Approximation and support theorem for a wave equation in two space dimensions

We prove a characterization of the support of the law of the solution for a stochastic wave equation with two-dimensional space variable, driven by a noise white in time and correlated in space. The result is a consequence of an approximation theorem, in the convergence of probability, for equations obtained by smoothing the random noise. For some particular classes of coef®cients, approximation in the L -norm for p > 1 is also proved.

Let E denote the inner product space of measurable functions j : R 2 3 R such that d yjj(x)j f (jx À yj)jj( y)j , I, endowed with the scalar product and let H denote the completion of E .Set H L 2 ([0, T ], H); notice that H and H need not be spaces of functions.For h P H, let fÖ h (t, x), (t, x) P [0, I) 3 R 2 g be the solution of We prove in Section 2 that the support of the law of fu(t, x), (t, x) P [0, T ] 3 Kg is the closure with respect to the norm k X k ã, K of the set of functions fÖ h , h P Hg, where fÖ h (t, x), (t, x) P [0, T ] 3 Kg is the solution of (1.4).The proof is based on an approximation result for equations more general than (1.3) constructed by smoothing the random noise F(t, x).We refer the reader to Millet and Sanz-Sole Â (1994a;1994b) and Bally et al. (1995) for a presentation of the method and applications to stochastic differential and stochastic partial differential equations.
In the framework of stochastic partial differential equations, the regularization of the noise raises technical dif®culties connected with the explosion of the corresponding integral (see, for instance, Bally et al. 1995).This problem does not appear here for the following reasons.The noise F is smoother than space-time white noise.On the other hand, the integrability condition (C1) and Lemma A.1 in Millet and Sanz-Sole Â (1999) yield (see (A.2) and (A.11) below).This fact prevents explosions, as is made explicit in the proofs.
We now introduce some preliminaries and notation used throughout this paper.Let fe j , j P Ng be a sequence of functions of E which is a complete orthonormal system of H and is taken to be ®xed.De®ne Clearly fW j , j P Ng is a sequence of independent Brownian motions.
Let H be the separable Hilbert space of functions k : [0, T ] 3 R N such that T 0 I j1 jk j (s)j 2 ds , I and endowed with the inner product Notice that the mapping provides an isometry between H and H . Let hj(s, Ã), e j (Ã)i H W j (ds), (1X8) t P [0, T ], so that the stochastic integral with respect to the martingale measure F can also be viewed as a stochastic integral with respect to the in®nite-dimensional Gaussian process fW j (t), t P [0, T ], j P Ng.We introduce smooth approximations of F constructed as follows.
Fix n P N and let P n be the partition of [0, T ] determined by iTa2 n , i 0, 1, F F F , 2 n .
Denote by Ä i the interval [iT a2 n , (i 1)T a2 n ) and by jÄ i j its length.We write W j (Ä i ) for the increment W j ((i 1)T where, for j .n, W n j 0, and, for 1 It is easy to check that almost surely, ù n P H and, more precisely, Moreover, for any 0 < t 1 < t 2 < T, similar computations imply Let ( " Ù, " F , " P) be the canonical space associated with a standard Brownian motion.Denote by (Ù, F , P) the product space ( " Ù N , " F N , " P N ), which will be our reference probability space.
Set " k(t) t 0 k(s)ds for k P H .For any integer n > 1, let T k n denote the transformation of Ù de®ned by Notice that T k n (ù) ù X 0 j n (s, ù)ds, where fj n (t, ù), t P [0, T ]g is an H -valued process adapted to the ®ltration generated by fW j (t), t P [0, 1], j P Ng.Therefore, by Girsanov's theorem, P (T k n ) À1 ( P. This fact will be used in the proof of Theorem 2.1.The paper is organized as follows.In Section 2 we prove the characterization of the support by means of an approximation in probability.In Section 3 we prove approximations in L p -norm under stronger hypotheses on the coef®cients.As usual, all constants are denoted by C, regardless of their values.

Approximation in probability and support theorem
The purpose of this section is to prove the following result: Theorem 2.1.Assume (C1)±(C3), ®x a compact set K & R 2 and let fu(t, x), t P [0, T ], x P Kg be the solution of (1.3).Then for any ã P (0, âa2(1 â)) the topological support of the law of u in the space C ã ([0, T ] 3 K) of ã-Ho Èlder continuous functions in (t, x) is given by the closure in C ã ([0, T ] 3 K) of the set of functions fÖ h , h P Hg, where fÖ h (t, x), t P [0, T ], x P Kg is the solution of (1.4).
The proof of Theorem 2.1 is a consequence of an approximation result, concerning convergence in probability, for an equation more general than (1.3).
More precisely, let us introduce the following hypothesis: (C39) The coef®cients A, B, D, b : R 3 R are globally Lipschitz functions.
We ®rst show that Theorem 2.1 is an easy consequence of this proposition.
Proof of Theorem 2.1.Assume that Proposition 2.1 has been proved.For n > 1, set Clearly, equations (2.5) and (2.6) are particular cases of (2.1), while equations (1.3) and (1.4) are particular cases of (2.2), obtained by choosing A D 0, B ó and A D ó, B Àó , respectively.Moreover, u n Ö ù n .Given h P H, set k J (h), where J is the isometry de®ned in (1.7).Then, by (1.8), equation (2.6) can be rewritten as follows: , where T k n is the absolutely continuous transformation on Ù de®ned by (1.14).
The convergence (2.4) implies, for any ç .0, These two convergences yield the characterization of the support stated in Theorem 2.1 (see, for instance, Millet and Sanz-Sole Â (1994a;or 1994b).Indeed, since ù n P H, the ®rst convergence implies that the support of u in C ã ([0, T ] 3 K) is included in the closure of fÖ h : h P Hg.Since ù n is adapted, P (T h n ) À1 ( P; the second convergence yields the converse inclusion.h The question of the existence and uniqueness of a solution to equations (2.1) and (2.2) is solved in Lemma A.1, which is a slight extension of Theorem 1.2 in Millet and Sanz-Sole Â (1999).We remark that the existence of a solution X n to (2.1) as well as upper estimates of X n require some localization, due to the term involving ù n which has an unbounded Hnorm.For this reason we localize ù n as follows: for any positive integer n, M P N and t P [0, T ], set and (2 ln 2) 1a2 and, for every n .0, set and Lemma 2.1.The following convergence holds: Proof.Let Z denote an N (0, 1) random variable.Then and, for any 0 < t < t9 < T, on A n (t9) we have Support theorem for a wave equation in two space dimensions Lemma A.1 yields the existence and uniqueness of the solution X n to (2.1) and Remark 2.2 shows that the trajectories of X n almost surely have ã-Ho Èlder continuous trajectories for ã , âa2(1 â); since X is a particular case of X n , it also has ã-Ho Èlder continuous trajectories.
Our next purpose is to check that the sequence of processes Y n (t, x) X X n (t, x) À X (t, x), n > 1, satis®es the requirements of Lemma A.2.To this end, we introduce some notation and prove several lemmas.For any n > 1, t P [0, T ], set For convenience' sake, we do not write explicitly the fact that the process X À depends on n.
In what follows, k k p denotes the L p (Ù)-norm.

X
The local property of stochastic integrals, the inclusion A n (s) ' A n (t) for s < t, Burkholder's and Ho Èlder's inequalities and (A.11) yield The Cauchy-Schwarz and Ho Èlder inequalities, along with (2.13) and (A.11), imply Similarly, using (A.3) for the last inequality, we have Thus (2.24)±(2.28)conclude the proof of (2.22).Similar computations can be carried out to prove upper estimates of the p moments of the increments jX k1 n (s, x) À X k n (s, x)j1 A n (s) and j " X k1 n (s, x) À " X k n (s, x)j1 A n (s) ; they yield lim Therefore, (2.22) and (2.29) yield (2.23).h We now prove the convergence of X À n (s, y) to X n (s, y).

Support theorem for a wave equation in two space dimensions
Proof.Consider the decomposition where 2 3

X
In the proof of Proposition 1.4 in Millet and Sanz-Sole Â (1999) we checked that (C2) implies that for ä âa(2(1 â)).Burkholder's and Ho Èlder's inequalities yield Using the operators ô n and ð n introduced in the proof of Lemma 2.4, we may rewrite the scalar product appearing in R 3 n as a stochastic integral; thus Burkholder's inequality implies The Cauchy-Schwarz and Ho Èlder inequalities, along with (2.11) and (2.31), imply We now prove that the processes fX n (t, x), n > 1g satisfy condition (P2) of Lemma A2.
Proposition 2.3.Suppose that conditions (C1), (C2) and (C39) are satis®ed.Then, for any p P [1, I), (t, x) P [0, T ] 3 K, where Support theorem for a wave equation in two space dimensions with X À n and X À de®ned in (2.15) and (2.16), respectively.For t P [0, T ], let K(t) fx P R 2 : d(x, K) < T À tg; ®x t P [0, T ] and x P K(t).Burkholder's and Ho Èlder's inequalities imply The Cauchy-Schwarz and Ho Èlder inequalities yield k(X n (s, y) À X (s, y))1 A n (s) k p p dsX Analogously, Burkholder's and Ho Èlder's inequalities easily yield Thus, (2.31) and (2.17) ensure that The Cauchy-Schwarz and Ho Èlder inequalities, along with (2.11) and (2.31), imply that Burkholder's and Ho Èlder's inequalities and (2.31) imply X By Burkholder's inequality and the fact that ð n is a contraction of H, we deduce that

Approximation in L p
In the previous section, we proved an approximation theorem in probability, by showing the L p convergence of the sequence X n localized by A n, M( n) .The aim of this section is to check that under a stronger growth assumption on the coef®cients, a slight modi®cation of the proof S(t À s, x À y) f (j y À zj)S(t À s, x À z)2 X 3 jh1 [0,s n ] ( X )S(s À X , y À Ã)D(X ( X , Ã)), hi H j 3 jh1 [0,s n ] ( X )S(s À X , z À Ã)D(X ( X , Ã)), hi H j (t À s, x À y) f (j y À zj)S(t À s, x À z) À r, z À ae)b(X (r, ae))dr dae (t À s, x À y) f (j y À zj)S(t À s, x À z) À r, y À ç)(A B)(X (r, ç))F(dr, dç) À r, z À ae)(A B)(X (r, ae))F(dr, dae) À r, y À ç)(A B)(X (r, ç))F(dr, dç),F ô and (A.11) imply that Support theorem for a wave equation in two space dimensions T 2 estimations and (A.15) imply, for p P [1, I[,E sup n kS(t À X , x À Ã)B(X À ( X , Ã))k p H , IXThus, by dominated convergence, the sequence ( Z n (t, x)) n>1 decreases to 0. Moreover, Z n (t, x) is jointly continuous in (t, x); consequently, by Dini's theorem, sup ( t,x)P[0,T ]3 K Z n (t, x) 5 0 as n 3 Remark 2.2.Proposition 2.2 establishes the ã-Ho Èlder continuity for the trajectories of X n on A n (T ), because the sets A n (t), t P [0, T ], are decreasing.In Lemma 2.1 we have shown lim á3I P(A n (T )) 1, so that the trajectories of X n are almost surely ã-Ho Èlder continuous on [0, T ] 3 K for any ã , âa2(1 â) and any compact subset K of R 2 .