THE SUBSTITUTION THEOREM FOR SEMILINEAR STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS. Salah-Eldin A. Mohammed and Tusheng Zhang

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1 THE SUBSTITUTION THEOREM FOR SEMILINEAR STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS Salah-Eldin A. Mohammed and Tusheng Zhang Abstract. In this article we establish a substitution theorem for semilinear stochastic evolution equations (see s) depending on the initial condition as an infinite-dimensional parameter. Due to the infinitedimensionality of the initial conditions and of the stochastic dynamics, existing finite-dimensional results do not apply. The substitution theorem is proved using Malliavin calculus techniques together with new estimates on the underlying stochastic semiflow. Applications of the theorem include dynamic characterizations of solutions of stochastic partial differential equations (spde s) with anticipating initial conditions and non-ergodic stationary solutions. In particular, our result gives a new existence theorem for solutions of semilinear Stratonovich spde s with anticipating initial conditions.. Introduction. Statement of the substitution theorem. The main objective of this article is to answer the following simple (but basic) question: Given a non-anticipating stochastic partial differential equation with its initial condition as an infinite-dimensional parameter, is it justified to replace the initial condition/parameter by an arbitrary random variable? An answer to the affirmative for the above question is well-known for a wide class of finitedimensional sde s via the substitution theorems in [Nu.-2] and [M-S.2]). However, the existing substitution theorems in ([Nu.-2],[M-S.2]) do not apply to infinite-dimensional systems. There are two serious obstructions to this approach: The substitution theorems are based largely on finite-dimensional selection techniques that are known to fail in infinite-dimensional settings, as indicated by the failure of Kolmogorov s continuity theorem for infinite-dimensional random fields ([Mo.-2], [M-Z-Z]) and the failure of Sobolev inequalities in infinite dimensions. The infinite-dimensionality of the dynamics renders the conditions of the substitution theorems in [Nu.-2] inapplicable (cf. Theorem [Nu.], Theorem [Nu.2]). Both obstructions are resolved using ideas and techniques of the Malliavin calculus together with new global estimates on the semiflow generated by the spde (Section 2) ([Ma]). The use of Malliavin June, 27. To appear in Journal of Functional Analysis, 27. The research of this author is supported in part by NSF Grants DMS , DMS and DMS The research of this author is supported in part by EPSRC Grant GR/R944. AMS 99 subject classifications. Primary 6H, 6H2; secondary 6H25. Key words and phrases. Malliavin calculus, stochastic semiflow, C k cocycle, stochastic evolution equation (see), anticipating stochastic partial differential equation (spde).

2 2 S.-E.A. MOHAMMED AND T.S. ZHANG calculus techniques in this context seems to be necessitated by the infinite-dimensionality of the underlying stochastic dynamics. The difficulty in proving the substitution theorem for stochastic systems with memory was pointed out by M. Scheutzow and one of the authors in ([M-S.], Part II); but no rigorous proof or counterexamples are known. The purpose of the discussion in [M-S.] is to provide a dynamic characterization of stable/unstable manifolds for stochastic systems with memory near hyerbolic stationary states. In work by Grorud, Nualart and Sanz-Solé ([G-Nu-S]) a substitution theorem for Stratonovich integrals in Hilbert space is developed under the restriction that the substituting random variable takes values in a relatively compact sen the Hilbert space. The substitution resuln [G-Nu-S] is obtained within the context of Hilbert-space-valued stochastic ordinary differential equations, using metric entropy techniques. Cf. also [A-I], where the substituting random variable takes values in a σ-compact space. In this article we establish a substitution theorem for semilinear spde s for a large class of infinite-dimensional Malliavin smooth random variables. We strongly believe that the techniques developed in this article will yield a similar substitution theorem for semiflows induced by sfde s. We expect the results in this article to be useful in establishing regularity in distribution of the invariant manifolds for semilinear spde s. equation (see): In order to formulate our results, consider the following semilinear Itô stochastic evolution du(t, x) = Au(t, x) dt + F ( u(t, x) ) dt + Bu(t, x) dw (t), t > u(, x) = x H in a separable real Hilbert space H. } (.) In the above equation A : D(A) H H is a closed linear operator on the Hilbert space H. Assume that A has a complete orthonormal system of eigenvectors {e n : n } with corresponding positive eigenvalues {µ n, n }; i.e., Ae n = µ n e n, n. Suppose A generates a strongly continuous semigroup of bounded linear operators T t : H H, t. Furthermore, we let F : H H be a (Fréchet) Cb non-linear map, thas F has a globally bounded continuous Fréchet derivative DF : H L(H). Let E be a separable Hilbert space and W (t), t, be an E-valued Brownian motion defined on the canonical filtered Wiener space (Ω, F, (F t ) t, P ) and with a separable covariance Hilbert space K. In particular, K E is a Hilbert-Schmidt embedding. Furthermore, Ω is the space of all continuous paths ω : R E such that ω() = with the compact open topology, F is its Borel σ-field, F s the sub-σ-field of F generated by all evaluations Ω ω ω(u) E, u t, and P is Wiener measure on Ω. The Brownian motion is given by W (t, ω) := ω(t), ω Ω, t R,

3 and may be represented by THE SUBSTITUTION THEOREM FOR SPDE S 3 W (t) = W k (t)f k, t R, (.2) where {f k : k } is a complete orthonormal basis of K, and W k, k, are standard independent one-dimensional Wiener processes ([D-Z.], Chapter 4). converges absolutely in E but non K. Note that, in general, the above series Denote by L 2 (K, H) L(K, H) the Hilbert space of all Hilbert-Schmidt operators S : K H, given the norm [ /2 S 2 := S(f k ) H] 2, where H is the norm on H. Suppose B : H L 2 (K, H) is a bounded linear operator. The stochastic integral in (.) is defined in the following sense ([D-Z.], Chapter 4): and such that Let ψ : [, a] Ω L 2 (K, H) be (B([, a]) F, B(L 2 (K, H))-measurable, (F t ) t -adapted a E ψ(t) 2 L 2 (K,H) dt <. Define a ψ(t) dw (t) := a ψ(t)(f k ) dw k (t), where the H-valued stochastic integrals on the right-hand side are with respect to the one-dimensional Wiener processes W k, k. Note that the above series converges in L 2 (Ω, H) because E a ψ(t)(f k ) dw k (t) 2 = a E ψ(t) 2 L 2 (K,H) dt <. Throughout the rest of the article, we will denote by θ : R Ω Ω the standard P - preserving ergodic Wiener shift on Ω: Hence (W, θ) is a helix: θ(t, ω)(s) := ω(t + s) ω(t), t, s R. W (t + t 2, ω) W (t, ω) = W (t 2, θ(t, ω)), t, t 2 R, ω Ω. As usual, we let L(H) be the Banach space of all bounded linear operators H H given the uniform operator norm L(H). Denote by L 2 (H) L(H) the Hilbert space of all Hilbert-Schmidt operators S : H H, furnished with the Hilbert-Schmidt norm: [ /2 S 2 := S(e n ) H] 2. n=

4 4 S.-E.A. MOHAMMED AND T.S. ZHANG A mild solution of (.) is a family of (B(R + ) F, B(H))-measurable, (F t ) t -adapted processes u(, x, ) : R + Ω H, x H, satisfying the following stochastic integral equation: u(t, x, ) = T t x + T t s F (u(s, x, )) ds + T t s Bu(s, x, ) dw (s), t, (.3) ([D-Z.-2]). The see (.) has the equivalent Stratonovich form du(t, x) = Au(t, x) dt + F ( u(t, x) ) dt 2 u(, x) = x H B 2 k u(t, x) dt + Bu(t, x) dw (t) (.4) where B k L(H) are given by B k (x) := B(x)(f k ), x H, k. Condition (A ): µ n B(e n) 2 L 2 (K,H) <. n= Conditions (A 2 ): (i) A is a trace class operator, i.e., n= µ n <. (ii) T t L(H), t, is a strongly continuous contraction semigroup. Conditions (B): (i) The operator B : H L 2 (K, H) can be extended to a bounded linear operator H L(E, H), which will also be denoted by B. (ii) The series B k 2 L(H) converges, where the bounded linear operators B k : H H, k, Remarks. are defined as in (.4). (i) Note that Condition (A ) is implied by the following two requirements: (a) The operator B : H L 2 (K, H) is Hilbert-Schmidt. (b) inf n µ n >. (ii) Requirement (b) above is satisfied if A =, where is the Laplacian on a compact smooth d-dimensional Riemannian manifold M with boundary, under Dirichlet boundary conditions. (iii) Suppose A = where is the Laplacian on a compact smooth d-dimensional manifold with Dirichlet boundary condition. Then Condition (A 2 ) implies that d =. This follows easily from the fact that µ n = O(n 2/d ) for large n ([T], Theorem 3., p. 89). (iv) Unlike Condition (A 2 ), note that Condition (A ) does not entail any restriction on the spatial dimension of the underlying spde.

5 THE SUBSTITUTION THEOREM FOR SPDE S 5 Under Conditions (B) together with either (A ) or (A 2 ), the see (.) (or (.4)) admits a perfect (B(R + ) B(H) F, B(H))-measurable cocycle (U, θ), U : R + H Ω H, with the following properties: (i) For each ω Ω, the map R + H (t, x) U(t, x, ω) H is continuous; and for fixed (t, ω) R + Ω, the map H x U(t, x, ω) H is C. (ii) U(t + s,, ω) = U(t,, θ(s, ω)) U(s,, ω) for all s, t R + and all ω Ω. (iii) U(, x, ω) = x for all x H, ω Ω. (U, θ) if For proofs of the above properties see ([M-Z-Z], Theorem.2.6); cf. [F.-2]. An F-measurable random variable Y : Ω H is said be a stationary point for the cocycle for all (t, ω) R + Ω. U(t, Y (ω), ω) = Y (θ(t, ω)) For any integer p 2, denote by D,p (Ω, H) the Sobolev space of all F-measurable random variables Y : Ω H which are p-integrable together with their Malliavin derivatives DY ([Nu.-2]). We now state the main substitution theorem in this article. Theorem.. Assume that the see (.) satisfies Conditions (B) together with either (A ) or (A 2 ). Suppose F is C b. Let Y D,4 (Ω, H) be a random variable, and U : R + H Ω H be the C cocycle generated by all mild solutions of the Stratonovich see (.4). Then U(t, Y ), t, is a mild solution of the (anticipating) Stratonovich see du(t, Y ) = AU(t, Y ) dt + F ( U(t, Y ) ) dt 2 U(, Y ) = Y. B 2 k U(t, Y ) dt + BU(t, Y ) dw (t), t >, In particular, if Y D,4 (Ω, H) is a stationary point of the see (.4), then U(t, Y ) = Y ( θ(t) ), t, is a stationary solution of the (anticipating) Stratonovich see dy (θ(t)) = AY (θ(t)) dt + F ( Y (θ(t)) ) dt 2 Y (θ()) = Y. B 2 k Y (θ(t)) dt + BY (θ(t)) dw (t), t >, Furthermore, assume that F is Cb 2. Then the linearized cocycle DU(t, Y ) is a mild solution of the linearized anticipating see ddu(t, Y ) = ADU(t, Y ) dt + DF ( U(t, Y ) ) DU(t, Y ) dt B 2 2 kdu(t, Y ) dt + { B DU(t, Y ) } dw (t), t >, DU(, Y ) = id L(H). (.5) (.6) (.7)

6 6 S.-E.A. MOHAMMED AND T.S. ZHANG In the subsequent sections we will detail the proof of the above theorem. In Section 2, we begin by offering a series of estimates on the cocycle U(t, x, ), its Fréchet derivative DU(t, x, ) and its Malliavin derivative DU(t, x, ). These estimates-interesting in their own right-will be used in the proofs of the substitution theorem (Theorem.) and its finite-dimensional version (Theorem 3.). In Section 3, we prove a special case of Theorem. in case the random variable Y is finitedimensional (Theorem 3.). This resuls then used to give a detailed proof of Theorem. in Section 4. Section 5 contains an alternative proof of one of the estimates in Section 2, using a chaos-type expansion in the Hilbert space L 2 (H). In Section 6, we show existence and regularity of solutions for semilinear spde s with anticipating initial conditions. 2. Moment estimates of the cocycle. In this section, we develop new estimates on the non-linear cocycle U : R + H Ω H, its spatial Fréchet derivative DU(t, x, ) and its Malliavin derivatives D u U(t, x, ) for u, t [, a] and x H. The derivations are based on results in [M.Z.Z], Gronwall s lemma and the fact that W has independenncrements. As before, assume the notation and hypotheses of Section. Let Φ(t, ω), ω Ω, t, be the linear cocycle associated with the see (.). Thas for each x H, Φ(t, )(x), t, is a mild solution of the linear see } dφ(t, )(x) = AΦ(t, )(x) dt + BΦ(t, )(x) dw (t), t > (2.) Φ(, )(x) = x H. Recall that (F t ) t is the filtration generated by Brownian motion W. Define V (t, ω) := Φ(t, ω) T t, t, ω Ω. Then V (t, ), t, is the continuous L 2 (H)-valued solution of the following stochastic integral equation in L 2 (H): V (t, ) = T t s BV (s, ) dw (s) + T t s BT s dw (s), t. (2.2) Fix s, and denote ˆV (t, ω) := V (t s, θ(s, ω)), t s. Then ˆV is a solution of the following integral equation: ˆV (t, ) = s T t u B ˆV (u, ) dw (u) + in L 2 (H). See the proof of Theorem.2.4 in [M-Z-Z]. We will need the following Gronwall-type lemma: s T t u BT u s dw (u), t s. (2.3)

7 Lemma 2.. THE SUBSTITUTION THEOREM FOR SPDE S 7 Fix a (, ). Let f, g : [, a] Ω R + be non-negative (B([, a]) F, B(R + ))-measurable processes and h : [, a] [, a] Ω R + an (B([, a] [, a]) F, B(R + ))-measurable random field satisfying the following hypotheses: (i) For a.a. ω Ω and all s [, a], the paths f(, ω), g(, ω), h(, s, ω) are continuous on [, a]. (ii) The process f is (F t ) t [,a] -adapted; and whenever < s < t a, the random variables h(t s, s, ) are measurable with respect to the σ-algebra generated by the Brownian increments W (s 2 ) W (s ), s s s 2 t. (iii) E sup t a g(t, ) + sup s a E sup t a h(t, s, ) <. Suppose that a.s. for all t [, a]. Then that for all t [, a]. f(t, ) g(t, ) + h(t s, s, )[ + f(s, )] ds (2.4) sup f(t, ) is integrable and there exist positive constants K, K 2 such t a E sup f(s, ) K e K 2t s t (2.5) Proof. Use Conditions (i), (iii), put t = t in (2.4), and take sup to obtain t t sup f(t, ) sup g(t, ) + t t x H t a a.s. for all t [, a]. sup u a h(u, s, ) ds + sup u a For each integer N, and any s [, a], define the events ( Ω s,n := sup f(s, ) < N s s ). h(u, s, ) Since f is is (F t ) t [,a] -adapted, then Ω s,n F s for all s [, a], N. Furthermore, Ω t,n Ω s,n, s t, N, sup f(s, ) ds s s (2.6) and Ωt,N Ωs,N, s t, N. (2.7) Since f has a.a. sample-paths bounded on [, a] (actually continuous), we have Ω s,n = Ω (2.8) N

8 8 S.-E.A. MOHAMMED AND T.S. ZHANG for each s [, a]. Define f N (t, ) := sup f(t, ) Ωt,N, t a, N t t Clearly f N (t, ) N a.s. and Ef N (t, ) N for all t [, a] and all N. Now multiply both sides of (2.6) by Ωt,N, use (2.7), take expectations, use hypothesis (iii) together with the independence of sup f(s, ) Ωs,N and sup h(u, s, ), to obtain s s u a Ef N (t, ) K + K 2 Ef N (s, ) ds, t a, (2.9) for all N. The positive constants K, K 2 in (2.9) are independent of N. By Gronwall s lemma, (2.9) gives for all N. Letting N in (2.), using the fact that Ef N (t, ) K e K 2t, t a, (2.) f N(t, ) = sup f(t, ) N t t a.s., and applying the Monotone Convergence Theorem, we get E sup f(t, ) K e K 2t t t for all t a. This proves (2.5). Now consider the random integral equation U(t, x, ) = Φ(t, )(x) + Φ(t s, θ(s, ))F ( U(s, x, ) ) ds, t, x H, (2.) where F : H H is C b (as in Section ). Theorem 2.2. Adopt the set-up of Section. Assume Hypotheses (B) and (A ) or (A 2 ). Let U : R + H Ω H be the cocycle generated by the mild solutions of the see (.). Fix any a (, ). Then the following assertions hold: (i) The estimate holds for all p. (ii) Let F be of class C b. Then U(t, x, ) 2p E sup t a ( + x 2p x H H ) < (2.2) E sup DU(t, x, ) 2p < (2.3) t a x H

9 THE SUBSTITUTION THEOREM FOR SPDE S 9 for all p. In the above estimate, D stands for the Fréchet derivative of U in the spatial variable x. (iii) Let F be C 2 b. Then Proof. for all p. Assume Hypotheses (B) and (A ) or (A 2 ). E sup D 2 U(t, x, ) 2p < (2.3 ) t a x H We will first prove the estimate (2.2). Fix any p. By a simple application of Gronwall s lemma, (2.3) gives for any fixed s [, a]; and hence, for each s [, a]. [ ] E sup V (t s, θ(s, )) 2p L 2 (H) s t a [ ] E sup Φ(u, θ(s, )) 2p L(H) u a By (2.) and the linear growth property of F, we get < (2.4) < (2.5) U(t, x, ) 2p Φ(t, ) p L(H) x 2p + C Φ(t s, θ(s, )) 2p L(H)( + U(s, x, ) 2p ) ds (2.6) a.s. for t a, x H, and C is a deterministic positive constant depending only on a. In (2.6), divide both sides of the inequality by ( + x 2p H ) and take sup to obtain x H U(t, x, ) 2p sup x H ( + x 2p Φ(t, ) 2p H ) a.s. for t a. Now set + C L(H) + C Φ(t s, θ(s, )) 2p L(H) ds Φ(t s, θ(s, )) 2p L(H) sup U(s, x, ) 2p x H ( + x 2p ds (2.7) H ) U(t, x, ) 2p f(t, ) := sup x H ( + x 2p H ), g(t, ) := Φ(t, ) 2p L(H), h(t, s, ) := Φ(t, θ(s, )) 2p L(H), a.s. for s t a. Then (2.7) becomes f(t, ) g(t, ) + a.s. for all t [, a]. h(t s, s, )[ + f(s, )] ds (2.8)

10 S.-E.A. MOHAMMED AND T.S. ZHANG We will now verify that the processes f, g, h satisfy all the conditions of Lemma 2.. First, note that f, g, h are finite a.s. ([M-Z-Z], Theorems.2.4,.2.6). Secondly, the processes f, g, h(, s) are sample-continuous for each s [, a] ([M-Z-Z], Theorems.2.,.2.2,.2.3,.2.6). Thirdly, the process f is (F t ) t [,a] -adapted. Fourthly, from (2.3), it follows that ˆV (t, ) = V (t s, θ(s, )) = Φ(t s, θ(s, )) T t s is measurable with respect to the σ-algebra generated by the Brownian increments W (s 2 ) W (s ), s s s 2 t, and hence so is h(t, s, ). Finally, Hypothesis (iii) of Lemma 2. is satisfied because of (2.5) and the measure-preserving property of θ. Hence the conditions of Lemma 2. are satisfied; thus (2.2) follows from (2.8). In fact, one gets E sup t t x H U(t, x, ) 2p ( + x 2p H ) K e K2t (2.9) for all t [, a] and some positive constants K, K 2 (depending possibly on a). To prove part (ii) of the theorem, assume Hypotheses (B) and (A 2 ) or (A ); and let F be of class Cb. Fix any p. Take Fréchet derivatives with respect to x H on both sides of the random integral equation This gives U(t, x, ω) = Φ(t, ω)(x) + Φ(t s, θ(s, ω))f ( U(s, x, ω) ) ds, t, x H, ω Ω. DU(t, x, ) = Φ(t, ) + Φ(t s, θ(s, ))(DF (U(s, x, )))(DU(s, x, )) ds, t. As in the proof of part (i), observe that Φ(t s, θ(s, )) is measurable with respect to the σ-algebra generated by the Brownian increments W (s 2 ) W (s ), s s s 2 t, while DU(, x, )) is (F t ) t [,a] -adapted (and DF is bounded). Using this observation together with the above equation and Lemma 2., one obtains E sup DU(t, x, ) 2p L(H) <. t a x H This proves the first assertion in (ii) for all p. The proof of the second assertion in (ii) follows by a similar argument. above. If F is Cb 2, assertion (iii) of the theorem may be proved by an argument similar to the The next theorem gives global spatial estimates on the Malliavin derivatives of the stochastic semiflow U : R + H Ω H and its linearization.

11 Theorem 2.3. THE SUBSTITUTION THEOREM FOR SPDE S Assume the setting of Section. In the see (.), assume Hypotheses (B) and (A ) or (A 2 ). Then the following assertions hold: (i) Let u, t [, a]. Then V (t, ) D,2p (Ω, L 2 (H)) and for all p. [ ] E sup D u V (t, ) 2p L 2 (H) u t a (ii) Suppose F is Cb. Then for all p, we have [ E sup t a x H where D stands for the Malliavin derivative. (iii) Let F be C 2 b. Then Proof. for all p. [ E sup <u,t a x H <. (2.2) DU(t, x, ) 2p ] H ( + x 2p H ) <, (2.2) D u DU(t, x, ) 2p ( + x 2p H ) L 2 (H) Assume Hypotheses (B) and (A ) or (A 2 ) throughout this proof. ] < (2.2 ) We prove the first assertion in part (i) of the theorem. Let p. Fix u [, a] and take Malliavin derivatives in (2.2) to get the following stochastic integral equation in L 2 (H): D u V (t, ) = T t u BV (u, ) + T t u BT u + Define the sequence of events ( ) Ω s,n := sup D u V (s, ) 2p u s L 2 (H) < N s u T t s BD u V (s, )dw (s), t u. (2.22) for u s a, N. Now, from (2.22) and Proposition 7.3 in [D-Z.], we obtain [ E for all t u. ] sup D u V (t, ) 2p u t L 2 (H) K Ωt,N E V (u, ) 2p L 2 (H) + K 2 t + K 3 u [ E ] sup D u V (s, ) 2p u s L 2 (H) ds Ωs,N s Using (2.4) together with Gronwall s lemma and the Monotone Convergence Theorem, the above inequality implies (2.2).

12 2 S.-E.A. MOHAMMED AND T.S. ZHANG equation To prove the first assertion in (ii) of the theorem, let F be Cb. Rewrite the random integral U(t, x, ω) = Φ(t, ω)(x) + in the form U(t, x, ω) = V (t, ω)(x)+t t (x)+ Taking the Malliavin derivative on both sides of (2.23), we get Φ(t s, θ(s, ω))f ( U(s, x, ω) ) ds, t, x H, ω Ω, (2.) [V (t s, θ(s, ω))+t t s ]F ( U(s, x, ω) ) ds, t, x H, ω Ω. (2.23) D u U(t, x, ) = D u V (t, )(x) + + D u V (t s, θ(s, ))(F (U(s, x, ))) ds [V (t s, θ(s, )) + T t s ](DF (U(s, x, )))(D u U(s, x, )) ds, t. (2.24) As in the proof of Theorem 2.2, observe that V (t s, θ(s, )), D u V (t s, θ(s, )) are measurable with respect to the σ-algebra generated by the Brownian increments W (s 2 ) W (s ), s s s 2 t, while U(, x, ), D u U(, x, ) are (F t ) t [,a] -adapted. Using this observation together with (2.24) and Lemma 2., one obtains [ E sup t a x H for all u [, a] and all p. This implies (2.2). above. Let F be C 2 b. D u U(t, x, ) 2p ] H ( + x 2p H ) <, (2.25) Then assertion (iii) of the theorem follows by a similar argument to the 3. Finite-dimensional substitutions. Assume the notation and hypotheses of Section. In this section, we will prove assertion (.5) of Theorem. in the special case when the random variable Y D,4 (Ω, H) is replaced by its finite-dimensional projections on H. The proof of (.7) (in this special case) is analogous to that of (.5) and is left to the reader. Relation (.6) follows immediately from (.5). Recall that {e n : n } is a complete orthonormal system of eigenvectors of A. For each integer n, denote by H n := L{e i : i n}, the n-dimensional linear subspace of H spanned by {e i : i n}. Define the sequence of projections P n : H H n, n, by n P n (x) := < x, e k, > e k, x H. (3.) Define Y n : Ω H n by Y n := P n Y, n. (3.2)

13 THE SUBSTITUTION THEOREM FOR SPDE S 3 Note that Y n Y a.s.. The main resuln this section is the following finite-dimensional substitution theorem (Theorem 3.). Note that the proof of this theorem still requires Malliavin calculus techniques, largely due to the underlying infinite-dimensional semi-group dynamics in {T t } t. Theorem 3.. Assume all the conditions of Theorem.. Then for each integer n, (.5) and (.7) hold when Y D,4 (Ω, H) is replaced by Y n. In particular, du(t, Y n ) = AU(t, Y n ) dt + F ( U(t, Y n ) ) dt 2 U(, Y n ) = Y n. for each n. BkU(t, 2 Y n ) dt + BU(t, Y n ) dw (t), t >, (3.3) Proof. In this proof, we denote by C i, i =, 2,, 9, positive dterministic constants. Rewrite (.4) in its mild form U(t, x) = T t (x) + T t s F ( U(s, x) ) ds 2 + T t s Bk 2 U(s, x) ds T t s BU(s, x) dw (s), t >. (3.4) Using the fact that each Y n D,4 (Ω, H n ) is a finite-dimensional random variable, we will show that x in (3.4) can be replaced by Y n to get U(t, Y n ) = T t (Y n ) + T t s F ( U(s, Y n ) ) ds 2 + T t s B 2 ku(s, Y n ) ds T t s BU(s, Y n ) dw (s), t >, n, (3.5) for each n (cf. [Nu.], Section 3.3.2, [M-S.2]). To justify (3.5), is sufficient to prove that the random field has a version H n Ω H satisfying T t s BU(s, x) dw (s), x H n, T t s BU(s, x) dw (s) = x=yn T t s BU(s, Y n ) dw (s) (3.6)

14 4 S.-E.A. MOHAMMED AND T.S. ZHANG a.s. for fixed t >. To prove (3.6), we will first establish some estimates on U(t, x), t, x H. Let x, x H and t [, a]. Then (3.4) implies E U(t, x) U(t, x ) 2p C T t (x) T t (x ) 2p { + C 2 E Tt s F ( U(s, x) ) T t s F ( U(s, x ) )} ds + C 3 E T t s B ( U(s, x) U(s, x ) ) 2p dw (s) C 4 x x 2p + C 5 E U(s, x) U(s, x ) 2p ds } p { + C 6 E U(s, x) U(s, x ) 2 ds C 4 x x 2p + C 7 E U(s, x) U(s, x ) 2p ds. Gronwall s lemma implies E U(t, x) U(t, x ) 2p C 8 x x 2p, x, x H, t [, a]. (3.7) 2p Fix t a <, and define S m (x) : = = T t s P m BU(s, x) dw (s) T t s P m BU(s, x) dw (s) + 2 T t s (P m B k ) 2 U(s, x) ds (3.8) for all x H. Since each H m is invariant under T t, t [, a], then S m (x) H m for all x H. Claim. Assume Condition (B) of Section. Fix t [, a] in (3.8). Then there is a constant C 9 > independent of m and t [, a] such that E S m (x) S m (x ) 2p C 9 x x 2p (3.9) for all x, x H and all m. Proof of Claim. Let x, x H and fix t [, a]. Assume Condition (B) of Section. Then E 2p T t s P m BU(s, x) dw (s) T t s P m BU(s, x ) dw (s) E T t s P m B [ U(s, x) U(s, x ) ] dw (s) C T t s P m B E U(s, x) U(s, x ) 2p ds C E U(s, x) U(s, x ) 2p ds C 2 x x 2p. 2p (3.)

15 THE SUBSTITUTION THEOREM FOR SPDE S 5 The series on the right-hand-side of (3.8) is absolutely convergent (uniformly in m) because of Condition (B) and the a.s. estimates Furthermore, E T t s (P m B k ) 2 U(s, x) ds C 3 T t s (P m B k ) 2 U(s, x) ds { C 4 { P m B k 2 U(s, x) ds C 3 U(s, x) ds B k 2 <. T t s (P m B k ) 2 U(s, x ) ds {E Tt s (P m B k ) 2[ U(s, x) U(s, x ) ] 2p} /2p ds } 2p B k 2 { E U(s, x) U(s, x ) 2p} /2p ds } 2p { C 4 C 8 B k } x 2 x 2p = C 5 x x 2p. (3.) 2p Therefore (3.) and (3.) imply (3.9). This proves our claim. We next show than (3.6), one can replace B by P m B: a.s. for all m, n. Let To prove (3.2), write T t s P m BU(s, x) dw (s) = x=yn T t s P m BU(s, x) dw (s) = R N (x) := (for fixed m ). Then N R N(x) = N T t s P m BU(s, Y n ) dw (s) (3.2) T t s P m B k U(s, x) dw k (s), x H. (3.3) T t s P m B k U(s, x) dw k (s), N, x H n (3.4) T t s P m B k U(s, x) dw k (s) (3.5)

16 6 S.-E.A. MOHAMMED AND T.S. ZHANG in L 2, because the series on the right-hand-side of (3.5) converges absolutely in L 2 (Ω, H m ). Also for x, x H, ( N { E R N (x) R N (x ) 2p E C 5 x x 2p T t s P m B k [U(s, x) U(s, x ) ] dw k (s) were C 5 is independent of m, N (by the proof of the Claim and Condition (B)). 2p} /2p ) 2p Now apply Lemma 4. ([M.S-2], or [Nu.2], Lemma 5.3.) to the sequence of random fields {R N (x) : x H n }, N. This implies the following in probability: N { RN (x) x=yn } = ( = T t s P m B k U(s, x) dw k (s)) x=y n T t s P m BU(s, x) dw (s). (3.6) x=yn We next show that, for each k, the following substitution rule holds: a.s. ([Nu.2], Theorem 5.3.3). T t s P m B k U(s, x) dw k (s) = x=yn T t s P m B k U(s, Y n ) dw k (s) (3.7) From (3.4), (3.5), (3.6), (3.7), and the finite-dimensional substitution theorem for Stratonovich integrals (Theorem (5.3.4) in [Nu.2]), we get the following its in probability: T t s P m BU(s, x) dw (s) = x=yn N = { RN (x) x=yn } N N = = = N N T t s P m B k U(s, x) dw k (s) x=yn T t s P m B k U(s, Y n ) dw k (s) T t s P m B k U(s, Y n ) dw k (s) T t s P m BU(s, Y n ) dw (s). (3.8) The last equality in (3.8) follows from the definition of the Stratonovich integral in infinitedimensions. This proves (3.2). Recall that S m (x) := T t s P m BU(s, x) dw (s) + 2 T t s (P m B k ) 2 U(s, x) ds. (3.9)

17 THE SUBSTITUTION THEOREM FOR SPDE S 7 Define We will show that S(x) := = T t s BU(s, x) dw (s) T t s BU(s, x)dw (s) + 2 in probability for all x H. Now, for every x H, a.s., T t s B 2 ku(s, x) ds. (3.2) S m(x) = S(x) (3.2) m P mb ( U(s, x) ) = B ( U(s, x) ), m P m B ( U(s, x) ) B ( U(s, x) ), and E 2p T t s P m BU(s, x)dw (s) T t s BU(s, x)dw (s) 2p = E T t s (P m B B)U(s, x)dw (s) { = E T t s Pm B ( U(s, x) ) B ( U(s, x) )} dw (s) C 6 E Pm B ( U(s, x) ) B ( U(s, x) ) 2p ds (3.22) 2p a.s. for all m and all x H. Then by the dominated convergence theorem, it follows that E m T t s P m BU(s, x)dw (s) T t s BU(s, x)dw (s) 2p =, x H. Also, for each m and any x H, we have E Tt s (P m B k ) 2 U(s, x) T t s BkU(s, 2 x) 2p = E { T t s (Pm B k ) 2( U(s, x) ) Bk 2 ( )} U(s, x) 2p C 7 E (Pm B k P m B k ) ( U(s, x) ) (P m B k B k ) ( U(s, x) ) 2p + C 8 E (Pm B k B k ) ( U(s, x) ) Bk 2 ( ) U(s, x) 2p C 7 B k E ( ( )) ( ) Pm Bk U(s, x) Bk U(s, x) 2p + C 8 E ( ( )) ( ) Pm B 2 k U(s, x) B 2 k U(s, x) 2p (3.23)

18 8 S.-E.A. MOHAMMED AND T.S. ZHANG and Tt s (P m B k ) 2 U(s, x) T t s B 2 ku(s, x) C 9 [ Pm B k 2 + B k 2] U(s, x) 2C 9 B k 2 U(s, x) (3.24) a.s.. The right-hand side of (3.23) converges to a.s. as m (for each fixed k ). Hence, by (3.24), Theorem 2.2, the convergence of B k 2 and the dominated convergence theorem, we obtain in L 2p for all x H. m T t s (P m B k ) 2 U(s, x) ds = T t s B 2 ku(s, x) ds (3.25) Using (3.9), (3.2), (3.22), and (3.25), we get (3.2). By (3.2), we may apply Lemma 5.3. in [Nu.2] (Lemma 4., [M-S.2]) to get for each n. Observe that S m(y n ) = S(Y n ) in probability (3.26) m Using (3.26), we may let m in (3.8) to get m in L 2( [, T ] Ω ). T t s P m BU(s, x) dw (s) = x=yn = m T t sp m BU(s, Y n ) = T t s BU(s, Y n ) m T t s P m BU(s, Y n ) dw (s) T t s BU(s, x) dw (s). (3.27) x=yn Using a truncation argument, one can show that the process [, t] s T t s BU(s, Y n ) L 2 (K, H) is Stratonovich integrable, and T t s P m BU(s, Y n ) dw (s) = m T t s BU(s, Y n ) dw (s). (3.28) Details of the truncation argument are given in Section 3 (replacing B by P m B). (Note that this truncation argument does not depend on (3.28)). Combining (3.27) and (3.28) gives T t s BU(s, x) dw (s) = x=yn This proves (3.6) and hence (3.5) holds. T t s BU(s, Y n ) dw (s). (3.29)

19 4. Proof of the substitution theorem. THE SUBSTITUTION THEOREM FOR SPDE S 9 In this section, we will complete the proof of the main substitution theorem (Theorem.) in Section. Our argument will appeal to the estimates in Section 2 on the cocycle U(t, x), t, x H, its Frèchet and Malliavin derivatives DU(t, x), DU(t, x), respectively Proof of Theorem.. Assume that Y D,4 (Ω, H), and the see (.) satisfies Hypothesis (B) together with either (A ) or (A 2 ). We will prove the equality (.5) in Section. Equality (.6) is a special case of (.5). The proof of (.7) is similar to that of (.5), and is left to the reader. Fix t > throughout this proof. To prove (.5) in Section, we will show that the anticipating process U(t, Y ) satisfies the Stratonovich integral equation U(t, Y ) = T t (Y ) + T t s F ( U(s, Y ) ) ds 2 + We start with the mild Stratonovich form of the see (.): U(t, x) = T t (x) + T t s F ( U(s, x) ) ds 2 + T t s Bk 2 U(s, Y ) ds T t s BU(s, Y ) dw (s). (4.) T t s Bk 2 U(s, x) ds T t s BU(s, x) dw (s). (4.2) Denote by L,2 the class of all processes v : [, t] Ω H such that v L 2 ([, t] Ω, H), v(s, ) D,2 (Ω, H) for almost all s [, t] and E[ D uv(s, ) 2 H du ds] <. We say that v belongs to L,2 loc if there exists a sequence (Ω m, v m ) F L,2 with the following properties: (i) Ω m Ω as m, (ii) v = v m on Ω m. We first show that the Stratonovich integral in (4.) is well-defined. To prove this, is sufficient to verify that the process v(s) := T t s BU(s, Y ), s t, belongs to L,2 loc ([Nu.2], Theorem 5.2.3). For any integer m, let φ m Cb 2(R, R) be a bump function such that φ m(z) = for z m and φ m (z) = for z > m +. Define v m (s) := v(s)φ m ( Y H ), s t. Clearly, v = v m on Ω m := {ω : Y (ω) H m} for each m. Thus v is Stratonovich integrable if we can show that v m L,2 for every m. To see this, note first that the estimate v m (s) H C sup U(s, x) H, s t, x H x H m+ together with Theorem 2.2 (i) imply that v m L 2 ([, t] Ω) for each m. On the other hand, D u v m (s) = T t s B[D u U(s, Y ) + DU(s, Y )D u Y ]φ m ( Y H ) + T t s BU(s, Y )φ m( Y H )D u Y H

20 2 S.-E.A. MOHAMMED AND T.S. ZHANG for all u, s [, t]. Therefore, D u v m (s) H C m sup D u U(s, x) H + C m sup DU(s, x) L(H) D u Y H x H m+ x H m+ + C m sup U(s, x) H D u Y H, u, s [, t]. (4.3) x H m+ Using the fact that Y D,4 (Ω, H), Theorem 2.3 (ii) and Theorem 2.2 (i),(ii), it follows from (4.3) that Hence, v m L,2 for each m. [ E ] D u v m (s) 2 H ds du <. Next we prove that U(t, Y ) satisfies the equation (4.). Y n := P n Y as in (3.2). Then by Theorem 3., we know that for every n, U(t, Y n ) = T t (Y n ) + T t s F ( U(s, Y n ) ) ds 2 + For any integer n, define T t s B 2 ku(s, Y n ) ds T t s BU(s, Y n ) dw (s), t >. (4.4) We wish to pass to the it a.s. as n in (4.4). To do this, first note the following easy a.s. its: n 2 n U(t, Y n) = U(t, Y ) n T t(y n ) = T t (Y ) n T t s F ( U(s, Y n ) ) ds = T t s B 2 ku(s, Y n ) ds = 2 Therefore, (4.) will hold provided we show that in probability. n T t s BU(s, Y n ) dw (s) = T t s F ( U(s, Y ) ) ds To prove (4.5), we use the following truncation argument: T t s B 2 ku(s, Y ) ds. By the local property of the Stratonovich integral ([Nu.]), we have T t s BU(s, Y n ) dw (s) = on Ω m := {ω : Y (ω) H m}, and T t s BU(s, Y ) dw (s) = T t s BU(s, Y ) dw (s) (4.5) T t s BU(s, Y n )φ m ( Y H ) dw (s), T t s BU(s, Y )φ m ( Y H ) dw (s),

21 THE SUBSTITUTION THEOREM FOR SPDE S 2 on Ω m for any fixed m. So, to establish (4.5), is enough to prove that n T t s BU(s, Y n )φ m ( Y H ) dw (s) = in probability for each m. To see this, fix m and let T t s BU(s, Y )φ m ( Y H ) dw (s) (4.6) g n (s) := T t s BU(s, Y n )φ m ( Y H ), g(s) := T t s BU(s, Y )φ m ( Y H ) for all s [, t]. We first show that g = g n in L,2. Since both g n (s) and g(s) are bounded by n C U(s, x) H, then by Theorem 2.2 and the Dominated Convergence Theorem, it follows sup x H m+ that the sequence {g n } n= converges to g in L2 ([, a] Ω, L 2 (K, H)) for each a (, ). Notice that D u g n (s) = T t s B[D u U(s, Y n ) + DU(s, Y n )D u Y n ]φ m ( Y H ) for all s [, t]. Since Y n H Y H and D u Y n H D u Y H, we have + T t s BU(s, Y n )φ m( Y H )D u Y H, (4.7) D u g n (s) H C m sup D u U(s, x) H + C m sup DU(s, x) L(H) D u Y H x H m+ x H m+ + C m sup U(s, x) H D u Y H (4.8) x H m+ Applying Theorem 2.2 (i), (ii), Theorem 2.3 (ii) and the Dominated Convergence Theorem again, we conclude that [ T T ] E D u g n (s) D u g(s) 2 n L 2 (K,H) du ds =. (4.9) For a given process v, recall the following notations from [O-P]: (D + v) u = s u+ D uv(s), (D v) u = s u D uv(s) ( v) u = (D + v) u + (D v) u We now find the expressions ( g n ) u and ( g) u. Replacing x by Y n in (2.24), we obtain s D u U(s, Y n ) = D u V (s, )(Y n ) + D u V (s l, θ(l, ))(F (U(l, Y n, ))) dl s ) + (V (s l, θ(l, )) + T s l (DF (U(l, Y n, )))(D u U(l, Y n, )) dl (4.) By (2.22), we have (D + V ) u = s u+ D uv (s, ) = BV (u, ) + BT u

22 22 S.-E.A. MOHAMMED AND T.S. ZHANG a.s.. Similarly, we obtain (D + V l (θ(l, ω))) u = s u+ D uv (s l, θ(l, ω)) for a.a. ω Ω. Thus, it follows from (4.) that = BV (u l, θ(l, ω)) + BT u l u (D + U) u (Y n ) = [BV u + BT u ](Y n ) + [BV (u l, θ(l, )) + BT u l ](F (U(l, Y n, ))) dl u ) + (V (u l, θ(l, )) + T u l (DF (U(l, Y n, )))(D u U(l, Y n, )) dl (4.) a.s.. Now taking its as s u+ in (4.7), we get (D + g n ) u = T t u B[(D + U) u (Y n ) + DU(u, Y n )D u Y n ]φ m ( Y H ) + T t u BU(u, Y n )φ m( Y H )D u Y H. (4.2) Note that D u U(s, Y n ) = when u > s. Therefore, letting s u in (4.7) gives (D g n ) u = T t u B[DU(u, Y n )D u Y n ]φ m ( Y H ) + T t u BU(u, Y n )φ m( Y H )D u Y H. (4.3) Because of the continuity of the functions involved, is easy to see from (4.2) and (4.3) that where (D + g) u and (D g) u are given by ( g n) u = [(D +g n ) u + (D g n ) u ] n n = ( g) u = (D + g) u + (D g) u, (4.4) (D + g) u = T t u B[(D + U) u (Y ) + DU(u, Y )D u Y ]φ m ( Y H ) + T t u BU(u, Y )φ m ( Y H)D u Y H, and (D g) u = T t u B[DU(u, Y )D u Y ]φ m ( Y H ) + T t u BU(u, Y )φ m( Y H )D u Y H. Now, (4.9) implies that n g n (s)dw (s) = g(s) dw (s) in probability, where the stochastic integral is the Skorohod integral. Therefore, (4.6) (and (4.5)) will hold, and hence the theorem, if we can show that g n (s) dw (s) = g n (s)dw (s) + 2 ( g n ) s ds, n, (4.5 )

23 and THE SUBSTITUTION THEOREM FOR SPDE S 23 g(s) dw (s) = g(s)dw (s) + 2 ( g) s ds (4.5) a.s.. We will prove (4.5). The proof of (4.5) is very similar. It seems difficult to verify the known sufficient conditions in the literature for proving (4.5) (cf. [Nu.] and [Nu.2]). Instead, we will prove (4.5) from first principles, using approximations by Riemann sums. Following [Nu.], choose any partition π = {t = < t <... < t n < t n = t} of [, t], with mesh π, and introduce the following step process: g π (r) = Consider the Riemann sums: S π := n i= n i= + + ( i+ ( i+ ) g(s)ds I (ti,+ ](r), From the definition of the Stratonovich integral, it follows that π Sπ = r [, t]. ) g(s) ds (W (+ ) W ( )). g(s) dw (s) whenever the above in probability exists. On the other hand, by (3.4) in [Nu.], we have S π = n g π (s)dw (s) + Since π gπ = g in L,2 (see [Nu.]), then π i= + g π (s) dw (s) = i+ i+ g(s) dw (s) D u g(s) du ds in probability. So, to complete the proof of (4.5), it remains to show that n π i= To simplify the notation, set Split I π into two integrals, I π = n i= + + I π := i+ n i= i+ i+ + D u g(s) du ds = i+ i+ u n du D u g(s) ds + i= D u g(s) du ds + i+ ( g) s ds. i+ du D u g(s) ds. u

24 24 S.-E.A. MOHAMMED AND T.S. ZHANG Denote the first and second term on the right-hand-side of the above equality by II π and III π, respectively. Write, II π = We will prove that n i= + n + i= i+ + du i+ u du T t s B[DU(s, Y )D u Y ]φ m ( Y H ) ds u T t s BU(s, Y )φ m ( Y H)D u Y H ds. n π i= + i+ du u = 2 T t s B[DU(s, Y )D u Y ]φ m ( Y H ) ds T t u B[DU(u, Y )D u Y ]φ m ( Y H ) du (4.7) and n π i= + i+ du u = 2 T t s BU(s, Y )φ m ( Y H)D u Y H ds T t u BU(u, Y )φ m ( Y H)D u Y H du. (4.8) We will prove (4.7). The proof of (4.8) is very similar. Rewrite the left-hand-side of (4.7) in the form n i= n = + i= + n + i= i+ du i+ + u du T t s B[DU(s, Y )D u Y ]φ m ( Y H ) ds u i+ Since the sequence of functions {T t s B[DU(s, Y )D u Y ]φ m ( Y H ) T t u B[DU(u, Y )D u Y ]φ m ( Y H )} ds (u )T t u B[DU(u, Y )D u Y ]φ m ( Y H ) du. (4.9) [, t] u n i= + (u )I (ti,+ ](u) R, n, converges weakly to the constant function 2 in L2 ([, t], R), then n π i= + i+ (u )T t u B[DU(u, Y )D u Y ]φ m ( Y H ) du = 2 T t u B[DU(u, Y )D u Y ]φ m ( Y H ) du. (4.2)

25 THE SUBSTITUTION THEOREM FOR SPDE S 25 For a given partition π := {t = < t <... < t n < t n = t} of [, t], and any u (, + ], denote u π :=, u π+ := +. We now estimate the first term of (4.9) as follows n i= + n = i= i+ i+ sup u π s u du u {T t s B[DU(s, Y )D u Y ]φ m ( Y H ) T t u B[DU(u, Y )D u Y ]φ m ( Y H )}ds sup { T t s B[DU(s, Y )D u Y ]φ m ( Y H ) T t u B[DU(u, Y )D u Y ]φ m ( Y H ) } du u π s u { T t s B[DU(s, Y )D u Y ]φ m ( Y H ) T t u B[DU(u, Y )D u Y ]φ m ( Y H ) } du. (4.2) By the continuity of T t s BDU(s, Y ) in s [, t], we see that sup π u π s u { T t s B[DU(s, Y )D u Y ]φ m ( Y H ) T t u B[DU(u, Y )D u Y ]φ m ( Y H ) } = for any fixed u (, + ], i n. On the other hand, sup { T t s B[DU(s, Y )D u Y ]φ m ( Y H ) T t u B[DU(u, Y )D u Y ]φ m ( Y H ) } u π s u 2 sup T t s B[DU(s, Y ) D u Y φ m ( Y H ) s t a.s. for all u [, t]. Applying the Dominated Convergence Theorem, we see that the right side of (4.2) tends to zero as π tends to. Thus, (4.7) follows from (4.9) and (4.2). This gives the a.s. it π IIπ = 2 To treat III π, write in the form III π = n i= We will prove that n π i= + n + i= + i+ + i+ = 2 T t u BDU(u, Y )φ m( Y H )D u Y H du + 2 T t u BU(u, Y )φ m ( Y H)D u Y H du. (4.22) i+ du T t s B[D u U(s, Y ) + DU(s, Y )D u Y ]φ m ( Y H ) ds u i+ i+ du T t s BU(s, Y )φ m( Y H )D u Y H ds. (4.23) u i+ du T t s B[D u U(s, Y ) + DU(s, Y )D u Y ]φ m ( Y H ) ds u T t u B[(D + U) u (Y ) + DU(u, Y )D u Y ]φ m ( Y H )du (4.24)

26 26 S.-E.A. MOHAMMED AND T.S. ZHANG and n π i= + i+ = 2 i+ du T t s BU(s, Y )φ m( Y H )D u Y H ds u T t u BU(u, Y )φ m( Y H )D u Y H du. (4.25) The proof of (4.25) is similar to that of (4.24). proving (4.24). To do this, consider We will complete the proof of the theorem by n i= + i+ i+ du T t s B[D u U(s, Y ) + DU(s, Y )D u Y ]φ m ( Y H ) ds u = J π + J π 2, where J π := n i= + i+ [ i+ du {T t s B[D u U(s, Y ) + DU(s, Y )D u Y ]φ m ( Y H ) u ] T t u B[(D + U) u (Y ) + DU(u, Y )D u Y ]φ m ( Y H )}ds and J π 2 n := i= + i+ (+ u)t t u B[(D + U) u (Y ) + DU(u, Y )D u Y ]φ m ( Y H ) du. As noted before, the sequence of functions [, t] u n i= + (+ u)i (ti,+ ](u) R converges weakly to the constant function 2 in L2 ([, t], R). Therefore, π J π 2 = 2 T t u B[(D + U) u (Y ) + DU(u, Y )D u Y ]φ m ( Y H )du. (4.26) Now we show that J π tends to zero as π. First note that ( J π sup { T t s B[D u U(s, Y ) + DU(s, Y )D u Y ]φ m ( Y H ) u s u π+ ) T t u B[(D + U) u (Y ) + DU(u, Y )D u Y ]φ m ( Y H ) } du. (4.27)

27 THE SUBSTITUTION THEOREM FOR SPDE S 27 Furthermore, there is a positive random constant C such that sup { T t s B[D u U(s, Y ) + DU(s, Y )D u Y ]φ m ( Y H ) u s u π+ T t u B[(D + U) u (Y ) + DU(u, Y )D u Y ]φ m ( Y H ) } 2 sup { T t s B[D u U(s, Y ) + DU(s, Y )D u Y ]φ m ( Y H ) } u s t C[ sup u s t D u U(s, Y ) + D u Y ] (4.28) a.s.. Let ĥ(u, a) := sup u s a D u U(s, Y ). Replacing x by Y (ω) in (2.24), there is a positive random constant c such that ( ĥ(u, a) c Y sup u s t D u V (s, ) + + c sup u s t a ) D u V (s l, θ(l, )) dl sup l s a { V (s l, θ(l, )) + T s l }ĥ(u, l) dl a.s. for all a [, t]. By Gronwall s inequality, the above inequality implies ( ĥ(u, t) c Y sup u s t D u V (s, ) + ) sup u s t D u V (s l, θ(l, )) dl ( exp Using the above estimate and Theorem 2.3 (i), is easy to see that ) sup { V (s l, θ(l, )) + T s l } dl. l s t (ĥ(u, t))2 du <. (4.29) By the definition of (D + U), the integrand in (4.27) approaches zero as π for any fixed u. Applying the Dominated Convergence Theorem, it follows from (4.28) and (4.29) that J π =. (4.3) π This together with (4.26) implies (4.24). The proof of equality (4.5) is now complete. 5. Alternative proof of Theorem 2.3 (ii). In this section we give an alternative proof of the estimate in Theorem 2.3 (ii). This proof is based on a chaos-type expansion in the Hilbert space L 2 (H). The argument we presens of independennterest. Proof of Theorem 2.3 (ii). In the see (.), assume Hypotheses (B) and (A ) or (A 2 ). Suppose F is C b. In this proof, C will denote a generic positive constant which may change from line to line.

28 28 S.-E.A. MOHAMMED AND T.S. ZHANG Recall the equation D u U(t, x, ) = D u V (t, )(x) + + for u [, a], a [, t], t. Fix any p. Set D u V (t s, θ(s, ))(F (U(s, x, ))) ds [V (t s, θ(s, )) + T t s ](DF (U(s, x, )))(D u U(s, x, )) ds, (2.24) D u U(t, x) H h(u, t) := sup x H ( + x H ), t. From (2.24) it follows that there are positive constants M, C such that Define and h(u, t) M D u V (t, ) L2 (H) + C + C g(u, t) := M D u V (t, ) L2 (H) + C D u V (t s, θ(s, )) L2 (H)ds ( ) V (t s, θ(s, )) L2 (H) + h(u, s)ds, t. (5.) D u V (t s, θ(s, )) L2 (H)ds L(s, t) := V (t s, θ(s, )) L2 (H) + s t. Iterating the inequality (5.) n times, we obtain n s sk h(u, t) g(u, t) + C k L(s, t) ds L(s 2, s ) ds 2 L(s k, s k )g(u, s k ) ds k + R n+, where s R n+ := C n+ L(s, t)ds (n + )! Cn+ almost surely as n. This implies that h(u, t) g(u, t) + ( sn L(s 2, s )ds 2 L(s n+, s n )h(u, s n+ ) ds n+ n+ sup L(u 2, u )) sup h(u, s) u 2 u t u s t s C k L(s, t)ds sk L(s 2, s )ds 2 L(s k, s k )g(u, s k )ds k (5.2) Next we estimate E[(h(u, t) 2p ]. First observe that [( s sk ) 2p ] E L(s, t) ds L(s 2, s ) ds 2 L(s k, s k )g(u, s k )ds k ( ) 2p ds k...ds <s k <...<s <t E[L(s, t) 2p L(s 2, s ) 2p L(s k, s k ) 2p g(u, s k ) 2p ] ds k ds. <s k <...<s <t (5.3)

29 THE SUBSTITUTION THEOREM FOR SPDE S 29 Since L(s i, s i ) is measurable with respect to the σ-algebra F si,s i := σ{w (l) W (s i ) : l [s i, s i ]} and W has independenncrements, it follows that the random variables L(s, t) 2p, L(s 2, s ) 2p,, L(s k, s k ) 2p, g(u, s k ) 2p are independent for < s k <... < s < t. Hence, E[L(s, t) 2p L(s 2, s ) 2p L(s k, s k ) 2p g(u, s k ) 2p ] Therefore, (5.3) gives where = E[L(s, t) 2p ]E[L(s 2, s ) 2p ] E[L(s k, s k ) 2p ]E[g(u, s k ) 2p ]. ( s sk ) 2p E L(s, t)ds L(s 2, s )ds 2 L(s k, s k )g(u, s k )ds k tk t k M k t E[g(u, s) 2p ]ds, (5.4) k! (k )! M t := Combining (5.2) with (5.4), we arrive at (E[(h(u, t) 2p ]) 2p (E[(g(u, t) 2p ]) 2p sup E[L(u 2, u ) 2p ]. u 2 u t + ( t C k k t k M k ) ( 2p t ) t E[g(u, s) 2p 2p ] ds, t. k! (k )! Hence, [ E sup x H DU(t, x) 2p ] H ( + x 2p H ) C E[h(u, t) 2p ] du { C E[g(u, t) 2p ] du + } E[g(u, s) 2p ] du ds, t. (5.5) We show now that the right side of (5.5) is finite. Is easy to see that V (t s, θ(s, )) = T t l BV (l s, θ(s, )) dw (l) + s s Thus D u V (t s, θ(s, )) = for u [s, t]; and for u [s, t], D u V (t s, θ(s, )) = s T t l BD u V (l s, θ(s, )) dw (l) T t l BT l s dw (l). + T t u BV (u s, θ(s, )) + T t u BT u s.

30 3 S.-E.A. MOHAMMED AND T.S. ZHANG By the Itô isometry we get s E[ D u V t s (θ(s, )) 2p L 2 (H) ]du C This implies that, for any T >, s s +C l s E[ D u V (t s, θ(s, )) 2p L 2 (H) ]du C + C and all t [, T ]. By Grownwall s inequality, we have for all t [, T ]. Now, E[g(u, t) 2p ] du C s s E[ T t l BD u V (l s, θ(s, )) 2p L 2 (H)] du dl E[ T t u BV (u s, θ(s, )) 2p L 2 (H) ] du s + C l s s E[ T t u BT u s 2p L 2 (H) ] du. E[ D u V (l s, θ(s, )) 2p L 2 (H)] du dl E[ D u V (t s, θ(s, )) 2p L 2 (H) ] C (5.6) E[ D u V (t, ) 2p L 2 (H) ]du + C E[ D u V (t s, θ(s, )) 2p L 2 (H)] ds du. So it follows from Theorem 2.3(i) and (5.6) that the right side of (5.5) is finite, which completes the proof of (2.2). 6. Anticipating semilinear spde s. As a corollary of Theorem., we show existence and regularity of solutions to a semilinear Stratonovich see with anticipating initial conditions. The proof is essentially a reformulation of the corresponding argument for Theorem.. Is not clear whether the solution of (6.) is unique. Corollary 6.. Assume that Conditions (B) together with either (A ) or (A 2 ). Suppose F is C b and let Y D,4 (Ω, H) be a random variable. Consider the following anticipating semilinear see dv(t) = Av(t) dt + F ( v(t) ) } dt + Bv(t) dw (t), t >, v() = Y. Then the anticipating semilinear see (6.) has a pathwise continuous (B(R + ) F, B(H))-measurable mild solution v : R + Ω H with the following properties: (i) v(t) D,2 (Ω, H) for all t. (ii) (iii) sup E Dv(t) 2 H < for all a (, ). t [,a] sup t [,a] v(t, ω) H K(ω)[ + Y (ω) H ] for a.a. ω Ω, where K is a random positive constant such that K L 2p (Ω, R + ) for all integers p. (6.)

31 Proof. THE SUBSTITUTION THEOREM FOR SPDE S 3 Assume all the conditions of the corollary. In (6.), replace the initial condition Y by a deterministic vector x H. Then with this replacement, (6.) is equivalent to the semilinear Itô see Set du(t, x) = Au(t, x) dt + F ( u(t, x) ) dt + 2 u(, x) = x. BkU(t, 2 x) dt + Bu(t, x)dw (t), t >, (6.2) F (u) := F (u) Bku 2 for all u H. By Condition (B), is easy to see that F : H H is Cb. Therefore the (adapted) see (6.2) satisfies all the requirements of Theorem.. In particular, its mild solutions generate a C cocycle U : R + H Ω H. Moreover, the cocycle U satisfies all the estimates in Section 2 (Theorem 2.2 (i),(ii), Theorem 2.3 (ii)). Now set v(t, ω) := U (t, Y (ω), ω) for all t, ω Ω. Using the substitution theorem is not hard to check that v is a mild solution of (6.) which satisfies all the estimates in Corollary 6.. Remark. A similar result for anticipating stochastic ordinary differential equations in a Hilbert space H is given in ([G-Nu-S], Theorem 4.4) under the restriction that the initial random variable takes values in some relatively compact sen H. Cf. also [A-I] where the substituting random variable takes values in a σ-compact space.

32 32 S.-E.A. MOHAMMED AND T.S. ZHANG REFERENCES [A-I] [D-Z.] [D-Z.2] [F.] Arnold, L., and Imkeller, P., Stratonovich calculus with spatial parameters and anticipative problems in multiplicative ergodic theory, Stochastic Processes and their Applications, Vol. 62 (996), Da Prato, G., and Zabczyk, J., Stochastic Equations in Infinite Dimensions, Cambridge University Press (992). Da Prato, G., and Zabczyk, J., Ergodicity for Infinite Dimensional Systems, Cambridge University Press (996). Flandoli, F., Regularity theory and stochastic flows for parabolic SPDE s, Stochastics Monographs, 9, Gordon and Breach Science Publishers, Yverdon, 995. [F.2] Flandoli, F., Stochastic flows for nonlinear second-order parabolic SPDE, Ann. Probab. 24 (996), no. 2, [G-Nu-S] [Ma] [Mo.] [Mo.2] [M-S.] [M-S.2] [M-Z-Z] [Nu.] Grorud, A., Nualart, D., and Sanz-Solé, M., Hilbert-valued anticipating stochastic differential equations, Annales de l institut Henri Poincaré (B) Probabilits et Statistiques, 3 no. (994), Malliavin, P., Stochastic calculus of variations and hypoelliptic operators, Proceedings of the International Conference on Stochastic Differential Equations, Kyoto, Kinokuniya, 976, Mohammed, S.-E.A., Stochastic Functional Differential Equations, Research Notes in Mathematics, no. 99, Pitman Advanced Publishing Program, Boston-London-Melbourne (984). Mohammed, S.-E. A., Non-Linear Flows for Linear Stochastic Delay Equations, Stochastics, Vol. l7 #3, (987), Mohammed, S.-E. A., and Scheutzow, M. K. R., The Stable Manifold Theorem for Nonlinear Stochastic Systems with Memory, Part I: Existence of the Semiflow, Journal of Functional Analysis, 25, (23), Part II: The Local Stable Manifold Theorem, Journal of Functional Analysis, 26, (24), Mohammed, S.-E. A., and Scheutzow, M. K. R., The stable manifold theorem for stochastic differential equations, The Annals of Probability, Vol. 27, No. 2, (999), Mohammed, S.-E. A., Zhang, T. S. and Zhao, H. Z., The stable manifold theorem for semilinear stochastic evolution equations and stochastic partial differential equations, Part : The Stochastic semiflow, Part 2: Existence of stable and unstable manifolds, pp. 98 (26), Memoirs of the American Mathematical Society (to appear). Nualart, D., The Malliavin Calculus and Related Topics, Probability and its Applications, Springer-Verlag (995). [Nu.2] Nualart, D., Analysis on Wiener space and anticipating stochastic calculus, Springer LNM, 69, Ecole d Et e de Probabilit es de Saint-Flour XXV-995, ed: P. Bernard (995).

33 THE SUBSTITUTION THEOREM FOR SPDE S 33 [N-P] [O-P] [T] Nualart, D., and Pardoux, E., Stochastic calculus with anticipating integrands, Analysis on Wiener space and anticipating stochastic calculus, Probab. Th. Rel. Fields, 78 (988), Ocone, D., and Pardoux, E., A generalized Itô-Ventzell formula. Application to a class of anticipating stochastic differential equations. Ann. Inst. Henri Poincaré, Probabilités et Statistiques 25: (989) Taylor, M. E., Partial Differential Equations II, Qualitative Studies of Linear Equations, Applied Mathematical Sciences 6, Springer (996). Salah-Eldin A. Mohammed Department of Mathematics Southern Illinois University at Carbondale Carbondale, Illinois salah@sfde.math.siu.edu Web page: Tusheng Zhang Department of Mathematics University of Manchester Oxford Road, Manchester M3 9PL UK., and CMA, University of Oslo P.O.Box 53 Blindern N-36 Oslo, Norway tzhang@math.man.ac.uk