergence theorem , that

T

n-+0 lim E l l ( f n ( 4 w , X n ( t ) ) - f ( 4 w, m)2 dtl = 0 (14)

Let ( X , Y, K 1 , K 2 ) be the unique solution to the backward SDE with re-

flecting barriers with g( t , w) = f ( t , w, X ( t ) ) replacing f ( t , w , .). Then

T

X ( t ) = I + 1 f ( s , w , X(s ) )ds + K1(T) - K1( t )

l T ( X ( t ) - L( t ) )dKl ( t ) = ( U ( t ) - X ( t ) ) d K 2 ( t ) = 0 , (17) LT

T

- (K2(T) - K 2 ( t ) ) - / Y' (s )dB(s) , b'0 5 t 5 T , (15) t

L( t ) 5 X ( t ) 5 U ( t ) , "0 5 t 5 T, (16)

almost surely.

Next we show that Xn( t ) converges to X ( t ) , and hence, X ( t ) = X ( t ) . From

the It6 rule,

(Xn (S) - X ( S ) ) ( Y ' ( S ) - YL(s))dB(s) - lY'(s) - Y;(s)l2ds. +2 LT LT

Keeping in mind that L(s) 5 Xn(s ) 5 U ( s ) and L(s) 5 X ( s ) 5 U ( s ) we

have

330

- 2 L T ( X n ( S ) - X(s ) )dK ' ( s )

4' = -2 1 (Xn(S) - L(s) )dK l (s ) + 2

= -2 1 (Xn(S) - L(s) )dK l (s )

T

( X ( s ) - L(s ) )dK l ( s )

T

1 0

Therefore] limn+mE[(Xn(t) - X( t ) ) ' ] = 0. So, X ( t ) = X ( t ) . Conse-

quently, (XI Y, K' , K 2 ) is a solution to the backward SDE with reflecting

barriers with coefficient f ( t , w , z), and the terminal condition (. This ends

the proof.

Similarly, it can be seen that

33 1

Acknowledgements: Financial support of the British EPSRC (grant no.

GR/R91144/01) is gratefully acknowledged.

References

1. J.Cvitanic and LKaratzas: Backward Stochastic Differential Equations With

Reflection and Dynkin Games, The Annals of Probability 21:4 (1996) 2024-

2056. 2. Duffie.D and Epstein.L, Stochastic differential utility, Econometrica 60

(1992) 353-394. 3. N.Ikeda and Swatanable, Stochastic Differential Equations and Diffusion

Processes, North-Holland and Kodansha,Amsterdam and Tokyo, 1981. 4. LKaratzas and S.E.Shreve, Brownian Motion and Stochastic Calculus,2nd

ed. Springer, New York, 1991. 5. N.E1 Karoui, C.Kapoudjian, E.Pardoux, S.Peng and M.C.Quenez, Re-

flected solutions of beckward SDEs and related obstacle problems for PEDs.

Preprint,l995.

6. E. Pardoux and S. Peng, Adapted solutions of backward stochastic differential equations, Systems and Control Letters 14 (1990) 55-61.

7. T.S.Zhang, On the strong solution of one-dimensional stochastic differential

equations with reflecting boundary, Stochastic Processes and Their Applica-

tions 50 (1994) 135-147. 8. T.S.Zhang, On the quasi-everywhere existence of the local time of the solution

of a stochastic differential equation, Potential Analysis 5 (1996) 231-240.

BURGERS EQUATION AND THE WKB-LANGER ASYMPTOTIC L2 APPROXIMATION OF

EIGENFUNCTIONS AND THEIR DERIVATIVES

A. TRUMAN

Department of Mathematics, University of Wales Swansea, Singleton Park, Swansea SA2 8PP, UK

Email: A . [email protected]

H.Z. ZHAO

Department of Mathematical Sciences, Loughborough University, Loughborough, L E l l 3TU, UK

Email: [email protected]

In this paper we study the WKB-Langer asymptotic expansion of the eigenfunc-

tions of a Schrodinger operator H = -+ti2& + V(z). Applying these asymptotic

formulae we prove that the exact L2 eigenfunction Q J I E ( N , ~ ) (and its derivative

hQ&(N, f i ) ) of the Schrodinger operator with a well-shaped analytic potential are

approximated up to arbitrary order hm by the semi-classical WKB-Langer approx-

imate eigenfunction QE, ( ~ , f i ) , ~ (and its derivative hQ& ( N , h ) , m ) respectively in

L2, i.e. IIwE(N,fi)-Q'E,(N,fi),mttL2 = o(hm+l), l l h Q & ( ~ , h ) -hw&n(~ , f i ) ,m I IL2 =

O(hm+l) uniformly for any N. Here Em(N,h) approximates E ( N , h ) up to m-th

order (in h) and satisfies the m-th order quantization condition. There are applications of this limit to Burgers equations, turbulence and the large scale structure

of the universe.

n

1. Introduction

Consider the following second order Schrodinger operator on IR1

1 a2 2 ax2

H = --h2- + V ( x )

We are interested in studying approximate eigenfunctions and their deriva-

tives by using our Hamilton Jacobi methods, extending the results in Tru-

man and Zhao 21 to more general potentials than linear harmonic oscillator

ones.

It is well known that the WKB method leads to approximate eigen-

functions. However, the WKB solution is not finite at the turning points.

Langer therefore introduced Bessel functions to study the wave functions

332

333

near the turning points (Langer g 1 1 0 7 1 1 ) . The WKB-Langer semi-classical

solution associated with the N-th Bohr-Sommerfeld quantization rule gives

the approximation for the exact eigenfunction associated with the N-th ex-

act eigenvalue. As we will explain soon, however, the exact eigenfunction

is not dominated by the WKB or WKB-Langer solution pointwise. The

approximation of the WKB or WKB-Langer semi-classical solution to the

eigenfunction needs to be justified mathematically. One way to do this is to

compare the error of the approximation with the modulus function M ( z ) of Airy functions (Olver 15). However, M ( z ) is related to an Airy function

which is exponentially large for large 1x1 and is excluded in the approxi-

mation of the eigenfunction. In this paper by using the Hamilton-Jacobi

continuity equations we will prove that the exact eigenfunction Q E ( N , R )

is approximated by the WKB-Langer semi-classical approximate solution

Q E , ( N , ~ ) , ~ in L2(R) up to m-th order in h. Moreover, we will prove its

derivative hQk(N,h) is also approximated by hQ’,m(N,h),m in L2(R) up to

m-th order in h. Here E,(N, h) approximates E ( N , h) up to rn-th order

(in h) and satisfies the rn-th order quantization condition. Note that the

modulus function used in Olver l5 is not in L2(R), so Olver’s result does

not lead to the approximation in L2(R) studied in this paper. Our re-

sults are valid for large quantum number N as long as N and f i satisfy a

rigid relation such that E ( N , h) is bounded and V(z) satisfies some minor

condition at the turning points. Our proof is simple. We do not need to

use the pseudo-differential operator theory of Helffer (Helffer, Martinez and

Robert 7, or the sophiscated canonical operator method of Maslov (Maslov

and Fedoriuk12). Moreover, our proof leads to the approximation of the

derivatives in L2(R). We should point out that if we use the WKB expan-

sion near the turning points although the first term in the WKB expansion

is in L2(R), the second term is definitely not in L2(R). This makes i t dif-

ficult to use the WKB expansion as a method of approximating the exact

eigenfunction in L2(R). The WKB-Langer expansion does not suffer from

this difficulty. We prove that the WKB-Langer semi-classical approximate

wave function (which is well defined at the turning points) actually gives

the correct approximation to the exact eigenfunction in L2(R). See Simon17

for the low lying eigenvalue case.

We should like to point out that the pointwise approximation of the

WKB-Langer semi-classical solution (which reduces to WKB for z being

sufficiently far away from the turning points) is not mathematically justi-

fied. To see this we refer to the asymptotic expansion formula (2.41) of

this paper. As the first term cos(k J,” b(z)dz - 2) has many zeros, b= (XI

334

4 sin( i s," b(z)dz - :)&(z)h is the dominant term at these zeros, not

1 cos(i s," b(z)dz - 2). However, the first term in the WKB-Langer

semiclassical asymptotic expansion approximates both the eigenfunction

and its derivative in L2(R), the natural norm of quantum mechanics.

There will be many applications of the results of this paper, e.g. quan-

tum probability, quantum tunnelling problems (Jona-Lasinio, Martinelli

and Scoppola ', Simon la) etc. The Hopf-Cole transformation applied in

this setting should also yield new results for Burgers equation and its in-

viscid limit. This is not so surprising since the Hamilton-Jacobi continuity

equations first arose in this context. We do not include these results here

due to the length of this paper. But we aim to study some of these appli-

cations in our future publications.

b y )

bz (T)

2. WKB-Langer asymptotic expansions

Let Qg be the eigenfunction of H with eigenvalue Eh. Then Qg and Eh satisfy the following time-independent Schrodinger equation:

d2 dx2

h2-Q;(z) + Q 2 ( ~ ) Q k ( z ) = 0.

Here Q2(z) = 2(E - V(z)). Let the real function be defined by

( d 2 ( E - V(z)) if V(z) < El

- E ) if V ( z ) > E. b(z) =

First we consider the simplest case where we have only one turning point

in order to obtain some useful formulae. In the next section we will apply

these formulae to more complicated but practical and useful situations. Let

T denote a turning point and for simplicity here we suppose r is a simple

zero of Q2(z). In this section we consider the case when V ( z ) > E if z < T

and V(x ) < E if x > 7- for a T E R1. Then Q(x) is simply

Define a complex single-valued function [(z) by

335

and a real valued function @(z) by

The following simple proposition tells us about the smoothness of @(z)

and the dependence of @E on E.

Lemma 2.1. Suppose that V ( z ) is C" and 0 < IV'(x)I < 00 for z E

[a, b] a small neighbourhood containing r , then the function @(x) is C" on [a, b] and for any fixed z, I@E(z) - @fi(z)l = O(IE - El) and I@&(z) -

@k(z)I = O(IE - 81) for any E , E E V ( [ a , 61). In particular I@(z)I > 0 for z E [r - 6, r + 61 for some 6 > 0. Furthermore, if V ( z ) is analytic for z E [r - 6, r + 61, then @(z) is analytic on [r - 6, r + 61. Proof. For y E [V(b),V(a)] , define z = V-'(y) to be the solution of

V(z) = y , assumed unique for z E [a, b]. Set

G-dy) = V- l (y) ,

Gn(y) = G k - l ( ~ ) , n = 0 , 1 , 2 , . . .,

and

&E(Y) = @E(V-'(y)).

Then for y E [E, V(a ) ] , changing the integration variable and integrating

by parts lead to

Ly Gl(y)(2(y - E))Qdy. 1 - 1

3 3 (2 (y -E ) )+ = --Go(y) + - .

By induction we can prove for each n 2 0,

336

Similarly, for y E [V(b), El, for each n 2 0,

I t follows that & ~ ( y ) is smooth in y and Lipschitz continuous in E. Then

the smoothness of @.E(x) in x and Lipschitz continuity in E follow from

the identity @E(x) = &E(V(X) ) . The Lipschitz continuity of S b ( x ) in E follows similarly due to the derivative formulae (2.4) and (2.5). From the

definition of Go(y ) , IGo(y)l > 0 for y E [V(b), V ( a ) ] . Therefore from (2.4) and (2.5) for n = 0, I6(y)I > 0 for y sufficiently close to E , which implies

I@(z)I > O for x sufficiently close to T , say x E [T - 6, r + S ] for a S > 0.

In the following we always write r* = T * 6, for a S > 0 such that 0 < IV'(x)I < 00 and I@(x)I > 0 for x E [ r - , r f ] . Denote S(z) = @-'(z)@''(z)

which is smooth for x E [T- , T+ ] by Lemma 2.1. Moreover e(x) is analytic

for z E [r-, r+] if V ( x ) is analytic. Define

Lemma 2.2. O n [ r - , r f ] , aj E CW,@ E C", are bounded. Therefore, for x E [r-, 7'1, the power series cj"=, aj(x)h2j , 1 + cj"=, ,6j(x)h2j, and their derivatives C,"=, ai(x)h2j , Cj"=, ,L?: (x)h2j are asymptotic expansions as ii -+ 0 f o r x E [r-, r+] in the sense of Poincare'. And if V ( x ) is analytic, then a j (x ) ,P j (x ) are also analytic for x E [T - ,T+] . Furthermore, a j ( x ) , &(z) and & ~ j ( x ) , &.Pj(x) are Lipschitz continuous with respect to E uniformly in x fo r x in [r-, r f ] .

Proof. Define for any function e,

Suppose for x E [r-, T+] , f i (x) is smooth and V ( x ) # 0. Define V-'(y) to be the solution of V ( x ) = y for y E [ V ( r f ) , V(r- ) ] and

G- l (x ) = lx &x)dx.

337

Let G-l(y), Go(y), Gl(y ) , . . . be defined by

~ - I ( Y ) = G-i(v-'(y))

G n ( y ) = en-l(y), n = O , I , 2 , . . .,

G(Y) = c.(v-'(Y)). and

Similar to (2.4), using the integration by parts formula and induction prin-

ciple, we have for y E [E , V(T- ) ] , and each n = 0 , 1 , 2 , . . .,

and for y E [V(T+), El, and each n = 0 , 1 , 2 , . . .,

Applying (2.8) and (2.9) to al, a2,. . . , we have the smoothness of aj and

pj in z and Lipschitz continuity in E. The rest of the lemma follows

from van der Corput's fundamental theorem on asymptotic series and its

consequence on asymptotic series with a parameter (Theorem 4.1 and P391,

We follow Langer 9110111 to define wave functions using Bessel functions

near the turning point. Alternatively one can use Airy functions (Olver

13,14,15, Heading 6 ) . Define

van der Corput 2z). See also 0lverl3>l4.

K"4

= C-ESJ-&) 1 6 +c+[+J+( , ) 6 (2.10)

e$i(Jz," b(z)dx)i x(C-J-g(- f i J,'b(z)dz) + C+J+( - i J , 'b (s )dz) ) , if x < 7 , = I x(C-J-+( iJ,"b(z)dz) +C+J+(kS,"b(z)dz)) , if z > 7,

where J-+(- ) and J+( - ) are two Bessel functions and C- and C+ are

some constants, and then, Langer's approximate wave function is defined

(J," b(z)dz)i

by

!P~(x) = @ ( ~ ) K ' ( Z ) . (2.11)

338

The following celebrated result was given in Langer g > l O z l l .

Proposition 2.3. (Langer) For any E, suppose V ( x ) is smooth near the turning point T : V ( T ) = El and V'(T) # 0 , then there exists 6 > 0 such that 0 < IV'(z)I < 00 and I@(z)I > 0 for x E [T- , T + ] , T* = T f 6 . The function Qo(x) defined by (2.11) satisfies the following differential equation for z E [T- ,T+]

(2.12) d2 1

-Qo(x) + jgQ2(x)Qo(x) = O(x)Qo(x). dx2

Furthermore, fo r x E [T-, T+] the solution of the equation (2.1) has the following representation

Q'(Z) = Q;(x)A'(x) + Qo(x)B'(x). (2.13)

Here A(x) , B(z ) satisfy the differential equations fo r x E [T- , T+]

(2ti28-2Q2)A'+(ti20' -2QQ')A+h2B"+ti20B = 0 , BA+A" +2B' = 0.

And moreover, A(x) , B(z ) have the asymptotic expansions fo r x E [T - , T + ] ,

as h 4 0 , 00 M

j=1 j=1

where c~ j ,p j given b y (2.6) are smooth and bounded functions for x E

[T- , T ' ] .

For x < T , recall some standard results about Bessel functions (c.f. e.g.

Whittaker and Watson 25)

-e2 " i JL ( - - ; L ' b ( z ) d x ) = e : ' J ~ ( ~ / ' b ( x ) d x ) = I + ( z L 1 ' b(x )dx ) ,

3 t i x

and

Then for x < T . we have

339

3 ;lr = (lT b(z)dz)s(-C-K1(- 7r b(x)dz) - (C+ - C - ) I ; ( i s,'b(z)ds)). 1 2 a

Notice that I ; ( JZr b(z)dz) - exp{ JzT b(z)dz} when f i is small for

z E [T- ,T - is]. In order to have a L2(R) solution, we have to choose

C- = C+. Recall for z E [T - , T - $61, when ti is small,

with M h ( x ) = 1 + O(h) having an asymptotic expansion. If we take C- =

C+ = &, then

and so when tL is small,

(2.16)

For x E [T + ? ~ S , T + ] , when f i is small,

(2.17)

with

j=1 j=1

and

(2.19)

Notice that L1 and L2 in (2.19) for J-; are the same as those in (2.17) for

So for z E [T + $5, T + ] , using the same C- and C+ as in the region

x E [T-,T - $71, i.e. C- = C+ = &, we have from (2.10) and (2.11), as

J; .

340

h is small.

Qo(x) = (2.20)

1

4

1

4 (x)dx - -.rr)LF(x) + b(x)dx - -.rr)Lg(x)).

It is noted that the term sin( sTx b(x)dx - ;7r)Lg(x) = O(h) should not be

neglected.

For simplicity in the following we only consider bound states where we

take C- = Cf = 6. So for r- 5 x < r ,

From P366 in Whittaker and Watson 2 5 ,

341

From P354 in Whittaker and Watson 2 5 ,

i t follows that

I t follows that

d dx -K"z)

So for T < rc 5 T + ,

(2.22)

and the term ( ~ ~ , " b ( z ) d z ) ~ ( J - ~ ( ~ ~ ~ b ( z ) d z ) - J+( iJ :b(z)dz)) is

bounded.

I t is evident that for z E [T - ,T+] , for any ti > 0, 90, h9&, h29&' are bounded. But we need some uniform (in h) estimate. We prove the

following lemma which will be used in the next section.

Lemma 2.4. Assume conditions of Lemma 2.1. Then for any interval [a, b ] , J'(90(x))2dz, Jab(hQ;(z))2dz and Jab(h2Qg(z))2dz are bounded uniformly in ti.

Proof. We only need to prove that (90(x)) ' is integrable uniformly in ti with respect to z if the turning point T E [a, b] . Note z+K+(z) , z i J ; ( z )

By similar argument we have

342

and , z i J -+ (z ) are bounded, so (I J,' b ( y ) d y l ) k K h ( z ) is bounded uniformly

in k. So by the definition of \ko and Lemma 2.1, we know

for a constant M > 0 wich is independent of h and z. But T is a simple

zero of V(z) - E, so the improper integral s, dx is convergent.

That is to say s , b ( \ k ~ ( x ) ) ~ d x is bounded uniformly in h. Then results for

J:(h\kb(z))2ddz and J:(h"@&'(~))~dx follows from (2.21), (2.22) and (2.12)

Away from the turning point, Langer's construction turns out to be a

simple formula. We first study the asymptotics of (2.13) for z E [T- , T - $1 and z E [T + $5, ~ f ] . Then we will extend the solution to the whole line

R1.

From the asymptotics (2.15) and K; for large argument (P367, Whit-

taker and Watson 2 5 ) 1 for x E [T - , T - +6] for small ti,

b

(I J,r b(Y)dYO

respectively.

with Rf i (z) = 1 + O(h) and having an asymptotic expansion. It turns out

from (2.16), (2.21) and (2.23) that for z E [T-,T - 461

(XI

(2.25)

where

F ( z , h) = M"z)B(x)+h-M @'(x) fL (x)T+b(z)Ryz)- A(%) A ( x ) = l+O(h) . @(x) h

It is obvious that P-(x) has an asymptotic expansion in powers of ti, say

(2.26)

Dc)

P-(z, h) N 1 + = - p i ) j $ j ( z ) , j=1

343

for some smooth and bounded functions $j on [T - , T - ;6] which are com-

binations of aj and ,Bj and the asymptotic expansions of M h ( x ) and Rh(x). In particular we have fked values of $j (T- ) .

Note if we take

rv1 WI A, = aj(z)h2j and Bm(z) = 1 + ,Bj(z)ti2j

and define

Then

P-(z , ti) - PJz , ti) = O(tirn+1).

For z E [T+ $S,T+] , when ti is small, again from P362 in Whittaker and

Watson 25 ,

(2.27)

and

Similarly

(2.28)

Note that R1 and R2 in (2.28) are the same as the ones in (2.27). It

turns out that

344

From (2.20), (2.22) and (2.29)

Clearly we have

and

Now by (2.13), we have for any fixed r + i6 I x I r f , for small f i ,

where

and therefore PI(z) = 1 + O(fi2) and Pz(x) = O(fi). Moreover, PI and

PZ have asymptotic expansions of even powers of h and odd powers of h respectively, say the following

for some smooth and bounded function 4j on [r + i6, r+]. In particular we

have fixed values for $j (r+). We study the WKB asymptotic expansion outside [r-, r f ] . From Theo-

rem 26.3 in Wasow 24, for x < r - :6, the Schrodinger equation (2.1) which

can be reduced to a system of 2-dimensional singular perturbed differential

equations possesses a solution of the form

(2.33)

345

and for x > r + i6, the Schrodinger equation possesses a solution of the

form

(2.34)

and P- (x) ,Pki (x) have asymptotic expansions in tr. for x < r - i6 and

x > r + i6 respectively. It is easy to prove the following lemma.

Lemma 2.5. For x < r - $6, the function P-(x) satisfies the following dafferentaal equation,

b+ (x) $.- (x) = - - 1 fi- d2 (b- + (x) P- (x)) , 2 dx2

and f o r x > r + i6, P*Z(x) satisfy

b+(X)-&P*'(X) d = &-hz-(b-qx)P*Z(x)). 1 d2 I

2 dx2

(2.35)

(2.36)

Proof. The proof is some simple elementary computations. We leave it to

the reader. 0 Define $O(Z) = 1, and

4 q - ) - ; s,' - b - ~ ( y ) ~ ( b - 3 ( y ) $ ~ - ~ ( y ) ) d y , ifx < 7-,

q + ( ~ + ) + $ J:+ b - + ( y ) ~ ( b - q ( y ) ~ ~ - ~ ( y ) ) d y , ifx > r+, (2.37) + j ( X ) = i j = 1 , 2 , . . . ,

where $j(r*) are defined in (2.26) and (2.32). It is evident that 4j(x, E ) is Lipschitz continuous with respect to E as r* is Lipschitz continuous

with respect to E. It turns out that $j ( j = 0,1, . . .) satisfy the following

iterated time-independent Hamilton Jacobi continuity equations (Truman

and Zhao 20), for x < r- and x > r+,

d 1 d2 1 bf(Z)-$j(X) = --(b-qZ)&I(x)),j = 0,l;

dx 2 dx2 (2.38)

with convention that 4-1 = 0. Note that $j(x) are bounded for any

x < r- and x > r+ and Lipschitz continuous with respect to E. There-

fore Cj"=,$j(x)(-fi)j, for x < r-, and Cj"=o$j(x)(ffii)j, for LC > r+, are asymptotic expansions in the sense of Poincare, as f i -+ 0 by the

van der Corput Theorem. It is a simple exercise to check that formally

Cj"=oq5j(x)(-h)j,for x < r-, and Cj"=04j(z)(ffii)j,for x > r+, satisfy

(2.35) and (2.36) respectively. However, since P- and P*i should have

unique asymptotic expansions,

00

P-(x) N C $j(x)(-fi)j, for x < r - , j =O

346

00

~ * z ( z ) N C ( ~ ~ ( z ) ( * h i ) J , for z > T+. (2.39) j = O

As P-(z ) has the asymptotic expansion (2.26) for z E [T - , T - ;a] and

satisfies (2.35), so it is easy to check that $j ( j = 0,1 ,2 , . . .) satisfy (2.38)

for z E [T- ,T - is]. Similarly (2.38) is satisfied for z E [T + it?, 7+].

Therefore $j ( j = 0 ,1 ,2 , . . .) are smooth for z < 7 - $5 and z > 7 + $5. In particular, $ j are smooth a t x = T*. To see the asymptotic behaviour

of P*i(z) for z a t infinity, we can easily show that l$j(x)I 5 cj1zl-j and

l$>(z)l 5 cjlx1-j for a cj. Therefore for large z, by the van der Corput oc) 00

Theorem, C $j(z)(-h)Jforz < 7-, and C $j(z)(fhi)jforz > T+ are also j =O j =O

asymptotic expansions of powers of z for large 1x1. In particular P-(z ) is

uniformly bounded and P- (z ) - 1 = O(fi) uniformly in z for z < T - . The

same conclusion about P*i is true for z > T+.

It is noted that $j(z) are analytic for z < 7- and z > T+ if V ( z ) is

analytic. We will only need this in the next section.

I t is important to note that linear combinations of 9+2 and 9-2 are also

solutions of the Schrodinger equation (2.1).

We have to choose the appropriate combination of V i and V i for

x > T + to match the solution from Langer’s construction. For the bound

state we set 9 = 9- for z < r-, i.e.

(2.40)

where P-(x) = 1 + O(fi) for small f i uniformly in z for z < 7-, and

9 = exp{-$i}Q+i +exp($ i}W for z > T+, i.e.

k LT *‘(z~) = b-+(z)exp{-- b(y)dy} x ~ - ( z ) ,

(2.41)

where

Pl(z) N 1 + Cj”=1(-l)j$~j(z)h2j and P2(x) N Cj”=1(-l)j$2j-l(z)h2j-1

and Pl(z) = 1 + O(h2) and P2(z) = O(h) for small h uniformly in z for

z > T+. And for z near 7, 9’(z) = Qh(z)A(z) + *o(z)B(z) as in Proposi-

tion 2.3. From our construction we know that 9 is smooth on R’. Similar

combinations were also used in Furry 4 , Heading to exploit and Berry

the Stokes’ phenomenon in physics literature. We formulate a proposition.

347

Proposition 2.6. Assume all conditions of Proposition 2.3. Then @(x ) which is given by: (2.40) for x < T - ; (2.13) for r- I x 5 r f ; (2.41) for x > I-+, is a smooth solution of the Schrodinger equation (2.1).

Remarks. (i) The asymptotic expansions (2.39) only make sense for fixed x # T . They give a pointwise WKB asymptotic expansion ((2.4O), (2.41) for x < I- and x > T respectively) of the wave function for x # r . Although the first term is in L2(R), the second and higher terms are not due to higher order singularities of q5j (j = 1,2, . . .) at x = r. The key to solve this problem is to use WKB-Langer semi-classical asymptotic expansions presented in this paper.

(ii) Equation (2.1) possesses another solution @+ which for x < r - 4 S is of the form

9 + ( x ) = b-$(x) exp{ - b(y)dy}P+(x), : L (2.42)

where P+ has an asymptotic expansion in powers of ti. We can choose appropriate C- and C+ different from before so that for x E [r-, r - is], !P;(x) given by Langer’s formula (2.13) has asymptotic expansion (2.42). A smooth extension to the whole interval (--00, m) can be done in the same way as before and by exploiting the asymptotic properties of the Bessel functions. This solution is linearly independent of 9 given in Proposition 2.6, but is exponentially large for x < I-.

3. Semi-classical approximation of eigenfunctions and their derivatives in L2

Consider a smooth well-shaped potential V ( x ) bounded below with

limlzl+mV(x) = +00. Then by the limit point criteria H = -:& + V ( x ) is a self-adjoint operator with discrete spectrum {E(N, h ) } ~ = ~ , l , ..., E ( N , Ti) -+ +m as N -+ +00 with corresponding orthonormal eigenfunc-

tions @ k ( x ) for any fixed ti > 0 (see Reed and Simon 16). Consider the

N-th eigenvalue E( N , h) and corresponding eigenfunction @k(x ) . Suppose

there are only two classical turning points r l (E) and r2(E), the only two

roots of V ( x ( E ) ) = E. Assume V ( x ) is smooth near q ( E ) and r2(E) and

V‘(r1) # 0 and V ‘ ( T ~ ) # 0. Therefore we can apply Langer’s construction

of the wave function near both r1 and 7 2 .

First by Lemma 2.1, there exists 6 > 0 such that if writing rf = ~j f 6, j = 1,2, then 0 < IV’(x)I < 00, lQT1(x)l > 0 for x E [rL,r1 ] and 0 < IV’(x)I < 00, 1QT2(x)I > 0 for x E [r;,rz]. Let 6 > 0 be small enough

such that r,’ < r p . We construct the wave function Jrk(x) by Proposition

J

348

2.6. In the following, Qk,m(z) denotes the first m terms of WKB-Langer

semi-classical asymptotic expansions in 5 different regions respectively. For

x _< 71, take

m

= Q E , O ( Z ) ( l + X(-h)j$j(Z) + 0(hrn+l))

= Qk,,(x) + QE,o(z)0(hm+l),

(3.1) j=1

with a uniform 0(hm+') for x E ( - w , T ~ ] . Hence Q;(x),Qk,,(x) and

Q;,,(x) are exponentially small for z < 71. For 71 5 x 5 T:, take

Langer's construction and apply Lemma 2.4,

Qk(4 = Q E , O b ) % 4 + QL,,(x)A(x)

IYl = QE,&)(l+ c Pj(Z)h2j + U ( P + l ) ) (3.2)

j=1

rw1 +(hQk,o(x))( c CYj(z)h2j-l + 0(hrn+l))

j=1

= Qk,,(z) + QE,o(x)O(II'"+').

For 71' < x < T;, set

(3.3)

= Qk,,(x) + 0(hrn+l),

where $ j are defined by (2.37) with $j(r:) derived from (3.2) as initial

conditions, i.e. $j(x) = $ j ( ~ ; ) f $ JTy+ b4(~/>A(b-+(y)$j-l(y))dy. On the

other hand we should also have

349

= Gk,&E) + o(hm+l)' (3.4)

where $j are defined by (2.37) with JJ(r;) derived from (3.5) below as

initial conditions, i.e. $ j ( z ) = $ j ( r F ) - $ szTz b:(y)A(b-i(y)Jj-1(y))dy.

And for 72 5 x 5 r2$, take Langer's construction

-

G k ( x ) = G,,o(x)B(x) + G.',,,(z)A(z)

r-1 = GE,O(Z)( l+ c &(z)h2j + O ( P + l ) )

j=1

IF1 +(hGL,o(x))( c clj(z)h2j-1 + O(h"f1)) (3.5)

j=1

= Gk,,(z) + GE,o(x)o(hm+l).

For x 2 r z , take

' k k ( x ) = b-i(z)exp{-- b(y)dy} x p - ( x ) : 1: m

= !i&,o(x)(l+ c $ j ( Z ) ( - h ) j + O(hrn+l))

= Gk,,(z) + GE,O(z)O(hm+l)l

j=1

with a uniform O(hm+l) in z for x > 7-2'. Here +k(x) and G;,,(x) are

both exponentially small when x > r2$ is large. Here P- in (3.1) and P - in (3.6) are defined as in Section 2. From Section 2, we know that 9 is

smooth for z E (-00, r;] and ?t is smooth for x E [r:, 00).

Remarks. (i) From Section 2, especially Remark (i i) following Proposi- t ion 2.6, Equation (2.1) also possesses a solution Q+ # L2(R) given by

(2.42), i e . 9 i ( x ) = b-*(x)exp{i szT1 b(y)dy}P+(s) , for x E (-ml~,-] .

350

The smooth extension of the solution to the whole space (-00, m) can be done by using the same method as (3.1)-(3*6) and (3.8). The solution is linearly independent of the L2 solution 9 given in (3.1)-(3.6) and (3.8). Furthermore any solution 9 1 of (2.1) is of the f o r m 9 1 = c19- +c29'+ for

constants c1 and c2. But for a L2 solution 91, c2 = 0 is satisfied, whence 91 = c1Q. That means any L2 solution Q1 of (2.1) is linearly dependent on 9, which is equivalent to the vanishing Wronskian determinant property for any x ,

d d -@1(x)9(x) d x - 91(2 ) -Q(x ) d x = 0.

In particular, we can choose c1 = 1/11911 so that 11Q111 = I. Therefore 9 is the unique L2 solution up to normalization. Here we state our results for

the WKB-Langer solution 9. One can give our results for the normalized wave function if one likes.

(ii) The semi-classical WKB-Langer approximate solution ~ E , ~ ( x ) is given by the first m terms of the series in (3.1)-(3.6) in five different regions respectively. Note

m

j = l

lim ~ E , ~ ( z ) ZTT;

and

lim 9 ~ , ~ ( x ) = b(y)dy}P;(r;, f i ) ZIT;

from (2.26). These two limits are different, but the difference is 0(fim+') as P-(z, f i ) - PG(x,fi) = 0(hm+'), so are limxfT; hQk,m(z) and

limxLT; k9b,m(x). A similar remark applies for x = I-:, 72, I-;. However, the discontinuity of Q E , ~ ( X ) and t i9&,m(x) at only four discrete points r1 , r t , 1-2, r$ does not give rise to any dificulty in L2(R) as ~ E , ~ ( x ) is differentiable on (-m, TT) , (I-;, I-:), (T?, I-;), (I-;, I-,') and (I-:, m) respectively. One can choose a continuous or even differentiable 9 ~ , ~ . But this is not necessary here and is not the point of the paper.

I n the following, Q E , ~ ( X ) and f i9k ,m(x) at I - ~ ( j = 1,2) are not necessarily defined. But one can define them by either the left limits or the right limits as these two limits are asymptotically close as f i -+ 0.

The semi-boundedness of V guarantees that the Schrodinger operator H has a unique L2 (R*) eigenfunction up to normalization. As V is assumed to

be smooth, this eigenfunction is also smooth. For the validity of the formula

for the exact wave function @ E , 9~ and &, must be linearly dependent

-

&

351

in x E [r:, r;]. That is to say the following Wronskian determinant must

vanish for any ti > 0, x E [T:, r;]:

(-Q;(x))@;(x) d - *;(x)-QE(x) d - i i = 0. dx dx (3.7)

We will see soon that quantization condition gives the exact eigenvalue E , i.e. { E ( N , ~ ) } N = o , I , .... Now we transform (3.7) to an explicit equation of

E. For this we first differentiate (3.3) and (3.4),

and

It is crucial here that the leading term in (3.8) has a different sign from

the leading term in (3.9). Substituting (3.3)-(3.4) and (3.8) and (3.9) into

(3.7) we obtain

= -H"x). (3.10)

352

Here the formula of H can be given explicitly if one wants to. We note

here that H " ( z ) is bounded for all z E [ T ~ , T ; ] uniformly in h. It turns

out that

7r sin(; b(y)dy - -) = 4hHh(z).

2 (3.11)

Recall that the Wronskian determinant (3.7) is vanishing for all z E [T:, T;]

is equivalent to that the Wronskian determinant (3.7) is vanishing at a

particular point (eg. see Hartman 5). Therefore (3.11) is equivalent to

1 J2(E - V(y))dy = ( N + -)7rh + harcsin(4hHfi(M)), (3.12)

for N = 0 ,1 , . . .. Here M E [T?, T;] is the minima of V(z). The solution

E = E ( N , h) of the above equation gives the exact N-th eigenvalue of the

Schrodinger operator.

We take the first m terms in the asmptotics expansions of Qi(z) and

*;(x), denoted by 9;,,(x) and *;,,(x). We require the Wronskian de-

terminant vanishes a t x = MI i.e.

6 2

(3.13)

Similar to (3.7), we will see that this gives discrete values

{E,(N, ~ ) } N = o , J , ... as follows. First we go through all the calculations

of (3.8)-(3.12) for Q;,,(z) and *;,,(x), then we derive

(y5,7r(4)*t,m(4 d - Q 5 , 7 J ( ~ ) ~ Q E , , ( ~ ) / s = M d - i i = 0.

7r b(y)dy - -) = 4hHk(M).

2 (3.14)

Similar to H ( z ) , H k ( M ) is also bounded uniformly in ti. It is followed

from (3.14) that

7 2 ( E ) 1 / 7 1 ( E ) /2 (E - V(y))dy = ( N + -)7rh + harcsin(4hHL(M)),

2 (3.15)

for N = 0,1, . . .. The solution of the above equation depends on m, denoted

by E,(N, h). That is to say we have

1 d 2 ( E m ( N ) - V(y))dy = (N+ -)nh+h arcsin (4hHk (Ad)) , (3.16)

2

for N = 0,1, . . .. It will be seen that that E,(N, h) is an approximation to

E ( N , h) for each N (see (3.26)).

Recall the Bohr-Sommerfeld quantization condition

353

Here TI(&) and 72(E0) are the only two roots of V(a:) = Eo. We will

analyze the solution in ascending order, setting E = Eo(N,h) for N =

0,1, . . .. The following result is uniform for all N if E ( N , h) is in a compact

subset of { E : JV(2)<E J- da: < +m}. For low lying eigenvalues 2(E-V(z))

( N is fixed), similar estimate was obtained by Simon (1983).

Lemma 3.2. Suppose the same conditions as in Lemma 3.1 and V i s analytic. Assume the E ( N , h ) and Ern(NIh) satisfies following travel

d y < +GO and 0 <

dy < +a and Eo(N, ti) is the solution of the

Bohr-Sommerfeld quantization equation (3.17) and 0 < (V’(x)( < 00 and IV”(a:)I < 00 fo r 2 between

T z (E ( N, ti)) 1 time inequality < ST1(E(N,h)) J2(E(N,fi)-qy))

JTl (Em ( N J 3 ) J2( Em (N,h) - V(y))

72 (Em ( N , h)) 1

min{n(E(N, h)) , 71(Em(N, h)) , 7i(Eo(N, h ) ) )

and

and between

and

Then

E ( N , h) = Eo(N, h) + 0(h2) , (3.18)

354

is O(hz) uniformly in N . E ( N , h) > Eo(N, h). Then the above gives

Without losing generality we assume that

That is

(3.21)

355

> 0.

Together with (3.20), we have

72(E(N,ft)) 1 dY(E(N, ti) - Eo(N, ti)) = 0 ( h 2 )

TI ( E ( N , W ) &(E(N, ti) - v(Y)) 0

We need a spectral gap result. This can be proved by using the Bohr-

Sommerfeld quantization rule and Lemmas 3 .2 . We first prove the following

lemma.

Lemma 3.2. I f 0 < JV'(x)) < 00 fo rx E [71(E0(N+l,ti)),71(Eo(N,h))]U [72(EO", ti)), 72(EO(N + 1, and 0 < S7:;'E": d m dx < 00 for

E = Eo(N, Ti)) and E = Eo(N + 1, h)) , then for suficiently small ti > 0,

s Then (3.18) follows. The proof of (3.19) is similar.

356

I Eo(N + 1, ti) - Eo(N, 27rh

(3.22)

Proof. By the Bohr-Sommerfeld quantization rule we know

But also

357

And

for sufficiently small ti. Here M2 in above is a constant. The lower bound

0 of Eo(N + 1, ti) - Eo(N, ti) in (3.22) follows.

358

It follows that there exist constants C1 > 0 and C2 > 0 such that for

sufficiently small h > 0

Clh 5 E ( N + 1, h) - E ( N , h) 5 Czh,

and

Clh 5 Em(N + 1, h) - Em(N, h) 5 Czh.

Therefore there exists a neighbourhood IN of Eo(N, h) of which the length

is O(h2), there exists one and only one E and Em which are E(N, h) and

E,(N, h) respectively. But it is easy to see that

d I d - h ( ~ Q E ( N , f i ) , m ( x ) \ E ( N , ~ ) , m (.) - ~ \ E ( N , f i ) , m ( x ) \ ~ ( ~ , f i ) , m ) ( x ) I ~ = M

+O(Pf l ) = 0. (3.23)

Recall

d I d -

d x = 0. (3.24)

But from the construction of Q m we know that there exist constants L1 > 0

and L2 > 0 such that

Q E , w,ft) ,m ( x ) Q E ~ ( N , R ) , ~ (x> - z QE, ( ~ , f i ) , m (x) %,,, ( N , R ) , ~ (x) L=M -

d d -

d d -

h2 I (-QE,m d x (x)*E,m(x) - z Q E , m ( Z ) Q E , m (x)) ~ z = M

- ( z Q E , , n ( Z ) * E , , , ,m(x) - -QEm,m ( z ) 8 ~ m ,m) (T) Iz=M I 2 (LI - Lzh)(E - Em(.

d x (3.25)

This can be seen from the fact that

n ( E i ) Q ( E z ) / d2(E1- V ( x ) ) d x - 1 ~ ( E z - V ( z ) ) d x 2 CIES - Ezl, TI (EI) 71 (Ez

for a constant C > 0 and Lipschitz continuity of $ j in E. This can be seen

easily from the proof of Lemma 3.2. It follows from (3.23)-(3.25) that

E ( N , h) - Em(N, h) = 0(hm+'). (3.26)

We are now in the position to prove the following lemma. Let QE,,~

be the WKB-Langer approximate eigenfunction corresponding to the ap-

proximate eigenvalue E, ( N , h).

Lemma 3.3. Assume conditions in Lemma 3.1. Then for small h

IIQ'E,m - SEm,mll~2(W) = 0(hm+'), (3.27)

359

and

Proof. Without any loss of generality we assume E( N , h) 2 Em ( N , h). We

begin by estimating

'&(N,h)(Z) - EE,(N,h)(z)

360

From the Lipschitz continuity of @ ~ ( x ) in E in Lemma 2.1 we know

@ E ( N , A ) (x) - @E,(N,A)( ” ) = O(E(N1 h) - Em(N, o(tim+’)

But by the definition

1 G ( N , h ) (.) = C-Ei(N,h) ( 4 J - S ( $ E ( N , h ) (XI )

G m ( N , t i ) ( 4 = c - E ~ _ ( N , h ) j Z ) J - ; ( ~ E E , , ( N , h ) ( ~ ) )

+C+<k,(N,h) (4J; (ZEE,”(N,h) (x)).

1 -tC+Ee(N,h)(x)’JQ(ilEE(N,h) (z)),

1

1

Recall z i $ - ( z ; J - k ( z ) ) = - z : J + ( z ) , is bounded and similarly to

z i $ ( z i J ~ ( z ) ) . Therefore by the Mean Value Theorem, there exists a

constant M > 0 and E* between E ( N , ti) and E,(N, ti) such that

Therefore there exists M I > 0

And similarly

for x E [T;, .,‘I. So from the Lipschitz continuity of cxj and

to E l we know for x E [T,,T,’],

with respect

36 1

for a constant M2 > 0. Similar to the proof of Lemma 2.4,

(3.32)

The same estimate is true for x E [T,, ~ $ 1 . For z < 71 and x > T$ we know

that the approximate wave functions Q E , ~ and QE,,~ are exponentially

small. Thus the L2 estimate (3.27) follows immediately. The derivative

estimate (3.28) can be proved by a similar argument. Here, similar to

(3.29), for r t < z < 72,

h%(N,h) ,m(X) - hQL,(N,h),rn(X) = 0(hrn+')

which can be easily proved by straightforward calculations. For 71 < z < r:, recall (3.31) and (2.12), and the Lipschitz continuity of aj, ,Bj, a;, in

E, similar to (3.32), we also have

(3.33)

Lemma 3.4. Suppose that V E C" and bounded below, and liml,l+oo V ( x ) = +m. Assume E is an exact eigenvalue of the Schrodinger operator H and 71 and 72 are the only two classical turning points, with V'(rj) # 0 , j = 1 ,2 . Then (3.7) is satisfied for any x E [7,f,72] which gives the exact eigenvalue E ( N , ti) in ascending order and the exact L2(R) wave function Q E i s approximated by the corresponding semi-classical approximate wave function Q E , ~ in L2(R) up to m- th order in h, i .e.

ll*E - QE,mllL2 = o(hmf l ) , (3.34)

dz < $00).

(3.35)

uniformly for E in a compact set of { E : JV(,)<E

Furthermore, i f V is analytic, then the derivatives of @E and @ E , ~ satisfy d- 2(E--V(z) )

IIhQllE: - hQ'E,mllL2 = 0(hrn+l).

Proof. We have shown (3.7) holds for any z E (7-1,7-2) . From the asymp-

totics * E in (3.1)-(3.6) in the different regions respectively, we have

ll*E - QE,mI ($ = ( ~ E , o ( z ) ~ ( h ~ + ' ) ) ~ d z (3.36)

-

362

(9 E , O ( x ) o ( 1: (Q E,O (z) 0 (ti"f1 ) > 2 d x . 7 2

Note that J?: ( QE,O (x ) 0 ( hmfl ) ) dx and JT' ( QE, 0 (x) 0 (ti"+')) dx are

exponentially small because of exponentially small integrand. Then using

Lemma 2.4, (3.34) follows easily.

To prove (3.35), calculate the derivative of Qk(z) in different regions

respectively. For x 5 71,

d hQ. lE (X) = hQb,o(x)P-(x) + h@E,o(z)zP-(s) .

But P-(x, h) is analytic in z and fL for x 5 71 and h # 0 as it satisfies

the differential equation (2.35) with analytic coefficients. Therefore we can

differentiate its asymptotic expansion term by term (Wasow 24 and van der

Corput 22), i.e. &P-(x) - C q!((x)(-h)j, therefore 00

j=1

hQh(2) - hQ&,"(2) = (k9/,,,(z))O(ti"+l) + QE,O(Z)O(h"+2). (3.37)

For 71 5 x 5 ~ f ,

hQ L ( z) = Ti@ k, ( x )B ( x) + f i 9 s , o (5)B' ( X) + fiQ z,o (x)A( X) + fLQL,, (x)A'( X) .

Here similarly] B'(z) - C,"=, /3;(x)h2j, and A'(z) N Cj"=, a i (z )h2j l there-

fore,

fiQ'lE(2) - hQ&&(Z) = (h";,,(x))O(h'+')

+ h Q / , , o ( ~ ) o ( h ~ + ~ ) + ~ ~ , o ( x ) o ( h ~ + ~ ) . ( 3 . 3 8 )

For 7-1' < x < 72,

Here similarly]

363

and

M

j = 1

therefore

~ Q & ( x ) - F L Q ~ , ~ ( X ) = b+(x)O(hm+')

d 1 1

dX bT(x) bT(x) +(-(r) + ,-))o(hm+2). (3.39)

For 72 5 x 6 T$, similarly we have

hQ'lE(2) - hQ&,,(x) = ( h 2 Q ~ , o ( x ) ) o ( h m + ~ )

+hQ&,o (z)O ( hm+' ) + QE,O ( x)O( hmf2) .( 3.40)

And for x 2 r;,

~ Q & ( x ) - h\IIL,,(x) = h\IIL,o(x))0(hm+') + Q E , O ( X ) O ( ~ ~ + ~ ) . (3.41)

Then (3.35) follows from (3.37)-(3.41) and similar argument as (3.36). Here

we use Lemma 2.4. 0

Remark. The quantity T = JV(x) lE d- ' dx in Lemma 3.4 i s the 2 ( E - - V ( x ) )

classical travel time between turning points. If {x : V ( x ) = E } consists of two simple zeros, then the classical travel time T is finite.

The main result of this paper is the following result.

Theorem 3.5. Assume conditions of Lemma 9.1, then the exact N-th eigenvalue E ( N , ti) of the Schrodinger operator H is approximated by the m-th order approximate N-th eigenvalue Em(N, h) which satisfies the m- th order quantization condition in the sense that E (N ,h ) - Em(N,h) =

O(hm+2), and the corresponding exact L2(R) wave function Q E ( N , ~ ) and its derivative T L Q & ( ~ , ~ ) are approximated by the WKB-Langer semi-classical approximate wave function Q E , ( N , ~ L ) , ~ associated with Em(N, h) and its derivative hQ",,(N,h),m in L2(R), i.e.

l lQE(N,h) - QEm(N,h) ,ml IL2(R) = o(hmf l ) i (3.42)

and

I I '%( N,h) - hQkm (N, f i ) ,m 1 I Lz (a) = ). (3.43)

T Z ( E O ) In particular, set Eo(F) to be the solution of JTl(Eo) (2(Eo - V ( y ) ) i d y = 7rF for any given F > 0, then the exact eigenvalue E (N ,h ) has semi-

364

classical limit Eo(F) in the sense that

asymptotic expansion Em up to m-th order in the sense that

lim E ( N , h) = Eo(F), and has h-0

N-CC ( N + $ ) h = F

1 lim - (E(N, ti) - Em(N, h ) ) = 0, h-0 ti"

N - m

(3.44) . .~

( N + i ) h = F

and the exact L2 (R) eigenfunction 9 ~ ( ~ , h ) has the semi-classical asymptotic expansion I ; I I E ~ ( F ) , ~ up to m-th order in the sense that

1 lim - km I I Q E ( N , W - QE,(N,fi),mIILz(R) = 0, (3.45) h-0

N-CC ( N + $ ) h = F

Prooj By the triangle inequality

I IQE(N,h) - qE,(N,h),mIIL2(R) 5 IIq'E(N,h) - 9E(N,h) ,ml lL2(R)

+I I Q E ( N, FL) ,m - @ E, (N,h) ,m I I L2 (R)

and applying Lemma 3.4 and Lemma 3.3 we have (3.42). Similarly we have

(3.43) by using

ll"L(N,/i) - hQLm(N,ti),mIIL2(R) 5 ll"L(N,h) - "L(N,h),mIIL2(R)

+ll'Qk(N,h),m - hQL, (N,h) ,m I ILZ(R).

The rest of the theorem follows immediately. 0

We have the following simple, but interesting corollary.

Theorem 3.6. Suppose that V is analytic and bounded below, and liml,l+oo V ( x ) = +oo. For any constant EO > minwl V ( x ) , let 71 and 7 2 be

the only two classical turning points and define F ( E ~ ) = + JT,(Eo) (~(EO -

V(y)) i d y . If V'(rj) # 0, j = 1 ,2 , and the following travel t ime inequal-

d y < t o o holds, then as h + 0, N -+ oo and Zty O < J n ( ~ o ) , / w j

( N + i ) h = F(Eo), the exact eigenvalue E ( N , h) of the Schriidinger operator H = -$h2A + V ( x ) has the semi-classical limit EO in the sense that E ( N , h ) -+ Eo, as ti -+ 0 , N -+ 00 but ( N + $)Ti = F(E0) and the

TZ(EO)

7 2 (Eo 1

365

exact L2(W) eigenfunction Qk(N,hl has the semi-classical limit Qko,o, the WKB-Langer solution, in L2(R),

lim I IQE(N,h) - QEo,011L2(R) = 0, (3.47) h-0

N-CC

( N + 4 t r = ~ ( E ~ )

and

(3.48)

Acknowledgement

It’s our great pleasure t o thank D.Elworthy, B.Simon, D.Williams and

W. Zheng for helpful conversations.

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