the law of the solution to a nonlinear hyperbolicspde

Journal o/" Theoretical Probability Vol. 9, No. 4. 1996

The Law of the Solution to a Nonlinear Hyperbolic SPDE

Caries Rovira 2 and Marta Sanz-Sol6 2

Receired March 10. 1995: revised September 5. 1995

Let { flz.,, (s. t } ~ } be a white noise on R-'+. We consider the hyperbolic stochastic partial differential equation

~ X , 0X,, ,

?'-X~"=alls, t ) ~ + a , ( s , t ) f~f +a~lA,s , t ) k[z,,+a4(X,s t) c%" c~ t "

The purpose of this paper is to study the law of the solution to this equation. We analyze the existence and smoothness of the density using the tools of Malliavin Calculus. Finally we prove a large deviation principle on the space of contint, ous functions, lbr the family of probabilities obtained by perturbation of the noise in the equation.

KEY WORDS: Hyperbolic SPDE's; strong and weak solution; Malliavin calculus: large deviations.

1. INTRODUCTION

Consider the following hyperbolic stochastic partial differential equation:

O'-X,., OX,., OX,., " =al (s , t)~+a,(s,_ t)--~-s a3(X,s,t) W,..,+a4(X,s,t) (1.1)

(s, t)6 E + , X~., = Xo on the axes. Here { W. ..... (s, t)6 R~_} denotes a white noise on (~_,.~/(R~_)), Xo is a ~.o-measurable random variable, where {.~,..,, (s, t)~R~_} is the natural filtration associated with the Brownian sheet { W,..,, (s , t )~R~_}; a i, i = 1 , 2 are real functions defined on R~_

This work has been partially snpported by the grant of the DGICYT No. PB 930052 and the EU Science project CT910459.

2 Facultat de Matemfitiques, Universitat de Barcelona, Gran Via, 585, 08007 Barcelona, Spain.

863

0894-9840/96,1000-0863509.50/0 !~ 1996 Plenum Publishing Corporation

864 Rovira and Sanz-Sol~

and a,, i = 3 , 4 real-valued functions defined on rC(R~_)xN~_. The case a~ = a 2 - 0 has been widely studied (see for instance, Refs. 3 and 22); the corresponding integral form is

X,, ,=Xo+ fR [a3(X, u, v) dW,,,+a4(X, u, v) dudv] (1.2)

where R.,.., is the rectangle (0, s] x (0, t] and the stochastic integral is the two-parameter It6 integral defined by Wong and Zakai I-'~) (see also Ref. 4).

In Ref. 8, Eq. (l.1) has been analyzed using the following approach. A solution to Eq.(l.1) is a continuous, o~,,-adapted process X={X, . , , (s, t) E N+ } satisfying the integral equation

X,., = Xo + fR [al(u, v) X(u, dr) du + a2(u, v) X(du, v) dv s. t

+ a3(X , u, l;) dW,,.,. + a4(X, u, v) du dv] (1.3)

The two first mixed stochastic integrals in Eq. (1.3) make sense whenever X belongs to the class of two-parameter processes called representable semi-martingales (see Ref. 9). Roughly speaking a process in this class can be written as the sum of stochastic and mixed integrals with respect to the Brownian sheet and a Lebesgue integral.

Theorem 2.1 in Ref. 8 establishes existence and uniqueness of solution to Eq. (1.3) in the class of representable semi-martingales. More precisely, under appropriate assumptions on the coefficients the unique solution to Eq. (1.3) satisfies

X.,.., = Xo + a3(X, u, v) dW,,.,, + fl,(w; u, v) dw dW, .... s . t ~ . t

+ flz(u,v;r) drdW,,.,.+ qo(u,v) dudv (1.4) s. t s . t

where

fll(w;u,v)=az(u,w)ll,,~,,. I a3(X,u,v)+ fll(W';U,v) dw' (1.5)

( ) flz(u,v;r)=at(r,v)ll,<.,. I a3(X,u ,v) -F f l2(u,v;r ')dr' (1.6)

The Law of a Nonlinear SPDE 865

cp(u,v)=a4(X,u,v)+al(u,o) q~(r,v)dr+a2(u,v) cp(u,w)dw

+a~(u, v) I /~,(v; r, w) dWr.w R u , i,

+a2(u, v) fR ~2(r' w; u) dWr.w (1.7)

U, U, W, r $ ~ + . The proof of this resul requires expertise in two-parameter stochastic

calculus. In the sequel the process {Xs,,, (s, t)~ Rz+} defined by Eq. (1.4) will be called the strong solution to Eq. (1.1). In view of the literature on parabolic and elliptic stochastic partial differential equations it seems also natural of study Eq. (1.1) following a different approach, inspired by Riemann's method for solving the deterministic analogue of Eq. (1.1). Indeed, consider the second order differential operator A ~ given by

s t) 02f(s' t) al(s, t)Of~t t) a2(s, t )Of~ t) Os Ot (1.8)

and denote by Z,..,(u, v), O<~u<~s, 0 <<. v<~ t its associated Green function (see the Appendix for the precise definition of this problem). By a weak solution to Eq. (1.1) we mean a continuous, ~ . ,-adapted process X = { Xs.,, (s, t) ~ R~_ } satisfying

Xs.,=Xo+fR y.~.t(u,o)[aa(X,u,v)dW,,,~+aa(X,u,v)dudv] (1.9) t

The aim of this paper is to study Eq. (1.1) under this point of view. In Ref. 19, we have proved existence and uniqueness of solution to the stochastic Eq. (1.9), we also have established a result on approximation of this solution. In Section 2, we state the equivalence between the two approaches presented before: the two notions of solutions, strong and weak, coincide. The representation in Eq. (1.9) seems to be more appropriate than in Eq. (1.4) for the purpose of studying properties of the process X. In Section 3, we apply the tools of Malliavin calculus to deduce the existence and smoothness of density for the law of Xs.,, s. t ~ 0. For a~ = a2---0 and multidimensional X, this problem has been studied in Refs. 17 and 18; the existence of density has also been proved in Ref. 7 using the expression in Eq. (1.4), with Eq. (1.9) this question becomes almost trivial. In Section4, we establish a large deviation principle for the family { X ~, e > 0} of solutions to Eq. (1.9) obtained by a perturbation of the noise.


For the solution to Eq. (1.2) (multidimensional X) this result has been studied in Ref. 5. In comparison with this last paper, the difficulty here rises from the fact that the process {JR,., y.~..,(u, v)a3(X, u, v ) d W , ..... (s, t ) e ~ + } does not have the martingale property, thus exponential inequalities for processes of this type have to be proved. Finally we include in Section 5 some properties concerning the Green function associated with Aa that are used throughout the paper.

Recently, Norris ~t6~ has proved an existence and uniqueness theorem for a class of hyperbolic two-parameter stochastic differential equations more general than Eq. (1.1). As in Ref. 8 the method uses two-parameter stochastic calculus for "semimartingales." His theorem is applied to con- struct some path-valued processes in a Riemannian manifold and, in parti- cular, a manifold-valued Brownian sheet. This interesting question provides a good motivation for getting some insight into this class of stochastic partial differential equations.

Throughout the paper all constants appearing in the proofs are called C, although they may vary from one part to another one. The points in ~+ will usually be denoted by z = (s, t), ~/= (u, v) or 0c = (r, w).

2. EXISTENCE AND UNIQUENESS OF SOLUTION

Let T = [ 0, 1 ] 2 endowed with the partial order defined coordinatewise. We consider functions a;: T ~ ff~, i = 1, 2 satisfying the assumptions

(H1) a;, i = 1, 2, are differentiable and bounded, with bounded first partial derivatives.

Fix (s, t) e Tand let 7.,..,(u, v) be the function defined on {(u, v): (0, 0) ~< (u, v) ~< (s, t)} satisfying

a'-ys.,(u, v) -~ a(a~(u, v) ys.,(u, v)) ~- a(a,_(u, v) 9,,..,(u, v)) = 0 (P) 3u Ov Ov 3u

O)'s.,(u, v) Ou t -a , (u ,v)~ ,s . , (u ,v)=O when v = t

O~,,.,(u, v) Ov

+ a z ( U , V ) G . , ( u , v ) = O when u = s

~,~.,(u,v)=l when u = s , v = t

The existence of such function is established in the Appendix. We will refer to ?s.,(u, v) as the Green function associated with the operator s defined in Eq. (1.8).


Set ~ = ~( T; R), the space of real-valued continuous functions defined on T. Consider measurable mappings a~: ~ x T ~ R, i=3 , 4, which are causal, that means, for any z e T, a~(., z) is measurable with respect to the a-algebra generated by the continuous functions g: [0, z] ~ R. Suppose

(H2) There exists a constant C > 0 such that, for all f , f ' e ~ f , (s, t )e T and i = 3 , 4

la,.(f, s, t ) - a i ( f ' , s, t)[ <<, C If(s , t) - f ' ( s , t)l

la;(f, s, t)l ~< C(1 + I f (s , t)[)

The purpose of this section is to check that, under hypotheses (HI) and (H2), the unique continuous, adapted process satisfying the stochastic equation in Eq. (1.9) coincides with the solution to Eq. (1.1) in the sense of distributions. In addition the solution to Eq. (1.9) satisfies Eq. (1.4). Conse- quently, both interpretations of the evolution equation in Eq. (1.1), by means of Eq. (1.3), as has been proposed in Ref. 8, or using Eq. (1.9), are equivalent.

First we recall the existence and uniqueness of solution to Eq. (1.9), which can be easily checked using a Picard iteration scheme (see Proposi- tion 2.1 in Ref. 19).

Theorem 1. Assume that the coefficients a~, i = 1, 2, 3, 4, satisfy hypotheses (HI), (H2) and Xo is some .~0.0-measurable random variable belonging to L 2e for some p >~ 1. Then, there exists a unique continuous adapted solution X = {X,.,, (s, t )e T} of Eq. (1.9) which is bounded in L2C In addition, if ,go e L z' for any p >/1, a.s. the paths of X are e-H61der continuous for any ~ e (0, �89

White noise { 14z.,..,, (s, t) e T} is the f o rmal derivative of the Brownian sheet (which is nowhere differentiable). Another meaning to the formal expression in Eq. (1.1) can be given throughout a weak formulation. We next make this notion precise and prove some results which explain why the unique process satisfying Eq. (1.9) has been called a weak solution of Eq. (1.1).

Let ca2 be the set of functions cp: T--+R such that (O~o(s, t)/Os), (&p(s, t)/Ot), (02~o(s, t)/OsOt) exist. Multiply both sides in Eq. (1.1) by a function ~0 in ca2 and integrate over R,,, ,

LIs. [a3(X, u, v) ~0(u, v) dl.V..v Wa4(X, u, v) r v) dudv]

( 82X U) ~- -a2(u , o)~-~--~ qa(u,v) dudv = f.,, \a-7 o -a'(u'


Integrating by parts the right-hand side of this equation we obtain

fR [a3(X" o) co(u, t)) dW,,.,, + a4(X , o) co(//, /~) du dr] u, u, s . t

fo =X(s,t) co(s,t)-Xoco(O,O)+Xo al(u,O) co(u,O)du

s; ( ) ) +Xo a2(O,v) co(O>v)dv- X(u,t) (u,t)+at(u>t)co(u,t) du

, (oco )av+fR, x(u ,v)(~ v) - fo X(s, v) ~v (s, v) + a2(s, v) co(s, v) . - (u,

O(al(u, V)Ovco(u, v)) + O(a2(u, V)Ouco(u, v))) du do + (2.1 )

We can now state the following result.

Proposition 1. Assume (H1) and (H2). There exists at most one continuous process {X~.,, (s, t )eT} satisfying Eq.(2.1), for any coeff2, and such that X~.., = ,go if s- t = 0.

Proof Fix (s, t) ~ T. In Proposition 10 we prove that the Green function Ys., associated with s belongs to ~g2. Let 2" be a process satisfying Eq. (2.1) for any co~cg2, with the value Xo on the axis. For co=-y.,., and taking into account properties (P) we obtain

fR )'.,.,(U, v)[a3(2", o) dWu.,,+a4(X, v)] dudv U, u, s , t

= X(s, t) - 2"oL..,(O, O)

+ X o al(u,O) y,.,(u,O)du+ a2(O,v) y,.,(O,v)dv (2.2)

Let X 1 and X 2 be two processes with the same properties as X. The identity in Eq. (2.2) implies

X i _ * [ s., X~,= y,.,(u,v)[(a3(Xi, u,v)-a3(X-,u,v))dl/V,,.v aR s . t

"k- (a4(X I, u, v) --a4(X "2, U, U)) du dr]


The isometry property of the stochastic integral, H61der's inequality, Eq. (5.5) and (H2) yield

E(IX~,,-X~.,I-)<<. C E( ' X~,ol )dudv Rx.t

Then, using a Gronwall's inequality and the continuity of X ~, X 2 we obtain I 2 X~.,=X,., for any (s, t )~T, a.s. []

Proposition 2. Suppose that (H1) and (H2) are satisfied. The unique solution to Eq. (1.9) satisfies Eq. (2.1).

Proof Let {X.,..,, (s, t ) e T} be the solution to Eq. (1.9). For any test function ~0 e cgz set

V(s, t )=I_ ~o(u, v)[a3(X, u, v) dW,,.v+a4(X, u, v) dudv] s.t

-- X(s, t) q~(s, t ) - X o q~(0,0)- al(u, O) ~o(u, O) du

fo ) - X(u,t) ~-s du

- X(s,v) ~v(S'v)+a2(s'v)q~(s'v) dv

( O'-q~ (u, v)+ O(a~(u' v) ~o(u, v)) f + _.IR~, X(u, v) \ O-Y~v Ov

Equation (1.9) and Fubini's theorem yields

V(s , t )=XoV~ I V,.. ,(u,v)[a4(X,u,v)dudv+a3(X,u,v)dWu.v] Rs, i

870

where

and

Rovira and Sanz-Sol6

V.,.,(u, v)= ~p(u, v)- Z,.,(u, v) cp(s, t)

+ ?,..,(u,v) -~-7(r, t)+aj(r, t)cp(r, t) dr

+ . y.,..,,.(u, v) ~ (s, w) + a2(s, w) cp(s, w) dw

f~f' (02~o(r,~,,) O(a,(,',w) cp(,',w)) - , .?"'"(u'v)\ OrOw + Ow

+ O(a2(r, w) cp(r, w))) dr dw Or

(2.3)

fj ~l 0 V~ t) = -cp(s, t) + cp(O, O) - at(u, O) cp(u, O) d u - a2(O, v) cp(O, v) dv

+ ~u (u, t) + aj(u, t) cp(u, t) du

+ ~ (s, v) + a2(s, v) cp(s, v) dv

,., \ Ou Ov + O(at(u, v) cp(u, v)) O(a2(u, v) cp(u, v))) du dv

Ov ~- c3u

Integrating by parts Eq. (2.3) (here we use Proposition 10),

f ( ) V,.,(u, v) = cp(r, v) adr, v) 7,..,.(u, v) Oy,..,.(u, v) dr t 0]"

( ) + cp(u, w) a~(u, w) ?,.,,.(u, v)--c~Y"'"'(u' v) dw �9 - 0 W

f," f,' (a~-~''"'(u' v) - cp(r, w) \ Or Ow I '

at(r, w) d~%,(u, v)

aw

- - a~(r , w ) aT'''''(u' v)) dw dr - ~ r


Properties in Eqs. (5.8)-(5.10) on the Green function ensure V,..,(u, v ) = 0 for all (0, 0)~< (u, v)~<(s, t). The same arguments show V~ t)=0. Thus V-- 0. Consequently, the process X satisfies Eq. (2.1). []

The last result of this section states the equivalence between the notions of weak and strong solutions to Eq. (1.1).

Proposition 3. Under (H1) and (H2) the solution of Eq. (1.9) satisfies Eq. (1.4).

Proof. Let X be a process satisfying Eq. (1.9), ill(w; u, v), P2(u, v; r) as in Eqs. (1.5) and (1.6), respectively. The integral equation Eq.(5.4) satisfied by the Green function, Eqs. (1.5) and (1.6) yield

fe,., ~',..,(u, v) a3(X, u, v) dW, ....

= a3(X, u, v) 1 + al(r, v) y~.,(r, v) dr Rs, t

f; ] + . a~(u, w) 7.,.,(u, w) dw dW, .... (2.4)

[,i a3( X, u, v) a2(tt , W)),s.,(t,, W)dw

= ~ (fl,(w; u, v)-a2(u, w) Ii"flt(w'; u, v) dw') z,~,(u, w) dw

= f,: flt(w; u, v) (Y.,..,(u, w ) - f.,i. a2(u, w') Ts.,(u, w') dw') dw

= ~,i fl,(w; u, v) (l + ~,: at(r, w) 7.~,,(r, w) dr) dw

= ~o ill(w; u, v ) d w + ~,i" ~i a,(,', w)),~.,(,', w)fl,(w; u, v )d r dw (2.5)

and

~.f a~(X, u, v) at(r, v) 7.~,,(r, v) dr t

fo- i f, , = fl2(u, v; r) dr + az(r, w) 7s.,(r, w) f12( u, v; r) dr dw I '

(2.6)


Replace in Eq. (1.9) the stochastic integral by the right-hand side of Eq. (2.4). Then, using Eq. (2.5) and Eq. (2.6)

Let

s,t

+ },~.,(r, w) al(r, w) ill(w; u, v) dr dw dW,,.~, s,t r

+ f.. (f: . . . . .

a(X ,u , v )=a4(X ,u , v )+a t (u , v )~ fl ,(v;r,w) dWr .... Ru.~.

+ a2(u, v) fR flz(r, w; u) dW~ ....

Equation (2.7) can be written as

X, . ,=Xo+ a3(X,u,v)+ fl,(w;u,v) dw+ fl2(u,v;r) dr dVr s.t

(2.7)

Set

+ fR y,,,(u, v) a(X, u, v) du dv (2.8) s.l

f

Z(s, t) =1_ ~,.,.,(u, v) a(X, u, v) du dv s,t

The results on the Green function proved in the Appendix show that for any o~ e O, Z(s, t) is the solution of the partial differential equation

02Z = a(X, s, t) + a l(s, t) ~ + a2(s, t)

O__ZZ Os Ot ot Os


Z(s, t) = 0 if s. t = O. The function r t) = (Ozz/os at) satisfies

~p(s, t )=a(X,s , t )+al(s , t ) ~o(u,t) du+a2(s,t) ~o(s,v) dv

Consequently, Eq. (2.8) shows that X clearly satisfies

a3(X , u, v) + ill(W; U, V) dw + fl~(u, v; r) dr dW.,,, x(s, t) = Xo + ;,,~,

+ fR,, cp(u, v) du dv

So, X is the strong solution of Eq. (1.1).

Remark 1. In Ref. 8 the existence and uniqueness of solution to Eq. (1.3) is established under the following assumptions, weaker that (HI) and (H2).

(HI) ' a t i = 1, 2, are bounded.

(H2)' There exists a constant C > 0 such that, for any f , f ' s ~ , (s , t )ET, i = 3, 4,

lai(f, s, t ) - a i ( f ' , s, t)l ~< C sup O<~(u,v)<~(s,t~

laj(f, s, t)l ~< C sup O<~(u.v) <~(s,t)

If(u, v ) - i f ( u , v)l

If(u, v)l

Since the process {~ R~., 7.,..,(U, V) a3(X, u, t~) dW, .... (s, t) ~ T} does not have the martingale property, we can not use maximal inequalities for moment estimates and therefore (H2)' is not suitable to study weak solutions.

3. REGULARITY OF THE LAW

Let E = {z = (s, t) e T: s. t = 0}. In this section we give sufficient conditions ensuring existence and smoothness of the density for the probability law of)(=, solution to Eq. (1.9) for any fixed zq~ T\E. Along this section we assume the following set of hypotheses on the coefficients:

(H1) a i : T ~ , i = 1 , 2 , are differentiable and bounded with bounded first partial derivatives.


(H2)' a~: R x T ~ R, i = 3, 4, are measurable, infinitely differentiable with respect to the first variable with uniformly bounded derivatives of any order. Moreover,

sup [a~(x, z)l ~< C(1 + Ixl) = E T

We also introduce additional assumptions needed to state the main result of this section.

(H3) sup [a3(x, u, v ) - a3(x, u', v') I ~< C( lu -u ' l + Iv-v ' l ) , , . ~

u ,u ' , v , v ' e [O , 1],

(H4) la4(x, u, v ) - a 4 ( x , u', v)l <~ C lu -u ' [ , u, u' e [0, 1], sup x ~ R e~[0.1]

sup x E ~

t , e [ 0 , 1 ]

sup xEff~

( u . v ) ~ T

1631a3(x, u, v ) - a la3(x, u', v)l ~< C l u - u ' h u, u '~ [0, 1],

IO~ a3(x, u, I))1 ~< C,

where 0ja3(x , u, 0), j = 1, 2, denote the partial derivative with respect to x and u of a3(x, u, v), respectively.

Fix z = (s, t) ~ T \E ,

(H5) a3(Xo, O,t)~O,

(H6) a3(X0, 0, v) = 0, for any v e (0, t],

f2 02a3(Xo, O,t)+Ola3(Xo, O,t) 70.,(O,w)a4(Xo, O,w) dwr

Our purpose here is to prove the following Theorem.

Theorem 2. Suppose that hypotheses (HI) , (H2) ' and (H3) are satisfied. Fix z=(s , t ) ~ T \ E and assume that either (H5) or (H4), (H6) hold. Then, the law of X~ is absolutely continuous with respect to Lebesgue's measure on R and its density is infinitely differentiable.

Remark 2. The conclusion of this theorem can also be obtained by exchanging the roles of t and s in assumptions (H4)-(H6) .

The proof of this theorem is carried out using Malliavin Calculus. Since X. is one-dimensional, 7_-( ) smooth and strictly positive on u = s or


on v = t (see the Appendix), the proof does not present the technical dif- ficulties of Refs. 17 and 18. Nevertheless, the method used here provides an alternative simplified approach to H6rmander's theorem, with restricted and unrestricted assumptions, for two-parameter processes in dimension one.

For the sake of completeness we briefly recall the main ingredients which are necessary in order to state Malliavin's result.

Let (s ~-, P) be the canonical space associated with the Wiener sheet W, IF a separable Hilbert space. An E-valued random vector F: s ~: is called smooth if

M

F = ~. f~(W_., ..... W~,,)v, (3.1) i = l

where f i e ~ ; ~ ( R " ) , zl ..... z , ,e T, VI,. . . , VM~.~-. Here ~g~(N") denotes the set of cg~ functions f : R " ~ N which are bounded, together with all their derivatives. The set of smooth random variables is denoted by 5 e. The Malliavin derivative o f F (given by Eq. (3.1)) is the L2(T, E)-valued random vector defined by

i= • k= I ~ ( W'H''''' W=.) 1 [0.~,](r/) v;

The kth derivative of F can be defined by iteration, D,~k ...,kF_- D,s~(...(D,~F)). For every p ~ [ 1, 0o) and any natural number N we denote by DN'P(E) the completion of 50 with respect to the norm

N

IlFIl~v.p = IIFII PCxa;E, + ~ E IlOkFll P-'~Tk: ~, k = l

Set D~-(n :) = Op.v DN'P(~-) �9 If IF = R we will write D ~ instead of D~ and DN'p(~:), respectively.

We can now state the main tool in our framework.

and D N'p

Theorem 3 (see e.g., Refs. 14 or 10). Let F: s ~ R be a random variable satisfying the following conditions:

(i) F ~ D ~,

(ii) (~rlD, ,Fl2 drl) - ' ~Dp>~ Lp.

Then, the law of F is absolutely continuous with respect to Lebesgue's measure on R and the density is continuously differentiable.

860,'9.,'4-5


We also recall a technical result suitable to check condition (i) of Theorem 3 for the Wiener functional Y= X~ (see e.g., Ref. I 1 ).

Lemma 1. Let {F,, n>/ I} be a sequence of random vectors in DI'P(E), p e [ 2 , oo), converging in LP(g2;E) to a random variable F. Assume that the sequence {DF,, n>_-1} is bounded in LP(I2;L2(T;E)), that is,

s u p E IlD, f ,,ll~dq < n T

Then, F e DI'P(E).

As a consequence we obtain Lemma 2.

Lemma 2. Let {F,,, n t> 1} be a sequence of random variables in D N'p, N~>I, p c [ 2 , ~ ) . Assume there exist F6D N-I'p such that { D N - IF. , rt >~ 1} converges to D N - IF in LP((2; L 2 ( T N - I ) ) a s n goes to infinity and, moreover, the sequence {DNF,, n>~ 1} is bounded in LP(Y2; L2(TN)). Then, Fe D N'p.

Indeed, set D~ by convention. For N = 1 the conclusion of Lemma 2 follows from Lemma 1 with ~:=R. For N > 1 set G.~.DN-JF., G = D r ' - tF. The sequence { G,,, n f> 1 } and the random vector G satisfy the assumptions of Lemma 1 with E=L2(TN-I), consequently, DN-IF6 [I~ I'P(L2( T N - i ) ) and hence, F e [I) N'p.

Using Lemma 2 we can now prove the following

Proposition 4. Assume (H1) and (H2)'. For every integer N>~ 1 and p c [ 2 , ~ ) , X. e D N'p.

Proof. Consider the Picard approximations

x'2-- x0

(3.2) f a dW, l+a4(X,, x ' : + ' = x o + :,=(q)[ 3(x".,q) " ~)d~], n ~>0

R:

We will now check, by induction on N, that the sequence {F,,-= X' , n i> 1 } satisfies the assumptions of Lemma 2.

Let N = 1. We have proved in Ref. 19 the convergence of {X_'_', n ~> 0} to 2"__ in LP(ff2), for any p e [2, oo ). We want to show the following facts

(i) fo reve ryn>~0 , X".eD t'p,

(ii) sup s u p E ( [ D a X n [ P ) < +o0. n ~,z~ T

The Law of a Nonlinear SPDE 8 7 7

Statement (i) is checked by induction on n, using the stochastic rules for Malliavin derivatives, the properties on the coefficients and Eq. (5.5). Moreover, the derivatives satisfy the equations

D~X ~ = 0

D=X,~+ t = ~,(~) a3(X-n, OC)

I, " + ),_.(r/) D~X,';[O,a3(X',;, q) dW~ +O,a4(X~, ~l) d~l] (3.3)

Set C o = s u p , sup:~rE(IX~lP). Then Eqs. (3.3) and (5.5) yield

E(ID~,X~I P) = E(ly_.(~) a3(Xo, ~)l ") ~< CCo

E(ID~XTI")<~ C {Co+ f, E(lD~X'~-' l ' )dtl}, n>~2 ~ . ,7 ]

Consequently, (ii) holds true. This shows that { X'__', n >~ 1 } satisfies the assumptions of Lemma 2 for N = 1 (and even a strengthening of them).

In order to deal with the derivatives of X~ and X__ of any order we borrow some notation from Ref. 2. Let ~=(0c~ . . . . . O~N) ETN; we denote by ["l the length of 0~, that means, N. Set 02~=(0c,,..., 0ci_~, ,~+~ ..... 0oN) , i = 1 ..... N. For a random variable YE D u'r, we denote by D ~ Y the mixed partial derivative D~,ND .... ,...D~, Y. Let f E cg~'~ x T), the space of continuous functions defined on R x T, infinitely differentiable with respect to the first variable with bounded derivatives. Set

A, f i . V f t ' " ) t X z) (P'IX F~(f, X_, z )= ~" ~. , , :, Dp, : m = 1 i = 1

(3.4)

where the second sum extends to all partitions p~ ..... p,,, of length m of and f"" ) denotes the inth derivative of f with respect to the first variable. In an analogous way F~(f, X':', z) can be defined.

Assume now that {F,,=X~, n>~0}, F = X ~ satisfy the assumptions of Lemma 2 for the integer N - 1 ( N > 1) and p ~ [2, 00, or, more precisely, the set of hypotheses (HN_ i)

~ X" D N- l.p (a) t ~ , n l > 0 } c

(b) DN-2X " , DN-2Xin LP(ff2, L2(TN-2)), t , ~ cas

(c) sup sup sup E([D~-IX"__IP)<+oo. n z e T I=I = N - I


Then, X__ e D N- I'p, by Lemma 2. We want to check (HN). The proof of (a) in (HN) is done using induction on n. Moreover, one

obtains

N 0 D~X: =0 N

D~,X'~ '+ = ~. F~,(a3, X,,," ~,) },:(~i) i=1

+I ),:(r/)[r~(a3, X,';,r/)dW.+r~(a4, X',;,r/)aV] (3.5) sup •. z]

where sup~:=0c) v . . . v 0c N, The convergence (b) of (HA,) can be easily checked taking into

account that DN-~X":, n i> 1, and DN-~X_ satisfy equations of the same type than Eq, (3.5). Set

A~,( f X":, z) = F~(f X~, z) - f'~V)(x'.', z) --DNX ' ' . --:

We can write, for n >/0,

N

--I-)N)("+'~ - - z = 2 / " ~ , ( a 3 , X~,, o~i) y _ ( a , ) + y:(r/)[A,(a 3, X,,, q) dW,, i = I sup :x. z]

+ A,(a4, X'~, 17) drl]

"3l-f ? Z ( q ) N n {N) n D, X~, t [a3 ( X , t , q ) dW, lwn(u)(Y'* " - - 4 " - - ' l ' q ) dr/] ( 3 . 6 ) sup ~t. z]

Notice that, in the first two terms of the right-hand side of Eq. (3.6) only derivatives of X'_' of order less or equal to N - 1 appear. Hence, condition (c) in (HN- , ) implies

sup sup supE(lA~(ai, X;i,r/)[e)~C, j = 3 , 4 " )~,l=N ,l (3.7)

sup sup sup E( [Fa,(a3, X~, 0~i)1 r) ~< C n.i I~ I=N q

Therefore, Eqs. (3.6) and (3.7) yield

N I p ) E(ID= X:I <~ C,

N,} E( _DNy"+'IP)<~ ____ 1+ E(ID=X,,I )dr/ sup ~, : ]

for all n>~ 1, z~ T and I~1 = N . This proves property (c) of the set of assumptions (HN) and completes the proof of the proposition. []

The Law of a Nonlinear S P D E 879

The Malliavin derivative of X_ satisfies the equation

D=X__ = 7=(0Q a3(X., 0c) + Ii ~ ] y~(q) D~X.[O,a3(X,, , rl) dWq

+Ola4(X.,rl)dq], O<~e<.z

Let { Y:(0t), 0 ~< e <~ z ~< ( 1, 1 )} be the solution to

Y:(e) - y:(oQ + Ic 7:(r/) Yn(ot)[O,a3(Xn, q) dWn+O,a4(Xq, q)drl ] (3.8)

It is easy to check that D,X~ = a3(X~, 0c) Y_(e), by uniqueness of solution. Fix e, fl, ~ e (0, 1 ), z = (s, t) ~ T\E and set D}(e) = (0, e/t) x (0, t), C}.,~(e) = (0, e I~) x (t--e '~, t).

The next estimates for p e [ 1, oo) will be used in the sequel:

sup E(IX~- Xol2p) <~ Ce/~p (3.9)

sup E(I Y_-(~) -7=(0QI 2p) ~< Ce ap (3.10) E Cit,,~(~:)

For proving Eq. (3.9) we use Burkholder's and H61der's inequality, the boundedness of y_.(r/), the linear growth of ai, i = 3, 4 and the LP-bounded - ness of the process {X_., z eT}. Obviously, SUpo~__E(lY_-(a)12P)<~C. Consequently Eq. (3.10) can be easily obtained from Eq. (3.8).

The next result provides the first step for the proof of Theorem 2.

Proposition 5. Assume (H1), (H2)', (H3). Fix z = ( s , t)e T\E and suppose (HS). Then, (IR: ID~X=I'- do~) -1 EL e, for any p o l l , oo).

Proof It suffices to show

P { ~ la3(X=,e) Y:(e)lZ de<~e} ~ : , p e [ 1 , oo) (3.11)

for any e---<eo, where eo depends on p, z and the coefficients of Eq. (1.1). Suppose e e (0, 1) and choose f le (�89 �89 Then,

P {~R: la3(X~" e) Y:(~ d~ <~ et <~ P'(e" fl) + p2(e" fl) (3.12)

880

wi th

Rovira and Sanz-Sol~

If':J' f ' p,(e, fl) =P ~.o ",-~p a3(X~''''' " w) Y](r, w) dr dw<~e,

f, ' dr dw>4e}

p2(e, fl) = P a3(X o, O, t) 7~(0, t) dr dw <<. 4e I i : f l

We want to show, for any q e [ 1, co),

sup E([a3(X,.,,.,r,w) Y=(r,w)-a3(Xo, O,t) y=(O,t)[Z'l)<~Ce ~q (,..,,.) ~ (o.~/b • , - ~/'.,) (3.13)

Indeed, the assumptions (H2)' and (H3) on a3 yield

]a3(Xr.,,., r, w) Y:(r, Iv) -a3(Xo, O, t) y:(0, t)l

~< C{(1 + IX,,..I)lY=(r. i v ) - 9,:(0, t)[

+ 19':(0, t)l r Ix,....-x,,I + I,1 + Iw- t l ] }

thus, Eq. (3.13) follows from the estimates of Eqs. (3.9) and (3.10) with fl = 5 that have been proved before.

Then, by Chebychev's inequality,

{(, p](t, f l ) ~ P J,_,:l[a3(X,..,,.,r,w) Y._(r,w)

O, t)y:(O, t)]Z dr dn, > t } a3(Xo,

<<eq~21~-t~ sup E(la3(X,..,,., r, w) Y_.(r, w) ( r . u ' ) e (0,,*:/t) x ( t - - e l l . t )

-a3(Xo, O, t) 7=(0, t)l 2q)

Since fl > �89 there exists e ) > 0 such that p ](e, fl)~< e p, for any e ~< e~. Set K : = la3(Xo, O, t)y.(0, t)l. Following (H5) and Eq. (5.11), K > 0 .

Consequently, since fl < �89 there exists e2 > 0 such that p_,(e, f l )= 0 for any e ~< e_~. Thus, the inequality (3.12) and the results proved so far for pj(e, fl), p2(e, fl) yield Eq. (3.11). []


The second step in the proof of Theorem 2 is given by the following

Proposition 6. Assume (HI), (H2)', (H3). Fix z ~ T\E and suppose that (H4) and (H6) hold. Then, (J R: [D ~ X._[ 2 do~) - t ~ D', for any p ~ [ 1, ~ ).

Before proving this proposition we want to check some auxiliary results under the set of assumptions of this statement. For a fixed z = (s, t) ~ T \E set

Z~ = Xo + f&,~ n: 7~ v)[a3(X"' tl) dW. + a4(X., q) dtl], ~ ~ T, t 1 = (u, v)

Notice that {J&~R: )'0.,( O, v) a3(X,1, q) dW,~, r <<. z} is a two-parameter martingale with respect to the filtration generated by the Brownian sheet. Moreover, supr ~ ~ E(IZ~[ r) ~< C. Fix fl, e ~ (0, 1); we want to check

sup E(lZ~-Xol2r+lXc-Xol2")<~Ce 2/~', p c [ l , oo) (3.14)

Indeed, (H6), the Lipschitz properties of a 3 and Eq.(3.9) imply, for

{( (" E([Z~-XoI'-P)<~C E f&),o,,(O, v)(a3(X,,,q)-a3(Xo, vl))dWe

+[~&)',,.,(O,v)(a3(Xo, rl)--a3(Xo, O,v))dW,,I 2v

+ ~&70.,(O,v) a4(X,,q)drlZP)}

Analogously, E(]Xr "-p) <~ Ce 2/~p. Consequently, Eq. (3.14) is proved. Furthermore, Eq. (3.14) yields

sup E(IXc-Z--cl2")<~Ce 2'', p c [ I , ~ ) (3.15)

882 Rovira and Sanz-Sol6

The following estimate will also be needed,

sup E(IA'r 'p) ~< C{e3aP+e ''-a+me} (3.16) ( ~ C~.,~(e)

Indeed, Burkholder's and H6lder's inequality yield

E([X~_ Z~12p) <~ C {E (ifr (~,~(q)_ yo.,(O, v)) a3(X,,, ~l) dW, ' 2p)

+ E( fR (~)~(rl)--Yo.t(O, v))a4(X,,,~l)drl 2P)}

~<C{}Rr lT~(q)-,o.,(O,v)12PE(l+lX,,lr)d~l} (3.17) r

Properties in Eqs. (5.6) and (5.7) of 7:(") ensure I),c(r/)- 7o.,(0, v)l ~ [(1[ + [~'2 - t[ + [u[, where ( = (('l, ~2). Consequently, for ~'e C~/~.6(e),

[yr v)l ~< C(e'~ + e '~)

and the last term of Eq. (3.17) is bounded by C(t3/~P+t ~26+p~p) showing Eq. (3.16).

Proof of Proposition 6. We have to check Eq.(3.11). For e, fl, 6 e (0, 1 ), we have

: ) ["C~,,s{~) <~ q~(t, fl, c~) + q2(t, fl, c~) (3.18)

with

= (u, v).


Next we show

sup E(]a3(X,1, rl) Yz(r/)--a3(Z~, u, t) 1%(u, t)[2q)~< C(e'~q+e3/J'O (3.19) ~1 E C"#.,~c)

Indeed, Eqs. (3.10), (5.5), (5.6), (3.16), and (H3) yield, for r/~ C~.,~(e)

E(Ia3(X,, rl) Y_-(rl)-as(Z~, u, t) ?:(u, t)[ 2q)

~< CE{(1 + IX.I z',) I Y=(r/)- ),=(r/)[ Zq

+ (1 + IX,1I-"0 I~'_-(rl)- 9'=(u, 012"+ IrAu, 012`' IX,~-Z~,l 2"

+ [~,..(u, t)[ 2q [a3(Z~, r / ) - a3(Z~, u, t)[ 2q}

C { B 6q .-~ e 2'~q + ~3flq -I"- e ( 2,~ +/J)q ...[_ e2,hl}

C(e '~q + e 3/~q)

Consequently, by Chebychev's inequality

q,(e, fl, fi) <~ P ,-t ff~,,~a.. . [a3(X,, , r /)Y=(rl)--as(Z:~, u, t)?:(u, t)]2 dq > e t

C(e,:12~ +/~- i~ + eq~s+41~- J)) (3.20)

We now apply the It6 formula to the process { Z~.,., u ~ [0, 1 ] },

a3(Z= u .... u, t )=a3 (Z o .... u, t) + f 63tas(Z ~ .... u, t) yo.t(O, w) Ru, i.

x {as()(,.,,., r, w) dW~.,,.+a4(X,. .... r, w) drdw}

+ �89 f 8~a3(Z ~ .... u, t) yo.,(0, w) a3(X,..,., r, W) dr dw I{u, r

A Taylor expansion yields a3(X o, u, t) = O2a3(Xo, 0, t)u + a2a3(Xo, r, t)u2/2, for some i:e (0, u), because a3(X o, 0, t ) = 0 . Thus,

a3(Z~,.~,,u,t)--u O2a3(Xo, O,t)+Ola3(Xo, O,t) ?o.,(O,w) a4(Xo, O,w)dw

4

= ~ Mj(u, v) (3.21) j= l

884

w h e r e ,


MI(u, v) =fR a'a3(ZL" u, t) yo.,(0, w) a3(X, ..... r, w) dW,.,. u r

M2(u, v )=fR yo.,(O, w)[Ola3(ZZ,. .... u, t)a4(X,.,,,., r, w) u r

-- O, a3(X'o, O, t) a4(Xo, O, w)] dr dw

1 f a~a3(Z~ .... u, t)),2,,(O, w) a~(X~ ..... r, w) drdw M3(U, v ) = ~ R,,,.

u 2

M4(u, v) = a_~a3(Xo, ~, t) T

For q = (u, v) ~ C~s.a(e),

4 E( IMj(rl)l 2q) ~ Ce 3/',,

j = l q ~ [ 1 , oo) (3.22)

Indeed, since a3(Xo, 0, w) = 0, for any w ~< t, the Lipschitz properties of a3 and Eq. (3.14) yield

E([M'(rl)12q)=E( fn 0ta3(Z: .... u't) Y~ .... r,w) u r

- a3 (Xo , 0, w)] dW,..,,. 2,1)

<~ CeP"I- ~ I_ (E( IX,.,,.- Xo[ 2q) + [r[ 2q) dr dw ~ Ct 3/sq u l ,

Moreover, the Lipschitz condit ions on 01 a3, a4 and Eq. (3.14) imply

E( I M2(r/)l 2,,)

<~ C {E( fR,,,.,o.,(O, w)a4(Xr. .... r, w)EOta3(Zr.v,u,, )

--Ola3(X o, O, t)] dr dw 12q

12%) -+- f&,,.)'o.t(O,W)Ola3(Xo, O,t)(a4(X r ..... r ,w)-a4(Xo, O,w))drdw ) I

Ce. 4ls"

The Law of a Nonlinear SPDE

Analogously,

E(IM3(q)I 2q) ~< Ce 21~q sup (r , It') E C~. j (~.)

~< Cc6pq

Finally E(IM4(q)I 2") <~ Ce 41~" and Eq. (3.22) identity Eq. (3.21) yields

q2(~, fl, 6) <<. qzt(e, fl, 6) + q2z(E, fl, 6)

with

qz,(e, fl, ~ ) = P y~(u, t) Mj(~l) >4e P . , ~ ( ~ ) - . i I

(E( lX~ . , , . -Xo l%+lr l%

885

is completely proved. The

(3.23)

q22(e ' f l '6)=Pffc:p~, , ) '~ ' (u ' t )ua[O~a3(X~176 : - _

1" } X a 4 ( X o , O, w) dw du dv < 16e

By Chebychev's inequality, Eqs. (5.5) and (3.22)

qRl( e, B, 6) <~ Ce 1~ + 4/~- i iq (3.24)

Set

q2z~(e, fl, fi)= P {fci~,,m: 7~(u, t) u2 I ala3(Xo, O, t)

]2 } x fL. ~o.,(0, w) a4(Xo, 0, w) dw du dv > 16e

q222(e, fl, 6 ) = P { ; c )'~(u,t)u2IO2a3(Xo, O,t)+O,aa(Xo, O,t)

]2 ) x fo yo.,(0, w) a4(Xo, O, w) dw du dv < 64e

Then,

q_~21(e, fl, 6) <<. Ce c3~ + 3p- I~q (3.25)


Furthermore, let

f 4 K= O,a3(Xo, O,t)+Ola3(Xo, O,t) ~o.,(O,w)a4(Xo, O,w) dw

Since

),:(u, t)=exp (fj at(r, t) dr)

(see Eq. (5.14)) and

) inf exp a)(r,t)dr > C > O O~<u~<s t

clearly

c )'~-(u't)u2[ 82a3(X~176 ,( .

• fJo ))~ W) a4(Xo, O, Iv) d w du dv

K 2

Consequently,

q222(e, fl, O)<~P{~<~64e'-'3/J+'~' t (3.26)

In conclusion, let ~, fl ~ (0, 1 ) satisfying 26 + f l - 1 > 0, 6 + 4fl - 1 > 0 and l - 3 f l - - ~ > 0 (for instance, d=�89 f l= l ) . Then, by Eq.(3.18), (3.20), (3.23)-(3.26) we check the existence of eo ~ (0, 1) such that Eq. (3.11) holds for any e ~ (0, eo). This ends the proof. []

Proof of Theorem 2. We apply Theorem 3 to the random variable F = X~. The results established in Propositions 4-6 show the validity of the assumptions of that theorem. []

Remark 3. If the coefficients a,, i = 3, 4, do not depend on z~ T, Theorem 2 can be stated under the following set of hypotheses: (H1), (H2)' and either aa(X0) ~ 0 or a3(Xo) = 0 and a'3(Xo) ag(Xo) v~ 0.


4. LARGE DEVIATIONS

The purpose of this section is to study small perturbations of Eq. ( 1.1 ). Let x0eR, we consider the family {X':, t > 0 } of processes indexed by T which satisfy the stochastic equation

t ' X~ = xo + J ?=(t/)[~a3(X~) dW, + a4(X~) drl] (4.1)

Rx

We seek to establish a large deviation principle (LDP) for { X ~', e > 0} on the space ~,.,,(T) of real continuous functions defined on T and taking the value x0 on the axes. That means, we will prove the existence of a lower semi-continuous function I: cg~0(T)~ [0, oo], called rate fimetion, such that {I<~a} is compact, for any aE[0 , c~), and

t 2 log P{ X': e O} > / - A(O), for each open set O

e 2 log P{ X ~" e F} <. --A(F), for each closed set F

where, for a given subset A ccg,.,,(T), A(A)=inff~t, I(f). The proof will follow the method set up by Azencott cl~ (see also Refs.

6 and 15). Let (Ei, di), i = 1, 2 be two Polish spaces and X~: g2 ~ E i , e > 0 , i = 1, 2, families of random variables. Assume that { X], e > 0} satisfies a LDP with rate function 7: El--* [0, oo]. Let F: {7< + m } -~E2 be a mapping such that its restriction to the compact sets { 7 ~< a}, a e [ 0, oo ), is continuous in the topology of EL. For any geE2 we set I(g)=inf{I(f): F( f )=g} . Suppose that for any R,p,a>O there exists 0t>0 and Co>0 such that for f e E t satisfying I ( f )~<a and e~e0 we have

(") P{dz(X~, F(f))>~p, d,(X~,f)<0t} ~exp - ~ (4.2)

then, the family {X[, e > 0} satisfies a LDP with rate function L Our aim is to apply this result to the random variables X~I = eW and

X ~ = X ~ given in Eq. (4.1). In the proof of inequality (4.2), which is the fundamental part of this section, we need some exponential inequalities for some type of stochastic integrals which are not martingales. The same problem appears in Ref. 20 in the context of parabolic spde's.

Proposition 7. Let g:(t/) be a real function defined on T 2 with go(" )=0 . Assume there exists a positive constant K~ such that for all z, z' ~ T,

~r(g~( tl) -- g~'(~l))2 dr 1 <~ Kgd(z, z')


where d denotes the Euclidean distance on T. Let a = {a(z), z~ T} be an ~:-adapted stochastic process. There exist positive constants K~ and K2 such that

Z 2

P {sup [frg:(q)a(q, o9)dW,,[>>. L} ~<exp ( K ~ . K j K z ) (4.3)

for any L~>0 with L/K.~/2K.>~KI, where K.=sup:~r, .... a [a(z, o9)1.

Proof Set

I: = frg:O1) a(q, o9) dW, l

Assume we can prove the existence of positive constants C~ and C2 and a random variable B such that

E(B) <<. ~ (4.4)

sup II_ I ~< C~K~/ZK,,[(ln+ B)~/2+ C2] (4.5) z ~ T

where In + z = max{ In z, 0}. Then, the estimate E(exp(ln + B)) < I + , , / ' 2 - which follows immediately from inequalities (4.4) and (4.5) and Chebychev's exponential inequality yield

{ ( )) L _ C2 - P{sup_.~r II:l >~L} < P (ln+ B)>>. C,K~./2K ~

f ( '~ )2) ~<E[exp(ln+ B)] exp - CtK~/2K~ C2

.~x~f ( ~ ~2) ~ ,n,~ K.~,2K~, + + V/2)}

= exp KeK~ - L C2

whenever L/K~/'-K~ >1 Ct C2.


Set K2:=1/8C~, K,=max{2C, C2, ~ / - 8 ( l n ( l + x / ~ ) ) ' a C , } , and assume L/K.~/2K~>~K~. The last term in Eq. (4.6) is equal to

L 2 KI/ZKa C2 CI - L2 C 2 exp K,K~ 8K2 1 - ' L

and is bounded by

L 2 L 2 K2 }

Consequently, Eq. (4.3) holds. We will now proceed to the proof of inequalities (4.4) and (4.5). This

is done in a similar way by Sowers. t2~ First, notice

= [ (g:(q) _g,(q))2 E(a2(rl, co)) dr l E(II:-I:,] 2) d T

<. GK]d(z, z')

Set p(y) =yl/221/4K~,/2K~, y >7 O, ~b(x) = exp(x2/4), x e R. Let

( / , ) We will check that B satisfies Eq. (4.4). Set

gl:.:,~(q) = (g__(r/) -- g:,(q)) p(d(z, z')/x/~ )

We consider the continuous ~,.t-martingale defined by

~I r = IRr, g~:.:,~(tl) dW,,

with associated quadratic variation

f " ) Rrt

890 Rovira and Sanz-Sol6

There exists a Brownian motion Z such that )14,. = Z<~-r Consequently,

I . - I . , E[~k(p(d~z,z--7-~x/~i)]=E[exp(-~)]

~< E [ exp {~ (o sup 'Z,.[2)} ] =x/ '2

and therefore E(B) < + oo. The Garsia-Rodemich-Rumsey

z, z' e T,

and since I0 = 0,

lemma yields, for any co-a.s, and

, , ~k - l dp( y ) " 0

sup II:l ~<sup ~b -l dp(y) z ~ T z ~ T aO

~<29/2 I f ? ( ( l n + B)l/2+(ln+ y-4)l/2)dp(y) (4.7)

Some easy calculations show ~o J2 (In+ y-4) , /2y - t /2dy=2x/~ . Hence, inequality (4.5) follows from inequality (4.7). In fact,

sup__~ T l/z[ ~< 29/2 [ (ln+ B) I/2 p(x//2)+ KI/ZK~2t/4~. - -o ~'/5 (In + y-4)l/Zy-l/2 dy]

<~ 25KIg/2K~,[ (In + B),/z + 2 -i/4 %/ /~]

= Cl KI./2K~[ (ln + B) 1,2 + C2 ]

with Ci = 25 and C2 = 2 - i / 4 N / ~ . [ ]

We also need a Gronwall type lemma, as follows.

L e m m a 3. Let h: T ~ R be a nonnegative continuous function satisfying

h(z)<<.K+fR fl(q)h(q)dq, z ~ T


where K 1> 0 and fl: T ~ ~ is an integrable, nonnegative function. Then, for any z e T,

h(z) <~K exp { In..fl(rl) drl }

The proof is the same as that of Lemma 4.13 in Ref. 13.

Consider the set of assumptions

(HI ) a;, i = 1, 2, are differentiable and bounded, with bounded first partial derivatives.

(H2)" a~: R ~ R, i = 3, 4 are bounded Lipschitz functions.

Denote by '~.,.,, the set of absolutely continuous functions f e cg.,.,,(T) with ~r If(z)l 2dz < + oo, where J'(z) denotes the derivative 02f(s, t)/Os Ot. We recall that {eW, e > 0 } satisfies a L D P on %(T) with rate function I ( f ) = �89 z Ilfll.,%, if f e ~ o and l ( f ) = +0% otherwise. For any f e ~ o we consider the function { S(f)(z), z ~ T} satisfying

S(f)(z)=Xo+ fR ?__(rl)[a4(S(f)(rl))+a3(S(f)(q))j'(~l) ] dq (4.8)

Let f, g e :~o, I[fll.,r0 + IIgll.n0 ~< a. By Lemma 3 and (H2)", IIS(f) - S(g)l[ C I1~11 ..... where ~(z) = IR-~:(r/) a3(S(g)(rl))(f(q)- g(q)) d~l. Assume first

that a 3 is a ~ - bounded function with bounded derivatives. Integrating by parts and using the boundedness properties of ? one can easily prove

[Ictl[ ~ ~ C I I f - gll ~. (4.9)

with a constant C depending on Ila311~+ Ila~ll~_. + ILa~l[~. In the general case, the estimate (4.9) still holds true, as can easily be checked by smoothing a 3 by convolution and passing to the limit. Thus f~-~, S(f) defines a continuous mapping from { [If[I .g0 ~< a} into cg~o(T) with respect to the topology of the uniform convergence. Our purpose is to prove the following result.

Theorem 4. Suppose (H1), (H2)". The family of processes {X ~, e >/0} solution of Eq. (4.1) satisfies a large deviations principle on ~ o ( T ) with rate function

I (g) = i n f (�89 ~7. 'J'(z)]2dz' g=S(f), fe~.,.o}

where S(f) is defined in Eq. (4.8).

,~60 9 4-6


This result is obtained by transferring some Ventzell-Freidlin type estimates stated in Theorem 5.

Theorem 5. Assume (HI), (H2)". For all f e ~,o, R, p > 0, e ~ [0, 1 ], there exists ~ > 0 such that

P{IIX~-S(f)[I,.>p, [leW-fl[~<o~} ~<exp - ~ (4.10)

Let { ~ , z e T} be the process defined by

Y:=xo+~R e)':(rl)a3(Yi'~)dW'~ + fR )':(rl)(a4(Y~i)+a3(Yi;) f(q))drl (4.11)

The proof of Theorem 5 reduces to establish the following result.

Theorem 6. For all f E ~ , R, p > 0, there exists ~ > 0 such that, for any ee [0 , 1]

P{llY~'-S(f)ll~>p, I l e W l l . ~ < ~ } ~ < e x p ( - R ) (4.12)

Indeed, let WS= W_-(I/e)f(z) . Girsanov's theorem ensures that { W_~, z c T} is a Wiener process with respect to the probability P~ given by deTaP = exp{ 1/~ IT/(~) aW__- 1/2~ -~ IT If(z)l-' ,t.-_}. Set, for any p, ~, e > 0,

B':= { llX':- S(f)]l~_ > p, IleW- fll~_ <~}

Then,

w h e r e a = Ilffl ~,,, a n d 2 ~ R.

(4.13)


Consider the continuous strong martingale ~n:j~dW,~, z~T. The exponential inequality established in Ref. 5, Proposition 5, yields the existence of constants k l , J~2 such that

P{Ifvf(z'dW: >~)<~Rlexp( R22_~a)

for any 2 > 0. Choose 2 z >f aF2z(R + In R~), then

P{ ~r,(z) dW: > ~ } ~ < e x p ( - eR---3) (4.14)

Furthermore, in terms of co~=og-(I/e)f, the process X~(o)) can be expressed as follows.

X'~( ~ ) = X: ( o)': + ~ f ) = xo + I R e~'~( ~l ) a.~ ( X:, ( o/" + ~ f ) ) dW;

Set Y~'(co':) = )("(co':+ (l/e) f) . Then, P':(B':) = P{ U y r . _ S ( f ) l [ ~ > p, UeWit~- <0c}, with Y'- satisfying Eq. (4.11).

Consequently, the estimates given in (4.13), (4.14), and (4.12) complete the proof of Theorem 5. The proof of Theorem 6 relies on some lemma and propositions.

Lemma 4. Fix f e ~ , , Ilfll ~,,, ~< a. There exists a nonnegative constant K depending only on the coefficients and a such that o)-a.s.

,, Y':- S(f)I,~ <~ K ~R e,:(rl) a3( Y~) ~ (4.15)

Proof The identities Eqs. (4.11) and (4.8) yield

~R ~ dW'l [Y'j_-S(f)(z)[<<. .e~,:(q)a3(Y,,) z r,,c~

+ C f IYii-g(f)(rl)l [1 + f ( q ) l drl ~ , R z

Therefore, Lemma 3 yields the estimate (4.15) with K = exp( C( 1 + a)). []

8 9 4 Rovira and S a n z - S o l ~

We now introduce a discretization of Y". To this end we first state some notation. For n e [M, k, j = 0, 1,..., n - 1, set s~ = k/n, t"j = j/n," zk/" =

. . . . . . . . . )x[ t j : , t j+~) . Let YS'"=Y~_, if (s, t)~zl],.j. (s k, t~) and/ Ik , j - [s k, s k + t _ .~.j

Proposition 8. For all R > 0 a n d / t > 0 there exists no ~ [~, depending on R a n d / t such that for all n >t no and e s (0, 1 ]

Proof

with

P{ [I Y " - Y""ll,_ > lZ} <<. exp ( - R )

Fix n and set F " = { II Y * - Y~'"II ~ >/~ }, then

n - - 1 I : , l l 17,,11 I: I I

F ~'c U ( F I . k i U F2.1~ i U F 3 : k i ) k,j=O

(4.16)

~ - z e d o

F~:~:/: f s u p [fR e(7--(tl'-)'=Zj(tl')as( Y:, dW,, > 3 }

F 3 j , : / s u p (1R (1/) 9,_.(t/) -- 1R4b(t/) ~,..~v(tl))(a4(Y,,) + as( ~" " L z e Ankj T

We want to give an exponential bound for the probabilities P(Fi'fj), i = 1, 2, 3 for fixed k,j. Let z=( s , t), set z /n=(s/n , t/n) and ~ ( r / ) = 1 R=b+:/,,\e=lj(r/) ?:~j+ :/,,(r/). Then

sup fR ey:(t/) as(Y:~) dW,, = s u p I x]f eg:(r/, as(Y,';) dW,, : r A~.j : -- R:~:/ : ~ T T

It is easy to check that ~ = 0. Moreover, by gq. (5.7)

fr(g~(tl) -g~,(tl)) 2 dr l <<. C 1 d(z, z') n

Then Proposi t ion 7 ensures the existence of positive constants Kt and K2 such that .... 2 3 2 P(FI.~j ) <~ exp(( - p n/9C e ) K2), whenever pni/2/3C3/2e >f g t. Analogously,

sup e(?:(t/) - - a3(Y~) dW,, = sup eg~(r/) a3(Y,~) dW.


with r = 1 R.Zj(r/)(y_-~j + ~/,,(r/) -- y.-~V(~/)). Then g~ = 0 and for all z, z' ~ T,

fT (g2( /~)- -g2"( /~))2 d/~ <~ C 1 d(z, Z')

Hence, by Proposition 7 there exist positive constants K~ and K2 such that P(F~I~i) <~ exp((-#2n2/9C3e2) K2), if bm/3C3/2e >1 K~.

Schwarz's inequality yields

:~'~sup [fr (1 R:(q) Y--(r/) -- 1R:'~J(~)Yr"~(rl))(a4(Y~)+a3( Y'~) f(rl)) ~ " drl <~x/"C

Consequently, for n>9C2//~ 2 the set F~2j is empty and therefore P(F3.kj) = 0.

Finally,

- - p 2 n 2 .~

\(exp \ 9-~-~e 2( -# 'n K2)+ exp ( 9--~e2 K- ) )} log e{F-} log l .

~< e2(log n2 + log 2) - ~< --R

for n big enough and e ~ [ O, I ]. This completes the proof of the proposition. []

Proposition 9. For all R > 0, p > 0, n s t~, there exist /~o > 0 (not depending on n) and ~o>0 (depending on n) such that, for all /~<Po, 0c~<0% and e~(0, 1],

P {fR: e~'=(q)a3(Y"i)dW" ~.. >p, [[eWIl,~ <~, IIY ~'- Y""I[~ ~</.t}

~ < e x p ( - R ) (4.17)

Proof We can write

+ P e~,=(r/) a3( Y . ) d W . > -~, R.-

896

Notice


gyz(r]) a3( Y. ) d W . = ea3( Y~-r ~,_(q) : J k 0 :r

j , k = 0 : "

We prove in the appendix that 7'_-() has bounded first order derivatives and mixed second order derivatives. Thus,

fs '2 ] `2 ),~(q) dW,, = W.,.,. ,, ?:(s2, t2) - W,.,. ,: y_(s,, t2) - W,.:. ,, ?_-(s2, t,) ?

I " t l

+ W,., .,,),:(s~, t~)

"" OY: (u, t2) du + ,- W,, ,, (u, tl) du - W,,.,2 T d

- W,.,_.,, "(s2, v) dv + W.,.,. '-~v ( s l , v) dv

+ IV,, ,, 02y: (u, v) du dv �9 " Ou Ov

Indeed, this formula can be checked by approximating the stochastic integral by Riemann sums and passing to the limit.

Then, for all e~(0, 1], cr and n such that a < p / 1 8 C 2 n 2 we have

P ey~(q)a3(Yi, ) >~,lleWll~<0~ =0 = , j

Consider g=(r/)= 1R:(r/) y__(r/). Proposition 7 yields the existence of constants K~ and K2 such that

P e~ ' - (r l ) (as(Y;) -a3(Y~'") )dW" > 3 ' 1[ ~ < P z

- - p -

if f l < p / 2 K i C 3/2. Hence the estimate (4.17) holds for o~<~p/18C2n 2 and I.t 2 <~ p2K2/4C3R. []


Using the auxiliary results established so far it is now possible to prove Theorem 6.

Proof of Theorem 6. Set Z ~ = { II Y~ - S(f)ll ~ > p, IleWll ~ < ~}. The inequality (4.15) yields for all n ~ N,/~ > 0, A~c U/2= I AT, with

a~ = { 1 1 Y ' : - Y':'"H ~ </~}

A ~ = { I t''(rl) " ~> PK' I,~WIl~<0~, , lY~- Y='"llo~>~}

The estimates provided in Propositions 8 and 9 complete the proof of the Theorem and, consequently the large deviation principle for {X% t > 0} is established. I-1

5. A P P E N D I X

In this section we give some results on the Green function 7__(r/) that have been used throughout the paper. Some of them have been proved in our previous work, ~19) in this case we will only write here the statements without their proofs.

Consider the partial differential equation

32f(s, t) Of(s, t) 3f(s, t) asOt at(s,t) O-m-f - a2(s,t) O------s~-b(s,t) (5.1)

(s, t) ~ T, f(s, t) = Xo, Xo ~ ~, if s- t = 0, where the coefficients at , a2, b are smooth. Fix (s, t) s T and let ~,,.,(u, v) be the function defined on {(u, v): (0, 0) <~ (u, v) ~< (s, t)} satisfying the properties (P) described at the begin- ning of Section 2.

Classical results on pde's ensure that f(s, t) = Xo + j ns., 7.,..,(u, v) b(u, v) du dv is the unique solution of Eq. (5.1).

The existence of ?.,.,(u, v) satisfying (P) can be established under (HI) using an iterative scheme, as follows. Set

Ho(s, t; u, v) = 1

H,,+l(s,t;u,v)= al(r,v) H,,(s , t;r ,v)dr+ a,(u,w) H,,(s,t;u,w)dw (5.2)

n >/0. Define

~s.,(u, v) = ~ H,,(s, t; u, v), (u, v) <<. (s, t) (5.3)

8 9 8 Rovira and S a n z - S o l ~

Proposition I0. (a) The series in Eq. (5.3) defining the function 7, is absolutely con-

vergent and satisfies

) , , . , ( u , v ) = l + at(r,v)),.,..,(r,v)dr+ a,_(u,w) 7.~.,(u,w)dw (5.4) t

(O,O)~(u,v)<~(s,t).

(b) For any (s, t )eT , ),,.,(.) has uniformly first order bounded derivatives and second order mixed partial derivatives on { (u, v): 0 < u < s, 0 < v < t } . By continuity we can extend these derivatives to {(u,v): O<u<<.s, 0<v~<t} .

(c) For any (u, v )e T, the function (s, t)~--*y.,..,(u, v) has uniformly bounded first order derivatives and second order mixed derivatives with respect to s, t on { (s, t) s T: 0 ~< u < s < 1, 0 ~< v < t < 1 }. By continuity we can extend these derivatives to { (s, t) e T: 0 ~ u ~ s < 1, 0 ~< v ~ t < 1 }.

Notice that the integral equation Eq. (5.4) yields properties (P). We refer the reader to Ref. 19, Propositions 3.1 and 3.2, for the proof of these results.

Proposition 11. There exists a universal constant C > 0 such that

sup sup 1),.,. ,(u, v)l ~< C (.v.l)~ T (u.c)~<(. , . / )

sup I)'.,. ,(u, v)-- ~,.,..,(ti, 6)1 ~< C([u-al + I v - ~1), Is, t) �9 T

sup b'.,..,(u, v ) - L~.i(u,v)l ~<C(Is-Yl + [ t - i[), (It, r ) e T

(5.5)

(u, v), (if, 6) ~< (s, t) (5.6)

(s, t), (L i)1> (u, v) (5.7)

Proof. Property (5.5) follows from

1, 1 ) IH,/s, t; u, v)l ~< C" max

proved inductively on n (see Eq. (3.4) in Ref. 19). Equations (5.6) and (5.7) are obvious consequences of parts (b) and (c) in Proposition I0, respectively. []

Proposition 12. Let ?.,..,(u, v) be the function defined by the series Eq. (5.3). The following properties are satisfied.

Tile Law of a Nonlinear SPDE 899

al(s, t) 7.,..,(u, v) 07.,.,(u, v) Os when t = v (5.8)

a~(s, t) 7.,. ,(u, v) 07.~,,(u, v) - = Ot when s = u (5.9)

02y.,. ,(u, v) O~,.,. ,(u, v) Or.,..,(u, v) Os at at(s, t) at a2(s, t) as = 0 (5.10)

7.,. ,(s, v) > 0, ~,,..,(u,t)>O, O<<.v<~t, O<~u<<.s (5.11

Proof We first check Eq. (5.8). The first step consists in proving

al(s, t) H,,(s, t; u, v ) - OH,,+ i(s, t; u, v)

OS t=v , n > ~ O (5.12)

by induction on n. This property is obvious for n = 0. Equation (5.2) yields

O H , , + l ( s , t; u, v) ,=,, - a l ( s ' t) H, , ( s , t; u, v)[,= 0S t"

=al(s ,v) H,,(s,t;s,v)l,=,, + al(r,v) OH''(s't;r'v) , ~s dr

I " dr - a l ( s , t ) al(r,v) H, ,_ j (s , t ; r ,v ) i I ~ l ,

(5.13)

Assume that Eq. (5.12) has been proved for any j ~< n - 1. Then, the right- hand side of Eq. (5.13) is equal to 0 and therefore Eq. (5.12) holds for any n~>l.

Property in Eq. (5.12) implies

at(s,v) ~ H, , ( s ,v ;u ,v )= ~ OH"+l(s 'v;u 'v) Os

thus Eq. (5.8) holds true. The proof of Eq. (5.9) is analogous. The identity in Eq. (5.10) can be

checked establishing first

OZH,,+ i(s, t; u, v) OH,,(s, t; u, v) OH,,(s, t; u, v) OsOt - a l ( s , t) Ot az(s, t) Os = 0


n>~0. This follows easily f rom Eqs. (5.2) and (5.12) and its ana logue

a~(s, t) H,,(s, t; u, v) - OH,,+ x(s, t; u, v)

Ot s = u , n>>.O

using, as before, induction on n. We finally show Eq. (5.11 ). Solving the differential equation satisfied

by y.,..,(u, v) (see (P)),

(0) , , ,(u, v)

IL. ,(u, v) : 1,

we obtain

a2(u, 0) ys.,(u, ~3), tt = s

t / = s ,

(S' ) 7.,.,(s, v ) = e x p , a2(s, w) dw

and, consequent ly y.,..,(s, v) > 0, 0 ~< v ~< t. Analogously ,

y.,..,(u, t ) = e x p a~(r, t) & t

and therefore y.,..,(u, t) > 0, 0 ~< u ~< s.

v = t

(5.14)

[]

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the law of the solution to a nonlinear hyperbolicspde

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