ultrametric pseudodifferential equations and applications
TRANSCRIPT
ULTRAMETRIC PSEUDODIFFERENTIAL
EQUATIONS AND APPLICATIONS
Starting from physical motivations and leading to practical applications, this bookprovides an interdisciplinary perspective on the cutting edge of ultrametricpseudodifferential equations. It shows the ways in which these equations linkdifferent ields, including mathematics, engineering, and geophysics. In particular,the authors provide a detailed explanation of the geophysical applications of p-adicdiffusion equations useful when modeling the lows of liquids through porous rock.p-Adic wavelets theory and p-adic pseudodifferential equations are also presented,along with their connections to mathematical physics, representation theory, thephysics of disordered systems, probability, number theory, and p-adic dynamicalsystems.Material that was previously spread across many articles in journals of many
different ields is brought together here, including recent work on the van der Putseries technique. This book provides an excellent snapshot of the fascinating ield ofultrametric pseudodifferential equations, including their emerging applications andcurrently unsolved problems.
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Theory and the Zel’manov Approach168 A. Yu. Khrennikov, S. V. Kozyrev and W. A. Zúñiga-Galindo Ultrametric Pseudodifferential Equations
and Applications
Encyclopedia of Mathematics and its Applications
Ultrametric Pseudodifferential
Equations and Applications
ANDREI YU. KHRENNIKOVLinnéuniversitetet, Sweden
SERGEI V. KOZYREVSteklov Institute of Mathematics, Moscow
W. A. ZÚÑIGA-GALINDOCentro de Investigación y de Estudios Avanzados
del Instituto Politécnico Nacional, Mexico
University Printing House, Cambridge CB2 8BS, United Kingdom
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Contents
Preface page xi
1 p-Adic Analysis: Essential Ideas and Results 1
1.1 The Field of p-Adic Numbers 11.2 Topology of QN
p 21.3 The Bruhat–Schwartz Space and the Fourier Transform 31.4 Distributions 41.5 Some Function Spaces 6
2 Ultrametric Geometry: Cluster Networks and Buildings 8
2.1 Introduction 82.2 Clustering, Trees, and Ultrametric Spaces 92.3 Family of Metrics and Multiclustering 122.4 Afine Bruhat–Tits Buildings and Cluster Networks 132.5 Groups Acting on Trees and the Vladimirov Operator 17
3 p-Adic Wavelets 20
3.1 Introduction 203.2 Basis of p-Adic Wavelets 263.3 Coherent States 283.4 Orbits of Mean-Zero Test Functions as Wavelet Frames 303.5 Multidimensional Wavelets and Representation Theory 333.6 Wavelets with Matrix Dilations 343.7 Wavelet Transform of Distributions 403.8 Relation to the Haar Basis on the Real Line 413.9 p-Adic Multiresolution Analysis 433.10 p-Adic One-Dimensional Haar Wavelet Bases 453.11 p-Adic Scaling Functions 483.12 Multiresolution Frames of Wavelets 493.13 Multidimensional Multiresolution Wavelet Bases 51
vii
viii Contents
3.14 p-Adic Shannon–Kotelnikov Theorem 533.15 Spectral Theory of p-Adic Pseudodifferential Operators 543.16 Wavelets and Operators for General Ultrametric Spaces 59
4 Ultrametricity in the Theory of Complex Systems 63
4.1 Introduction 634.2 p-Adic Parametrization of the Parisi Matrix 654.3 Dynamics on Complex Energy Landscapes 674.4 Actomyosin Molecular Motor 704.5 2-Adic Model of the Genetic Code 73
5 Some Applications of Wavelets and Integral Operators 76
5.1 Pseudodifferential Equations 765.2 Non-linear Equations and the Cascade Model of Turbulence 785.3 p-Adic Brownian Motion 81
6 p-Adic and Ultrametric Models in Geophysics 83
6.1 Tree-like Structures in Nature 846.2 p-Adic Coniguration Space for Networks of Capillaries and
Balance Equations for Densities of Fluids 856.3 Non-linear p-Adic Dynamics 89
7 Recent Development of the Theory of p-Adic Dynamical
Systems 94
7.1 Van der Put Series and Coordinate Representations ofDynamical Maps 96
7.2 Recent Results about Measure-Preserving Functions andErgodic Dynamics 99
7.3 Ergodic Dynamical Systems Based on 1-Lipschitz Functions 105
8 Parabolic-Type Equations, Markov Processes, and Models of
Complex Hierarchical Systems 114
8.1 Introduction 1148.2 OperatorsW , Parabolic-Type Equations, and Markov
Processes 1158.3 Elliptic Pseudodifferential Operators, Parabolic-Type
Equations and Markov Processes 1218.4 Non-Archimedean Reaction–Ultradiffusion Equations and
Complex Hierarchic Systems 123
9 Stochastic Heat Equation Driven by Gaussian Noise 133
9.1 Introduction 1339.2 p-Adic Parabolic-Type Pseudodifferential Equations 1349.3 Positive-Deinite Distributions and the Bochner–Schwartz
Theorem 1369.4 Stochastic Integrals and Gaussian Noise 138
Contents ix
9.5 Stochastic Pseudodifferential Equations Driven by aSpatially Homogeneous Noise 148
10 Sobolev-Type Spaces and Pseudodifferential Operators 155
10.1 Introduction 15510.2 The Spaces H∞ 15610.3 A Hörmander–Łojasiewicz-Type Estimation 16210.4 The Spaces W∞ 16510.5 Pseudodifferential Operators on W∞ 16810.6 Existence of Fundamental Solutions 17010.7 Igusa’s Local Zeta Functions and Fundamental Solutions 17210.8 Local Zeta Functions and Pseudodifferential Operators inH∞ 175
11 Non-Archimedean White Noise, Pseudodifferential Stochastic
Equations, and Massive Euclidean Fields 177
11.1 Introduction 17711.2 Preliminaries 17811.3 Pseudodifferential Operators and Green Functions 17911.4 The Generalized White Noise 18711.5 Euclidean Random Fields as Convoluted Generalized White
Noise 19011.6 The p-Adic Brownian Sheet on QN
p 195
12 Heat Traces and Spectral Zeta Functions for p-Adic Laplacians 198
12.1 Introduction 19812.2 A Class of p-Adic Laplacians 19912.3 Lizorkin Spaces, Eigenvalues, and Eigenfunctions for Aβ
Operators 20112.4 Heat Traces and p-Adic Heat Equations on the Unit Ball 20312.5 Analytic Continuation of Spectral Zeta Functions 210
References 214Index 236
Preface
The present book aims to provide an interdisciplinary perspective of the state of theart of the theory of ultrametric equations and its applications, starting from physicalmotivations and applications of the ultrametric geometry, and covering connectionswith probability, functional analysis, number theory, etc. in a novel form. In recentyears the connections between non-Archimedean mathematics (mainly analysis) andmathematical physics have received a lot of attention, see e.g. [53]–[60], [63], [90],[90], [132]–[137], [164]–[166], [168], [190], [191], [220]–[228], [322]–[328], [336],[346]–[350], [366], [373], [413]–[411], [423]–[435] and the references therein. Allthese developments have been motivated by two physical ideas. The irst is the con-jecture (due to Igor Volovich) in particle physics that at Planck distances space-timehas a non-Archimedean structure, see e.g. [438]–[435], [413], [412]. The second ideacomes from statistical physics, more precisely, in connection with models describ-ing relaxation in glasses, macromolecules, and proteins. It has been proposed thatthe non-exponential nature of those relaxations is a consequence of a hierarchicalstructure of the state space which can in turn be related to p-adic structures. GiorgioParisi introduced the idea of hierarchy for spin glasses (disordered magnetics) in amore precise form in 1979, then the idea was extended to other physical problemsand combinatorial optimization problems, see [336]. Then in the 1980s effects ofslow non-exponential relaxation and aging were observed in deeply frozen proteins,implying the occurrence of a glass transition similar to that in spin glasses. Thus in themiddle of the 1980s the idea of using ultrametric spaces to describe the states of com-plex biological systems, which naturally possess a hierarchical structure, emerged inthe works of Frauenfelder, Parisi, Stain, and among others see e.g. [164]. In proteinphysics, it is regarded as one of the most profound ideas put forward to explain thenature of distinctive attributes of life.
For replica symmetry breaking in spin glasses the p-adic models were proposedindependently by Avetisov et al. [53] and Parisi and Sourlas [373]. The idea of usingp-adic diffusion equation to describe protein relaxation was proposed in [53].
xi
xii Preface
From a mathematical point of view, in these models the time-evolution of a com-plex system is described by a p-adic master equation (a parabolic-type pseudodiffer-ential equation) which controls the time-evolution of a transition function of aMarkovprocess on an ultrametric space, and this stochastic process is used to describe thedynamics of the system in the space of conigurational states which is approximatedby an ultrametric space (Qp). This is a central motivation for developing a theory ofultrametric reaction–diffusion equations or, more generally, a theory of pseudodiffer-ential equations on ultrametric spaces.
The simplest ultrametric diffusion equation is the one-dimensional p-adic heatequation. This equation was introduced in the book of Vladimirov, Volovich, andZelenov [434, Section XVI]. Kochubei [275, Chapters 4 and 5] presented a generaltheory for one-dimensional parabolic-type pseudodifferential equations with variablecoeficients, whose fundamental solutions are transition density functions forMarkovprocesses in the p-adic line, see also [11], [12], [104], [101], [386], [464], [411]. Ap-adic diffusion equation was also considered by Albeverio and Karwowski [10]–[11]. Zúñiga-Galindo and his collaborators have developed a very general theory oflinear pseudodifferential equations, based on the work of Kochubei, over p-adics andadeles, see [470]. At this point it is important tomention the differences between [470]and this book. The book [470] was written from the perspective of “pure mathemat-ics,” while this book has been written from an interdisciplinary perspective. There isa small intersection, namely the material presented in Sections 8.1–8.3, which corre-sponds to some basic results on p-adic parabolic-type equations and the associatedMarkov processes; this material is summarized here without proofs.
The tree-like structure of coniguration spaces was widely used in applicationsto cognitive science and psychology, see, e.g., the pioneering works of Khrennikov[222], [223]; see also [141], [14]. Recently Khrennikov and Oleschko proposed usingthis class of coniguration spaces in geology [253], [252]. This is a new area ofresearch and very important for applications, especially because of the possibilityof being able to couple the output of theoretical modeling with applied petroleumresearch (performed by the research team of Oleschko working on the Mexican oilields). For the moment, only the irst steps in this direction have been taken.
This book does not enlighten the reader concerning advanced research devoted tothe models of mathematical physics with p-adic-valued wave functions (in particu-lar, p-adic-valued probabilities), see [222] for details. We present only some resultsabout the theory of p-adic dynamical systems, concerning iterations of maps in theields of p-adic numbers. The development of this theory was partially motivated bymathematical physics, but later this theory was mainly explored in applications tomodeling of cognition, see e.g. [222], [223], [141], [14], [20] and in cryptography,see e.g. [35], [39].
The book is organized as follows. In Chapter 1, we review, without proofs, thebasic deinitions and results on p-adic functional analysis of complex-valued func-tions of p-adic arguments, for an in-depth discussion of these results, the readermay consult [18], [402], [434]. Chapter 2 aims to present the essential ideas of
Preface xiii
ultrametrics in connection with clustering and trees. The material presented includesafine Bruhat–Tits buildings and multiclustering, groups acting on trees, and theVladimirov operator. Chapter 3 is dedicated to the theory of p-adic wavelets and itsapplications. This chapter presents an in-depth discussion of p-adic multiresolutionanalysis and wavelet techniques for solving several types of general ultrametric equa-tions. In addition, connections with mathematical physics and representation theory(the theory of coherent states) are also discussed. This material is not covered in ref-erences such as [18]. Chapter 4 aims to give a short review of some applications ofp-adic and more general ultrametric methods in the statistical physics of disorderedsystems, dynamics of macromolecules, and genetics. A well-known and acceptedscientiic paradigm in the physics of complex systems (such as glasses and proteins)asserts that the dynamics of a large class of complex systems is described as a randomwalk on a complex energy landscape, see e.g. [164]–[166], [440], and [294, and ref-erences therein]. A landscape is a continuous real-valued function that represents theenergy of a system. The term complex landscape means that the energy function hasmany local minima. In the case of complex landscapes, in which there are many localminima, a “simpliication method” called interbasin kinetics is applied. The idea isto study the kinetics generated by transitions between groups of states (basins). Akey idea is that the dynamics on a complex energy landscape is approximated by afamily of Arrhenius transitions between local energy minima. Moreover, the set oflocal minima and transition states between the minima is given by a “disconnectivitygraph” of basins (a tree) and by functions on this graph that describe the distributionsof energies of the minima and activation energies of the transition states. The p-adicmodels introduced by Avetisov, Kozyrev et al. have master equations of the followingform:
∂ f (x, t )
∂t=
∫
Qp
[w(x|y) f (y, t ) − w(y|x.) f (x, t )]dy, (1)
where x ∈ Qp, t ≥ 0. The function f (x, t ) : Qp × R+ → R+ is a probability densitydistribution, so
∫
Bf (x, t )dx is the probability of inding the system in a domain B ⊂
Qp at the instant t. The function w(x|y) : Qp × Qp → R+ is the probability rate ofthe transition from state y to state x per unit of time.In Chapter 5, we give applications of wavelet techniques for solving certain integral
equations, and also applications to the construction of the p-adic one-dimensionalversion of Brownian motion.
In Chapter 6, we present a new conceptual approach for modeling of luid lows inrandom porous media based on explicit exploration of the tree-like geometry of com-plex capillary networks. Such patterns can be representedmathematically as ultramet-ric spaces and the dynamics of luids by ultrametric diffusion. In this model the porousbackground is treated as the environment contributing to the coeficients of evolu-tionary equations. For the simplest trees, these equations are signiicantly less com-plicated than those with fractional differential operators which are commonly appliedin geological studies looking for some fractional analogs to conventional Euclidean
xiv Preface
space but with anomalous scaling and diffusion properties. The systems of ultramet-ric reaction–diffusion equations can be used to model the process of extraction ofoil from an extended network of capillaries. This process is especially important forthe design of oil recovery programs and especially for the selection of enhanced oilrecovery (EOR) methods, where the luid low from the solid matrix is stimulated.In Chapter 6 a new non-linear p-adic pseudodifferential equation, which is the non-Archimedean counterpart of the porous medium equation, is introduced.
Chapter 7 describes recent developments in p-adic dynamical systems (see themonographs [222], [35]) and their connections with cryptography. Discrete dynami-cal systems based on iterations of functions belonging to the special functional class,namely, 1-Lipschitz functions, are considered. The importance of this class for the the-ory of p-adic dynamical systems was emphasized in a series of pioneering works byV. Anashin [31], [32], [33]. Then some interesting results about such discrete dynam-ics were obtained in joint works by V. Anashin, A. Khrennikov, and E. Yurova, see,e.g., [34], [35], [452].
Chapter 8 has two goals. The irst is to present general results for a large class ofpseudodifferential equations, which contains equations of type (1). These equationsare related to models of complex systems. In the second part, we introduce a newclass of non-linear p-adic pseudodifferential equations. Chapter 9 is dedicated to thestudy of general p-adic diffusion equations driven by Gaussian noise.
Chapter 10 aims to present the basic results about the Sobolev-type spaces overQNp and to show the existence of fundamental solutions for pseudodifferential equa-
tions over these spaces. We consider two types of spaces, denoted H∞ and W∞.Both spaces are countably Hilbert nuclear spaces, withW∞ continuously embeddedin W∞. These spaces are invariant under the action of a large class of pseudodif-ferential operators. The spaces H∞ were introduced by Zúñiga-Galindo in [472]. Inthe spaces W∞ we show the existence of fundamental solutions for pseudodiffer-ential operators whose symbols involve general polynomials. This result is the non-Archimedean counterpart of Hörmander’s solution of the problem of the division ofa distribution by a polynomial, see [202], [316]. We also summarize the results of[471], without proofs. In this work the existence of fundamental solutions for pseu-dodifferential equations using local zeta functions is established in the spaces H∞.
In Chapter 11 we present a new class of non-Archimedean Euclidean quantumields, in arbitrary dimension, which are constructed as solutions of certain covari-ant p-adic stochastic pseudodifferential equations (SPDEs), by using techniques ofwhite-noise calculus. The connection between quantum ields and SPDEs has beenstudied intensively in the Archimedean setting, see e.g. [9]–[30] and the referencestherein. A massive non-Archimedean ield � is a random ield parametrized byH∞
(
QNp ; R
)
, the nuclear countably Hilbert spaces introduced in Chapter 10. Heuris-tically, � is the solution of
(
Lα + m2)
� = , where Lα is a pseudodifferential oper-ator,m > 0, and is a generalized Lévy noise. This type of noise is introduced in thischapter. Finally, as an application,we give a general construction of a p-adic Brown-ian sheet on QN
p .
Preface xv
In Chapter 12, we commence the study of p-adic spectral zeta functions. In the realsetting, the spectral zeta function attached to the Laplacian (under a suitable hypoth-esis) is the Riemann zeta function. This spectral zeta function is studied by using thetechniques of heat equations. There are many types of p-adic heat equation, and thusmany types of p-adic Laplacian. It is natural to study the spectral zeta functions ofthese p-adic Laplacians. Of course there are very serious arithmetical motivations forthis study. In Chapter 12, we study heat traces and spectral zeta functions attached tocertain p-adic Laplacians, like the ones introduced in Chapter 8, which are denotedAβ . By using an approach inspired by the work of Minakshisundaram and Pleijel,see [340]–[342], we ind a formula for the trace of the semigroup e−tAβ acting on thespace of square integrable functions supported on the unit ball with average zero.The trace of e−tAβ is a p-adic oscillatory integral of Laplace–type. We do not knowthe exact asymptotics of this integral as t tends to ininity, however, we obtain a goodestimation for its behavior at ininity.
Two of the authors (AKH and WAZ-G1) wish to thank the Consejo Nacional deCiencia y Tecnología de México (CONACYT) for supporting their research activitiesthrough several grants.
1 Latest grant no. 250845 and through the program Sistema Nacional de Investigadores (SNI III).
1
p-Adic Analysis: Essential Ideas and Results
In this chapter, we present, without proofs, the essential aspects of, and basic resultson, p-adic functional analysis needed in the book. For a detailed exposition on p-adicanalysis the reader may consult [18], [402], [434].
1.1 The Field of p-Adic Numbers
Throughout this book pwill denote a prime number. The field of p-adic numbers Qp
is defined as the completion of the field of rational numbers Q with respect to thep-adic norm | · |p, which is defined as
|x|p =⎧
⎨
⎩
0 if x = 0
p−γ if x = pγa
b,
where a and b are integers coprime with p. The integer γ := ord(x), with ord(0) :=+∞, is called the p-adic order of x. We extend the p-adic norm to QN
p by taking
||x||p := max1≤i≤N
|xi|p, for x = (x1, . . . , xN ) ∈ QNp .
We define ord(x) = min1≤i≤N{ord(xi)}, then ||x||p = p−ord(x). The norm || · ||psatisfies
‖x+ y‖p ≤ max(‖x‖p, ‖y‖p),the strong triangle inequality. The metric space (QN
p , || · ||p) is a complete ultrametricspace. As a topological space Qp is homeomorphic to a Cantor-like subset of the realline, see e.g. [18], [434]. In Chapter 2 only, we will work with more general norms,namely, norms of type
Nq1,...,qN (x) = max1≤i≤N
qi|xi|p,
where the q1, . . . , qN are fixed positive numbers.
1
2 p-Adic Analysis: Essential Ideas and Results
Any p-adic number x = 0 has a unique expansion of the form
x = pord(x)∞∑
j=0
xi pj,
where x j ∈ {0, 1, 2, . . . , p− 1} and x0 = 0. By using this expansion, we define thefractional part of x ∈ Qp, denoted {x}p, as the rational number
{x}p ={
0 if x = 0 or ord(x) ≥ 0
pord(x)∑−ord(x)−1
j=0 x j pj if ord(x) < 0.
In addition, any p-adic number x = 0 can be represented uniquely as x = pord(x)ac(x)with |ac(x)|p = 1; ac(x) is called the angular component of x.
1.1.1 Additive Characters
Set χp(y) := exp(2π i{y}p) for y ∈ Qp. The map χp(·) is an additive character on Qp,i.e. a continuous map from (Qp,+) into S (the unit circle considered as a multi-plicative group) satisfying χp(x0 + x1) = χp(x0)χp(x1), x0, x1 ∈ Qp. We notice thatχp satisfies the following relations:
χp(0) = 1, χp(−x) = χp(x) = χ−1p (x),
χp(mx) = χmp (x), m ∈ Z.
The additive characters ofQp form an Abelian group which is isomorphic to (Qp,+);the isomorphism is given by ξ → χp(ξx), see e.g. [18, Section 2.3].Wewill call χp(·)the standard additive character of Qp.
1.2 Topology of QNp
For r ∈ Z, denote by BNr (a) = {x ∈ QNp ; ||x− a||p ≤ pr} the ball of radius pr with
center at a = (a1, . . . , aN ) ∈ QNp , and take BNr (0) := BNr . Note that B
Nr (a) = Br(a1)
× · · · × Br(aN ), where Br(ai) := {x ∈ Qp; |xi − ai|p ≤ pr} is the one-dimensionalball of radius pr with center at ai ∈ Qp. The ball BN0 equals the product of N copiesof B0 = Zp, the ring of p-adic integers. We also denote by SNr (a) = {x ∈ QN
p ; ||x−a||p = pr} the sphere of radius pr with center at a = (a1, . . . , aN ) ∈ QN
p , and takeSNr (0) := SNr . We notice that S10 = Z×
p (the group of units of Zp), but (Z×p )
N � SN0 .The balls and spheres are both open and closed subsets in QN
p . In addition, two ballsin QN
p are either disjoint or one is contained in the other.As a topological space (QN
p , || · ||p) is totally disconnected, i.e. the only connectedsubsets of QN
p are the empty set and the points. A subset of QNp is compact if and only
if it is closed and bounded in QNp , see e.g. [434, Section 1.3], or [18, Section 1.8].
The balls and spheres are compact subsets. Thus (QNp , || · ||p) is a locally compact
topological space.
1.3 The Bruhat–Schwartz Space and the Fourier Transform 3
Wewill use(p−r||x− a||p) to denote the characteristic function of the ballBNr (a).We will use the notation 1A for the characteristic function of a set A.
1.3 The Bruhat–Schwartz Space and the Fourier Transform
A complex-valued function ϕ defined on QNp is called locally constant if for any
x ∈ QNp there exists an integer l(x) ∈ Z such that
ϕ(x+ x′) = ϕ(x) for all x′ ∈ BNl(x). (1.1)
The C-vector space of locally constant functions will be denoted as E (QNp ).
A function ϕ : QNp → C is called a Bruhat–Schwartz function (or a test function)
if it is locally constant with compact support. Any test function can be represented asa linear combination, with complex coefficients, of characteristic functions of balls.The C-vector space of Bruhat–Schwartz functions is denoted by D(QN
p ) := D. Forϕ ∈ D(QN
p ), the largest number l = l(ϕ) satisfying (1.1) is called the exponent oflocal constancy (or the parameter of constancy) of ϕ. We also say that pl is the diam-eter of constancy of ϕ. We denote by DR(QN
p ) the R-vector space of real-valued testfunctions.We denote by Dl
M (QNp ) := Dl
M the finite-dimensional space of test functions fromD(QN
p ) having supports in the ball BNM and with parameters of local constancy ≥ l.The following embeddings hold: Dl
M ⊂ Dl′M′ for M ≤ M′, l′ ≤ l.
Any function ϕ ∈ DlM can be represented in the following way:
ϕ(x) =pN(M−l)∑
v=1
ϕ(aν )(p−l‖x− aν‖p), x ∈ QNp , ϕ ∈ D
(
QNp
)
, (1.2)
where (p−l‖x− aν‖p) is a characteristic function of ball BNl (aν ), and the points
aν = (aν1, . . . , aνN ) ∈ BNM do not depend on ϕ and are such that the balls BNl (a
ν ),ν = 1, . . . , pN(M−l) are disjoint and cover BNM , see e.g. [18, Lemma 4.3.1].
Convergence in D(QNp ) is defined in the following way: ϕk → 0, k → ∞, in
D(QNp ) if and only if (i) ϕk ⊂ BlM withM and l independent of k; (ii) ϕk uniformly−−−−−−→ 0
in QNp .
Set DM (QNp ) := DM = lim indl→−∞Dl
M . Then D(QNp ) = lim indM→∞DM . The
space D(QNp ) is a complete locally convex topological vector space. If U is an open
subset of QNp ,D(U ) denotes the space of test functions with supports contained inU ,
then D(U ) is dense in
Lρ (U ) ={
ϕ : U → C; ‖ϕ‖ρ ={∫
|ϕ(x)|ρdNx}
1ρ
< ∞}
,
where dNx is the Haar measure on QNp normalized by the condition vol(BN0 ) = 1, for
1 ≤ ρ < ∞, see e.g. [18, Section 4.3]. We will also use the simplified notation Lρ ,1 ≤ ρ < ∞, if there is no danger of confusion.Wewill denote byDR(U ), theR-space
4 p-Adic Analysis: Essential Ideas and Results
of test functions with support in U , and by LρR(U ), 1 ≤ ρ < ∞, the real counterpart
of Lρ (U ), 1 ≤ ρ < ∞.
1.3.1 The Fourier Transform of Test Functions
Given ξ = (ξ1, . . . , ξN ) and y = (x1, . . . , xN ) ∈ QNp , we set ξ · x := ∑N
j=1 ξ jx j. TheFourier transform of ϕ ∈ D(QN
p ) is defined as
(Fϕ)(ξ ) =∫
QNp
χp(ξ · x)ϕ(x)dNx for ξ ∈ QNp ,
where dNx is the normalized Haar measure on QNp . The Fourier transform is a linear
isomorphism from D(QNp ) onto itself satisfying (F (Fϕ))(ξ ) = ϕ(−ξ ), see e.g. [18,
Section 4.8]. More precisely, if ϕ ∈ DlM then ϕ ∈ D−M
−l . We will also use the notationFx→ξϕ and ϕ for the Fourier transform of ϕ.
In the definition of the Fourier transform, the bilinear form ξ · x can be replacedfor any symmetric non-degenerate bilinear form B(ξ, x). We will use such Fouriertransforms in Chapter 11. In Chapter 3, we will also use the symbols F , · to denotethe Fourier transform in Rn.
1.4 Distributions
LetD′(QNp ) denote the C-vector space of all continuous functionals (distributions) on
D(QNp ). We endow D′(QN
p ) with weak topology. We denote by D′R(Q
Np ) the real ana-
log of D′(QNp ). The natural pairing D′(QN
p )×D(QNp ) → C is denoted as (T, ϕ) for
T ∈ D′(QNp ) and ϕ ∈ D(QN
p ). Convergence inD′(QNp ) is defined as the weak conver-
gence Tk → 0, k → ∞, in D′(QNp ) if (Tk, ϕ) → 0, k → ∞, for all ϕ ∈ D(QN
p ). ThespaceD′(QN
p ) agrees with the algebraic dual ofD(QNp ), i.e. all functionals onD(QN
p )are continuous. In addition,D′(QN
p ) is complete, i.e. if Tk − Tj → 0, k, j → ∞, thenthere exists a functional T ∈ D′(QN
p ) such that Tk − T → 0, k → ∞ in D′(QNp ), see
e.g. [18, Section 4.4].Let U be an open subset of QN
p . A distribution T ∈ D′ (U ) vanishes on V ⊂ U if(T, ϕ) = 0 for all ϕ ∈ D(V ). Let UT ⊂ U be the maximal open subset on which thedistribution T vanishes. The support of T is the complement ofUT inU . We denoteit by supp T .Given a fixed test function θ and a distribution T ∈ D′(QN
p ), we define the distri-bution θT by the formula (θT, ϕ) = (T, θϕ) for ϕ ∈ D(QN
p ). We say that a distribu-tion T ∈ D′(QN
p ) has compact support if there exists a k ∈ Z such that �kT = T inD′(QN
p ), where �k(x) := (p−k‖x‖p).Every f ∈ E (QN
p ), or more generally in L1loc, defines a distribution f ∈ D′(QNp ) by
the formula
( f , ϕ) = ∫
QNp
f (x)ϕ(x)dNx.
Such distributions are called regular distributions.
1.4 Distributions 5
1.4.1 The Fourier Transform of a Distribution
The Fourier transform F[T ] of a distribution T ∈ D′(QNp ) is defined by
(F[T ], ϕ) = (T,F[ϕ]) for all ϕ ∈ D(
QNp
)
.
The Fourier transform T → F[T ] is a linear (and continuous) isomorphism fromD′(QN
p ) onto D′(QNp ). Furthermore, T = F[F [T ](−ξ )].
Let A be a matrix, detA = 0 and b ∈ QNp . Then for a distribution T ∈ D′(QN
p ) onehas
F[T (Ax+ b)](ξ ) = |detA|−1p χp
(−A−1b · ξ)F[T (x)](
(A∗)−1ξ)
, (1.3)
where A∗ is the transpose matrix.Let T ∈ D′(QN
p ) be a distribution. Then suppT ⊂ BNN if and only ifF[T ] ∈ ˜E (QNp ),
where the exponent of local constancy of F[T ] is ≥ −N. In additionF[T ](ξ ) = (T (y),�N (y)χp(ξ · y)),
see e.g. [18, Section 4.9].
1.4.2 The Direct Product of Distributions
Given F ∈ D′(QNp ) and G ∈ D′(Qm
p ), their direct product F × G is defined by theformula
(F (x)× G(y), ϕ(x, y)) = (F (x), (G(y), ϕ(x, y))) for ϕ(x, y) ∈ D(
QN+mp
)
.
The direct product is commutative: F × G = G× F . In addition the direct product iscontinuous with respect to the joint factors.
1.4.3 The Convolution of Distributions
Given F,G ∈ D′(QNp ), their convolution F ∗ G is defined by
(F ∗ G, ϕ) = limk→∞
(F (y)× G(x),�k(x)ϕ(x+ y))
if the limit exists for all ϕ ∈ D(QNp ). We recall that, if F ∗ G exists, then G ∗ F exists
and F ∗ G = G ∗ F , see e.g. [434, Section 7.1]. If F,G ∈ D′(QNp ) and supp G ⊂ BNN ,
then the convolution F ∗ G exists, and it is given by the formula
(F ∗ G, ϕ) = (F (y)× G(x),�N (x)ϕ(x+ y)) for ϕ ∈ D(
QNp
)
.
In the case in which G = ψ ∈ D(QNp ), F ∗ ψ is a locally constant function given by
(F ∗ ψ )(y) = (F (x), ψ (y− x)),
see e.g. [434, Section 7.1].
6 p-Adic Analysis: Essential Ideas and Results
1.4.4 The Multiplication of Distributions
Set δk(x) := pNk(pk‖x‖p) for k ∈ N. Given F,G ∈ D′(QNp ), their product F · G is
defined by
(F · G, ϕ) = limk→∞
(G, (F ∗ δk )ϕ)
if the limit exists for all ϕ ∈ D(QNp ). If the product F · G exists then the productG · F
exists and they are equal.We recall that the existence of the product F · G is equivalent to the existence of
F[F] ∗ F[G]. In addition, F[F · G] = F[F] ∗ F[G] and F[F ∗ G] = F[F] · F[G],see e.g. [434, Section 7.5].
1.5 Some Function Spaces
1.5.1 p-Adic Lizorkin Spaces
In papers [314], [315] the (real) Lizorkin spaces invariant with respect to action ofreal fractional operators were introduced. By [15], [16], the p-adic Lizorkin space oftest functions is defined as the space �(QN
p ) ⊂ D(QNp ) of mean-zero test functions.
Topology on this set is defined by restriction of the topology onD(QNp ). Equivalently
the space �(QNp ) can be defined as the Fourier image of the space �(QN
p ) of testfunctions equal to zero in zero.Let us denote by �′(QN
p ) and �′(QN
p ) respectively the spaces topologically dualto �(QN
p ) and �(QNp ). We call �′(QN
p ) the Lizorkin space of p-adic distributions.Let �⊥(QN
p ) and �⊥(QNp ) respectively be the subspaces of functionals in D′(QN
p )orthogonal to �(QN
p ) and �(QNp ), i.e. �
⊥(QNp ) = {T ∈ D′(QN
p ) : T = Cδ, C ∈ C}and �⊥(QN
p ) = {T ∈ D′(QNp ) : T = C, C ∈ C}. Then
�′(QNp
) = D′(QNp
)/
�⊥(QNp
)
, � ′(QNp
) = D′(QNp
)/
�⊥(QNp
)
, (1.4)
cf. [15].Therefore the space �′(QN
p ) is obtained from D′(QNp ) by factorization over con-
stants (two distributions in D′(QNp ) which differ by a constant are equal as elements
of �′(QNp )).
The Fourier transform of distributionsF ∈ �′(QNp ) andG ∈ � ′(QN
p ) is given by theformulae (F[F], ψ ) = (F,F[ψ]), for all ψ ∈ �(QN
p ), and (F[G], φ) = (G,F[φ]),for all φ ∈ �(QN
p ). One can see that F[�′(QNp )] = � ′(QN
p ) and F[� ′(QNp )] =
�′(QNp ), [15].
The Vladimirov operator is defined by the formula
(Dαϕ)(x) = F−1(|ξ |αpFx→ξϕ)
.
1.5 Some Function Spaces 7
This operator maps the space �(QNp ) into itself. For α > 0 (in the one-dimensional
case) there is the integral representation
Dαϕ(x) = 1
�p(−α)∫
Qp
ϕ(x)− ϕ(y)
|x− y|1+αp
dy, (1.5)
where �p(−α) = (pα − 1)/(1− p−1−α ) is the p-adic �-function, cf. [434, Sec-tion IX, (1.1)].
2
Ultrametric Geometry: Cluster Networksand Buildings
2.1 Introduction
In the present chapter we discuss how to find ultrametric structures in applications;in particular, we discussmultidimensional ultrametric structures. Further applicationswill be discussed in Chapter 4.Hierarchy is a natural feature in ultrametric spaces which mathematically can be
expressed as a duality between ultrametric spaces and trees of balls in these spaces[310], [294]. In the p-adic case there exists also multidimensional hierarchy which isdescribed by the Bruhat–Tits buildings [95], [173].The general approach to ultrametricity in applications is related to a clustering
procedure, i.e. hierarchical classification of objects using similarity (in particular,proximity in metric spaces), see for example [353], [354], [196]. Hierarchicalclassification of data using the “tree of life” was extensively used in biology startingfrom the eighteenth century [313].Clustering generates classification trees of clusters with the partial order defined by
inclusion, thus the border of a classification tree will be an ultrametric space. For anexplanation of the relation of ultrametricity and clustering and other applications ofdata analysis see [196], [398]. Duality between trees and ultrametric spaces was con-sidered in particular in [310]. Hierarchical classification can be considered as ultra-metric approximation of a complex system.In applications the classification metric is usually ambiguously defined. In com-
puter science clustering with respect to a family of metrics results in a network ofclusters (this network is not necessarily a tree) [400], [68]. This approach is referredto as multiclustering, multiple clustering, or ensemble clustering. One of the possibleapplications of this approach is to phylogenetic networks needed for descriptionof hybridization and horizontal gene transfer in biological evolution [283]. Fora discussion of mathematical methods used in the investigation of phylogeneticnetworks see [203], [139].It was found that a network of clusters can be considered as a simplicial complex
which in the case of a family of metrics on multidimensional p-adic spaces is directlyrelated to the Bruhat–Tits buildings [293], [27], [295], [296].
8
2.2 Clustering, Trees, and Ultrametric Spaces 9
For some other applications of p-adic, hierarchical, and wavelet methods in dataanalysis and related fields see [357, 356, 358, 355], [88, 89]. One of the most promis-ing applications of hierarchical methods is to deep learning [200, 77].In this chapter we discuss clustering and multiclustering in relation to the Bruhat–
Tits buildings. At the end of the chapter we discuss certain relations between groupactions on trees and pseudodifferential operators.
2.2 Clustering, Trees, and Ultrametric Spaces
Clustering is a method of hierarchical classification of objects using trees. In thepresent section we consider clustering in metric spaces (namely we discuss the exam-ple of nearest-neighbor clustering). Let us recall that an ultrametric space is a metricspace X where the metric d satisfies the strong triangle inequality
d(x, y) ≤ max(d(x, z), d(y, z)), for any x, y, z ∈ X.
Duality between trees and ultrametric spaces gives the most transparent example ofclustering.
2.2.1 Duality between Trees and Ultrametric Spaces
LetX be a complete locally compact ultrametric space. Let us consider the set T (X ) ofballs in X which contains all balls with non-zero diameter and balls of zero diameterwhich are isolated points in X . We endow T (X ) with a graph structure by attaching anedge to each pair of balls (ball, maximal subball1). Then T (X ) is a partially orderedtree (the tree of balls in ultrametric space X ; the partial order is defined by inclusionof balls).For convenience let us recall the definition of a partially ordered set.
Definition 2.1 A partially ordered set S is a set endowed with binary relation≤ suchthat, for any pair (x, y) ∈ S× S, it holds that x, y are not comparable, or x ≥ y, orx ≤ y, moreover
(i) a ≤ a (reflexivity);(ii) if a ≤ b and b ≤ a then a = b (antisymmetry);(iii) if x ≤ y and y ≤ z then x ≤ z (transitivity).
Map f is a map of partially ordered sets (or order-preserving map, or monotonemap) f : M → N, if the image of the ordered pair is ordered, i.e. for any x, y ∈ M,with x ≤ y, it holds that f (x) ≤ f (y) in N.
The tree T (X ) of balls in a locally compact ultrametric space satisfies the followingcriteria [294].
1 A maximal subball I in ball J is a ball I ⊂ J, which is maximal, i.e. there are no balls in between Iand J.
10 Ultrametric Geometry: Cluster Networks and Buildings
(i) The set of vertices in T (X ) is finite or countable.(ii) For any vertex I the set of vertices connected with I by edges is finite. An edge
(I, J) is called increasing if J > I; if I > J it is called decreasing. Given a vertexI, the number of increasing edges beginning in I is equal to one (if the ball I isnot equal to X) or is equal to zero (if the ball I is equal to X); the number ofdecreasing edges beginning in I is larger than one (if the ball I is not an isolatedpoint in X) or is equal to zero (if the ball I is an isolated point in X).
(iii) Any path (without self-intersections) in the tree T (X ) which connects two ver-tices is finite and contains a unique maximal vertex.
(iv) The diameter d(I) of balls2 is a function on the set of vertices of the tree T (X )which takes non-negative values and increases monotonically with respect to thepartial order. The function d(I) tends to zero for infinite decreasing sequences {I}of nested balls and is equal to zero for vertices corresponding to isolated pointsof X (vertices of T (X ) incident to only one edge).
We will denote by sup(S) the minimal ball which contains the set S ⊂ X , in par-ticular sup(x, y), x, y ∈ X denotes the minimal ball containing points x, y.Conversely, let T be a tree with partial order and let d(·) be a function on the set
of vertices in T , and let the above four conditions for the tree, the partial order, andthe function d(·) be satisfied.Let us define a metric in T in the following way. For vertices I, J in the tree T there
exists a unique finite path without self-intersections which connects these vertices.Let us denote by sup(I, J) the maximal vertex in this path (this vertex is unique) andconsider the function d(I, J) = d(sup(I, J)). This function is an ultrametric on theset of vertices of the tree T .Let us consider a completion of the set of vertices of the tree T with respect to the
metric d(·, ·). This completion contains the tree T itself. Let us eliminate all internalvertices of the tree T from the completion. The result of this operation will be denotedby X (T ) (internal vertices of the tree are vertices incident to more than one edge).Then X (T ) will be a locally compact ultrametric space, and we will also denote thecorresponding metric on this space by d(·, ·). The space X (T ) is called the boundaryof the tree T .
If we fix the partial order in T and choose different functions d(·) of diametersof balls (increasing with respect to the same partial order) then the correspondingultrametric spaces X (T ) will be equivalent (metrics in X (T ) for different functionsd(·) will generate the same set of balls).
2.2.2 Comparison with the Construction of the Absolute of a Tree
Let us discuss the construction of the absolute of a tree, see e.g. [391], [392]. Let Tbe a tree. Let us consider paths in the tree without self-intersection which are either
2 The diameter of a subset in metric space is the supremum of distances between the points of this set.
2.2 Clustering, Trees, and Ultrametric Spaces 11
infinite or stop at a vertex which is incident to a single edge. Two paths are equivalentif they coincide starting from some vertex.The absolute of a tree is the set of equivalence classes of paths. Let us fix some path
in the tree. This path defines a point at the absolute, and we call this point the infinitepoint. We denote this path by O∞, where O is the vertex where the path begins and∞ is the infinite point of the absolute.Let us assume that the tree T is a p-tree, i.e. each vertex of the tree is incident
to p+ 1 edges. Let us define a metric at the absolute (without the infinite point)in the following way. For two non-infinite points x, y of the absolute we consider(infinite) path xy in the tree which connects these points. Let γ be the distancebetween the point O and the path xy (the number of edges in the finite path OI whichconnects O and xy; this path is unique). We say that the distance between points ofabsolute x and y is equal to p±γ , where we take the distance to be equal to p−γ if thepaths OI and O∞ intersect only in O, and the distance is equal to pγ if OI belongsto O∞.This construction (for prime p) defines a metric on the absolute without the infi-
nite point equivalent to the metric on Qp. The whole absolute (with infinite point) isequivalent to the projective line over Qp.It is easy to see that this construction of the absolute of a tree T is a particular case
of the construction of X (T ) described above. Here the partial order on the tree T isdefined by the point∞ (I > J if I belongs to the path J∞) and the function describedabove taking values p±γ increases with respect to the partial order.
2.2.3 Clustering and Ultrametrics
In a more general situation one can consider clustering in metric spaces which gives(in some sense) an approximation of a metric space by a simpler ultrametric space.Here we consider the example of nearest-neighbor clustering.Let (M, ρ) be some metric space (for example a set of points in Rn with the stan-
dard Euclidean metric). A sequence of points {xi}, i = 1, . . . ,L in (M, ρ) is calledan ε-chain if for any two consequent points of this sequence one has ρ(xi, xi+1) ≤ ε.Points a, b ∈ (M, ρ) are called ε-connected if there exists an ε-chain {xi}, i =1, . . . ,L which connects these points, i.e. the points of this chain satisfy a = x1,b = xL.The chain distance d(a, b) between a and b is defined as the infimum of ε satisfying
the criterion that a and b are ε-connected. The chain distance is a pseudoultrametric(i.e. it satisfies the axioms of an ultrametric except for non-degeneracy: the chaindistance between two different points can be zero). Thus the chain distance definesthe ultrametric on the set of equivalence classes of points in (M, ρ), where two pointsare equivalent if the chain distance between the points is equal to zero.For example, for an ultrametric space the chain distance coincides with the ultra-
metric; for a connected domain in Rn the chain distance between any two points isequal to zero.
12 Ultrametric Geometry: Cluster Networks and Buildings
A ball in (M, ρ) with respect to the chain distance d(·, ·) is called a cluster.Clustering of (M, ρ) is some covering of M by clusters satisfying the followingproperties.
(i) Any point belongs to at least one cluster.(ii) For any pair a, b of points there exists the minimal cluster sup(a, b) which
contains both points.(iii) For any pair A ⊂ C of nested clusters there exists the maximal cluster B, A ⊂
B ⊂ C.
The clustering procedure generates a partially ordered tree of clusters (or dendro-gram) in the following way:
� vertices of the tree are clusters;� the partial order is defined by inclusion of clusters;� edges connect clusters nested without intermediaries.
Let us consider a function on the tree of clusters which puts in correspondenceto a cluster the (chain) diameter of this cluster. This function takes non-negativevalues and increases monotonically with respect to the partial order. Moreover,for any finite path in the tree (a path is a sequence of vertices of the tree wheresubsequent vertices in the path are connected by edge) there exists a unique maximalvertex.
Example 2.2 Let {dk} be an increasing sequence of positive numbers without con-densation points except for zero, for example {q−k}, 0 < q < 1, k ∈ Z. Then one canconsider a clustering in (M, ρ) which contains clusters with diameters q−k.
2.3 Family of Metrics and Multiclustering
The clustering procedure described in the previous section can be generalized for thecase where instead of a single clustering metric one has a family of metrics. In thiscase we obtain a family of clustering trees which can be combined in a single networkwith the structure of a simplicial complex.The idea is as follows. Let us consider a set X with a family of metrics S. Let
us denote by T (X, s) the tree of clusters in X with respect to the metric s ∈ S. Itmight happen that for two different metrics r, s ∈ S some clusters (correspondingto vertices in the trees T (X, s), T (X, r)) coincide as sets in X . In this case we canidentify these vertices in the different trees (i.e. glue the trees at these vertices). Wewill obtain a graph with a partial order (the partial order is defined by inclusion ofclusters).We will denote by T (X, S) the resulting partially ordered graph for a family S of
metrics. In this graph we fix a pair of S-clusters I, J (i.e. clusters with respect to allmetrics s ∈ S), where one of the clusters belongs to the other without intermediaryS-clusters (i.e. there can exist intermediaries between I, J vertices in T (X, S) but these
2.4 Affine Bruhat–Tits Buildings and Cluster Networks 13
ABCAB
AA B
B C
C
Figure 2.1. Cluster tree A.
vertices are clusters with respect to some subsets in S which do not coincide with allof the set S of metrics).Let us choose a metric s ∈ S and consider a set of all intermediary s-clusters
between I, J. We will denote this set I = J0 ⊂ J1 ⊂ · · · ⊂ Jk = J (we will consideronly the case when this set is finite). Let us call this set a simplex. We also considerarbitrary subsets of this set as simplices. This procedure defines the structure of asimplicial complex on T (X, S).
Example 2.3 Figures 2.1 and 2.2 describe two cluster trees A, B for a set of threepoints. Figure 2.3 describes the union of the trees of clusters in a cluster network Ccorresponding to two metrics. The simplices in C correspond to paths between theminimal and the maximal vertices of a cycle in C and to subsets of these paths (thereare three such cycles).
2.4 Affine Bruhat–Tits Buildings and Cluster Networks
In the present section we will show that for some network of balls in QNp there exists
a natural structure of a simplicial complex which is related to the affine Bruhat–Titsbuilding. A good discussion of buildings can be found in [173]. For a discussion ofp-adic geometry (in particular lattices) see [442].
2.4.1 Affine Bruhat–Tits Building
The vertices of the building are equivalence classes of lattices. A lattice in QNp is an
open compact Zp-module in QNp . Any lattice can be put in the form
⊕Ni=1Zpei,
where {ei} is a basis in QNp .
ABCBC
AB C
B C
A
Figure 2.2. Cluster tree B.
14 Ultrametric Geometry: Cluster Networks and Buildings
ABCBC
ABA B
A BC
C
Figure 2.3. Cluster tree C.
Two lattices are equivalent if one is a scalar multiple of the other. Two lattices L1and L2 are adjacent (connected by an edge) if some representatives from equivalenceclasses L1 and L2 satisfy
pL1 ⊂ L2 ⊂ L1.
(k − 1)-Simplices are defined as equivalence classes of k adjacent lattices, i.e. thechains
pLk ⊂ L1 ⊂ L2 ⊂ · · · ⊂ Lk.
Here 1 ≤ k ≤ N.An apartment in the affine building is the subcomplex corresponding to a fixed
basis {ei} in QNp which contains the equivalence classes of lattices ⊕N
i=1Zppaiei,ai ∈ Z.
2.4.2 Multidimensional p-Adic Metric
A norm in QNp is a functionN (·) taking values in [0,∞) and satisfying the following
conditions:
(i) non-degeneracy: N (x) = 0 ⇔ x = 0;(ii) linearity: N (ax) = ‖a‖pN (x), x ∈ QN
p , a ∈ Qp;(iii) strong triangle inequality: N (x+ y) ≤ max [N (x),N (y)].
Let us consider the norm
Nq1,...,qN (z) = max1≤i≤N
(qi|zi|p), qi ≥ 0, (2.1)
and the metric sq1,...,qN (x, y) in QNp defined by
sq1,...,qN (x, y) = Nq1,...,qN (x− y). (2.2)
Dilations pkZNp , k ∈ Z are balls with respect to all such normsNq1,...,qN if p
−1 < qi ≤1. In particular, for a norm Nq1,...,qN of the form (2.1), with
p−1 < q1 < · · · < qN ≤ 1, (2.3)
2.4 Affine Bruhat–Tits Buildings and Cluster Networks 15
the set of intermediary Nq1,...,qN -balls between pZNp and ZN
p contains the balls
B( j) = Zp × · · · × Zp × pZp × · · · × pZp
with j components Zp and N − j components pZp, j = 0, . . . ,N.By factorizing these balls (as Zp-modules) over pZN
p we represent the abovesequence of balls as a complete flag over the finite field Fp (a residue field with pelements).A general norm (U-rotation of Nq1,...,qN ) is defined as
NUq1,...,qN (z) = Nq1,...,qN (Uz), (2.4)
whereU is amatrix fromGL(Qp,N). Ametric sUq1,...,qN is defined by the normNUq1,...,qN
as above (2.2). Using multiplications of zi by integer powers of p, one can put a norm(2.4) in the form with p−1 < qi ≤ 1, i = 1, . . . ,N, and, by then applying transposi-tions of the coordinates, one can get
p−1 < q1 ≤ · · · ≤ qN ≤ 1.
2.4.3 Simplicial Complex of Balls
Let us define the structure of a simplicial complex on the network C of balls withrespect to the above-defined family of metrics sUq1,...,qN ,U ∈ GL(Qp,N).
Let s be a metric from the described family and I an s-ball containing zero (ans-ball is a ball with respect to s; zero is the vector in QN
p with zero coordinates). Thenthe dilation pI is also an s-ball. The s-balls I and J (containing zero) are adjacent ifpI ⊂ J ⊂ I. (k − 1)-Simplices are defined as families of k adjacent s-balls
pIk ⊂ I1 ⊂ I2 ⊂ · · · ⊂ Ik.
Let us consider the maximal sequence of nested intermediary s-balls between pI andI. If the parameters qi of the norm are generic (any two parameters cannot be madeequal by multiplication by powers of p, for example when the parameters satisfy(2.3)) then the above sequence contains N + 1 balls and defines an (N − 1)-simplex.Subsets of this simplex are simplices of lower rank.General simplices in the simplicial complex of balls with respect to ametric sUq1,...,qN
are defined as translations of the simplices described above (translations as familiesof sets in QN
p ).The simplicial complex C of balls with respect to the family {sUq1,...,qN } of metrics
is defined as a union of complexes of balls for different metrics sUq1,...,qN . Here weidentify an s-ball and an r-ball which coincide as sets (and identify s- and r-simpliceswhich coincide as sets of balls).
2.4.4 Relation between Norms in QNp and Simplices in the Affine Building
Any ball centered at zero with respect to a norm in QNp is a lattice. Let us consider,
in the above-defined simplicial complex C of balls, the subcomplex C0 of balls which
16 Ultrametric Geometry: Cluster Networks and Buildings
contain zero. For any ball I containing zero a dilation pkI, k ∈ Z, is also a ball (withrespect to the same norm). The same holds for simplices. Therefore the factor C0/�defined by the group of dilations by pk, k ∈ Z, is a simplicial complex.There exists a natural simplicial map from the simplicial complex C0/� to
the affine Bruhat–Tits building which corresponds to a ball on the correspondinglattice.The defined map is an embedding of the complex C0/� into the affine building. Let
us show that this map is surjective (i.e. it is an isomorphism of simplicial complexes).We say that two norms are equivalent if they generate the same family of balls. Let
us show that, to any maximal simplex in the affine Bruhat–Tits building, one can putin correspondence an equivalence class of norms in QN
p .Let L be a lattice in QN
p and
pL = L0 ⊂ L1 ⊂ · · · ⊂ LN = L (2.5)
be a maximal sequence of (different) embedded lattices (equivalently, a maximalsimplex in the affine building).For any pair Lj ⊃ Lj−1, j = 1, . . . ,N of consecutive lattices in the above sequence
let us choose an element f j ∈ Lj, f j /∈ Lj−1. This gives a set { f1, . . . , fN} of vectorsin QN
p . One has the following lemma.
Lemma 2.4 (i) The set { f1, . . . , fN} defined above is a basis in QNp . (ii) The lattices
L j have the form
Lj = ⊕ ji=1Zp fi ⊕⊕N
i= j+1pZp fi. (2.6)
Let us introduce a norm in QNp as follows. Let us put in correspondence to lattices
Lj from (2.5) some positive numbers q j, p−1 < q1 < · · · < qN ≤ 1.Let us define a function N (x) on QN
p in the following way. For x ∈ Lj\Lj−1, j =1, . . . ,N we define N (x) = qj. For x = 0 we put N (x) = 0. We define N (·) in allQNp using the condition N (pkx) = p−kN (x), k ∈ Z.
Lemma 2.5 The functionN (·) defined as above will be a norm in QNp satisfying the
strong triangle inequality. The sequence (2.5) of lattices will be a maximal sequenceof balls with respect to N (·) which lie between the balls pL and L.
The introduced normN belongs to the family (2.4) with the parameters satisfyingp−1 < q1 ≤ · · · ≤ qN ≤ 1. In particular, the matrix U can be chosen as the matrixwhich maps the basis { f1, . . . , fN} from the above lemma to the coordinate basis inQNp and N (·) = NU
q1,...,qN (·). Any two norms defined in this way will be equivalent(will generate the same set of balls).We have constructed a norm of the form (2.4), (2.3) starting from a maximal sim-
plex in the affine building. Analogously, let us consider for a normNUq1,...,qN (·) defined
2.5 Groups Acting on Trees and the Vladimirov Operator 17
by (2.4), (2.3) the set of lattices (2.6) where the basis { f1, . . . , fN} is defined by thematrixU as above. This set defines a simplex in the affine building.We have shown that there exists a one-to-one correspondence between the equiv-
alence classes of norms of the form (2.4), (2.3) and maximal simplices in the affineBruhat–Tits building.In the above construction it is important to consider normswith generic parameters.
Let us consider a general norm N of the form (2.1), (2.4) and take an N -ball L. Itis possible (say, if some parameters qi in (2.1) are equal) that a set of intermediaryballs between L and pL contains fewer than N + 1 balls and therefore cannot definean (N − 1)–simplex in the affine building.
2.5 Groups Acting on Trees and the Vladimirov Operator
Let us discuss some properties of groups acting on trees and their relation to theVladimirov fractional operator, following the article [26]. For general discussion ofgroups acting on trees see [370], [362], [98], [99].Let us consider the p-tree, where p is prime, see the discussion in Section 2.2,
the absolute of this tree, and the selected infinite point ∞ of the absolute. We willconsider the following subgroups of the group of automorphisms of the tree.The parabolic subgroup contains automorphisms of the tree which conserve some
fixed point of the absolute. This point can identified with the infinite point ∞ of theprojective line. Therefore the parabolic group can be identified with the group of ball-morphisms of Qp, i.e. the group of one-to-one maps of Qp for which the image andthe inverse image of any ball are balls.The orispheric subgroup in the parabolic group contains ball-morphisms φ which
not only conserve the point ∞ of the absolute of the tree, but also have the propertythat for any φ in the orispheric subgroup there exists a path in the tree in the equiv-alence class of ∞ such that φ conserves the tail of the path (i.e. maps any vertex inthe path to itself starting from some vertex in the path). Equivalently, the orisphericmap φ conserves some ball I in Qp and the sequence of increasing balls startingin I.
Lemma 2.6 The orispheric group coincides with the group of isometries in Qp.
The following statement can be found in [370].
Lemma2.7 The orispheric group is a normal subgroup in the parabolic group, more-over the factor-group is isomorphic to Z.
For the p-adic case, the factor-group mentioned above can be identified with thegroup of multiplications by powers of of p.
Lemma 2.8 An arbitrary transformation belonging to the parabolic group can beuniquely expressed as a product of the multiplication by a power of p and an isometry(i.e. a map from the orispheric group).
18 Ultrametric Geometry: Cluster Networks and Buildings
In contrast to the real case, in the p-adic case the group of isometries doesnot possess a finite set of generators. The parabolic group (i.e. the group of ballmorphisms) for p-adic analysis should play the role of the group of diffeomorphismsin real analysis.One can consider p-adic diffeomorphisms. It is easy to check that there exist dif-
feomorphisms which do not belong to the group of ball-morphisms. Also there existnon-differentiable ball-morphisms.The following lemma can be found in [390].
Lemma 2.9 (i) If the function f : Qp → Qp is differentiable at a and the derivativeat a is not equal to zero, then there exists a ball with a sufficiently small diameterand the center at a satisfying the following property: for any x in this ball
| f (x)− f (a)|p = | f ′(a)|p|x− a|p.
(ii) If the function f : Qp → Qp is differentiable at a and the derivative at a is equalto zero, then for any C > 0 there exists a ball with a sufficiently small diameterand the center at a satisfying the following property: for any x in this ball
| f (x)− f (a)|p ≤ C|x− a|p.
Corollary 2.10 For a continuously differentiable isometry on Qp the norm of thederivative is equal to one.
2.5.1 Pseudodifferentiation of a Composite Function
Let us investigate transformations of pseudodifferential operators with respect to theaction of parabolic maps. Let us consider a parabolic map φ which is a compositionof multiplication by pγ and isometry. One can check that isometry commutes withthe Vladimirov fractional operator Dα , see (1.5).
This implies the following formula for the pseudodifferentiation of a compositefunction:
Dα ◦� f (x) = p−γα � ◦ Dα f (x).
Here � is a map acting on functions which corresponds to the parabolic map φ:� f (x) = f (φ(x)).In the case where the parabolic ball-morphism φ is continuously differentiable on
Qp, by Corollary 2.10 the multiplier p−γα can be expressed as the degree of the normof the derivative of the map φ:
Dα ◦� f (x) = |φ′(x)|αp � ◦ Dα f (x).
2.5 Groups Acting on Trees and the Vladimirov Operator 19
The above formula can be compared with the formula for differentiation of a com-posite function in real analysis,
d
dxf (φ(x)) = df (y)
dy
∣
∣
∣
∣
y=φ(x)
dφ(x)
dx.
We have proved that in p-adic analysis for the non-local Vladimirov operator Dα offractional differentiation there exists an analog of the formula for differentiation ofa composite function. Let us note that the p-adic parabolic group does not possess afinite set of generators. In this sense this group is infinite-dimensional.
3
p-Adic Wavelets
3.1 Introduction
3.1.1 p-Adic Wavelets
In the present chapter we give a review of the theory of p-adic wavelets, see also thereview article [302]. Wavelets are widely used in many areas of application. The firstwavelet basis was introduced in 1910 by Haar. In [194] he introduced an orthonormalbasis in L2(R) consisting of dyadic translations and dilations of a single function:
ψHjn(x) = 2− j/2ψH(2− jx− n), x ∈ R, j ∈ Z, n ∈ Z, (3.1)
where
ψH(x) =⎧
⎨
⎩
1 if 0 ≤ x < 12
−1 if 12 ≤ x ≤ 1
0 if x /∈ [0, 1)= 1[0, 12 )(x)− 1[ 12 ,1](x), x ∈ R, (3.2)
is called a Haar wavelet. Here 1A denotes the characteristic function of a set A ⊂ R.Although the Haar basis (3.1) and some generalizations of it were discussed in
many articles, until the 1990s there were no new examples of wavelet functions(functions which generate orthonormal bases by translations and dilations). At thebeginning of the 1990s the multiresolution analysis (MRA) method for constructingwavelet bases was proposed by Y. Meyer [332] and S. Mallat [321], [320]. With thisapproach many examples of wavelet functions were constructed, in particular, impor-tant examples of wavelet bases were constructed by I. Daubechies [117]. For a reviewof the wavelet theory and related subjects see the books [217], [282], [117], [363].Let us mention that the translations and dilations used for the construction of real
wavelet bases do not constitute a group.p-Adic wavelet theory started in 2002 when S. V. Kozyrev [286] introduced a basis
of complex-valued wavelets with compact support in L2(Qp). This basis is an analogof the Haar basis and has the form
ψk; jn(x) = p− j/2χp(
p−1k(pjx− n))
(|pjx− n|p
)
, x ∈ Qp, (3.3)
20
3.1 Introduction 21
where k ∈ {1, 2, . . . , p− 1}, j ∈ Z, and n can be considered as an element of thefactor group Qp/Zp understood as a fraction of the form
n =−1∑
i=ani p
i, (3.4)
where a is a negative integer, ni ∈ {0, . . . , p− 1} (in the following this system ofrepresentatives in Qp/Zp will be denoted Ip; equivalently, these representatives canbe identified with p-adic numbers with zero integer part), χp is the standard additivecharacter of Qp, and (t ) is a characteristic function of [0, 1] ⊂ R.Let us note that functions ψk; jn of the form (3.3) depend on the choice of repre-
sentatives in n ∈ Qp/Zp (on taking different representatives one gets functions whichdiffer by multiplication by pth roots of unity). To avoid this inconvenience we alwaysfix the choice of representatives n ∈ Qp/Zp by the rule (3.4).
The basis (3.3) is given by translations and dilations of the wavelet function
ψ (x) = χp(
p−1x)
(|x|p
)
, x ∈ Qp. (3.5)
Moreover, an orbit of the above function with respect to all translations and dilationsfrom the affine group
f (x) �→ |a|−12
p f
(
x− b
a
)
, a, b ∈ Qp, a = 0
coincides with the set of all products of elements of the basis (3.3) and pth rootsof unity. In this simple example we see already the difference of real and p-adicwavelets. In the p-adic case, wavelet systems are related to representations of groups.This connection can be generalized to different wavelet functions and different groupsof transformations (in the multidimensional case).The basis of p-adic wavelets (3.3) was generalized to the case of locally compact
ultrametric spaces in [290], [233], [298].J. J. Benedetto and R. L. Benedetto [72] and R. L. Benedetto [76] proposed a
method for constructing wavelet bases on locally compact Abelian groups, with com-pact open subgroups, based on the theory of wavelet sets. This method allows one toconstruct wavelet functions with Fourier transforms equal to characteristic functionsof some sets [72, Proposition 5.1.], in particular the basis (3.3).
3.1.2 p-Adic Multiresolution Analysis
The scaling (or refinement) equation for the p-adic MRA (see Definition 3.39) hasthe form (see [256] and Section 3.9.2)
φ(x) =p−1∑
r=0
φ
(
1
px− r
p
)
, x ∈ Qp. (3.6)
This equation reflects the natural self-similarity of the space Qp. The solution φ ofthis equation (the scaling function) is a characteristic function(|x|p) of the unit ball.
22 p-Adic Wavelets
Equation (3.6) for p = 2 takes the form
φ(x) = φ
(
1
2x
)
+ φ
(
1
2x− 1
2
)
, x ∈ Q2, (3.7)
which is a direct analog of the scaling equation
φ(t ) = φ(2t )+ φ(2t − 1), t ∈ R, (3.8)
related to the Haar MRA and the Haar wavelet basis (3.1), (3.2). The 2-adic scalingfunction φ(x) =
(|x|2)
is a characteristic function of a unit ball in Q2 while the realscaling function (the solution of (3.8))
φH(t ) ={
1 if t ∈ [0, 1]0 if t /∈ [0, 1]
= 1[0,1](t ), t ∈ R, (3.9)
is a characteristic function of the unit interval [0, 1].p-Adic MRA in L2(Qp) (see Definition 3.39 in Section 3.9.1 and Section 3.10)
was introduced in [393]. Unlike in the real Haar MRA in L2(R), in the p-adic case,there exist infinitely many different orthogonal Haar bases in L2(Qp) generated bythe same MRA, see [393], [261] and Theorems 3.43 and 3.42 in Section 3.10.3. Thisfamily contains basis (3.3) (see (3.68) and (3.66) later), but the majority of bases inthis family cannot be obtained by Benedetto’s construction [72].In [257], [258], [259] examples of non-Haar wavelet bases were constructed.In [262] (see Section 3.11) p-adic scaling equations and scaling functions (solu-
tions of scaling equations) were studied. A class of p-adic scaling functions and corre-sponding MRAwas described. These functions are 1-locally constant and their trans-lations are mutually orthogonal. In [4], [5], [6] it was shown that there do not existcompactly supported locally constant scaling functions with orthogonal translationsdifferent from the described in [262].Moreover, in [4], [5] all these functions generatethe same p-adic Haar MRA. Also in [4] there no orthogonal multiresolution waveletbases different from those described in [393], [261] (see Theorems 3.43 and 3.42 ofthe latter).In Section 3.13 we describe the p-adic analog of the Meyer–Mallat [363, Section
2.1] multidimensional MRA as a tensor product of one-dimensional p-adic MRA. In[364] multiresolution Haar analysis on various spaces was discussed.
3.1.3 Wavelet Frames as Systems of Coherent States
In the real case the standard approach to wavelets is the multiresolution analysis. Inthe p-adic case bases and frames of wavelets are related to systems of coherent statesfor the affine group in the one-dimensional case, and, in the multidimensional case,are related to actions of various groups of transformations.A system of generalized coherent states for the action of group G in the Hilbert
space H (see [374]) is an orbit of an unitary representation of this group in H.Coherent states were studied in quantum mechanics, see [268]; in p-adic quantummechanics, coherent states were considered by E. I. Zelenov [434].
3.1 Introduction 23
For p-adic groups of transformations (for instance subgroups of linear transforma-tions and affine groups) an orbit of a locally constant function (the function f fromthe spaceD(QN
p )) will be a discrete set since a small transformation will map a locallyconstant function into itself.By the theory of coherent states, see e.g. [374], in the case of irreducible represen-
tations and under some condition of integrability, an orbit will be a tight homogeneousframe, i.e. the orbit { f (l)} (where l parametrizes the orbit of the function f ) will satisfythe following condition: there exists A > 0 such that for any g ∈ H
∑
l
|〈g, f (l)〉|2 = A‖g‖2,
where 〈·, ·〉 is the scalar product in H, and ‖ · ‖ denotes the corresponding norm.The integrability condition mentioned above will be satisfied for orbits of functionsf from the space �(QN
p ) of mean-zero test functions of p-adic argument.Consequently, since the continuous wavelet transform is the expansion over coher-
ent states of the affine group, in the p-adic case, continuous and discrete waveletanalysis can be considered as an application of representation theory [21].In Section 3.4 (see also [22]) we investigate the orbit of a function f ∈ �(Qp) with
respect to the one-dimensional p-adic affine group. We show that if the function f isgeneric (see the definition in Section 3.4) the corresponding tight frame possesses aparametrization similar to the parametrization of the p-adic wavelet basis. This factallows us to obtain naturally the structure of the multiresolution wavelet bases in thep-adic case. In this way one can construct very general wavelet frames includingframes which cannot be obtained by the multiresolution analysis. In addition, wewill see that to perform the multiresolution construction one needs a structure ofrepresentation of the group Qp/Zp on the space V0 of the multiresolution expansion.Therefore, in the p-adic case the multiresolution analysis is a particular example ofthe method of coherent states.The multidimensional p-adic basis obtained as a direct product of the one-
dimensional wavelet bases (3.3) was considered in [15] (see also Section 3.13). In[23] it was shown that this basis can be considered as a system of coherent states forthe group generated by translations, dilations, and norm-preserving linear transfor-mations, see Section 3.5.In the multidimensional case metrics on p-adic spaces can be introduced in differ-
ent ways. Different metrics will generate the different trees of balls in QNp . Automor-
phisms of trees of balls with respect to the aforementioned metrics will be relatedto matrix dilations of multidimensional wavelets known in the real wavelet analysis[188]. In the p-adic case matrix dilations were considered in [267]. It was shownthat the quincunx wavelet basis (which in the real case contains functions with sup-port on fractals) in the p-adic case consists of test functions of p-adic argument.In Section 3.6, see also [25], we construct different examples of p-adic waveletbases with matrix dilations which correspond to groups of automorphisms of trees ofballs in QN
p with respect to different multidimensional metrics. Therefore, different
24 p-Adic Wavelets
multidimensional p-adic wavelet bases are related to systems of coherent states fordifferent groups.
3.1.4 p-Adic Pseudodifferential Operators
Models of p-adic mathematical physics use pseudodifferential operators insteadof differential operators. The theories of p-adic pseudodifferential operators andwavelets are related. In the first article on p-adic wavelets [286], it was men-tioned that wavelets are eigenvectors of the Vladimirov p-adic fractional derivativeoperator1
Dαψk; jn = pα(1− j)ψk; jn.
On the other hand, the Monna map allows one to define the action of p-adic pseu-dodifferential operators on functions of real argument, see [286] and Section 3.8. Inthis way the real Haar wavelets can be considered as eigenvectors of the Vladimirovoperator.Further development of wavelet methods in application to spectral theory of p-
adic pseudodifferential operators takes place in the articles [15, 16, 24, 17, 19],[28], [233, 298, 299, 237], [256, 259, 257], [261], [294, 286, 288, 287, 291, 290] (seeSection 3.15). Different examples of p-adic pseudodifferential operators with appli-cations, in particular to wavelets, were studied in [100], [103], [386], [405], [472],[470], [468], [467], [466], [464].Besides this, it was shown (see Section 3.15) that, in addition to the standard family
of pseudodifferential operators (which can be diagonalized by the Fourier transform)in the p-adic case [287], [288], [303] (and for general locally compact ultrametricspaces [298], [233], [290]), there exists a new class of integral operators which canbe diagonalized by the wavelet transform: wavelets are eigenvectors of the operatorsand the spectra can be explicitly computed. These operators have the form
T f (x) =∫
XT (sup(x, y))( f (x)− f (y))dν(y).
Here X is a complete locally compact ultrametric space (for example Qp), ν is aBorel measure, sup(x, y) is the minimal ball containing the points x and y, and theintegration kernel T (I) is a function on the set of balls in X .
In particular, wavelet bases on general locally compact ultrametric spaces wereintroduced [290], [233], [298]. Wavelets and pseudodifferential operators on adeleswere discussed in [285], [284], [263].
3.1.5 p-Adic Analysis and Mathematical Physics
Mathematical physics was developed on the basis of real and complex numbers. Start-ing from the 1980s the field of p-adic numbers Qp began to be used in theoretical and
1 The existence of bases of compactly supported eigenvectors for p-adic pseudodifferential operators wasmentioned in [433], but the basis constructed in this work was not a wavelet basis.
3.1 Introduction 25
mathematical physics. For the results of this development see the books [434], [275],[18], [222], [294], [278], [470] (for analysis in the fields of positive characteristic)and the review article [135].The initial idea of application of p-adic analysis in mathematical physics is as
follows: the problems of compatibility of quantum mechanics and gravitation the-ory are related to the use of an infinitely divisible real continuum as a base formathematical models of physical space. It was proposed that in cosmology and stringtheory the structure of space-time at the Planck scale (around 10−33 cm) has a non-Archimedean nature (the Archimedean axiom will not be satisfied), see Volovich[435], [436], [437], [438] (and the books by Vladimirov, Volovich, and Zelenov [434]and by Khrennikov [225]).The foundations of p-adic quantum mechanics were established in articles by V.
S. Vladimirov, I. V. Volovich, B. Dragovich, and A. Yu. Khrennikov, see e.g. [419],[433], [430], [432], [44], [46], [42], [45], [43], [221], [222], [434], [127], [125], [129],[137], [168], [458] (where adelic models of decoherence were considered); p-adicstring amplitudes were considered in [423], [422], [421], [424]. Summation of p-adicseries with applications to quantum mechanics was considered in [128]. Non-linearequations related to p-adic strings were investigated in [431, 425, 426, 427, 428, 429].p-Adic pseudodifferential operators and their applications were investigated by V.
S. Vladimirov [419], [420], [433]. Fundamental solutions for p-adic pseudodifferen-tial operators with symbols | f |αp , α > 0, and f an arbitrary polynomial were estab-lished by W. A. Zúñiga-Galindo in [462], see also [470]. In Chapter 10, we present aproof of the existence of fundamental solutions inspired by the ideas of Hörmanderand Łojasiewicz, see [202], [316], which does not require the machinery of local zetafunctions. Important contributions to the theory of non-Archimedean pseudodifferen-tial equations were made by A. N. Kochubei [270, 269, 270, 273, 275, 274, 277, 279]and W. A. Zúñiga-Galindo [101], [100], [104]–[105], [172], [386], [406], [405][462]–[473]. Bases of eigenvectors with compact support for p-adic pseudodifferen-tial operators were constructed by V. S. Vladimirov [433] and A. N. Kochubei [270],[269].A generalization of these bases is the basis of p-adic wavelets introduced by
S. V. Kozyrev [286]. Further results on p-adic wavelets were obtained in [294], [288],[287], [15], [233], [298], [256], [259], [257], [258], [261], [262], [393]. Wavelet the-ory is important for applications of p-adic analysis and allows one to investigate notonly linear but also non-linear p-adic pseudodifferential equations [17], [259], [294],[291] and equations with singular potentials [28], [305]. In [245] expansion of p-adicrandom walk over wavelets was used for computation of correlation functions of thisstochastic process.
3.1.6 Relation to the Walsh Theory
Walsh functions can be considered as characters of Cantor dyadic group C. In thearticles [307], [306] the first examples of multiresolution orthogonal wavelets on the
26 p-Adic Wavelets
group C were constructed, the multifractal structure of these wavelets was discussed,and conditions to generate unconditional bases in the spaces Lq(C), 1 < q < ∞, werefound. In the article [153] for finite scaling functions in the space L2(C) (analogous tothe Daubechies scaling functions) an algorithm of expansion in lacunar Walsh serieswas proposed and estimates for themoduli of smoothness were found (see also [385]).In [153] orthogonal wavelets on locally compact Abelian group G equal to the weakdirect product of a countable set of cyclic groups of order p were constructed (forp = 2 the group G is isomorphic to the Cantor group and for p > 2 the group Gis a Vilenkin group). In [154] a review on multiresolution analysis and orthogonalwavelets on Vilenkin groups was given and in [155] the biorthogonal case was con-sidered.Multiresolution analysis was also developed for the half-line R+ with dyadic addi-
tion. In [375] for any natural n at the half-line R+ scaling functions with masks equalto the Walsh polynomials of order 2n − 1 were introduced. For the correspondingscaling (or refinement) equations the Strang–Fix condition, the partition of unity,linear independence, and other properties for integer translations were investigated.Necessary and sufficient conditions in terms of blocking sets for the solutions ofthe aforementioned scaling equations to generate multiresolution analyses in L2(R+)were found. It was proven that the finite scaling function on R+ is either a dyadicinteger or has a finite smoothness which can be estimated from above. In [157] peri-odic wavelets and frames on the half-line R+ related to the Dirichlet–Walsh kernelswere studied, and in [156] a method to construct p-wavelets with compact supportson R+ was described and generalizations of corresponding results from [375] werefound (here p ∈ N, p ≥ 2, is the scaling coefficient).For a review of the Walsh theory see [184]. In [180, 181, 182, 183] pseudo-
differential operators on the dyadic half-line R+ were investigated.
3.2 Basis of p-Adic Wavelets
In this section we discuss the construction of a basis of p-adic wavelets and its rela-tions with the spectral theory of p-adic pseudodifferential operators [286] and withthe action of the affine group [22], [21]. Unlike in the real case, in the p-adic case thewavelet theory is related to the spectral theory and the representation theory.The basis of p-adic wavelets is constructed as a set of translations and dilations of
a finite set of wavelets related to a unit ball. Let us consider the following complex-valued function of p-adic argument
ψk(x) = ψ (kx) = χp(p−1kx)(|x|p), x ∈ Qp, (3.10)
where |k|p = 1, (|x|p) is a characteristic function of the unit ball B0 ⊂ Qp, and χpis the standard additive character of Qp.
There exist exactly p− 1 different functions of the form (3.10) (considered as func-tions of x) due to the local constancy of ψk(x) with respect to k. Taking the represen-tatives k = 1, . . . , p− 1 in maximal subballs of the sphere |k|p = 1, we get the set of
3.2 Basis of p-Adic Wavelets 27
wavelets
ψk(x) = χp(p−1kx)(|x|p), k = 1, . . . , p− 1.
Thus the orbit of the group of dilations from the unit sphere applied to the waveletψ = χp(p−1·)(| · |p) is exactly the set of wavelets ψk.
The wavelet basis is introduced by translations and dilations of the functions ψk.Unlike in the real case we cannot use translations by integers, since in the p-adiccase integers constitute a dense set in a unit ball. Instead we will use translationsby representatives from equivalence classes of the factor group Qp/Zp of the form(3.12).
Theorem 3.1 ([286]) (i) The set of functions {ψk; jn} (p-adic wavelets), obtained from{ψk} by dilations by integer powers of p and by translations by elements of the form(3.12),
ψk; jn(x) = p−j2ψk(p
jx− n), x ∈ Qp, j ∈ Z, n ∈ Qp/Zp, (3.11)
n =−1∑
i=βnip
i, ni = 0, . . . , p− 1, β ∈ Z, with β < 0, (3.12)
forms an orthonormal basis in L2(Qp).(ii) The elements of this basis are eigenvectors of the Vladimirov operator (1.5)
(the p-adic fractional derivative) Dα:
Dαψk; jn = pα(1− j)ψk; jn.
If we replace the set of representatives (3.12) of Qp/Zp by others, we get a basisof wavelets whose elements differ from (3.11) by multiplication by p-roots of theunity.Let us discuss the connection of the above basis and the representation theory of
the p-adic affine group. The affine group acts in L2(Qp) by translations and dilations:
G(a, b) f (x) = |a|−12
p f
(
x− b
a
)
, a, b ∈ Qp, a = 0, for f ∈ L2(Qp).
Orbits of functions from the space �(Qp) (of locally constant compactly supportedmean-zero functions) with respect to actions of p-adic groups of transformations (i.e.systems of generalized coherent states for these groups in the sense of [374]) will bewavelet frames (see Definition (3.4) for a definition of frame).In particular we have the following result describing this orbit.
Lemma 3.2 ([21]) The orbit of the wavelet ψ (x) = χp(p−1x)(|x|p) with respect tothe action of the affine group is a frame of wavelets which contains all products ofvectors from the basis (3.11) and p-roots of the unity:
{e2π ip−1mψk; jn}, j ∈ Z, n ∈ Qp/Zp, k = 1, . . . , p− 1, m = 0, 1, . . . , p− 1.
28 p-Adic Wavelets
This implies that the continuous p-adic wavelet transform
F (a, b) = |a|−12
p
∫
Qp
f (x)ψ
(
x− b
a
)
dx
coincides with the expansion over the (discrete) basis of p-adic wavelets.Let us discuss [23] the multidimensional basis of p-adic wavelets. We consider a
set of functions
ψk(x) = χp(p−1k · x)(‖x‖p), x, k ∈ QN
p , (3.13)
where ‖k‖p = 1, k · x = ∑Nl=1 klxl , and ‖x‖p = maxi |xi|p is the norm in QN
p .There exist pN − 1 different functions ψk(x) (as functions of x). Let us choose
the following representatives for SN0 /(pZp)N (here SN0 is the N-dimensional sphere ofradius 1):
k = (k1, . . . , kN ) , kl = 0, . . . , p− 1, (3.14)
where at least one of the kl is non-zero. The index k enumerates the set of maximalN-dimensional subballs in the N-dimensional sphere.We construct the basis {ψk; jn} ofN-dimensional p-adic wavelets by applying to the
set of functions {ψk} dilations, by integer powers of p, and translations by elementsof the factor group QN
p /ZNp (understood as representatives (3.16)):
ψk; jn(x) = p−N j2 ψk(p
jx− n), x ∈ QNp , j ∈ Z, n ∈ QN
p /ZNp , (3.15)
n = (
n(1), . . . , n(N ))
, n(l) =−1∑
i=βln(l)i pi, n(l)i = 0, . . . , p− 1, βl ∈ Z, βl < 0.
(3.16)
Theorem 3.3 ([23]) The set of functions {ψk; jn} defined by (3.15), (3.16) forms anorthonormal basis in L2(QN
p ).
This multidimensional basis of wavelets coincides with the multiresolution multi-dimensional wavelet basis, see Section 3.13.
3.3 Coherent States
Let us recall the construction of generalized coherent states [374]. Let G be a locallycompact topological group and let T (g) be a unitary representation of this group in aHilbert space H. Let us fix a vector ψ0 ∈ H. Let H ⊂ G be the stabilizer of ψ0 (thesubgroup which leaves ψ0 invariant).
A system of coherent states {G,T, ψ0} for the unitary representation T of the groupG in H is defined as the orbit T (g)ψ0, g ∈ G, with respect to the representation T .Therefore coherent states correspond to points of the factor space G/H.We use the following notations. We denote the group action
T (g)|0〉 = |x(g)〉, |0〉 = ψ0.
3.3 Coherent States 29
Completeness of the system of coherent states follows from irreducibility of the rep-resentation T . Let μ denote the left invariant Haar measure of the group G. Thismeasure induces a measure on the factor space G/H, if the restriction to H of themodular function of G coincides with the modular function of H (the modular func-tion is the continuous homomorphism ofG to the multiplicative group of positive realnumbers generated by right translations of the left invariant Haar measure).Let us consider the integral over the projection operators
B =∫
G/H|x〉〈x|dμ(x).
Here |x〉〈x| is the projection on vector |x〉.By the invariance of μ the operator B commutes with all operators T (g):
T (g)BT−1(g) = B.
Therefore, by the irreducibility of T , the operator is proportional to the unit operator:
B = d id, d = 〈0|B|0〉 =∫
G/H|〈0|x〉|2 dμ(x). (3.17)
In particular the convergence of the above integral is necessary for existence of theoperator B.If the orbit {|x〉} is discrete the formula for 〈ψ0,Bψ0〉 takes the form
∑
x∈G/H|〈ψ0|x〉|2 = d‖ψ0‖2.
Here the measure on G/H is normalized to make the measure of each element equalto one. This implies that this system of coherent states is a tight homogeneous framewith the bound d.Let us recall the definition.
Definition 3.4 A frame {en} in the Hilbert spaceH is a set of vectors inH satisfyingthe following criterion: there exists A, B > 0 such that, for any g ∈ H,
A‖g‖2 ≤∑
n
|〈g, en〉|2 ≤ B‖g‖2.
The constants A and B respectively are called the lower and upper bounds ofthe frame. If A = B the frame is called tight. If ‖en‖ is constant, the frame ishomogeneous.
Example 3.5 Lemma 3.2 states that the system of coherent states for the p-adic affinegroup which corresponds to vector ψ (wavelet) is a tight homogeneous frame withthe bound p. This frame is related to the p-adic wavelet basis.
30 p-Adic Wavelets
3.4 Orbits of Mean-Zero Test Functions as Wavelet Frames
In this section, see also [22], we study orbits with respect to the one-dimensionalaffine group of mean-zero test functions of p-adic argument. The space D(Qp) ofp-adic test functions contains complex-valued compactly supported locally constantfunctions. The space �(Qp) is the subspace in D(Qp) of mean-zero test functions.
The orbit of a function from �(Qp) is a tight homogeneous frame and for such afunction the integral (3.17) converges (for non-mean-zero functions fromD(Qp) thisintegral diverges). In addition, we will show that an orbit of generic function from�(Qp) (see the definition below) possesses a parametrization [22] which coincideswith the one used in the multiresolution analysis. Consequently, in the p-adic case,the multiresolution analysis arises naturally and it is related to representation theoryof p-adic groups.The next lemma describes orbits and stabilizers of characteristic functions of balls
in Qp with respect to the affine group G.
Lemma 3.6 (i) The orbit of (| · |p) with respect to the affine group G has the form
G(p− j, p− jn)(|x|p) = p−j2(|pjx− n|p), n ∈ Qp/Zp, j ∈ Z.
(ii) The stabilizer Gp−
j2 (|pj ·−n|p)
of the function p−j2(|pj · −n|p) contains all g=
(a, b) with |a|p = 1 and b of the form
b = p− j(n(1− a)+ z), z ∈ Zp.
In particular the stabilizer G(·) contains b ∈ Zp and a with |a|p = 1. Here elementsof the factor group Qp/Zp are given by the representatives (3.12).
The next lemma describes the stabilizer of the wavelet χp(p−1x)(|x|p) withrespect to action of the affine group, see Lemma 3.2.
Lemma 3.7 The stabilizer Gψk; jn of the wavelet ψk; jn contains all g= (a, b) with
a ≡ 1mod p, pjb ≡ n(1− a)mod p.
Any function f from �(Qp) can be uniquely represented as a finite linear combi-nation of wavelets:
f =∑
k jn
Ck jnψk; jn, Ck jn ∈ C. (3.18)
The action of the affine group maps a wavelet to a wavelet. Thus, if an element ofthe affine group maps f ∈ �(Qp) into itself then either this transformation preservesall wavelets in expansion (3.18) or it acts as a reshuffling of wavelets in this expan-sion. The second option is possible only in the special case when the correspondingcoefficients in the expansion differ by a p-root of the unity.We will say that a function f ∈ �(Qp) is generic if the stabilizer of f with respect
to the action of the affine group coincides with intersection of the stabilizers ofwavelets in the expansion (3.18) for f .
3.4 Orbits of Mean-Zero Test Functions 31
Example 3.8 We give an example of a non-generic function. Consider the waveletψk;−1,p−1 (where k can take values 1, . . . , p− 1) supported in the ball |x− 1|p ≤ p−1.Take a transformation G(a, b) from the affine group with b = 0, and a satisfying|a|p = 1, |a− 1|p = 1, ap−1 ≡ 1 mod p2, such that a exists. Indeed, by Fermat’slittle theorem ap−1 ≡ 1 mod p. Since the derivative (ap−1 − 1)′ = (p− 1)ap−2 ≡0 mod p for |a|p = 1, by the Hensel lemma, there exists an a, with |a|p = 1, sat-isfying ap−1 ≡ 1 mod p2.Consider
f (x) =p−2∑
i=0
Gi(a, 0)ψk;−1,p−1 (x) =p−2∑
i=0
ψ(
kp−1( x
ai− 1
))
(3.19)
i.e. f is a sum of wavelets supported in maximal subballs of the sphere |x|p = 1obtained by iterations of the action of G(a, 0) on ψk;−1,p−1 . Since the functionψ
(
kp−1 (x− 1))
is locally constant with diameter of constancy p−2, this proves that fis invariant with respect to the transformation G(a, 0), but this transformation reshuf-fles wavelets in the expansion (3.19). Thus f is non-generic.
Lemma 3.9 Let f ∈ �(Qp) given by (3.18) be a generic function. Then the stabilizerGf of the action of the affine group on f consists of all g= (a, b) in G satisfying thefollowing conditions: a belongs to the ball
|1− a|p ≤ p− jA = min
[
p−1,max(pji−1, pjl−1)
|p− ji ni − p− jl nl |p
]
, (3.20)
where the minimum is taken over all ( ji, ni) and ( jl, nl ) in (3.18) which satisfy
p− ji ni − p− jl nl = 0,
and b satisfies
|b− p− j0n0(1− a)|p ≤ pj0−1, (3.21)
where j0 is the minimal j in the expansion (3.18) and n0 is the corresponding n.
The index n0 can be defined non-uniquely, but in this case the stabilizer does notdepend on the choice of n0.
Remark 3.10 In the proof of the above lemma, it helps a lot that it is easy to computeintersections in ultrametric spaces. In particular, the intersection of any number ofp-adic balls can be a ball, a point, or an empty set.
The next theorem describes frames of p-adicwavelets generated as orbits of genericfunctions from �(Qp) with respect to the action of the affine group (see Definition3.4).
Theorem 3.11 Let f ∈ �(Qp) be a generic function with the wavelet expansion
f =∑
k jn
Ck jnψk; jn; j ∈ Z, n ∈ Qp/Zp, k = 1, . . . , p− 1. (3.22)
32 p-Adic Wavelets
Then, the orbit of f with respect to the action of the affine group possesses the fol-lowing properties:
(i) the orbit coincides with the set of functions
f (k jn) = G(pjk, pjkn) f , j ∈ Z, n ∈ Qp/p1− j0Zp,
k = k0 + k1p+ k2p2 + · · · + k jA−1p
jA−1,
k0 = 1, . . . , p− 1, ki = 0, . . . , p− 1, i > 0; (3.23)
where j0, jA, n0 are introduced in Lemma 3.9;(ii) the orbit is a tight homogeneous frame in L2(Qp): the norms of all f (k jn) are equal
and for any g ∈ L2(Qp) one has
∑
k jn
∣
∣
⟨
g, f (k jn)⟩∣
∣
2 = ‖g‖2∑
k jn
|Ck jn|2pjA− j0+ j. (3.24)
The indices k in (3.23) enumerate balls with the diameter p− jA in the unit sphereand constitute a group with respect to multiplication mod pjA .On the left-hand side (LHS) of (3.24) the summation runs over the elements of the
orbit of the function f and on the right-hand side (RHS) of this formula the summationruns over the elements of the wavelet basis (i.e. the index k on the RHS and that onthe LHS of this formula have different meanings; the same thing happens in formulae(3.22), (3.23)).
Example 3.12 This example was discussed in Lemma 3.2. For the wavelet ψ (x) =χ (p−1x)(|x|p) the orbit coincides with the set of products
{
e2π ip−1mψk; jn
}
of wavelets from the basis (3.11) and p-roots of the unity. The frame bound is equalto p.
Example 3.13 Let us consider a generic function of the form
f =∑
kn
Cknψk; jn (3.25)
(i.e. we fix the scale j in this expansion). In this case, by (3.24) we have the followingexpression for the bound of the frame { f (k jn)}:
pjA∑
kn
|Ckn|2 = pjA‖ f‖2,
where jA ≥ 1 (the minimal jA = 1 was considered in the previous example).
3.5 Multidimensional Wavelets and Representation Theory 33
3.5 Multidimensional Wavelets and Representation Theory
In this section (based on [23]) we study the relations of the multidimensional waveletbasis (3.15), (3.16) considered in Theorem 3.3 and representation theory of some p-adic group of transformations. For representation theory of p-adic groups see [174],[98, 99, 391, 392, 370, 362].We have the following lemma about the action of the affine group on balls with
respect to the norm ‖x‖p = maxi |xi|p.Lemma 3.14 (i) The group QN
p /ZNp consisting of the elements of the form
n = (
n(1), . . . , n(d))
, n(l) =−1∑
i=βln(l)i pi, n(l)i = 0, . . . , p− 1, βl ∈ Z, βl < 0,
(3.26)
with component-wise addition modulo one, acts transitively on the set of balls ofdiameter one in QN
p .(ii) A characteristic function of any ball in QN has a unique representation of the
form
(∥
∥pjx− n∥
∥
p
)
, x ∈ QNp , n ∈ QN
p /ZNp , j ∈ Z.
Definition 3.15 The group GL(Zp,N) is the group of all linear transformations inQNp which conserve the p-adic norm ‖·‖p, i.e. if g ∈ GL(Zp,N), then ‖gx‖p = ‖x‖p
for any x ∈ QNp .
This group can be considered as the p-adic analog of the group of orthogonal lineartransformations in RN . The group GL(Zp,N) is the stabilizer of the unit ball ZN
p inthe group of non-degenerate linear transformations.
Lemma 3.16 The group GL(Zp,N) coincides with the set of matrices with entries inZp and |det(·)|p = 1. Equivalently GL(Zp,N) can be defined as a group of matricesg such that g and g−1 have entries in Zp.
Let us consider the group G of transformations, generated by matrices inGL(Zp,N), arbitrary translations, and homogeneous dilations:
x �→ pjx, j ∈ Z, x ∈ QNp .
We discuss the representation of the group G which acts in the space L2(QNp ) by uni-
tary transformations, i.e. matrices from GL(Zp,N) act as f (x) �→ f (gx), translations
act as f (x) �→ f (x+ b), and dilations by a power of p act as f (x) �→ p−N j2 f (pjx).
The next theorem gives the interpretation of the N-dimensional wavelet basis (3.15),(3.16) as a system of coherent states (the orbit of the representation described above)for the group G.
Theorem 3.17 The orbit ofψ (1)(x) = χp(p−1x)(‖x‖p), x ∈ QNp , with respect to the
unitary representation of group G defined above is a frame in L2(QNp ), which consists
34 p-Adic Wavelets
of all the products formedwith elements of the basis of p-adic wavelets {ψk; jn} definedby (3.15), (3.16) and p-roots of the unity.
Remark 3.18 Let us note that, unlike in the real case, the construction of the p-adicmultidimensional wavelet basis involves an action of the group GL(Zp,N) of norm-preserving linear transformations. Therefore, p-adic wavelet bases are generated notonly by translations and dilations but also by more general linear transformations.
3.6 Wavelets with Matrix Dilations
In the multidimensional case, norms on p-adic spaces can be introduced in severalnon-equivalent ways. Different metrics will generate different trees of balls in QN
p
with different automorphism groups, and will correspond to different wavelet basesand frames, and, in general, these wavelet bases will have matrix dilations.Let us recall that the tree of balls in a locally compact ultrametric space is defined as
follows: vertices are non-trivial balls (balls of non-zero diameter or isolated points),edges connect pairs of vertices (ball, maximal subball), and, in addition, a partialorder is defined by the embedding of balls.Matrix dilations have been used in real wavelet analysis [188]. In the p-adic case,
matrix dilations were considered in [267]. In this article it was shown that the quin-cunx wavelet basis, which in the real case contains wavelets with supports on frac-tals, consists of p-adic test functions. In this section, see also [25], we build bases ofp-adic wavelets with matrix dilations and discuss the corresponding groups of auto-morphisms of trees of balls.
3.6.1 Multidimensional Metrics and Dilations
Multidimensional Metric Let us recall examples of metrics in QNp ; see also Section
2.4. The standard ultrametric has the form
d(x, y) = ‖x− y‖p = max1≤l≤N
(|xl − yl |p), x = (x1, . . . , xN ), y = (y1, . . . , yN ).
(3.27)
A deformed metric depending on positive weights q1, . . . , qN is defined as
s(x, y; q) := s(x, y) = max1≤l≤N
(ql|xl − yl |p), where q := (q1, . . . , qN ) . (3.28)
We denote the corresponding norm on QNp as ‖x− y‖. The reader should keep in
mind that ‖ · ‖ denotes a family of norms depending on a weight q = (q1, . . . , qN ),and that in the particular case (1, . . . , 1) the corresponding norm ‖ · ‖agrees with‖ · ‖p. More general ultrametrics can be obtained from (3.28) by non-degenerate lin-ear transformations:
r(U, x, y) := r(x, y) = s(Ux,Uy), U ∈ GL(Qp,N). (3.29)
3.6 Wavelets with Matrix Dilations 35
It is easy to see that, if B is a ball (centered at zero) with respect to some metric fromthe above family, then pjB, j ∈ Z, will also be balls with respect to this metric. Therealso might be intermediary balls between B and pB.
In particular, the sets pjZNp , j ∈ Z, are balls with respect to all ultrametrics (3.28),
(3.29) if p−1 < ql ≤ 1, l = 1, . . . ,N, and the matrixU in (3.29) belongs to the groupGL(Zp,N) described in Lemma 3.16, i.e. the matrix elements of U are in Zp and|det(U )|p = 1.For metric (3.28) with parameters
p−1 < q1 < q2 < · · · < qN ≤ 1, (3.30)
the sequence of embedded balls in between pZNp and ZN
p has the form
a components︷ ︸︸ ︷
Zp × · · · × Zp ×N−a components
︷ ︸︸ ︷
pZp · · · × pZp (3.31)
with a = 0, . . . ,N.
Dilation
Definition 3.19 A dilation with respect to ultrametric r onQNp is a linear mapQN
p →QNp which maps an arbitrary r-ball centered in zero to a maximal r-subball (with a
center in zero) of this ball.
Here an r-subball of r-ball J (a ball with respect to metric r) is an r-ball I containedin J. The maximal r-subball of J is an r-ball I which is maximal (i.e. there no r-ballsin between I and J).
Let us note that for a given metric a dilation does not necessarily exist. For metric(3.27) dilation is given by multiplication by p; see the previous section. Since any r-ball is a translation of a ball centered at zero, a dilation is an automorphism of the treeT (QN
p , r) of balls in QNp with respect to metric r. When the metric r given by (3.28),
(3.29) is generic (any two parameters ql cannot be made equal by multiplication bya degree of p) a dilation should satisfy |det(·)|p = p−1.
Lemma 3.20 Let A be a dilation in QNp with respect to metric r given by (3.28),
(3.29). Then the set of characteristic functions of balls with respect to metric (3.29)is in one-to-one correspondence with the set of functions
(‖Ajx− n‖), j ∈ Z, n ∈ QNp /B1,
where (·) is a characteristic function of unit interval [0, 1]. The unit ball is definedas
B1 = {
x ∈ QNp : ‖x‖ ≤ 1
}
,
and QNp /B1 denotes a fixed set of representatives of this factor group.
When B1 = ZNp then n ∈ QN
p /ZNp is given by (3.26). In the general case, the
unit ball can be expressed as the image of a ZNp by a linear map, i.e. B1 = VZN
p ,
36 p-Adic Wavelets
V ∈ GL(Qp,N). The representatives in QNp /B1 can be taken in the form Vn, where
n is given by (3.26). In particular, for metric (3.28), (3.29) with p−1 < ql ≤ 1,l = 1, . . . ,N, one has B1 = U−1ZN
p .Let us consider the following matrix in QN
p :
A =
⎛
⎜
⎜
⎜
⎜
⎜
⎝
0 1 0 . . . 00 0 1 . . . 0...
...0 0 . . . 0 1p 0 . . . 0 0
⎞
⎟
⎟
⎟
⎟
⎟
⎠
=
⎛
⎜
⎜
⎜
⎜
⎜
⎝
1 0 . . . 00 1 . . . 0...
...0 . . . 1 00 . . . 0 p
⎞
⎟
⎟
⎟
⎟
⎟
⎠
⎛
⎜
⎜
⎜
⎜
⎜
⎝
0 1 0 . . . 00 0 1 . . . 0...
...0 0 . . . 0 11 0 . . . 0 0
⎞
⎟
⎟
⎟
⎟
⎟
⎠
. (3.32)
Lemma 3.21 Matrix A is a dilation in QNp with respect to ultrametric s defined in
(3.28) when the parameters q1, . . . , qN satisfy p−1 < q1 < q2 < · · · < qN ≤ 1.
3.6.2 Examples of Two-Dimensional Wavelet Bases
We now describe two examples of wavelet bases in L2(Q22) which are constructed
using matrix dilations with respect to different metrics. Consider the matrix
S =(
0 12 0
)
, (3.33)
and the metric s of form (3.28) in Q22 with parameters q2 = 1, q1 = q, p−1 < q < 1.
Lemma 3.22 The matrix S is a dilation with respect to the metric s.
It is straightforward to describe all dilations with respect to s.
Lemma 3.23 A matrix
A =(
a bc d
)
is a dilation with respect to the metric s described above if and only if
a ≡ 0 mod 2, b ≡ 1mod 2, c ≡ 2mod 4, d ≡ 0mod 2. (3.34)
Let us consider a wavelet as a difference of two characteristic functions of s-balls:
θ (x) = (‖S−1x‖)−
(∥
∥
∥
∥
S−1x−(
1/20
)∥
∥
∥
∥
)
. (3.35)
The next theorem describes the corresponding wavelet basis.
3.6 Wavelets with Matrix Dilations 37
Theorem 3.24 The set of functions θ jn(x) = p−j2 θ (S jx− n), j ∈ Z, n ∈ Q2
2/Z22 is
an orthonormal basis in L2(Q22).
The next example of wavelet basis (the quincunx basis in L2(Q22)) was considered
in [267], [25]. The analogous real basis [188] contains wavelets supported on frac-tals. Let us describe the 2-adic quincunx basis using the language of dilations anddeformed metrics [25]. The quincinx matrix Q, detQ = 2, has the form
Q =(
1 −11 1
)
. (3.36)
The next lemma shows that the quincunx matrix is a dilation with respect to ametric q on Q2
2, which is a rotation of the metric s described above.
Lemma 3.25 ([25]) Consider the metric q in Q22 of the form
q(x, y) = s(Ux,Uy) = max(q|x1 − y1|p, |x1 + x2 − y1 − y2|p), (3.37)
where p−1 < q < 1 and
U =(
1 01 1
)
∈ GL(Z2, 2). (3.38)
Then quincunx matrix Q is a dilation in Q22 with respect to the metric q.
Let us consider a wavelet of the form
ψ (x) = (‖Q−1x‖)−
(∥
∥
∥
∥
Q−1x−(
1/21/2
)∥
∥
∥
∥
)
. (3.39)
Theorem 3.26 ([267]) The set of functions ψ jn(x) = p−j2ψ (Qjx− n), j ∈ Z, n ∈
Q22/Z
22, is an orthonormal basis in L
2(Q22) (the quincunx basis).
3.6.3 Wavelets and Pseudodifferential Operators
In this section we consider more general examples of wavelet bases attached todeformed metrics and discuss relations with the spectral theory for pseudodifferentialoperators.
Conjugate Metric Let us consider spaceV = QNp with coordinates x = (x1, . . . , xN )
and conjugate space W = QNp with coordinates k = (k1, . . . , kN ). The space W is a
space of Qp-valued linear functionals on V defined by k · x = ∑Ni=1 kixi.
Let r be a generic ultrametric of the form (3.28), (3.29) in QNp . In an abuse of
notation, we denote by ‖ · ‖ a norm such that r(x, y) = ‖x− y‖. The reader shouldtake into account that the norm ‖ · ‖ = ‖ · ‖q,U depends on the set of parametersq,U .We denote by
B1 = {
x ∈ QNp ; ‖x‖ ≤ 1
}
38 p-Adic Wavelets
the unit ball with respect to the metric r. Let us consider the sequence of intermediaryballs with respect to metric r in V ,
B1 ⊃ B2 ⊃ · · · ⊃ BN ⊃ BN+1 = pB1
with the diameters2
1 ≥ q1 > q2 > · · · > qN > p−1.
Then we have the sequence of conjugate balls inW ,
B∗1 ⊂ B∗
2 ⊂ · · · ⊂ B∗N ⊂ B∗
N+1 = p−1B∗1,
where
B∗l =
{
k ∈ QNp : |k · x|p ≤ 1, for any x ∈ Bl, with k · x =
N∑
i=1
kixi
}
.
This sequence {B∗l } constitutes a maximal simplex consisting of lattices in the conju-
gate space (with the parameter k) and therefore defines a metric in this space (usingthe approach of Section 2.4). Here we take the diameters of B∗
l equal to
1 ≤ q−11 < q−1
2 < · · · > q−1N < p.
Thus, given a metric r in the initial space, we construct a new metric, denoted as r∗,in the conjugate space.
Lemma 3.27 Let A be a dilation with respect to metric r (this dilation will satisfy|detA|p = p−1). Then A∗ (the matrix transpose to A) is a dilation in the conjugatespace with respect to metric r∗.
We now consider the wavelet with matrix dilation A:
�k(x) = χp(
k · A−1x)
(‖x‖), k ∈ B∗1/A
∗B∗1\{0}, (3.40)
where k is not congruent to zero in B∗1/A
∗B∗1. As usual, we understand that k is running
through a fixed set of representatives for the equivalence class, and that this set ofrepresentatives is used in the definition of a wavelet basis.There exist p− 1 wavelets of the above form (since there are p representatives in
B∗1/A
∗B∗1). This formula is analogous to the definition of the one-dimensional wavelet
χp(kp−1x)(|x|p), k = 1, . . . , p− 1.In particular, for metric (3.28) with parameters satisfying p−1 < q1 < q2 < · · · <
qN ≤ 1 and dilation A given by (3.32), the set of k ∈ ZNp /A
∗ZNp in (3.40) takes
2 Here we use the opposite ordering of ql in comparison to (3.30).
3.6 Wavelets with Matrix Dilations 39
the form⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
010...00
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
,
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
001...00
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
, . . . ,
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
000...01
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
.
The next lemma generalizes the constructions given in (3.35) and (3.39) to the caseof more general wavelets with matrix dilations.
Lemma 3.28 The family of wavelets defined in (3.40) can be equivalently describedas
�k(x) =p−1∑
l=0
χp(
k · A−1ml)
(‖A−1(x− ml )‖
)
, (3.41)
where ml, l = 0, . . . , p− 1, are representatives in B1/AB1. The coefficientsχp
(
k · A−1ml)
are p-roots of the unity satisfying χp(
k · A−1ml) = χp
(
k · A−1ml′)
forl = l′. In addition, the wavelet �k is a locally constant function with mean zero.
The wavelet basis is constructed by translations and matrix dilations of wavelet�k:
�k; jn(x) = p−j2�k(A
jx− n), j ∈ Z, n ∈ QNp /B1, (3.42)
see Lemma 3.20 and the remark after this lemma for the discussion of the indices n.
Theorem 3.29 The set of functions {�k; jn} defined by (3.40) and (3.42) forms anorthonormal basis in L2(QN
p ).
Lemma 3.30 The set of all products of elements from the basis {�k; jn} in (3.42) andp-roots of the unity constitutes an orbit of a wavelet from the basis {�k; jn}with respectto the group of transformations generated by translations in QN
p and dilations Aj.
Let us consider the following generalization of the Vladimirov fractional derivationoperator
Dα f (x) = F−1 (‖k‖αF[ f ]) (x), (3.43)
where ‖ · ‖ is the r∗-norm (corresponding to the metric r∗) in the conjugate space QNp
and F is the Fourier transform. The reader should take into account that we use thenotation ‖ · ‖ for norms in the space V and in the conjugate spaceW .
Lemma 3.31 The basis {�k; jn} is a basis of eigenvectors of operator Dα:
Dα�k; jn(x) = ‖A∗( j−1)k‖α�k; jn(x), (3.44)
where k ∈ B∗1/A
∗B∗1\{0}, j ∈ Z, n ∈ QN
p /B1. Here A∗ is the matrix transpose to A.
40 p-Adic Wavelets
Proof The Fourier transform of the wavelet �k(x) in (3.40) is given by
F [�k] (ξ ) =∫
QNp
χp
((
ξ + A∗−1k)
· x)
(‖x‖)dNx.
There appears on the right-hand side a mean-zero function, unless
ξ = −A∗−1kmodB1.
In this case, the integral is equal to μ(B1). Consequently, the Fourier transform isequal to the characteristic function of −A∗−1k + B1. The multiplication by ‖k‖α in(3.43) gives rise to the coefficient ‖A∗−1k‖α and the inverse Fourier transform givesrise to the initial wavelet. This proves (3.44) in the particular case j = 0, n = 0. Moregeneral cases can be checked analogously. �
3.7 Wavelet Transform of Distributions
In the present section, see also [24], we discuss the existence of expansions for test anddistributions from spaces D(QN
p ), D′(QNp ), �(QN
p ), �′(QN
p ) using p-adic wavelets.The space �(QN
p ) of mean-zero test functions is the linear span of p-adic wavelets,and the space D(QN
p ) of test functions is the linear span of p-adic wavelets and acharacteristic function of some (any non-zero diameter) ball.
Lemma 3.32 (i) Any function φ ∈ �(QNp ) admits an expansion of the form
φ =∑
k jn
φk jnψk; jn, φk jn ∈ C, (3.45)
φk jn = φ(ψk; jn) = 〈ψk; jn, φ〉, (3.46)
where ψk; jn are elements of the wavelet basis (3.15), indices are as in (3.16),namely k = (k1, . . . , kN ) (at least one of the kl is non-zero), j ∈ Z, n ∈ QN/ZN,and 〈·, ·〉 is the scalar product of L2(QN
p ).(ii) Let us fix some ball with characteristic function (‖pj0 · −n0‖p), j0 ∈ Z, n0 ∈
QNp /Z
N. Any function φ ∈ D(QNp ) can be represented as a finite sum
φ = p−N j0(∥
∥pj0 · −n0∥
∥
p
)
∫
QNp
φ(x)dNx+∑
k jn
ηk jnψk; jn,
ηk jn =⟨
ψk; jn, φ − p−N j0(∥
∥pj0 · −n0∥
∥
p
)
∫
QNp
φ(x)dNx
⟩
.
Since the space �(QNp ) of mean-zero test functions is the linear span of p-adic
wavelets, the space�′(QNp ) of linear functionals on�(QN
p ) can be identified with thespace of formal series over p-adic wavelets, where the action on the space �(QN
p ) isgiven by the following lemma.
3.8 Relation to the Haar Basis on the Real Line 41
Lemma 3.33 Any distribution f ∈ �′(QNp ) can be represented as a series of the
form
f =∑
k jn
fk jnψk; jn, fk jn = f (ψk; jn), (3.47)
where ψk; jn are elements of the wavelet basis (3.15), and the indices are as in (3.16):k = (k1, . . . , kN ) (at least one of the kl is non-zero), j ∈ Z, n ∈ QN/ZN.For φ ∈ �(QN
p ) of the form (3.45) and f ∈ �′(QNp ) of the form (3.47) the action
of f on φ has the form
f (φ) =∑
k jn
fk jnφk jn. (3.48)
Let us construct similar expansions over wavelets for distributions from D′(QNp ).
Lemma 3.34 For a ball with a characteristic function (‖pj0 · −n0‖p), j0 ∈ Z,n0 ∈ QN
p /ZNp , and a set of complex numbers {u0, uk jn} (where the indices k jn of the
set correspond to the indices of the basis (3.15)) there exists a unique distributionu ∈ D′(QN
p ) satisfying
u(
(∥
∥pj0 · −n0∥
∥
p
)) = u0pN j0 , u(ψk; jn ) = uk jn. (3.49)
Lemma 3.35 A series of the form
u = u0 +∑
k jn
uk jn(
ψk; jn − p−N j0ψk; jn(∥
∥pj0 · −n0∥
∥
p
))
, (3.50)
where the ψk; jn are the wavelets defined in (3.15), is a distribution u ∈ D′(QNp )
satisfying the conditions (3.49).
Since the space of test functions is generated by wavelets and a single characteristicfunction of a ball, there is a one-to-one correspondence between distributions andexpansions of the form (3.50).The convergence of series (3.50) is understood in a weak sense, meaning that,
actually, the action of this series (distribution) on any test function contains only afinite number of terms. Here
ψk; jn(∥
∥pj0 · −n0∥
∥
p
) =∫
ψk; jn(x)(∥
∥pj0x− n0∥
∥
p
)
dμ(x).
In [24] the statements similar to Lemmata 3.32, 3.33, 3.34, 3.35 were formulated forthe case of spaces of the Lizorkin type on ultrametric spaces. In [236] the expansionof distributions over wavelets was applied to the construction of a random field ofp-adic argument.
3.8 Relation to the Haar Basis on the Real Line
Let us discuss the relation between the basis {ψk; jn} of p-adic wavelets in L2(Qp) andthe basis of Haar wavelets in the space L2(R+) of quadratically integrable functionson the real half-line, see [286].
42 p-Adic Wavelets
The real Haar wavelet ψH was defined in (3.2) and the Haar basis in L2(R) gen-erated by translations and dilations of this wavelet was defined in (3.1). The waveletbasis on the real half-line is obtained from (3.1) by restriction to non-negative n.Let us consider a generalization of the Haar basis which corresponds to an arbitrary
p. This basis in L2(R+) contains vectors of the form
ψ(p)k; jn(x) = p−
j2ψ
(p)k (p− jx− n), j ∈ Z, n ∈ N � {0} , (3.51)
ψ(p)k (x) =
p−1∑
l=0
e2π ikl p−11[l p−1,(l+1)p−1](x), k = 1, . . . , p− 1. (3.52)
The basis (3.1) can be obtained from (3.51) by taking p = 2.The surjective map
ρ : Qp → R+∞∑
i=γxi p
i �→∞∑
i=γxi p
−i−1, xi = 0, . . . , p− 1, γ ∈ Z, (3.53)
is called the Monna map.
Lemma 3.36 The map ρ is one-to-one almost everywhere, continuous, and 1-Lipschitz, i.e.
|ρ(x)− ρ(y)| ≤ |x− y|p, x, y ∈ Qp.
In addition, it preserves the measure (i.e. maps the p-adic Haar measure to theLebesgue measure on the real half-line).
Theorem 3.37 The map ρ sends the orthonormal basis of wavelets (3.51) in L2(R+)to the basis of wavelets (3.11) in L2(Qp):
ψ(p)k; jρ(n)(ρ(x)) = ψk; jn(x). (3.54)
The above formula is valid up to a finite number of points (for each wavelet it neednot be satisfied on a finite number of points).
Remark 3.38 Using the map ρ one can define the action of the Vladimirov operatorin L2(R+) as follows:
∂αp f (x) =pα − 1
1− p−1−α
∫ ∞
0
f (x)− f (y)
|ρ−1(x)− ρ−1(y)|1+αp
dy, (3.55)
where ρ−1 is the inverse to ρ. Since ρ is not one-to-one, the map ρ−1 is multivalued,but ρ−1 is multivalued on a set of measure zero which makes definition (3.55) correct.The Haar wavelets (3.51), (3.52) will be eigenvectors of the integral operator ∂αp.
3.9 p-Adic Multiresolution Analysis 43
3.9 p-Adic Multiresolution Analysis
3.9.1 Definition of the p-Adic MRA
Let us consider the set Ip = {x ∈ Qp; {x}p = x}. We will identify this set with
{
n = p−γ(
n0 + n1p+ · · · + nγ−1pγ−1
); γ ∈ N, n j = 0, . . . , p− 1,
j = 0, 1, . . . , γ − 1} . (3.56)
The set Ip enumerates unit balls in Qp (these balls do not intersect). The set Ip can beidentified with the factor-group Qp/Zp and can be used as a set of translations in thedefinition of wavelet bases in Qp.
Let us introduce the p-adic version of the multiresolution analysis (MRA). For thereal version, the reader may consult [117], [363, Section 1.3].
Definition 3.39 ([393]) A set of closed subspaces Vj ⊂ L2(Qp), j ∈ Z, is called(MRA) in L2(Qp) if the following postulates are satisfied:
(a) Vj ⊂ Vj+1 for all j ∈ Z;(b)
⋃
j∈ZVj is dense in L2(Qp);
(c)⋂
j∈ZVj = {0};(d) f (·) ∈ Vj ⇐⇒ f (p−1·) ∈ Vj+1 for all j ∈ Z;(e) there exists a function φ ∈ V0 such that the system {φ(· − n), n ∈ Ip} constitutes
an orthonormal basis in V0.
The function φ from postulate (e) is called the scaling function. Postulates (d) and(e) imply that functions pj/2φ(p− j · −n), n ∈ Ip, form orthonormal bases inVj, j ∈ Z.
Following the standard scheme (see for example [117], [334], [363, Section 1.3])for the construction of multiresolution wavelets, for each j we define the spaceWj (awavelet space) as the orthogonal complement to Vj in Vj+1, i.e.
Vj+1 = Vj ⊕Wj, j ∈ Z, (3.57)
whereWj ⊥ Vj, j ∈ Z. One can see that
f ∈Wj ⇐⇒ f (p−1·) ∈Wj+1, for all j ∈ Z, (3.58)
andWj ⊥Wk, j = k. By the postulates (b) and (c) we have
⊕
j∈ZWj = L2(Qp) (orthogonal direct sum). (3.59)
If we find a finite number of functions ψν ∈W0, ν ∈ A, satisfying the criterion thatthe system {ψν (x− n), n ∈ Ip, ν ∈ A} will be an orthonormal basis for W0, then by(3.58) and (3.59) the system
{pj/2ψν (p− j · −n), n ∈ Ip, j ∈ Z, ν ∈ A}
44 p-Adic Wavelets
will be an orthonormal basis in L2(Qp). The functions ψν , ν ∈ A, are called thewavelet functions, and the corresponding basis is called the wavelet basis.
3.9.2 p-Adic Refinement Equation
Let φ be the scaling function for theMRA. By Definition 3.39 the system of functions{p1/2φ(p−1 · −n), n ∈ Ip} constitutes a basis of V1. The postulate (a) implies that
φ =∑
n∈Ipαnφ(p
−1 · −n), αn ∈ C. (3.60)
Then function φ is a solution of a special functional equation. Equations of thistype are called refinement (or scaling) equations; their solutions are called scalingfunctions.The natural way to construct the MRA (see [363, Section 1.2]) is as follows. We
choose the function φ in the following way: Ip-translations form an orthonormal sys-tem, and consider the set
Vj = span{
φ(
p− j · −n) : n ∈ Ip}
, j ∈ Z. (3.61)
The postulates (d) and (e) of the Definition 3.39 will be satisfied.Of course it is not the case that any arbitrary function φ of this form satisfies the
postulate (a). In reality the condition V0 ⊂ V1 is satisfied if and only if the scalingfunction satisfies the scaling equation. In the p-adic case the situation is different. Ingeneral the scaling equation (3.60) does not imply the condition of inclusionV0 ⊂ V1.Actually we need the following: all functions φ(· − b), b ∈ Ip, belong to the spaceV1,i.e. the conditions
φ(x− b) =∑
n∈Ipαn,bφ(p
−1x− n)
should be satisfied for all b ∈ Ip. Since in general p−1b+ n does not belong to Ip, wecannot say that
φ(x− b) =∑
n∈Ipαnφ(p
−1x− p−1b− n) ∈ V1
is automatically satisfied for all b ∈ Ip. But it is possible that for some scaling func-tions φ the inclusion condition holds.This condition will be satisfied if the group Qp/Zp is represented in the space V0
by translations, i.e.
φ(n+ n′) = φ(n+ n′ mod 1)
for n, n′ ∈ Ip. Here on the left-hand side of the formula the addition of representativesfrom Ip is taken inQp (i.e. the sum should not belong to Ip), and on the right-hand sideof the formula the representatives from Ip are summed as fractions modulo one. Thisgives the following constraint on the form of the scaling function φ: the function
3.10 p-Adic One-Dimensional Haar Wavelet Bases 45
φ should be locally constant with a diameter of constancy one (equivalently, it is1-periodic).Analogously in the spaceW0 we have a representation of the group Qp/pZp.The natural scaling equation (3.6) (reproduced below in (3.63)) reflects the self-
similarity of the space Qp (see [434, I.3, Examples 1, 2]): the unit ball B0(0) ={x : |x|p ≤ 1} has the form of a union of p smaller non-intersecting balls B−1(r) ={x : |x− r|p ≤ p−1} with the diameters p−1:
B0(0) = B−1(0) ∪ ∪p−1r=1B−1(r). (3.62)
By (3.62) the characteristic function φ(x) = (|x|p
)
of the unit ballB0(0) is equal to asum of p characteristic functions of non-intersecting balls B−1(r), r = 0, 1, . . . , p−1, i.e. it satisfies the scaling equation (3.6):
(|x|p
) =p−1∑
r=0
(
p|x− r|p) =
p−1∑
r=0
(
∣
∣
∣
∣
1
px− r
p
∣
∣
∣
∣
p
)
, x ∈ Qp. (3.63)
3.10 p-Adic One-Dimensional Haar Wavelet Bases
3.10.1 Construction of the p-Adic MRA Generated by theScaling Function φ(x) =
(|x|p)
For the construction of the p-adic MRA we use the scaling equation (3.6). We definethe set of closed subspacesVj ⊂ L2(Qp), j ∈ Z, by the formula (3.61), where φ(x) =(|x|p
)
(the scaling function) is a solution of equation (3.6).
Theorem 3.40 There exists an MRA in L2(Qp) generated by the scaling functionφ(x) =
(|x|p)
.
In the case p = 2, p-adic scaling equation (3.6) is the analog of the real scalingequation (3.8). Therefore the MRA constructed in Theorem 3.40 is the p-adic analogof the HaarMRA.We call thisMRA the p-adic HaarMRA. Unlike in the real case, thescaling function φ(x) =
(|x|p)
which generates this MRA is locally constant withdiameter one. We will show that there exists an infinite set of different orthonormalwavelet bases for the same p-adic Haar MRA (see Sections 3.10.2 and 3.10.3 below).
3.10.2 p-Adic Haar Wavelet Basis
Using the scheme described above, we introduce the spaceW0 as the orthogonal com-plement to V0 in V1. We define
ψ(0)k (x) =
p−1∑
r=0
e2π ikrp φ
(
1
px− r
p
)
, x ∈ Qp, k = 1, 2, . . . , p− 1. (3.64)
46 p-Adic Wavelets
Theorem 3.41 The set {ψ (0)k (· − n), k = 1, 2, . . . , p− 1; n ∈ Ip} consisting of
translations of all of the functions of type (3.64) constitutes an orthonormal basisin W0.
By using the defintion of χp and (3.62), the functions (3.64) can be expressed as
ψ(0)k = χp(p
−1kx)(|x|p
)
, k = 1, 2, . . . , p− 1, x ∈ Qp. (3.65)
Thus, the Haar wavelets (3.65) and the corresponding Haar wavelet basis
ψ(0)k; jn(x) = p− j/2ψ
(0)k (pjx− n)
= p− j/2χp
(
k
p(pjx− n)
)
(|pjx− n|p
)
, j ∈ Z, n ∈ Ip, (3.66)
coincide with the wavelets (3.5) and the wavelet basis (3.3).For p = 2 the scaling equation (3.6) takes the form (3.7), where φ(x) =
(|x|2)
isthe solution. In this case, the wavelet function (which defines an orthonormal basisin the spaceW0) is equal to
ψ (0)(x) = φ( x
2
)
− φ
(
x
2− 1
2
)
= χp(2−1x)(|x|2), x ∈ Q2. (3.67)
The corresponding Haar basis is given by
ψ(0)jn (x) = 2− j/2ψ (0)(2 jx− n)
= 2− j/2χp(
2−1(2 jx− n))
(|2 jx− n|2
)
, j ∈ Z, n ∈ I2. (3.68)
3.10.3 Description of p-Adic Wavelet Functions
Unlike in the real case, in the p-adic case a wavelet basis generated by the HaarMRA is not unique. In [393] for p = 2 and in [261] for arbitrary p, it was shownthat in the space W0 there exists an infinite family of wavelet functions ψν , ν =1, . . . , p− 1 (generated by the same Haar MRA), which generate different waveletbases in L2(Qp).
In the following, we will put the fraction n = p−s(
n0 + n1p+ · · · + ns−1ps−1) ∈
Ip, n j = 0, 1, . . . , p− 1, j = 0, 1, . . . , s− 1, in the form n = k/ps, where k = n0 +n1p+ · · · + ns−1ps−1.
The next theorems give the explicit description for the families of waveletfunctions.
Theorem 3.42 ([261]) The set of all wavelet functions with compact support is givenby the formula
ψμ(x) =p−1∑
ν=1
ps−1∑
k=0
αμ
ν;kψ(0)ν
(
x− k
ps
)
, μ = 1, 2, . . . , p− 1, (3.69)
3.10 p-Adic One-Dimensional Haar Wavelet Bases 47
where wavelet functions ψ (0)ν are given in (3.65), s = 0, 1, 2, . . . , and
αμ
ν;k =
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
−p−s∑ps−1m=0 e
−2π i− νp+mps k
σμmzμμ if μ = ν
p−2s∑ps−1m=0
∑ps−1n=0 e
−2π i− νp+mps k 1− e2π i
μ−νp
e2π iμ−νp +m−nps − 1
σνmzνμ if μ = ν.(3.70)
Here |σμm| = 1, zμν are the entries of some unitary (p− 1)× (p− 1) matrix Z.
An orthonormal wavelet basis of L2(Qp) is generated by taking translations anddilations of all the wavelet functions (3.69)–(3.70):
ψμ; jn(x) = p− j/2ψμ(pjx− n), μ = 1, 2, . . . , p− 1, j ∈ Z, n ∈ Ip.
(3.71)On putting p = 2 in Theorem 3.42, we get the following result.
Theorem 3.43 ([393]) Let ψ (0) be the wavelet function given in (3.67). For anys = 0, 1, 2, . . . , the function
ψ (s)(x) =2s−1∑
k=0
αkψ(0)
(
x− k
2s
)
(3.72)
is a compactly supported wavelet function for the Haar MRA if and only if
αk = 2−s2s−1∑
r=0
γre−iπ 2r−1
2s k, k = 0, . . . , 2s − 1, (3.73)
where γr ∈ C, |γr| = 1.
In [257], [258], [259] some different examples of wavelet bases were constructed(these examples were called there “non-Haar”). In particular, one of these bases hasthe form
θ(m)s; jn(x) = p− j/2χp
(
s(pjx− n))
(|pjx− n|p
)
, x ∈ Qp. (3.74)
Here (t ) is a characteristic function of the segment [0, 1] ⊂ R, j ∈ Z, n ∈ Ip,s = (s0, . . . , sm−1) ∈ Jp;m, and this set is defined as
{s = p−m(
s0 + s1p+ · · · + sm−1pm−1
); sl = 0, . . . , p− 1;l = 0, . . . ,m− 1; s0 = 0},
where m ≥ 1 is a fixed natural number. The basis (3.74) is generated by translationsand dilations of the family of wavelet functions
θ (m)s (x) = χp(sx)(|x|p
)
, s ∈ Jp;m, x ∈ Qp. (3.75)
The number of generating wavelet functions (3.75) for the basis (3.74) is not minimaland is equal to (p− 1)pm−1 instead of (p− 1) as for the basis (3.3). For m = 1 thebasis (3.74) coincides with the basis (3.3).
48 p-Adic Wavelets
3.11 p-Adic Scaling Functions
3.11.1 Construction of Scaling Functions
p-Adic refinement (scaling) equations and their solutions were considered in [262].Scaling equations (3.60) with a finite number of terms on the right-hand side wereconsidered:
φ(x) =ps−1∑
k=0
βkφ
(
1
px− k
ps
)
. (3.76)
If φ ∈ L2(Qp), then, by taking the Fourier transform and using (1.3), see Chapter 1,we can put (3.76) in the form
φ(ξ ) = m0
(
ξ
ps−1
)
φ(pξ ), (3.77)
where
m0(ξ ) = 1
p
ps−1∑
k=0
βkχp(kξ ) (3.78)
is a trigonometric polynomial (called themask). It is clear thatm0(0) = 1 ifφ(0) = 0.
Proposition 3.44 ([262]) If φ ∈ L2(Qp) is a solution of the scaling equation (3.76),φ(ξ ) is continuous in 0 and φ(0) = 0, then
φ(ξ ) = φ(0)∞∏
j=1
m0
(
ξ
ps− j
)
. (3.79)
We recall that any test function φ ∈ D(Qp) is pM-locally constant for someM ∈ Z
and supported in BL(0) for some L ∈ Z, i.e φ ∈ DML (Qp), see Chapter 1.
The following theorem summarizes the results of [262] (see also [4]).
Theorem 3.45 Let φ be the function defined in (3.79):
φ(ξ ) = φ(0)∞∏
j=1
m0
(
ξ
pL− j
)
, φ(0) = 1,
where m0 is the trigonometric polynomial (3.78):
m0(ξ ) = 1
p
pL+1−1∑
k=0
βkχp(kξ ), m0(0) = 1.
If m0(k/pL+1) = 0 for all k = 1, . . . , pL+1 − 1 which are not divisible by p, then φ ∈D0L. If moreover |m0(k/pL+1)| = 1 for all k = 1, . . . , pL+1 − 1 which are divisible by
p, then {φ(x− n) : n ∈ Ip} is an orthonormal system.Conversely, if suppφ ⊂ B0(0) and the system {φ(x− n) : n ∈ Ip} is orthonormal,
then |m0(k/pL+1)| = 0 for all k not divisible by p, |m0(k/pL+1)| = 1 for all k divisibleby p, k = 1, 2, . . . , pL+1 − 1, and |φ(x)| = 1 for all x ∈ B0(0).
3.12 Multiresolution Frames of Wavelets 49
3.11.2 Scaling Functions and MRA
In the articles [4]–[5] a description of scaling functions for the p-adic MRA wasgiven. It was found that the class of scaling functions with orthogonal Ip-translationsis comparably small.
Theorem 3.46 ([4], [5]) Let φ be a scaling test function (for some p-adic MRA) withorthogonal Ip-translations and φ(0) = 0. Then suppφ ⊂ B0(0).
Since the support of the Fourier transform of the above scaling function is a subsetof B0(0), this function is locally constant with diameter of constancy one. Functionsof this kind were described in Theorem 3.45. It was shown that there exists an infinitefamily of scaling functions which generate the same Haar MRA.
Theorem 3.47 ([4], [5]) There exists a unique MRA generated by a scaling testfunction with orthogonal Ip-translations. This MRA coincides with the Haar MRA ofTheorem 3.40, which is generated by the scaling function φ = (| · |p) (the solutionof the scaling equation (3.6)).
Therefore Theorems 3.42 and 3.43 describe all p-adic Haar bases of compactlysupported wavelets.
3.12 Multiresolution Frames of Wavelets
In [4] the following generalization of Definition 3.39 of the p-adic MRA was consid-ered. The postulate (e) in Definition 3.39 was substituted by the following statement:there exists a function φ ∈ V0 satisfying V0 = span{φ(x− n) : n ∈ Ip}.
Let us assume that the spaces {Vj} j∈Z constitute an MRA in L2(Q2). Let usdefine for any j ∈ Z the space of wavelets Wj as the orthogonal complement toVj in the space Vj+1, i.e. Vj+1 = Vj ⊕Wj. One can see that f ∈Wj ⇐⇒ f (pj·) ∈W0 for all j ∈ Z and Wj ⊥Wk for all j = k. In this case the condition (3.59) issatisfied.Let us also assume the existence of the set of wavelet functions ψ (ν ) ∈ L2(Qp),
ν = 1, . . . , r, satisfying
W0 = span{ψ (ν )(x− n), ν = 1, . . . , r, n ∈ Ip}.Then the corresponding wavelet system is
{
pj/2ψ (ν )(p− jx− n), ν = 1, . . . , r, n ∈ Ip, j ∈ Z}
. (3.80)
The following theorem describes the scale functions which generate the MRA.
Theorem 3.48 ([4]) Let DM;L(Qp) be the set of all test functions with diameterof local constancy pM and supports in BL(0) (see Section 3.11.1). Then a func-tion φ ∈ DM;L(Qp), M,L ≥ 0 which satisfies φ(0) = 0 generates a MRA if andonly if
50 p-Adic Wavelets
(i) φ satisfies the scaling equation (3.60) and(ii) there exist at least pM+L − pL integers l satisfying 0 ≤ l < pM+L and φ(l/pL) =
0.
The following theorem gives a construction of p-adic wavelet frames, i.e. framesin L2(Qp) which consist of functions of the form pj/2ψ (ν )(p− jx− n), n ∈ Ip,ν = 1, . . . , r.
Theorem 3.49 ([4, Theorem 5.2]) Let ψ (ν ), ν = 1, . . . , r, be a set of wavelet func-tions with compact support generated by the MRA {Vj} j∈Z. Then the set of functions(3.80) is a frame in L2(Qp).
The following algorithm describes the construction of a set of wavelet functionsψ (ν ), ν = 1, . . . , r, for the given MRA [4]. Let the MRA {Vj} j∈Z be generated by thescaling function φ ∈ DM;L(Qp), φ(0) = 0 with the mask
m0(ξ ) = 1
p
pL+1−1∑
k=0
hkχp(kξ ).
The function ψ (ν ) is defined as
ψ (ν )(ξ ) = n(ν )0
(
ξ
pL
)
φ(pξ ),
where
n(ν )0 (ξ ) = 1
p
pL+1−1∑
k=0
g(ν )k χp(kξ )
is a trigonometric polynomial which is the wavelet mask, ν = 1, . . . , r. Then weprove that
span{ψ (ν )(x− n), ν = 1, . . . , r, n ∈ Ip} ⊂ V1.
At the next step we choose the mask n(ν )0 satisfying: if φ(l/pM ) = 0 for somel = 0, 1, . . . , pM+L − 1 then n(ν )0 (l/pM+L) = 0 (see Theorem 3.48). This impliesψ (ν )(l/pM ) = 0 when 0 ≤ l < pM+L and φ(l/pM ) = 0. Therefore
span{ψ (ν )(x− n) : ν = 1, . . . , r, n ∈ Ip} ⊥ V0.
One can also prove that
span{ψ (ν )(x− n) : ν = 1, . . . , r, n ∈ Ip} ⊂W0. (3.81)
3.13 Multidimensional Multiresolution Wavelet Bases 51
By the construction
φ
(
x− n
pL
)
=pL+1−1∑
k=0
hknφ
(
x
p− k
pL+1
)
, n = 0, 1, . . . , pL − 1; (3.82)
ψ (ν )
(
x− n
pL
)
=pL+1−1∑
k=0
g(ν )kn φ
(
x
p− k
pL+1
)
, n = 0, 1, . . . , pL − 1,
ν = 1, . . . , r. (3.83)
If the functions on the right-hand sides of (3.82) and (3.83) can be expressed as linearcombinations of the functions on the left-hand sides, then the following inclusionholds:
W0 ⊂ span{ψ (ν )(x− n) : ν = 1, . . . , r, n ∈ Ip}. (3.84)
Then by (3.81) this proves that ψ (ν )(x− n), ν = 1, . . . , r, n ∈ Ip constitute a set ofwavelet functions.The functions on the right-hand sides of relations (3.82) and (3.83) can be
expressed as linear combinations of the functions on the left-hand sides of these rela-tions only if the linear system of equations
pL+1−1∑
k=0
hklxk = 0,pL+1−1∑
k=0
g(ν )kl xk = 0, l = 0, 1, . . . , pL − 1, ν = 1, . . . , r
possesses a non-trivial solution.
3.13 Multidimensional Multiresolution Wavelet Bases
3.13.1 p-Adic Separable Multidimensional MRA
Let us describe, following the approach by Y. Meyer [333], [363, Section 2.1],the construction of multidimensional wavelet bases by tensor products of a one-dimensional MRA.Let {V (ν )
j } j∈Z , ν = 1, . . . ,N be a one-dimensional MRA (see Section 3.9.1). Letus introduce subspaces Vj, j ∈ Z, in the space L2(Qp) as follows:
Vj =N
⊗
ν=1
V (ν )j = span
{
F = f1 ⊗ · · · ⊗ fN, fν ∈ V (ν )j
}
. (3.85)
Let φ(ν ) be the scaling function of the νth MRA {V (ν )j } j. Let us put
φ = φ(1) ⊗ · · · ⊗ φ(N ). (3.86)
Since the system {φ(ν )( · −n)}nν∈Ip constitutes an orthonormal basis in V (ν )0 for any
ν = 1, . . . ,N, we have
Vj = span{
φ(p− j · −n) : n = (n1, . . . , nN ) ∈ INp}
, j ∈ Z, (3.87)
52 p-Adic Wavelets
where INp = Ip × · · · × Ip is a direct product of N sets Ip and the system φ( · −n),n ∈ INp , forms an orthonormal basis in V0, i.e. the following statement holds.
Theorem 3.50 Let {V (ν )j } j∈Z, ν = 1, . . . ,N, be an MRA in L2(Qp). Then subspaces
Vj in the space L2(QNp ) defined by the relations (3.85) satisfy the following conditions:
(a) Vj ⊂ Vj+1 for all j ∈ Z;(b) ∪ j∈ZVj is dense in L2(QN
p );(c) ∩ j∈ZVj = {0};(d) f (·) ∈ Vj ⇐⇒ f (p−1·) ∈ Vj+1 for all j ∈ Z;(e) the set {φ(x− n), n ∈ INp } forms an othonormal basis in V0, where φ ∈ V0 is
defined by relation (3.86).
As in Definition 3.39, the set of spaces Vj, j ∈ Z, satisfying the conditions (a)–(e)of Theorem 3.50 is called a multiresolution analysis in L2(QN
p ), and the function φin the condition (e) is called a scaling function. A multidimensional scaling functionhas the form (3.86).Following the standard scheme (see for example [363, Section 2.1]), we define
wavelet spacesWj as orthogonal complements to Vj in Vj+1, i.e.
Wj = Vj+1 �Vj, j ∈ Z.
Thus
Vj+1 =N
⊗
ν=1
V (ν )j+1 = Vj ⊕
⊕
e⊂{1,...,N}, e=∅Wj,e,
Wj,e =⎛
⎝
⊗
ν∈eW (ν )
j
⎞
⎠
⎛
⎝
⊗
μ ∈eV (μ)j
⎞
⎠ .
Therefore the space Wj is a direct sum of 2N − 1 subspaces Wj,e, e ⊂ {1, . . . ,N},e = ∅.Let ψ (ν )
k , k = 1, . . . , p− 1, be a set of wavelet functions, i.e. the translations ofthese functions (with respect to n ∈ Ip) constitute an orthonormal basis inW (ν )
0 . Thenthe translations (with respect to n ∈ INp ) of the functions
ψe;{kν } =⎛
⎝
⊗
ν∈eψ
(ν )kν
⎞
⎠
⎛
⎝
⊗
μ ∈eφ(μ)
⎞
⎠ , e ⊂ {1, . . . ,N}, e = ∅, (3.88)
form an orthonormal basis in W0,e. Hence the functions p−N j/2ψe;{kν }(pj · −n), e ⊂
{1, . . . ,N}, e = ∅, j ∈ Z, n ∈ INp , constitute an orthonormal basis in L2(QNp ).
3.13.2 Multidimensional p-Adic Haar Wavelet Bases
The above discussion implies the following result.
Theorem 3.51 The following set of translations and dilations of wavelet functions(3.88), where one-dimensional wavelets and scaling functions are the Haar wavelets
3.14 p-Adic Shannon–Kotelnikov Theorem 53
described in Section 3.10, constitutes the orthonormal p-adic Haar wavelet basis inL2(QN
p ):
p−N j/2ψe;{kν }(
pjx− n)
, x ∈ QNp , (3.89)
where e ⊂ {1, . . . ,N}, e = ∅, kν = 1, 2, . . . , p− 1, ν = 1, 2, . . . ,N; j ∈ Z; n ∈ INp .
Remark 3.52 The set of bases (3.89) contains the basis generated by the one-dimensional wavelet basis (3.3). This basis coincides, see [23], with the multidimen-sional wavelet basis (3.15) which was introduced as an orbit of action of the groupgenerated by translations, dilations, and norm-conserving linear maps inQN
p . Henceone-dimensional and multidimensional p-adic multiresolution wavelet bases are par-ticular cases of p-adic wavelet frames generated by groups of transformations.
3.14 p-Adic Shannon–Kotelnikov Theorem
The Shannon–Kotelnikov theorem allows us to reproduce a function with boundedFourier spectrum using values of the function at equidistant points of the real lineR (see for example [208, Section 5.1]) as follows. Let f ∈ L2(R) and F[ f ](ξ ) = 0for |ξ | > M. Then one can reconstruct the function f (t ) using values at the pointstn = (n/2M), n ∈ Z, with the help of the interpolation formula
f (t ) =∑
n∈Zf (tn)
sin (2πM (t − tn))
2πM (t − tn), t ∈ R, (3.90)
where { sin(2πM(t−tn ))2πM(t−tn ) ; n ∈ Z} is the Shannon–Kotelnikov basis. It is known that con-
vergence of this series is rather slow.The p-adic version of the Shannon–Kotelnikov theorem has the following form.
Theorem 3.53 ([255], [18, Section 8.13]) Let f ∈ L2(QNp ) and suppF[ f ] ⊂ BNj (0).
Then a function f may be reconstructed using its values at the points xa = pja, a ∈ INp ,by the formula
f (x) =∑
a∈INpf (pja)
(∥
∥p− jx− a∥
∥
p
)
, x ∈ QNp . (3.91)
The series converges inVj ⊂ L2(QNp ) and the spaceVj is defined by (3.87). The family
of functions
�a,− j(x) = pjN/2�(p− jx− a)
= pjN/2(‖p− jx− a‖p), a ∈ INp , x ∈ QNp , (3.92)
constitutes the p-adic Shannon–Kotelnikov basis in the space Vj (here �(x) =(‖x‖p) is the scaling function of the multidimensional p-adic Haar MRA). For anypoint x ∈ QN
p the series in (3.91) contains only one term.
54 p-Adic Wavelets
Unlike in the real case, see (3.90), in the p-adic case the series, see (3.91), containsonly one term at any point x ∈ Qp. Therefore in the p-adic case the convergence isfaster.
Corollary 3.54 (i) The set of spaces Vj = F[
L2(BNj )]
form the multidimensionalHaar MRA.
(ii) f ∈ L2(QNp ) ∩Vj ⇐⇒ suppF[ f ] ⊂ BNj (0), j ∈ Z.
(iii) For any f ∈ Vj there exists the representation (3.91), j ∈ Z.
In the real case an analog of Corollary 3.54 was proved in [363, Theorem 1.4.1.].
Remark 3.55 The function
φS(t ) = sin(πt )
πt, t ∈ R,
used in the representation (3.90) is the scaling function of the (real) Shannon–Kotelnikov MRA. This MRA is related to the corresponding system of wavelets (see[363, Theorem 1.4.3.]). Since
F[
φS]
(ξ ) ={
1 if |ξ | ≤ π
0 if |ξ | > π= 1[−π,π](ξ ), ξ ∈ R,
and in the real case the Haar basis (3.1), (3.2) is generated by the MRA with thescaling function φH(t ) = 1[0,1](t ) given by (3.9), the real Haar MRA is very differ-ent from the Shannon–Kotelnikov MRA (the functions φS and φH are related by theFourier transform). In the p-adic case the Fourier transform maps a characteris-tic function of the unit ball centered at zero into itself and the Haar MRA and theShannon–Kotelnikov MRA coincide.
3.15 Spectral Theory of p-Adic Pseudodifferential Operators
3.15.1 Two Approaches to p-Adic Pseudodifferential Operators
In the real case spherical functions are eigenvectors of the Laplace operator. In thep-adic case the same property holds for wavelet bases and some families of integraloperators. In [433] compactly supported eigenvectors for the Vladimirov operator ofp-adic fractional derivation were built in analogy with spherical functions.There are twomain structures on the fieldQp of p-adic numbers – the field structure
and the structure of ultrametric space. Both these structures can be used to introducethe notion of a pseudodifferential operator. The standard way to introduce pseudodif-ferential operators on functions on an Abelian group is to define such an operator asdiagonalizable by the Fourier transform. Equivalently, pseudodifferential operatorscan be considered as integral operators
T f (x) =∫
T (x− y)( f (x)− f (y))dμ(y). (3.93)
3.15 Spectral Theory of p-Adic Pseudodifferential Operators 55
Here μ is the Haar measure on the Abelian group; the integration kernel T (x) is acomplex valued function.An alternative approach to pseudodifferential operators on Qp (with application
of the structure of ultrametric space) was proposed in [287]. The following class ofintegral operators was considered:
T f (x) =∫
Qp
T (sup(x, y))( f (x)− f (y))dy, (3.94)
where sup(x, y) is the minimal ball in Qp containing both x and y, and the integrationkernel T (I) is a complex-valued function on the set of balls in Qp.It is clear that operators of this kind cannot in general be diagonalized by the Fourier
transform. It was shown that, under some convergence condition, the basis of p-adicwavelets is a basis of eigenvectors for an operator of the form (3.94).
Theorem 3.56 Let the series∞∑
j=0
pj|T (I j0)| < ∞ (3.95)
converge. Then the operator (3.94) is densely defined in the space L2(Qp) and isdiagonal in the basis of p-adic wavelets (3.11) with the following eigenvalues:
λ jn = pjT (I jn)+(
1− p−1)∞∑
j′= j+1
pj′T(
I j′,pj′− jn
)
. (3.96)
Here Ijn is a ball with characteristic function(|pjx− n|p). The summation in (3.96)runs over the increasing sequence of balls which contain the ball Ijn.
Eigenvalue λ jn of the operator T , see (3.94), in the wavelet basis (which corre-sponds to wavelets ψk; jn) depends on two indices j and n (the diameter and the posi-tion of the ball which supports the wavelet). For operators of the form (3.93) thecorresponding eigenvalue depends only on the scale j of the support.
The above construction of pseudodifferential operators diagonal in the waveletbases was generalized to the case of general locally compact ultrametric spaces([233], [298], [290]).
3.15.2 Finite-Diagonal Integral Operators in the Wavelet Basis
In the present section, we show a wide family of integral operators in L2(Qp) pos-sessing a finite-diagonal form in terms of the basis of wavelets, i.e. the matrices ofthese operators are non-zero only in a finite number of diagonals (namely the maindiagonal of the matrix and several adjacent diagonals), [301].We study integral operators of the form
T f (x) =∫
Qp
T (x− y)( f (x)− f (y))dy, (3.97)
56 p-Adic Wavelets
where dy is the Haar measure. T is a complex-valued kernel satisfying T (z) = T (−z).Moreover, T (z) is locally constant for z = 0 and satisfies the following properties.
(i) the function T (z) is absolutely integrable at infinity, i.e.∫
|z|p≥r|T (z)|dz < ∞
for some r > 0.(ii) Domains of local constancy of the integration kernel T (z) have the following
form: the function T (z) is constant on balls with the diameter p−kR, k ≥ 1, whichlie in spheres of radius Rwith center at zero. Any one of these concentric spheresof radius R is a disjoint union of pk(1− p−1) subballs with diameter p−kR, andT (z) is constant on any of these subballs.
In this section, we use the duality between locally compact ultrametric spaces andpartially ordered trees. In particular, the tree of balls T (Qp) is a tree where verticesare balls in Qp (with non-zero diameter), the partial order is defined by inclusion ofballs, and edges are pairs (ball, maximal subball).The space �(Qp) of mean-zero test functions is filtered by subspaces related to
subtrees in the tree of balls. The space of test functions D(S ) (related to a subtreeS of the tree of balls T (Qp)) is a linear span of balls corresponding to vertices inS, and �(S ) is the subspace in D(S ) of mean-zero test functions. Here the tree Sshould satisfy the property: if S contains a pair (a ball A, a maximal subball in A),then S also contains all maximal subballs in the ball A. The space of mean-zero testfunctions�(S ) is a linear span of wavelets with the supports being non-minimal ballsin S.We consider the action of the above integral operators in the space�(Qp) of mean-
zero test functions. The following theorem describes the action of operators (3.97) onp-adic wavelets from the basis (3.11).
Theorem 3.57 The action of the operator T of the form (3.97) (which satisfies theconditions (i) and (ii) above) on a p-adic waveletψ with support in ball I is a functionfrom the space�(S ), where the tree S is a union of trees TL,k, I ≤ L ≤ J, the diameterof the ball J ⊃ I is equal to pk−1 diam I, and the tree TL,k contains all subballs in theball L with diameter greater than or equal to p−k diam L.
The theorem will also be satisfied in the case k = 0 (when the integration kernel isconstant on spheres centered at zero). In this case, the wavelets will be eigenvectors ofthis integral operator. Therefore the above theorem generalizes Theorem 3.56 aboutthe relation between wavelet theory and spectral theory of p-adic pseudodifferentialoperators.
Corollary 3.58 The matrix of the operator T of form (3.97) in the wavelet basispossesses a finite-diagonal form, i.e. the non-zero matrix elements of this matrix areconcentrated on a finite number of main diagonals. Namely, for wavelets ψI , ψJ with
3.15 Spectral Theory of p-Adic Pseudodifferential Operators 57
supports in balls I and J respectively, the matrix element 〈ψJ,TψI〉 can be non-zeroonly if |IJ| ≤ 3k − 3, where |IJ| is the distance between the balls I and J in the treeT (Qp) (the number of edges on the path between vertices corresponding to the balls).
Corollary 3.59 Operator (3.97) is a filtered operator in the space �(Qp), i.e. thisoperator maps the spaces of filtration of the space �(Qp) by finite-dimensional sub-spaces �(S ) (see above) into spaces of the same filtration.
This statement follows from Theorem 3.57 and description of filtration spaces�(S ) as linear spans of sets of wavelets.
3.15.3 Multidimensional Pseudodifferential Operators
Let us consider p-adic pseudodifferential operators on the space�(QNp ) of mean-zero
test functions defined by the formula
(Aφ)(x) = F−1[
A(ξ )F[φ](ξ )]
(x)
=∫
QNp
∫
QNp
χp(
(y− x) · ξ)A(ξ )φ(y) dNξ dNy, φ ∈ �(QNp ), (3.98)
where A(ξ ) ∈ E (QNp \ {0}) is a symbol of the operator A. If we define the conjugate
operator AT on �(QNp ) by the formula
(ATφ)(x) = F−1[A(−ξ )F [φ](ξ )](x), (3.99)
then operator A in the space of distributions �′(QNp ) can be defined as follows: for
f ∈ �′(QNp ) one has
(A f , φ) = (
f ,ATφ)
, for anyφ ∈ �(
QNp
)
. (3.100)
This implies
A f = F−1[AF [ f ]], f ∈ �′(QNp
)
. (3.101)
Lemma 3.60 Spaces �(Qdp) and �
′(Qdp) are invariant with respect to the action of
operators (3.98).
In particular, on taking in (3.101) A(ξ ) = ‖ξ‖αp , ξ ∈ QNp , we get the multidimen-
sional fractional operator Dα as follows:(
Dα f)
(x) = F−1[ ‖·‖αp F[ f ](·)](x), f ∈ �′(QN
p
)
. (3.102)
An operator of this kind was introduced by Taibleson in [401, Section 2], [402, Sec-tion III.4.] on the space of distributions D′(QN
p ) for α ∈ C, α = −N.A considerable contribution to the study of this operator was made by V. S.
Vladimirov [420, 433, 434]. In particular, in [420] spectral theory was studied andbases of eigenvectors with compact support were built. Generalization of these basesresults in the introduction of p-adic wavelets [286].
58 p-Adic Wavelets
The class of operators under consideration contains also pseudodifferential oper-ators with symbols A(ξ ) = |f(ξ1, . . . , ξN )|αp, α > 0, where f is an elliptic quadraticform, i.e. f(ξ1, . . . , ξN ) = 0⇐⇒ ξ1 = 0, . . . , ξN = 0 (see [275], [274]). On the otherhand, this class does not contain pseudodifferential operators with symbols involv-ing arbitrary polynomials. In this case, the existence of fundamental solutions is adelicate matter. In [462], see also [470], W. A. Zúñiga-Galindo showed the existenceof fundamental solutions using the existence of meromorphic continuation for localzeta functions, [205]. In Chapter 10, we will construct a class of Sobolev-type spaceswhich are invariant under the action of pseudodifferential operators involving arbi-trary polynomial symbols, and show the existence of fundamental solutions, withoutusing local zeta functions.
3.15.4 Pseudodifferential Operators and Wavelets
Here we discuss the conditions that wavelets built in Section 3.10 should satisfy inorder for them to be eigenvectors of pseudodifferential operators (3.98).
Theorem 3.61 ([17]) Let A be a pseudodifferential operator (3.98) with symbolA(ξ ) ∈ E (QN
p \ {0}); and let k ∈ JNp0, j ∈ Z, n ∈ INp , where JNp0 is defined by (3.14).
Then N-dimensional wavelet (3.15),
ψk; jn(x) = p−dj/2χp(
p−1k · (pjx− n))
( ∥
∥pjx− n∥
∥
p
)
, x ∈ QNp ,
is an eigenvector of A if and only if
A(
pj(−p−1k + η)) = A
(−pj−1k)
, η ∈ ZNp . (3.103)
The corresponding eigenvalue is given by λ = A(−pj−1k), i.e.
Aψk; jn(x) = A(−pj−1k
)
ψk; jn(x).
It is easy to see that A(ξ ) = |ξ |αp satisfies (3.103):
A(−p−1k + η) = ∥
∥−p−1k + η∥
∥
α
p = | − p−1k|αp = A(−p−1 j) = pα, η ∈ ZNp ,
k = (k1, . . . , kN ) ∈ JNp0. Then by Theorem 3.61 we have the following result.
Corollary 3.62 The N-dimensional wavelet (3.15) is an eigenvector of theVladimirov fractional derivative operator (3.102):
Dαψk; jn = pα(1− j)ψk; jn(x), α ∈ C, x ∈ QNp ,
j ∈ Z, n ∈ INp , k = (k1, . . . , kN ) ∈ JNp0. (3.104)
The same will hold for arbitrary p-adic Haar wavelets (3.89).
3.16 Wavelets and Operators for General Ultrametric Spaces 59
3.16 Wavelets and Operators for General Ultrametric Spaces
3.16.1 Wavelets on Ultrametric Spaces
Let us consider a complete locally compact ultrametric space X with the correspond-ing tree T (X ) of balls, see Section 2.2. Let ν be a positive Borel σ -additive measureon X , and let L2(X, ν) be the Hilbert space of quadratically integrable with respect toν complex valued functions on X .For a ball I of non-zero diameter in X , V (I) denotes the space of complex-valued
functions on X generated by characteristic functions of maximal subballs in I, andV0(I) denotes the subspace in V (I) of mean-zero functions with respect to ν. Thedimension ofV0(I) is less than or equal to pI − 1, where pI is the number of maximalsubballs in I.The proof following the next lemma is straightforward.
Lemma 3.63 Spaces V0(I), V0(J) for different I, J are orthogonal in L2(X, ν).
Let us introduce in the spaceV0(I) some orthonormal basis {ψIk}. We will call thisbasis the set of wavelets corresponding to ball I. The next theorem shows that theunion of bases {ψIk} in spaces V0(I) over all balls I with non-zero diameter gives anorthonormal basis in L2(X, ν), which we call the basis of ultrametric wavelets.
Theorem 3.64 (i) If the measure ν(X ) of the space X is infinite, then the set of func-tions {ψIk}, where I runs over all balls in X with non-zero diameter, is an orthonormalbasis in L2(X, ν).(ii) If the measure ν(X ) = A > 0, then the set of functions {ψIk,A− 1
2 }, where I runsover all balls in X with non-zero diameter, is an orthonormal basis in L2(X, ν).
The basis of p-adic wavelets is a particular case of the above basis. The construc-tion of the basis announced in Theorem 3.64 uses only the hierarchy of balls in anultrametric space; in particular, it does not need any group structure.Let us introduce the space D(X ) of test functions as the set of locally constant
compactly supported complex-valued functions on X . This space is filtered by finite-dimensional subspaces D(S ), where S ⊂ T (X ) is a finite subtree in T (X ) satisfyingthe following condition: for any pair (ball I, maximal subball in I) inS this tree shouldcontain also all other maximal subballs in I.The space D(S ) is defined as the linear span of characteristic functions of balls in
S. The space �(S ) ⊂ D(S ) contains mean-zero functions from D(S ).The space �(S ) is the space of finite linear combinations of wavelets correspond-
ing to balls in S\Smin, i.e. non-minimal balls in S (where a wavelet belonging toV0(I)corresponds to ball I).
3.16.2 Pseudodifferential Operators on Ultrametric Spaces
We now discuss the construction of pseudodifferential operators on general locallycompact ultrametric spaces ([233], [298], [290]). These operators are introduced as
60 p-Adic Wavelets
integral operators of the form
T f (x) =∫
XT (sup(x, y))( f (x)− f (y)) dν(y), (3.105)
where X is a complete locally compact ultrametric space, ν is a Borel measure onX , sup(x, y) is the minimal ball which contains both x and y, and T (I) is a complex-valued function on the tree T (X ) of balls in X .When the condition of convergence (3.106) given below is satisfied, the operator
(3.105) possesses a dense domain in L2(X, ν), which contains compactly supportedlocally constant mean-zero functions. The following theorem generalizes the resultsof Section 3.15.1.
Theorem 3.65 Assume that the following series converges for some ball R with non-zero diameter:
∑
J>R
|T (J)|ν(S(J,R)) < ∞, (3.106)
where S(J,R) is the set obtained by elimination from ball J the maximal subballcontaining ball R (a sphere with center in R corresponding to the ball J). Then theoperator (3.105)
T f (x) =∫
XT (sup(x, y))( f (x)− f (y)) dν(y)
is densely defined in L2(X, ν), diagonal in the basis of ultrametric wavelets describedin Theorem 3.64:
TψIk(x) = λIψIk(x) (3.107)
with the eigenvalues
λI = T (I)ν(I)+∑
J>I
T (J)ν(S(J, I)). (3.108)
The integrable constant functions are eigenfunctions corresponding to the eigenvaluezero.
Proof Let us apply the operator T to wavelet ψIk
TψI j(x) =∫
T (sup(x, y))(ψIk(x)− ψIk(y))dν(y)
and consider the following cases.
Case 1. Let x /∈ I. Then
TψIk(x) = −T (sup(x, I))∫
ψIk(y)dν(y) = 0.
We recall that
1I j (x) ={
1 if x ∈ Ij0 if x /∈ I j.
3.16 Wavelets and Operators for General Ultrametric Spaces 61
Case 2. Let x ∈ I. Then
TψI j(x) =(∫
d(x,y)>diam(I)+
∫
d(x,y)=diam(I)+
∫
d(x,y)<diam(I)
)
T (sup(x, y))
× (ψIk(x)− ψIk(y)) dν(y)
=(∫
d(x,y)>diam(I)+
∫
d(x,y)=diam(I)
)
T (sup(x, y))
× (ψIk(x)− ψIk(y))dν(y)
= ψIk(x)∫
d(x,y)>diam(I)T (sup(x, y))dν(y)
+∫
d(x,y)=diam(I)T (sup(x, y))(ψIk(x)− ψIk(y))dν(y)
= ψIk(x)∫
d(I,y)>diam(I)T (sup(I, y))dν(y)+ T (I)ν(I)ψIk(x).
Here diam(I) is the diameter of the ball I.To prove the last equality let us compute for ψ ∈ V0(I) the integral
∫
d(x,y)=diam(I)T (sup(x, y))(ψ (x)− ψ (y))dν(y)
= T (I)∫
d(x,y)=diam(I)(ψ (x)− ψ (y))dν(y)
= T (I)pI−1∑
j=0
1I j (x)
[ ∫
I\I j(ψ (x)− ψ (y))dν(y)
]
= T (I)pI−1∑
j=0
1I j (x)
[
ψ (x)(ν(I)− ν(I j ))−∫
I\I jψ (y)dν(y)
]
= T (I)pI−1∑
j=0
1I j (x)
[
ψ (x)(ν(I)− ν(I j ))+∫
I j
ψ (y)dν(y)
]
= T (I)ν(I)ψ (x).
Here I j are maximal subballs in I.We get
TψIk(x) = λIψIk(x),
where
λI = T (I)ν(I)+∫
d(I,y)>diam(I)T (sup(I, y)) dν(y).
62 p-Adic Wavelets
Since any two increasing paths in the partially ordered tree of balls T (X ) coincidestarting from some vertex, condition (3.106) guarantees convergence of the integral
∫
d(I,y)>diam(I)T (sup(I, y)) dν(y).
For J > I one has∫
d(I,y)=diam(J)dν(y) = ν(S(J, I)).
This implies
λI = T (I)ν(I)+∑
J>I
T (J)ν(S(J, I)),
which finishes the proof. �
4
Ultrametricity in the Theory of Complex Systems
4.1 Introduction
In the present chapter we give a short review of some of the applications of p-adicand more general ultrametric methods in the statistical physics of disordered systems,dynamics of macromolecules, and genetics. The application of p-adic analysis tomathematical physics was initiated in [436]. These methods were used in stringtheory, high-energy physics, cosmology, and other fields [434], [135], [275], [35],[90], [322], [323], [324], [42], [411], [412], [349], [325], [326], [152], [191], [198],[190], [470].Applications to the physics of complex systems and other related areas were
discussed in [336], [403], [162], [83], [376]. The starting point of applicationsof ultrametric methods to complex systems was the replica symmetry-breakingapproach [336], [372], [335], where ultrametric spaces were used to describe spacesof states of spin glasses. In the replica symmetry-breaking approach the ultramet-ric was a result of the branching process in a space of high dimension, see formathematical discussion [359], [61], [461].The relation between replica symmetry-breaking and p-adic analysis (p-adic
parametrization of the Parisi matrix) was found in [53], [373]. In particular, in [373]correlation functions of the replica approach were expressed in the form of p-adicintegrals. Generalizations of these results to more general ultrametric spaces wereconsidered in [234, 235, 239]. In [97, 118] the application of the Fourier transformon some Abelian groups to the diagonalization of Parisi matrices was discussed (thep-adic case is treated in a similar way).The p-adic Potts model was considered in [352, 246, 351, 350]. Hierarchical
models of quantum statistical mechanics related to trees were considered in [186],[329], [327], [328], [311], [347], [346]. In [10, 11], [3] the p-adic random walk wasconsidered. p-Adic Brownian motion was considered in [146], [148]. In [359] arandom walk on the border of the tree in the complex plane was considered (the treewas related to the energy landscape of the Dedekind function). In [170] the quantumdynamics on a complex energy landscape was discussed in relation to Anderson
63
64 Ultrametricity in the Theory of Complex Systems
localization. The application of p-adic methods to fractals was discussed in particularin [211].The statistical mechanics of complex systems is also related to the dynamics on
complex energy landscapes, in particular in applications to the dynamics of macro-molecules [41], [366], [86], [201], [166], [399], [70], [165]. The approximationof basin-to-basin kinetics appears when the dynamics is described by a hierarchyof transitions between the local energy minima [70]. Local energy minima andtransition states between energy minima constitute a tree (called the disconnectivitygraph), and the Arrhenius kinetics for transitions between local energy minimais described by a system of equations with a matrix of transition rates equal toa generalization of the Parisi matrix, see for example [366]. An example of thisdynamics is given by the p-adic diffusion equation generated by the Vladimirovoperator of p-adic fractional differentiation [53].In [57], it was shown that predictions of this model are in agreement with the
results of spectroscopical experiments on protein dynamics (myoglobin–CO rebind-ing), and in [60] this observation was extended to the area of low temperatures. In[56], [55], [51], [58], [52] different ultrametric models of protein dynamics wereconsidered. It was shown that general basin-to-basin kinetics is related to ultrametricdiffusion with drift [292]. In [59], [244], systems of p-adic integral equations formodeling molecular motors were discussed. A construction of molecular machinesas crumpled hierarchical polymer globules was discussed in [62].In [360], [361] an approach for describing the hierarchical structure of proteins
was proposed. It was shown that the domain structure of proteins is one of the levelsof the hierarchy. In [297] this approach was generalized as a model which unifies thefragment approach to proteins (construction of protein conformations as sequencesof conformations of short fragments, see [408, 337, 92]) and the method of statisticalpotentials.A model of biological evolution by gene duplication in which the genome is con-
sidered as an algorithm was considered in [243].To understand the structure of chromatin (organization of DNA in living cells) the
structure of a hierarchical crumpled globule was discussed [189]. This structure ischaracterized by the absence of entanglement of polymer chains, which is of partic-ular importance for the functioning of the genome. One of the tricks to avoid entan-glement is the ring topology for polymers [195], [206]. Another trick is the hierar-chical organization of a polymer globule similar to known examples of Peano curves(space-filling curves) [62]. This hierarchical organization was observed experimen-tally [169], [87], [119]. The chromatin is organized in a hierarchy of levels of folding.Some of these levels include ring structures, and this hierarchy is related to regulationof gene expression.Another example discussed in this chapter is the p-adic parametrization of the
genetic code (a way of encoding of amino acids by triples of nucleotides). The one-dimensional version of this construction was introduced in [133], [134], [226] anddiscussed in [132, 130, 131, 136]. We discuss the two-dimensional parametrization
4.2 p-Adic Parametrization of the Parisi Matrix 65
of the genetic code on the basis of the following observation: using the natu-ral parametrization of nucleotides by their chemical properties, the resulting two-dimensional 2-adic parametrization of codons can describe the degeneracy of thegenetic code (the degeneracy becomes local constancy of the map of the code atsmall distances with respect to the two-dimensional 2-adic metric) [238, 240, 300,242, 241].p-Adic dynamical systems with applications in cryptography were considered in
[35], [39], [31], [33], [38].The structure of this chapter is as follows. In Section 4.2, the p-adic parametriza-
tion of the Parisi matrix in replica theory of spin glasses is discussed. In Section 4.3,the relation between models of basin-to-basin kinetics and general equations of ultra-metric diffusion with drift is considered. In Section 4.4, the two-dimensional 2-adicmodel of the genetic code is described.
4.2 p-Adic Parametrization of the Parisi Matrix
In the present section (see also [53]), we discuss the p-adic parametrization of theParisi matrix from replica symmetry-breaking theory of spin glasses and considerthe application to protein dynamics. This parametrization allows one to express thecorrelation functions of the Sherrington–Kirkpatrick model of a spin glass in the formof integrals over p-adic parameters.The Parisi matrix is the n× n matrix Q = (Qab) with entries defined as
Qaa = 0, Qab = qi,
⌈
a
mi
⌉
=⌈
b
mi
⌉
;⌈
a
mi+1
⌉
=⌈
b
mi+1
⌉
, (4.1)
where the mi are natural numbers such that their ratios pi = mi+1/mi are also naturalnumbers, and
1 = m0 < m1 < · · · < mk < mk = n.
The qi are non-negative real parameters, and �x� denotes the ceiling function,which gives the smallest integer larger than or equal to x.For instance, the Parisi matrix for pi = 2 for all i has the form
Q =
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
0 q1 q2 q2 q3 q3 q3 q3 · · ·q1 0 q2 q2 q3 q3 q3 q3 · · ·q2 q2 0 q1 q3 q3 q3 q3 · · ·q2 q2 q1 0 q3 q3 q3 q3 · · ·q3 q3 q3 q3 0 q1 q2 q2 · · ·q3 q3 q3 q3 q1 0 q2 q2 · · ·q3 q3 q3 q3 q2 q2 0 q1 · · ·q3 q3 q3 q3 q2 q2 q1 0 · · ·...
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
. (4.2)
66 Ultrametricity in the Theory of Complex Systems
In papers [53], [373] it was shown that for the case mi = pi (i.e. all pi = p) for-mula (4.1) is equivalent to the following p-adic parametrization formula: after thecorresponding enumeration of the indices of the Parisi matrix one has
Qab = q(|l(a)− l(b)|p), (4.3)
where | · |p is the p-adic norm and the function q(x) is defined by the conditionsq(pi) = qi, q(0) = 0.
The enumeration of the indices is as follows:
l : {1, 2, . . . , pk} → p−kZ/Z,
l−1 : x =−1∑
i=−kxi p
i �→ 1+k
∑
i=1
xi pi−1, 0 ≤ xi ≤ p− 1.
This is a particular case of the Monna map [348].
4.2.1 Hierarchical Kinetics and p-Adic Diffusion
Let us consider a system of kinetic equations with a matrix of transition rates that isequal to a Parisi matrix:
d
dtf (a, t ) = −
pk∑
b=1
Qab [ f (a, t )− f (b, t )] . (4.4)
This is a model of dynamics on a complex energy landscape, as we shall see in thenext section.Upon applying the p-adic parametrization of Qab, the above system takes the form
∂
∂tf (a, t ) = −
∫
p−MZp/pLZp
q(|a− b|p) [ f (a, t )− f (b, t )] dμ(b),
where the integration is with respect to the Haar measure on the group p−MZp/pNLZp,and M + L = k.By taking the simultaneous limit M, L → ∞ in the above equation (with the nor-
malization∫
Zpdy = 1) we obtain
∂
∂tf (x, t ) = −
∫
Qp
q(|x− y|p) [ f (x, t )− f (y, t )] dy. (4.5)
For q(|x|p) = �−1p (−α)|x|−1−α
p this equation takes the form of the p-adic diffusionequation attached to the Vladimirov fractional operator, which was studied in [434].From amathematical perspective the limit process takes an ordinary differential equa-tion in Rm, with m = m(M,L) and m → ∞ when M, L → ∞, see (4.4), and pro-duces a pseudodifferential equation over Qp, see (4.5). Recently, Zúñiga-Galindoshowed for a large class of non-linear p-adic equations that includes (4.5) that theabove-mentioned limit makes sense mathematically, see [469], Chapter 8, and thereferences therein.
4.3 Dynamics on Complex Energy Landscapes 67
4.3 Dynamics on Complex Energy Landscapes
In the present section, we discuss (on the physical level of rigor) a model of hier-archical kinetics on complex energy landscapes and show that this kinetics can bedescribed by a model of ultrametric diffusion with drift.Let h be a smooth generic function defined on a connected domain inRn, and taking
positive values (a Morse function). Any such h will be called an energy (or an energylandscape). Let S(x, y) be a space of paths (smooth curves) between x and y, wheres ∈ S(x, y) is a path between x and y, and t ∈ s is a point at the path s. Consider thefunction of two variables
d(x, y) = infs∈S(x,y)
supt∈s
h(t ), x = y.
For x = y we define d(x, x) = 0. The value of d(x, y) is equal to the maximal energybarrier which should be overcome in order to travel between x and y. If the functionh is non-monotonic along any path between x and y, then d(x, y) is equal to the valueof the function h at some critical point (at a critical point the gradient of the functionis equal to zero).The function d(x, y) is an ultrametric.Wewill consider this function to be restricted
on the set X of local minima of the function h (in this case h is non-monotonic alongany path between local minima in X).Clustering of the set X of local minima of hwith respect to ultrametric d(·, ·) gener-
ates a tree of clusters. We will call this tree a tree of basins. This tree describes the setof local minima of h and critical points (energy barriers) between local minima. Thediameters of the clusters will be equal to the energy barriers for transitions betweenlocal minima.
Example 4.1 Let the function h have three local minima A, B,C, a barrier h1 betweenA and B, and a barrier h2 between the pair (A, B) and C, where h1 < h2. The matrixof distances between local minima (where lines and columns are enumerated by theminima A, B, C) takes the form
⎛
⎝
0 h1 h2h1 0 h2h2 h2 0
⎞
⎠ .
Example 4.2 Let us consider the case of four local minima A, B, C, D, when thebarriers are as follows: barrier h1 between A and B, the same barrier betweenC andD, and barrier h2 > h1 between the pairs (A, B) and (C, D). The matrix of distancesbetween the minima takes the form
⎛
⎜
⎜
⎝
0 h1 h2 h2h1 0 h2 h2h2 h2 0 h1h2 h2 h1 0
⎞
⎟
⎟
⎠
.
68 Ultrametricity in the Theory of Complex Systems
Matrices of barriers have a hierarchical form analogous to the Parisi matrix fromthe theory of spin glasses.
Let us construct a system of kinetic equations for dynamics on a complex landscapeusing the Eyring–Polanyi formula. According to this formula the rate of transitionsbetween two local energy minima (the reaction rate) is proportional to
e−β�F ,
where β is the inverse temperature, and�F = F1 − F0 is the difference of free ener-gies of the transition state and the initial state of the reaction. The initial state is oneof local minima of energy and the transition state is a saddle point of the energylandscape between two local minima. The analogous formula where we use energyinstead of free energy (i.e. we do not take into account entropy) is called the Arrheniusformula.Let us recall that free energy is equal to
F = E − TS,
where E is energy, T = β−1 is temperature, and S is entropy (the logarithm of thevolume of a vicinity of a local minimum or a transition state).Consider, for the tree of basins of the energy landscape, the system of kinetic equa-
tions with transition rates given by the Eyring formula:
d
dtg(i, t ) = −
∑
j =iC(i, j)
[
eβ(F (i)−F (sup(i, j)))g(i, t )− eβ(F ( j)−F (sup(i, j)))g( j, t )]
.
Here g(i, t ) is the population of the ith local minimum, F (i) is the free energy of theith local minimum, and sup(i, j) is the transition state between local minima i and j(corresponding to the maximal vertex at the path between vertices i and j in the treeof clusters corresponding to the energy landscape). The value F (sup(i, j)) is the freeenergy of this transition state.The valuesC(i, j) constitute a set of positive numbers symmetric with respect to the
permutation of i and j. The choice of these coefficients fixes the model of Arrheniuskinetics. We will use
C(i, j) = #(i)#( j)
{#(sup(i, j))}2 ,
where #(i) and #( j) are volumes of vicinities of the local minima, and #(sup(i, j)) isthe volume of the vicinity of the transition state (this choice is scale-invariant).Let f (i, t ) = g(i, t )/#(i) be the population density of the ith local minimum. Then
the system of kinetic equations takes the form
d
dtf (i, t ) = −
∑
j =i
e−βE(sup(i, j))
#(sup(i, j))
[
eβE(i) f (i, t )− eβE( j) f ( j, t )]
#( j).
Here E(i) and E( j) are energies of local minima, and E(sup(i, j)) is the energy of thetransition state.
4.3 Dynamics on Complex Energy Landscapes 69
This system of equations is equivalent to the equation of ultrametric diffusion withdrift on an ultrametric space X containing a finite number of points (corresponding tolocal minima of the energy landscape). One can generalize this equation to the case ofan arbitrary locally compact ultrametric space (which corresponds to an infinite com-plex energy landscape). In this way, we obtain the equation of ultrametric diffusionwith drift:
∂
∂tf (x, t )+
∫
X
e−βE(sup(x,y))
ν(sup(x, y))
[
eβE(x) f (x, t )− eβE(y) f (y, t )]
dν(y) = 0. (4.6)
Here ν is a Borel measure on X , E(x) is a real-valued function on X (the depth ofpotential wells), and E(sup(x, y)) is the distribution of energies of transition states (areal-valued function monotonically increasing with respect to the partial order on thetree T (X ) of basins, where E(sup(x, y)) > E(x), E(sup(x, y)) > E(y)).
Remark 4.3 Let us note that in the above equation (4.6) of ultrametric diffusionwith drift, the generator is the composition of an ultrametric pseudodifferential oper-ator and the drift term. The drift term is the operator of multiplication by eβE(x) anddescribes distributions of depths of potential wells. In the presence of the drift termthe stationary state of the diffusion equation is given by a Gibbs distribution e−βE(x).For comparison, the equation of diffusion in a potential U (in a real domain) has
the form
∂ f
∂t= � f + β ∇ f · ∇U + β f �U,
or, equivalently,
∂ f
∂t= div
[
e−βU grad[
f eβU]]
.
In [304] the spectral asymptotics for the generator of this equation was investi-gated, and it was shown that the application of spectral asymptotics related to thetunneling phenomenon for some Schrödinger operator allows one to give a proofof the Arrhenius formula of kinetic theory (i.e. the classical Arrhenius formula pos-sesses a quantum proof). This approach generalizes Witten’s spectral asymptoticsapproach to Morse theory [443], [84] (i.e. the Arrhenius formula can be consideredas a second-order correction to Betti numbers in the spectral asymptotics of someSchrödinger operator).
Example 4.4 In the case when X = Qp, the measure ν is the Haar measure, thedepths of local minima are constant at E (x) = constant, and the energies of transitionstates satisfy E(sup(x, y)) = ln |x− y|κp, the above equation of ultrametric diffusiontakes the form
∂
∂tf (x, t )+ Dα f (x, t ) = 0, (4.7)
where Dα is the Vladimirov operator of p-adic fractional differentiation, α = βκ .This p-adic diffusion equation was studied in [434], and the corresponding stochastic
70 Ultrametricity in the Theory of Complex Systems
process was considered. Generalizations of p-adic diffusion equations of types (4.5)and (4.7) are described in Chapter 9.
Example 4.5 Ultrametric reaction–diffusion equation for myoglobin (Mb)–CO bind-ing. The results of spectroscopical experiments by Frauenfelder [41, 165] on Mb–COrebinding can be reproduced by the p-adic model (4.8) below (both the time depen-dence and the temperature dependence of the relaxation can be reproduced [57]).Mb–CO binding is effective when the myoglobin molecule is in a special set of
conformations (where histidine in myoglobin interacts with iron). We describe this setof conformations by a unit ball in Qp. The model of Mb–CO binding is described bythe equation of p-adic diffusion with a sink (ultrametric reaction–diffusion equation)as follows:
[
∂
∂t+ Dλβ
x +(|x|p)]
f (x, t ) = 0. (4.8)
Here(|x|p) – the characteristic function of the unit ball |x|p ≤ 1, f (x, t ) – is a distri-bution function over conformations of myoglobin molecules not bound to CO, the sinkterm(|x|p) describes binding of CO to myoglobin, and β is the inverse temperature.
4.4 Actomyosin Molecular Motor
This section is based on [244]. The actomyosin molecular motor drives motility incells [281]. The motor operates by converting ATP to ADP and using the energy ofthis conversion to perform power strokes. The working cycle of the motor is shown(very schematically) in the diagram below and includes two major conformationalrearrangements of myosin, indicated by horizontal arrows:
1Power stroke−−−−−−→ 2
6
⏐
⏐
⏐
⏐
8
4 ←−−−−− 3
The state of a motor is described by a pair f1(x, t ), f2(x, t ) of distribution functions,where t is time, x (a p-adic argument) describes the conformation of myosin (thetree of basins of the energy landscape for myosin corresponds to the tree of balls inQp). The function f1(x, t ) ( f2(x, t )) is the distribution over conformations x of myosinbound to actin (not bound to actin).The operation of the motor will be described by the system of two ultrametric
reaction–diffusion equations with drift:
∂
∂tf1(x, t ) = − [
k1(x)+ Dλβx eβU1(x)
]
f1(x, t )+ k2(x) f2(x, t ); (4.9)
∂
∂tf2(x, t ) = k1(x) f1(x, t )−
[
k2(x)+ Dλβx eβU2(x)
]
f2(x, t ). (4.10)
4.4 Actomyosin Molecular Motor 71
Here the diffusion terms (which contain Dλβx ) describe conformational transforma-
tions corresponding to the horizontal arrows in the scheme above and the reactionterms (which contain ki(x), i = 1, 2) describe the binding of myosin to actin and itsunbinding, where β is the inverse temperature.The equations above imply the conservation law (conservation of number of
myosin molecules)∫
Qp
( f1(x, t )+ f2(x, t ))dμ(x) = constant. (4.11)
The term k1(x) describes the conformationally dependent reaction rate of the fol-lowing sequence of transformations of myosin: ATP binding, actin unbinding, ATPhydrolysis. The term k2(x) is the reaction rate of actin binding, followed by ADP andphosphate release. TheU1(x) is the energy landscape of myosin bound to actin. TheU2(x) is the energy landscape of a complex of myosin, ADP, and phosphate wheremyosin is not bound to actin.The operatorDλβ
x eβU (x) of ultrametric diffusion with drift has the form of the prod-uct of the ultrametric diffusion operatorDλβ
x and the operator of multiplication by thefunction eβU (x), see Section 4.3.
In order to specify the above system of equations we make the following assump-tions.(1) All reaction rates in the above system are proportional to characteristic func-
tions of some balls:
k1(x) = k1(|x|p), k2(x) = k2(|x− a|p), |a|p > 1, (4.12)
where k1, k2 are positive. Thus the reactions run in the two non-intersecting balls.(2)We construct the potentialsU1(x),U2(x) as follows. These potentials correspond
to potential wells proportional to the same characteristic functions of balls as thecorresponding reaction rates k1(x), k2(x), and the potentials outside some ball aresufficiently large (for example, see below, they are constant and equal to U∞). Wechoose
U1(x) = U1(|x|p)+U∞(1−(|pγ x|p)), (4.13)
U2(x) = U2(|x− a|p)+U∞(1−(|pγ x|p)), (4.14)
whereU1,U2 are negative (i.e. describe potential wells), pγ ≥ |a|p > 1, andU∞ > 0.This choice corresponds to the following picture: the drift (conformational dynam-
ics) goes in the direction of the potential well. The reaction takes place in this potentialwell. After the reaction the myosin molecule arrives at a different potential surface(which corresponds to the different complex of myosin with actin) and performs thesecond part of the cycle. The application of ultrametric diffusion to describe the con-formational dynamics of myosin allows one to take into account different possiblepaths of conformational rearrangements.
72 Ultrametricity in the Theory of Complex Systems
4.4.1 Stationary Solution for the Molecular Motor
Let us investigate the stationary solution for the reaction–diffusion equations (4.9),(4.10) for the molecular motor. The sum of these equations implies for the stationarystate
Dλβx
[
eβU1(x) f1(x)+ eβU2(x) f2(x)] = 0,
hence
eβU1(x) f1(x)+ eβU2(x) f2(x) = constant. (4.15)
Another equation for the stationary solution,
Dλβx eβU1(x) f1(x) = k2(x) f2(x)− k1(x) f1(x), (4.16)
implies that k2(x) f2(x)− k1(x) f1(x) is a mean-zero function, i.e.∫
f1(x)k1(x)dμ(x) =∫
f2(x)k2(x)dμ(x). (4.17)
The above condition has the physical meaning of the coincidence of the flows (i.e.the reaction rates of binding to actin and unbinding from actin) for the two stages ofwork of the molecular motor.Let us take into account the above choice (4.12), (4.13), (4.14) of the potential and
the reaction rates. For simplicity we also take pγ = |a|p = p, i.e. the reactions run inmaximal subballs of a ball of diameter p. Then, up to multiplication by a constant,the right-hand side of (4.16) takes the form
k2(x) f2(x)− k1(x) f1(x) = (|x− a|p)−(|x|p). (4.18)
The above choice of the normalization constant corresponds to unit flow (see (4.17)).The sketch of the proof is as follows. Being a mean-zero function, the function
k2(x) f2(x)− k1(x) f1(x) possesses an expansion over wavelets. The wavelet with thelargest support in this expansion has the form (|x− a|p)−(|x|p). The expansionalso may contain wavelets with smaller supports inside the balls with characteristicfunctions(|x− a|p),(|x|p). On substituting this expansion into (4.15), (4.16) andtaking into account that for the operators of multiplication by functionsU1(x),U2(x)the wavelets with sufficiently small supports are eigenfunctions, we get (4.18).The function (4.18) is an eigenvector of the diffusion operator Dλβ
x with theeigenvalue 1. Equation (4.16) with the normalization (4.18) implies the followingproposition.
Proposition 4.6 Assume that the potential and the reaction rates for the system (4.9)–(4.10) of reaction–diffusion equations have the form (4.12)–(4.14). Then we have thefollowing expression for the stationary state of the system:
f1(x) = e−βU1(x)(
(|x− a|p)−(|x|p)+ 1+ eβU1k−11
)
, (4.19)
f2(x) = e−βU2(x)(
(|x|p)−(|x− a|p)+ 1+ eβU2k−12
)
. (4.20)
4.5 2-Adic Model of the Genetic Code 73
In the limit U∞ → +∞ we get the solution for f1, f2 localized in the ball with diam-eter pγ with its center at zero.
4.5 2-Adic Model of the Genetic Code
Proteins and nucleic acids (DNA and RNA) are linear polymers, i.e. finite sequencesof monomers or finite words with letters from finite alphabets. For nucleic acidsthere are four kinds of monomers, and proteins are copolymers of amino acids of 22possible kinds (of which selenocysteine and pyrrolysine are incorporated into pro-teins differently from the other 20 amino acids, and will not be considered here). Thegenetic (or amino acid) code is a way of encoding of proteins by nucleic acids. Thiscode has a triplet structure – any one of 20 amino acids is encoded by a codon, a tripleof consecutive nucleotides. There are 64 possible codons and 21 values of the geneticcode (20 amino acids and the stop codon). Thus the genetic code is degenerate. In[238] a model of the genetic code based on two-dimensional 2-adic parametrizationof the space of codons was introduced and investigated. Here we reproduce thisconstruction, which can be considered as an example of multidimensional hierarchy.The two-dimensional 2-adic parametrization of the genetic code is based on the
following observations.(1) Nucleotides can be parametrized by pairs of zeros and ones using two proper-
ties: nucleotides can be purines or pyrimidines; nucleotides have two or three hydro-gen bonds in pairing with complementary nucleotides in the double helix. For RNAone has the following two-dimensional parametrization of nucleotides
A GU C
= 00 0110 11
. (4.21)
(2) Using the above parametrization of nucleotides one can define the two-dimensional parametrization of codons (triples of nucleotides). Any nucleotide inthe triple will define the parametrization on the corresponding scale with respect tothe two-dimensional 2-adic metric. Namely, we consider the 2-adic plane with thecoordinates (x, y):
x = (x0x1x2) = x0 + 2x1 + 4x2, xi = 0, 1,
y = (y0y1y2) = y0 + 2y1 + 4y2, yi = 0, 1.
Let us consider the map ρ which maps codons to the 2-adic plane,
ρ : C1C2C3 �→ (x, y) = (x0x1x2, y0y1y2) (4.22)
according to the following rules:C2 defines the pair (x0, y0),C1 defines the pair (x1, y1),C3 defines the pair (x2, y2).
74 Ultrametricity in the Theory of Complex Systems
Thus nucleotides define pairs (xi, yi) using the rule (4.21) and the order ofnucleotides in codons is defined by the rule
2 > 1 > 3,
i.e. more important nucleotides in codons define larger in the 2-adic norm terms in(x, y).We define the following metric on the 2-adic plane:
d(U,V ) = max(|x− z|2, q|y− t|2), 1/2 < q < 1. (4.23)
HereU = (x, y), V = (z, t ) are points in the 2-adic plane.For visualization of the 2-adic plane we use the following kind of Monna map:
η : x0 + 2x1 + 4x2 �→ 1+ 4x0 + 2x1 + x2;η : y0 + 2y1 + 4y2 �→ 1+ 4y0 + 2y1 + y2.
After this map the 2-adic plane becomes an 8× 8 square (with lines and columnsenumerated by the image of the map η for the coordinates x and y correspondingly):
AAA AAG GAA GAG AGA AGG GGA GGGAAU AAC GAU GAC AGU AGC GGU GGCUAA UAG CAA CAG UGA UGG CGA CGGUAU UAC CAU CAC UGU UGC CGU CGCAUA AUG GUA GUG ACA ACG GCA GCGAUU AUC GUU GUC ACU ACC GCU GCCUUA UUG CUA CUG UCA UCG CCA CCGUUU UUC CUU CUC UCU UCC CCU CCC
Application of the genetic code to this table (i.e. we put in correspondence to acodon the corresponding amino acid) gives the following table of amino acids at the2-adic plane:
LysAsn
GluAsp
TerSer
Gly
TerTyr
GlnHis
TrpCys
Arg
MetIle
Val Thr Ala
LeuPhe
Leu Ser Pro
In particular, codons AAA and AAG map to Lys, codons AAU and AAC map toAsn, codons CCA, CCG, CCU, and CCC map to Pro, and inverse images of theseamino acids are balls with respect to the metric (4.23).Here we use the mitochondrial genetic code (there are several versions of the
genetic code which differ by their values at a few codons). The above table shows
4.5 2-Adic Model of the Genetic Code 75
that almost all of the degeneracy of the genetic code corresponds to local constancyof the map (4.22) with respect to the metric (4.23). The domains of constancy of themap (4.22) (which contain two or four codons) are balls with respect to the metric(4.23) (with diameters of q/4 or 1/4 correspondingly). There are three exceptions tothis rule – the inverse images of Leu, Ser, and the stop codon Ter are pairs of balls.Let us consider hydrophobic amino acids at the 2-adic plane:
TrpCys
MetIle
Val
LeuPhe
Leu
One can see that hydrophobic amino acids are clustered with respect to themetric (4.23) (i.e. they belong to two balls with diameters 1/2 and 1/4). Thus thetwo-dimensional 2-adic parametrization introduced here is related to the chemicalproperties of amino acids.
5
Some Applications of Wavelets andIntegral Operators
5.1 Pseudodifferential Equations
Pseudodifferential equations on p-adic domains have been discussed in many arti-cles and books, in particular in [434], [275], [212], [470]. This section is based on[24], [299], [237], [17], [256], [259]. An integrable non-linear equation related tocascade models of turbulence was investigated in [291].
5.1.1 Localization of Solutions
For p-adic integral equations one can mention the following effect of localization ofsolutions. For the p-adic Schrödinger equation
i ∂tψ (x, t ) = Dαxψ (x, t ), (5.1)
where Dαx is the Vladimirov operator, there exist solutions of the form
ψ (x, t ) = e−iωtψk; jn(x), ω = pα(1− j),
where ψk; jn is a p-adic wavelet and the frequency ω is equal to the correspondingeigenvalue of the Vladimirov operator.In the general case, taking the initial condition in the space �(S ) described in
Section 3.16.1, one will obtain a solution of (5.1) in the form of a linear combinationof wavelets multiplied by oscillating time exponents. This solution will belong to thesame space �(S ). This effect can be discussed in relation to Anderson localization[170]. This effect can be generalized for non-linear equations, in particular the p-adicanalog of the non-linear Schrödinger equation, using the fact that the p-adic waveletssatisfy
|ψ (x)|2ψ (x) = ψ (x).
76
5.1 Pseudodifferential Equations 77
5.1.2 Cauchy Problems for Ultrametric Integral Equations
Consider an ultrametric pseudodifferential operator T onQNp , diagonal in somemulti-
dimensional wavelet basis {ψk; jn} with the eigenvalues λ jn, j ∈ Z, n ∈ QNp /Z
Np ,
Tψk; jn = λ jnψk; jn.
Let us recall the definition of the derivative of a distribution (or generalized function)in real analysis: u′(φ) = −u(φ′), where u is a generalized function and φ is a testfunction on the real line.The action of ultrametric pseudodifferential operators on generalized functions can
be defined in the same way. Moreover, since pseudodifferential operators kill con-stants, one can consider the pseudodifferential operator T as acting to bring aboutD′(QN
p ) → �′(QNp ).
Definition 5.1 A distribution u ∈ D′(QNp ) is a solution of the Cauchy problem for the
equation
Tu = f , (5.2)
where f ∈ �′(QNp ), if u satisfies this equation and the following initial condition:
u(
(∥
∥pj0 · −n0∥
∥
p
)) = u0pdj0 , u0 ∈ C, (5.3)
for the characteristic function (∥
∥pj0 · −n0∥
∥
p
)
of a ball.
It is possible, in particular in multidimensional cases for operators of the formT = Dα
x − Dαy , that some eigenvalues of the operator T vanish. In this case we will
obtain some necessary conditions [24] for the existence of solutions of equationscontaining this kind of operator: if λ jn = 0 for some ball ( j, n) with characteristicfunction (
∥
∥pj0 · −n0∥
∥
p), then for corresponding wavelets ψk; jn
fk jn = f (ψk; jn) = 0. (5.4)
Lemma 3.35 about the expansion of generalized functions over wavelets impliesthe following theorem.
Theorem 5.2 Assume that the distribution f ∈ �′(QNp ) on the right-hand side of
equation (5.2) satisfies the necessary conditions (5.4). Then there exists a solution ofthe Cauchy problem (5.2)–(5.3) given by
u = u0 +∑
k jn
1
λ jnfk jn
(
ψk; jn − p−dj0ψk; jn((∥
∥pj0 · −n0∥
∥
p
)
)
+∑
k jn
uk jn
(
ψk; jn − p−dj0ψk; jn((∥
∥pj0 · −n0∥
∥
p
)
)
. (5.5)
The summation in the first line runs over ( j, n) satisfying λ jn = 0, and that in thesecond line runs over ( j, n): λ jn = 0. Here fk jn = f (ψk; jn) and uk jn are arbitrary.
78 Some Applications of Wavelets and Integral Operators
The subset in the tree T (QNp ) containing vertices corresponding to balls ( j, n) with
λ jn = 0 for the operator T will be called the characteristic set of T .
5.2 Non-linear Equations and the Cascade Model of Turbulence
In this section, see also [291], we discuss an exactly solvable non-linear integral equa-tion related to Richardson energy-cascading models in developed turbulence. Thismodel is a modification of the model considered in [161]. The equation under con-sideration has the following quadratic non-linearity:
∂
∂tv(x, t )+
∫∫
F (sup(x, y, z))v(y, t )(v(z, t )− v(x, t ))dν(y)dν(z)
+∫
G(sup(x, y))(v(x, t )− v(y, t ))dν(y) = 0. (5.6)
Here x, y, z belong to (locally compact) ultrametric space X with measure ν, seeSection 3.16.2, and the time t is real.The complex-valued functions F and G on the tree T (X ) of balls in X satisfy the
convergence conditions∑
J>I
|F (J)|ν(S(J, I)) < ∞,∑
J>I
|G(J)|ν(S(J, I)) < ∞, (5.7)
where S(J, I) is the sphere described in Section 3.16.2.It is easy to see that equation (5.6) has solutions of the form of the product of a
wavelet and a time exponent,
v(x, t ) = eωtψIk(x).
Wewill construct a unique solution of the Cauchy problemwith an initial condition inthe space �(X ) of mean-zero test functions using a recurrent hierarchical procedurerelated to cascade models of turbulence. We have the following lemma.
Lemma 5.3 Let us consider ultrametric wavelets φ,ψ supported in balls I, J respec-tively. Let the integration kernel F in (5.6) satisfy (5.7). Then the integral
I[φ,ψ](x) =∫∫
F (sup(x, y, z))φ(y)(ψ (z)− ψ (x)) dν(y) dν(z) (5.8)
converges and takes the form
I[φ,ψ](x) = ψ (x)φ(x)�IJ,
with the coefficient �IJ which is non-zero only for J < I (i.e. the ball J is a strictsubset of the ball I). In this case
�IJ = ν2(I, J)F (I)− ν2(J)F (J)−∑
L : J<L<I
(ν2(L)− ν2(L, J))F (L). (5.9)
5.2 Non-linear Equations and the Cascade Model 79
Here ν(L, J) is the measure of the maximal subball in L containing J.
Remark 5.4 If we fix the integration order in (5.8) as follows:∫ [ ∫
F (sup(x, y, z))φ(y)(ψ (z)− ψ (x))dν(y)
]
dν(z),
then the integral (5.8) will converge for arbitrary φ, ψ from �(X ) even without con-dition (5.7).
Using Lemma 5.3 one can find a particular solution of equation (5.6) of the form
v(x, t ) = e−ηI tψ (x),
where ψ is a wavelet supported in ball I and ηI is the corresponding eigenvalue ofthe pseudodifferential operator in (5.6) (since I[ψ,ψ] vanishes):
ηI = G(I)ν(I)+∑
J>I
G(J)ν(S(J, I)).
Let us consider a more complex example of the solution of (5.6) in the form of asum of two wavelets,
v(x, t ) = v1(t )ψ (x)+ v2(t )φ(x).
Here ψ and φ are wavelets supported in balls J and I respectively.If balls I and J have zero intersection then I[φ,ψ] = I[ψ, φ] = 0 and the solution
of (5.6) takes the form
v(x, t ) = v1(0)e−ηJtψ (x)+ v2(0)e
−ηI tφ(x).
Let the balls I and J be comparable and different: J < I. Then the solution of (5.6)takes the form
v(x, t ) = v1(0)e−ηJt+φ(x)�IJv2(0)η−1
I (e−ηI t−1)ψ (x)+ v2(0)e−ηI tφ(x).
Here �IJ is given by (5.9) and we assume ηI = 0.This gives a non-trivial solution of the non-linear equation (5.6) with a double
exponent (an exponent of an exponent) of time.Let us consider the Cauchy problem for equation (5.6) with v = v(x, t ) ∈ �(X )⊗
C1([0,∞)). The space �(X )⊗C1([0,∞)) is the inductive limit of spaces �(S )⊗C1([0,∞)).1 Thus any function v ∈ �(X )⊗C1([0,∞)) lies in some space �(S )(where S does not depend on t) and v(x, t ) has a continuous derivative with respectto t for any x.The function v ∈ �(X )⊗C1([0,∞)) is a solution of the Cauchy problem for equa-
tion (5.6) with initial condition v0 ∈ �(X ) if
v(x, 0) = v0(x)
and v satisfies (5.6) for t > 0.
1 See the definitions in Section 3.16.1.
80 Some Applications of Wavelets and Integral Operators
Theorem 5.5 The Cauchy problem with initial condition v0 ∈ �(X ) for equation(5.6) satisfying (5.7) possesses the unique solution v ∈ �(X )⊗C1([0,∞)).This solution can be constructed explicitly using the recurrent procedure described
below.
Proof Any function v ∈ �(X )⊗C1([0,∞)) has the form of a finite linear combina-tion of wavelets
v(x, t ) =∑
Ik
vIk(t )ψIk(x),
where the vIk lie in C1([0,∞)) and the I belong to S \ Smin, S ⊂ T (X ). On substi-tuting this expansion into (5.6) we get
∑
Ik
ψIk(x)
[
d
dtvIk(t )+ ηIvIk(t )+ vIk(t )
∑
Jk′ : J>I
vJk′ (t )ψJk′ (x)�JI
]
= 0.
Since wavelets are linearly independent and wavelets of larger scale are constant onsupports of wavelets of smaller scales, the above equation is equivalent to the systemof ordinary differential equations
d
dtvIk(t ) = −vIk(t )
[
ηI +∑
Jk′ : J>I
vJk′ (t )ψJk′ (x)�JI
]
. (5.10)
Here the summation runs over an increasing sequence of balls larger than ball I. Thissystem is non-linear (quadratic). Since x ∈ I and wavelet ψJl for J > I is constant onI, the coefficient ψJk′ (x) does not depend on the choice of x ∈ I. Since the sequencecontains only a finite number of non-zero coefficients vJk′ , the right-hand side of theequation contains a finite number of terms.Let us describe the recurrent procedure for the construction of a solution of sys-
tem (5.10). The initial condition for (5.6) as a function in �(X ) has the followingexpansion over wavelets:
v(x, 0) =∑
Ik
vIk(0)ψIk(x). (5.11)
Take a maximal I in the above expansion with vIk(0) = 0 (this I can be non-unique).Since for a maximal I the corresponding equation in system (5.10) is linear, we getthe following exponential solution vIk(t ) of the Cauchy problem:
vIk(t ) = vIk(0)e−ηI t .
Then, taking maximal subballs I′ < I, we substitute the obtained solution into thecorresponding equations in (5.10) and get equations for vI′k′ . These equations will belinear and will depend on the function vIk(t ) obtained in the previous step.
Then we iterate this procedure and obtain solutions for all pairs (I, k):
vIk(t ) = vIk(0) exp
{
−ηIt −∫ t
0
∑
Jk′ : J>I
vJk′ (τ )ψJk′ (x)�JI dτ
}
. (5.12)
5.3 p-Adic Brownian Motion 81
If vIk(0) = 0 then vIk(t ) = 0. Thus, since the initial condition is a finite linear com-bination (5.11) of wavelets, the recurrent procedure described above gives all of thenon-zero vIk(t ) in a finite number of steps.Any equation in (5.10) is linear just like the equation for vIk(t ) and depends only
on vJk′ (t ) related to larger scales. This implies the uniqueness of the solution of theCauchy problem and finishes the proof of the theorem. �Remark 5.6 If the initial condition v0 for (5.6) belongs to some finite-dimensionalspace �(S), S ⊂ T (X ), then the solution v(x, t ) of the Cauchy problem for (5.6) willalso belong to the space �(S). This is an example of the general phenomenon of theexistence of localized solutions for ultrametric integral equations (see Section 5.1).
5.3 p-Adic Brownian Motion
p-Adic Brownian motion was introduced and the corresponding correlation func-tions were computed in [82]. Fractional p-adic Brownian motion was investigated in[81]. In Chapter 11 a general construction of p-adic Brownian motion using white-noise calculus is presented. In [236] Brownian motion on a general locally compactultrametric space was described using the wavelet transform. In [245] a quadraticcorrelation function for discretized fractional p-adic Brownian motion defined onQp/Zp was computed. For other models of p-adic stochastic processes see [216],[446], [447], [450], [276]. Brownian motion where both space and time are p-adicwas considered in [460], [459].The fractional p-adic Brownian motion is a solution of the equation
Dα f (x) = φ(x), (5.13)
where φ ∈ �′(Qp) is the white noise (a delta-correlated Gaussian mean-zero gener-alized complex-valued stochastic process).The white noise possesses the expansion over p-adic wavelets
φ(x) =∑
k; jndk; jnψk; jn(x),
where dk; jn are mean-zero Gaussian independent delta-correlated random variables.By Lemma 3.35 the solution of (5.13) in D′(Qp) is given by the expansion over
wavelets
f (x) = f0 +∑
k; jnp−α(1− j)dk; jn
(
ψk; jn(x)−∫
Zp
ψk; jn(x)dx
)
, (5.14)
where f0 is a mean-zero Gaussian random variable independent from dk; jn.Wewill consider the solution with f0 = 0, i.e. the solution which satisfies the initial
condition∫
Zp
f (x)dx = 0, (5.15)
with α > 0.
82 Some Applications of Wavelets and Integral Operators
Let us consider p-adic multiresolution spaces Vj (see Definition 3.39) – spaces ofpj-locally constant functions (i.e. functions satisfying f (·) = f (· + pj )) with com-pact support. The completion of the space V0 can be identified with l2(Qp/Zp).The projection in L2(Qp) to the completion of the spaceVj is given by the formula
(� j f )(x) = pj∫
|y|p≤p− j
f (x+ y)dy, (5.16)
where μ is the Haar measure, x ∈ Qp/pjZp (i.e. x can be considered as given by theterms in expansion of x over degrees of p with degrees i < j).Condition (5.15) after application of the projection to space V0 takes the form
(�0 f )(0) = 0.
The following theorem describes a discretization of the p-adic Brownian motion.
Theorem 5.7 The quadratic correlation function for the discretization F = �0 f of(5.14) with the initial condition F (0) = 0 is given by
〈F (x)F (y)〉 = ρ(x)+ ρ(y)− ρ(x− y), x, y ∈ Qp/Zp,
where ρ(0) = 0 and for x = 0 the function ρ(x) is given by
ρ(x) = 1− p−1
1− p2α−1+ |x|2α−1
p
p−2α − 1
1− p2α−1.
6
p-Adic and Ultrametric Models in Geophysics
The cooperation between the research groups of K. Oleschko (applied geophysicsand petroleum research) and A. Khrennikov (p-adic mathematical physics) led to thecreation of a new promising field of research [245], [367]: p-adic and more generallyultrametric modeling of the dynamics of flows (of, e.g., water, oil, and oil-in-waterand water-in-oil droplets) in capillary networks in porous randommedia. The startingpoint of this project is the observation that tree-like capillary networks are very com-mon geological structures. Fluids move through such trees of capillaries and, hence,it is natural to try to reduce the configuration space to these tree-like structures and theadequate mathematical model of such a configuration space is given by an ultrametricspace.The simplest tree-like structure of a capillary network can bemodeled as the field of
p-adic numbers on the ring of p-adic integersQp (or the ring of p-adic integersZp). Inp-adic modeling the variable x ∈ Qp and the real time variable t ∈ R are used. Herex is the “pore network coordinate,” meaning that each pathway of pore capillariesis encoded by a branch of the p-adic tree. The center of this tree is selected as anarbitrary branching point of the pore network. For the moment, it plays the role of thecenter of coordinates, i.e. it is a purely mathematical entity. Thus, by assigning thep-adic number x to a system, one gets to know in which pathway of capillaries it islocated, nothing more. Hence, the p-adic model provides a fuzzy description of porenetworks. In particular, the size of capillaries is not included in the geometry. It can beintroduced into the model with the aid of the coefficients of the anomalous diffusion–reaction equation playing the role of the master equation. From the dynamics, one canknow the concentration of fluid (oil, water, or emulsions and droplets) in capillaries.However, the model does not give the concentration of fluid at any precisely fixedpoint of Euclidean physical space.This modeling heavily explores the theory of p-adic pseudodifferential equations,
equations with fractional differential operators Dα (Vladimirov’s operators), see e.g.[18], [259], [275], [286], [287], [302], [233], [232], [290], [470], and more generalpseudodifferential operators. In particular, to find solutions of p-adicmaster equations
83
84 p-Adic and Ultrametric Models in Geophysics
of the diffusion type, the theory of p-adic wavelets established by S. Kozyrev [286],[287] is used, see [18], [259], [302] for its further development.In spite of its mathematical beauty, the p-adic model does not reflect completely
the complex branching structure of trees of capillaries in random porous media.Therefore the use of general ultrametric spaces is very important for concrete geo-logical applications. Here the theory of ultrametric wavelets and pseudodifferentialoperators is applied. This theory was established by S. Kozyrev and A. Khrennikov[233], [290]. For the moment, this theory is about linear equations. However, the realequations describing geological flows are non-linear both in the Euclidean system[379], [380], [410] and ultramentric models [367]. There are only three works aboutthe existence of a solution of ultrametric pseudodifferential equations, [289], [19],[229]. Geological studies have strongly stimulated the development of a theory ofnon-linear ultrametric pseudodifferential equations [229].The idea of selecting an appropriate subspace of the three-dimensional Euclidean
space R3 as the configuration space within which to model fluids’ flows in a capillarynetwork has already been explored by using fractal configuration spaces, see e.g.Oleschko et al. [369], [368] for geological applications.Another mathematical representation of the tree-like structure of the configuration
space is based on the use of equations with fractional differential operators acting onthe real space R3, see e.g. [330], [319] for geological applications.
All these representations can be treated as special representations for anomalousdiffusion processes.
6.1 Tree-like Structures in Nature
A detailed review on tree-like capillary structures in nature (in physics, geology,and biology) can be found in [253, Section 1.1]. Our presentation is based on thisreference.Capillary phenomena were discovered by Leonardo da Vinci, B. Pascal, and J.
Jurin in experiments with glass tubes, and the theory was developed by P. S. Laplace,Th. Young, and J. W. Gibbs. In [108] Churaev gives a very general definition ofcapillary phenomena which can be seen as the set of complex processes occurringon the interfaces between the unmixed media, derived from the surface tensionappearing at their boundary. During the early 1970s, the tree-like structure of thecapillary networks in porous media was intuitively accepted by several physicists.However, the absence of high-precision imaging techniques was the main obstaclehindering the direct observation and imaging of such tree-like structures. Thatlimitation notwithstanding, several physicists have manually painted their virtualcapillary nets with such exactness that the p-adic pictures extracted from these graphscoincide surprisingly well with the expected number distributions. For instance,Kachinskiy [207] has schematically represented the capillary distribution of waterfrom the groundwater in soil as a system with a clear hierarchical tree-like structureof capillary nets, see Figure 6.1 (a modified version of a figure from [207], see [253]).
6.2 p-Adic Configuration Space and Balance Equations 85
Figure 6.1. Capillary distribution of water from the groundwater in soil. (From [253].)
In spite of the lack of high-precision imaging tools, some very imaginativeresearchers found other ways to document the tree-like ramification patterns of dif-ferent physical and chemical phenomena. For instance, Georg Christoph Lichtenberg(born in 1777) produced the most well-known pictures of tree-like ramification ofelectric discharges inside dielectric materials, imaging these by colored powders andprinting the pattern of their distribution on paper, see [369]. These representationsare called Lichtenberg figures, see Figure 6.2.Tree-like capillary networks are present in a variety of geological structures, see
[253] for a selection of geological tree-like structures (which were collected inpetroleum research performed by the group led by K. Oleschko).
6.2 p-Adic Configuration Space for Networks of Capillaries andBalance Equations for Densities of Fluids
As above, the variable x belongs to the field of p-adic numbers Qp and the time vari-able t is real. Here x is the “pore network coordinate,” meaning that each pathway ofpore capillaries is encoded by a branch of the p-adic tree. From the dynamics on thep-adic configuration space, one gets to know the concentration of fluid (oil, water, oremulsions and droplets) in capillaries. However, one cannot get to know the concen-tration of fluid at any concrete point of physical space (the latter is represented asR3).
86 p-Adic and Ultrametric Models in Geophysics
Figure 6.2. Lichtenberg figure (©Bert Hickman, www.teslamania.com. Fromhttps://es.wikipedia.org/wiki/Figuras_de_Lichtenberg.)
Each p-adic ball represents a bundle of capillaries: the longer the common root ofsuch a bundle, the smaller the ball’s radius.Consider fluid moving in the network of capillaries; denote its concentration over
capillaries by ρ(x, t ). The dynamics of ρ(x, t ) is described by the Cauchy problemfor the following master equation:
∂ρ
∂t(x, t ) =
∫
Qp
[v(x|y)ρ(y, t )− v(y|x)ρ(x, t )]dy, ρ(x, 0) = ρ0(x).
Here v(x|y) is the probability of transition (per unit time) of fluid from the capillaryy to the capillary x. If the amount of fluid in the tree-like network of capillaries underconsideration is preserved, then the conservation law
∫
Qp
ρ(x, t )dx = constant =∫
Qp
ρ0(x)dx
has to hold true. This is equivalent to the condition
0 = ∂
∂t
∫
Qp
ρ(x, t )dx =∫
Qp
dx∫
Qp
dy[v(x|y)ρ(y, t )− v(y|x)ρ(x, t )],
and the integral on the right-hand side is zero.If the transition probability is symmetric, v(x|y) = v(y|x), then the dynamics is
essentially simpler. For a fluid’s dynamics in a capillary network, this symmetryof transition probabilities is natural. It is also natural to assume that the probabilityof transition increases with decreasing p-adic distance between two capillaries. Thesimplest mathematical model for such a behavior is given by transition probabilities
6.2 p-Adic Configuration Space and Balance Equations 87
depending on the distance,
v(x|y) = v(|x− y|p).In modeling dynamics on complex energy landscapes it is assumed that the transitionprobability is determined by an Arrhenius-type potential, i.e.
v(|x− y|p) = A(T )
|x− y|p exp(
U (|x− y|p)kT
)
.
The most commonly used transition probability has the form v(|x− y|p) = 1/|x−y|α+1p (it can be obtained by selecting the appropriate energy potentialU (x)). It leads
to Vladimirov’s fractional differential operators and the equation of p-adic fractionaldiffusion:
∂ρ
∂t(x, t ) = Dαρ(x, t ), ρ(x, 0) = ρ0(x).
However, in geophysical applications there is no reason to restrict transition proba-bilities to ones leading to Vladimirov’s operators. Thus one has to develop a theoryfor arbitrary dynamics of the type
∂ρ
∂t(x, t ) =
∫
Qp
v(|x− y|p)[ρ(y, t )− ρ(x, t )]dy, ρ(x, 0) = ρ0(x). (6.1)
The Cauchy problem (6.1) and several generalizations have been studied intensivelyby Kochubei, Albeverio et al., and Zúñiga-Galindo, among others, see Chapter 8,[18], [19], [275], [470, Chapters 2 and 3], [406], and the references therein. However,there are still several aspects to study, for instance the existence and uniqueness ofstationary solutions.Suppose now that there are a few types of fluid in different states; they are labeled
by i = 1, 2, . . . ,N − 1 (here we distinguish, e.g., free oil and oil bound to the surfaceof capillaries, free and bound water). Denote their concentrations over capillaries bythe symbols ρi(x, t ). The corresponding master equation has the form
∂ρi
∂t(x, t ) =
∑
j =i(ki j(x)ρ j(x, t )− k jiρi(x, t ))
+∫
Qp
vi(|x− y|p)[ρi(y, t )− ρi(x, t )]dy, (6.2)
ρi(x, 0) = ρ0i(x). (6.3)
Some of the transfer factors ki j are zero; strictly positive transfer factors correspondto non-trivial state transfers. For example, free oil can be caught by the surfaces ofcapillaries and oil bound to a surface can leave it by moving through a capillary.The following result is simple, but important for the model’s justification:
Proposition 6.1 The equation (6.2) with (6.3) implies a conservation law, namelyconservation of the total number of “particles”:
∫
Qp
∑
i
ρi(x, t )d(x) = constant =∫
Qp
∑
i
ρ0i(x)dx. (6.4)
88 p-Adic and Ultrametric Models in Geophysics
To the best of our knowledge, there is no theory for systems of parabolic-type equa-tions of the form (6.2). It seems plausible that the method introduced in [270], see also[275, Chapter 4], [470, Chapters 3], by Kochubei for parabolic-type equations withvariable coefficients can be extended to systems of type (6.2). In addition, we think itwould be relevant to study the following problems: (i) the existence and uniquenessof a solution in special spaces of functions; (ii) representation of solutions with theaid of wavelet series; and (iii) the existence and uniqueness of stationary solutions.For geological applications, stationary solutions and especially their capillary
supports are particularly interesting. For example, consider the dynamics of oil inrandom porous media. The p-adic dynamics for molecular motors (see Chapter 4)can be applied to model the flows of free and surface-bound oil, see [252], [367] fordetails. Denote the concentration of fluid over capillaries by ρ1(x, t ). This is fluidwhich is not bound to the interfaces. The concentration of fluid which is bound to theinterface with a solid is denoted by ρ2(x, t ). We arrive at the following two coupledbalance equations (see also Section 4.4):
∂
∂tρ1(x, t ) = − [
K1(x)+ Dλx e
βU1(x)]
ρ1(x, t )+ K2(x)ρ2(x, t ), (6.5)
∂
∂tρ2(x, t ) = K1(x)ρ1(x, t )−
[
K2(x)+ Dλx e
βU2(x)]
ρ2(x, t ), (6.6)
where λ > 0 and β > 0 are the parameters of the model describing, respectively,the degree of fractionality of the diffusion and the strength of coupling with thepotentialsUj(x).We now discuss the interpretation of other coefficients in the systemof equations (6.5)–(6.6). We start with the transfer factors Kj, j = 1, 2. Here thetransfer factor K1(x) encodes the reaction rate of absorption of fluid by the interfacewith a solid; the transfer factor K2(x) encodes the reaction rate of release of absorbedfluid. Now we turn to the potentialsUj, j = 1, 2. HereU1(x) is the potential functioninside the xth pore andU2(x) is the potential function describing the binding of fluidto the interface with the solid in the xth pathway of capillaries.It is possible to show that (under some restrictions on coefficients, transfer factors,
and potentials) this system has a stationary solution concentrated on a p-adic ballwhose radius is determined by the coefficients of this system of equations.Now we discuss a concrete application of our model – to the reaction–diffusion
dynamics of oil, see [252], [367] for details. Geologically this behavior means con-centration of oil droplets in the emulsion in a sub-bundle of the network of capillariesin random porousmedia. In terms of Euclidean geometry this is nothing other than thedescription of the process of formation of a cluster of capillaries, where oil dropletsare concentrated and are growing in size, generating, step by step, the petroleummicro-scale “reservoir.”This kind of concentration in a p-adic ball corresponds to the process of creation
of a cluster of oil droplets in a sub-tree of capillaries in random porous media, e.g.the creation of an oil “reservoir” in such a medium. In future the present model willbe updated to be applied to two important geological processes.
6.3 Non-linear p-Adic Dynamics 89
One of them is the process of formation of oil reserves in porous media duringgeological evolution. Such a model is of mainly theoretical value.Another possible application may have a real practical outcome. In fact, our system
of reaction–diffusion equations can be used to model the process of extraction ofoil from an extended network of capillaries. This process is especially important forthe design of oil recovery programs and especially for the selection of enhanced oilrecovery (EOR) methods, whereby fluid flow from the solid matrix is stimulated. Byapplying properly selected pressures it is possible to concentrate oil in a restrictedsub-network of capillaries which is centered with respect to one selected pathwayof capillaries. In our model it is denoted by the symbol a. Here the “centering” iswith respect to the p-adic topology, namely the tree-structure. A proper potentialstructure can be achieved, in particular, by the creation of an appropriate distributionof pressure in the network of pores, e.g. by means of pumping of water in the network.The latter process will be based on a more complex model, since the distribution ofwater in the network of pores also has to be taken into account. The derivation ofsystems of such balance equations and the analysis of their stationary solutions is aninteresting mathematical problem.
6.3 Non-linear p-Adic Dynamics
6.3.1 Quasilinear Diffusion
As was pointed out, non-linear effects play an important role in the modeling of thetransport of fluids in random porous media. However, the theory of non-linear p-adicdiffusion equations has not yet been developed. One of the first steps towards such atheory is consideration of a quasilinear diffusion equation.We shall present concreteresults from [19], [18] about the following quasilinear diffusion equation:
∂ρ
∂t(x, t ) = Dαρ(x, t )+ ρ(x, t )ρ(x, t )2m(x, t ), ρ(x, 0) = ρ0(x). (6.7)
By expanding ρ(x, t ) with respect to the wavelet basis
ρ(x, t ) =∑
s, j,a
�s, j,a(t )ψs, j,a(x) (6.8)
we obtain the infinite system of ordinary differential equations for the wavelet coef-ficients:
d�s, j,a(t )
dt+ pα(1− j)|s|p�s, j,a(t )+ p−mj�2m+1
s, j,a (t ) = 0. (6.9)
By integrating (6.9), we obtain
�2ms, j,a(t )
pα(1− j)|s|p + p−mj�2ms, j,a(t )
= Es, j,ae−2mpα(1− j)t, (6.10)
90 p-Adic and Ultrametric Models in Geophysics
where Es, j,a are constants which can be found from the wavelet expansion of theinitial function ρ0(x). Here the key point is that wavelets are the eigenfunctions ofthe Vladimirov operator Dα . By resolving (6.10) we obtain the following result.
Proposition 6.2 Any solution of the quasilinear diffusion equation (6.7) can be rep-resented by wavelet series (6.8) with the coefficients given by
�s, j,a(t ) = (Es, j,apα(1− j)|s|p) 12m e−2mpα(1− j)t
(1− p−mjEs, j,ae−2mpα(1− j)t )12m
. (6.11)
Thus we were able to find analytic expressions for the wavelet coefficients of thisquasilinear fractional diffusion equation. This result (although simple) encouragesus to seek analytic solutions, e.g. by using p-adic wavelet theory, for more complexnon-linear fractional diffusion equations.
6.3.2 Porous Medium Equation
In [229] there was considered a p-adic analog of one of the most important classicalnon-linear equations, the porous medium equation (see [414]), that is the equation
∂u
∂t+ aDα (ϕ(u)) = 0, u = u(t, x), t > 0, x ∈ Qp, (6.12)
whereDα , α > 0, is Vladimirov’s fractional differentiation operator, and ϕ is a strictlymonotone increasing continuous real function, satisfying the following inequality:
|ϕ(s)| ≤ C|s|m
for s ∈ R (C > 0, m ≥ 1). A typical example of such a function is the function
ϕ(u) = u|u|m−1, m > 1.
In [229] the following strategy for studying equation (6.12) was used. There existsan abstract theory of the equations
∂u
∂t+ A(ϕ(u)) = 0 (6.13)
developed by Crandall and Pierre [113] and based on the theory of stationary equa-tions
u+ Aϕ(u) = f (6.14)
developed by Brézis and Strauss [93]. In equations (6.13) and (6.14), A is a linearm-accretive operator in L1(), where is a σ -finite measure space. Under somenatural assumptions, the non-linear operator Aϕ = A ◦ ϕ is accretive and admits anm-accretive extension Aϕ , the generator of a contraction semigroup of non-linearoperators. This result implies the generalized solvability of equation (6.13), thoughthe available description of Aϕ is not quite explicit.
6.3 Non-linear p-Adic Dynamics 91
In order to use this method for equation (6.12), one needs an L1-theory of theVladimirov operator Dα . This is a subject of independent interest, see [229] fordetails.In the real case, where = Rn and A is the Laplacian, there are stronger results
based on the study of equation (6.14) (see [78, 109]) showing that Aϕ is m-accretiveitself. This approach employs some delicate tools of local analysis of solutions, suchas embedding theorems for Marcinkiewicz and Sobolev spaces in bounded domains.For the p-adic case, the result obtained in [229] is weaker: we havem-accretivity of
the closure of the operatorAϕ. The main tool used to obtain this result is the L1-theoryof the Vladimirov-type operator on a p-adic ball developed in [229].The Cauchy problem for the heat-like equation
∂u
∂t+ Dαu = 0, u(0, x) = ψ (x), x ∈ Qp, t > 0,
possesses many properties resembling classical parabolic equations. If ψ is regularenough, for example, ψ ∈ D(Qp), then a classical solution is given by the formula
u(t, x) =∫
Qp
Z(t, x− ξ )ψ (ξ )dξ,
where Z is, for each t, a probability density and
Z(t1 + t2, x) =∫
Qp
Z(t1, x− y)Z(t2, y)dy, t1, t2 > 0, x ∈ Qp. (6.15)
Explicitly, for x = 0,
Z(t, x) =∞∑
m=1
(−1)m
m!· 1− pαm
1− p−αm−1tm|x|−αm−1
p . (6.16)
The “heat kernel” Z satisfies the estimate
0 < Z(t, x) ≤ Ct(t1/α + |x|p)−α−1, t > 0, x ∈ Qp, C > 0. (6.17)
Using the fundamental solution Z, we define the operator family
(S(t )ψ )(x) =∫
Qp
Z(t, x− ξ )ψ (ξ )dξ, ψ ∈ L1(Qp), t > 0.
It follows from (6.15), (6.17), and the Young inequality that S is a contraction semi-group in L1(Qp).
Proposition 6.3 S(t ) has the C0-property.
In further considerations the realization A of Dα in space L1(Qp) is defined asthe generator of the semigroup S(t ). Let us return to equation (6.12) interpreted asequation (6.13) on L1(Qp), where the linear operator A is a generator of the semigroup
92 p-Adic and Ultrametric Models in Geophysics
S(t ). As before, ϕ is a strictly monotone increasing continuous real function, |ϕ(s)| ≤C|s|m, m ≥ 1. Below we re-interpret equation (6.12) as the equation
∂u
∂t+ Aϕ(u) = 0, (6.18)
where Aϕ is the closure of Aϕ. It can be shown [229] that the operator Aϕ is alreadydensely defined, all the more so since it is valid for Aϕ.Recall that amild solution of the Cauchy problem for a non-linear equationwith the
initial condition u(0, x) = u0(x) is defined as a function given by a limit, uniformlyon compact time intervals, of solutions of the problem for the difference equationsapproximating the differential one. This is the usual “non-linear version” of the notionof a generalized solution; see [64] for the details.
Theorem 6.4 The operator Aϕ is m-accretive, so that, for any initial function u0 ∈L1(Qp), the Cauchy problem for equation (6.18) has a unique mild solution.
6.3.3 Example of Explicit Solution for Porous Medium Equation
In applications, including applications of p-adics to geology [245], [367], it is impor-tant to get explicit solutions. Here we present the concrete explicit solution of theporous medium equation, see [229].Let us consider equation (6.12) with α > 0, ϕ(u) = |u|m, m > 1. We look for a
solution of the form
u(t, x) = ρ
( |x|γpt0 − t
)ν
, 0 < t < t0, x ∈ Qp, (6.19)
where t0 > 0, γ > 0, ν > 0, 0 = ρ ∈ R. We have
∂u
∂t= νρ|x|γ νp (t0 − t )−ν−1,
Dα (|u|m) = |ρ|m(t0 − t )−νmDα(|x|γ νmp
)
.
On comparing powers of t0 − t we find that ν = (m− 1)−1.In concrete calculations it is convenient to represent the Vladimirov operator Dα
as the convolution operator defined in the framework of the p-adic theory of distri-butions, see e.g. [18] for details.Let u ∈ L1(Qp). Then Dαu can be defined (as a distribution from D′(Qp)) by the
convolution
Dαu = u ∗ f−α,
where
f−α (x) =|x|−α−1
p
�p(−α)
6.3 Non-linear p-Adic Dynamics 93
and
�p(z) = 1− pz−1
1− p−z.
We remark that fβ is defined (with the aid of analytic continuation [18]) for all realβ except β = 1. If β = 1, −α + β = 1, then
f−α ∗ fβ = f−α+β.
The above convolution identity can be written as
Dα (|x|β−1p ) = �p(β )
|x|−α+β−1p
�p(β − α)
or, if we substitute β + 1 for β, as
Dα (|x|βp ) =�p(β + 1)
�p(β − α + 1)|x|β−αp , β = α. (6.20)
On calculating Dα (|u|m) by (6.20), substituting into (6.19), and canceling out powersof t0 − t one comes to the identity
ρ
m− 1|x|
γ
m−1p = −|ρ|m �p(γm/(m− 1)+ 1)
�p(γm/(m− 1)− α + 1)|x|
γmm−1−αp
implying γ = α,
ρ
m− 1= −|ρ|m�p(αm/(m− 1)+ 1)
�p(α/(m− 1)+ 1). (6.21)
Both the numerator and denominator on the right-hand side of (6.21) are negative.Thus ρ < 0, and the solution can be represented in the form
u(t, x) = ρ(t0 − t )−1
m−1 |x|α
m−1p , t > 0, x ∈ Qp, (6.22)
where
ρ = −[
�p(1+ α/(m− 1))
(m− 1)�p(1+ αm/(m− 1))
]1
m−1
.
7
Recent Development of the Theory of p-AdicDynamical Systems
The theory of p-adic (and more generally non-Archimedean) dynamical systemsarose from the mixing of various (and very different) research flows (see e.g. [20],[14], [31], [32], [33], [34], [35], [37], [38], [39], [36], [49], [73], [74], [75], [110],[142], [122], [141], [149], [151], [150], [192], [193], [223], [227], [249], [250],[247], [248], [264], [265], [266], [312], [382], [383], [384], [453], [454], [452], [455],and the references therein):
� number-theoretic methods in the study of monomial dynamical systems1
� theory of ergodic dynamical systems� p-adic (and more generally non-Archimedean) mathematical physics� p-dynamical systems in cryptography� p-adic modeling of cognition and psychology.
The aim of this chapter is to present some recent results about p-adic dynamics.Here discrete dynamical systems based on iterations of functions belonging to a spe-cial functional class, namely 1-Lipschitz functions, will be considered. The impor-tance of this class for theory of p-adic dynamical systems was emphasized in a seriesof pioneering works by V. Anashin [31], [32], [33]. Then some interesting resultsabout such discrete dynamics were obtained in joint works by V. Anashin, A. Khren-nikov, and E. Yurova [34], [35], [37], [38], [39], [36], [265], [266], [453], [454], [452],[455].Let Zp be a ring of p-adic integers. We recall that the space Zp is equipped with a
natural probabilitymeasure, namely theHaarmeasureμp normalized asμp(Zp) = 1.Balls B−r(a) of non-zero radii constitute the base of the corresponding σ -algebraof measurable subsets, μp(B−r(a)) = p−r. The measure μp is a regular Borel mea-sure, so all continuous transformations f : Zp → Zp are measurable with respectto μp. As usual, a measurable mapping f : Zp → Zp is called measure-preserving
1 We remark that the theory of p-adic dynamical systems is a part of the general theory of arithmeticdynamical systems, see e.g. [395], [417], [416].
94
Recent Development of the Theory of p-Adic Dynamical Systems 95
if μp( f−1(S)) = μp(S) for each measurable subset S ⊂ Zp. A measure-preservingmapping f : Zp → Zp is called ergodic if f−1(S) = S implies either μp(S) = 0 orμp(S) = 1.Consider a map f : Zp → Zp. The theory of (discrete) dynamical system studies
trajectories (orbits), i.e. sequences of iterations of f ,
x0, x1 = f (x0), . . . , xi+1 = f (xi) = f (i+1)(x0), . . . .
If the map f generating a dynamical system is measure-preserving (ergodic), then thedynamical system is called measure-preserving (ergodic).In what follows functions f : Zp → Zp which satisfy Lipschitz condition with con-
stant 1 will be considered:
| f (x)− f (y)|p ≤ |x− y|p .
The 1-Lipschitz property may be re-stated in terms of congruences, in the followingway.Given a, b ∈ Zp and k ∈ N = {1, 2, 3, . . .}, the congruence a ≡ b mod pk is well
defined. The congruence just means that the images of a of b under the action ofthe ring epimorphism (mod pk ) : Zp → Z/pkZ of the ring Zp onto the residue ringZ/pkZ modulo pk coincide.Recall that by definition the epimorphism (mod pk ) sends a p-adic integer that has
a canonical representation
∞∑
i=0
αi pi, αi ∈ {0, 1, . . . , p− 1} , i = 0, 1, 2, . . . ,
to
k−1∑
i=0
αi pi ∈ Z/pkZ.
Note also that if necessary one can treat elements from Z/pkZ as numbers from{0, 1, . . . , pk − 1}.
Now it is obvious that the congruence a ≡ b (mod pk ) is equivalent to the inequal-ity |a− b|p ≤ p−k. Therefore the transformation f : Zp → Zp is 1-Lipschitz if andonly if it is compatible, i.e. f (a) ≡ f (b) (mod pk ) once a ≡ b (mod pk ).For a 1-Lipschitz transformation f : Zp → Zp, we consider its reduced map-
ping modulo pk, namely fk : Z/pkZ → Z/pkZ, z �→ f (z) (mod pk ). The mappingfk is well defined ( fk does not depend on the choice of representative z in the ballz+ pkZp).
The property of a function being 1-Lipschitz is significant. For example, letf = (x2 + x)/2 be a 2-adic function. This function is not 1-Lipschitz since | f (2)−f (0)|2 = 1 but |2− 0|2 = 2−1. Now let us define a function f1 as f1(x (mod 2)) =f1(x) (mod 2). Then f1(0) = 0 and f1(2) = 1. However, 2 ≡ 0 (mod 2).
96 Recent Development of the Theory of p-Adic Dynamical Systems
A 1-Lipschitz transformation f : Zp → Zp is called bijective modulo pk if thereduced mapping f (mod pk ) is a permutation on Z/pkZ. Furthermore, f is calledtransitive modulo pk if f (mod pk ) is a permutation that is a cycle of length pk.Also there the definition of uniformly differentiable modulo ps functions, see e.g.
Section 3.7, [35], will be used. A function f : Zp → Zp is said to be uniformly dif-ferentiable modulo ps on Zp if there exists a positive integer N and ∂s f (x) ∈ Qp suchthat for any k > N the congruence
f (x+ pkh) ≡ f (x)+ pk · h · ∂s f (x) (mod pk+s)
holds simultaneously for all x, h ∈ Zp. The smallest of these N is denoted by Ns( f ).For example, functions defined by polynomials over Zp[x] are uniformly differen-tiable modulo p.Also the function f is differentiable at the point x, if there exists
limk→∞
f (x+ pkh)− f (x)
pkh= ∂s f (x),
i.e. the congruence ( f (x+ pkh)− f (x))/(pkh)− ∂s f (x) ≡ 0 (mod pn) holds for anynatural number n. For the function f to be differentiable at the point x modulo ps thelast congruence must be satisfied only modulo ps for fixed s.
7.1 Van der Put Series and Coordinate Representations of Dynamical Maps
In this section we introduce two representations of p-adic functions which will beused to describe the measure-preservation and ergodicity of discrete dynamical sys-tems, namely the van der Put series and the coordinate representation of 1-Lipschitzp-adic functions.The construction of the van der Put series is described in detail in [317] and
[390]. Consider a continuous function f : Zp → Zp. There exists a unique sequenceV0,V1,V2, . . . of p-adic integers such that
f (x) =∞∑
m=0
Vmχ (m, x) (7.1)
for all x ∈ Zp, where
χ (m, x) ={
1, if |x− m|p ≤ p−n
0, otherwise
and n = 1 if m = 0; n is uniquely defined by the inequality pn−1 ≤ m ≤ pn − 1 oth-erwise. The series on the right-hand side in (7.1) is called the van der Put series ofthe function f .Note that the sequenceV0,V1,V2, . . . of the van der Put coefficients of the function
f tends (p-adically) to 0 as m → ∞, and the series converges uniformly on Zp. Viceversa, if a sequenceV0,V1,V2, . . . of p-adic integers tends p-adically to 0 asm → ∞,
7.1 Van der Put Series and Coordinate Representations 97
then the series on the right-hand side of (7.1) converges uniformly on Zp and thusdefines a continuous function f : Zp → Zp.
The number n in the definition of χ (m, x) has a very natural meaning. It is just thenumber of digits in a base-p expansion of m ∈ N. Since
⌊
logp m⌋ = (the number of digits in a base-p expansion for m)− 1,
therefore n = ⌊
logp m⌋+ 1 for all m ∈ N� {0} and ⌊
logp 0⌋ = 0.
Recall that α! for a real α denotes the integral part of α, that is, the nearest rationalinteger to α which does not exceed α. Note that χ (m, x) is merely a characteristicfunction of the ball
B− logp m!−1(m) = m+ p− logp m!−1Zp
of radius p− logp m!−1 centered at m ∈ N = {0, 1, 2 . . .}:
χ (m, x) ={
1, if x ≡ m(
mod p logp m!+1)
0, otherwise
={
1, if x ∈ B− logp m!−1(m)
0, otherwise.
The coefficients Vm are related to values of the function f in the following way.Let m = m0 + · · · + mn−2pn−2 + mn−1pn−1 be a base-p expansion for m, i.e. mj ∈{0, . . . , p− 1}, j = 0, 1, . . . , n− 1, and mn−1 = 0. Then
Vm ={
f (m)− f (m− mn−1pn−1), if m ≥ pf (m), otherwise.
Nowwe consider a simple example of the van der Put series for the function g(x) = x2
with p = 2. Denote q(m) = mn−1pn−1.Here we calculate step by step the van der Put coefficients Vm for the function g.
From this table we can write the van der Put series:
g(x) = 0 · χ (x, 0)+ 1 · χ (x, 1)+ 4 · χ (x, 2)+ 8 · χ (x, 3)+ 16 · χ (x, 4)+ 24 · χ (x, 5)+ 32 · χ (x, 6)+ 40 · χ (x, 7)+ · · · .
Let x = 5 = 1 · 20 + 0 · 21 + 1 · 22. The characteristic function χ (x,m) is equal to 1for m = 1 and m = 5, as one can see considering segments of x.
Therefore,
g(5) = 1 · χ (5, 1)+ 24 · χ (5, 5) = 25.
It is worth noticing that to calculate the concrete value of a function one shouldknow
98 Recent Development of the Theory of p-Adic Dynamical Systems
Table 7.1. For p = 2 and g(x) = x2
Canonical expansion m g(m) g(m− q(m)) Vm
0 = 0 · 20 + 0 · 21 0 01 = 1 · 20 + 0 · 21 1 101 = 0 · 20 + 1 · 21 2 4 0 411 = 1 · 20 + 1 · 21 3 9 1 8001 = 0 · 20 + 0 · 21 + 1 · 22 4 16 0 16101 = 1 · 20 + 0 · 21 + 1 · 22 5 25 1 24011 = 0 · 20 + 1 · 21 + 1 · 22 6 36 4 32111 = 1 · 20 + 1 · 21 + 1 · 22 7 49 9 40...
......
......
(i) an “infinite number” of coefficients for the power series f (x) = ∑∞k=0 akx
k;(ii) “m” coefficients for the Mahler series f (x) = ∑∞
i=0 ai(xi
)
, ai ∈ Zp,(xi
) =x(x− 1) . . . (x− i+ 1)/i!;
(iii) “logp m” coefficients for the van der Put series g(x) = ∑∞m=0Vmχ (m, x).
1-Lipschitz functions in terms of the van der Put series were described in [390].
Theorem 7.1 Let a function f : Zp → Zp be represented via the van der Put series(7.1); then f is 1-Lipschitz if and only if |Vm|p ≤ p− logp m! for all m = 0, 1, 2, . . . .In other words, f is 1-Lipschitz if and only if it can be represented as
f (x) =∞∑
m=0
p logp m!bmχ (m, x)
for suitable bm ∈ Zp,m = 0, 1, 2, . . . .
To describe ergodic discrete dynamical systems based on 1-Lipschitz functions, itis also useful to use the coordinate representation of p-adic functions, see e.g. [35,Proposition 3.35]. Let f : Zp → Zp be a 1-Lipschitz function. The function f has thefollowing coordinate representation:
f (x0 + px1 + · · · + pkxk + · · · ) = ϕ0(x0)+ pϕ1(x0, x1)+ · · ·+ pkϕk(x0, x1, . . . , xk )+ · · · , (7.2)
where
ϕk : Z/pZ × · · · × Z/pZ︸ ︷︷ ︸
k+1
→ Z/pZ
are p-valued functions that depend on p-valued variables x0, x1, . . . , xk, k =0, 1, 2, . . . . Each function ϕk can be set by a polynomial of k + 1 variables over thefield Z/pZ.
7.2 Recent Results about Measure-Preserving Functions 99
In other words, coordinate functions ϕk of 1-Lipschitz p-adic functions dependonly on variables x0, x1, . . . , xk, k = 0, 1, 2, . . . . Denote ϕk,x as a subfunctionobtained from ϕk by fixing variables x = (x0, x1, . . . , xk−1).
7.2 Recent Results about Measure-Preserving Functions andErgodic Dynamics
7.2.1 Measure-Preserving 1-Lipschitz Functions
Let us consider a ring of p-adic integers Zp. Let f : Zp → Zp be a 1-Lipschitz func-tion of this ring to itself.We give the description of the large classes of measure-preserving functions, see,
for example, [31], [32], [35].Let f : Zp → Zp be a 1-Lipschitz function represented by theMahler series
f (x) =∞∑
k=0
ak
(
x
i
)
,
where ak ∈ Zp,(xi
) = x(x− 1) . . . (x− i+ 1)/i!.
Theorem 7.2 (Theorem 4.40, [35]) The function f defines a 1-Lipschitz measure-preserving transformation on Zp whenever the following conditions hold simultane-ously:
a1 = 0 (mod p);ai ≡ 0 (mod p) logp i!+1, i = 2, 3, . . . .
Moreover, in the case p = 2 these conditions are necessary: namely, if f is 1-Lipschitzand measure-preserving then the conditions hold simultaneously.
Theorem 7.3 (Lemma 4.41, [35]) Given a 1-Lipschitz function f : Zp → Zp and p-adic integers c, d, c ≡ 0 (mod p), the function g(x) = d + cx+ p · f (x) preservesmeasure.
Theorem 7.4 (Theorem 4.45, [35]) Let the function f : Zp → Zp be uniformly dif-ferentiable modulo p, and let all partial derivatives modulo p of the function f beinteger-valued, i.e. f is a 1-Lipschitz function.The function f is measure-preserving if and only if f is bijective modulo pk for
some k ≥ N1( f )+ 1 (the value of N1( f ) is defined by the condition of uniform differ-entiability modulo p).
7.2.2 General Criterion of Measure-Preserving 1-Lipschitz Functions
Using the representation of p-adic functions in the van der Put series, it is possible toobtain necessary and sufficient conditions for 1-Lipschitz functions to be measure-preserving.
100 Recent Development of the Theory of p-Adic Dynamical Systems
Theorem 7.5 (Theorem 2.1, [265]) Let f : Zp → Zp be a 1-Lipschitz functionand let
f (x) =∞∑
m=0
p logp m!bmχ (m, x)
be the van der Put representation of this function, where bm ∈ Zp, m ∈ {0, 1, 2, . . .}.Then f (x) preserves Haar’s measure if and only if
(i) {b0, b1, . . . , bp−1} establish a complete set of residues modulo p, i.e. the functionf (x) is bijective modulo p;
(ii) for every k ∈ {2, 3, . . .} and m ∈ {0, . . . , pk − 1}, the elements in the set{
bm+pk , bm+2pk , . . . , bm+(p−1)pk}
are all non-zero residues modulo p.
From this theorem a description of 2-adic measure-preserving functions in termsof the van der Put basis follows immediately.
Theorem 7.6 (Theorem 2.2, [455]) The function f : Z2 → Z2 is 1-Lipschitz and pre-serves the measure μp if and only if it can be represented as
f (x) = b0χ (0, x)+ b1χ (1, x)+∞∑
m=2
2 log2 m!bmχ (m, x),
where bm ∈ Z2 for m = 0, 1, 2, . . . , and (i) b0 + b1 ≡ 1 (mod 2), (ii) |bm|2 = 1, ifm ≥ 2.
Note that in the proof of this theorem we use the representation of the function ffrom Theorem 7.3.
7.2.3 “Additive-Form” Criterion of Measure-Preserving 1-Lipschitz Functions
In the paper [265] a characterization of measure-preserving functions by using the“additive form representation” was presented. In the additive-form criterion, a 1-Lipschitz measure-preserving function is decomposed into a sum of two functions.The first is an arbitrary 1-Lipschitz function – the “free” part – and the second is a1-Lipschitz function of a special type. This “special” function is given by the van derPut basis, where the coefficients are defined via an arbitrary set of permutations on theset of non-zero residues modulo p and one permutation modulo p. Thus a “special”function ξ (x) provides the condition of measure-preservation.
Theorem 7.7 (Theorem 4.1, [265]) A 1-Lipschitz function f : Zp → Zp preservesmeasure if and only if it can be represented as
f (x) = ξ (x)+ p · h(x),
7.2 Recent Results about Measure-Preserving Functions 101
where h : Zp → Zp is an arbitrary 1-Lipschitz function and the function ξ (x) repre-sented via the van der Put series is such that
ξ (x) =p−1∑
i=0
G(i)χ (i, x)+∞∑
k=1
pk−1∑
m=0
p−1∑
i=1
gm(i)pk · χ (m+ i · pk, x)
=p−1∑
i=0
G(i)χ (i, x)+∞∑
k=1
pk−1∑
m=0
p−1∑
i=1
i · pk · χ(m+ g−1m (i) · pk, x),
where gm is a permutation on the set {1, . . . , p− 1} and G is a permutation on theset {0, 1, . . . , p− 1}.As an example of a measure-preserving p-adic function constructed by using the
additive form of representation, we show that the function
f(
x0 + x1p+ · · · + xk pk + · · · ) = (p− 1)(1+ x0)+
∞∑
k=1
pk · xsk mod p
+∞∑
k=1
p2k+1 · xk
= (p− 1)(1+ x0)+∞∑
k=1
(
xs2k+1 mod p+ xk) · p2k+1
+∞∑
k=1
p2k · xs2k modp
preserves the measure.
7.2.4 Criterion of Measure-Preserving 1-Lipschitz FunctionsRepresented in the Coordinate Form
The criterion on measure-preservation can be stated in terms of the coordinate formof representation of the function f .
Theorem 7.8 (Theorem 3.1, [266]) Let the p-adic 1-Lipschitz function f : Zp → Zp
have the coordinate representation (7.2). The function f preserves measure if andonly if
(i) ϕ0 is bijective on the set of residues modulo p;(ii) ϕk,xk−1 (xk ) is bijective on the set of residues modulo p for any k = 1, 2, . . . and
fixed values (x0, x1, . . . , xk−1) = xk−1.
In fact, this theorem states that the condition of measure-preservation is equivalentto bijectivity for all subfunctions ϕk,x of the coordinate functions ϕk over the field ofresidues modulo p, k ≥ 0.
102 Recent Development of the Theory of p-Adic Dynamical Systems
7.2.5 Criterion of Measure-Preserving 1-Lipschitz Functions for p = 3
Now let us consider the case p = 3. In contrast to the general case for p-adic functionsZp → Zp, the case p = 3 has the following characteristic. All transformations of thefield of residues Z/3Z can be set as linear polynomials of the type ax+ b, a = 0. Inparticular, all bijective subfunctions ϕk,x are set via such polynomials.
Theorem 7.9 (Theorem 2, [453]) Let f : Z3 → Z3 be a 1-Lipschitz function and let
f (x) =∞∑
m=0
3 log3 m!bmχ (m, x)
be the van der Put representation of this function, where bm ∈ Z3,m ∈ {0, 1, 2, . . .}.Then f (x) preserves the Haar measure if and only if the following conditions holdsimultaneously:
(i) bm ≡ 0 (mod 3) for m ≥ 3;(ii) bm+3k + bm+2·3k ≡ 0 (mod 3) for 0 ≤ m ≤ 3k − 1, k ≥ 2, where m is an integer
variable;(iii) {b0, b1, b2} is a complete set of residues modulo 3, or, in other words,
{
b0 + b1 + b2 ≡ 0 (mod 3)b20 + b21 + b22 ≡ 2 (mod 3).
Theorem 7.10 (Theorem 5, [453]) Let g: Z3 → Z3 be an arbitrary 1-Lipschitz func-tion. A 1-Lipschitz function f : Z3 → Z3 preserves measure if and only if it can berepresented as
f (x) = ξ (x)+ 3 · g(x),
where
(i) for bm ∈ {1, 2}
ξ (x) = b0χ (x, 0)+ b1χ (x, 1)+ b2χ (x, 2)
+∞∑
k=1
3k ·⎛
⎝
3k−1∑
m=0
bm · (χ (x, m+ 3k )− χ (x, m+ 2 · 3k ))⎞
⎠+ φ(x);
(ii) for b0, b1, b2 ∈ {0, 1, 2} we have b0 + b1 + b2 ≡ 0 (mod 3) and b20 + b21 + b22 ≡2 (mod 3);
(iii) φ(x) = φ(x0 + 3x1 + · · · + 3kxk + · · · ) = ∑∞k=1 3
k · xk(xk − 1)/2.
The representation of the function φ(x) is due to the possibility of representingany p-adic number in the following form. For each s = 0, 1, 2, . . . , p− 1 we define
7.2 Recent Results about Measure-Preserving Functions 103
functions
�: Z2 → Zp, �(x) = �(x0 + 2x1 + · · · + 2kxk + · · · )
=∞∑
k=0
pkxk (xk ∈ {0, 1});
ψs: Zp → Z2, ψs(x) = ψs(x0 + px1 + · · · + pkxk + · · · )
=∞∑
k=0
2kI(δk(x) = s);
�: Zp → Z2 × · · · × Z2︸ ︷︷ ︸
p
, �(x) = (
ψ0(x), . . . , ψp−1(x))
,
where δk: Zp → {0, 1, . . . , p− 1}, δk(x0 + px1 + · · · + pkxk + . . .) = xk, k ≥ 0, and
I(δk(x) = s) ={
1, if δk(x) = s;
0, if δk(x) = s.
Mapping � is a bijective embedding Zp in
Z2 × · · · × Z2︸ ︷︷ ︸
p
.
Then the elements of the image of Zp have the following property. Set x ∈ Zp and�(x) = (ψ0(x), . . . , ψp−1(x)). Then for any k ≥ 0
δk(ψ0(x))+ δk(ψ1(x))+ · · · + δk(ψp−1(x)) = 1, (7.3)
and, in particular, ψ0(x)+ ψ1(x)+ · · · + ψp−1(x)+ 1 = 0 for any x ∈ Zp.Thus, if x ∈ Zp, then
x = �(ψ1(x))+ 2�(ψ2(x))+ · · · + (p− 1)�(ψp−1(x)).
Vice versa, suppose that we are given 2-adic numbers d0, d1, . . . , dp−1 ∈ Z2 suchthat
δk(d0)+ δk(d1)+ · · · + δk(dp−1) = 1, k ≥ 0.
This set of 2-adic numbers uniquely defines a p-adic number D,
D = �(d1)+ 2�(d2)+ · · · + (p− 1)�(dp−1).
104 Recent Development of the Theory of p-Adic Dynamical Systems
Let us consider a numerical example. Given 2-adic numbers
d0 = (0000001000000111 . . .),
d1 = (0000000001000000 . . .),
d2 = (0110010100000000 . . .),
d3 = (1001000010010000 . . .),
d4 = (0000100000101000 . . .).
The numbers d0, d1, d2, d3, d4 satisfy conditions (7.3) and uniquely define a 5-adicnumber
D = �(d1)+ 2�(d2)+ · · · + 4�(d4) = (3223420231434000 . . .).
Let us re-state a criterion of measure-preservation when a “constant” term in therepresentation of the function ξ can be set as the function x.
Theorem 7.11 (Theorem 4.18, [266]) The 1-Lipschitz function f : Z3 → Z3 pre-serves measure if and only if f can be represented as
f (x) = ξ (x)+ 3 · h(x),where h : Z3 → Z3 is 1-Lipschitz function and
ξ (x) = ξ(
x0 + 3x1 + · · · + 3kxk + · · · )
= b+ c · x0 + x+∞∑
k=1
3k(
IMk
(
x0 + · · · + 3k−1xk−1) · xk · (5− 3xk )
2
)
,
c ∈ {0, 1}, b ∈ {0, 1, 2}, and for Mk ⊆ {0, 1, . . . , 3k − 1}
IMk
(
x0 + · · · + 3k−1xk−1) =
{
1, if x0 + · · · + 3k−1xk−1 ∈ Mk
0, otherwise
(in other words, IMk is the characteristic function of Mk, k ≥ 1).
From this theorem in particular, we get that functions of the form
f (x) = b+ x+ 3h(x) (7.4)
if we set c = 0 and Mk = ∅, k ≥ 1; or
f (x) = b+ 2x+ 3h(x) (7.5)
if we set c = 1 and Mk = {0, 1, . . . , 3k − 1}, k ≥ 1, preserve the measure. Suchclasses of 1-Lipschitz measure-preserving functions were obtained byV. Anashin, seee.g. [35]. However, Theorem 7.11 shows that the class of 1-Lipschitz 3-adic measure-preserving functions is bigger than Anashin’s classes (7.4) and (7.5).
7.3 Ergodic Dynamical Systems Based on 1-Lipschitz Functions 105
7.3 Ergodic Dynamical Systems Based on 1-Lipschitz Functions
In this section we describe the widest classes of 1-Lipschitz functions generatingdiscrete ergodic dynamical systems, see e.g. [31], [32], [33], [35].Let f : Zp → Zp be a 1-Lipschitz function represented by theMahler series f (x) =
∑∞k=0 ak
(xi
)
, where ak ∈ Zp,(xi
) = x(x− 1) . . . (x− i+ 1)/i!.
Theorem 7.12 (Theorem 4.40, [35]) The function f defines a 1-Lipschitz ergodictransformation on Zp whenever the following conditions hold simultaneously:
a0 = 0 (mod p);a1 ≡ 1 (mod p), for p odd;a1 ≡ 1 (mod p), for p = 2;ai ≡ 0
(
mod p logp(i+1)!+1), i = 2, 3, . . . .
Moreover, in the case p = 2 these conditions are necessary: namely, if f is 1-Lipschitzand ergodic then conditions hold simultaneously.
In particular, the following result holds.
Theorem 7.13 (Lemma 4.41, [35]) Given a 1-Lipschitz function v : Zp → Zp anda p-adic integer c ≡ 0 (mod p), the function h(x) = c+ x+ p ·�v(x) is ergodic.(Recall that � is a difference operator: �v(x) = v(x+ 1)− v(x) by the definition).
The following theorem describes 1-Lipschitz uniformly differentiable functionsgenerating ergodic dynamical systems.
Theorem 7.14 (Theorem 4.55, [35]) Let a function f : Zp → Zp be uniformlydifferentiable modulo p2, and let the derivative modulo p2 of the function f be integer-valued. Then f is ergodic if and only if it is transitive modulo pn for some (equiva-lently, for every) n ≥ N2( f )+ 1 whenever p is odd or, respectively, for some (equiv-alently, for every) n ≥ N2( f )+ 2 whenever p = 2.
Note that 1-Lipschitz functions represented by polynomials from Zp belong tothe classes mentioned above. In particular, in the case p = 3 ergodic functions aredescribed in terms of the coefficients in the canonical representation of polynomialsfrom Zp[x], see [142]. However, the problem of how to describe 1-Lipschitz ergodicfunctions for p = 2 remains unsolved. Moreover, it remains unknown how to findnecessary and sufficient conditions of ergodicity for 1-Lipschitz uniformly differen-tiable modulo p functions, see Open Question 4.60 in [35].
It is a challenge to describe an ergodic function without an analytical expressionof its iterations (which is hard to present explicitly).It turns out that these difficulties are characterized by the non-commutativity of
the symmetric group Sp of permutations on Z/pZ. Let p = 2 and let the functionbe represented by some series (for example, a Mahler series, van der Put series, etc.).
106 Recent Development of the Theory of p-Adic Dynamical Systems
Then one can obtain the conditions of ergodicity in terms of coefficients of such seriesbecause S2 is an Abelian group.In cases for which the symmetric group Sp is an Abelian group of permutations on
Z/pZ we can obtain a compact description of ergodic functions.
7.3.1 Criterion of Ergodicity of Discrete Dynamical Systems Generated by1-Lipschitz Functions
For an arbitrary prime number p > 1, the following criterion of ergodicity for 1-Lipschitz functions represented in the coordinate form was obtained in [266], [454].
Theorem 7.15 (Theorem 3.2, [266]) Let the p-adic 1-Lipschitz function f : Zp →Zp be presented in the coordinate form (7.2), where ϕ0 and ϕk,xk−1 , k = 1, 2, . . . , arepermutations on the set of residues modulo p. The function f is ergodic if and only if
(i) the map ϕ0 is transitive on the set of residues modulo p;(ii) the permutation
Fk,xk−1 = ϕk, f (p
k−1)k−1 (xk−1 )
◦ ϕk, f (p
k−2)k−1 (xk−1 )
◦ · · · ◦ ϕk,xk−1
is transitive on the set of residues modulo p for any k = 1, 2, . . . , where
f (s)k (xk−1) = fk( fk(. . . fk(xk−1)) . . .)︸ ︷︷ ︸
s
,
fk ≡ f (mod pk+1), and f0 = ϕ0.
It was shown that the conditions of this criterion do not depend on the choice ofxk−1. And, in particular, we can set xk−1 = (x0, x1, . . . , xk−1) = (0, . . . , 0).
To check that a function generates the ergodic dynamical system, one should checkthe transitivity of the permutations Fk,xk−1 , k = 1, 2, . . . . Each permutation is a prod-uct of permutations ϕk,xk−1 , xk−1 ∈ {0, . . . , pk−1 − 1}. The order of their appearancein the resulting product (i.e. in the Fk,xk−1 ) is defined by the sequence of residuesmodulo pk
τk ={
xk−1, fk−1(xk−1), . . . , f(pk−2)k−1 (xk−1), f
(pk−1)k−1 (xk−1)
}
.
In other words, to check the transitivity of the function fk ≡ f (mod pk+1) oneshould construct the sequence τk, find Fk,xk−1 , and verify its transitivity.
Note that the order of residues modulo pk in the sequence τk is significantly impor-tant. This is due to the fact that the symmetric group Sp (permutations on Z/pZ) isnon-Abelian. Therefore, in general by determining Fk,xk−1 we cannot change the orderof the permutations ϕk,xk−1 .
Of course, in some special cases the permutations ϕk,xk−1 commute. Then one canexpect to find compact conditions of ergodicity for the corresponding class of func-tions. For example, the following statement holds.
7.3 Ergodic Dynamical Systems Based on 1-Lipschitz Functions 107
Theorem 7.16 (Corollary 3.10, [266]) Let the p-adic 1-Lipschitz function f : Zp →Zp be presented in the coordinate form (7.2), where the subfunctions (of the coordi-nate functions) ϕ0, ϕk,xk−1 , k = 1, 2, . . . are permutations on Z/pZ. Suppose that thesubfunctions have the form
ϕk,xk−1 = gn(xk−1 )k , (7.6)
where gk is a permutation on Z/pZ and n(xk−1) is a positive integer (g0k-identitypermutation), k = 1, 2, . . . . Then the function f is ergodic if and only if
(i) ϕ0, gk are transitive permutations;(ii)
∑pk−1xk−1=0 n(xk−1) ≡ 0 (mod p), k = 1, 2, . . . .
In Theorem 7.16 there was considered the case such that, for each k = 1, 2, . . . ,permutations ϕk,xk−1 ∈ Sp belong to the cyclic group generated by a permutation gk.In this case, all ϕk,xk−1 ∈ Sp commute, Fk,xk−1 does not, depending on the order ofelements in the sequence τk, and, moreover, Fk,xk−1 = gαk for some α depend only onk. Therefore, to verify the transitivity of Fk,xk−1 , there is no need to build the sequenceτk. This simplifies substantially the verification of the ergodicity of f .
7.3.2 Ergodic 1-Lipschitz Functions (Coordinate Functions – LinearUnitary Polynomial)
Another important example is the case when the permutations ϕk,xk commute, i.e.the corresponding subfunctions of the coordinate functions constitute a linear unitarypolynomial from Z/pZ[x].
Theorem 7.17 (Theorem 4.1, [266]) Let the p-adic (p = 2) 1-Lipschitz functionf : Zp → Zp be presented in the coordinate form (7.2), where the coordinate func-tions have the form
ϕk,xk (xk ) = xk + α(xk ),
k = 1, 2, . . . . The function f is ergodic if and only if
(i) ϕ0 is a transitive (monocycle) permutation on the set of residues modulo p;(ii) 2p−2 + (1/pk )
∑pk−1i=0 f (i) ≡ 0 (mod p), k = 2, 3, . . . .
Note that the sums∑pk−1
i=0 f (i) appear in the definition of Volkenborn’s integral, seee.g. [390, Definition 55.1]. However, the precise nature of the connection between thetheory of Volkenborn integration and the ergodicity of p-adic dynamical systems hasnot yet been clarified.It turns out that the representation of the function ϕk,xk as a linear unitary polyno-
mial is equivalent to representation of the function f : Zp → Zp as
f (x0 + px1 + · · · + pkxk + · · · ) = ϕ0(x0)+ (x− x0)+ p · g(x),
108 Recent Development of the Theory of p-Adic Dynamical Systems
where g: Zp → Zp is an arbitrary 1-Lipschitz function. Thus Theorem 7.17 can berestated in the following way.
Theorem 7.18 (Theorem 4.4, [266]) Let a p-adic (p = 2) 1-Lipschitz functionf : Zp → Zp have the form f (x) = f (x0 + px1 + · · · + pkxk + · · · ) = ϕ0(x0)+ (x−x0)+ pg(x), where g: Zp → Zp is a 1-Lipschitz function and ϕ0 : Z/pZ �−→ Z/pZ.The function f is ergodic if and only if
(i) ϕ0 is a transitive (monocycle) permutation on the set of residues modulo p;(ii) (1/pk−1)
∑pk−1i=0 g(i) ≡ 2p−2 (mod p), k = 2, 3, . . . .
Note that Theorem 7.17 describes a wider class of ergodic functions than the func-tions of the form f (x) = c+ x+ p�g(x), see e.g. [35].From the applications point of view, see e.g. [35, Chapter 9], we state the theorem,
which describes ergodic functions of the form
f(
x0 + px1 + · · · + pkxk + · · · ) = c+ a0 · x0 + a1 · px1 + · · · + ak · pkxk + · · ·for the case p = 2. The ergodicity of such functions in the case p = 2 is proved ine.g. [35, Theorem 9.20 (part 2)].
Theorem 7.19 (Theorem 4.7, [266]) Let f : Zp → Zp (p = 2) be a 1-Lipschitz func-tion of the form
f(
x0 + px1 + · · · + pkxk + · · · ) = c+ a0 · x0 + a1 · px1 + · · · + ak · pkxk + · · · ,where ak ∈ Zp, k = 0, 1, . . . , c ∈ Zp. The function f is ergodic if and only if c ≡ 0(mod p) and ak ≡ 1 (mod p), k = 0, 1, . . . .
7.3.3 Ergodic 1-Lipschitz Functions (Coordinate Functions – Linear Polynomial)
Now we consider the case when the coordinate functions have the form
ϕk(x0, . . . , xk ) = xkAk(x0, . . . , xS−1)+ αk(x0, . . . , xk−1)
= xkAk(xS−1)+ αk(xk−1)
for some fixed integer S. In this case Ak(xS−1) is considered as some polynomialover the field Z/pZ. Note that in the general case p = 2 the subfunctions ϕk,x of thecoordinate function need not commute as permutations on Z/pZ. Thus
Theorem 7.20 (Theorem 4.8, [266]) Let a p-adic (p = 2) 1-Lipschitz measure-preserving function f : Zp → Zp be presented in the coordinate form (7.2), whereϕk are coordinate functions such that
ϕk(x0, . . . , xk ) = xkAk(x0, . . . , xS−1)+ αk(x0, . . . , xk−1)
= xkAk(xS−1)+ αk(xk−1),
k = S, S+ 1, . . . for some fixed integer S.
7.3 Ergodic Dynamical Systems Based on 1-Lipschitz Functions 109
The function f is ergodic if and only if it holds simultaneously that
(i) fS−1 ≡ f (mod pS) is a transitive (monocycle) permutation on the set of residuesmodulo pS;
(ii)∏pS−1
i=0 Ak(i) ≡ 1 (mod p), k = S, S+ 1, . . .;(iii) for k = S+ 1, S+ 2, . . .
2−1pS−1∑
i=0
τ(i)k −
pS−1∑
i=0
τ(i)k
pk
⎡
⎣pk−S(
f (i+1)(0) mod pS)
−pk−S−1∑
β=0
f(
f (i)(0) mod pS + pSβ)
⎤
⎦ ≡ 0 (mod p),
and for k = S
pS−1∑
i=0
τ(i)S
pS[
f(
f (i)(0) mod pS)− f (i+1)(0) mod pS
] ≡ 0 (mod p),
where τ(i)k = ∏pS−1
j=i+1 Ak( f( j)(0)), i = 0, 1, . . . , pk − 2 and τ
(i)k = 1 for i =
pk − 1.
A description of ergodic uniformly differentiable modulo p 1-Lipschitz functionsfollows from this theorem.
Theorem 7.21 (Corollary 4.11, [266]) Let f : Zp → Zp (p = 2) be a 1-Lipschitzuniformly differentiable modulo p function. The function f is ergodic if and only if
(i) fS ≡ f mod pS+1 is a transitive (monocycle) permutation on the set of residuesmodulo pS+1;
(ii)∏pS−1
i=0 ∂ f (i) ≡ 1 mod p;(iii) for k = S+ 1, S+ 2, . . .
2−1pS−1∑
i=0
τ (i) −pS−1∑
i=0
τ (i)
pk
⎡
⎣pk−S(
f (i+1)(0) mod pS)
−pk−S−1∑
β=0
f(
f (i)(0) mod pS + pSβ)
⎤
⎦ = 0 mod p,
where τ (i) = ∏pS−1j=i+1 ∂ f ( f
( j)(0)), i = 0, 1, . . . , pS − 2 and τ (i) = 1 for i =pS − 1.
Thus, Open Question 4.60 in [35] is fully resolved. Note that 1-Lipschitz uniformlydifferentiable modulo p2 ergodic functions are described by V. Anashin, see e.g.Theorem 4.55 in [35].
110 Recent Development of the Theory of p-Adic Dynamical Systems
7.3.4 Construction of Ergodic Functions from GivenMeasure-Preserving Functions
Now let us consider the following problem. Let f be a measure-preserving 1-Lipschitz function. How much should one change such a function to get an ergodicfunction? It turns out that the amount of change required, in general, is surprisinglysmall.Let us consider the coordinate function ϕk(x0, x1, . . . , xk ). We then set ϕk,xk−1 , a
subfunction of the coordinate function ϕk. This subfunction is obtained by fixingthe values of the variables x0, . . . , xk−1. (Denote the vector (x0, . . . , xk−1) with fixedvalues of their coordinates as xk−1. The subfunction ϕk,xk−1 depends on the values ofthe variable xk ∈ {0, . . . , p− 1}).All ϕk,xk−1 are permutations onZ/pZ because the function f preserves the measure.
Then, for the function f to be ergodic it suffices merely to choose just one series ofpermutations in a special way, for example, ϕk,0, k = 1, 2, . . . . All other permuta-tions may be arbitrary. The choice of the permutations ϕk,0, k = 1, 2, . . . , imposesthe following restrictions.Let Gk = ϕ
k, f (pk−1)
k−1 (0)◦ · · · ◦ ϕk, fk−1(0), where the order of the permutations is sig-
nificant. We can assume that this permutation is arbitrary on Z/pZ (because ϕk,xk−1 ,
xk−1 = 0, can be set arbitrary). Then we choose ϕk,0 in such a way that Gk ◦ ϕk,0 is atransitive permutation on Z/pZ.As p is a prime number, one can write this condition analytically. Namely, ϕk,0
is a solution in permutations (Gk ◦ X )p = e, where e is the identical permutation.This means that ϕk,0 can be chosen in (p− 1)! variants, i.e. the proportion of suitablepermutations is 1/p of all permutations on Z/pZ.
Theorem 7.22 (Corollary 3.8, [266]) Let the p-adic 1-Lipschitz function f : Zp →Zp be presented in the coordinate form (7.2), where the subfunctions of thecoordinate functions, ϕ0 and ϕk,xk−1 , k = 1, 2, . . . , are permutations on the setZ/pZ and ϕ0 is transitive. Then it is always possible to construct permutationsg1, g2, . . . , gk, . . . such that by setting ϕk,0 = gk, k = 1, 2, . . . , the correspondingfunction f which is defined with the aid of subfunctions ϕk,xk−1 , xk−1 = 0, and ϕk,0 isergodic.
Ergodic 1-Lipschitz Functions for p = 3 Now let us consider the case p = 3, whichhas the following characteristic. In the general criteria of ergodicity for any primep restrictions are imposed on the subfunctions ϕk,xk of the coordinate functions ϕk,which set the function f : Zp → Zp in the coordinate form. The functions ϕk,xk arepermutations on Z/pZ.In the case p = 3 all permutations on Z/3Z can be set as linear polynomials of the
type ax+ b, a = 0. Besides that, transitive permutations are set as polynomials x+ 1and x+ 2.
7.3 Ergodic Dynamical Systems Based on 1-Lipschitz Functions 111
Theorem 7.23 (Theorem 4.14, [266]) Let f : Z3 → Z3 be a 1-Lipschitz 3-adicfunction represented via coordinate form (7.2), where ϕk,x(xk ) = αk(x) · 3xk + βk(x),x = (x0, x1 . . . , xk−1), k ≥ 0 (and besides ϕ0(x0) = α0 · px0 + β0).Then the function f is ergodic if and only if
(i) α0 ≡ 1 (mod 3) and β0 = 0;(ii)
∏3k−1x=0 αk(x) ≡ 1 (mod 3), k ≥ 1;
(iii)∑3k−1
i=0
(∏3k−1
j=i αk( f( j)k−1(0))
) · βk(
f (i)k−1(0)) ≡ 0 (mod 3), k ≥ 1, where
f (s)k (0) = fk( fk(. . . ( fk(0) . . .)))︸ ︷︷ ︸
s
and fk(x) ≡ f (x) (mod 3k+1).
As in the general case (p = 2) the conditions of ergodicity from Theorem7.23 depend on the order of appearance of elements in the sequence τk ={0, fk−1(0), . . . , f
(pk−1)k−1 (0)}. However, in the case p = 3 such a dependence appears
only for arbitrary coefficients of the polynomial Bk, by which the functions Fk,0(xk ) =Ak · xk + Bk are determined. The values of the coefficients Ak do not depend on theorder of the elements of τk.
Let us rewrite Theorem 7.22 for the case p = 3.
Theorem 7.24 (Theorem 4.16, [266]) Let f : Z3 → Z3 be a 1-Lipschitz 3-adic func-tion represented via the coordinate form (7.2), where ϕk,x(xk ) = αk(x) · xk + βk(x),x = (x0, x1 . . . , xk−1), k ≥ 0 (and besides ϕ0(x0) = α0 · x0 + β0) such that (i) α0 ≡ 1(mod 3) and β0 = 0 and (ii)
∏3k−1x=0 αk(x) ≡ 1 (mod 3), k ≥ 1. Then there exists a
number c ∈ Z3 such that, if f (0) = c, then f is an ergodic function.
As we see, these conditions do not depend on the order of appearance of the ele-ments of the sequence
τk ={
0, fk−1(0), . . . , f(3k−1)k−1 (0)
}
.
Then to satisfy an ergodic property we need to restrict the value of the function fonly at the point 0 (with already defined values at other points).Let us re-state Theorem 7.24 in terms of the van der Put series.
Theorem 7.25 (Corollary 4.17, [266]) Let f : Z3 → Z3 be a 1-Lipschitz measure-preserving 3-adic function represented via the van der Put series f (x) =∑∞
m=0 3 log3 m!bmχ (x,m) satisfying (i) f (mod 3) is transitive on Z/3Z and (ii)
∏3k−1x=0 bx+3k ≡ 1 (mod 3), k ≥ 1. Then there exists a number c ∈ Z3 such that, if
f (0) = c, then f is an ergodic function.
Now let us describe 1-Lipschitz 3-adic ergodic functions in terms of additiverepresentation.
112 Recent Development of the Theory of p-Adic Dynamical Systems
Theorem 7.26 (Theorem 4.19, [266]) Let f , h : Z3 → Z3 be 1-Lipschitz functionswith f (x) = ξ (x)+ 3 · h(x) and
ξ (x) = ξ (x0 + 3x1 + · · · + 3kxk + · · · )
= b+ x+∞∑
k=1
3k(
IMk
(
x0 + · · · + 3k−1xk−1) · xk · (5− 3xk )
2
)
,
where IMk is the characteristic function of the set Mk ⊆ {0, 1, . . . , 3k − 1}, andbesides (i) b ∈ {1, 2} and (ii) Mk contains an even number of elements, i.e. |Mk| ≡ 0(mod 2), k ≥ 1. Then there exists a number c ∈ Z3 such that, if f (0) = c, then f is aergodic function.
Let us illustrate this theorem with an example. Let a 1-Lipschitz function f : Z3 →Z3 be in additive form f (x) = ξ (x)+ 3 · h(x), where h : Z3 → Z3 is a 1-Lipschitzfunction and f (mod 3) is transitive onZ/3Z. Set the function ξ (x) in such a way that∅ = M1 = M2 = . . . . In this case the function f has the form f = b+ x+ 3h(x), b =1, 2. Furthermore, the function f is ergodic if and only if b+ (1/3k−1)
∑3k−1i=0 h(x) ≡
0 (mod 3), k ≥ 2.
Ergodic 1-Lipschitz Functions for p = 2 In this case p = 2 criteria of ergodicity of1-Lipschitz functions f : Z2 → Z2 in terms of the Mahler series have been obtainedby V. Anashin, see e.g. [32], [35]. Also in these works a convenient form of ergodic2-adic functions has been obtained, that is f (x) = 1+ x+ 2�h(x), where �h(x) =h(x+ 1)− h(x). Using this representation we obtained criteria of ergodicity for 1-Lipschitz 2-adic functions in terms of the van der Put series.
Theorem 7.27 (Theorem 7, [39]) A 1-Lipschitz function f : Z2 → Z2 is ergodic ifand only if it can be represented as
f (x) = b0χ (0, x)+ b1χ (1, x)+∞∑
m=2
2 log2 m!bmχ (m, x)
for suitable bm ∈ Z2 that satisfy the following conditions:
(i) b0 ≡ 1 (mod 2);(ii) b0 + b1 ≡ 3 (mod 4);(iii) |bm|2 = 1, m ≥ 2;(iv) b2 + b3 ≡ 2 (mod 4);(v)
∑2n−1m=2n−1 bm ≡ 0 (mod 4), n ≥ 3.
For an example, using this theorem it is easy to check ergodicity of the functionf (x) = f (x0 + 2x1 + · · · ) = 1+ x0 + 6x1 +
∑∞k=2(1+ 2(x (mod pk )) · 2kxk.
We describe 1-Lipschitz 2-adic ergodic functions by general criteria of ergodicity.
7.3 Ergodic Dynamical Systems Based on 1-Lipschitz Functions 113
Theorem 7.28 (Theorem 4.22, [266]) Let f : Z2 → Z2 be a 1-Lipschitz measure-preserving 2-adic function. Then the function f is ergodic if and only if the followingconditions hold:
(i) f (0) ≡ 1 (mod 2);(ii) f (0)+ f (1) ≡ 3 (mod 4);(iii) (1/2k−1)
∑2k−1i=0 f (i) ≡ 1 (mod 4), k ≥ 2.
7.3.5 Criteria of Ergodicity on 2-Adic Spheres
At the end of this chapter let us consider the criteria of ergodicity on 2-adic spheres.Dynamical systems on p-adic spheres are an interesting and nontrivial example ofthe dynamics. The first result in this direction, namely the ergodicity criterion formonomial dynamical systems on p-adic spheres, was obtained in [192], [193]. It isnoteworthy that, although these dynamical systems are a p-adic counterpart of classi-cal dynamical systems, circle rotations, in the p-adic case the dynamics exhibit quitea different behavior than the classical one. Later the case of monomial dynamicalsystems on p-adic spheres was significantly extended. In [33], ergodicity criteria forlocally analytic dynamical systems on p-adic spheres were obtained, for arbitraryprime p.
Theorem 7.29 (Theorem 3.1, [38]) A 1-Lipschitz function f : Z2 → Z2, representedby the van der Put series
f (x) =∞∑
m=0
2 log2 m!b f (m)χ (m, x),
where b f (m) ∈ Z2, m = 0, 1, 2, . . . , is ergodic on the sphere S2−r (a) if and only ifthe following conditions hold simultaneously:
(i) f (a+ 2r ) ≡ a+ 2r + 2r+1 (mod 2)r+2;(ii) |b f (a+ 2r + m · 2r+1)|2 = 1, for m ≥ 1;(iii) b f (a+ 2r + 2r+1) ≡ 1 (mod 4);(iv) b f (a+ 2r + 2r+2)+ b f (a+ 2r + 3 · 2r+1) ≡ 2 (mod 4);(v)
∑2n−1m=2n−1 b f (a+ 2r + m · 2r+1) ≡ 0 (mod 4), for n ≥ 3.
Then we present the necessary and sufficient conditions for ergodicity of perturbedmonomial dynamical systems on 2-adic spheres around 1 in the case when perturba-tions are 1-Lipschitz and 2-adically small.
Theorem 7.30 (Theorem 4.1, [38]) Let u : Z2 → Z2 be an arbitrary 1-Lipschitzfunction, and let s, r ∈ N. The function f (x) = xs + 2r+1u(x) is ergodic on thesphere S2−r (1) =
{
1+ 2r + 2r+1x : x ∈ Z2}
if and only if s ≡ 1 (mod 4) and u(1) ≡1 (mod 2).
8
Parabolic-Type Equations, Markov Processes, andModels of Complex Hierarchical Systems
8.1 Introduction
During the last 30 years there has been a strong interest in stochastic processes onultrametric spaces mainly due to their connections with models of complex systems,such as glasses and proteins. These processes are very convenient for describing phe-nomena whose spaces of states display a hierarchical structure, see e.g. [54]–[52],[135], [214], [231], [275], [336], [376], [411], [434], [468], [464], Chapter 4 andreferences therein. Avetisov, Kozyrev et al. constructed a wide variety of models ofultrametric diffusion constrained by hierarchical energy landscapes, see [54]–[52].From a mathematical point of view, in these models the time-evolution of a complexsystem is described by a p-adic master equation (a parabolic-type pseudodifferen-tial equation) which controls the time-evolution of a transition density function of aMarkov process on an ultrametric space. This process describes the dynamics of thesystem in the space of configurational states which is approximated by an ultrametricspace (Qp). This is the main motivation for developing a general theory of parabolic-type pseudodifferential equations. For an in-depth discussion of the above matters thereader may consult Chapter 4.This chapter has two goals. The first is to review some basic results about N-
dimensional linear parabolic-type equations and their associated Markov processes.These equations are the main tool in the construction of many models of complexsystems, and they have been studied extensively, see e.g. [275], [434], [470] and thereferences therein. For further details the reader may consult [470] and the refer-ences therein. The second goal is to present some of the results from [469], with-out proofs. This work initiates the study of non-Archimedean reaction–ultradiffusionequations and their connections with models of complex hierarchical systems. Froma mathematical perspective, these equations are the p-adic counterpart of the integro-differential models for phase separation introduced by Bates and Chmaj, and gener-alizations of the ultradiffusion equations on trees studied in the 1980s by Ogielski,Stein, Bachas, and Huberman, among others, and also generalizations of the masterequations of the Avetisov, Kozyrev et al. models, which describe certain complex
114
8.2 OperatorsW, Parabolic-Type Equations, and Markov Processes 115
hierarchical systems. These equations are gradient flows of non-Archimedean free-energy functionals, and their solutions describe the macroscopic density profile of abistable material whose space of states has an ultrametric structure.
8.2 OperatorsW, Parabolic-Type Equations, and Markov Processes
8.2.1 A Class of Non-local Operators
TakeR+ := {x ∈ R; x ≥ 0}, and fix a functionw : QNp → R+ satisfying the following
properties:
(i) w(y) is a radial (i.e. w(y) = w(‖y‖p)), continuous, and increasing function of‖y‖p;
(ii) w(y) = 0 if and only if y = 0;(iii) there exist constants C0 > 0, M ∈ Z, and α1 > N such that
C0‖y‖α1p ≤ w(‖y‖p), for ‖y‖p ≥ pM.
Note that condition (iii) implies that∫
‖y‖p≥pM
dNy
w(‖y‖p) < ∞. (8.1)
In addition, since w(y) is a continuous function, (8.1) holds for any M ∈ Z.We define
(Wϕ)(x) = κ
∫
QNp
ϕ(x− y)− ϕ(x)
w(y)dNy, for ϕ ∈ D,
where κ is a positive constant. Then, for 1 ≤ ρ ≤ ∞,
D(
QNp
) → Lρ(
QNp
)
,
ϕ → Wϕ
is a well-defined linear operator. Furthermore, F[Wϕ](ξ ) = −κAw(ξ )F [ϕ](ξ ),where
Aw(ξ ) :=∫
QNp
1− χp(y · ξ )w(y)
dNy,
cf. [470, Lemma 4]. The function Aw(ξ ) has the following properties: (i) for ‖ξ‖p =p−γ = 0, with γ = ord(ξ ),
Aw(p−γ ) = (1− p−N )
∞∑
j=γ+2
pN j
w(pj )+ pNγ+N
w(pγ+1);
(ii) it is radial, positive, and continuous, and Aw(0) = 0, and (iii) Aw(p−ord(ξ ) ) is adecreasing function of ord(ξ ), cf. [470, Lemma 5].
116 Parabolic-Type Equations and Models of Complex Hierarchical Systems
By using standard ideas and results from semigroup theory, see e.g. [470,Proposition 7] and the references therein, one gets the following. (i) (Wϕ)(x) =−κF−1
ξ→x(Aw(‖ξ‖p)Fx→ξϕ) for ϕ ∈ D(QNp ), andWϕ ∈ C(QN
p ) ∩ Lρ (QNp ), for 1 ≤
ρ ≤ ∞. The operator W extends to an unbounded and densely defined operator inL2(QN
p ) with domain
Dom(W ) = {
ϕ ∈ L2;Aw(‖ξ‖p)Fϕ ∈ L2}
.
(ii) (−W ,Dom(W )) is a self-adjoint and positive operator. (iii) W is the infinites-imal generator of a contraction C0 semigroup (T (t ))t≥0. Moreover, the semigroup(T (t ))t≥0 is bounded holomorphic with angle π/2.In the study of the evolution equations attached to operators W it is completely
necessary to impose certain growth conditions on the function w(‖ξ‖p). Assume thatthere exist positive constants α1,α2,C0, C1, with α1 > N, α2 > N, and α3 ≥ 0, suchthat
C0‖ξ ′‖α1p ≤ w(‖ξ ′‖p) ≤ C1‖ξ ′‖α2p eα3‖ξ′‖p , for any ξ ′ ∈ QN
p . (8.2)
Then there exist positive constants C2, C3, such that
C2‖ξ‖α2−Np e−α3p‖ξ‖−1p ≤ Aw(‖ξ‖p) ≤ C3‖ξ‖α1−Np , for any ξ ∈ QN
p , (8.3)
with the convention that e−α3p‖0‖−1p := lim‖ξ‖p→0 e−α3p‖ξ‖
−1p = 0. Furthermore, if
α3 > 0, then α1 ≥ α2, and if α3 = 0, then α1 = α2, cf. [470, Lemma 8].Assuming hypothesis (8.2) and using estimation (8.3), one gets that
e−tκAw (‖ξ‖p) ∈ Lρ(
QNp
)
for 1 ≤ ρ < ∞ and t > 0, (8.4)
cf. [470, Lemma 10].
Definition 8.1 We say that W (or Aw) is of exponential type if inequality (8.2) ispossible only for α3 > 0 with α1,α2, C0, C1 positive constants and α1 > N, α2 > N.If (8.2) holds for α3 = 0 with α1,α2, C0, C1 positive constants and α1 > N, α2 > N,we say thatW (or Aw) is of polynomial type.
We note that, ifW is of polynomial type, then α1 = α2 > N andC0,C1 are positiveconstants with C1 ≥ C0.
8.2.2 p-Adic Description of Characteristic Relaxation in Complex Systems
In [56] Avetisov et al. developed a new approach to the description of relaxationprocesses in complex systems (such as glasses, macromolecules, and proteins) on thebasis of p-adic analysis. The dynamics of a complex system is described by a randomwalk in the space of configurational states, which is approximated by an ultrametricspace (Qp). Mathematically speaking, the time-evolution of the system is controlled
8.2 OperatorsW, Parabolic-Type Equations, and Markov Processes 117
by a master equation of the form
∂ f (x, t )
∂t=
∫
Qp
{v(x | y) f (y, t )− v(y | x) f (x, t )}dy, x ∈ Qp, t ∈ R+, (8.5)
where the function f (x, t ) : Qp × R+ → R+ is a probability density distribution, andthe function v(x | y) : Qp × Qp → R+ is the probability of transition from state yto state x per unit time. The transition from a state y to a state x can be perceivedas overcoming the energy barrier separating these states. In [56] an Arrhenius-typerelation was used:
v(x | y) ∼ A(T ) exp
(
−U (x | y)kBT
)
,
where U (x | y) is the height of the activation barrier for the transition from the statey to state x, kB is the Boltzmann constant and T is the temperature. This formulaestablishes a relation between the structure of the energy landscape U (x | y) and thetransition function v(x | y). The case v(x | y) = v(y | x) corresponds to a degenerateenergy landscape. In this case the master equation (8.5) takes the form
∂ f (x, t )
∂t=
∫
Qp
v(|x− y|p
) { f (y, t )− f (x, t )}dy,
where
v(|x− y|p
) = A(T )
|x− y|p exp{
−U(|x− y|p
)
kT
}
.
By choosing U conveniently, several energy landscapes can be obtained. Following[56], there are three basic landscapes: (i) logarithmic,
v(|x− y|p
) = 1
|x− y|p lnα(
1+ |x− y|p) , α > 1;
(ii) linear,
v(|x− y|p
) = 1
|x− y|α+1p
, α > 0;
(iii) exponential,
v(|x− y|p
) = e−α|x−y|p
|x− y|p , α > 0.
Thus, it is natural to study the following Cauchy problem:⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
∂u(x, t )
∂t= κ
∫
QNp
u(x− y, t )− u(x, t )
w(y)dNy, x ∈ QN
p , t ∈ R+
u(x, 0) = ϕ ∈ D(
QNp
)
,
(8.6)
118 Parabolic-Type Equations and Models of Complex Hierarchical Systems
where w(y) is a radial function belonging to a class of functions that containsfunctions like
(i) w(‖y‖p) = �(N )p (−α)‖y‖α+Np , here �(N )
p (·) is the N-dimensional p-adic gammafunction, and α > 0;
(ii) w(‖y‖p) = ‖y‖βpeα‖y‖p , α > 0.
By imposing condition (8.2) onw, we include the linear and exponential energy land-scapes in our study. On the other hand, takew(‖y‖p) satisfying (8.2) and take h(‖y‖p)a continuous and increasing function such that
0 < infy∈QN
p
h(‖y‖p) < supy∈QN
p
h(‖y‖p) < ∞.
Then h(‖y‖p)w(‖y‖p) satisfies (8.2). This fact shows that the class of operatorsW isvery large.We note that ‖y‖βp lnα (1+ ‖y‖p), β > N, α ∈ N, does not satisfies ‖y‖α1p ≤
‖y‖βp lnα (1+ ‖y‖p) for any y ∈ QNp . Hence the logarithmic landscapes are not
included in our discussion. In our terminology, the linear landscapes of Avetisov et al.correspond to operatorsW of polynomial type.
8.2.3 Heat Kernels
In this section we assume that the function w satisfies conditions (8.2). We define
Z(x, t;w, κ ) := Z(x, t ) =∫
QNp
e−κtAw (‖ξ‖p)χp(−x · ξ )dNξ for t > 0 and x ∈ QNp .
Note that by (8.4), Z(x, t ) = F−1ξ→x[e
−κtAw (‖ξ‖p)] ∈ C ∩ L2 for t > 0. We call such afunction a heat kernel. When considering Z(x, t ) as a function of x for t fixed we willwrite Zt (x). The following are the main properties of the heat kernels.
Theorem 8.2 ([470, Theorem 13]) The function Z(x, t ) has the following properties:
(i) Z(x, t ) ≥ 0 for any t > 0;(ii)
∫
QNpZ(x, t )dNx = 1 for any t > 0;
(iii) Zt (x) ∈ C(QNp ,R) ∩ L1(QN
p ) ∩ L2(QNp ) for any t > 0;
(iv) Zt (x) ∗ Zt ′ (x) = Zt+t ′ (x) for any t, t ′ > 0;(v) limt→0+ Z(x, t ) = δ(x) in D′(QN
p ), where δ denotes the Dirac distribution.
8.2.4 Markov Processes over QNp
We consider (QNp , ‖ · ‖p) as a complete non-Archimedean metric space and use the
terminology and results of the theory of Markov processes on metric spaces, see e.g.[143, Chapters 2, and 3]. Let B denote the Borel σ -algebra of QN
p . Thus (QNp ,B, dNx)
is a measure space.
8.2 OperatorsW, Parabolic-Type Equations, and Markov Processes 119
We set
p(t, x, y) := Z(x− y, t ) for t > 0, x, y ∈ QNp ,
and
P(t, x,B) ={
∫
B p(t, y, x)dNy for t > 0, x ∈ QN
p , B ∈ B1B(x) for t = 0.
Theorem 8.3 (cf. [470, Theorem 16]) Z(x, t ) is the transition density of a time- andspace-homogeneous Markov process which is bounded, right-continuous, and has nodiscontinuities other than jumps.
8.2.5 The Cauchy Problem
Consider the following Cauchy problem:
⎧
⎪
⎨
⎪
⎩
∂u
∂t(x, t )−Wu(x, t ) = g(x, t ), x ∈ QN
p , t ∈ [0,T ],T > 0,
u(x, 0) = u0(x), u0(x) ∈ Dom(W ).
(8.7)
We say that a function u(x, t ) is a solution of (8.7), if u(x, t ) belongs toC([0,T ),Dom(W )) ∩C1
(
[0,T ],L2(QNp ))
and if u(x, t ) satisfies equation (8.7) fort ∈ [0,T ]. By using standard results of the semigroup theory, see e.g. [102, Proposi-tion 4.1.6], one shows that
Theorem 8.4 ([470, Theorem 21]) Assume that u0 ∈ Dom(W ) and g ∈C(
[0,∞),L2(QNp )) ∩ L1((0,∞),Dom(W )). Then the Cauchy problem (8.7) has a
unique solution given by
u(x, t ) =∫
QNp
Z(x− ξ, t )u0(ξ )dNξ +
∫ t
0
∫
QNp
Z(x− ξ, t − θ )g(ξ, θ )dNξ dθ.
8.2.6 The Taibleson Operator and the p-Adic Heat Equation
We set
�(N )p (α) := 1− pα−N
1− p−α, for α ∈ R\{0}.
This function is called the p-adic gamma function. The function
kα (x) =||x||α−Np
�(N )p (α)
, α ∈ R\{0,N}, x ∈ QNp ,
120 Parabolic-Type Equations and Models of Complex Hierarchical Systems
is called the multidimensional Riesz kernel; it determines a distribution on D(QNp ) as
follows. If α = 0, N, and ϕ ∈ D(QNp ), then
(kα (x), ϕ(x)) = 1− p−N
1− pα−Nϕ(0)+ 1− p−α
1− pα−N
∫
||x||p>1||x||α−Np ϕ(x)dNx
+ 1− p−α
1− pα−N
∫
||x||p≤1||x||α−Np (ϕ(x)− ϕ(0))dNx. (8.8)
Then kα ∈ D′(QNp ), for R\{0,N}. In the case α = 0, by passing to the limit in (8.8),
we obtain
(k0(x), ϕ(x)) := limα→0
(kα (x), ϕ(x)) = ϕ(0),
i.e., k0(x) = δ(x), the Dirac delta function, and therefore kα ∈ D′(QNp ), for R\{N}.
It follows from (8.8) that, for α > 0,
(k−α (x), ϕ(x)) = 1− pα
1− p−α−N
∫
QNp
||x||−α−Np (ϕ(x)− ϕ(0))dNx. (8.9)
Definition 8.5 The Taibleson pseudodifferential operator DαT , α > 0, is defined as
(
DαTϕ
)
(x) = F−1ξ→x
(||ξ ||αpFx→ξϕ)
, for ϕ ∈ D(QNp ).
By using (8.9) and the fact that (Fk−α ) (x) equals ||x||αp, α = −N, in D′(QNp ), we
have
(
DαTϕ
)
(x) = (k−α ∗ ϕ)(x)
= 1− pα
1− p−α−N
∫
QNp
||y||−α−Np (ϕ(x− y)− ϕ(x))dNy. (8.10)
Then the Taibleson operator belongs to the class of operatorsW introduced before.The right-hand side of (8.10) makes sense for a wider class of functions, for example,for locally constant functions ϕ(x) satisfying
∫
||x||p≥1||x||−α−Np |ϕ(x)|dNx < ∞.
A similar observation is valid in general for operators ofW type. The equation
∂u(x, t )
∂t+ κ (Dα
T u)(x, t ) = 0, x ∈ QNp , t ≥ 0,
where κ is a positive constant, is a multidimensional analog of the p-adic heat equa-tion introduced in [434].
8.3 Elliptic Pseudodifferential Operators 121
8.3 Elliptic Pseudodifferential Operators, Parabolic-Type Equations andMarkov Processes
In this section we consider the following Cauchy problem:⎧
⎪
⎨
⎪
⎩
∂u(x, t )
∂t+ (A(∂, f, β )u)(x, t ) = 0, x ∈ QN
p , N ≥ 1, t > 0
u(x, 0) = ϕ(x),
(8.11)
where A(∂, f, β ) is an elliptic pseudodifferential operator of the form
(A(∂, f, β )φ)(x, t ) = F−1ξ→x
(|f(ξ )|βpFx→ξφ(x, t ))
.
Here β is a positive real number, and f(ξ ) ∈ Qp [ξ1, . . . , ξN] is a homogeneous poly-nomial of degree d satisfying the property f(ξ ) = 0 ⇔ ξ = 0. We establish the exis-tence of a solution to the Cauchy problem (8.11) in the case in which ϕ(x) is a contin-uous and an integrable function. Under these hypotheses we show the existence of asolution u(x, t ) that is continuous in x, for a fixed t ∈ [0,T ], bounded, and integrablefunction. In addition the solution can be presented in the form
u(x, t ) = Z(x, t ) ∗ ϕ(x),where Z(x, t ) is the fundamental solution (also called the heat kernel) to Cauchy’sproblem (8.11):
Z(x, t, f, β ) := Z(x, t ) = ∫
QNp
χp(−x · ξ )e−t|f(ξ )|βp dNξ, ξ ∈ QN
p , t > 0. (8.12)
The fundamental solution is a transition density of a Markov process with space stateQNp .
8.3.1 Elliptic Operators
Definition 8.6 Let f(ξ ) ∈ Qp [ξ1, . . . , ξN] be a non-constant polynomial. We say thatf (ξ ) is an elliptic polynomial of degree d, if it satisfies the criteria (i) f(ξ ) is a homo-geneous polynomial of degree d, and (ii) f(ξ ) = 0 ⇔ ξ = 0.
Any elliptic polynomial satisfies
C0‖ξ‖dp ≤ |f(ξ )|p ≤ C1‖ξ‖dp, for every ξ ∈ QNp , (8.13)
where C0 = C0(f),C1 = C1(f) are positive constants, cf. [470, Lemma 25].Throughout this section f(ξ ) will denote an elliptic polynomial of degree d. Now,
since c f (ξ ) is elliptic for any c ∈ Q×p when f(ξ ) is elliptic, we will assume that all
the elliptic polynomials have coefficients in Zp.
Definition 8.7 If f(ξ ) ∈ Zp[ξ ] is an elliptic polynomial of degree d, then we say that|f|βp is an elliptic symbol, and that A(∂, f, β ) is an elliptic pseudodifferential operatorof order d.
122 Parabolic-Type Equations and Models of Complex Hierarchical Systems
By [470, Lemma 24], the Taibleson operator is elliptic for p = 2. However, thereare elliptic symbols which are not radial functions. For instance,
∣
∣ξ 21 − pξ 22∣
∣
β
p=
[
max{|ξ1|2p , p−1 |ξ2|2p
}]β. Then, there are two different generalizations of the Taible-
son operator: the W operators which are pseudodifferential operators with radialsymbols, and the elliptic operators which include pseudodifferential operators withnon-radial symbols.
8.3.2 Some Properties of the Fundamental Solution
Theorem 8.8 ([470, Theorems 32, 34]) (i) For any x ∈ QNp and any t > 0,
|Z(x, t )| ≤ At(
‖x‖p + t1βd
)−dβ−N,
where A is a positive constant.(ii) Z(x, t ) ≥ 0 for every x ∈ QN
p and every t > 0.
We denote by Cb := Cb(
QNp ,R
)
the R-vector space of all functions ϕ : QNp → R
which are continuous and satisfy ‖ϕ‖L∞ = supx∈QNp|ϕ(x)| < ∞.
Proposition 8.9 ([470, Proposition 35]) The fundamental solution has the followingproperties:
(i)∫
QNpZ(x, t ) dNx = 1, for any t > 0;
(ii) if ϕ ∈Cb, then lim(x,t )→(x0,0)∫
QNpZ(x− y, t )ϕ(y)dNy = ϕ(x0);
(iii) Z(x, t + t ′) = ∫
QNpZ(x− y, t )Z(y, t ′)dNy, for t, t ′ > 0.
Theorem 8.10 ([470, Theorem 39]) Z(x, t ) is the transition density of a time- andspace-homogeneous Markov process which is bounded, right-continuous, and has nodiscontinuities other than jumps.
8.3.3 The Cauchy Problem
Theorem 8.11 ([470, Theorem 38]) If ϕ ∈ L1 ∩Cb, then the Cauchy problem⎧
⎪
⎨
⎪
⎩
∂u(x, t )
∂t+ (A(∂, f, β )u)(x, t ) = 0, x ∈ QN
p , t > 0
u(x, 0) = ϕ(x),
has a classical solution given by
u(x, t ) = ∫
QNpZ(x− y, t )ϕ(y)dNy.
Furthermore, the solution has the following properties:
(i) u(x, t ) is a continuous function in x, for every fixed t ≥ 0;(ii) sup(x,t )∈QN
p×[0,+∞) |u(x, t )| ≤ ‖ϕ‖L∞ ;(iii) u(x, t ) ∈ Lρ , 1 ≤ ρ ≤ ∞, for any fixed t > 0.
8.4 Non-Archimedean Reaction–Ultradiffusion Equations 123
8.4 Non-Archimedean Reaction–Ultradiffusion Equations andComplex Hierarchic Systems
In the middle of the 1980s the idea of using ultrametric spaces to describe thestates of complex biological systems, which naturally possess a hierarchical struc-ture, emerged in the works of H. Frauenfelder, G. Parisi, and D. Stein, among others,see e.g. [135], [164], [336], [376]. A central paradigm in the physics of complex sys-tems (for instance proteins) asserts that the dynamics of such systems can be modeledas a random walk in the energy landscape of the system, see e.g. [164], [294], andthe references therein. In protein physics, it is regarded as one of the most profoundideas put forward to explain the nature of the distinctive attributes of life. Typicallythese landscapes have a huge number of local minima. It is clear that a descriptionof the dynamics on such landscapes requires an adequate approximation. Interbasinkinetics offers an acceptable solution to this problem. Using this approach, an energylandscape is approximated by an ultrametric space (a disconnectivity graph which isa rooted tree) and a function on this space describing the distribution of the activa-tion barriers, see e.g. [70]. After that, a model of hierarchical dynamics based on theultrametric space is constructed, and by using the postulates of interbasin kinetics onegets that the transitions between basins are described by the following equations:
∂u(i, t )
∂t= −
∑
j
{T (i, j)u(i, t )− T ( j, i)u( j, t )}v( j), (8.14)
where the indices i, j number the states of the system (which correspond to localminima of energy), T (i, j) ≥ 0 is the probability per unit time of a transition from ito j, and the v( j) > 0 are the basin volumes. At this point it is relevant to mention thatequations of type (8.14) are a generalization of the ultradiffusion equations on treeswhich were studied intensively in the 1980s, see e.g. [63] and the references therein,and that these equations appeared in models of protein folding, see e.g. [474]. Thereader may also consult Chapter 4 for an in-depth discussion of the above matters.Around 2000, V. Avetisov et al. discovered, among several other things, that, under
suitable physical and mathematical hypotheses, the ultradiffusion equations on treesstudied by Ogielski, Stain, Bachas, and Huberman, among several others, see e.g.[63], have a “continuous p-adic limit.” This fact is a consequence of the existence ofp-adic parametrizations of the Parisi matrices. Avetisov et al. introduced a new classof models for complex hierarchical systems based on p-adic analysis, see [54]–[58].The p-adic limit of the master equation (8.14) has the form:
∂u(x, t )
∂t=
∫
QNp
J(‖x− y‖p)[u(y, t )− u(x, t )]dNy, (8.15)
x ∈ QNp , t ≥ 0. The function u(x, t ) : QN
p × R+ → R+ is a probability density dis-tribution, so that
∫
B u(x, t )dNx is the probability of finding the system in a domain
B ⊂ QNp at the instant t. The function J(‖x− y‖p) : QN
p × QNp → R+ is the probabil-
ity of the transition from state y to state x per unit time. It is known that, for many
124 Parabolic-Type Equations and Models of Complex Hierarchical Systems
values of J, equations of type (8.15) are ultradiffusion equations, i.e. they are p-adiccounterparts of the classical heat equations. More precisely, the fundamental solutionof (8.15) is the transition density of a bounded right-continuousMarkov process with-out discontinuities of the second kind, see e.g. [434], [275], [464], [468], [405], [406],[386]. The original models of Avetisov et al.were formulated in dimension one, moreprecisely, these models were constructed by using “exactly one” cross section of anenergy landscape. However, an argument given in [164, p. 98 and Figures 11.3 and11.4] by Frauenfelder et al. strongly suggests that the p-adic master equations shouldbe N-dimensional just like those of type (8.15).The terminology “reaction–diffusion equations” has been used in connection with
the models of Avetisov et al. to mean (linear) parabolic-type equations with variablecoefficients. A general theory for this type of equation is given in [470] and the refer-ences therein. Here “reaction–diffusion equations” means non-linear equations as inthe Euclidean case, see e.g. [160], [187], [397]. We use the term ultradiffusion insteadof diffusion due to the fact that in classical probability the term diffusion is used onlyin connection with stochastic processes with continuous paths, whereas in the p-adicsetting, the paths cannot be continuous. In this section we consider equations of thetype
∂u(x, t )
∂t=
∫
QNp
J(‖x− y‖p
)
[u(y, t )− u(x, t )]dNy− λ f (u(x, t )), (8.16)
where J(‖x‖p) ≥ 0,∫
QNpJ(‖x‖p)dNx = 1, λ > 0 is sufficiently large, and f is (for
instance) a polynomial having roots in −1, 0, 1. Formally, equation (8.16) is the L2-gradient flow of the following non-Archimedean Helmholtz free-energy functional:
E[ϕ] = 1
4
∫
QNp
∫
QNp
J(‖x− y‖p
) {ϕ(x)− ϕ(y)}2 dNxdNy+ λ
∫
QNp
W (ϕ(x))dNx,
(8.17)
where ϕ is a function taking values in the interval [−1, 1] and W is a double-wellpotential. Equations of type (8.17) can be well approximated in finite-dimensionalreal spaces by ordinary differential equations (ODEs). In a suitable basis, where theunknown function is identified with the column vector [u(i, t )]i∈Gn
M, these equations
have the form
∂
∂t[u(i, t )]i∈GN
M= −A(M)[u(i, t )]i∈GN
M− λ[ f (u(i, t ))]i∈GN
M, (8.18)
where A(M) is the matrix representation of a linear operator that approximates, ina suitable finite-dimensional vector space, the integral operator involving the func-tion J on the right-side of (8.16). Equation (8.18) is the L2-gradient flow of a “finite”Helmholtz energy functional. In Section 8.4.3, we present some results about the con-vergence of Helmholtz functionals. Equations of type (8.18) are generalizations of theultradiffusion equations on trees considered in [63]. The setGN
M is a finite ultrametricspace, and this class of spaces contains as particular cases the finite rooted trees. Thischapter is dedicated to studying the interplay between all of the above-mentioned
8.4 Non-Archimedean Reaction–Ultradiffusion Equations 125
objects and their physical significance. We determine the spaces and conditions forwhich the Cauchy problems for equations (8.16)–(8.18) are well-posed, see Theo-rems 8.23, 8.24. We show that equations (8.16)–(8.18) have stationary solutions with“arbitrary interfaces”, this means, in the case of equation (8.16), the following. Givena ball BNN0
(x0) of radius pN0 centered at x0, Qnp can be divided into three disjoint sets
M, BNN0(x0) �M, and QN
p � BNN0(x0). Equation (8.16) admits a stationary solution
u(x) satisfying α+ ≤ u(x) ≤ 1 for x ∈ M,−1 ≤ u(x) ≤ α− for x ∈ BNN0(x0) �M, and
lim‖x‖p→∞ u(x) = 0, for some suitable constants α+, α−, see Theorems 8.20, 8.22.We also show that the solution of the Cauchy problem attached to (8.18) convergesto the solution of the Cauchy problem attached to (8.16) in the case in which theinitial condition for equation (8.18) is a continuous function taking values in theinterval [−1, 1], see Theorem 8.26. Matrix A(M) in equation (8.18) is the Q-matrixof a finite homogeneous Markov chain with state space GN
M , and equation (8.18)with f = 0 is the Kolmogorov backward equation attached to this Markov chain, seeTheorem 8.16.From a physical perspective equations (8.16)–(8.18) model phase separation of
bistable materials whose spaces of states have an ultrametric structure. Our modelsare the p-adic counterparts of the integro-differential models for phase separation dueto Bates and Chmaj, see [66], [67], and [1]–[2]. The function u(x, t ), or [u(i, t )]i∈GN
M,
the order parameter, represents the macroscopic density profile of a material whichhas two equilibrium states u(x, t ) ≡ −1, u(x, t ) ≡ 1, and−1 < u(x, t ) < 1 representsthe “interface.” and equations (8.16)–(8.18) model a transition between the equilib-rium phases. Theorems 8.20 and 8.22 show that our models of a bistable system candevelop arbitrary stable interfaces.
8.4.1 Basic Function Spaces and Operators
We define X∞(QNp ) := X∞ = (
DR(QNp ), ‖ · ‖∞
)
, where the overbar means the com-pletion with respect the metric induced by ‖ · ‖∞ and ‖φ‖∞ = supx∈QN
p|φ(x)|. We
also use ‖ · ‖∞ to denote the extension of ‖ · ‖∞ to X∞. Notice that all the functionsin X∞ are continuous and that
X∞ ⊂ C0 :=({
f : QNp → R; f continuous with lim
‖x‖p→∞f (x) = 0
}
, ‖ · ‖∞)
.
On the other hand, since DR(QNp ) is dense in C0, cf. [402, Chapter II, Proposition
1.3], we conclude that X∞ = C0. In a more general case, ifU is an open subset of Qnp,
we define X∞(U ) = (DR(U ), ‖ · ‖∞).We set
XM := (
D−MM
(
QNp
)
, ‖ · ‖∞)
for M ≥ 1.
Any ϕ ∈ XM has support in BNM = (p−MZp)N , and ϕ satisfies ϕ(x+ x′) = ϕ(x) forx′ ∈ BN−M = (pMZp)N . In addition, BN±M are additive subgroups and GN
M := BNM/BN−M
is a finite group with #GNM := p2MN elements. Any element i = (i1, . . . , iN ) of GN
M
126 Parabolic-Type Equations and Models of Complex Hierarchical Systems
can be represented as
i j = a j−Mp−M + a j−M+1p
−M+1 + · · · + a j0 + a j1p+ · · · + a jM−1pM−1 (8.19)
for j = 1, . . . ,N, with ajk ∈ {0, 1, . . . , p− 1}. From now on, we fix a set of represen-tatives in QN
p for GNM of form (8.19). We denote by(pL‖x− x0‖p) the characteristic
function of the ball x0 + (pLZp)N . We notice that any non-zero function ϕ in XM hasan index of local constancy
lϕ ∈ {−M,−M + 1, . . . , 0, 1, . . . ,M},and that BNlϕcan be covered by a finite disjoint union of balls of the form BN−M ( j),with j ∈ GN
M . Hence {(pM‖x− i‖p)}i∈GNMis a basis of D−M
M , see also e.g. [18,Lemma 4.3.1]. If ϕ(x) = ∑
i∈GNMϕ(i)(pM‖x− i‖p), with ϕ(i) ∈ R, then ‖ϕ‖∞ =
maxi |ϕi|. Hence XM is isomorphic as a Banach space to (R#GNM , ‖ · ‖R), where
‖(t1, . . . , t#GNM)‖R = max1≤ j≤#GN
M|t j|.
We now define, for M ≥ 1, PM : X∞ → XM as
PMϕ(x) =∑
i∈GNM
ϕ(i)(
pM‖x− i‖p)
.
Therefore PM is a linear bounded operator. Indeed, ‖PM‖ ≤ 1, and
limM→∞
‖ϕ − PMϕ‖∞ = 0
for any ϕ ∈ X∞, cf. [469, Lemma 1].We denote by EM , M ≥ 1, the embedding XM → X∞. The following result is a
consequence of the above observations. If Z, Y are real Banach spaces, we denote byB(Z,Y ) the space of all linear bounded operators from Z into Y .
Lemma 8.12 (Condition A) With the above notation, the following assertions hold:
(i) X∞, XM for M ≥ 1, are real Banach spaces, all with the norm ‖ · ‖∞;(ii) PM ∈ B (X∞,XM ) and ‖PMϕ‖∞ ≤ ‖ϕ‖∞ for any M ≥ 1, ϕ ∈ X∞;(iii) EM ∈ B (XM,X∞) and ‖EMϕ‖∞ = ‖ϕ‖∞ for any M ≥ 1, ϕ ∈ XN;(iv) PMEMϕ = ϕ for M ≥ 1, ϕ ∈ XM.
The Operators AM , A Set R+ := {x ∈ R; x ≥ 0}. We fix a continuous function J :R+ → R+, and take J(x) = J(||x||p) for x ∈ QN
p , then J(x) is a radial function onQNp . In addition, we assume that
∫
QNpJ(||x||p)dNx = 1.
We define, for M ≥ 1,
AM : XM → XMφ(x) → − ∫
BNM
JM (||x− y||p){φ(y)− φ(x)}dNy,
and
A : X∞ → X∞ϕ(x) → − {
J(‖x‖p
) ∗ ϕ(x)− ϕ(x)}
.
8.4 Non-Archimedean Reaction–Ultradiffusion Equations 127
The operator A : X∞ → X∞ is linear and bounded. In addition, the spectrum of A,σ (A), is contained in the interval [0, 2], cf. [469, Lemmata 3 and 4].
8.4.2 The Matrix Representation of Operators AM and Markov Chains
By using the basis {(pM‖x− i‖p)}i∈GNMwe can identifyXM with (R#GN
M , ‖ · ‖R). Thusthe operator AM is given by a matrix. This matrix is computed by means of the fol-lowing two lemmata.
Lemma 8.13 ([469, Lemma 13]) We set a(x, i) := JM (‖x‖p)∗(pM‖x− i‖p) for x ∈BNM, i ∈ GN
M. Let x denote the image of x under the canonical map BNM → GN
M. Then
a(x, i) = a(x, i) =
⎧
⎪
⎪
⎨
⎪
⎪
⎩
p−MNJ(
p−ord(x−i)) if ord(x− i) = +∞∫
(pMZp)NJ(‖y‖p
)
dNy if ord(x− i) = +∞.
Remark 8.14 Notice that a(x, i) = a(i, x) = a(‖x− i‖p), where a(‖x− i‖p) meansthat there exists a function g: R+→ R such that a(i, x) = g(‖x− i‖p), i.e. a(i, x) isa radial function of x− i.
Lemma 8.15 ([469, Lemmata 15 and 16]) (i) The matrix for the operator AM actingon XM is A(M) = [
A(M)ki
]
k,i∈GNM= [ jNδki − aki]k,i∈GN
M, where aki := a(k, i) and δki
denotes the Kronecker delta.(ii) −A(M) is a Q-matrix, i.e. −A(M)
i j ≥ 0 for i = j with i, j ∈ GNM, and A(M)
ii =−∑
j =i A(M)i j .
A real matrix A is called non-negative if each of its entries is greater than or equalto zero. In this case we use the notation A ≥ 0. Similarly, we say that a real matrix isnon-positive if each of its entries is less than or equal to zero. In this case we use thenotation A ≤ 0. We denote by E the identity matrix and by 1 the unit vector, namelya vector having all its entries equal to one.
Theorem 8.16 ([469, Theorem 17]) (i) Set P(M)(t ) := e−tA(M), t ≥ 0. Then P(M)(t ) is
a semigroup of non-negative matrices with P(M)(0) = E, the identity matrix,which satisfies
∂P(M)(t )
∂t+ A(M)P(M)(t ) = 0
and P(M)(t )1 = 1 for t ≥ 0.(ii) The function P(M)(t − s), t ≥ s ≥ 0, is the transition function of a homogeneous
Markov chain with state space GNM. Furthermore, this stochastic process has
right-continuous piecewise-constant paths.
128 Parabolic-Type Equations and Models of Complex Hierarchical Systems
8.4.3 Non-Archimedean Helmholtz Free-Energy Functionals
We define for ϕ ∈ XM , λ > 0,
EM (ϕ) = 1
4
∫
BNM
∫
BNM
JM(‖x− y‖p
) {ϕ(x)− ϕ(y)}2dNx dNy+ λ
∫
BNM
W (ϕ(x))dNx,
(8.20)
where JM(‖x‖p
)
is as before, ϕ is a scalar density function defined on BNM that takesvalues in [−1, 1], andW : R → R, with derivative f ∈ C2(R), is a double-well poten-tial having (not necessarily equal) minima at ±1. The functional EM (ϕ) is a non-Archimedean version of a non-local Helmholtz free-energy functional. The func-tion ϕ, the order parameter, represents the macroscopic density profile of a systemwhich has two equilibrium pure phases described by the profiles ϕ ≡ 1 and ϕ ≡ −1,and −1 < ϕ < 1 represents the “interface.” The function JN is a positive, possi-bly anisotropic, interaction potential which vanishes at infinity. If ϕ is an energy-minimizing configuration, the second term in EM forces the minimizer ϕ to take val-ues close to the pure states, while the first term in EM represents an interaction energywhich penalizes the spatial inhomogeneity of ϕ.
In the classical Archimedean setting (i.e. RN), the L2-gradient of functionals oftype (8.20) leads to the non-local versions of Allen–Cahn equations, see [1], [2],[66] The next result shows that a similar situation occurs in the non-Archimedeansetting.
Lemma 8.17 ([469, Lemma 8]) (i) By identifying ϕ(x)with the vector [ϕ(i)]i∈GNM, i.e.
by identifying XM with R#GNM , we have
EM(
[ϕ(i)]i∈GNM
)
= jM p−MN
2
∑
i∈GNNM
ϕ2(i)− p−MN
2
∑
i, j∈GNM
ai jϕ(i)ϕ( j)
+ λp−MN∑
i∈GNM
W (ϕ(i)),
where, with jM := ∫
BNMJ(||y||p)dNy, [ai j]i, j∈GN is the matrix defined in Lemma
8.15.(ii) We assume that ϕ depends on i ∈ GN
M and t ≥ 0. The gradient flow in theEuclidean space R#GN
M of the functional EM : R#GNM → R is the evolution in R#GN
M
given by
∂
∂t[ϕ(i, t )]i∈GN
M= −∇EM
(
[ϕ(i, t )]i∈GNM
)
= −p−MNA(M)[ϕ(i, t )]i∈GNM− λp−MN[ f (ϕ(i, t ))]i∈GN
M, (8.21)
where A(M) is the matrix defined in Lemma 8.15.
8.4 Non-Archimedean Reaction–Ultradiffusion Equations 129
Remark 8.18 Notice that, in XM, (8.21) can be written as
∂
∂tϕ(x, t ) = −AMϕ(x, t )− λ f (ϕ(x, t )). (8.22)
Consider (GNM, ‖ · ‖p) as a finite ultrametric space. Then (8.21) is the reaction–
ultradiffusion equation in (GNM, ‖ · ‖p), which is the L2-gradient of an energy func-
tional defined on (GNM, ‖ · ‖p). These equations are generalizations of the ultradif-
fusion equations studied in [366], [63]. The limit M → ∞ of some ultradiffusionequations of type (8.21) with f ≡ 0 was considered by Avetisov et al. in [53] for thecase when the matrix A(M) comes from a Parisi-type matrix. More precisely, in thespecial case f ≡ 0 in [53] it was established, by a physical argument involvingthe parametrization of Parisi matrices by p-adic numbers, that the “limit” of an equa-tion of type (8.21) as M tends to infinity is
∂
∂tϕ(x, t ) = −Aϕ(x, t )− λ f (ϕ(x, t )), x ∈ QN
p , t ≥ 0. (8.23)
In [469] it was established that the solutions of the Cauchy problem attached to equa-tion (8.22) converge to the solutions of the Cauchy problem attached to equation(8.23), see Theorem 8.26, in the case that f ∈ C2 with three zeros at −1, 0, 1. Equa-tion (8.23) is formally the L2-gradient of the following energy functional:
E(ϕ) = 1
4
∫
QNp
∫
QNp
J(‖x− y‖p
) {ϕ(x)− ϕ(y)}2dNx dNy+ λ
∫
QNp
W (ϕ(x))dNx
where ϕ is a scalar density function defined on QNp that takes values in [−1, 1], and
W is a double-well potential having minima at ±1 as before.
8.4.4 Stationary Solutions
We take J(x) = J(||x||p) for x ∈ QNp as in Section 8.4.1. We fix a function f : R → R
having the following properties:
f ∈ C2(R); (H1)
f has exactly three zeros at − 1, 0, 1; (H2)
f ′(−1) > 0, f ′(0) < 0, f ′(1) > 0; (H3)
and (H4) the function g(u) := u+ λ f (u) has three zeros and exactly three intervalsof monotonicity for any sufficiently large λ > 0. We denote by u−λ , u
+λ the extreme
roots of g(u) = 0, then u−λ < u+λ , u−λ > −1, u+λ < 1, and u±λ → ±1 as λ → ∞.
The following technical conditions always hold under hypotheses (H1)–(H4) ontaking λ sufficiently large.(C5) We take α− ∈ (−1, 0), α+ ∈ (0, 1) such that f ′(u) ≥ δ > 0 for u in
[−1, α−] ∪ [α+, 1]. In addition, we assume that α+, α− satisfy u−λ < α− < 0 <α+ < u+λ ;
(1+ α+)+ λ f (α+) ≤ 0; (C6)
α− + λ f (α−) ≥ 0. (C7)
130 Parabolic-Type Equations and Models of Complex Hierarchical Systems
Notice that conditions (C6) and (C7) hold if
λ ≥ max
{ −α−
f (α−),1+ α+
− f (α+)
}
.
Remark 8.19 Notice that polynomial u3 − u satisfies hypotheses (H1)–(H4). Iff (u) ∈ C2(R) and f ′′(u) has exactly a zero β ∈ (−1, 1) and f ′′(u) is positive to theright of β and negative to the left of β, then g(u) = u+ λ f (u) satisfies hypothesis(H4).
Theorem 8.20 ([469, Theorem 2]) Assume that f satisfies hypotheses (H1)–(H4).Then, for any measurable subset M ⊂ BNN0
and for λ sufficiently large, the equation
{
u ∈ X∞Au(x)+ λ f (u(x)) = 0, x ∈ QN
p ,(8.24)
has a unique solution u satisfying
α+ ≤ u(x) ≤ 1 for x ∈ M and − 1 ≤ u(x) ≤ α− for x ∈ BNN0�M. (8.25)
Remark 8.21 In Theorem 8.20, ball BNN0can be replaced by a compact subset.
However, for the sake of simplicity we use a ball centered at the origin.
Theorem 8.22 ([469, Theorem 3]) Fix N0 ≥ 1 and assume that supp J(‖x‖p) ⊂ BNN0,
and that f satisfies hypotheses (H1)–(H4). Then, for any open and compact subset Mcontained in BNN0
and for λ sufficiently large, the equation
{
u ∈ XN0
AN0u(x)+ λ f (u(x)) = 0, x ∈ BNN0,
(8.26)
has a unique solution u satisfying
α+ ≤ u(x) ≤ 1 for x ∈ M and − 1 ≤ u(x) ≤ α− for x ∈ BNN0�M. (8.27)
8.4.5 The Cauchy Problem
Set u(x, t ) := u(x)+ εe−βt , u(x, t ) := u(x)− εe−βt , where u(x) is the function givenin Theorem 8.20, ε, β > 0, and x ∈ QN
p , t ≥ 0.
Theorem 8.23 ([469, Theorem 5]) We consider the Cauchy problem⎧
⎨
⎩
u(x, t ) ∈ C (
[0,T ],X∞(
QNp
)) ∩C1(
[0,T ],X∞(
QNp
))
, T > 0∂u(x, t )/∂t = J
(‖x‖p) ∗ u(x, t )− u(x, t )− λ f (u(x, t )), t ∈ [0,T ]
u(x, 0) = u0(x),(8.28)
with u0(x) ∈ X∞(
QNp
)
satisfying u(x)− ε ≤ u0(x) ≤ u(x)+ ε, with λ, u(x) as inTheorem 8.20, and with ε sufficiently small. Then the initial-value problem (8.28)has a unique solution satisfying u(x, t ) ≤ u(x, t ) ≤ u(x, t ) for (x, t ) ∈ QN
p × [0,∞).In addition, u(x, t ) then satisfies limt→∞ ‖u(x, t )− u(x)‖∞ = 0.
8.4 Non-Archimedean Reaction–Ultradiffusion Equations 131
By using the same reasoning we obtain the following finite-dimensional version ofTheorem 8.23.
Theorem 8.24 ([469, Theorem 6]) We consider the Cauchy problem⎧
⎨
⎩
u(x, t ) ∈ C ([0,T ],XM ) ∩C1 ([0,T ],XM ) , T > 0∂u(x, t )/∂t = JM
(‖x‖p) ∗ u(x, t )− u(x, t )− λ f (u(x, t )), x ∈ BNM, t ∈ [0,T ]
u(x, 0) = u0(x),(8.29)
with M sufficiently large, u0(x) ∈ XM satisfying u(x)− ε ≤ u0(x) ≤ u(x)+ ε, withλ, u(x) as in Theorem 8.22, and with ε sufficiently small. Then the initial-valueproblem (8.29) has a unique solution satisfying u(x) ≤ u(x, t ) ≤ u(x) for (x, t ) ∈BNM × [0,∞). In addition, u(x, t ) satisfies limt→∞ ‖u(x, t )− u(x)‖∞ = 0.
Remark 8.25 If in Theorem 8.23 the hypotheses on the initial conditions arechanged to read “with u0(x) ∈ X∞
(
Qnp
)
satisfying−1 ≤ u0(x) ≤ 1,” then there existsa unique solution satisfying −1 ≤ u(x, t ) ≤ 1 for x ∈ QN
p and t ≥ 0. A similar resultis obtained if in Theorem 8.24 the hypotheses on the initial conditions are changedto read “with M sufficiently large, u0(x) ∈ XM satisfying −1 ≤ u0(x) ≤ 1.”
8.4.6 Finite Approximations
In this section we study finite approximations to the solutions of{
∂u(x, t )/∂t + Au(x, t ) = −λ f (u(x, t )), x ∈ QNp , t ≥ 0
u(x, 0) = u0(x),(8.30)
where the function f (u) satisfies all of the conditions given in Section 8.4.4. Our goalis to approximate the solution u(x, t ) of the Cauchy problem (8.30) in X∞ using onlythat u0(x) ∈ X∞ and −1 ≤ u0(x) ≤ 1. The techniques for constructing such approxi-mations are well known; here we use [338, Section 5.4]. It is possible to approximateu(x, t ) without using any a priori information on the initial solution; however, thisrequires one to impose that the non-linearity f be globally Lipschitz, and this con-dition greatly reduces the range of types of potential W to which we can apply ourresults.The discretization of the Cauchy problem (8.30) in the spaces XM takes the follow-
ing form:{
duM (t )/dt + AMuM (t ) = −λPM f (EMuM (t ))uM (0) = PMu0.
(8.31)
By taking PMu0(x) =∑
i∈GNMu0(i)(pN‖x− i‖p) and identifying uM (t ) with the col-
umn vector [uM (i, t )]i∈GNM, we can rewrite the Cauchy problem (8.31) as
{
d[uM (i, t )]i∈GNMdt + A(M) [uM (i, t )]I∈GN
M= −λ [ f (uM (i, t ))]i∈GN
M
[uM (i, 0)]i∈GNM= [u0(i)]i∈GN
M,
(8.32)
cf. Lemma 8.15.
132 Parabolic-Type Equations and Models of Complex Hierarchical Systems
Theorem 8.26 ([469, Theorem 7]) (i) −A is the generator of a strongly continuoussemigroup {e−tA}t≥0 on X∞. Moreover, ‖e−tA‖ ≤ 1 for t ≥ 0 and
limM→∞
supt≥0
ebt∥
∥EMe−AMtPNϕ − e−tAϕ
∥
∥
∞ = 0 for all ϕ ∈ X∞, b ∈ (0,∞).
(ii) Take u0(x) ∈ X with−1 ≤ u0(x) ≤ 1. Let u be the solution of (8.30) and let uNbe the solution of (8.31). Then
limM→∞
sup0≤t≤T
‖EMuM (t )− u(t )‖∞ = 0.
9
Stochastic Heat Equation Driven by Gaussian Noise
9.1 Introduction
In this chapter we study a new class of stochastic pseudodifferential equations inR+ × QN
p , driven by a spatially homogeneous Gaussian noise. More precisely, weconsider pseudodifferential equations of the type
Lu(t, x) = σ (u(t, x))·W (t, x)+ b(u(t, x) = 0, t ≥ 0, x ∈ QN
p ,
where L = ∂/∂t + A(∂, a, β ), β > 0, with A(∂, β ) := A(∂, β ) a pseudodifferentialoperator of the form Fx→ξ (A(∂, β )ϕ) = |a(ξ )|βpFx→ξ (ϕ) and a(ξ ) an elliptic poly-nomial. The coefficients σ and b are real-valued functions and
·W (t, x) is the formal
notation for a Gaussian random perturbation defined on some probability space. Weassume that it is white in time and with a homogeneous spatial correlation given by afunction f , see Section 9.4.2. Themain result, see Theorem 9.22, asserts the existenceand uniqueness of mild random-field solutions for these equations. The equationsstudied here are the non-Archimedean counterparts of the Archimedean stochasticheat equations studied for instance in [114], [116], and [441].The pseudodifferential equations of the form
∂u(t, x)
∂t+ A(∂, β )u(t, x) = 0
are the p-adic counterparts of the Archimedean heat equations. Indeed, the funda-mental solutions of these equations (i.e. the heat kernels) are transition density func-tions of Markov processes on QN
p , see Section 9.2.2. The one-dimensional p-adicheat equation was introduced in [434, Section XVI]. Ever since then the theoryof such equations has been steadily developing, see Chapter 8 and the referencestherein.Stochastic equations over p-adics have been studied intensively by many authors,
see e.g. [81], [82], [147], [212], [209], [210], [275], [236], [228]. The p-adic Gaus-sian noise and the corresponding stochastic integrals were studied in [82], [147],[209], [210], [228]. All these articles consider processes and stochastic integrals
133
134 Stochastic Heat Equation Driven by Gaussian Noise
depending on p-adic variables. Here, we introduced a non-Archimedean, spatiallyhomogeneous Gaussian noise parametrized by a non-negative real variable, the timevariable, and by a p-adic vector, the position variable. On the other hand, in [270] and[275] Kochubei introduced stochastic integrals with respect to the “p-adic Brownianmotion” generated by the one-dimensional heat equation. This is a non-Gaussianprocess parametrized by a non-negative real variable and by a p-adic variable.The chapter is organized as follows. In Section 9.2, we review some aspects of the
parabolic-type pseudodifferential equations needed for other sections. In Section 9.3,we prove a p-adic version of the Bochner–Schwartz theorem, see Theorem 9.4. InSection 9.4, we review the stochastic integration with respect to Hilbert-space-valuedWiener processes, and introduce the Gaussian noiseW and its associated cylindricalprocess, see Proposition 9.11. We also give some results about the spectral measureof W , see Theorem 9.15. Finally, we give a result, Proposition 9.16, which givesus examples of random distributions that can be integrated with respect to W . It isinteresting to note that the proof of Proposition 9.16 is much more involved than thecorresponding result in the Archimedean setting, see e.g. the proof of Proposition 3.3in [115]. This is due to the fact that, in the p-adic setting, the smoothing of a processrequires “cutting” and convolution operations, whereas in the Archimedean setting,it requires only a convolution operation. In Section 9.5, we prove the main result, seeTheorem 9.22. Just as in [114] we prove Theorem 9.22 under “Hypotheses A and B,”here we give explicit and sufficient conditions to satisfy these hypotheses in terms ofthe spectral measure ofW , see Theorem 9.15 and Lemma 9.21.
9.2 p-Adic Parabolic-Type Pseudodifferential Equations
We use the following standard notation:
(i) C(I,X ) denotes the space of continuous functions u on a time interval I withvalues in X ;
(ii) C1(I,X ) denotes the space of continuously differentiable functions u on a timeinterval I such that u′ ∈ X ;
(iii) L1(I,X ) denotes the space of measurable functions u on I with values in X suchthat ‖u‖ is integrable;
(iv) W 1,1(I,X ) denotes the space of measurable functions u on I with values in Xsuch that u′ ∈ L1(I,X ).
9.2.1 Elliptic Pseudodifferential Operators
Let a(ξ ) ∈ Qp[ξ1, . . . , ξN] be an elliptic polynomial. We have that a(ξ ) satisfies
C0‖ξ‖dp ≤ |a(ξ )|p ≤ C1‖ξ‖dp, for every ξ ∈ QNp , (9.1)
for some positive constantsC0 = C0(a),C1 = C1(a), see Section 8.3. Without loss ofgenerality we will assume that a(ξ ) ∈ Zp[ξ1, . . . , ξN].
9.2 p-Adic Parabolic-Type Pseudodifferential Equations 135
Lemma 9.1 With the above notation the following assertions hold.
(i) D → C(
QNp ,C
) ∩ L2(
QNp , d
Nx)
φ → A(∂, β )φ;(ii) the closure of the operator A(∂, β ), β > 0 (let us denote it by A(∂, β ) again)
with domain
Dom(A(∂, β )) := Dom(A) = {
f ∈ L2 : |a|βpf ∈ L2}
(9.2)
is a self-adjoint operator;(iii) −A(∂, β ) is the infinitesimal generator of a contraction C0 semigroup
(T (t ))t≥0;(iv) we can set
�(t, x) := F−1ξ→x
(
e−t|a(ξ )|βp)
for t > 0, x ∈ QNp . (9.3)
Then
(T (t ) f )(x) ={
(�(t, ·) ∗ f )(x) for t > 0
f (x) for t = 0,
for f ∈ L2.
Proof The results follow from the properties of the heat kernels given in [470,Section 2.3] by using well-known techniques of semigroup theory, see e.g. [102].Alternatively, the reader may consult [405, Lemma 3.21, Lemma 3.23, Lemma 7.4,Theorem 7.5], or [470, Chapter 4] for the same results in a more general setting. �
9.2.2 p-Adic Heat Equations
Consider the following Cauchy problem:{
∂u(t, x)/∂t + A(∂, β )u(t, x) = f (t, x), x ∈ QNp , t ∈ [0,T ]
u(0, x) = u0(x) ∈ Dom(A).(9.4)
We say that a function u(x, t ) is a solution of (9.4) if u ∈ C([0,T ],Dom(A)) ∩C1([0,T ],L2) and u satisfies equation (9.4) for all t ∈ [0,T ].
Theorem 9.2 Let β > 0 and let f ∈ C([0,T ],L2). Assume that at least one of thefollowing conditions is satisfied:
(i) f ∈ L1((0,T ),Dom(A));(ii) f ∈W 1,1((0,T ),L2).
Then the Cauchy problem (9.4) has a unique solution given by
u(t, x) =∫
QNp
�(t, x− y)u0(y)dNy+
∫ t
0
{
∫
QNp
�(t − τ, x− y) f (τ, y)dNy
}
dτ,
where � is defined in (9.3).
136 Stochastic Heat Equation Driven by Gaussian Noise
Proof The result follows fromLemma 9.1 by application of somewell-known resultsin semigroup theory, see e.g. [102]. Alternatively, the reader may consult [405, The-orem 7.9] or [470, Chapter 4] for the same result in a more general setting. �
Theorem 9.3 The heat kernel (or fundamental solution of (9.4)) �(t, x), t > 0, sat-isfies the following:
(i) �(t, x) ≥ 0 for any t > 0;(ii)
∫
QNp�(t, x)dNx = 1 for any t > 0;
(iii) �(t, ·) ∈ L1(
QNp
)
for any t > 0;(iv) (�(t, ·) ∗ �(t ′, ·))(x) = �(t + t ′, x) for any t, t ′ > 0;(v) limt→0+ �(t, x) = δ(x) in D′;(vi) �(t, x) ≤ At
(
t1dβ + ‖x‖p
)−dβ−Nfor any x ∈ QN
p and t > 0;(vii) �(x, t ) is the transition density of a time- and space-homogeneous Markov pro-
cess which is bounded, right-continuous, and has no discontinuities other thanjumps.
Proof See Theorem 1, Theorem 2, Proposition 2, and Theorem 4 in [464] or [470,Section 2.3]. �
9.3 Positive-Definite Distributions and the Bochner–Schwartz Theorem
In this section, we establish a p-adic version of the Bochner–Schwartz theoremon positive-definite distributions following to Gel’fand and Vilenkin [177,Chapter II].
9.3.1 The p-Adic Bochner–Schwartz Theorem
Throughout this section we work with complex-valued test functions. A distributionF ∈ D′(QN
p
)
is called positive if (F, ϕ) ≥ 0 for every positive test function ϕ, i.e. ifϕ(x) ≥ 0 for every x. In this case we will use the notation F ≥ 0. We say that F ismultiplicatively positive if (F, ϕϕ) ≥ 0 for every test function ϕ, where ϕ denotes thecomplex conjugate of ϕ. A distribution F is positive-definite if, for every test functionϕ, the inequality (F, ϕ ∗ ϕ) ≥ 0 holds, where ϕ(x) = ϕ(−x).
Theorem 9.4 (p-adic Bochner–Schwartz theorem) Every positive-definite distribu-tion F on QN
p is the Fourier transform of a regular Borel measure μ on QNp , i.e.
(F, ϕ) =∫
QNp
ϕ(ξ )dμ(ξ ) for ϕ ∈ D(
QNp
)
.
Conversely, the Fourier transform of any regular Borel measure gives rise to apositive-definite distribution on QN
p .
9.3 Positive-Definite Distributions 137
Proof (⇒) By the Riesz–Markov–Kakutani theorem every positive distribution F onQNp has the form
(F, φ) =∫
QNp
φ(ξ )dμ(ξ ) for φ ∈ D(
QNp
)
,
where μ is a regular Borel measure. Conversely, every regular Borel measure μ
defines a positive linear functional on D(
QNp
)
. On the other hand, since F is a mul-tiplicatively positive distribution if and only if F is a positive distribution, we canreplace positive by multiplicatively positive in the above assertion. We now note thatthe Fourier transform carries positive-definite distributions into multiplicatively pos-itive distributions, and every multiplicatively positive distribution can be obtained inthis manner. Indeed,
(F, ϕϕ) = (F, ϕ ∗ ϕ)
= (F,ϕ ∗ ϕ) = (F (ξ ), (ϕ ∗ ϕ)(−ξ ))
= (F, ϕ ∗ ϕ),
since (ϕ ∗ ϕ)(−ξ )= (ϕ ∗ ϕ)(ξ ). Now, let F ∈ D′(QNp
)
be a multiplicatively positive
distribution, i.e. (F, ψψ ) ≥ 0 for everyψ ∈ D(
QNp
)
. Then, there exist a distribution T
and a test function φ satisfyingT = F andψ = φ, because the Fourier transform is anisomorphism on D′(QN
p
)
and on D(
QNp
)
. From this observation we have (F, ψψ ) =(T, ϕ ∗ ϕ) ≥ 0.(⇐) It follows from this calculation:
∫
QNp
F ((ϕ ∗ ϕ))(ξ )dμ(ξ ) =∫
QNp
(F−1ϕ)(ξ )(F−1ϕ)(ξ )dμ(ξ )
=∫
QNp
|(F−1ϕ)(ξ )|2dμ(ξ ) ≥ 0. �
9.3.2 Positive-Definite Functions
We recall that a continuous function g: QNp → C is positive-definite, if for any
p-adic numbers x1, . . . , xm and any complex numbers σ1, . . . , σm, it holds that∑
j
∑
i g(x j − xi)σ jσ i ≥ 0. Such a function g satisfies the following: g is positive-definite, g(−x) = g(x), g(0) ≥ 0, and |g(x)| ≤ g(0). We associate with g the distri-bution
∫
QNpg(x)ϕ(x)dNx, while Gel’fand and Vilenkin attach to g the distribution
∫
QNpg(x)ϕ(x)dNx. For this reason our definition of a positive-definite distribution is
slightly different from, but equivalent to, the one given in [177, Chapter II]. Finally,we recall that g satisfies (g, ϕ ∗ ϕ) ≥ 0 for any test function ϕ, i.e. g generates apositive-definite distribution, see e.g. [79, Proposition 4.1].
138 Stochastic Heat Equation Driven by Gaussian Noise
9.4 Stochastic Integrals and Gaussian Noise
In this section we introduce stochastic integration with respect to a spatially homoge-neous Gaussian noise. Our exposition has been strongly influenced by [115]. Thereare two distinct approaches (or schools) of study for stochastic partial differentialequations, based on different theories of stochastic integration: the Walsh theory[441], which uses integration with respect to suitable martingale measures, and atheory of integration with respect to Hilbert-space-valued processes [116]. In [115]the authors discuss the connections between these theories. In this article we use theHilbert-space approach. In this section we present the non-Archimedean counterpartof this theory.
9.4.1 Stochastic Integrals with Respect to a SpatiallyHomogeneous Gaussian Noise
LetV be a separable Hilbert space with inner product 〈·, ·〉V . Following [115] and thereferences therein, we define the general notion of a cylindrical Wiener process in Vas follows.
Definition 9.5 Let Q be a symmetric and non-negative definite bounded linear oper-ator on V . A family of random variables B = {Bt (h), t ≥ 0, h ∈ V } is a cylindricalWiener process if the following conditions hold:
(i) for any h ∈ V, {Bt (h), t ≥ 0} defines a Brownian motion with variance t〈Qh, h〉V ;(ii) for all s, t ∈ R+ and h, g ∈ V,
E(Bs(h)Bt (g)) = (s ∧ t )〈Qh, g〉V ,where s ∧ t := min{s, t}.
If Q = IV is the identity operator in V , then B will be called a standard cylindricalWiener process. We will refer to Q as the covariance of B.
Let Ft be the σ -field generated by the random variables
{Bs(h), h ∈ V, 0 ≤ s ≤ t}and the P-null sets. We define the predictable σ -field in [0,T ]× generated by thesets
{(s, t]× A,A ∈ Fs, 0 ≤ s < t ≤ T }.We denote by VQ the completion of the Hilbert space V endowed with the inner
semi-product
〈h, g〉VQ := 〈Qh, g〉V , h, g ∈ V.We define the stochastic integral of any predictable square-integrable process
with values in VQ as follows. Let (v j ) j be a complete orthonormal basis of VQ. For
9.4 Stochastic Integrals and Gaussian Noise 139
any predictable process g ∈ L2(× [0,T ];VQ), the following series converges inL2(,F,P) and the sum does not depend on the chosen orthonormal basis:
g · B :=∞∑
j=1
∫ T
0〈gs, v j〉VQdBs(v j ). (9.5)
We note that each summand in the above series is a classical Itô integral with respectto a standard Brownian motion, and the resulting stochastic integral is a real-valuedrandom variable. The stochastic integral g · B is also denoted by
∫ T0 gs dBs. The inde-
pendence of each of the terms in the series (9.5) leads to the isometry property
E((g · B)2) = E
(
(∫ T
0gs dBs
)2)
= E
((∫ T
0‖gs‖VQds
))
.
9.4.2 Spatially Homogeneous Gaussian Noise
Let (,F,P) be a complete probability space.We denote by I (R) theR-vector spaceof functions of the form
∑mk=1 ck1Ik (x), where c1, . . . , ck are real numbers and each
Ik is a bounded interval (open, closed, half-open). It is well known that I (R) is densein
LρR
(
QNp , d
Nx)
:= LρR= {
f : QNp → R; ‖ f‖Lρ < ∞}
for 1 ≤ ρ < ∞.
We denote by I (R)⊗alg DR(QNp ) the algebraic tensor product of the R-vector spaces
I (R) and DR(QNp ), the space of R-valued test functions. Notice that I (R)⊗alg
DR(QNp ) is the R-vector space spanned by
∑
j∈Jc j(t )
(
pm∥
∥x− x j∥
∥
p
)
,
where c j(t ) ∈ I (R), m ∈ Z, and J is a finite set.On (,F,P), we consider a family of mean-zero Gaussian random variables
{
W (ϕ), ϕ ∈ I (R)⊗alg DR
(
QNp
)}
(9.6)
with covariance
E(W (ϕ)W (ψ )) =∫ ∞
0
∫
QNp
∫
QNp
ϕ(t, x) f (x− y)ψ (t, y)dNx dNy dt
=∫ ∞
0
∫
QNp
f (z)(ϕ(t ) ∗ ˜ψ (t ))(z)dNz dt, (9.7)
where ˜ψ (t )(z) = ψ (t,−z) and f is a non-negative continuous function on QNp � {0}.
This function induces a positive distribution onQNp and then f is the Fourier transform
of a regular Borel measure μ on QNp , see Theorem 9.4. This measure is called the
140 Stochastic Heat Equation Driven by Gaussian Noise
spectral measure ofW . In this case
E(W (ϕ)W (ψ )) =∫ ∞
0
∫
QNp
Fϕ(t )(ξ )Fψ (t )(ξ )dμ(ξ )dt.
SomeExamples of Kernels The basic example of a kernel function is the white-noisekernel: f (x) = δ(x), dμ(ξ ) = dNξ . Here are some typical examples.
Example 9.6 If dμ(ξ ) = ‖ξ‖−αp dNξ , 0 < α < N, then f (x) = Rα (x) =(
(1− p−α )/(
1− pα−N))‖x‖α−Np , the Riesz kernel, see e.g. [402, Chapter III, Section 4].
Example 9.7 If dμ(ξ ) = e−‖ξ‖βp , β > 0, then f (x) = Fξ→x(
e−‖ξ‖βp ) is the p-adicheat kernel. Notice that we can replace ‖ξ‖βp by |a(ξ )|βp, where a(ξ ) is an ellipticpolynomial.
Before presenting our next example, we recall the following result.
Lemma 9.8 ([402, Lemma 5.2]) Suppose that α > 0. Define
Kα (x) =⎧
⎨
⎩
1− p−α
1− pα−N(‖x‖α−Np − pα−N
)
(‖x‖p) if α = N
(1− p−N ) logp(p/‖x‖p)(‖x‖p) if α = N.
Then Kα ∈ L1 and FKα (ξ ) = max(1, ‖ξ‖p)−α .The distribution Kα is called the Bessel potential of order α, see e.g. [402,
Chapter III, Section 5].
Example 9.9 If dμ(ξ ) = max(1, ‖ξ‖p)−α , α > 0, then f (x) = Kα (x), the Besselpotential of order α.
A Cylindrical Wiener Process Associated with W Let U be the completion of theBruhat–Schwartz space DR(QN
p ) endowed with semi-inner product
〈ϕ,ψ〉U :=∫
QNp
Fϕ(ξ )Fψ (ξ )dμ(ξ ), ϕ, ψ ∈ DR
(
QNp
)
,
where μ is the spectral measure ofW . We denote by ‖ · ‖U the corresponding norm.ThenU is a separable Hilbert space (because DR(QN
p ) is separable) that may containdistributions.We fix a time interval [0,T ] and setUT := L2([0,T ];U ). This set is equipped with
the norm given by
‖g‖2UT :=∫ T
0‖g(s)‖2Uds.
Wenow associate a cylindricalWiener process withW as follows. A direct calculationusing (9.7) shows that the generalized Gaussian random fieldW (ϕ) is a random lin-ear functional, in the sense thatW (aϕ + bψ ) = aW (ϕ)+ bW (ψ ), almost surely, andϕ →W (ϕ) is an isometry from
(
I ([0,T ])⊗alg DR(QNp ), ‖ · ‖UT
)
into L2(,F,P).
9.4 Stochastic Integrals and Gaussian Noise 141
The following lemma identifies the completion of I ([0,T ])⊗alg DR(QNp ) with
respect to ‖ · ‖UT .Lemma 9.10 The space I ([0,T ])⊗alg DR(QN
p ) is dense in UT = L2([0,T ];U ) for‖ · ‖UT .Proof Let C denote the closure of I ([0,T ])⊗alg DR(QN
p ) in UT for ‖ · ‖UT . Sup-pose that we are given ϕ1 ∈ L2([0,T ];R) and ϕ2 ∈ DR(QN
p ). We show that ϕ1ϕ2 ∈ C.Indeed, let
{
ϕ(n)1
}
n∈N ⊂ I (R) such that, for all n, the support of ϕ(n)1 is contained
in [0,T ] and ϕ(n)1 → ϕ1 in L2([0,T ];R). Now ϕ
(n)1 ϕ2 ∈ I ([0,T ])⊗alg DR(QN
p ) ⊂ Cand ϕ(n)
1 ϕ2 ‖ · ‖UT−−−→ ϕ1ϕ2, therefore ϕ1ϕ2 ∈ C.Suppose that ϕ ∈ UT .We show that ϕ ∈ C. Indeed, let (e j ) j be a complete orthonor-
mal basis ofU with e j ∈ DR(QNp ) for all j. Then, since ϕ(s) ∈ U for any s ∈ [0,T ],
‖ϕ‖2UT =∫ T
0‖ϕ(s)‖2Uds =
∑∞j=1
∫ T
0〈ϕ(s), e j〉2Uds.
We now note that, for any j ≥ 1, the function s → 〈ϕ(s), e j〉U belongs toL2([0,T ];R). Thus, by the above considerations,
ϕ(n)(·) :=∑n
j=1〈ϕ(·), e j〉Ue j ∈ C.
Finally, since limn→∞ ‖ϕ − ϕ(n)‖2UT = 0, we conclude that ϕ ∈ C. �
Using the above lemma, we can extendW toUT by following the standard methodsfor extending an isometry. This establishes the following result.
Proposition 9.11 For t ≥ 0 and ϕ ∈ U, set Wt (ϕ) :=W (1[0,t](·)ϕ(�)). Then theprocess W = {Wt (ϕ), t ≥ 0, ϕ ∈ U} is a cylindrical Wiener process as in Defini-tion 9.5, with V there replaced by U and Q = IV . In particular, for any ϕ ∈ U,{Wt (ϕ), t ≥ 0} is a Brownian motion with variance t‖ϕ‖U and, for all s, t ≥ 0 andϕ,ψ ∈ U, E(Wt (ϕ)Wt (ψ )) = (s ∧ t )〈ϕ,ψ〉U .Remark 9.12 This proposition allow us to use the stochastic integration defined inSection 9.4.1. This defines the stochastic integral g ·W for all g ∈ L2(× [0,T ];U ).In order to use the stochastic integral of Section 9.4.1, let (e j ) j ⊂ DR(QN
p ) be a com-plete orthonormal basis of U, and consider the cylindrical Wiener process {Wt (ϕ)}defined in Proposition 9.11. For any predictable process g in L2(× [0,T ];U ), thestochastic integral with respect to W is
g ·W =∫ T
0gs dWs :=
∑∞j=1
∫ T
0〈gs, e j〉UdWs
(
e j)
,
and the isometry property is given by
E((g ·W )2) = E
(
(∫ T
0gs dWs
)2)
= E
(∫ T
0‖gs‖2Uds
)
. (9.8)
142 Stochastic Heat Equation Driven by Gaussian Noise
We also use the notation∫ T0
∫
QNpg(s, y)W (ds, dy) instead of
∫ T0 gs dWs. In later sec-
tions we use the notation E( ∫ T
0 ‖g(s)‖2Uds)
for E((g ·W )2).
9.4.3 The Spectral Measure
Recall that μ is the spectral measure of W . In the following we use a function �
satisfying the following hypothesis.
Hypothesis A. The function � is defined on R+ := [0,+∞) with values inD′R(Q
Np )
such that, for all t > 0, �(t ) is a positive distribution satisfying∫ T
0dt
∫
QNp
|F�(t )(ξ )|2dμ(ξ ) < ∞, (9.9)
and � is associated with a measure �(t, dNx) such that, for all T > 0,
sup0≤t≤T
�(
t,QNp
)
< ∞. (9.10)
We now set �(t, x) = F−1ξ→x
(
e−t|a(ξ )|βp)
, for t > 0, and �(0, x) := δ(x), i.e. � is thefundamental solution of (9.4), and since �(t, x) ∈ L1R
(
QNp , d
Nx)
for t > 0, it definesan element ofD′
R(QNp ). In addition, �(t, d
Nx) := �(t, x)dNx, and by Theorem 9.3 (ii)and (v),
sup0≤t≤T
�(
t,QNp
) = sup0<t≤T
∫
QNp
�(t, x)dNx = 1.
Hence �(t, dNx) satisfies (9.10).
Remark 9.13 If H(t, x) is a function on R × QNp , we use H(t ) instead of H(t, ·). If
G(t, x, ω) is a function on R × QNp ×, we use G(t, x) instead of G(t, x, ω), as it is
customary in probability, in certain special cases we will use G(t, x)(ω).
On the other hand, by using Fubini’s theorem and inequality (9.1), it is easy tocheck that condition (9.9) is equivalent to
∫
QNp
dμ(ξ )
max(1, ‖ξ‖p)dβ < ∞. (9.11)
Lemma 9.14 With the notation of Lemma 9.8, assuming (9.11) and∫
QNp
f (x)Kdβ (x)dNx < ∞,
we have∫
QNp
dμ(ξ )
max(1, ‖ξ‖p)dβ =∫
QNp
f (x)Kdβ (x)dNx.
9.4 Stochastic Integrals and Gaussian Noise 143
Proof Set
δn(x) := pNn(
pn‖x‖p)
, for n ∈ N. (9.12)
Then∫
QNpδn(x)dNx = 1 for any n, δn D′
−→ δ and Fδn(ξ ) = (
p−n‖ξ‖p)
pointwise−−−−→ 1.
Notice that (Kdβ ∗ δn)(x) = Kdβ (x), for x ∈ QNp � {0} and for any n > N(x), since
Kdβ is radial, then
f (x)(Kdβ ∗ δn)(x) = f (x)Kdβ (x), for x ∈ QNp � {0} and n big enough. (9.13)
Now, by the Riesz–Markov–Kakutani theorem, μ is an element of D′(QNp
)
andsince Kdβ ∗ δn ∈ D
(
QNp
)
, we have
(μ,F (Kdβ ∗ δn)) = (Fμ,Kdβ ∗ δn) = ( f ,Kdβ ∗ δn)
=∫
QNp
f (x)(Kdβ ∗ δn)(x)dNx.
Then, by applying the dominated convergence theorem and using (9.13) and thehypothesis
∫
QNpf (x)Kdβ (x)dNx < ∞, we get
limn→∞
∫
QNp
f (x)(Kdβ ∗ δn)(x)dNx =∫
QNp
f (x)Kdβ (x)dNx.
On the other hand, by the Riesz–Markov–Kakutani theorem,
(μ,F (Kdβ ∗ δn)) =(
μ,(Fδn)(ξ )
max(1, ‖ξ‖p)dβ)
=∫
QNp
(Fδn)(ξ )
max(1, ‖ξ‖p)dβ dμ(ξ ),
so that now, by the dominated convergence theorem and hypothesis (9.11),
limn→∞
∫
QNp
(Fδn)(ξ )
max(1, ‖ξ‖p)dβ dμ(ξ ) =∫
QNp
dμ(ξ )
max(1, ‖ξ‖p)dβ . �
From Lemmas 9.8 and 9.14, we obtain the following result.
Theorem 9.15∫
QNp
dμ(ξ )
max(1, ‖ξ‖p)dβ < ∞
⇔
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
1− p−dβ
1− pdβ−N
∫
‖x‖p≤1
(‖x‖dβ−Np − pdβ−N)
f (x)dNx < ∞ if dβ = N
(1− p−N )∫
‖x‖p≤1logp(p/‖x‖p) f (x)dNx < ∞ if dβ = N.
144 Stochastic Heat Equation Driven by Gaussian Noise
9.4.4 Examples of Integrands
The main examples of integrands are provided by the following result.
Proposition 9.16 Assume that � satisfies Hypothesis A. Let
Y = {
Y (t, x), (t, x) ∈ [0,T ]× QNp
}
be a predictable process such that
CY := sup(t,x)∈[0,T ]×QN
p
E(|Y (t, x)|2) < ∞.
Then, the random element G = G(t, x) = Y (t, x)�(t, x) is a predictable process withvalues in L2(× [0,T ];U ). Moreover,
E(‖G‖2UT
) = E
[
∫ T
0
∫
QNp
|F (�(t )Y (t )(ξ ))|2dμ(ξ )dt]
≤ CY
∫ T
0
∫
QNp
|F (�(t )Y (t )(ξ ))|2dμ(ξ )dt
and
E(|G ·W |2) ≤∫ T
0
(
supx∈QN
p
E(|Y (s, x)|2))
∫
QNp
|F (�(s)(ξ ))|2dμ(ξ )ds. (9.14)
Remark 9.17 The integral of G with respect to W will be also denoted by
G ·W =∫ T
0
∫
QNp
�(s, y)Y (s, y)W (ds, dNy).
Proof The proof will be accomplished through several steps.§1. Assertion A: G(t ) ∈ L1
(
QNp
)
, for t ∈ (0,T ] almost surely (a.s.).Indeed, by the Hölder inequality,
∫
QNp
E(|Y (t, x)|2)�(t, x)dNx ≤ CY‖�(t )‖L1(QNp
), for t ∈ (0,T ],
cf. Theorem 9.3 (iii). Hence,
∫
∫
QNp
|Y (t, x)(ω)|2�(t, x) dNx dP(ω) < ∞,
9.4 Stochastic Integrals and Gaussian Noise 145
and, by Fubini’s theorem, |Y (t, x)|2�(t, x) ∈ L1(
QNp
)
, for t ∈ (0,T ] a.s. Now,
∫
QNp
|Y (t, x)|�(t, x)dNx =∫
|Y (t,x)|>1|Y (t, x)|�(t, x)dNx
+∫
|Y (t,x)|≤1|Y (t, x)|�(t, x)dNx
≤∫
QNp
|Y (t, x)|2�(t, x)dNx+∫
QNp
�(t, x)dNx
< ∞ for t ∈ (0,T ] a.s.
By using the above reasoning, one verifies that
∫ T
0
∫
QNp
|Y (t, x)|h(t, x)dNx dt < ∞,
for every h ∈ L1(
[0,T ]× QNp
)
satisfying h ≥ 0, therefore
Y (t, x) ∈ L∞(
[0,T ]× QNp
)
a.s. (9.15)
As a consequence of Assertion A, we have G(t ) ∈ D′R
(
QNp
)
, for t ∈ (0,T ] a.s. Wenow proceed to regularize this distribution. We set �l (x) :=
(
p−l‖x‖p)
, δk(x) =pkN
(
pk‖x‖p)
for k, l ∈ N. Then∫
δk(x)dNx = 1, F (�k ) = δk, δk D′R−→ δ (the Dirac
distribution), and �lpointwise−−−−−−→1, as before.
We also set Gk,l (t ) := (�lY (t )�(t )) ∗ δk, k, l ∈ N, t ∈ (0,T ], and �k(t ) := �(t )∗δk. Then, Gk,l (t ) ∈ DR
(
QNp
)
for t ∈ (0,T ], or, more precisely,
Gk,l (t ) =∑
j
c j(t; k, l)(
pm(k)‖x− x j‖p)
.
Now, since
|(�lY (t )�(t )) ∗ δk| ≤ |Y (t )|�(t ) ∗ δk ≤ ‖Y‖L∞
(
[0,T ]×QNp
)‖�(t ) ∗ δk‖L1(QNp
)
≤ pNk‖Y‖L∞
(
[0,T ]×QNp
) a.s.,
cf. (9.15), we have c j(t; k, l) ∈ L∞([0,T ]) a.s. Therefore
Gk,l (t ) ∈ L2([0,T ])⊗alg DR
(
QNp
)
a.s. (9.16)
146 Stochastic Heat Equation Driven by Gaussian Noise
§2. A bound for supk,l≥1 E‖Gk,l‖2UT .By using the definition of the convolution and the uniform bound for the square
moments of Y , we get
supk,l≥1
E∥
∥Gk,l
∥
∥
2UT
= supk,l≥1
E∫ T
0
∫
QNp
∫
QNp
Gk,l (t, x) f (x− y)Gk,l (t, y)dNx dNy dt
= supk,l≥1
E∫ T
0
∫
QNp
∫
QNp
[
∫
QNp
�l (z)Y (t, z)�(t, z)δk(x− z)dNz
]
× f (x− y)
[
∫
QNp
�l(
z′)
Y(
t, z′)
�(
t, z′)
δk(
y− z′)
dNz′]
dNx dNy dt
≤ CY supk≥1
∫ T
0
∫
QNp
∫
QNp
�k(t, x) f (x− y)�k(t, y)dNx dNy dt
= CY supk≥1
∫ T
0
∣
∣F�k(t )(ξ )∣
∣
2dμ(ξ )dt ≤ CY
∫ T
0|F�(t )(ξ )|2dμ(ξ )dt,
because F�k(t )(ξ ) = F�(t )(ξ ) ·(
p−k‖ξ‖p) ≤ F�(t )(ξ ). Therefore
supk,l≥1
E(‖Gk,l‖2UT
) ≤ CY
∫ T
0|F�(t )(ξ )|2dμ(ξ )dt < ∞ (9.17)
for t ∈ (0,T ] a.s.As a consequence, we get that Gk,l ∈ L2(× [0,T ];U ) for k, l ≥ 1, since, by
(9.16), Gk,l (t ) ∈ L2([0,T ])⊗alg DR
(
QNp
)
a.s.§3. limk→∞Gk,l ∈ L2(× [0,T ];U ).We set Gl (t ) := �lY (t )�(t ), l ∈ N, t ∈ (0,T ]. By using the reasoning given in §2,
we get
E∫ T
0
∫
QNp
|FGl (t )(ξ )|2dμ(ξ )dt ≤ CY
∫ T
0
∫
QNp
|F�(t )(ξ )|2dμ(ξ )dt < ∞,
(9.18)
for any l ∈ N.We now assert that Gk,l UT−→ Gl as k → ∞. Indeed,
E∫ T
0
∫
QNp
|FGk,l (t )(ξ )− FGl (t )(ξ )|2dμ(ξ )dt
= E∫ T
0
∫
QNp
|FGl (t )(ξ )|2|�k(ξ )− 1|2dμ(ξ )dt → 0 as k → ∞,
9.4 Stochastic Integrals and Gaussian Noise 147
by the dominated convergence theorem and (9.18). Hence Gl ∈ L2(× [0,T ];U )and, by (9.18),
supl≥1
E(‖Gl‖2UT
) ≤ CY
∫ T
0|F�(t )(ξ )|2dμ(ξ )dt. (9.19)
§4. Gl UT−→ G, i.e. liml→∞ E(‖G− Gl‖2UT
) = 0.
Indeed,
E(‖G− Gl‖2UT
) = E∫ T
0
∫
QNp
|FG(t )(ξ )− FGl (t )(ξ )|2dμ(ξ )dt
≤ 2E∫ T
0
∫
QNp
|FG(t )(ξ )|2dμ(ξ )dt
+ 2E∫ T
0
∫
QNp
|FGl (t )(ξ )|2dμ(ξ )dt
≤ 4E∫ T
0
∫
QNp
|FG(t )(ξ )|2dμ(ξ )dt
≤ 4CY
∫ T
0
∫
QNp
|F�(t )(ξ )|2dμ(ξ )dt < ∞,
where the last inequality was obtained by using the reasoning given in §2. On the otherhand,
E(‖G− Gl‖2UT
) = E∫ T
0
∫
QNp
|FG(t )(ξ )− FG(t )(ξ ) ∗ δl (ξ )|2dμ(ξ )dt.
Now, by using the dominated convergence theorem and the fact that
liml→∞
FG(t )(ξ ) ∗ δl (ξ ) = FG(t )(ξ ) almost everywhere,
cf. [402, Theorem 1.14], we get that liml→∞ E(‖G− Gl‖2UT
) = 0, which impliesG ∈ L2(× [0,T ];U ).Moreover, we deduce that
E(‖G‖2UT
) = E
(
∫ T
0
∫
QNp
|FG(t )(ξ )|2dμ(ξ )dt)
= liml→∞
E(‖Gl‖2UT
)
≤ CY
∫ T
0
∫
QNp
F�(t )(ξ )|2dμ(ξ )dt,
cf. (9.19).
148 Stochastic Heat Equation Driven by Gaussian Noise
§5. A bound for E(|G ·W |2).The announced bound for E(|G ·W |2) is obtained from (9.8) by using a reasoning
similar to that used in §2. �
Remark 9.18 Let Y be a process as in Proposition 9.16. Consider the processes ofthe form
{
Y (t, x), (t, x) ∈ [T0,T ]× QNp
}
where 0 ≤ T0 < T , then
E(|G ·W |2) ≤∫ T
T0
(
supx∈QN
p
E(|Y (s, x)|2))
∫
QNp
|F (�(s)(ξ ))|2dμ(ξ )ds. (9.20)
9.5 Stochastic Pseudodifferential Equations Driven by aSpatially Homogeneous Noise
In this section we introduce a new class of stochastic pseudodifferential equations inQNp driven by a spatially homogeneous noise. More precisely, we study the following
class of stochastic equations:
{
(∂u/∂t )(t, x)+ A(∂, β )u(t, x) = σ (u(t, x))·W (t, x)+ b(u(t, x))
u(0, x) = u0(x), t ≥ 0, x ∈ QNp ,
(9.21)
where the coefficients σ and b are real-valued functions and·W (t, x) is the formal
notation for the Gaussian random perturbation described in Section 9.4.2.Recall that we are working with a filtered probability space (,F, (Ft ),P), where
(Ft )t is a filtration generated by the standard cylindrical Wiener process of Proposi-tion 9.11. We fix a time horizon T > 0.
Definition 9.19 A real-valued adapted stochastic process
{
u(t, x), (t, x) ∈ [0,T ]× QNp
}
is a mild random field solution of (9.21), if the following stochastic integral equationis satisfied:
u(t, x) = (
�(t ) ∗ u0)
(x)+∫ t
0
∫
QNp
�(t − s, x− y)σ (u(s, y))W (ds, dNy)
+∫ t
0ds
∫
QNp
�(s, y)b(u(t − s, x− y))dNy, a.s., (9.22)
for all (t, x) ∈ [0,T ]× QNp .
9.5 Stochastic Pseudodifferential Equations 149
The stochastic integral on the right-hand side of (9.22) is as defined in Remark 9.12.In particular, we need to assume that for any (t, x) the fundamental solution �(t − ·,x− �) satisfies Hypothesis A, and we require that
s → �(t − s, x− �)σ (u(s, �)), for s ∈ [0, t],
defines a predictable process taking values in the spaceU such that
E
(∫ t
0‖�(t − s, x− �)σ (u(s, �))‖2Uds
)
< ∞,
see Section 9.4.4. These assumptions will be satisfied by imposing that b and σ areLipschitz continuous functions (see Theorem 9.22). The last integral on the right-hand side of (9.22) is considered in the pathwise sense.The aim of this section is to prove the existence and uniqueness of a mild random-
field solution for the stochastic integral equation (9.22). We are interested in solutionsthat are L2()-bounded and L2()-continuous.
Lemma 9.20 Assume that u0 : QNp → R is measurable and bounded. Then
(t, x) → I0(t, x) := (�(t ) ∗ u0)(x)
is continuous and sup(t,x)∈[0,T ]×QNp|I0(t, x)| < ∞.
Proof Notice that
|I0(t, x)| ≤{
‖u0‖L∞‖�(t )‖L1 for t > 0
‖u0|L∞ for t = 0,(9.23)
and
sup(t,x)∈(0,T ]×QN
p
|I0(t, x)| ≤ ‖u0|L∞ sup(t,x)∈(0,T ]×QN
p
‖�(t )‖L1 = ‖u0|L∞ . (9.24)
By combining (9.23) and (9.24), we get sup(t,x)∈[0,T ]×QNp|I0(t, x)| ≤ ‖u0|L∞ .
The continuity of I0(t, x) at a point of the form (t0, x0), with t0 > 0, followsby the dominated convergence theorem and Theorem 9.3 (vi). The continuity ofI0(t, x) at (0, x0) is a consequence of the fact that lim(t,x)→(0,x0 ) I0(t, x) = u0(x0) =I0(0, x0). �
Hypothesis B. Let � be the fundamental solution of (9.4) as before. We assumethat
limh→0+
∫ T
0
∫
QNp
sup|r−t|<h
|F�(r)(ξ )− F�(t )(ξ )|2dμ(ξ )dt = 0. (9.25)
Lemma 9.21 If∫
QNp‖ξ‖dβp dμ(ξ ) < ∞, then Hypothesis B holds. Furthermore, the
condition∫
QNp‖ξ‖dβp dμ(ξ ) < ∞ also implies (9.9).
150 Stochastic Heat Equation Driven by Gaussian Noise
Proof By applying the mean-value theorem to the function e−t|a(ξ )|βp , we have
sup|r−t|<h
|F�(r)(ξ )− F�(t )(ξ )|2 ≤ h2|a(ξ )|2βp e−2(t−h)|a(ξ )|βp
≤ C2β1 h2‖ξ‖2dβp e−2Cβ
0 (t−h)‖ξ‖dβp ,
cf. (9.1), and thus
∫ T
0
∫
QNp
sup|r−t|<h
|F�(r)(ξ )− F�(t )(ξ )|2dμ(ξ )dt
≤ h2∫ T
0
∫
QNp
‖ξ‖2dβp e−2Cβ
0 (t−h)‖ξ‖dβp dμ(ξ )dt.
In order to prove the result, it is sufficient to show that
limh→0+
∫ T
0
∫
QNp
‖ξ‖2dβp e−2Cβ
0 (t−h)‖ξ‖dβp dμ(ξ )dt < ∞. (9.26)
Now, if
∫ T
0
∫
QNp
‖ξ‖2dβp e−2Cβ
0 t‖ξ‖dβp dμ(ξ )dt < ∞, (9.27)
then (9.26) follows on applying the dominated convergence theorem. On the otherhand, it is easy to check that
∫
QNp‖ξ‖dβp dμ(ξ ) < ∞ implies
∫
QNp
‖ξ‖2dβp
∫ T
0e−2Cβ
0 t‖ξ‖dβp dt dμ(ξ ) < ∞. (9.28)
Now (9.27) follows from (9.28) by using the Fubini theorem. The last assertion in thestatement follows from the fact that (9.9) is equivalent to (9.11). �
Theorem 9.22 Assume that b, σ are Lipschitz continuous functions, that u0 is a mea-surable and bounded function, and that
∫
QNp
‖ξ‖dβp dμ(ξ ) < ∞.
Then, there exists a unique mild random-field solution
{
u(t, x), (t, x) ∈ [0,T ]× QNp
}
of (9.22). Moreover, u is L2()-continuous and
sup(t,x)∈[0,T ]×QN
p
E(|u(t, x)|2) < ∞. (9.29)
9.5 Stochastic Pseudodifferential Equations 151
Proof The proof involves similar techniques and ideas to those of [114], [115], [339].We use the following Picard iteration scheme:
u0(t, x) = I0(t, x), (9.30)
un+1(t, x) = u0(t, x)+∫ t
0
∫
QNp
�(t − s, x− y)σ (un(s, y))W (ds, dNy)
+∫ t
0
∫
QNp
b(un(t − s, x− y))�(s, y)dNy ds
=: u0(t, x)+ In(t, x)+ J n(t, x), (9.31)
for n ∈ N.The proof will be accomplished through several steps.§1. un(t, x) is a well-defined measurable process.We prove by induction on n that
{
un(t, x), (t, x) ∈ [0,T ]× QNp
}
is a well-definedmeasurable process satisfying
sup(t,x)∈[0,T ]×QN
p
E(|un(t, x)|2) < ∞, (9.32)
for n ∈ N. By Lemma 9.20, u0(t, x) satisfies (9.32), and the Lipschitz property of σimplies that
sup(t,x)∈[0,T ]×QN
p
σ (u0(t, x)) < ∞.
By Proposition 9.16, the stochastic integral
I0(t, x) =∫ t
0
∫
QNp
�(t − s, x− y)σ (u0(s, y))W (ds, dNy)
is well defined and
E(|I0(t, x)|2) ≤ C∫ t
0supz∈QN
p
(1+ |u0(s, z)|2)∫
QNp
|F�(t − s)(ξ )|2dμ(ξ )ds
≤ C sup(s,z)∈[0,T ]×QN
p
(1+ |u0(s, z)|2)∫ T
0J(s)ds, (9.33)
where
J(s) =∫
QNp
|F�(s)(ξ )|2dμ(ξ ).
We now consider the pathwise integral
J 0(t, x) =∫ t
0
∫
QNp
b(u0(t − s, x− y))�(s, y)dNy ds.
152 Stochastic Heat Equation Driven by Gaussian Noise
By applying the Cauchy–Schwarz inequality with respect to the finite measure�(s, y)dNy ds on [0,T ]× QN
p and by using the Lipschitz property of b, one gets
|J 0(t, x)|2 ≤ C∫ t
0
∫
QNp
(1+ |u0(t − s, x− y)|2)�(s, y)dNy ds, (9.34)
which is uniformly bounded with respect to t and x. This fact together with (9.33)implies that
{
u1(t, x), (t, x) ∈ [0,T ]× QNp
}
is a well-defined measurable process, cf.Proposition 9.16. In addition, by (9.33), (9.34), and Hypothesis A,
sup(t,x)∈[0,T ]×QN
p
E(|u1(t, x)|2) < ∞.
Consider now the case n > 1 and assume that{
un(t, x), (t, x) ∈ [0,T ]× QNp
}
is awell-defined measurable process satisfying (9.32). By the same arguments as above,one proves that
E(|In+1(t, x)|2) ≤ C∫ t
0supz∈QN
p
E(1+ |un(s, z)|2)∫
QNp
|F�(t − s)(ξ )|2dμ(ξ )ds,
(9.35)
and that
E(|J n+1(t, x)|2) ≤ C∫ t
0supy∈QN
p
E(1+ |un(t − s, x− y)|2)∫
QNp
�(s, y)dNy ds.
(9.36)
Hence the integrals In+1(t, x) and J n+1(t, x) exist, so that un+1 is a well-definedmeasurable process satisfying (9.32).§2. We now show that
supn≥0
sup(t,x)∈[0,T ]×QN
p
E(|un(t, x)|2) < ∞. (9.37)
Indeed, by using the estimates (9.35) and (9.36), we have
E(|un+1(t, x)|2) ≤ C(
1+∫ t
0
(
1+ supz∈QN
p
E(|un(s, z)|2))(J(t − s)+ 1))
ds.
Now (9.37) follows from the version of Gronwall’s lemma presented in [114,Lemma 15].§3. un(t, x) L2()−−−→ u(t, x) uniformly in x ∈ QN
p , t ∈ [0,T ].
Following the same ideas as in the proof of [114, Theorem 13], we take
Mn(t ) := sup(s,x)∈[0,t]×QN
p
E({un+1(s, x)− un(s, x)}2).
9.5 Stochastic Pseudodifferential Equations 153
By using Proposition 9.16, the Lipschitz property of b and σ , and by applying thesame arguments as above, one gets
Mn(t ) ≤ C∫ t
0Mn−1(s)(J(t − s)+ 1)ds.
Now, by applying Gronwall’s lemma presented in [114, Lemma 15], we get
limn→∞
(
sup(s,x)∈[0,t]×QN
p
E(|un+1(s, x)− un(s, x)|2)
)
= 0.
Hence {un(t, x)}n∈N converges uniformly in L2() to a limit u(t, x). From this fact,we get
limn→∞ sup
(s,x)∈[0,T ]×QNp
E(|un(t, x)− u(t, x)|2) = 0. (9.38)
Finally, by (9.38) and (9.37),
E(|u(t, x)|2) = limn→∞E(|un(t, x)|2) ≤ sup
n≥0sup
(t,x)∈[0,T ]×QNp
E(|un(t, x)|2) < ∞.
§4. The process{
u(t, x), (t, x) ∈ [0,T ]× QNp
}
is L2()-continuous and has ajointly measurable version.The proof of this fact is based on the following result. LetL be a complete separable
metric space, with B(L) the σ -algebra of Borel sets of L, and let Xs, s ∈ L, be a realstochastic process on (,F,P), where real means [−∞,+∞]-valued. The processXs, s ∈ L, is jointly measurable if the map (s, ω) → Xs(ω) is B(L)× F-measurable.Let M be the space of all real random variables on (,F,P) with the topology ofconvergence in probability. Then Xs, s ∈ L, has a jointly measurable modification ifand only if the map fromL toM taking s to [Xs], the class of Xs inM, is measurable,see [107] and [111, Theorem 3].In our case, L = ([0,T ]× QN
p , d) with
d(
(t, x), (t ′, x′))
:= max{∣
∣t − t ′∣
∣, ‖x− x′|p}
. (9.39)
Then B([0,T ]× QNp ) = B([0,T ])× B(QN
p ). It is sufficient to show that the mapfrom [0,T ]× QN
p to M taking (t, x) to u(t, x) is continuous in L2(). Furthermoresince the convergence of un+1(t, x) to u(t, x) is uniform in L2(), it is sufficient toshow that un+1(t, x) is L2-continuous. In order to do this, we have to verify that
limh→0
E(|un+1(t, x)− un+1(t + h, x)|2) = 0 (9.40)
and
limx→y
E(|un+1(t, x)− un+1(t, y)|2) = 0. (9.41)
Indeed, (9.40) implies that x → un+1(t, x) is uniformly continuous in L2()and (9.41) implies that t → un+1(t, x) is continuous in L2(), therefore
154 Stochastic Heat Equation Driven by Gaussian Noise
(t, x) → un+1(t, x) is continuous in L2(). The proof of this fact follows fromHypotheses A and B by using the technique given in [114] to prove Lemma 19.§5. u(t, x) is a solution of (9.22).We set
I (t, x) :=∫ t
0
∫
QNp
�(t − s, x− y)σ (u(s, y))W(
ds, dNy)
and
J (t, x) :=∫ t
0
∫
QNp
b(u(t − s, x− y))�(s)dNy ds.
In order to to establish §5, it is sufficient to show that
limn→∞ sup
(t,x)∈[0,T ]×QNp
E(|In(t, x)− I (t, x)|2) = 0, (9.42)
and that
limn→∞ sup
(t,x)∈[0,T ]×QNp
E(|J n(t, x)− J (t, x)|2) = 0. (9.43)
To show (9.42) we proceed as follows. By the Lipschitz property of σ , Proposi-tion 9.16, and Hypothesis A,
E(|In(t, x)− I (t, x)|2)
≤ E
⎛
⎝
∣
∣
∣
∣
∣
∫ t
0
∫
QNp
�(t − s, x− y)[σ (un−1(s, y))− σ (u(s, y))]W (ds, dNy)
∣
∣
∣
∣
∣
2⎞
⎠
≤ C∫ t
0supz∈QN
p
(|un−1(s, z)− u(s, z)|2)∫
QNp
|F�(t − s)(ξ )|2dμ(ξ )ds
≤ C supz∈QN
p
(|un−1(s, z)− u(s, z)|2),
Where the last term tends to zero as n tend to infinity. The case (9.43) can be treatedin a similar form. According to the results of §4, the process {u(t, x), (t, x) ∈ [0,T ]×QNp } has a measurable version that satisfies (9.22).§6. u(t, x) is the unique solution of (9.22) satisfying (9.29).This fact can be checked by using standard arguments. �
10
Sobolev-Type Spaces andPseudodifferential Operators
10.1 Introduction
This chapter aims to present the basic results about Sobolev-type spaces over QNp
and to show the existence of fundamental solutions for pseudodifferential equationsover these spaces. We consider two types of spaces, denoted H∞ and W∞. Both ofthese spaces are countably Hilbert nuclear spaces, withW∞ continuously embeddedin W∞. These spaces are invariant under the action of a large class of pseudodif-ferential operators. The spacesH∞ were studied by Zúñiga-Galindo in [472]. Theseare locally convex spaces constructed from D by using a countable family of Hilber-tian seminorms, see Section 10.2. The spaces W∞ were introduced in [473]. Theseare locally convex spaces constructed from a Lizorkin-type space of test functions byusing a countable family of Hilbertian seminorms, see Section 10.4. In the spacesW∞we show the existence of fundamental solutions for pseudodifferential equations oftype (A(∂, f, α)g)(x) := F−1
ξ→x(|f(ξ )|αpFx→ξg), see Section 10.6. This result is valid innon-Archimedean local fields of arbitrary characteristic, for instance in Fp((t )). Thisresult is the non-Archimedean counterpart of Hörmander’s solution of the problemof the division of a distribution by a polynomial, see [202], [316]. The key ingredientis a non-Archimedean version of the Hörmander–Łojasiewicz inequality developedin Section 10.3. In [404, Theorems 4.1 and 4.2], Taylor, Varadarajan, Virtanen, andWeisbart also proved this inequality. The rest of the chapter is dedicated to connec-tions between local zeta functions and fundamental solutions. In the Archimedeansetting, the local zeta functions were introduced in the 1950s by Gel’fand and Shilov.The main motivation was that the meromorphic continuation of Archimedean localzeta functions implies the existence of fundamental solutions for differential ope-rators with constant coefficients. This result has a non-Archimedean counterpart. In[462], see also [470] and the references therein, Zúñiga-Galindo noticed that the clas-sical argument showing that the analytic continuation of local zeta functions impliesthe existence of fundamental solutions also works in non-Archimedean fields of char-acteristic zero, and that for particular polynomials the Gel’fand–Shilov method gives
155
156 Sobolev-Type Spaces and Pseudodifferential Operators
explicit formulae for fundamental solutions. We review these ideas in Section 10.7,without proofs. In Section 10.8, we summarize the results of [471], without proofs.In this work the existence of fundamental solutions for pseudodifferential equa-tions using local zeta functions is established in the spacesH∞. In addition, methodsinvolving pseudodifferential operators are used to study non-Archimedean local zetafunctions.
10.2 The Spaces H∞The Bruhat–Schwartz space D(QN
p ) is not invariant under the action of pseudo-differential operators. In this chapter, we present a class of nuclear countably Hilbertspaces introduced by Zúñiga-Galindo in [472], see also [471]. These spaces are invari-ant under the action of a large class of pseudodifferential operators. The notation hereis slightly different from the notation used in [472]. For an in-depth discussion aboutnuclear countably Hilbert spaces, the reader may consult [176], [177], [199], [365].
Notation 10.1 We setR+ := {x ∈ R : x ≥ 0}. We denote byN the set of non-negativeintegers. We set [ξ ]p := max(1, ‖ξ‖p). We recall that D := D(QN
p ) denotes the C-vector space formed by the complex-valued test functions. We denote by DR :=DR(QN
p ) the R-vector space formed by the real-valued test functions. We denote byC(QN
p ,C) := C the C-vector space of all the complex-valued functions which arecontinuous. We set
C0(QNp ,C) := C0
(
QNp
)
={
f : QNp → C; f is continuous and lim
x→∞ f (x) = 0}
,
where limx→∞ f (x) = 0 means that for every ε > 0 there exists a compact subsetB(ε) such that | f (x)| < ε for x ∈ QN
p � B(ε). We recall that (C0(QNp ), ‖ · ‖L∞ ) is a
Banach space.Given r ∈ [0,+∞), we denote by Lr(QN
p , dNx) := Lr the C-vector space of all
the complex-valued functions g satisfying∫
QNp|g(x)|rdNx < ∞. We denote byCunif :=
Cunif (QNp ,C) the C-vector space of all the complex-valued functions which are uni-
formly continuous. We denote by LrR, CunifR
the corresponding R-vector spaces.
We define for ϕ, in D(QNp ) the following scalar product:
〈ϕ, 〉l :=∫
QNp
[ξ ]lpϕ(ξ ) (ξ )dNξ, (10.1)
for l ∈ N, where the bar denotes the complex conjugate. We also set ‖ϕ‖2l = 〈ϕ, ϕ〉l .Notice that ‖ · ‖l ≤ ‖ · ‖m for l ≤ m. Then Hm ↪→ Hl (continuous embedding) forl ≤ m. We denote by Hl (QN
p ,C) = Hl (C) = Hl the completion of D(QNp ) with
respect to 〈·, ·〉l . We set
H∞(QNp ,C) = H∞(C) = H∞ := ⋂
l∈NHl .
10.2 The SpacesH∞ 157
Notice thatH0 = L2 and thatH∞ ⊂ L2. With the topology induced by the family ofseminorms ‖ · ‖l∈N,H∞ becomes a locally convex space, which ismetrizable. Indeed,
d( f , g) := maxl∈N
{
2−l‖ f − g‖l
1+ ‖ f − g‖l
}
, with f , g ∈ H∞,
is a metric for the topology of H∞ considered as a convex topological space. Asequence { fl}l∈N in (H∞, d) converges to f ∈ H∞ if and only if { fl}l∈N convergesto f in the norm ‖ · ‖l for all l ∈ N. From this observation it follows that the topologyon H∞ coincides with the projective limit topology τP. An open neighborhood baseat zero of τP is given by the choice of ε > 0 and l ∈ N, and the set
Uε,l := { f ∈ H∞; ‖ f‖l < ε}.Remark 10.2 We denote by Hl (QN
p ,R) = Hl (R) := Hl , H∞(QNp ,R) = H∞(R) =
H∞ the R-vector spaces constructed from DR(QNp ). In the case in which the ground
field is clear, we shall use simply Hl , H∞. All the above results are valid for thesespaces. We shall also use d to denote the metric ofH∞(R). These spaces will be usedlater on.
Lemma 10.3 H∞(C) endowed with the topology τP is a countably Hilbert space inthe sense of Gel’fand and Vilenkin, see e.g. [177, Chapter I, Section 3.1] or [365, Sec-tion 1.2]. Furthermore (H∞(C), τP) is metrizable and complete and hence a Fréchetspace.
Proof According to the previous considerations, it suffices to show that 〈·, ·〉l∈N is asystem of compatible scalar products, i.e. if a sequence { fl}l∈N of elements ofH∞(C)converges to zero in the norm ‖ · ‖m and is a Cauchy sequence in the norm ‖ · ‖n,then it also converges to zero in the norm ‖ · ‖n. We may assume without loss ofgenerality that m ≤ n and thus ‖ · ‖m ≤ ‖ · ‖n. By using fl ‖ · ‖m−−−→ 0 ∈ Hm(C) and fl
‖ · ‖n−−→ f ∈ Hn(C) ⊂ Hm(C), we conclude that f = 0. �
Lemma 10.4 (i) Set (D(QNp ), d) for the completion of the metric space (D(QN
p ), d).
Then (D(QNp ), d) = (H∞(C), d).
(ii) (H∞(C), d) is a nuclear space.
Proof (i) Set f ∈ (D(QNp ), d), then there exists a sequence { fn}n∈N in (D(QN
p ), d)
such that fn ‖ · ‖l−−→ f for each l ∈ N, i.e. f ∈ ∩l∈NHl . Hence (D(QNp ), d) ⊂
(H∞(C), d). Conversely, set g ∈ H∞(C). By using the density of D(QNp ) in
Hl (C), and the fact that ‖ · ‖m ≤ ‖ · ‖n ifm ≤ n, we construct a sequence {gn}n∈Nin D(QN
p ) satisfying
‖gn − g‖l ≤ 1/(n+ 1) for 0 ≤ l ≤ n.
Then d(gn, g) ≤ max{1/(n+ 1), 2/(n+ 1), . . . , 2−n/(n+ 1), 2−(n+1)} → 0 asn → ∞. This fact shows that g ∈ (D(QN
p ), d).
158 Sobolev-Type Spaces and Pseudodifferential Operators
(ii) We recall that D(QNp ) is a nuclear space, cf. [94, Section 4], and since the com-
pletion of a nuclear space is also nuclear, see e.g. [407, Proposition 50.1], by (i)above, H∞(C) is a nuclear space. �
Remark 10.5 (i) Lemma 10.4 is valid if we replace D(QNp ) by DR(QN
p ) andH∞(C)byH∞(R).
(ii) By using the Cauchy–Schwarz inequality, if l > N/2 and f ∈Hl (C), then f ∈ L1,and thus f ∈ Cunif . Indeed, Take f ∈ Hl with l > N, then, by using the Cauchy–Schwarz inequality,
∥
∥f∥
∥
L1 =∫
QNp
∣
∣f∣
∣ dNξ =∫
QNp
{
[ξ ]l2p
∣
∣f∣
∣
}
⎧
⎨
⎩
1
[ξ ]l2p
⎫
⎬
⎭
dNξ
≤∥
∥
∥
∥
∥
∥
1
[ξ ]l2p
∥
∥
∥
∥
∥
∥
L2
∥
∥
∥[ξ ]
l2p
∣
∣f∣
∣
∥
∥
∥
L2≤ C(N, l)‖ f‖l,
where C(N, l) is a positive constant, which shows that f ∈ L1. Consequently f ∈L1 for f ∈ H∞(C).
ThereforeH∞(C) ⊂ L2 ∩Cunif . A similar assertion is valid forH∞(R).
From Lemmas 10.3 and 10.4 we obtain the main result of this section.
Theorem 10.6 H∞(C) is a nuclear countably Hilbert space.
Remark 10.7 (i) As a nuclear Fréchet spaceH∞(C) admits a sequence of definingHilbertian norms | · |m∈N such that (1) |g|m ≤ Cm|g|m+1, g ∈ H∞(C), with someCm >
0; (2) the canonical map in,n+1 : Hn+1 (C) → Hn(C) is of Hilbert–Schmidt type, whereHn(C) is the Hilbert space associated with | · |n, cf. [365, Proposition 1.3.2].(iii) Theorem 10.6 is valid forH∞(R). This is due to the fact that any subspace of
a nuclear space is also nuclear, see e.g. [407, Proposition 50.1].
Lemma 10.8 With the above notation, the following assertions hold:
(i) Hl(
QNp
) = { f ∈ L2(
QNp
); ‖ f‖l < ∞} = {T ∈ D′(QNp
); ‖T‖l < ∞};(ii) H∞
(
QNp
) = { f ∈ L2(
QNp
); ‖ f‖l < ∞, for any l ∈ N};(iii) H∞
(
QNp
) = {T ∈ D′(QNp
); ‖T‖l < ∞, for any l ∈ N}.
The equalities in (i)–(iii) are in the sense of vector spaces.
We omit the proof of this result because is similar to the proof of Lemma 10.20.
Notation 10.9 If T is a distribution a formula of type ‖T‖l < ∞ requires implicitlythat T be a function.
10.2 The SpacesH∞ 159
10.2.1 The Dual Space ofH∞For m ∈ N and T ∈ D′(QN
p
)
, we set
‖T‖2−m := ∫
QNp
[ξ ]−m�
∣
∣T (ξ )∣
∣
2dNξ .
ThenH−m := H−m(
QNp
) = {T ∈ D′(QNp
); ‖T‖2−m < ∞} is a complex Hilbert space.Denote byH∗
m the strong dual space ofHm. It is useful to suppress the correspondencebetween H∗
m and Hm given by the Riesz theorem. Instead we identify H∗m and H−m
by associating T ∈ H−m with the functional on Hm given by
[T, g] :=∫
QNp
T (ξ )g(ξ )dNξ . (10.2)
Notice that
|[T, g]| ≤ ‖T‖−m‖g‖m. (10.3)
Now by a well-known result in the theory of countable Hilbert spaces, see e.g. [177,Chapter I, Section 3.1],H∗
0 ⊂ H∗1 ⊂ . . . ⊂ H∗
m ⊂ . . . and
H∗∞(
QNp
)=H∗∞=
⋃
m∈NH−m={T ∈ D′(QN
p
); ‖T‖−l < ∞, for some l ∈ N} (10.4)
as vector spaces. We mention that, sinceH∞ is a nuclear space, the weak and strongconvergence are equivalent in H∗
∞, see e.g. [176, Chapter I, Section 6, Theorem6.4]. We consider H∗
∞ endowed with the strong topology. On the other hand, letB : H∗
∞ ×H∞ → C be a bilinear functional. Then B is continuous in each of itsarguments if and only if there exist norms ‖ · ‖(a)m in H∗
m and ‖ · ‖(b)l in Hl such that|B(T, g)| ≤ M‖T‖(a)m ‖g‖(b)l with M a positive constant independent of T and g, seee.g. [177, Chapter I, Section 1.2] and [176, Chapter I, Section 4.1]. This implies that(10.2) is a continuous bilinear form on H∗
∞ ×H∞, which we will use as a pairingbetween H∗
∞ and H∞.
Remark 10.10 The spaces H∞ ⊂ L2 ⊂ H∗∞ form a Gel’fand triple (also called a
rigged Hilbert space), i.e. H∞ is a nuclear space which is densely and continuouslyembedded in L2 and ‖g‖2L2 = [g, g]. This Gel’fand triple was introduced in [472].
10.2.2 Pseudodifferential Operators Acting onH∞(C)
Definition 10.11 We say that a function a : QNp → R+ is a smooth symbol if it sat-
isfies the following properties:
(i) a is a continuous function;(ii) there exists a positive constant C = C(a) such that a(ξ ) ≥ C for any ξ ∈ QN
p ;(iii) there exist positive constants C0, C1, α, m0, with m0 ∈ N, such that
C0‖ξ‖αp ≤ a(ξ ) ≤ C1‖ξ‖αp for ‖ξ‖p ≥ pm0 .
160 Sobolev-Type Spaces and Pseudodifferential Operators
Given a smooth symbol a(ξ ), we attach to it the following pseudodifferentialoperators:
A : D(
QNp
) → L2 ∩Cg(x) → F−1
ξ→x(a(ξ )Fx→ξg),
A−1 : D(
QNp
) → L2 ∩Cg(x) → F−1
ξ→x(Fx→ξg/a(ξ )).
Notation 10.12 For t ∈ R, we set �t� := min{m ∈ Z;m ≥ t} (the ceiling function)and t! := max{m ∈ Z;m ≤ t} (the floor function). Notice that, for t ≥ 0,
�t� − t! ={
0 t ∈ Z
1 t /∈ Z.
Lemma 10.13 For any l ∈ N, the mapping A : Hl+�2α�(C) → Hl (C) is a well-defined continuous mapping between Banach spaces.
Proof Let g ∈ D(
QNp
)
, then
‖Ag‖2l ≤ ∫
BNm0
[ξ ]lp|a(ξ )|2 |g(ξ )|2dNξ +C21
∫
QNp�BNm0
‖ξ‖l+�2α�p |g(ξ )|2dNξ
≤(
supξ∈BNm0
[ξ ]lp|a(ξ )|2)
‖g‖20 +C21‖g‖2l+�2α� ≤ C2‖g‖2l+�2α�.
Now, by Lemma 10.8 (ii),Ag ∈ Hl (C). The result follows from the density ofD(
QNp
)
inHl+�2α�(C). �Lemma 10.14 For any l ∈ N, the mapping A−1 : Hl (C) → Hl+�2α�(C) is a well-defined continuous mapping between Banach spaces.
Proof Take g ∈ D(
QNp
) ⊂ Hl+1(C) and set u := F−1(g(ξ )/a(ξ )) ∈ L2 ∩C. Then uis the unique solution of Au = g. Now
‖u‖2l+�2α� = ∫
BNm0
[ξ ]l+�2α� |g(ξ )|2|a(ξ )|2 d
Nξ + ∫
QNp�BNm0
‖ξ‖l+�2α�p
|g(ξ )|2|a(ξ )|2 d
Nξ
≤(
supξ∈BNm0
[ξ ]l+�2α�
|a(ξ )|2)
‖g‖20 +1
C20
∫
QNp�BNm0
‖ξ‖l+2(�α�− α!)p |g(ξ )|2dNξ
≤ C′‖g‖20 +1
C20
‖g‖2l+1 ≤ C′′‖g‖2l+1.
By Lemma 10.8 (ii), u ∈ Hl+�2α�(C) and sinceD(
QNp
)
is dense inHl+1(C), the map-
ping A−1 : Hl (C) → Hl+�2α�(C) is well defined and continuous. �Theorem 10.15 (i) The mapping A : H∞(C) → H∞(C) is a bicontinuous isomor-phism of locally convex spaces. (ii) H∞(C) ⊂ L∞ ∩Cunif ∩ L1 ∩ L2. (iii) H∞
(
QNp
)
10.2 The SpacesH∞ 161
is continuously embedded in C0(QNp ,C). This is the non-Archimedean analog of
Sobolev’s embedding theorem.
Proof (i) By Lemma 10.13, A is a well-defined mapping. In addition, byLemma 10.14, A is a bijection fromH∞(C) onto itself. To verify the continuityofA, we take a sequence {gn}n∈N inH∞(C) such that gn d−→ g, with g ∈ H∞(C),i.e. gn ‖ · ‖m−−−→ g, for all m ∈ N. Take m = l + �2α�, gn ∈ Hl+�2α�(C), then byLemma 10.13,Agn ‖ · ‖l−−→ Ag ∈ Hl (C), for any l ∈ N. ThereforeAgn d−→ Ag. Wenow show that A−1 is continuous. Take a sequence {gn}n∈N inH∞(C) as before.By Lemma 10.14, there exists a unique sequence {un}n∈N such that Aun = gn,with gn ∈ Hl (C) and un ∈ Hl+�2α�(C), for any l ∈ N. By verifying that {un}n∈Nis Cauchy in ‖ · ‖m for any m ∈ N, there exists u ∈ H∞(C) such that un d−→ u,i.e. A−1gn d−→ u. By the continuity of A, we have Au = g, now by using that Ais a bijection onH∞(C), we conclude that A−1gn d−→ A−1g.
(ii) In Remark 10.5 (ii), we already noted that, for any function f in HC(∞), f isintegrable, and thus f ∈ Cunif, and by the Riemann–Lebesgue theorem f ∈ L∞.We now define for α > 0, the operator
(
˜Dα f)
(x) = F−1ξ→x
(
[ξ ]αpFx→ξ ( f ))
.
By part (i), ˜Dα gives rise to a bicontinuous isomorphism of H∞(C). Then, for
any f ∈ H∞(C), ˜Dα f ∈ H∞(C), and thus
˜Dα f = [ξ ]αpf ∈ L1, which impliesthat there are positive constants C0,C1 such that
∣
∣f (ξ )∣
∣ ≤ C0
[max(1,C1‖ξ‖p)]α . (10.5)
For k ∈ N, we set
Ak( fχp(·, ξ )) :=∫
QNp(
p−k‖x‖p)
f (x)χp(x · ξ )dNx,
where(p−k‖x‖p) is the characteristic function of the ball BNk . By using the factthat f (y) = f (−y), and that
Fx→y((p−k‖x‖p)χp(−x · ξ )) = pkN(pk‖y− ξ‖p),we have
Ak( fχp(·, ξ )) =∫
QNp(p−k‖x‖p)χp(x · ξ )f (−x)dNx
= ∫
QNpFx→y((p−k‖x‖p)χp(−x · ξ ))f (y)dNx
= pkN∫
‖y−ξ‖p≤p−kf (y)dNy.
Now, from (10.5),
pkN∫
‖y−ξ‖p≤p−k
∣
∣f (y)∣
∣dNy ≤ C0
[max(1,C1‖ξ‖p)]α
162 Sobolev-Type Spaces and Pseudodifferential Operators
for k big enough. Therefore limk→∞ Ak( fχp(·, ξ )) exists for every ξ ∈ QNp , in
particular limk→∞∫
‖x‖p≤pk f (x)dNx exists, which means that f ∈ L1.
(iii) By Remark 10.5 (ii), f ∈ L1 for any f ∈ H∞. Then, f is continuous and, by theRiemann–Lebesgue theorem (see e.g. [402, Theorem 1.6]), f ∈ C0
(
QNp
)
. On the
other hand, ‖ f‖L∞ ≤ ‖f‖L1 ≤ C(N, l)‖ f‖l , which shows thatHl is continuouslyembedded in C0
(
QNp
)
for l > N. Thus H∞ ⊂ C0(
QNp
)
. Now, if fm d−→ f in H∞,
i.e. if fm ‖ · ‖l−−→ f inHl for any l ∈ N, then fm ‖ · ‖L∞−−−→ f inC0(
QNp
)
. �
Remark 10.16 (i) Let f(ξ ) ∈ Qp[ξ1, . . . , ξN] be a non-constant polynomial of degreeD. The pseudodifferential operator with symbol |f(ξ )|αp, α > 0, is defined asA(∂, f, α)φ(x) = F−1
ξ→x(|f(ξ )|αpFx→ξφ) for φ ∈ D. By using a simple modifica-tion of the argument given to prove the first part in Theorem 10.15 and the factthat |f(ξ )|αp ≤ C[ξ ]αDp , one can show that A(∂, f, α) : H∞(C) → H∞(C) givesrise to a well-defined continuous operator. The problem of finding an inverse forA(∂, f, α), i.e. the problem of the existence of fundamental solutions, will be stud-ied in the next section.
(ii) It follows from the second part of Theorem 10.15 that f ∈ C0(
QNp
)
for f ∈H∞(C).
10.3 A Hörmander–Łojasiewicz-Type Estimation
In this section, we establish a non-Archimedean version of a Hörmander-Łojasiewiczinequality, see [202], [316]. This inequality plays a crucial role in our proof of theexistence of fundamental solutions for pseudodifferential operators of typeA(∂, f, α).In [404, Theorems 4.1 and 4.2], Taylor, Varadarajan, Virtanen, and Weisbart provedthis inequality. We present here our proof, see [473], which covers a particular case,but that gives an explicit description of the constants appearing in the inequality. Thisfact is very important in our discussion. Our proof and the proof of Taylor et al. workin arbitrary characteristic. For the sake of simplicity, we formulate the Hörmander–Łojasiewicz inequality in QN
p .Let g(ξ ) be a non-constant polynomial in Zp[ξ1, . . . , ξN] such that g(0) = 0. We
denote
V (g;Qp) := V (g) = {
x ∈ QNp ; g(x) = 0
}
,
Vsing(g;Qp) := Vsing(g) = {x ∈ V (g); ∇g(x) = 0},and
Vreg(g;Qp) := V (g) �Vsing(g).
We set
d(x,V (g)) = infy∈V (g)
‖x− y‖p
10.3 A Hörmander–Łojasiewicz-Type Estimation 163
for the distance from x to V (g). This is a continuous function from QNp into R+ =
{z ∈ R; z ≥ 0}.Theorem 10.17 ([185, Theorem 1], [69, Theorem 2.1]) Assume that g(x) ∈Zp[x1, . . . , xN] � Zp. There exists a positive integer α, depending on g, such that forany positive integer l and y ∈ ZN
p satisfying g(y) ≡ 0 mod pαl , there exists y ∈ ZNp
satisfying g(y) = 0 and y ≡ y mod pl.
We call α := α(g) the Greenberg constant of g.
Lemma 10.18 Let g(ξ ) be a non-constant polynomial in Zp[ξ1, . . . , ξN] such thatg(0) = 0. Then
|g(x)|p ≥ pα2+α−1d(x,V (g))α for x ∈ ZN
p . (10.6)
Proof
ClaimA Let g(ξ ) be a non-constant polynomial as in the statement of Lemma 10.18.Then
|g(x)|p ≥ pα2+α−1d
(
x,V (g) ∩ ZNp
)αfor x ∈ ZN
p . (10.7)
The result follows from Claim A on making the following observation:
d(x,V (g)) = d(
x,V (g) ∩ ZNp
)
for x ∈ ZNp . (10.8)
Since V (g) ∩ ZNp is compact, there exists y0 ∈ V (g) ∩ ZN
p such that d(x,V (g) ∩ ZNp )
= ‖x− y0‖p. By using the fact that 0 ∈ V (g), one obtains that ‖x− y0‖p ≤ ‖x‖p. Thisinequality implies (10.8). Indeed, for x ∈ ZN
p ,
d(x,V (g)) = inf{‖x− y‖p; y ∈ V (g)}= inf
({‖x‖p; y ∈ V (g) ∩ (
QNp � ZN
p
)} ∪ {‖x− y‖; y ∈ V (g) ∩ ZNp
})
= inf{‖x− y‖p; y ∈ V (g) ∩ ZN
p
}
.
Proof of Claim A We first note that, for x ∈ ZNp , ord (g(x)) ∈ {0, . . . , α − 1} ∪
{αl + j; l ∈ N � {0}, j = 0, . . . , α − 1} ∪ {∞}. Consider first the case ord (g(x)) =αl + j, i.e. that |g(x)|p = p−αl− j. In this case g(x) ≡ 0 mod pαl and by Greenberg’stheorem there exists y ∈ ZN
p such that x ≡ y mod pl , i.e. ord(x− y) ≥ l, from which
supy∈V (g)∩ZNp
ord(x− y) = maxy∈V (g)∩ZNp
ord(x− y) ≥ l,
since V (g) ∩ ZNp is compact. Then
d(
x,V (g) ∩ ZNp
)α = p−αmaxy∈V (g)∩ZNp ord(x−y) ≤ p−lα ≤ |g(x)|ppα−1. (10.9)
Now, we consider the case |g(x)|p = p− j, j ∈ {0, 1, . . . , α − 1}, i.e. g(x) = pju j,u j ∈ Z×
p . This implies that 0 ≤ ord(x− y) < α for any y ∈ V (g) ∩ ZNp . Indeed,
164 Sobolev-Type Spaces and Pseudodifferential Operators
suppose that there exists y0 ∈ V (g) ∩ ZNp such that ord(x− y0) = m ≥ α, i.e.
x = y0 + pmw, with w ∈ Z×p . Now
g(x) = g(y0)+ pmz = pmz, with z ∈ ZNp ,
implies that pm divides g(x) in Zp, for some m ≥ α, which is impossible. Therefore,0 ≤ maxy∈V (g)∩ZNp ord(x− y) < α, which implies that
p−α2< d(x,V (g) ∩ ZN
p )α ≤ 1.
Now
|g(x)|pd(x,V (g))α
≥ p− j
p−α2 ≥ p−(α−1)
p−α2 = pα2+α−1 for any j ∈ {0, 1, . . . , α − 1}.
(10.10)Finally, estimation (10.7) follows from (10.9) and (10.10). �
Theorem 10.19 Let g(ξ ) be a non-constant polynomial in Zp[ξ1, . . . , ξN] of degreeD satisfying g(0) = 0. Then
|g(x)|p ≥ pα2+α−1d(x,V (g))α
[x]γpfor x ∈ QN
p , (10.11)
where α is the Greenberg constant of g and γ = α2 + (D+ 1)(α − 1).
Proof If x ∈ ZNp , then estimation (10.11) is a consequence of Lemma 10.18. In the
case, x ∈ QNp � ZN
p , by using that
d(x,Vreg(g)) ≥ d(x,V (g)),
it is sufficient to show that
|g(x)|p ≥ pα2+α−1d(x,Vreg(g))α
[x]γpfor x ∈ QN
p � ZNp .
Thus, we may assume that x ∈ Vreg(g) when considering the case x ∈ QNp � ZN
p . Takex ∈ QN
p � ZNp , then (∂g/∂ξi)(x) = 0 for some index i. We assume that i = 1, and,
upon applying the non-Archimedean implicit function theorem, see e.g. [205, Theo-rem 2.1.1], there is a coordinate system (ξ ′
1, . . . , ξ′N ) around x, such thatV (g) ∩ BNe =
{(ξ ′1, . . . , ξ
′N ) ∈ BNe ; ξ ′
1 = g(x)}, thus d(x,V (g)) = d(0,V (g)) ∩ BNe = |g(x)|p. Now,by using the fact that |g(x)|p ≤ [x]Dp for x ∈ QN
p � ZNp , estimation (10.11) follows
from
pα2+α−1d(x,V (g))α
[x]γp |g(x)|p = pα2+α−1|g(x)|α−1
p
[x]γp≤ pα
2+α−1[x](α−1)Dp
[x]γp
≤ supx∈QN
p�ZNp
pα2+α−1
|x|γ−(α−1)Dp
= 1,
by taking γ = α2 + (D+ 1)(α − 1). �
10.4 The SpacesW∞ 165
10.4 The Spaces W∞From now on, we fix a polynomial f(ξ ) ∈ Zp[ξ1, . . . , ξN] � Zp such that f(0) = 0.We set
Lf
(
QNp
)
:= {
ϕ ∈ D(
QNp
); ϕ |V (f) ≡ 0}
.
Here the symbol “≡” is used to mean identically zero. We call Lf
(
QNp
)
the p-adicLizorkin space of test functions along V (f). In the special case f(ξ ) = ∏N
i=1 ξi, thespace Lf
(
QNp
)
is called the p-adic Lizorkin space of test functions of the first kind.These spaces were introduced by Albeverio, Khrennikov, and Shelkovich, see [18]and the references therein. Now, for l,m ∈ N and ϕ, θ ∈ Lf
(
QNp
)
, we define the scalarproduct
〈ϕ, θ〉l,m,f :=∫
QNp
[ξ ]lpd(ξ,V (f))m
ϕ(ξ )θ (ξ )dNξ,
where the bar denotes the complex conjugate. We also set
‖ϕ‖2l,m,f = 〈ϕ, ϕ〉l,m,f. (10.12)
For m ∈ N and ϕ, θ ∈ Lf
(
QNp
)
, we define the scalar product
〈ϕ, θ〉m,f :=∑
i+ j≤mi≥0, j≥0
〈ϕ, θ〉i, j,f,
and set
‖ϕ‖2m,f =∑
i+ j≤mi≥0, j≥0
‖ϕ‖2i, j,f.
Notice that, if
"m(ξ ; f) := "m(ξ ) =∑
i+ j≤mi≥0, j≥0
[ξ ]ipd(ξ,V (f)) j
,
then
‖ϕ‖2m,f =∫
QNp
"m(ξ )|ϕ(ξ )|2dNξ .
In addition, ‖ · ‖m ≤ ‖ · ‖r for r ≥ m. Denote by Wm(
QNp ; f
)
:= Wm the complexHilbert space obtained by completing L f
(
QNp
)
with respect to ‖ · ‖m. Then Wr ↪→Wm (continuous embedding) for r ≥ m. We setW∞
(
QNp ; f
)
:= W∞ = ∩l∈NWl . Withthe topology induced by the family of norms ‖ · ‖m∈N,W∞ becomes a locally convexspace, which is metrizable. Indeed,
df(h, g) := maxm∈N
{
2−m‖h− g‖m,f
1+ ‖h− g‖m,f
}
, h, g ∈ W∞, (10.13)
is a metric for the topology of W∞ considered as a convex topological space.
166 Sobolev-Type Spaces and Pseudodifferential Operators
Notice that, since ‖g‖l ≤ ‖g‖l,f for l ∈ N and g ∈ Lf
(
QNp
)
, we haveWl ↪→ Hl forany l ∈ N and W∞ ↪→ H∞.
Lemma 10.20 With the above notation, the following assertions hold:
(i) Lf
(
QNp
)
is dense in L2(QNp , d
Nx);(ii) Wl
(
QNp
) = {g ∈ L2(
QNp
); ‖g‖l,f < ∞} = {T ∈ D′(QNp
); ‖T‖l,f < ∞};(iii) W∞
(
QNp
) = {g ∈ L2(
QNp
); ‖g‖l,f < ∞, for every l ∈ N};(iv) W∞
(
QNp
) = {T ∈ D′(QNp
); ‖T‖l,f < ∞, for every l ∈ N}.The equalities in (ii)–(iv) are in the sense of vector spaces.
Proof (i) Since D(
QNp
)
is dense in L2(
QNp
)
, it is sufficient to show that, for anyϕ ∈ D
(
QNp
)
, there is a sequence {ϕm}m∈N of functions in Lf
(
QNp
)
such that ϕm‖ · ‖L2−−−→ ϕ. We fix ϕ ∈ D
(
QNp
)
. For m ∈ N, we set
�m,f(ξ ) :={
1 if d(ξ,V (f)) < p−m
0 if d(ξ,V (f)) ≥ p−m,
and
ϕm(ξ ) := {1−�m,f(ξ )}ϕ(ξ ) for m ∈ N.
Claim. ϕm ∈ Lf
(
QNp
)
for m ∈ N.By using the claim and the dominated convergence theorem,
limm→∞‖ϕ − ϕm‖2L2 = lim
m→∞
∫
QNp
∣
∣
(
ϕ − ϕm)
(ξ )∣
∣
2dNξ
= limm→∞
∫
QNp
�m,f(ξ )|ϕ(ξ )|2dNξ =∫
QNp
1V (f)(ξ )|ϕ(ξ )|2dNξ = 0
since V (f) has dNξ -measure zero.
Proof of the claim. It is sufficient to show that �m,f(ξ ) is a locally constantfunction. Take ξ0 ∈ QN
p such that�m,f(ξ0) = 1, then there is a sequence {yl}l∈Nof points of V (f) ∩ BN−(m+1)(ξ0) such that
liml→∞
‖ξ0 − yl‖p = d(ξ0,V (f)) = d(ξ0,V (f) ∩ BN−(m+1)(ξ0))
and d(ξ0,V (f) ∩ BN−(m+1)) ≤ ‖ξ0 − yl‖p < p−m for any l ∈ N. Consider a pointy such that ‖ξ0 − y‖p < p−m, then ‖y− yl‖p < p−m for any l, this implies that
d(y,V (f) ∩ BN−(m+1)(ξ0)) = infy∈V (f)∩BN−(m+1)(ξ0 )
‖ξ0 − y‖p ≤ infl∈N
‖ξ0 − yl‖p < p−m,
and thus�m,f |BN−(m+1)(ξ0 )≡ 1. Now consider ξ0 ∈ QN
p such that�m,f(ξ0) = 0, i.e.
d(ξ0,V (f)) ≥ p−m. Then there is a ball BN−l (ξ0) such that BN−l (ξ0) ∩V (f) = ∅,i.e. �m,f |BN−l (ξ0 )≡ 0.
10.4 The SpacesW∞ 167
(ii) In order to prove the first equality, it is sufficient to show that if g ∈ L2 and
‖g‖l,f < ∞ then g ∈ Wl . The condition ‖g‖l < ∞ is equivalent to [ξ ]l2p g ∈
L2(
QNp
)
, which implies that{[ξ ]
l2p g
}
(−ξ ) ∈ L2(
QNp
)
. By the density of Lf
(
QNp
)
in L2(
QNp
)
, there is a sequence {gk}k∈N in Lf
(
QNp
)
such that gk(ξ ) ‖ · ‖L2−−−→{[ξ ]
l2p g
}
(−ξ ), which implies that gk ‖ · ‖L2−−−→ [ξ ]l2p g, which is equivalent to
F−1(gk/[ξ ]l2p )‖ · ‖l,f−−−→ g with gk/[ξ ]
l2p ∈ Lf
(
QNp
)
for any k ∈ N. To establish the
second equality, we note that, since ‖ · ‖0 ≤ ‖ · ‖l,f for any l ∈ N, if T ∈ Wl ,thenT ∈ L2, and thus T ∈ D′(QN
p
)
and ‖T‖l,f < ∞. Conversely, if T ∈ D′(QNp
)
and ‖T‖l,f < ∞, then T ∈ L2 and ‖T‖l,f < ∞.(iii), (iv) follow from (ii). �
10.4.1 The Dual Space ofW∞For m ∈ N and ϕ ∈ Lf
(
QNp
)
, we set
‖ϕ‖2−m,f :=∑
i+ j≤mi≥0, j≥0
‖ϕ‖2−i,− j,f,
and denote byW−m(QNp ; f) := W−m the complex Hilbert space obtained as the com-
pletion ofLf
(
QNp
)
with respect to ‖ · ‖−m,f. Denote byW∗m(Q
Np ; f) := W∗
−m the strongdual space ofWm. We identifyW∗
m andW−m by associating T ∈ W−m with the func-tional on Wm given by
[T, g] :=∫
QNp
∫
T (ξ )g(ξ )dNξ . (10.14)
Notice that
|[T, g]| ≤(
1
m+ 1
)2
‖T‖−m,f‖g‖m,f. (10.15)
In addition, the strong dual space of W∞ is
W∗∞(
QNp ; f
) = W∗∞ =
⋃
m∈NW−m
= {
T ∈ D′(QNp
); ‖T‖l,f < ∞, for some l ∈ N}
, (10.16)
as vector spaces. SinceH∞ is a nuclear space and any subspace of a nuclear space isalso nuclear,W∞ is nuclear and thus weak and strong convergence are equivalent inW∗
∞. We consider W∗∞ to be endowed with the strong topology.
Lemma 10.21 H∗∞ ↪→ W∗
∞ (continuously embedded).
Proof A sequence of functionals {Tk}k∈N in H∗∞ (in W∗
∞), converges weakly to afunctional T if and only if all the Tk are functionals on the same normed space Hl
168 Sobolev-Type Spaces and Pseudodifferential Operators
(Wl), and converge weakly in this space, i.e. for every g in Hl (in Wl), the relation[Tk, g] → [T, g] holds, see [176, Section 5.6]. Using the fact that H∗
∞ ⊂ W∗∞, this
result implies thatH∗∞ is continuously embedded inW∗
∞ for the weak topology. Thenext step consists of showing thatH∞ andW∞ are perfect spaces (i.e. spaces where abounded and closed subset is compact), because in spaces of this type the strong andweak topologies agree, see [176, Section 6.3]. The nuclear spaces are perfect spaces,see [177, Section 3.4]. Finally, since H∞ is a nuclear space, and any subspace of anuclear space is also nuclear, W∞ is nuclear. �
10.5 Pseudodifferential Operators on W∞In this section we fix a non-constant polymonial f with coefficients in Zp such that
f(0) = 0 and |f(ξ )|p ≥ Cd(ξ,V (f))β
[ξ ]γp(10.17)
for some positive constants C > 1, β, γ , with β, γ ∈ N.We attach to f and α > 0 the following pseudodifferential operator:
Lf
(
QNp
) → Lf
(
QNp
)
g→ A(∂, f, α)g
with (A(∂, f, α)g)(x) := F−1ξ→x(|f(ξ )|αpFx→ξg). In the case α = 1, we use the notation
A(∂, f, 1) = A(∂, f).
Theorem 10.22 Assume that polynomial f satisfies hypothesis (10.17). Then, themapping
W∞ → W∞
u → A(∂, f, α)u
gives rise to a well-defined isomorphism of locally convex spaces. Furthermore,A−1(∂, f, α)g= F−1
ξ→x((1/|f(ξ )|αp )Fx→ξg) for g ∈ W∞.
Proof For the sake of simplicity, we prove the result for operators A(∂, f), since theproof of the general case is a simple variation of this particular case.We first show that A(∂, f) : W∞ → W∞ is a well-defined linear continuous oper-
ator. This fact follows from the following claim.
Claim 1. Let d denote the degree of f. Then, for u ∈ Wl+2d , with l ∈ N,
‖A(∂, f)u‖l,f ≤ ‖u‖l+2d,f. (10.18)
Indeed, according to Claim 1, A(∂, f) : Wl+2d → Wl is a well-defined linearbounded operator, which implies that A(∂, f) : W∞ → W∞ is a well-defined linearoperator. Since (W∞, df) is a complete metric space, the continuity ofA(∂, f) followsfrom (10.18) using a standard argument based on sequences.
10.5 Pseudodifferential Operators onW∞ 169
Proof of Claim 1. By using that |f(ξ )|Qp ≤ [ξ ]dp, we have
‖A(∂, f)u‖2l,f =∫
QNp
"l (ξ )|f(ξ )|2Qp|u(ξ )|2dNξ
≤∑
i+ j≤li≥0, j≥0
∫
QNp
[ξ ]i+2dp
d(ξ,V (f)) j|u(ξ )|2dNξ ≤ ‖u‖2l+2d,f,
which implies that ‖A(∂, f)u‖l,f < ∞, and thus A(∂, f)u ∈ Wl by Lemma 10.20 (ii).
Claim 2. A(∂, f) : W∞ → W∞ is an injective operator.The condition A(∂, f)u = 0 implies that u ≡ 0 almost everywhere, but since
W∞(QNp ; f) ↪→ H∞(C) ↪→ C0
(
QNp
)
, we have u ≡ 0, cf. Theorem 10.15 (iii).Finally, we show that A(∂, f) : W∞ → W∞ is a surjective operator with a contin-
uous inverse. This fact follows from the following claim.
Claim 3. Given v ∈ W∞, the equation A(∂, f)u = v has a unique solution u ∈ W∞satisfying
‖u‖l,f ≤ C‖v‖l+2(β+γ ),f, (10.19)
for any l ∈ N with β and γ given in (10.17).By Claim 3, the operator
A−1(∂, f) : Wl+2(β+γ ) → Wl
v → u
is a well-defined continuous operator for any l ∈ N, which implies thatA−1(∂, f) : W∞ → W∞ is a well-defined linear operator. Now take a sequence{vm}m∈N inW∞ such that vm df−→ v, i.e. vm ‖ · ‖l,f−−−→ v for any l ∈ N, with u = A−1(∂, f)v.
Then, by (10.19), ‖A−1(∂, f)(vm − v)‖l,f ≤ C‖vm − v‖l+2(β+γ ),f, which implies thatA−1(∂, f) is a continuous operator.
Proof of Claim 3. Take v ∈ Wl+2(β+γ ) and u = v/|f|p. Then, by using hypothesis(10.17),
‖u‖2l,f =∫
QNp
"l (ξ )|v(ξ )|2|f(ξ )|2p
dNξ ≤ C−2∫
QNp
"l (ξ )[ξ ]2γp
d(ξ,V (f))2β|v(ξ )|2dNξ
= C−2∑
i+ j≤li≥0, j≥0
∫
QNp
[ξ ]i+2γp
d(ξ,V (f)) j+2β|v(ξ )|2dNξ ≤ C−2‖v‖2l+2(β+γ ),f,
by Lemma 10.20 (ii), with u ∈ Wl , which implies u ∈ W∞ if v ∈ W∞. The unique-ness of the solution follows from Claim 2. �
170 Sobolev-Type Spaces and Pseudodifferential Operators
10.5.1 Adjoint Operators onW∞By using the fact that A(∂, f, α) : W∞ → W∞ is a continuous operator and someresults on adjoint operators in the setting of locally convex spaces, see e.g.[451, Chapter VII, Section 1], one gets that there exists a continuous operatorA∗(∂, f, α) : H∗
∞ → H∗∞ satisfying
[A∗(∂, f, α)T, g] = [T,A(∂, f, α)g]
for any T ∈ H∗∞ and any g ∈ H∞. We call A∗(∂, f, α) the adjoint operator of
A(∂, f, α). We denote the adjoint operator of A−1(∂, f, α) by A−∗(∂, f, α).
10.6 Existence of Fundamental Solutions
Remark 10.23 By the proof of Remark 10.5 (ii), if g ∈ H∞(C), then g ∈ L1 ∩ L2
and, by the dominated convergence theorem, g(0) = ∫
g|dNξ |Qp . Consequently,
[
1, g] =
∫
QNp
g(ξ )|dNξ |Qp = g(0),
and thus1 defines an element of H∗∞(C) ↪→ W∞, which we identify with the Dirac
distribution δ, i.e. [δ, g] = g(0). In addition, δ ∗ g= g for any g ∈ H∞(C). Indeed,take gn ‖ · ‖l−−→ g for any l ∈ N, with {gn}n∈N in D and g ∈ H∞(C). Then ‖δ ∗ gn −g‖l = ‖gn − g‖l → 0, since δ ∗ gn = gn, for any l ∈ N, which means that g→ δ ∗ gis continuous in D, which is dense inH∞(C).
Lemma 10.24 There exists E = E(f, α) ∈ W∗∞ such that A∗(∂, f, α)E = δ inW∗
∞.
Proof By using that A−∗(∂, f, α)T ∈ W∗∞ for any T ∈ W∗
∞, we have
[A−∗(∂, f, α)T, g] = [T,A−1(∂, f, α)g] =∫
QNp
T (ξ )g(ξ )
|f(ξ )|αpdNξ . (10.20)
Now we take T = 1, and then the right-hand side of (10.20) gives rise to elementE = E(f, α) ∈ W∗
∞, i.e.
[E, g] :=∫
QNp
g(ξ )
|f(ξ )|αpdNξ, (10.21)
which satisfies
[A∗(∂, f, α)E, g] = [E,A(∂, f, α)g] =∫
QNp
g(ξ )dNξ = g(0) = [δ, g].
�
10.6 Existence of Fundamental Solutions 171
Remark 10.25 (i) Notice that A∗(∂, f, α) is an extension of A∗(∂, f, α), since by(10.16)W0 ⊂ W∗
∞ andW0 = L2 ⊃ W∞. In addition,
[E, g] =∫
QNp
g(ξ )
|f(ξ )|αpdNξ =
[
1
|˜f|αp, g
]
inW∗∞,
where˜f(ξ ) := f(−ξ ). In particular E =1/|f|αp ∈ L′f
(
QNp
)
.
(ii) E = E(x) = Fξ→x[1/|f|αp] defines a distribution fromD′(QNp
)
. Indeed, by usingthe fact that the Fourier transform acts isomorphically from Lf
(
QNp
)
to D(QNp �
V (f)), and that(
1
|˜f|αp, φ
)
=(
1
|˜f|αp,Fφ
)
for φ ∈ Lf
(
QNp
)
,
we get that E ∈ D′(QNp �V (f)) ↪→ D′(QN
p
)
.
(iii) For ϕ∈D(
QNp
)
, we set ϕ(x) := ϕ(−x), and for T ∈ D′(QNp
)
, we set (˜T , ϕ) =(T, ϕ). We recall that F (F (T )) = ˜T . Then, the following formula holds:
Fx→ξ [E(x) ∗ φ(x)] = Eφ =φ
|f(ξ )|αpfor φ ∈ Lf
(
QNp
)
.
Lemma 10.26 The mapping
(Lf
(
QNp
)
, df) → (W∞(QNp ), df)
g(x) → E(x) ∗ g(x) (10.22)
gives rise to a well-defined continuous operator. Since (Lf
(
QNp
)
, df) is dense in(W∞
(
QNp
)
, df), the operator (10.22) has a continuous extension to (W∞(
QNp
)
, df),which we denote as E ∗ g.
Proof According to Remark 10.25 (iii), E ∗ g is a locally constant function, and thusit defines a distribution from L′
f
(
QNp
)
. Then
‖u‖2l,f =∫
QNp
"l (ξ )|g(ξ )|2|f(ξ )|2αp
dNξ ≤ C−2∫
QNp
"l (ξ )[ξ ]2γαp
d(ξ,V (f))2βα|g(ξ )|2dNξ
= C−2∑
i+ j≤li≥0, j≥0
∫
QNp
[ξ ]i+2γαp
d(ξ,V (f)) j+2βα|g(ξ )|2dNξ .
Now, by using∫
QNp
· dNξ =∫
d(ξ,V (f))≥1
· dNξ +∫
d(ξ,V (f))<1
· dNξ,
172 Sobolev-Type Spaces and Pseudodifferential Operators
we have
‖u‖2l,f ≤ C−2
⎧
⎪
⎪
⎨
⎪
⎪
⎩
∑
i+ j≤li≥0, j≥0
∫
QNp
[ξ ]i+�2γα�p
d(ξ,V (f)) j+ 2βα! |g(ξ )|2dNξ
+∑
i+ j≤li≥0, j≥0
+∫
QNp
[ξ ]i+�2γα�p
d(ξ,V (f)) j+�2βα� |g(ξ )|2dNξ
⎫
⎪
⎪
⎬
⎪
⎪
⎭
≤ 2C−2∑
i+ j≤l+�2γα�+�2βα�i≥0, j≥0
∫
QNp
[ξ ]ipd(ξ,V (f)) j
|g(ξ )|2dNξ
= 2C−2‖g‖2l+�2γα�+�2βα�,f, (10.23)
and consequently, by Lemma 10.20 (iv), E ∗ g ∈ W∞. Now, take a sequence {gm}m∈Nin (Lf
(
QNp
)
, d) satisfying gm df−→ 0, i.e. gm ‖ · ‖l,f−−−→ 0 for any l ∈ N. Then, from (10.23),
we obtain that ‖Em ∗ g‖l,f ‖ · ‖l,f−−−→ 0 for any l ∈ N, which establishes the continuity of
(10.22). �
Theorem 10.27 With the above notation, the following assertions hold: (i) the func-tion u = E ∗ g ∈ W∞ ⊂ L′
f
(
QNp
)
is the unique solution of the equation A(∂, f, α)u =g, with g ∈ W∞ inW∞; (ii) A∗(∂, f, α)E = δ inW∗
∞.
Proof (i) Consider first the equation
A(∂, f, α)um = gm,with gm ∈ Lf
(
QNp
)
. (10.24)
Then by Remark 10.25, um = E ∗ gm is a solution of (10.24) in L′f
(
QNp
)
. As aconsequence of the density of Lf
(
QNp
)
in W∞, there is a sequence {gm}m∈N inLf
(
QNp
)
such that gm df−→ g. Then, there is a sequence {um}m∈N in Lf
(
QNp
)
satisfy-
ing A(∂, f, α)um = gm. By using the fact that A(∂, f, α) : W∞ → W∞ is continu-ous, we get that A(∂, f, α)u = g, where u = limm→∞ um ∈ W∞. In addition, u =A−1(∂, f, α)g since A−1(∂, f, α) : W∞ → W∞ is a well-defined operator. Now, byusing Lemma 10.26, and taking the limits in um = E ∗ gm, we get u = E ∗ g=A−1(∂, f, α)g, i.e. u considered as an element of L′
f(QNp ) is the unique solution of
A(∂, f, α)u = g.(ii) follows from Lemma 10.24. �
10.7 Igusa’s Local Zeta Functions and Fundamental Solutions
We set for a > 0 and s ∈ C, as := es ln a. Let f(ξ ) ∈ Qp[ξ1, . . . , ξN] be a non-constantpolynomial. The p-adic complex power |f|sp associated with f (also called the Igusa
10.7 Igusa’s Local Zeta Functions and Fundamental Solutions 173
zeta function of f) is the distribution
(|f|sp, φ) :=∫
QNp�f−1(0)
φ(ξ )|f(ξ )|spdNξ,
where s ∈ C, Re(s) > 0, φ ∈ D. The Igusa local zeta functions are connected withthe number of solutions of polynomial congruences mod pm and with exponentialsums mod pm. There are many intriguing conjectures connecting the poles of localzeta functions with the topology of complex singularities, see e.g. [120], [205]. Thelocal zeta functions can be defined on any locally compact field K, i.e. for R, C, orfinite extensions of Qp or of Fp((t )), with Fp the finite field with p elements. In theArchimedean case K = R or C, the study of local zeta functions was initiated byI. M. Gel’fand and G. E. Shilov [175]. The main motivation was the constructionof fundamental solutions for partial differential operators with constant coefficients.Indeed, the meromorphic continuation of the local zeta functions implies the exis-tence of fundamental solutions. This fact was established, independently, by Atiyah[50] and Bernstein [80], see also [205, Theorem 5.5.1 and Corollary 5.5.1]. On theother hand, in the mid 1960s, A. Weil initiated the study of local zeta functions, in theArchimedean and non-Archimedean settings, in connection with the Poisson–Siegelformula. In the 1970s, Igusa developed a uniform theory for local zeta functions overlocal fields of characteristic zero [205].
Theorem 10.28 ([205, Theorem 8.2.1]) For Re(s) > 0, |f|sp defines a D′-valuedholomorphic function, and it has a meromorphic continuation to the whole complexplane such that (|f|sp, φ) is a rational function of p−s. More precisely, there exists afinite collection of pairs of non-negative integers {(NE , vE ) ∈ N × (N�{0});E ∈ T }depending only on f, such that
∏
E∈T(1− p−vE−NEs) · |f|sp
becomes a D′-valued holomorphic function on the whole complex plane.
Remark 10.29 (i) Notice that the possible poles of |f|sp have the form s =−vE/NE + (2π i/(NE ln p))Z.
(ii) Let f : QNp −→ Qp be a polynomial mapping satisfying f(0) = 0. Let
{(NE , vE ) ∈ N × N�{0};E ∈ T }
be as in Theorem 10.28. Set λ := λ(f) = minE vE/NE. Then −λ is the real partof a pole of (|f|sp, φ) for some φ ∈ D, cf. [415, Theorem 2.7] or [204]. Noticethat −λ < 0. This result implies that
∫
QNp�f−1(0)
φ(ξ )
|f(ξ )|βpdNξ < +∞ for any β satisfying 0 < β < λ(f).
174 Sobolev-Type Spaces and Pseudodifferential Operators
(iii) Given γ ∈ C, there exists a Laurent expansion of |f|sp around s = γ of the form
|f|sp =∑
k∈Zck(s+ γ )k, with ck ∈ D′.
For a proof of this fact, in the classical setting, the reader may consult [205,pp. 65–67]. This argument also works in the p-adic case, see e.g. [470]. We saythat γ is a pole of |f|sp if ck = 0 only for a finite number of values of k < 0. If |f|spdoes not have poles, we say that |f|sp is a holomorphic distribution on D.
The existence of a meromorphic continuation for |f|sp implies the existence of fun-damental solutions for operators of the form
(A(∂, f, α)φ)(x) = F−1ξ→x(|f(ξ )|αpFx→ξφ), for φ ∈ D,
see [462]; see also [470] and the references therein. In addition, for particular polyno-mials the Gel’fand–Shilov method gives explicit formulae for fundamental solutions.In this framework, we say that Eα ∈ D′, with α > 0, is a fundamental solution for
A(∂, f , α)u = φ, with φ ∈ D, (10.25)
if u = Eα ∗ φ is a solution of (10.25) in D′. It is important to mention that we cannotuse the standard definition of fundamental solution, i.e. A(∂, f, α)Eα = δ, because Dis not invariant under the action of A(∂, f, α). By using the meromorphic continua-tion of |f|sp and the Gel’fand–Shilov method one shows that Eα|f|αK = 1 in D′, whichimplies the following result.
Theorem 10.30 There exists a fundamental solution for A(∂, f, α)u = φ, withφ ∈ D.
We give another application of Theorem 10.28 to the existence of fundamentalsolutions. Let E(x) = Fξ→x[1/|˜f(ξ )|αp] be the fundamental solution introduced in
Lemma 10.24, according to Remark 10.25 (ii), Fξ→x[1/|˜f(ξ )|αp] ∈ D′ and by The-
orem 10.28, |˜f(ξ )|sp has a meromorphic continuation to the whole complex plane, if
−α is not a pole of |˜f(ξ )|sp, then Fξ→x[1/|f(ξ )|αp] can be extended to a distributionfrom D′.
Theorem 10.31 Let {(ME , lE ) ∈ N × (N�{0});E ∈ T } be a finite collection of pairsof non-negative integers {(ME , lE ) ∈ N × (N�{0});E ∈ T } depending only on ˜f,such that
∏
E∈T(1− p−lE−MEs) · |˜f|sp
becomes aD′-valued holomorphic function on the whole complex plane.Then E(x) =Fξ→x[1/|f(ξ )|αp] can be extended to a distribution fromD′(QN
p
)
for α /∈ {lE/ME;E ∈T }.
This result was established by Kochubei, see [275, Theorem 2.8], for the case inwhich˜f is a quadratic form.
10.8 Local Zeta Functions and Pseudodifferential Operators inH∞ 175
10.8 Local Zeta Functions and Pseudodifferential Operators inH∞In this section we present a new type of local zeta function and summarize, with-out proofs, some connections with pseudodifferential operators for the framework ofnon-Archimedean fields. For a detailed presentation the reader may consult [471].The new local zeta functions are defined by integrating complex powers of normsof polynomials multiplied by functions in H∞. The new local zeta functions admitmeromorphic continuations to the whole complex plane, but they are not rationalfunctions in p−s. The real parts of the possible poles have a description similar to thatfor the poles of Archimedean zeta functions, except that they can be irrational realnumbers, whereas those in the classical case are rational numbers. We also presentseveral results relating local zeta functions and the existence of fundamental solutionsfor pseudodifferential equations in the spaces H∞.As before, we fix a non-constant polynomial f in Zp[ξ1, . . . , ξN]. We set˜f(ξ ) :=
f(−ξ ). Then |˜f|sK , Re(s) > 0, defines a distribution from D′ satisfying
|˜f|sK = |f|sK in
D′ for Re(s) > 0. In addition, |˜f|sK , with Re(s) > 0, gives rise to a holomorphicH∗∞-
valued function in s, cf. [471, Section 3.3]. In this section we discuss the existence ofa meromorphic continuation of H∗
∞-valued functions of the type
Zg(s, f) :=[
|˜f|sK, g]
=∫
QNp�
˜f−1(0)
|f(ξ )|sKg(ξ )dNξ, (10.26)
where Re(s) > 0, g ∈ H∞(
QNp
)
, to the whole complex plane. Since D(
QNp
) ⊂H∞
(
QNp
)
, integrals of type (10.26) are generalizations of the classical Igusa localzeta functions.We start by presenting an example that illustrates the analogies and differences
between the classical Igusa zeta functions and the local zeta functions on H∞.
10.8.1 Example
We denote by “·” the reduction modulo p, i.e. the canonical mapping ZNp →
(Zp/pZp)N = FNp . If f is a polynomial with coefficients in Zp, we denote by f the
polynomial obtained by reducing modulo p the coefficients of f.We take g(x) = F−1
ξ→x(e−‖ξ‖αK ), with α > 0. By Lemma 10.8 (iii), g ∈ H∞(Kn).
We also pick a homogeneous polynomial f with coefficients in Zp � pZp of degreed, satisfying f(a) = ∇f(a) = 0 implies a = 0. We set
[|˜f|sK, g] =∫
QNp�f−1(0)
|f(ξ )|spe−‖ξ‖αp dNξ for Re(s) > 0.
Claim 10.32 ([471, Section 4.1]) [|˜f|sp, g] admits meromorphic continuation to thewhole complex plane and the real parts of the possible poles belong to the set {−1} ∪∪l∈N{−(n+ αl)/d}.
176 Sobolev-Type Spaces and Pseudodifferential Operators
10.8.2 Local Zeta Functions inH∞Definition 10.33 Let {γi}i∈N�{0} be a sequence of positive real numbers such thatγ1 ≥ 1. The generalized arithmetic progression generated by {γi}i∈N is the sequenceM = {mi}i∈N of real numbers defined as (1) m0 = 0 and m1 = γ1 − 1; (2) ml =∑l
j=1 γ j for l ≥ 2.
Theorem 10.34 ([471]) Assume that K is a non-Archimedean local field of char-acteristic zero and let f denote an arbitrary element of Zp[ξ1, . . . , ξN] � Zp; take
g ∈ H∞(
QNp
)
, s ∈ C, withRe(s) > 0. Then [|˜f|sp, g] defines anH∗∞(
QNp
)
-valued holo-morphic function of s, which admits a meromorphic continuation, denoted again as
[|˜f|sp, g], to the whole complex plane. Furthermore, there exists a finite collection ofpairs of non-negative integers {(NE , vE ) ∈ N × (N�{0});E ∈ T } depending only on˜f, such that the possible real parts of the poles of [|˜f|sp, g] are negative real numbersbelonging to the set
⋃
E∈E
−(vE +ME )
NE,
where each ME is a generalized arithmetic progression.
Theorem 10.35 ([471]) Let f be a non-constant polynomial with coefficients in Zp.Then, the following assertions are equivalent.
(i) There exists E ∈ H∗∞ such that E|f|K = 1 in L2.
(ii) Set A(∂, f)g= F−1(|f|KF (g)) for g ∈ Dom (A(∂, f)) := {g ∈ L2; |f|Kg ∈ L2}.There exists E ∈ H∗
∞ such that A∗(∂, f)E = δ inH∗∞.
(iii) There exists E ∈ H∗∞ such that E ∗ h ∈ H∗
∞ for any h ∈ H∞, and u = E ∗ g isa solution of A∗(∂, f)u = g inH∗
∞, for any g ∈ H∞.
Definition 10.36 The functional E ∈ H∗∞ is called a fundamental solution for
A∗(∂, f).
Theorem 10.37 ([471]) Let f be a non-constant polynomial with coefficients in Zp.Then there exists a fundamental solution for operator A∗(∂, f).
11
Non-Archimedean White Noise,Pseudodifferential Stochastic Equations,
and Massive Euclidean Fields
11.1 Introduction
There are general arguments that suggest that one cannot make measurements inregions of extent smaller than the Planck length ≈ 10−33 cm, see e.g. [413] and thereferences therein. The construction of physical models at the level of the Planck scaleis a relevant scientific problem and a very important area of mathematical research.In [436]–[438], I. Volovich conjectured the non-Archimedean nature of space-timeat the level of the Planck scale. This conjecture has given rise to a lot of research, forinstance, in quantum mechanics, see e.g. [222], [430], [433], [456], [457], in stringtheory, see e.g. [91], [163], [175], [431], [425], [427], and in quantum field theory,see e.g. [280], [344], [396]. On the other hand, the interaction between quantum fieldtheory and mathematics is very fruitful and deep, see e.g. [167], [171], [213], [220],[219], [444], [445], among several articles. Let us mention explicitly the connectionwith arithmetic, see e.g. [213], [308], [371]. From this perspective the investigationof quantum fields in a non-Archimedean setting is quite a natural problem.In this chapter we present a class of non-Archimedean Euclidean fields, in arbitrary
dimension, which are constructed as solutions of certain covariant p-adic stochas-tic pseudodifferential equations, by using techniques involving white-noise calculus.This chapter is based on [472]. The connection between quantum fields and SPDEshas been studied intensively in the Archimedean setting, see e.g. [9]–[30] and the ref-erences therein. A massive non-Archimedean field � is a random field parametrizedby H∞
(
QNp ;R
)
, the nuclear countably Hilbert spaces introduced in Chapter 10,which depends on (q, l,m, α), where q is an elliptic quadratic form, l is an ellipticpolynomial, and m and α are positive numbers. Here m is the mass of �. Heuristi-cally, � is the solution of (Lα + m2)� = �, where � is a generalized Lévy noise.This type of noise is introduced in this chapter. Here Lα (·) = F−1
ξ→x
(|l(ξ )|αpFx→ξ (·))
,where Fq := F is the Fourier transform on QN
p defined using the bilinear symmetricform corresponding to the quadratic form q. However, in this chapter we work withFourier transforms defined by using arbitrary bilinear forms. The operator Lα + m2 isa non-Archimedean analog of a fractional Klein–Gordon operator. At this point, it is
177
178 Non-Archimedean White Noise
useful to compare our construction with the classical one. In the Archimedean case,see e.g. [7], [30], the elliptic quadratic form b(ξ ) = ξ 21 + · · · + ξ 2N ∈ R[ξ1, . . . , ξN] isused to define the pseudodifferential fractional Klein–Gordon operator (−�+ m2)α ,m, α > 0, and also b(ξ ) is the quadratic form associated with the bilinear formξ1η1 + · · · + ξNηN , which is used in the definition of the Fourier transform on RN .This approach cannot be carried out in the p-adic setting because, when b(ξ ) is con-sidered as a p-adic quadratic form, it is not elliptic for N ≥ 5, and the ellipticityof b(ξ ) is essential to establish the non-negativity of the Green functions in the p-adic setting. For this reason, we have to replace ξ 21 + · · · + ξ 2N by an elliptic polyno-mial, which is a homogeneous polynomial that vanishes only at the origin. There areinfinitely many of them. The symmetries of our Green functions and fields dependon the transformations that preserve q and l. Thus, in order to have large groups ofsymmetries, we cannot fix q, instead, we work with pairs (q, l) that have a large groupof symmetries. These are two important differences between the p-adic case and theArchimedean one. On the other hand, the spaces H∞
(
QNp ;R
)
are completely neces-sary to carry out a construction similar to the one presented in [30], [7], For instance,the Green function attached to Lα + m2 gives rise to a continuous mapping fromH∞
(
QNp ;R
)
into itself. This fact is not true, if we replace H∞(
QNp ;R
)
by the spaceof test functions DR
(
QNp
)
, which is also nuclear.The chapter is organized as follows. In Section 11.3, we introduce non-
Archimedean analogs of the Klein–Gordon fractional operators and study the prop-erties of the corresponding Green functions, see Proposition 11.1. We also study thesolutions of the p-adic Klein–Gordon equations in take over H∞
(
QNp ;R
)
, see The-orem 11.2. In Section 11.4, we introduce a new class of non-Archimedean Lévynoises, see Theorem 11.6 and Definition 11.10. Section 11.5 is dedicated to thenon-Archimedean quantum field and its symmetries, see Proposition 11.12, Defini-tions 11.13 and 11.14 and Proposition 11.17. Finally, as an application,we constructa p-adic Brownian sheet on QN
p , see Theorem 11.23.
11.2 Preliminaries
In this chapter we will use all of the notation and results given in Section 10.2. Inparticular, we will use that H∞
(
QNp ;R
) = H∞(R) and H∞(
QNp ;C
) = H∞(C) arenuclear countably Hilbert spaces.
11.2.1 Fourier Transforms
Set χp(y) = exp(2π i{y}p) for y ∈ Qp. The map χp(·) is an additive character onQp, i.e. a continuous map from Qp into the unit circle satisfying χp(y0 + y1) =χp(y0)χp(y1), y0, y1 ∈ Qp.
Let B(x, y) be a symmetric non-degenerate Qp-bilinear form on QNp × QN
p . Thusq(x) := B(x, x), x ∈ QN
p is a non-degenerate quadratic form on QNp . We recall that
B(x, y) = 1
2{q(x+ y)− q(x)− q(y)}. (11.1)
11.3 Pseudodifferential Operators and Green Functions 179
We identify theQp-vector spaceQNp with its algebraic dual
(
QNp
)∗bymeans ofB(·, ·).
We now identify the dual group (i.e. the Pontryagin dual) of(
QNp ,+
)
with(
QNp
)∗by
taking x∗(x) = χp(B(x, x∗)). The Fourier transform is defined by
(Fqg)(ξ ) := (Fg)(ξ ) =∫
QNp
g(x)χp(B(x, ξ ))dμ(x), for g ∈ L1,
where dμ(x) is a Haar measure on QNp . Let L
(
QNp
)
be the space of continuous func-tions g in L1 whose Fourier transform Fg is in L1. The measure dμ(x) can be nor-malized uniquely in such a manner that (F (Fg))(x) = g(−x) for every g belongingto L
(
QNp
)
. We say that dμ(x) is a self-dual measure relative to χp([·, ·]). Notice thatdμ(x) = C(q)dNx, whereC(q) is a positive constant and dNx is the Haar measure onQNp normalized by the condition vol
(
BN0) = 1. For further details about the material
presented in this section the reader may consult [442].We will use the notation Fx→ξg and g for the Fourier transform of g as before. The
Fourier transform F[T ] of a distribution T ∈ D′(QNp
)
is defined by
(g,F[T ]) = (F[g],T ) for all g ∈ D(
QNp
)
.
11.2.2 Pseudodifferential Operators Acting onH∞(C)
In this chapter we work with pseudodifferential operators of the form
A : D(
QNp
) → L2 ∩Cg(x) → F−1
ξ→x(a(ξ )Fx→ξg), (11.2)
with F = Fq and a(ξ ) a smooth symbol. The results given in Section 10.2 arevalid in this general setting. In some cases, the proofs should be slightly modified.For instance, in Theorem 10.15, we established that (i) the mapping A : HC(∞) →HC(∞) is a bicontinuous isomorphism of locally convex spaces and (ii) HC(∞) ⊂L∞ ∩Cunif ∩ L1 ∩ L2. This result is still valid for operators of the form (11.2). Theproof of part (ii) requires a small modification, see [472, Theorem 3.15].
11.3 Pseudodifferential Operators and Green Functions
We take l(ξ ) ∈ Zp[ξ1, . . . , ξN] to be an elliptic polynomial of degree d. This meansthat l is homogeneous of degree d and satisfies l(ξ ) = 0 → ξ = 0. There are infi-nitely many elliptic polynomials. We consider the following elliptic pseudodifferen-tial operator:
(Lαϕ)(x) = F−1ξ→x
(|l(ξ )|αpFx→ξϕ)
,
where α > 0 and ϕ ∈ D(
QNp
)
.We shall call a fundamental solution G(x;m, α) of the equation
(Lα + m2)u = h, with h ∈ D(
QNp
)
, m > 0, (11.3)
180 Non-Archimedean White Noise
a Green function of Lα . As a distribution on D(
QNp
)
, the Green function is given by
G(x;m, α) = F−1ξ→x
( 1
|l(ξ )|αp + m2
)
. (11.4)
Notice that, since
Cα0 ‖ξ‖αdp ≤ |l(ξ )|α ≤ Cα
1 ‖ξ‖αdp , (11.5)
for some positive constants C0,C1, cf. Section 8.3,
1
|l(ξ )|αp + m2∈ L1
(
QNp , d
Nξ)
for αd > N,
and, in this case, G(x;m, α) is an L∞-function.
11.3.1 Green Functions on D′(QNp
)
Proposition 11.1 The Green function G(x;m, α) satisfies the following properties:
(i) the function G(x;m, α) is continuous on QNp � {0};
(ii) if αd > N, then the function G(x;m, α) is continuous;(iii) for 0 < αd ≤ N, the function G(x;m, α) is locally constant on QN
p � {0}, and
|G(x;m, α)| ≤{
C‖x‖2αd−Np for 0 < αd < N
C0 −C1 ln ‖x‖p for N = αd,
for ‖x‖p ≤ 1, where C, C0, C1 are positive constants,(iv) |G(x;m, α)| ≤ C2‖x‖−αd−Np as ‖x‖p → ∞, where C2 is a positive constant;(v) G(x;m, α) ≥ 0 on QN
p � {0}.
Proof (i) We first notice that
G(x;m, α) =∞∑
l=−∞g(l)(x;m, α), (11.6)
where
g(l)(x;m, α) =∫
p−l SN0
χp(−B(x, ξ ))|l(ξ )|αp + m2
dNξ,
here SN0 = {ξ ∈ ZNp : ‖ξ‖p = 1}. For x ∈ QN
p � {0}, we take x = pord(x)x0, x0 =(x0,1, . . . , x0,N ) with ‖x0‖p = 1. By using the standard basis of QN
p we have
B(x, ξ ) = pord(x)B(x0, ξ ) = pord(x)N
∑
i=1
ξi
⎡
⎣
N∑
j=1
Bi jx0, j
⎤
⎦ ,
11.3 Pseudodifferential Operators and Green Functions 181
where the Bi j ∈ Qp, and det[Bi j] = 0. On considering [Bi j] as a vector in QN2
p
we have [Bi j] = pβ[
˜Bi j]
, where the vector[
˜Bi j] ∈ QN2
p has norm 1. Then
B(x, ξ ) = pord(x)+βN
∑
i=1
ξi
⎡
⎣
N∑
j=1
˜Bi jx0, j
⎤
⎦ = pord(x)+βN
∑
i=1
ξi˜Ai, (11.7)
with∥
∥
(
˜A1, . . . ,˜AN)∥
∥
p = 1. Notice that β is a constant that depends only on B.We now assert that series (11.6) converges in D′(QN
p
)
. Indeed,
1
|l(ξ )|αp + m2= lim
l→∞
l∑
j=−l
1SNj (ξ )
|l(ξ )|αp + m2in D′(QN
p
)
,
now, since the Fourier transform is continuous on D′(QNp
)
, one gets
G(x;m, α) = liml→∞
l∑
j=−lF−1ξ→x
( 1SNj (ξ )
|l(ξ )|αp + m2
)
= liml→∞
l∑
j=−lg( j)(x;m, α) =
∞∑
l=−∞g(l)(x;m, α).
Consider now x = 0, with ‖x‖p = pk, k ∈ Z. By making the change of variablesξ = p−l z in g(l)(x;m, α), one gets
g(l)(x;m, α) = plN∫
SN0
χp(−p−lB(x, z))pldα|l(z)|αp + m2
dNz.
There exists a covering of SN0 of the form
SN0 =M⊔
i=1
zi +(
pLZp)n
(11.8)
with zi ∈ SN0 for i = 1, . . . ,M, and L ∈ N�{0}, such that
|l(z)|αp |zi+(pLZp)n = |l(zi)|αp for i = 1, . . . ,M, (11.9)
cf. [464, Lemma 3], or [470, Lemma 26]. From this fact and (11.7), one gets
g(l)(x;m, α)
= plN−LNM∑
i=1
χp(− p−lB
(
x, zi))
pldα∣
∣l(
zi)∣
∣
α
p + m2
∫
ZNp
χp
(
−p−l+L−k+βN
∑
i=1
yi˜Ai
)
dNy.
182 Non-Archimedean White Noise
Notice that g(l)(x;m, α) is locally constant for ‖x‖p = pk. We now recall that
∫
ZNp
χp
(
−p−l+L−k+βN
∑
i=1
yi˜Ai
)
dNy ={
1 if l ≤ L− k + β
0 if l ≥ L− k + β + 1.(11.10)
Hence,
g(l)(x;m, α) = 0 if l ≥ 1+ L− k + β (11.11)
and
∣
∣g(l)(x;m, α)∣∣ ≤ plN−LNMpldαγ + m2
, if l ≤ L− k + β, (11.12)
where γ = mini |l(zi)|αp > 0, and, since
Mp−LNL−k+β∑
l=−∞
plN
pldαγ + m2<Mp−LN
m2
L−k+β∑
l=−∞plN < ∞,
we have that the series (11.6) converges uniformly on the sphere ‖x‖p = pk,and equivalently (11.6) converges uniformly on compact subsets of QN
p � {0}.Therefore, G(x;m, α) is a continuous function on QN
p � {0}.(ii) If N < αd, the estimate (11.12) implies uniform convergence on QN
p and thecontinuity of G(x;m, α) on the whole of QN
p .(iii) If 0 < αd < N, then by (11.12), for ‖x‖p ≤ 1,
|G(x;m, α)| ≤ CL−k+β∑
l=−∞
plN
pldαγ + m2≤ C
γ
L−k+β∑
l=−∞pl(N−dα)
= C
γp(L−k+β )(N−αd)
∞∑
j=0
p− j(N−αd)
≤ C
γ
p(L+β )(N−αd)
1− pαd−N‖x‖αd−Np .
If αd = N, then, for ‖x‖p ≤ 1, from (11.12),
|G(x;m, α)| ≤ C
m2
0∑
l=−∞plN + C
γ
L−k+β∑
l=1
1 = C0 −C1 ln ‖x‖p.
(iv) Let ‖x‖p = pk. We notice that, if ‖ξ‖p ≤ p−k+β , then χp(−B(x, ξ )) = 1.Therefore
G(x;m, α) = G(1)(x;m, α)+ G(2)(x;m, α),
11.3 Pseudodifferential Operators and Green Functions 183
where
G(1)(x;m, α) =−k+β∑
l=−∞plN
∫
SN0
dNz
pldα∣
∣l(z)∣
∣
α
p + m2,
G(2)(x;m, α) =L−k+β∑
l=−k+β+1
plN∫
SN0
χp(−p−lB(x, z))pldα
∣
∣l(z)∣
∣
α
p + m2dNz.
By using the formula
1
m2 + t= m−2 − m−4t + O(t2), t → 0, (11.13)
we get
plN∫
SN0
dNz
pldα|l(z)|αp + m2
= plN (1− p−N )m−2 − pl(N+dα)m−4Z(α)+ (
pl(N+2dα))
,
as t → 0, where Z(α) := ∫
SN0|l(z)|αpdNz. Hence
G(1)(x;m, α) = (1− p−N )m−2−k+β∑
l=−∞plN − Z(α)m−4
−k+β∑
l=−∞pl(N+dα)
+O
(−k+β∑
l=−∞pl(N+2dα)
)
= pNβm−2‖x‖−Np − Z(α)m−4p(N+dα)β
1− p−N−dα ‖x‖−N−dαp
+O(‖x‖−N−2dα
p
)
, as ‖x‖p → ∞.
We now consider G(2)(x;m, α), by using (11.13),
plN∫
SN0
χp(−p−lB(x, z))pldα|l(z)|αp + m2
dNz = m−2plN∫
SN0
χp(−p−lB(x, z))dNz
−m−4pl(dα+N )∫
SN0
|l(z)|αpχp(−p−lB(x, z))dNz
+O(
pl(N+2dα))
.
184 Non-Archimedean White Noise
By using
∫
SN0
χp(−p−lB(x, z))dNz =∫
SN0
χp
(
− p−l−k+βN
∑
j=1
z j˜Aj
)
dnz
=
⎧
⎪
⎪
⎨
⎪
⎪
⎩
1− p−N if l + k − β ≤ 0
−p−N if l + k − β = 1
0 if l + k − β ≥ 2,
we get
m−2L−k+β∑
l=−k+β+1
plN∫
SN0
χp(−p−lB(x, z))dNz = −m−2pNβ‖x‖−Np ,
and by using (11.8) and (11.9)
−m−4pl(dα+N )∫
SN0
|l(z)|αpχp(−p−lB(x, z))dNz
= −m−4pl(dα+N )−LNM∑
i=1
∣
∣l(
zi)∣
∣
α
pχp(− p−lB
(
x, zi))
×∫
ZNp
χp(−p−l+L−kB(x0, y))dNy,
where x = p−kx0, ‖x0‖p = 1. By using (11.10).
−m−4p−LNM∑
i=1
∣
∣l(
zi)∣
∣
α
p
L−k+β∑
l=−k+β+1
χp(−p−lB(x0, zi))pl(dα+N )
= −m−4pdα+N−LN(
M∑
i=1
∣
∣l(
zi)∣
∣
α
p
L−1∑
j=0χp
(− pj−k+β+1B(
x, zi))
pj(dα+N ))
× pβ(dα+N )‖x‖−dα−Np .
Therefore
G(x;m, α) = G(1)(x;m, α)+ G(2)(x;m, α)= −m−4A(x)pβ(dα+N )‖x‖−dα−Np
+O(‖x‖−N−2dα
p
)
, as ‖x‖p → ∞,
11.3 Pseudodifferential Operators and Green Functions 185
where
A(x) := Z(α)
1− p−N−dα
+ pdα+N−LN(
M∑
i=1
|l(zi)|αp
L−1∑
j=0χp
(− pj−k+β+1B(
x, zi))
pj(dα+N ))
.
(iv) By (i) we have that G(x;m, α) is a continuous function on Qnp � {0} having
expansion (11.6). Thus, it is sufficient to show that
g(l)(x;m, α) ≥ 0 on Qnp � {0}. (11.14)
Take t ∈ Q×p , thenVt := {ξ ∈ SNj : l(ξ ) = t} is a p-adic compact submanifold of
the sphere SNj . Then, there exists a differential form ω0, known as the Gel’fand–Leray form, such that dξ1 ∧ · · · ∧ dξn = ω0 ∧ dl. Denote the measure corre-sponding to ω0 as dξ/dl, then
g(l)(x;m, α) =∫
Q×p
1
|t|αp + m2
⎧
⎨
⎩
∫
Vt
χp(−B(x, ξ ))dξdl
⎫
⎬
⎭
dt.
Thus, in order to establish (11.14), it is sufficient to show that
∫
Vt
χp(−B(x, ξ ))dξdl
=∫
Vt
χp
(
−pord(x)+ β
N∑
i=1
ξi˜Ai
)
dξ
dl≥ 0 for all x.
This last inequality is established as in the proof of Theorem 2 in [464], alterna-tively see [470, Section 2.3.3], by using the non-Archimedean implicit functiontheorem and by performing a suitable change of variables. �
11.3.2 Green Functions onH∞(R)
Theorem 11.2 Let α > 0, let m > 0, and let Lα be an elliptic operator. (i) Thereexists a Green function G(x;m, α) for the operator Lα , which is continuous and non-negative on Qn
p � {0}, and tends to zero at infinity. Furthermore, if h ∈ D(
QNp
)
, thenu(x) = G(x;m, α) ∗ h(x) is a solution of (11.3) in D′(QN
p
)
. (ii) The equation
(Lα + m2)u = g, (11.15)
with g ∈ H∞(R), has a unique solution u ∈ H∞(R).
Proof (i) We first notice that, for any ϕ ∈ D(
QNp
)
, u(x) := G(x;m, α) ∗ ϕ(x) isa locally constant function because it is the inverse Fourier transform ofϕ(ξ )/
(|l(ξ )|αp + m2)
, which is a distributionwith compact support. Taking u(x) =G(x;m, α) ∗ ϕ(x) ∈ D′(QN
p
)
, we have (Lα + m2)u = ϕ in D′(QNp
)
. The otherresults follow from Proposition 11.1.
186 Non-Archimedean White Noise
(ii) Take ϕ ∈ DR
(
QNp
)
, then, by (i), u(x) = G(x;m, α) ∗ ϕ(x) is a real-valued, locallyconstant function which is a solution of (11.15) in D′(QN
p
)
. Now, since u(ξ ) =ϕ(ξ )/
(|l(ξ )|αp + m2) ∈ L2, by using (11.5),
‖u‖2l+ 2αd! ≤ 1
m4‖ϕ‖20 +
∫
QNp�BN0
‖ξ‖l+ 2αd!p |ϕ(ξ )|2dNξ(|l(ξ )|αp + m2
)2
≤ 1
m4‖ϕ‖20 +
1
C2α0
∫
QNp�BN0
‖ξ‖l+ 2αd!p
|ϕ(ξ )|2‖ξ‖2αdp
dNξ
≤ 1
m4‖ϕ‖20 +
1
C2α0
‖ϕ‖2l
≤ C′‖ϕ‖2l , for any l ∈ N.
Then u ∈ Hr(R), for r = l + 2αd! ≥ 2αd!. In the case, 0 ≤ r ≤ 2αd! − 1,one gets ‖u‖r ≤ C′′‖ϕ‖0. Therefore u ∈ Hr(R), for any r ∈ N, i.e. u ∈ H∞(R).We now take a sequence gn ∈ DR
(
QNp
)
such that ϕn d−→ g ∈ H∞(R) and (Lα +m2)un = gn with un ∈ H∞(R). By the continuity of (Lα + m2)−1 |H∞(R) and thedensity of DR
(
QNp
)
in H∞(R), we have un d−→ u ∈ H∞(R) and (Lα + m2)u =g, cf. Chapter 10, Theorem 10.15. The uniqueness of u also follows from thistheorem. �
Corollary 11.3 The mapping
H∞(R) → H∞(R)
g(x) → G(x;m, α) ∗ g(x)
is continuous.
Proof By virtue of the proof of Theorem 11.2 (ii), and Theorem 10.15,
H∞(R) → H∞(R)
g→ (Lα + m2)−1g
is a well-defined continuous mapping. �
Remark 11.4 (i) It is worthmentioning that the Archimedean and non-ArchimedeanGreen functions share similar properties, cf. [179, Proposition 7.2.1] and Propo-sition 11.1 and Theorem 11.2.
(ii) Proposition 11.1 and Theorem 11.2 generalize to arbitrary dimension someresults established by Kochubei for Klein–Gordon pseudodifferential operatorswith elliptic quadratic forms attached, cf. [275, Proposition 2.8 and Theorem2.4].
11.4 The Generalized White Noise 187
(iii) Another possible definition for the p-adic Klein–Gordon operator, with real massm > 0, is the following:
ϕ → F−1((|l|p + m2
)αFϕ)
.
But, since Proposition 11.1 and Theorem 11.2 remain valid for this type of oper-ator, we prefer to work with (11.3) because operators of this type have beenstudied extensively in the p-adic setting.
11.4 The Generalized White Noise
11.4.1 Infinitely Divisible Probability Distributions
We recall that an infinitely divisible probability distribution P is a probability distri-bution having the property that for each n ∈ N there exists a probability distributionPn such that P = Pn ∗ · · · ∗ Pn (n times). By the Lévy–Khinchine theorem, the char-acteristic function CP of P satisfies
CP(t ) =∫
R
eist dP(s) = e�(t ), t ∈ R, (11.16)
where� : R → C is a continuous function, called the Lévy characteristic of P, whichis uniquely represented as follows:
�(t ) = iat − σ 2t2
2+ ∫
R�{0}
(
eist − 1− ist
1+ s2
)
dM(s), t ∈ R, (11.17)
where a, σ ∈ R, with σ ≥ 0, and the measure dM(s) satisfies∫
R�{0}min(1, s2)dM(s) < ∞. (11.18)
On the other hand, given a triple (a, σ, dM) with a ∈ R, σ ≥ 0, and dM a measureon R�{0} satisfying (11.18), there exists a unique infinitely divisible probability dis-tribution P such that its Lévy characteristic is given by (11.17).
Remark 11.5 From now on, we work with infinitely divisible probability distribu-tions which are absolutely continuous with all finite moments. This fact is equivalentto all the moments of the corresponding Ms are finite, cf. [30, Theorem 2.3].
Let N ∈ N be as before. Let H∞(R) and H∗∞(R) be the spaces introduced in Sec-
tion 10.2. We denote by [·, ·] the dual pairing between H∗∞(R) and H∞(R) as in
Chapter 10. Let B be the σ -algebra generated by cylinder sets of H∗∞(R). Then
(H∗∞(R),B) is a measurable space.By a characteristic functional onH∞(R), we mean a functionalC : H∞(R) → C
satisfying the following properties:
(i) C is continuous onH∞(R);(ii) C is positive-definite;(iii) C(0) = 1.
188 Non-Archimedean White Noise
Now, since H∞(R) is a nuclear space, by the Bochner–Minlos theorem (see e.g.[343]) there exists a one-to-one correspondence between the characteristic function-als C and probability measures P on (H∗
∞(R),B) given by the following relation:
C( f ) = ∫
H∗∞(R)ei[T, f ] dP(T ), f ∈ H∞(R).
Theorem 11.6 Let � be a Lévy characteristic defined by (11.16). Then there existsa unique probability measure P� on (H∗
∞(R),B) such that the Fourier transform ofP� satisfies
∫
H∗∞(R)ei[T, f ] dP� (T ) = exp
{
∫
QNp
�( f (x))dNx}
, f ∈ H∞(R).
The proof is based on [177, Theorem 6, p. 283] like in the Archimedean case, cf.[30, Theorem 1.1]. However, in the non-Archimedean case the result does not followdirectly from [177]. We need some additional results.
Lemma 11.7∫
QNp�( f (x))dNx < ∞ for any f ∈ H∞(R).
Proof By formula (11.17), we have to show that (i)∫
QNpf (x)dNx < ∞; (ii)
∫
QNpf 2(x)dNx < ∞; (iii)
∫
QNp
∫
R�{0}(eis f (x) − 1− is f (x)/(1+ s2))dM(s)dNx < ∞
for any f ∈ HR(∞). (i), (ii) follow from the facts that f ∈ L1 and f ∈ L2 respec-tively, cf. Theorem 10.15 (ii). To verify (iii) we use the fact that |eis f (x) − 1| ≤ |s f (x)|,s ∈ R, x ∈ QN
p , then integral (iii) is bounded by 2( ∫
R�{0} |s|dM(s))( ∫
QNp| f (x)|dNx),
which is finite because M has finite moments and f ∈ L1. �
Lemma 11.8 The function f → ∫
QNp�( f (x))dNx is continuous onH∞(R).
Proof By (11.17) and using the density of DR
(
QNp
)
in H∞(R), see Lemma 10.4 (i),it is sufficient to show that
f → D( f ) := ∫
QNp
∫
R�{0}
(
eis f (x) − 1− is f (x)
1+ s2
)
dM(s)dNx (11.19)
is continuous on(
DR
(
QNp
)
, d)
. Take f ∈ DR
(
QNp
)
and a sequence { fn}n∈N inDR
(
QNp
)
such that fn d−→ f , i.e. fn ‖ · ‖m−−−→ f for every m. By contradiction assume that fn −D( f ) does not converge to 0, then there exist ε > 0 and a subsequence gk = fnk suchthat |gk − D( f )| > ε. On the other hand, taking m = 0, we have gk L2−→ f , and thusthere is a subsequence {gkj }k j such that gkj → f almost uniformly. Now, since thesupport of f is contained in a ball, say BnN0
, the support of each gkj is contained in BnN0
almost everywhere. Hence |gkj (x)| ≤ C0‖ f‖∞1BnN0 (x) almost everywhere, for some
positive constant C0. By using the facts that |eis( f (x)−gk j (x)) − 1| ≤ |s|| f (x)− gkj (x)|,
11.4 The Generalized White Noise 189
s ∈ R, x ∈ QNp , and that M is a bounded measure with finite moments, we have
|gkj − D( f )| ≤ 2
{
∫
R�{0}s2dM(s)
}
∫
QNp
| f (x)− gkj (x)|dNx
≤ C∫
QNp
| f (x)− gkj (x)|dNx.
Then by the dominated convergence theorem, using the facts that f ∈ L1 and that|gkj (x)| ≤ C‖ f‖∞1BnN0 (x) almost everywhere, i.e. |gkj (x)| ≤ |g(x)| ∈ L1, we concludethat D(gkj ) → D ( f ), which contradicts |gk − D( f )| > ε. �
Set L( f ) := exp{ ∫
QNp�( f (x))dNx
}
for f ∈ H∞(R). Notice that, by Lemma 11.7,
this function is well-defined.
Proposition 11.9 The function L( f ) is positive-definite if and only if es�(t ) is positive-definite for every s > 0.
Proof Suppose that L( f ) is positive-definite, i.e.
m∑
j,k=1L( f j − fk )z jzk ≥ 0 for f j, fk ∈ HR(∞), z j, zk ∈ C, j, k = 1, . . . ,m. (11.20)
Take f j(x) = t j(p−l‖x− x0‖p), t j ∈ R, for j = 1, . . . ,m, l ∈ Z, and x0 ∈ QNp .
Then
m∑
j,k=1L( f j − fk )z jzk =
m∑
j,k=1exp
(
∫
‖x−x0‖p≤pl�(t j − tk )d
Nx
)
z jzk
=m∑
j,k=1
{
exp[
pNl�(t j − tk )]}
z jzk ≥ 0.
This proves that epNl�(t ) is a positive-definite function. Now, t → (
1− epNl�(t )
)
/pNl isnegative-definite, cf. [79, Corollary 7.7], for every l ∈ N. Furthermore, by [79, Propo-sition 7.4 (i)], liml→∞
(
1− epNl�(t )
)
/pNl = −�(t ) is a negative-definite function,and, since the negative-definite functions form a cone, −s�(t ) is negative-definitefor every s > 0. Finally, by the Schoenberg theorem, cf. [79, Theorem 7.8], es�(t ) ispositive-definite for every s > 0.We now assume that es�(t ) is positive-definite for every s > 0. In order to prove
(11.20), by Lemma 11.8, it is sufficient to take f j, fk ∈ DR
(
QNp
)
, for j, k = 1, . . . ,m.Consider the matrix A = [ai j] with
ai j := exp
(
∫
QNp
�( fi(x)− f j(x))dNx
)
.
We have to show that A is positive-definite. Take BNn such that supp fi ⊆ BNn fori = 1, . . . ,m. Then ai j = exp
( ∫
BNn�( fi(x)− f j(x))dNx
)
. Because each fi(x) is a
locally constant function and BNn is an open compact set, there exists a finite covering
190 Non-Archimedean White Noise
BNn = &Ll=1BNn′ (xl ) such that fi(x) |BN
n′ (xl )= fi (xl ). Hence ai j =
∏Ll=1 exp(p
Nn′αl ) withαl := �( fi (xl )− f j (xl )). According to Schur’s theorem, cf. [177, Theorem in p. 277],A is positive-definite if the matrix [exp(pNn
′αl )] is positive-definite, which follows
from the fact that es�(t ) is positive-definite for every s > 0. �Proof of Theorem 11.6 By [177, Theorem 1, p. 273], L( f ) ≡ 0 is the characteristicfunctional of a generalized random process (a random field in our terminology) withindependent values at every point, if and only if (A) L is positive-definite and (B),for any functions f1(t ), f2(t ) ∈ H∞(R) whose product vanishes, it holds that L( f1 +f2) = L( f1)L( f2). The verification of condition (B) is straightforward. Condition (A)is equivalent to es�(t ) being positive-definite for every s > 0, cf. Proposition 11.9. By[177, Theorem 4 on p. 279 and Theorem 3 on p. 189], the last condition turns out tobe equivalent to the fact that � has the form (11.17). �
11.4.2 Non-Archimedean Generalized White-Noise Measures
Definition 11.10 We call P� in Theorem 11.6 a generalized white-noise measurewith Lévy characteristic � and (H∗
∞(R),B,P� ) the generalized white-noise spaceassociated with �. The associated coordinate process
� : HR(∞)× (H∗R(∞),B,P� ) → R
defined by �( f ,T ) = [T, f ], f ∈ HR(∞), T ∈ H∗R(∞), is called generalized white
noise.
The generalized white noise � is composed by three independent noises: constant,Gaussian, and Poisson (with jumps given by M) noises, see Remark 1.3 in [7].
11.5 Euclidean Random Fields as Convoluted Generalized White Noise
11.5.1 Construction
Definition 11.11 Let (,F,P) be a given probability space. By a generalized ran-dom field � on (,F,P) with parameter spaceH∞(R), we mean a system
{�(g, ω) : ω ∈ }g∈H∞(R),
of random variables on (,F,P) having the following properties:
(i) P{ω ∈ : �(c1g1 + c2g2, ω) = c1�(g1, ω)+ c2�(g2, ω)} = 1, for c1, c2 ∈ R,g1, g2 ∈ H∞(R);
(ii) if gn → g inH∞(R), then �(gn, ω) → �(g, ω) in law.
The coordinate process in Definition 11.10 is a random field on the generalizedwhite-noise space (H∗
∞(R),B,P� ), because property (i) is satisfied pointwise andproperty (ii) follows from the fact that
limn→∞P�{T ∈ H∗
∞(R) : |[T, gn − g]| < ε} = 1. (11.21)
11.5 Euclidean Random Fields 191
Indeed, since H∗∞(R) is the union of the increasing spaces H∗
l (R), there exists l0 ∈N such that T ∈ H∗
l0(R), and thus |[T, gn − g]| ≤ ‖T‖−l0‖gn − g‖l0 ≤ ‖T‖−l0 , for n
big enough. Now, (11.21) follows by the dominated convergence theorem. Thereforelimn→∞ P�{T ∈ H∗
∞(R) : |[T, gn − g]| ≥ ε} = 0.We now recall that (G f )(x) := G(x;m, α) ∗ f (x) gives rise to a continuous map-
ping from H∞(R) into itself, cf. Corollary 11.3. Thus, the conjugate operator˜G : H∗
∞(R) → H∗∞(R) is a measurable mapping from (H∗
∞(R),B) into itself. Thegeneralized white-noise measure P� on (H∗
∞(R),B) associated with a Lévy charac-teristic � was introduced in Definition 11.10. We set P� to be the image probabilitymeasure of P� under ˜G, i.e. P� is the measure on (H∗
∞(R),B) defined by
P�(A) = P�(
˜G−1(A))
, for A ∈ B. (11.22)
Proposition 11.12 The Fourier transform of P� is given by
∫
H∗∞(R)
ei[T, f ] dP�(T ) = exp
⎧
⎪
⎨
⎪
⎩
∫
QNp
�
⎧
⎪
⎨
⎪
⎩
∫
QNp
G(x− y;m, α) f (y)dNy
⎫
⎪
⎬
⎪
⎭
dNx
⎫
⎪
⎬
⎪
⎭
,
for f ∈ H∞(R).
Proof For f ∈ H∞(R), by (11.22) and Theorem 11.6, we get that∫
H∗∞(R)
ei[T, f ] dP�(T ) =∫
H∗∞(R)
ei[˜GT, f ]dP� (T ) =
∫
H∗R(∞)
ei[T,G f ] dP� (T )
= exp
⎧
⎪
⎨
⎪
⎩
∫
QNp
�
⎧
⎪
⎨
⎪
⎩
∫
QNp
G(x− y;m, α) f (y)dNy
⎫
⎪
⎬
⎪
⎭
dNx
⎫
⎪
⎬
⎪
⎭
. �
By Proposition 11.12, the associated coordinate process
� : HR(∞)× (H∗R(∞),B) → R
given by �( f ,T ) = [T,G f ], f ∈ H∞(R), T ∈ H∗∞(R), is a random field on
(H∗∞(R),B,P�). In fact, � is nothing but ˜G�, which is defined by
˜G�( f ,T ) = �(G f ,T ), f ∈ H∞(R), T ∈ H∗∞(R).
It is useful to see � as the unique solution, in law, of the stochastic equation
(Lα + m2)� = �,
where (Lα + m2)�( f ,T ) := �((Lα + m2) f ,T ), for f ∈ H∞(R), T ∈ H∗∞(R). We
note that the correctness of this last definition is a consequence of Corollary 11.3 andTheorem 11.2 (ii).
192 Non-Archimedean White Noise
11.5.2 Symmetries
Given a polynomial a(ξ ) ∈ Qp[ξ1, . . . , ξn] and g ∈ GLN (Qp), we say that g preservesa if a(ξ ) = a(gξ ), for all ξ ∈ QN
p . For simplicity, we use gx to mean [gi j]xT, x =(x1, . . . , xN ) ∈ QN
p , where we identify gwith the matrix [gi j].
Definition 11.13 Let q(ξ ) be the elliptic quadratic form used in the definition of theFourier transform, see Section 11.2.1, and let l(ξ ) be the elliptic polynomial thatappears in the symbol of the operator Lα , see Section 11.3. We define the homoge-neous Euclidean group of QN
p relative to q(ξ ) and l(ξ ), denoted by E0(
QNp ; q, l
)
:=E0
(
QNp
)
, as the subgroup of GLN (Qp) whose elements preserve q(ξ ) and l(ξ ) simul-taneously. We define the inhomogeneous Euclidean group, denoted by E
(
QNp ; q, l
)
:= E(
QNp
)
, to be the group of transformations of the form (a, g)x = a+ gx, fora, x ∈ QN
p , g ∈ E0(
QNp
)
.
We notice that E(
QNp ; q, l
)
preserves the Haar measure. Indeed, if we take x = a+gy with g ∈ E0
(
QNp ; q, l
) ⊂ O(q), the orthogonal group of q, then dNx = dN (gy) =| det g|pdNy = dNy. In addition, it is not a straightforward matter to decide whetheror not E0
(
QNp ; q, l
)
is non-trivial. We also notice that (a, g)−1x = g−1(x− a).Let (a, g) be a transformation in E
(
QNp
)
. The action of (a, g) on a function f ∈HR(∞) is defined by
((a, g) f )(x) = f(
(a, g)−1x)
, for x ∈ QNp ,
and that on a functional T ∈ H∗R(∞) is defined by
[(a, g)T, f ] := [T, (a, g)−1 f ], for f ∈ H∞(R).
The action on a random field � is defined by
((a, g)�)( f ,T ) = �((a, g)−1 f ,T ), for f ∈ H∞(R), T ∈ H∗∞(R).
Definition 11.14 By Euclidean invariance of the random field � we mean that thelaws of � and (a, g)� are the same for each (a, g) ∈ E
(
QNp
)
, i.e. the probabilitydistributions of
{�( f , ·) : f ∈ H∞(R)} and {((a, g)�)( f , ·) : f ∈ H∞(R)}coincide for each (a, g) ∈ E
(
QNp
)
.
We say that G is (a, g)-invariant for some (a, g) ∈ E(
QNp
)
, if (a, g)G = G(a, g). IfG is invariant under all (a, g) ∈ E
(
QNp
)
, we say that G is Euclidean invariant.
Remark 11.15 Let f ∈ D(
QNp
)
and let (a, g) ∈ E(
QNp
)
. Then
Fx→ξ
[
f(
(a, g)−1x)] = χp(B(a, ξ ))Fx→ξ [ f ](g−1ξ ). (11.23)
Indeed, by taking g−1(x− a)x = y, we obtain∫
QNp
f(
g−1(x− a)x)
χp(B(x, ξ ))dμ(x) = χp(B(a, ξ ))∫
QNp
f (y)χp(B(gy, ξ ))dμ(y),
11.5 Euclidean Random Fields 193
where dμ(y) = C(q)dNy. The formula follows from
B(gy, ξ ) = 1
2{q(gy+ ξ )− q(gy)− q(ξ )}
= 1
2{q(y+ g−1ξ )− q(y)− q(g−1ξ )} = B(y, g−1ξ ).
By virtue of the density of D(
QNp
)
in D′(QNp
)
, formula (11.23) holds in D′(QNp
)
.
Lemma 11.16 G is Euclidean invariant.
Proof We first notice that the mapping f (x) → f ((a, g)−1x) is continuous fromH∞(R) into H∞(R), for any (a, g) ∈ E
(
QNp
)
. Hence G(a, g), (a, g)G : H∞(R) →H∞(R) are continuous, cf. Corollary 11.3, and it suffices to take f ∈ DR
(
QNp
)
inorder to show that
((a, g)G)( f ) = (G(a, g))( f ).
Now, since
((a, g)G)( f )(x) = (a, g)(G(x;m, α) ∗ f (x)) = (G ∗ f )((a, g)−1x)
and
(G(a, g))( f )(x) = G( f ((a, g)−1x)) = G(x;m, α) ∗ f ((a, g)−1x),
we should show that (G ∗ f )((a, g)−1x) = G(x;m, α) ∗ f ((a, g)−1x). We establishthis formula in D′(QN
p
)
by using the Fourier transform. Indeed,
F[
(G ∗ f )((a, g)−1x)] = χp(B(a, ξ ))F[(G ∗ f )](g−1ξ )
= χp(B(a, ξ ))F[ f ](g−1ξ )
|l(g−1ξ )|αp + m2
= χp(B(a, ξ ))F[ f ](g−1ξ )
|l(ξ )|αp + m2
and
F[
G(x;m, α) ∗ f (g−1x)] = 1
|l(ξ )|αp + m2Fx→ξ [ f (g−1x)]
= χp(B(a, ξ ))F[ f ](g−1ξ )
|l(ξ )|αp + m2,
cf. Remark 11.15. �
Proposition 11.17 The random field � = ˜G� is Euclidean invariant.
Proof According to the Bochner–Minlos theorem, it suffices to show that
C�( f ) = C(a,g)�( f ), for f ∈ H∞(R) and for every (a, g) ∈ E(
QNp
)
. (11.24)
194 Non-Archimedean White Noise
Indeed,
C(a,g)�( f ) =∫
H∗∞(R)
ei((a,g)�)( f ,T ) dP�(T )
=∫
H∗∞(R)
ei[˜GT,(a,g)−1 f ]dP� (T )
=∫
H∗∞(R)
ei[T,G((a,g)−1 f )]dP� (T )
= exp
⎧
⎪
⎨
⎪
⎩
∫
QNp
�(G((a, g)−1 f )(x))dNx
⎫
⎪
⎬
⎪
⎭
,
cf. Theorem 11.6 and Corollary 11.3. Then, since G((a, g)−1 f )(x) =(a, g)−1(G( f ))(x) (cf. Lemma 11.16) and (a, g) preserves dNx, we have
C(a,g)�( f ) = exp
⎧
⎪
⎨
⎪
⎩
∫
QNp
�(G( f )((a, g)x))dNx
⎫
⎪
⎬
⎪
⎭
= exp
⎧
⎪
⎨
⎪
⎩
∫
QNp
�(G( f )(x))dNx
⎫
⎪
⎬
⎪
⎭
= C�( f ),
cf. Proposition 11.12. �
11.5.3 Some Additional Remarks and Examples
In the Archimedean case the symmetric bilinear form used in the definition of theFourier transform has associated with it a quadratic form which is exactly the symbolof the Laplacian when it is considered as a pseudodifferential operator. This approachcannot be carried out in the p-adic setting. Indeed, the quadratic form ξ 21 + · · · + ξ 2Nassociated with the bilinear form
∑
i ξixi does not give rise to an elliptic operator ifN ≥ 5. This is the reasonwhy in the p-adic settingwe need two different polynomials,q(ξ ) and l(ξ ). In order to have a “non-trivial” group of symmetries, i.e.E0
(
QNp , q, l
) =1, the polynomials q(ξ ) and l(ξ ) should be related “nicely.” To illustrate this idea wegive two examples.
Example 11.18 In the case N = 4 there is a unique elliptic quadratic form,up to linear equivalence, which is l4(ξ ) = ξ 21 − sξ 22 − pξ 23 + sξ 24 , where s ∈Z�{0} is a quadratic non-residue, i.e. s/p = −1. We take q4(ξ ) = l4(ξ ), i.e.
11.6 The p-Adic Brownian Sheet on QNp 195
B4(x, ξ ) = ξ1x1 − sξ2x2 − pξ3x3 + sξ4x4. In this case, E0(
Q4p, l4, l4
)
equals
O(l4) =
⎧
⎪
⎪
⎨
⎪
⎪
⎩
g ∈ GL4(Qp) : gT
⎡
⎢
⎢
⎢
⎣
1 0 0 00 −s 0 00 0 −p 00 0 0 s
⎤
⎥
⎥
⎥
⎦
g=
⎡
⎢
⎢
⎣
1 0 0 00 −s 0 00 0 −p 00 0 0 s
⎤
⎥
⎥
⎦
⎫
⎪
⎪
⎬
⎪
⎪
⎭
,
the orthogonal group of l4.
Example 11.19 Take N = 5, B5(x, ξ ) = ξ1x1 − sξ2x2 − pξ3x3 + sξ4x4 + ξ5x5, andl5(ξ ) =
(
ξ 21 − sξ 22 − pξ 23 + sξ 24)2 − τξ 45 , where s is as in Example 11.19 and τ /∈
[
Q×p
]2, where
[
Q×p
]2denotes the group of squares of Q×
p . We notice that{
[
g 00 [1]1×1
]
5×5
∈ GL5(Qp) : g ∈ O(l4)
}
⊆ E0(
Q5p, q5, l5
)
.
Example 11.20 TakeB(x, ξ ) = ∑Ni=1 ξixi. Then the theory developed so far can be
applied to pseudodifferential operators of type
g→ F−1(‖ξ‖αpFg
)
, or g→ F−1
((
N∑
i=1
|ξi|p)α
Fg)
, with α > 0.
We notice that the group of permutations of the variables ξ1, . . . , ξN preserves ‖ξ‖αpand
(∑N
i=1 |ξi|p)α.
11.6 The p-Adic Brownian Sheet on QNp
As an application of the results developed in this section, we present a constructionof the Wiener process with its time variable in QN
p . In this section we take � witha = 0 andM = 0 in (11.17). Thus, the generalized white noise � in Definition 11.10is Gaussian with mean zero. Given t = (t1, . . . , tN ) ∈ QN
p , we set, for x ∈ QNp ,
1[0,t]N (x) :={
1 if ‖x‖p ≤ ‖t‖p0 otherwise.
We also setW (t ) := {W (t, ·)}t∈QNp= {�(1[0,t]N , ·)}t∈QN
p.We call the processW (t ) with
values in R the p-adic Brownian sheet on QNp .
Remark 11.21 Gâteaux derivatives Let y ∈ H∞(R) be fixed. Assume thatF : H∞(R) → R. For x ∈ H∞(R) consider λ → F(x+ λy) on R. If this functionis differentiable at λ = 0, we say that F is Gâteaux differentiable at x in the direc-tion y and denote DyF (x) := (d/dλ)F(x+ λy) |λ=0. Dy acts linearly and admits theusual chain and product rules. We define the partial derivative of the functional Fwith respect to g ∈ HR(∞) as
∂
∂gF = DgF. (11.25)
196 Non-Archimedean White Noise
Lemma 11.22 (i) Let
CT� : H∞(R) → R
g→∫
QNp
�(g(x))dNx.
Then partial derivatives of all orders of CT� exist everywhere on H∞(R). Forg1, . . . , gm ∈ H∞(R), we have
1
im∂m
∂gm · · · ∂g1CT�
∣
∣
∣
∣
0
= cm
∫
QNp
g1 · · · gm dNx,
where
c1 := a+∫
R�{0}
s3
1+ s2dM(s),
c2 := σ 2 +∫
R�{0}s2 dM(s),
cm :=∫
R�{0}sm dM(s), for m ≥ 3. (11.26)
(ii) Take C� = expCT�. Then C� has partial derivatives of any order, and themoments of � satisfy
M�m (g1 ⊗ · · · ⊗ gm) = 1
im∂m
∂gm · · · ∂g1C�
∣
∣
∣
∣
0
. (11.27)
Proof (i) See the proof of Lemma 3.1 in [7].(ii) The formula follows from Theorem 11.6 and (i), by
∂m
∂gm · · · ∂g1∫
H∗R(∞)
ei[T,g] dP� (T ) = imM�m (g1 ⊗ · · · ⊗ gm).
The last formula follows from the dominated convergence theorem on using thefacts that |ei[T,g]| = 1 and that P� is a probability measure. �
The following result is a consequence of Theorem 11.6 and Lemma 11.22.
Theorem 11.23 The process W (t ) has the following properties:
(i) W (0) = 0 almost surely;(ii) the process W (t ) is Gaussian with mean zero;(iii)
E[W (t )W (s)] ={
min(‖t‖p, ‖s‖p) if t = 0 and s = 00 if t = 0 or s = 0;
11.6 The p-Adic Brownian Sheet on QNp 197
(iv) let t1, t2, t3, t4 in QNp such that ‖t1‖p ≤ ‖t2‖p < ‖t3‖p ≤ ‖t4‖p, then W (t2)−
W (t1) and W (t4)−W (t3) are independent.
Bikulov and Volovich [82] and Kamizono [209], constructed Brownian motionwith p-adic time. In the case N = 1, our covariance function does not agree withthe one given in [82], [209]. Thus, in this case our result gives a different stochasticprocess.
12
Heat Traces and Spectral Zeta Functions forp-Adic Laplacians
12.1 Introduction
The connections between the Archimedean heat equations with number theory andgeometry are well known and deep. Let us mention here the connection with theRiemann zeta function which leads naturally to trace-type formulae, see e.g. [48] andthe references therein, and the connection with the Atiyah–Singer index theorem, seee.g. [178] and the references therein. The study of non-Archimedean counterparts ofthe above-mentioned matters is quite relevant, especially taking into account that theConnes and Deninger programs to attack the Riemann hypothesis lead naturally tothese matters, see e.g. [112], [121], [309] and the references therein. For instance,several types of p-adic trace formula have been studied, see e.g. [13], [96], [449] andthe references therein.In this chapter we study heat traces and spectral zeta functions attached to certain
p-adic Laplacians, denoted as Aβ , following [105]. Using an approach inspired bythe work of Minakshisundaram and Pleijel, see [340]–[342], we find a formula forthe trace of the semigroup e−tAβ acting on the space of square integrable functionssupported on the unit ball with average zero, see Theorem 12.13. The trace of e−tAβ isa p-adic oscillatory integral of Laplace–type. We do not know the exact asymptoticsof this integral as t tends to infinity; however, we can obtain a good estimation for itsbehavior at infinity, see Theorem 12.13 (ii). Several unexpected mathematical situa-tions occur in the p-adic setting. For instance, the spectral zeta functions are p-adicIgusa-type integrals, see Theorem 12.18. The p-adic spectral zeta functions studiedhere may have infinitely many poles on the boundary of their domain of holomor-phy. Thus, to the best of our knowledge, the standard Ikehara Tauberian theoremscannot be applied to obtain the asymptotic behavior for the function encompassingthe eigenvalues of Aβ less than or equal to T ≥ 0. However, we are still able to findgood estimates for this function, see Theorem 12.18, Remark 12.19, and Conjec-ture 12.20. The proofs require several results on certain “boundary-value problems”attached to p-adic heat equations associated with operators Aβ , see Proposition 12.5,Theorem 12.11, and Proposition 12.12.
198
12.2 A Class of p-Adic Laplacians 199
12.2 A Class of p-Adic Laplacians
Take R+ = {x ∈ R; x ≥ 0}, and fix a function
A : QNp → R+
satisfying the following properties:
(i) A(ξ ) is a radial function, i.e. A(ξ ) = g(‖ξ‖p) for some g: R+ → R+, forsimplicity we use the notation A(ξ ) = A(‖ξ‖p);
(ii) there exist constants C0, C1 > 0, and β > 0 such that
C0‖ξ‖βp ≤ A(ξ ) ≤ C1‖ξ‖βp, for x ∈ QNp . (12.1)
Taking into account that β in (12.1) is unique, we use the notation Aβ (‖ξ‖p) =A(‖ξ‖p).We define the pseudodifferential operator Aβ by
(Aβϕ)(x) = F−1ξ→x[Aβ (ξ )Fx→ξϕ], for ϕ ∈ D
(
QNp
)
. (12.2)
We will call Aβ (ξ ) the symbol of Aβ . The operator Aβ extends to an unbounded anddensely defined operator in L2
(
QNp
)
with domain
Dom(Aβ ) = {ϕ ∈ L2;Aβ (ξ )Fϕ ∈ L2}. (12.3)
In addition, we have the following properties:
(i) (Aβ,Dom(Aβ )) is a self-adjoint and positive operator;(ii) −Aβ is the infinitesimal generator of a contraction C0-semigroup, cf. [103,
Proposition 3.3].
We attach to operator Aβ the following “heat equation”:{
∂u(x, t )/∂t + Aβu(x, t ) = 0, x ∈ QNp , t ∈ [0,∞)
u(x, 0) = u0(x), u0(x) ∈ Dom(Aβ ).
This initial-value problem has a unique solution given by
u(x, t ) =∫
QNp
Z(x− y, t )u0(y) dNy,
where
Z(x, t;Aβ ) := Z(x, t ) =∫
QNp
χp(−ξ · x)e−tAβ (ξ ) dNξ, for t > 0, x ∈ QNp ,
cf. [103, Theorem 6.5]. The function Z(x, t ) is called the heat kernel associated withoperator Aβ .
200 Heat Traces and Spectral Zeta Functions
12.2.1 Operators Wα
The class of operators Aβ includes the class of operatorsWα studied by one of theauthors in [103], see also [104] and Chapter 8. In addition, most of the results onWα operators are valid for Aβ operators. We review briefly the definition of theseoperators. Fix a function
wα : QNp → R+
satisfying the following properties:
(i) wα (y) is a radial function, i.e. wα (y) = wα (‖y‖p);(ii) wα (‖y‖p) is a continuous and increasing function of ‖y‖p;(iii) wα (y) = 0 if and only if y = 0;(iv) there exist constants C0,C1 > 0, and α > N such that
C0‖y‖αp ≤ wα (‖y‖p) ≤ C1‖y‖αp, for x ∈ QNp .
We now define the operator
(Wαϕ)(x) = κ
∫
QNp
ϕ(x− y)− ϕ(x)
wα (‖y‖p) dNy, for ϕ ∈ D(
QNp
)
,
where κ is a positive constant. The operatorWα is pseudodifferential, more precisely,if
Awα(ξ ) :=
∫
QNp
1− χp(y · ξ )wα (‖y‖p) dNy,
then
(Wαϕ)(x) = −κF−1ξ→x[Awα
(ξ )Fx→ξϕ], for ϕ ∈ D(
QNp
)
.
The function Awα(ξ ) is radial (so we use the notation Awα
(ξ ) = Awα(‖ξ‖p)), continu-
ous, and non-negative, with Awα(0) = 0, and it satisfies
C′0‖ξ‖α−Np ≤ Awα
(‖ξ‖p) ≤ C′1‖ξ‖α−Np , for x ∈ QN
p ,
cf. [103, Lemmas 3.1, 3.2, 3.3]. The operator Wα extends to an unbounded anddensely defined operator in L2
(
QNp
)
.
12.2.2 Examples
Example 12.1 The Taibleson operator is defined as(
Dβ
Tφ)
(x) = F−1ξ→x
(‖ξ‖βpFx→ξφ)
, with β > 0 and φ ∈ D(
QNp
)
,
cf. [386], [18, Section 9.2.2].
12.3 Lizorkin Spaces, Eigenvalues, and Eigenfunctions 201
Example 12.2 Take Aβ (ξ ) = ‖ξ‖βp{B− Ae−‖ξ‖p} with B > A > 0. Then Aβ (ξ ) sat-isfies all the requirements announced at the beginning of this section. In general, iff : QN
p → R+ is a radial function satisfying
0 < infξ∈QN
p
f (‖ξ‖p) < supξ∈QN
p
f (‖ξ‖p) < ∞,
then Aβ (‖ξ‖p) f (‖ξ‖p) satisfies all the requirements announced at the beginning ofthis section.
12.3 Lizorkin Spaces, Eigenvalues, and Eigenfunctions for Aβ Operators
We set L0(
QNp
)
:= {ϕ ∈ D(
QNp
); ϕ(0) = 0}. The C-vector space L0 is called the p-adic Lizorkin space of the second class. We recall that L0 is dense in L2, cf. [18,Theorem 7.4.3], and that ϕ ∈ L0
(
QNp
)
if and only if∫
QNp
ϕ(x)dNx = 0. (12.4)
Consider the operator (Aβϕ)(x) = F−1ξ→x[Aβ (ξ )Fx→ξϕ] on L0
(
QNp
)
. Then Aβ isdensely defined on L2, and Aβ : L0
(
QNp
) → L0(
QNp
)
is a well-defined linearoperator.We set L0
(
ZNp
)
:= {
ϕ ∈ L0(
QNp
); suppϕ ⊆ ZNp
}
, and define
L20(
ZNp , d
Nx)
:= L20(
ZNp
) ={
f ∈ L2(
ZNp , d
Nx);
∫
ZNp
f (x)dNx = 0
}
.
Notice that, since L20(
ZNp
)
is the orthogonal complement in L2(
ZNp
)
of the space gen-erated by the characteristic function of ZN
p , we have that L20
(
ZNp
)
is a Hilbert space.Then L0
(
ZNp
)
is dense in L20(
ZNp
)
. Indeed, set
δk(x) := pNk(pk‖x‖p), for k ∈ N.
Then∫
QNpδk(x)dNx = 1 for any k, and we take f ∈ L20
(
ZNp
)
. Then fk = f ∗ δk ∈L0
(
ZNp
)
, and fk ‖ · ‖L2−−−→ f .
Set
ωγ bk(x) := p−Nγ2 χp(p
−1k · (pγ x− b))(‖pγ x− b‖p),where γ ∈ Z, b ∈ (Qp/Zp)N , k = (k1, . . . , kN ) with ki ∈ {0, . . . , p− 1} for i =1, . . . ,N, and k = (0, . . . , 0).
Lemma 12.3 With the above notation,
(Aβωγ bk )(x) = λγ bkωγ bk(x)
with
λγ bk = Aβ (p1−γ ).
202 Heat Traces and Spectral Zeta Functions
Moreover,∫
QNpωγ bk(x)dNx = 0 and {ωγ bk(x)}γ bk forms a complete orthogonal basis
of L2(
QNp , d
Nx)
.
Proof The result follows from Theorems 9.4.5 and 8.9.3 in [18], by using the factthat Aβ satisfies Aβ
(‖pγ (−p−1k + η)‖p) = Aβ
(‖pγ−1k‖p) = Aβ (p1−γ ), for all
η ∈ ZNp . �
Remark 12.4 (i) Notice that Aβ has eigenvalues of infinite multiplicity. Now, if weconsider only eigenfunctions satisfying supp ωγ bk(x) ⊂ ZN
p , then necessarilyγ ≤ 0 and b ∈ pγZN
p /ZNp . For γ fixed there are only finitely many eigenfunctions
ωγ bk satisfying Aβωγ bk = λγ bkωγ bk, i.e. the multiplicities of the λγ bk are finite.Therefore we can number these eigenfunctions and eigenvalues in the form ωm,λm with m ∈ N�{0} such that λm ≤ λm′ for m ≤ m′.
(ii) Notice that any ωm(x) is orthogonal to (‖x‖p), thus {ωm(x)}m∈N�{0} is nota complete orthonormal basis of L2
(
ZNp , d
Nx)
. We now recall that L0(
ZNp
)
is dense in L20(
ZNp
)
, and, since the algebraic span of {ωm(x)}m∈N�{0} con-tains L0
(
ZNp
)
, we have that {ωm(x)}m∈N�{0} is a complete orthonormal basis ofL20
(
ZNp
)
.
Proposition 12.5 Consider(
Aβ,L0(
ZNp
))
and the eigenvalue problem
Aβu = λu, λ > 0, u ∈ L0(
ZNp
)
. (12.5)
Then the function u(x) = ωm(x) is a solution of (12.5) corresponding to λ = λm, form ∈ N�{0}. In addition, the spectrum has the form
0 < λ1 ≤ λ2 ≤ · · · ≤ λm ≤ · · · with λm ↑ +∞,
where all the eigenvalues have finite multiplicity, and {ωm(x)}, with m ∈ N�{0}, is acomplete orthonormal basis of L20
(
ZNp , d
Nx)
.
Proof The result follows from Lemma 12.3, Remark 12.4 and (12.1). �
Definition 12.6 We define the spectral zeta function attached to the eigenvalue prob-lem (12.5) as
ζ(
s;Aβ,L0(
ZNp
))
:= ζ (s;Aβ ) =∞∑
m=1
1
λsm, s ∈ C.
Later on we will show that ζ (s;Aβ ) converges if Re(s) is sufficiently big, and itdoes not depend on the basis {ωm(x)} used in its computation. By an abuse of lan-guage (or following the classical literature, see [439]), we will say that ζ (s;Aβ ) isthe spectral zeta function of operator Aβ .
12.4 Heat Traces and p-Adic Heat Equations 203
12.3.1 Example
We compute ζ(
s;Dβ
T
)
. We first note that
Dβ
Tωγ bk = p−(γ−1)βωγ bk.
We now recall that if supp ωγ bk ⊂ ZNp then γ ≤ 0 and b ∈ pγZN
p /ZNp . We now
take −γ + 1 = m, with m ∈ N � {0}. Then b ∈ p−m+1ZNp /Z
Np and λm = pmβ , and
the multiplicity of λm is equal to (pN − 1)pN(m−1) = pNm(1− p−N ) for m ∈ N � {0}.Hence
ζ(
s;Dβ
T
) =∞∑
m=1
pNm(1− p−N )pmβs
=∫
QNp�ZNp
dNξ
‖ξ‖βsp= (1− p−N )
pN−βs
1− pN−βs ,
for Re(s) > N/β. Then ζ(
s;Dβ
T
)
admits a meromorphic continuation to the wholecomplex plane as a rational function of p−s with poles in the setN/β + 2π iZ/(β ln p).
12.4 Heat Traces and p-Adic Heat Equations on the Unit Ball
From now on, (Aβ,Dom(Aβ )) is given by
(Aβϕ)(x) = F−1ξ→x(Aβ (ξ )Fx→ξϕ) for ϕ ∈ Dom(Aβ ) = L0
(
ZNp
)
. (12.6)
12.4.1 p-Adic Heat Equations on the Unit Ball
We introduce the following function:
K(x, t ) =∫
QNp \ZNp
χp(−x · ξ )e−tAβ (ξ ) dNξ, for t > 0, x ∈ QNp .
We note that, according to (12.1), e−tAβ (ξ ) ≤ e−tC0‖ξ‖βp ∈ L1 for t > 0, which impliesthat K(x, t ) is well–defined for t > 0 and x ∈ QN
p .
Lemma 12.7 With the above notation, the following formulae hold:
K(x, t ) = (‖x‖p)⎧
⎨
⎩
(1− p−N )ord(x)∑
j=1
e−tAβ (pj )pN j − pord(x)Ne−tAβ (p
ord(x)+1 )
⎫
⎬
⎭
for x = 0 and t > 0; and
K(0, t ) = (1− p−N )∞∑
j=1
e−tAβ (pj )pN j
for any t > 0.
204 Heat Traces and Spectral Zeta Functions
Proof Take x = pord(x)x0, with ‖x0‖p = 1, then
K(x, t ) =∞∑
j=1
e−tAβ (pj )
∫
‖ξ‖p=pjχp(−x · ξ )dNξ
=∞∑
j=1
e−tAβ (pj )pN j
∫
‖y‖p=1
χp(−p− j+ord(x)x0 · y)dNy
=∞∑
j=1
e−tAβ (pj )pN j
⎧
⎪
⎪
⎨
⎪
⎪
⎩
1− p−N j ≤ ord(x)
−p−N j = ord(x)+ 1
0 j ≥ ord(x)+ 2.
ThenK(x, t ) = 0 for ‖x‖p > 1 and t > 0. Finally, we note that the announced formulais valid if x = 0. �
We identify L20(
ZNp
)
with an isometric subspace of L2(
QNp
)
by extending the func-tions of L20
(
ZNp
)
as zero outside of ZNp . We define {T (t )}t≥0 as the family of operators
L20(
ZNp
) → L20(
ZNp
)
f → T (t ) f
with
(T (t ) f )(x) ={
f (x) if t = 0
(K(·, t ) ∗ f )(x) if t > 0.
Lemma 12.8 With the above notation the following assertions hold:
(i) operator T (t ), t ≥ 0, is a well-defined bounded linear operator;(ii) for t ≥ 0,
(T (t ) f )(x) = F−1ξ→x
[
1QNp \ZNp (ξ )e
−tAβ (ξ )f (ξ )
]
,
where f (ξ ) denotes the Fourier transform in L2(
QNp
)
of f ∈ L20(
ZNp
)
;(iii) T (t ), for t > 0, is a compact, self-adjoint, and non-negative operator.
Proof (i) We recall that K(·, t ) ∈ L1(
QNp
)
for t > 0. Then, if f ∈ L20(
ZNp
) ⊂L2
(
QNp
)
, we have, by the Young inequality,
u(x, t ) := (K(·, t ) ∗ f )(x) ∈ L2(
QNp
)
, for t > 0.
Now, by Lemma 12.7, supp u(x, t ) ⊂ ZNp for t > 0, i.e. u(x, t ) ∈ L2
(
ZNp
)
, fort > 0. Again by the Young inequality,
‖u(x, t )‖L20
(
ZNp
) = ‖u(x, t )‖L2(
QNp
) ≤ ‖K(x, t )‖L1(
QNp
)‖ f (x)‖L2(
QNp
)
= C(t )‖ f (x)‖L20
(
ZNp
), for t > 0.
12.4 Heat Traces and p-Adic Heat Equations 205
We finally show that
∫
ZNp
u(x, t )dNx = 0, for t > 0.
Indeed, for t > 0, by using Fubini’s theorem,
∫
ZNp
u(x, t )dNx =∫
ZNp
{
∫
ZNp
K(y, t ) f (x− y)dNy
}
dNx
=∫
ZNp
K(y, t )
{
∫
ZNp
f (x− y)dNx
}
dNy
=∫
ZNp
K(z2, t )
{
∫
ZNp
f (z1)dNz1
}
dNz2 = 0.
(ii) Since f (x), u(x, t ) ∈ L1(
ZNp
) ∩ L2(
ZNp
)
for t > 0, because L2(
ZNp
) ⊂ L1(
ZNp
)
,we have
Fx→ξ (u(x, t )) = 1QNp \ZNp (ξ )e
−tAβ (ξ )f (ξ ).
The last function belongs to L1(
QNp
)
, indeed, by the Cauchy–Schwarz inequality,
‖1QNp \ZNp (ξ )e
−tAβ (ξ )f (ξ )‖
L1(
QNp
) ≤ ‖1QNp \ZNp (ξ )e
−tAβ (ξ )‖L2(
QNp
)‖f (ξ )‖L2(
QNp
)
≤ ‖e−tAβ (ξ )‖L2(
QNp
)‖ f (ξ )‖L2(
QNp
)
= ‖e−tAβ (ξ )‖L2(
QNp
)‖ f (ξ )‖L20
(
ZNp
) < ∞
because∫
QNpe−2tAβ (ξ ) dNξ ≤ ∫
QNpe−2C0t‖ξ‖βp dNξ < ∞, cf. (12.1). Finally,
(T (0) f )(x) =∫
QNp \ZNp
χp(−ξ · x)f (ξ )dNξ
=∫
QNp
χp(−ξ · x)f (ξ )dNξ −∫
ZNp
χp(−ξ · x)f (ξ )dNξ
= f (x)− F−1x→ξ ((‖ξ‖p)f (ξ )) = f (x)−(‖x‖p) ∗ f (x)
= f (x)−(‖x‖p)∫
ZNp
f (x)dNx = f (x).
206 Heat Traces and Spectral Zeta Functions
(iii) Since T (t ), for t > 0, is bounded and 〈T (t ) f , g〉 = 〈 f , T (t )g〉, for f , g ∈L2
(
ZNp
)
, where 〈·, ·〉 denotes the inner product of L2(
QNp
)
, T (t ) is self-adjointfor t > 0. To prove its compactness we show a sequence of bounded opera-tors Tl (t ) with finite range such that Tl (t ) ‖ · ‖−−→T (t ) for t > 0. For l ∈ N, weset Gl :=
(
Zp/plZp)N. We fix representatives, denoted by i, of Gl in ZN
p . Inparticular, for i ∈ Gl , ‖i‖p makes sense. Set L(l) to be the C-vector spacespanned by
{
(
pl‖x− i‖p)}
i∈Gl. Notice that ϕ ∈ L(l) if and only if supp
ϕ ⊂ BN0 and ϕ |i+(plZp)N = ϕ(i). On the other hand, by Lemma 12.7, K(x, t ) =(‖x‖p)h(‖x‖p, t ) with h(0, t ) defined and h(‖x‖p, t ) bounded on the unitball for t > 0. Set, for l ∈ N and t > 0, Kl (x, t ) :=
∑
i∈Glh(‖i‖p, t )(pl‖x−
i‖p) and Tl (t ) f := Kl (·, t ) ∗ f for f ∈ L20(
ZNp
)
. Then Tl (t ) : L20(
ZNp
) → L(l) ⊂L20
(
ZNp
)
is a bounded operator with finite range. Indeed, (
pl‖x− i‖p) ∗ f
has support in BN0 and (
pl‖x− i‖p) ∗ f is a constant function on the ball
i+ (
plZp)N. Finally, for t > 0, ‖Tl (t )− T (t )‖ ≤ ‖Kl (·, t )− K(·, t )‖
L1(
ZNp
) → 0
as l → ∞, by virtue of the dominated convergence theorem and the facts thatKl (x, t ) → K(x, t ) as l → ∞ and that supp Kl (·, t ), supp K(·, t ) ⊂ BN0 . �
Lemma 12.9 The one-parameter family {T (t )}t≥0 of bounded linear operators fromL20
(
ZNp
)
into itself is a contraction semigroup.
Proof The lemma follows from the following claims.
Claim 1. ‖T (t )‖L20
(
ZNp
) ≤ 1 for t ≥ 0. In addition, ‖T (t )‖L20
(
ZNp
) < 1 for t > 0.
Consider t > 0, by Lemma 12.8 and (12.1),
‖T (t ) f‖2L20
(
ZNp
) = ‖T (t ) f‖2L2(
QNp
) = ‖ T (t ) f (ξ )‖2L2(
QNp
)
≤∫
QNp \ZNp
e−2tAβ (ξ )|f (ξ )|2 dNξ
≤∫
QNp \ZNp
e−2C0t‖ξ‖βp |f (ξ )|2dNξ
≤ supξ∈QN
p \ZNpe−2C0t‖ξ‖βp
∫
QNp \ZNp
|f (ξ )|2 dNξ
<
∫
QNp \ZNp
|f (ξ )|2 dNξ ≤ ‖ f‖2L2(
QNp
)
= ‖ f‖2L20
(
ZNp
),
12.4 Heat Traces and p-Adic Heat Equations 207
where we used the fact that
supξ∈QN
p \ZNpe−2C0t‖ξ‖βp < 1.
Claim 2. T (0) = I.
Claim 3. T (t + s) = T (t )T (s) for t, s ≥ 0.This claim follows from Lemma 12.8 (ii).
Claim 4. For f ∈ L20(
ZNp
)
, the function t → T (t ) f belongs to
C(
[0,∞),L20(
ZNp
))
.
Notice that, since L20
(
ZNp
)
is dense in L20(
ZNp
)
for ‖ · ‖L2 norm, it suffices to showClaim 4 for f ∈ L2
0
(
ZNp
)
. Indeed,
limt→t0
‖T (t ) f − T (t0) f‖2L20
(
ZNp
) = limt→t0
‖T (t ) f − T (t0) f‖2L2(
QNp
)
= limt→t0
∥
∥T (t ) f − T (t0) f∥
∥
2
L2(
QNp
)
= limt→t0
∫
QNp \ZNp
|f (ξ )|2|e−tAβ (ξ ) − e−t0Aβ (ξ )|2 dNξ,
now, since 1QNp \ZNp (ξ )|f (ξ )|2|e−tAβ (ξ ) − e−t0Aβ (ξ )|2 ≤ 4|f (ξ )|2, which is an integrable
function, by applying the dominated convergence theorem, we have limt→t0 ‖T (t ) f −T (t0) f‖2L20(ZNp ) = 0. �
Lemma 12.10 The infinitesimal generator of semigroup {T (t )}t≥0 restricted toL0
(
ZNp
)
agrees with(− Aβ,L0
(
ZNp
))
.
Proof We show that
limt→0+
∥
∥
∥
∥
T (t ) f − f
t+ Aβ f
∥
∥
∥
∥
L20
(
ZNp
)
= 0, for f ∈ L0(
ZNp
)
.
Indeed, by Lemma 12.8 (ii),∥
∥
∥
∥
T (t ) f − f
t+ Aβ f
∥
∥
∥
∥
L20
(
ZNp
)
=∥
∥
∥
∥
T (t ) f − f
t+ Aβ f
∥
∥
∥
∥
L2(QNp )
=∥
∥
∥
∥
∥
T (t ) f − f
t+ Aβ f
∥
∥
∥
∥
∥
L2(
QNp
)
=∥
∥
∥
∥
∥
{
1QNp \ZNp (ξ )e
−tAβ (ξ ) − 1
t+ Aβ (ξ )
}
f (ξ )
∥
∥
∥
∥
∥
L2(
QNp
)
.
Now we note that{
1QNp \ZNp (ξ )e
−tAβ (ξ ) − 1}
f (ξ ) = f (ξ ){e−tAβ (ξ ) − 1} − 1ZNp (ξ )e−tAβ (ξ )
f (ξ ),
208 Heat Traces and Spectral Zeta Functions
and, since supp f ⊂ ZNp , we have that f (ξ + ξ0) = f (ξ ) for any ξ0 ∈ ZN
p . This fact
implies that 1ZNp (ξ )e−tAβ (ξ )f (ξ ) = e−tAβ (ξ )f (0) = 0 because f ∈ L0
(
ZNp
)
. Hence∥
∥
∥
∥
∥
{
1QNp \ZNp (ξ )e
−tAβ (ξ ) − 1
t+ Aβ (ξ )
}
f (ξ )
∥
∥
∥
∥
∥
L2(
QNp
)
=∥
∥
∥
∥
∥
{e−tAβ (ξ ) − 1}f (ξ )t
+ Aβ (ξ )f (ξ )
∥
∥
∥
∥
∥
L2(
QNp
)
= ‖Aβ (ξ )f (ξ ){1− e−τAβ (ξ )}‖L2(
QNp
) (for some τ ∈ (0, t )).
Therefore, by virtue of the fact that Aβ (ξ )f (ξ ) ∈ D(
QNp
)
and according to the domi-nated convergence theorem,
limt→0+
∥
∥
∥
∥
T (t ) f − f
t+ Aβ f
∥
∥
∥
∥
L20
(
ZNp
)
= limt→0+
‖Aβ (ξ )f (ξ ){1− e−τAβ (ξ )}‖L2(
QNp
) = 0.
�
Theorem 12.11 The initial-value problem⎧
⎪
⎪
⎨
⎪
⎪
⎩
u(x, t ) ∈ C([0,∞),Dom(Aβ )) ∩C1(
[0,∞),L20(
ZNp
))
∂u(x, t )/∂t + Aβu(x, t ) = 0, x ∈ QNp , t ∈ [0,∞)
u(x, 0) = ϕ(x) ∈ Dom(Aβ ),
(12.7)
where(
Aβ,Dom(
Aβ
))
is given by (12.6) has a unique solution given by u(x, t ) =T (t )ϕ(x).
Proof By Lemmas 12.9 and 12.10 and the Hille–Yosida–Phillips theorem, see e.g.[102, Theorem 3.4.4], the operator
(− Aβ,Dom(
Aβ
))
is m-dissipative with densedomain in L20
(
ZNp
)
. Hence the announced theorem now follows from [102, Theo-rem 3.1.1 and Proposition 3.4.5]. �
12.4.2 Heat Traces
Proposition 12.12 Let {ωm}m∈N�{0} be the complete orthonormal basis of L20(
ZNp
)
as above. Then
K(x− y, t ) =∞∑
m=1
e−λmtωm(x)ωm(y),
where the convergence is uniform on ZNp × ZN
p × [ε,∞), for every ε > 0.
Proof Upon applying the Hilbert–Schmidt theorem to T (1), see e.g. [377, Theo-rem VI.16], which is self-adjoint and compact, cf. Lemma 12.8 (iii), there exists acomplete orthonormal basis {φm}, m ∈ N�{0}, of L20
(
ZNp
)
consisting of eigenfunc-tions of T (1). Let {μm}, m ∈ N�{0}, be the sequence of corresponding eigenvalues.
12.4 Heat Traces and p-Adic Heat Equations 209
In addition, μm → 0 as m → ∞. By using the fact that {T (t )}t≥0 forms a semigroup,T (l/k)φm = μ
l/km φm, for every positive rational number l/k. Now, from the continuity
of {T (t )}t≥0, we get
T (t )φm = μtmφm, for t ∈ R+.
We note that μm > 0 for every m, since
φm = limt→0+
T (t )φm = φm limt→0+
μtm
implies that limt→0+ μtm = 1 because φm = 0. Hence μm = e−λm , with λm > 0,
because ‖T (t )‖L20
(
ZNp
) < 1 for t > 0, cf. Lemma 12.9 (i), implies that μm < 1 and
limm→∞ λm = ∞, since limm→∞ μm = 0.By using Mercer’s theorem, see e.g. [126] and the references therein, and [381],
K(x− y, t ) =∞∑
m=1
e−λmtφm(x)φm(y). (12.8)
Now, since T (t )φm(x) = e−λmtφm(x) is a solution of problem (12.7) with initial datumφm, cf. Theorem 12.11, and
−λme−λmtφm(x) = ∂
∂t(e−λmtφm(x)) = −Aβ (e
−λmtφm(x))
= −e−λmtAβφm(x),
φm(x) is an eigenfunction of Aβ with supp φm ⊂ ZNp . Now, by using that Aβωm =
λmωm, see Proposition 12.5, we get that u = e−λmtωm is a solution of the followingboundary-value problem:
{
∂u(x, t )/∂t = −Aβu(x, t ), u(x, t ) ∈ L20(
ZNp
)
, for t ≥ 0
u(x, 0) = ωm(x), ωm(x) ∈ L0(
ZNp
)
.
Then, by Theorem 12.11, the above problem has a unique solution, which impliesthat
u(x, t ) = T (t )ωm(x) = e−λmtωm.
Hence we can replace {φm} by {ωm} in (12.8). �
In the next result, we will use the classical notation e−tAβ for the operator T (t ) toemphasize the dependency on operator Aβ .
Theorem 12.13 The operator e−tAβ , for t > 0, is of trace class and satisfies(i)
Tr(e−tAβ ) =∞∑
m=1
e−λmt =∫
QNp�ZNp
e−tAβ (ξ ) dNξ, (12.9)
for t > 0;
210 Heat Traces and Spectral Zeta Functions
(ii) there exist positive constants C, C′ such that
Ct−Nβ ≤ Tr
(
e−tAβ) ≤ C′t−
Nβ ,
for t > 0.
Proof By virtue of Proposition 12.12 and the definition of K(x, t ), we have that
K(0, t ) =∫
QNp�ZNp
e−tAβ (ξ ) dNξ =∞∑
m=1
e−λmt |ωm(x)|2, (12.10)
for t > 0. By using the dominated convergence theorem and the fact that∑
m e−λmt
converges for t > 0, we can integrate both sides of (12.10) with respect to the variablex over ZN
p to get
∫
QNp�ZNp
e−tAβ (ξ ) dNξ =∞∑
m=1
e−λmt, for t > 0. (12.11)
We recall that
e−C1t‖ξ‖βp ≤ e−tAβ (ξ ) ≤ e−C0t‖ξ‖βp , (12.12)
cf. (12.1), and that e−Ct‖ξ‖βp ∈ L1, for t > 0 and for any positive constant C. Thus the
series on the right-hand side of (12.11) converges. Now
Tr(e−tAβ ) =∞∑
m=1
〈e−tAβωm, ωm〉 =∞∑
m=1
e−λmt‖ωm‖2L2
=∞∑
m=1
e−λmt < ∞, for t > 0,
i.e. e−tAβ is of trace class and the formula announced in (i) holds. The estimation forTr
(
e−tAβ)
follows from (12.12), by using∫
QNpe−Ct‖ξ‖
βp dNξ ≤ Dt−
Nβ for t > 0. �
12.5 Analytic Continuation of Spectral Zeta Functions
Remark 12.14 (i) We set, for a > 0, as := es ln a. Then as becomes a holomorphicfunction on Re(s) > 0.
(ii) We recall the following fact, see e.g. [205, Lemma 5.3.1]. Let (X, dμ) denote ameasure space, U a non-empty open subset of C, and f : X ×U → C a measur-able function. Assume that (1) if C is a compact subset of U, there exists an inte-grable function φC ≥ 0 on X satisfying | f (ξ, s)| ≤ φC(ξ ) for all (ξ, s) ∈ X × C;(2) f (ξ, ·) is holomorphic on U for every x in X. Then
∫
X f (ξ, s)dμ is a holo-morphic function on U.
12.5 Analytic Continuation of Spectral Zeta Functions 211
Proposition 12.15 The spectral zeta function for Aβ is a holomorphic function onRe(s) > N/β and satisfies
ζ (s;Aβ ) =∫
QNp�ZNp
dNξ
Asβ (ξ )for Re(s) >
N
β. (12.13)
In particular, ζ (s;Aβ ) does not depend on the basis of L20(
ZNp
)
used in Definition12.6.
Proof On using Proposition 12.5 and Remark 12.4, the eigenvalues have the formAβ (p1−γ ), with γ ≤ 0, and the corresponding multiplicity is the cardinality ofpγZN
p /ZNp times the cardinality of the set of ks, i.e. p−γN (pN − 1). Then
ζ (s;Aβ ) =∑
γ≤0
p−γN (pN − 1)
Asβ (p1−γ )
=∞∑
m=1
pmN (1− p−N )Asβ (p
m)
=∞∑
m=1
∫
‖ξ‖p=pm
dNξ
Asβ (‖ξ‖p)
=∫
QNp�ZNp
dNξ
Asβ (ξ ),
and, by (12.1),
|ζ (s;Aβ )| ≤ 1− p−N
CRe(s)
∞∑
m=1
pm(N−βRe(s)) < ∞ for Re(s) >N
β.
To establish the holomorphy on Re(s) > N/β we use Remark 12.14 (ii). Take X =QNp � ZN
p , dμ = dNξ ,U = {s ∈ C;Re(s) > N/β} and f (ξ, s) = A−sβ (‖ξ‖p). We now
verify the two conditions established in Remark 12.14 (ii). Take C a compact subsetofU , by (12.1),
∣
∣
∣
∣
∣
1
Asβ (‖ξ‖p)
∣
∣
∣
∣
∣
≤ 1
CRe(s)‖ξ‖β Re(s)p
,
where C is a positive constant. Since Re(s) belongs to a compact subset of{
s ∈ R;Re(s) > N
β
}
,
we may assume without loss of generality that Re(s) ∈ [γ0, γ1] with γ0 > N/β. Then
1
CRe(s)‖ξ‖β Re(s)p
≤ B(C) 1
‖ξ‖βγ0p
∈ L1,
where B(C) is a positive constant. Condition (2) in Remark 12.14 (ii) followsfrom Remark 12.14 (i) on noting that (Aβ (‖ξ‖p))−s = exp(−s lnAβ (‖ξ‖p)) withAβ (‖ξ‖p) > 0 for ‖ξ‖p > 1. �
212 Heat Traces and Spectral Zeta Functions
Remark 12.16 We notice that formula (12.13) can be obtained by taking the Mellintransform in (12.9). Indeed,
∞∫
0
⎧
⎪
⎨
⎪
⎩
∫
QNp�ZNp
e−tAβ (‖ξ‖p)ts−1dNξ
⎫
⎪
⎬
⎪
⎭
dt =∞∫
0
{ ∞∑
m=1
e−λmtts−1
}
dt = �(s)ζ (s;Aβ ),
for Re(s) > 1, where �(s) denotes the Archimedean gamma function. Now, by achange of variables to y = Aβ (‖ξ‖p)t, with ξ fixed, we have
ζ (s;Aβ ) =∫
QNp�ZNp
dNξ
Asβ (‖ξ‖p)for Re(s) > max
{
1,N
β
}
.
Lemma 12.17 ζ (s;Aβ ) has a simple pole at s = N/β.
Proof Set σ ∈ R+. Then, since
ζ (σ ;Aβ ) ≤ 1
C0
∫
QNp�ZNp
dNξ
‖ξ‖βσp
= (1− p−N )p−βσ+N
C0(1− p−βσ+N )for σ >
N
β,
we have
limσ→ N
β
(1− p−βσ+N )ζ (σ ;Aβ ) > 0. (12.14)
The assertion follows from (12.14), by using the fact that 1− p−βσ+N has a simplezero at N/β. Indeed,
1− p−βσ+N = 1− exp{(−βσ + N) ln p}
= {β ln p}(
σ − N
β
)
+ O
(
(
σ − N
β
)2)
,
where O is an analytic function satisfying O(0) = 0. �
Theorem 12.18 The spectral zeta function ζ (s;Aβ ) satisfies the following criteria.
(i) ζ (s;Aβ ) is a holomorphic function onRe(s) > N/β, and in this region it is givenby formula (12.13).
(ii) ζ (s;Aβ ) has a simple pole at s = N/β. However, this pole is not necessarilyunique.
(iii) Set N(T ) := ∑
λm≤T 1, for T ≥ 0. Then N(T ) = O(
TNβ
)
.
Proof (i) See Proposition 12.15.(ii) The first part was established in Lemma 12.17. TakeAβ to be the Taibleson oper-
ator DβT . Then ζ
(
s;Dβ
T
)
has a meromorphic continuation to the whole complexplane as a rational function of p−s having poles in the set N/β + 2π iZ/(β ln p),see Example 12.3.1.
12.5 Analytic Continuation of Spectral Zeta Functions 213
(iii) The result follows from the formulae
λm = Aβ (pm) and mult(λm) = pNm(1− p−N ), for m ∈ N�{0}. �
Remark 12.19 The fact that ζ (s;Aβ ) may have several poles on the line Re(s) =N/β prevents us from using the classical Ikehara Tauberian theorem to obtain theasymptotic behavior of N(T ), see e.g. [106, Appendix A], [394, Chapter 2, Section14]. Anyway, we expect that the following conjecture holds.
Conjecture 12.20 N(T ) ∼ CTNβ , for some suitable positive constant C.
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Index
actomyosin molecular motor, 70–71additive character, 2
balance equations for densities of luids, 85–89Bruhat–Tits buildingsafine, 13
cascade model of turbulence, 78–81characteristic relaxation in complex systems, 116clusteringcluster networks, 12ensemble clustering, 8multiclustering, 8, 12
coherent state, 22, 28complex energy landscapes, 67complex system, 63
disconnectivity graph, 64distributions, 4convolution, 5direct product, 5Fourier transform, 5multiplication, 6positive-deinite, 136wavelet transform, 40
ellipticoperator, 121polynomial, 121
energybarrier, 67Helmholtz-energy functionals, 124
ergodicconstruction of ergodic functions, 110dynamical systems, 1051-Lipschitz functions, 108
Euclidean random ields, 190
Fourier transformof a distribution, 5of a test function, 4
functionBruhat–Schwartz, 3Green function, 1801-Lipschitz functions, 94locally constant, 3positive-deinite, 137scaling, 21, 43, 44test, 3
fundamental solution, 121,174
genetic code, 73
Haar measure, 94heatkernel, 118, 121trace, 203
hierarchical kinetics, 66
Markovchains, 127processes, 115, 118, 121
measure preserving 1-Lipschitz functions,99–104
noiseGaussian, 138white, 81
operatorintegral operators, 55Klein–Gordon pseudodifferential operator,
186on ultrametric space, 59
236
Index 237
pseudodifferential operators, 54, 159, 168,179
Taibleson operator, 120Vladimirov operator, 6W , 122
p-adicBochner–Schwartz theorem, 136Brownian motion, 81Brownian sheet, 195diffusion, 66dynamical systems, 94fractional part, 2heat equations, 135multiresolution analysis, 21norms, 1numbers, 1order, 1parabolic-type pseudodifferential equations,
134parametrization of the Parisi matrix, 65scaling function, 48Shannon–Kotelnikov theorem, 53wavelets, 20, 46
Parisi matrix, 65porous medium equation, 90–92explicit solution, 92–93
quasilinear diffusion, 89–90
reaction–ultradiffusion equations, 123
spaceLizorkin spaces, 6H∞, 156–158
dual space ofH∞, 159–168W∞, 165–167
dual space ofW∞, 167spin glass, 65stochasticheat equations, 133integrals, 138pseudodifferential equations, 148
treeabsolute of a tree, 11groups acting on trees, 17of basins, 67
ultrametric models in geophysics, 83–84ultrametric space, 9, 59
Van der Put Series, 96
waveletHaar basis, 20, 41multidimensional wavelets bases, 33, 51multiresolution frames, 49on ultrametric spaces, 59
zeta functionIgusa’s local zeta function, 172spectral zeta function, 202, 210