Weak convergence rates of spectral Galerkin approximations for SPDEs with nonlinear diffusion coefficients

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1 Weak convergence rates of spectral Galerkin approximations for SPDEs with nonlinear diffusion coefficients Daniel Conus, Arnulf Jentzen, and Ryan Kurniawan Lehigh University, USA and EH Zurich, Switzerland arxiv: v math.pr 6 Sep 17 September 1, 18 Abstract Strong convergence rates fortemporal, spatial, and noise numerical approximations of semilinear stochastic evolution equations SEEs with smooth and regular nonlinearities are well understood in the scientific literature. Weak convergence rates for numerical approximations of such SEEs have been investigated for about two decades and are far away from being well understood: roughly speaking, no essentially sharp weak convergence rates are known for parabolic SEEs with nonlinear diffusion coefficient functions; see Remark.3 in A. Debussche, Weak approximation of stochastic partial differential equations: the nonlinear case, Math. Comp. 8 11, no. 73, for details. n this article we solve the weak convergence problem emerged from Debussche s article in the case of spectral Galerkin approximations and establish essentially sharp weak convergence rates for spatial spectral Galerkin approximations of semilinear SEEs with nonlinear diffusion coefficient functions. Our solution to the weak convergence problem does not use Malliavin calculus. Rather, key ingredients in our solution to the weak convergence problem emerged from Debussche s article are the use of appropriately modified versions of the spatial Galerkin approximation processes and applications of a mild tô type formula for solutions and numerical approximations of semilinear SEEs. his article solves the weak convergence problem emerged from Debussche s article merely in the case of spatial spectral Galerkin approximations instead of other more complicated numerical approximations. Our method of proof extends, however, to a number of other kinds of spatial and temporal numerical approximations for semilinear SEEs. Contents 1 ntroduction 1.1 Sketch of the proof of the main weak convergence result Examples Parabolic Anderson model and nonlinear heat-type SPDEs A Cahn-Hilliard-Cook type equation Notation Setting Auxiliary lemmas

2 Weak convergence for Galerkin projections of SEEs 1.1 Setting A weak convergence result Weak convergence for Galerkin approximations of SEEs with mollified nonlinearities Regularity properties for solutions of infinite dimensional Kolmogorov equations in Hilbert spaces Setting Weak convergence results Strong convergence of mollified solutions for SEEs Setting A strong convergence result Weak convergence for Galerkin approximations of SEEs Setting A weak convergence result Weak convergence rates for SEEs 51 7 Lower bounds for the weak error of Galerkin approximations for SEEs Setting Lower bounds for the weak error ntroduction Both strong and numerically weak convergence rates for numerical approximations of finite dimensional stochastic ordinary differential equations SODEs with smooth and regular nonlinearities are well understood in the literature; see, e.g., the monographs Kloeden & Platen 31 and Milstein 37. he situation is different in the case of possibly infinite dimensional semilinear stochastic evoluation equations SEEs. While strong convergence rates for temporal, spatial, and noise numerical approximations of semilinear SEEs with smooth and regular nonlinearities are well understood in the scientific literature, weak convergence rates for numerical approximations of such SEEs have been investigated since about 14 years ago and are far away from being well understood: roughly speaking, no essentially sharp weak convergence rates are known for parabolic SEEs with nonlinear diffusion coefficient functions see Remark.3 in Debussche 19 for details. n this article we solve the weak convergence problem emerged from Debussche s article in the case of spectral Galerkin approximations and establish essentially sharp weak convergence rates for spatial spectral Galerkin approximations of semilinear SEEs with nonlinear diffusion coefficient functions. o illustrate the weak convergence problem emerged from Debussche s article and our solution to the problem we consider the following setting as a special case of our general setting in Section 5 below. Let H,, H, H and U,, U, U be separable R-Hilbert spaces, let,, let Ω,F,P,F t t, be a stochastic basis, let W t t, be an d U - cylindrical Ω,F,P,F t t, -Wiener process, let e n n N H be an orthonormal basis of H, let λ n n N, be an increasing sequence, let A: DA H H be n N λ n e n,v H < } and a closed linear operator such that DA = {v H: n N: Ae n = λ n e n, let H r,, Hr, Hr, r R, be a family of interpolation spaces associated to A cf., e.g., 44, Section 3.7, let ι, 1 /4, ξ H ι, γ, 1 /,

3 and let F r<ι γ C 4 b H ι,h r, B r<ι γ/c 4 b H ι,hsu,h r. For two R-Hilbert spaces V 1,, V1, V1 and V,, V, V we denote by C 4 b V 1,V the R-vector space of all four times continuously Fréchet differentiable functions from V 1 to V with globally bounded derivatives and by HSV 1,V the R-Hilbert space of all Hilbert-Schmidt operators from V 1 to V. We also note that the hypothesis that H r,, Hr, Hr, r R, is a family of interpolation spaces associated to A ensures for all r, that H r = D A r and H 1 = DA H = H. he above assumptions imply cf., e.g., Da Prato et al. 14, Proposition 3, Brzeźniak 8, heorem 4.3, Van Neerven et al. 47, heorem 6. the existence of a continuous mild solution process X:, Ω H ι of the SEE dx t = AX t +FX t dt+bx t dw t, t,, X = ξ. 1 As an example for 1, we think of H = U = L,1;R being the R-Hilbert space of equivalence classes of Lebesgue-Borel square integrable functions from, 1 to R and A being an appropriate linear differential operator on H. n particular, in Subsection 1..1 we formulate the continuous version of the one-dimensional parabolic Anderson model as an example for 1 in this example the parameter γ, which controls the regularity of the operators F and B, satisfies γ = 1 / and in Subsection 1.. we formulate a fourth-order stochastic partial differential equation as an example for 1 in this second example the parameter γ satisfies γ = 1 /4. Strong convergence rates for temporal, spatial, and noise numerical approximations for SEEs of the form 1 are well understood. Weak convergence rates for numerical approximations of SEEs of the form 1 have been investigated for about two decades; cf., e.g., 45, 4, 18,,, 5, 19, 3, 1, 35, 36, 33, 5, 34, 6, 48, 4, 5, 7, 49. Except for Debussche & De Bouard 18, Debussche 19, and Andersson & Larsson 5, all of the above mentioned references assume, beside further assumptions, that the considered SEE is driven by additive noise. n Debussche & De Bouard 18 weak convergence rates for the nonlinear Schrödinger equation, which dominant linear operator generates a group see 18, Section instead of only a semigroup as in the general setting of the SEE 1, are analyzed. he method of proof in Debussche & De Bouard 18 strongly exploits this property of the nonlinear Schrödinger equation see 18, Section 5.. herefore, the method of proof in 18 can, in general, not be used to establish weak convergence rates for the SEE 1. n Debussche s seminal article 19 see also Andersson & Larsson 5, essentially sharp weak convergence rates for SEEs of the form1 are established under the hypothesis that the second derivative of the diffusion coefficient B satisfies the smoothing property that there exists a real number L, such that for all x,v,w H it holds that 1 B xv,w LH L v H 1/4 w H 1/4. As pointed out in Remark.3 in Debussche 19, assumption is a serious restriction for SEEs of the form 1. Roughly speaking, assumption imposes that the second derivative of the diffusion coefficient function vanishes and thus that the diffusion coefficient function is affine linear. Remark.3 in Debussche 19 also asserts that assumption is crucial in the weak convergence proof in 19, that assumption is used in an essential way in Lemma 4.5 in 19, and that Lemma 4.5 in 19, in turn, is used at many points in the weak convergence proof in 19. o the best of our knowledge, it remained an open problem to establish essentially sharp weak convergence rates for any type of temporal, spatial, or noise numerical approximation of the SEE 1 without imposing Debussche s assumption. n this article we solve this problem in the case of spatial 1 Assumption above slightly differs from the original assumption in 19 as we believe that there is a small typo in equation.5 in 19; see inequality 4.3 in the proof of Lemma 4.5 in 19 for details. 3

4 spectral Galerkin approximations for the SEE 1. his is the subject of the following theorem heorem 1.1, which follows immediately from Corollary 6.1 below. heorem 1.1. Assume the setting in the first paragraph of Section 1, let ϕ Cb 4H ι,r, let P N N N LH 1 satisfy for all N N, v H that P N v = N n=1 e n,v H e n, and for every N N let X N :, Ω P N H be a continuous mild solution of the SEE dx N t = P N AX N t +P N FX N t dt+p N BX N t dw t, t,, X N = P Nξ. 3 hen for every ε, there exists a real number C ε, such that for all N N it holds that E ϕx E ϕx N Cε λ N 1 γ ε. 4 Let us add a few comments regarding heorem 1.1. First, we would like to emphasize that in the general setting of heorem 1.1, the weak convergence rate established in heorem 1.1 can essentially not be improved. More specifically, in Corollary 7.5 in Section 7 below we give for every ι, 1 /4 and every γ, 1 / examples of A: DA H H, ξ H ι, F r<ι γ Cb 4H ι,h r, U,, U, U, B r<ι γ/cb 4H ι,hsu,h r, and ϕ Cb 4H ι,r such that there exists a real number C, such that for all N N it holds that E ϕx E ϕx N C λn 1 γ. 5 naddition, weemphasize thatinthesetting ofheorem 1.1it iswell known cf., e.g., Cox et al. 11, Corollary 3.3 that for every ε, there exists a real number C ε, such that for all N N it holds that E X X N H ι 1/ C ε λ N 1 γ ε. 6 he weak convergence rate 1 γ ε established in heorem 1.1 is thus twice the wellknown strong convergence rate 1 γ ε in 6. Moreover, heorem 1.1 is to the best of our knowledge the first result in the scientific literature which establishes an essentially sharp weak convergence rate for numerical approximations of the continuous version of the one-dimensional parabolic Anderson model see Subsection 1..1 for details. We also would like to point out that the weak convergence result in heorem. in Debussche 19 assumes that holds see.5 in 19, that B maps from H to LH instead of from H to HSU,H r for r, 1 /, and that ϕ, F, and B are three times continuously Fréchet differentiable with globally bounded derivativesinstead of four times continuously Fréchet differentiable as in heorem 1.1 above but restricts to the irregular case γ = 1/ in the above framework. he weak convergence result in heorem 1.1 above does not assume and does assume that ϕ, F, and B are four times continuously Fréchet differentiable but also establishes essentially sharp weak convergence rates in the more regular cases γ, 1 / such as in several cases of trace class noise. n the very regular case of finite dimensional SEEs it is typically assumed that F and B and ϕ are four times continuously differentiable cf., e.g., Kloeden & Platen 31, heorem Next weaddthatthe proofof heorem 1.1caninastraightforwardway beextended tothecase where ϕ has at most polynomially growing derivatives. t is, however, not clear to us how to treat the case where F and B are globally Lipschitz continuous but with the first four derivatives growing polynomially. Furthermore, we emphasize that heorem 1.1 solves the weak convergence problem emerged from Debussche s article see.5 and Remark.3 in Debussche 19 merely in the case of spatial spectral Galerkin approximations instead of other more complicated numerical approximations for the SEE 1. he method of proof of our weak convergence results, however, can be extended to a number of other kind of spatial and temporal numerical approximations for SEEs of the form 1. n particular, in our proceeding article 9 we extend the method of proof developed here to 4

5 establish essentially sharp weak convergence rates for different types of temporal numerical approximations such as exponential Euler {see 9, Subsection 1.5.1} and linear-implicit Euler {see 9, Subsection 1.5.} approximations for SPDEs for SPDEs with possibly non-constant diffusion coefficients without neither assuming nor that B maps from H to LH. Next we point out that the proof in Debussche s article 19 as well as many other proofs in the above mentioned weak convergence articles use Malliavin calculus. Our method of proof does not use Malliavin calculus but uses in some sense merely elementary arguments as well as the mild tô formula in Da Prato et al. 14. he paper is organized as follows. n Section 1.1 below we give a rough sketch of the proof of heorem 1.1 without technical details. However, the main ideas that we use to obtain an essentially sharp rate of convergence are highlighted in Section 1.1 below. n Section 1. we illustrate heorem 1.1 and Corollary 6.1, respectively, by two simple examples. Sections 1.3 and 1.4 present the notation and the framework used in this paper. Section studies weak convergence rates for the spectral Galerkin projections P N X, N N, associated to the solution process X t, t,, of the SEE 1. he result of this section is then used in Section 3 to obtain the weak convergence of the Galerkin approximation 3 to the solution of 1 in the case where the drift operator F, the diffusion operator B, and the initial condition are mollified in an appropriate sense. his provides a less general version of heorem 1.1. Section 4 is devoted to the proof of an elementary strong convergence result. n Section 5 the weak convergence result from Section 3 and the elementary strong convergence result from Section 4 are used to establish weak convergence see Corollary 5.3 for general drift and diffusion operators. Section 6 specializes the weak convergence result from Section 5 to the framework of this introductory section. Finally, in Section 7 we consider the case F = and provide examples of constant additive noise functions B which show that the weak convergence rate established in heorem 1.1 can essentially not be improved. 1.1 Sketch of the proof of the main weak convergence result n the following we give a brief sketch of our method of proof of heorem 1.1 and Corollary 6.1, respectively, in the case where ξ H ι+ the case where ξ H ι then follows from a standard mollification procedure; see 39 in the proof of Proposition 5. in Section 5 for details. n our weak convergence proof we intend to work as it is often the case in the case of weak convergence for SPDEs; see, e.g., Rößler 43 and Debussche 19 with the Kolmogorov backward equation associated to 1. n the case of an SEE with a general nonlinear diffusion coefficient it is, however, not clear whether the solutions of the SEE 1 also provide strong solutions of the Kolmogorov backward equation associated to 1; cf. 1, item iv of heorem 1.1, 6, Corollary 1., and 13, pages We therefore work with suitable mollified versions of 1 and 3. More formally, for every κ, let F κ : H ι H ι+ and B κ : H ι HSU,H ι+ be the functions which satisfy for all x H ι that F κ x = e κa Fx and B κ x = e κa Bx. For every κ,, x H ι let ˆX x,κ :, Ω H ι be a continuous mild solution of the SEE d ˆX x,κ t = A ˆX x,κ t +F κ x,κ ˆX t x,κ dt+b κ ˆX t dw t, t,, ˆXx,κ = x. 7 For every κ, let u κ :, H ι R be the function which satisfies for all t,x, H ι that u κ t,x = E x,κ ϕ ˆX t. nparticular, notice that forall κ, and all nonrandom x H ι it holds that u κ,x = ϕx. hen, for every κ,, N N let X N,κ :, Ω H ι be a continuous mild solution of the SEE dx N,κ t = P N AX N,κ t +P N F κ X N,κ t dt+p N B κ X N,κ t dw t, t,, X N,κ = P N ξ. 8 5

6 he first key idea in our proof is then to bring certain modified versions of the SEEs 3 and 8 respectively into play to analyze the weak approximation errors E ϕ ˆXξ,κ E ϕx N,κ for N N, κ,. More specifically, for every κ,, N N let Y N,κ :, Ω H ι+ be a continuous mild solution of the SEE dy N,κ t = AY N,κ t +F κ PN Y N,κ t dt+b κ PN Y N,κ t dw t, t,, Y N,κ = ξ. 9 t is crucial in 9 that P N appears inside the arguments of F κ and B κ instead of in front of F κ and B κ as in 8 and 3. Moreover, notice the projection P N Y N,κ t = X N,κ t P-a.s. for all N N, κ,, t,. o estimate the weak approximation errors E ϕ ˆXξ,κ E ϕx N,κ for N N, κ, we then apply the triangle inequality to obtain that for all κ,, N N it holds that E ϕ ˆXξ,κ E ϕx N,κ E ϕ ˆXξ,κ E ϕy N,κ = uκ,ξ E u κ,y N,κ = E uκ,y N,κ u κ,y N,κ + E ϕy N,κ E ϕx N,κ + E ϕy N,κ + E ϕy N,κ E ϕp N Y N,κ E ϕp N Y N,κ. Roughly speaking, the processes Y N,κ, N N, κ,, are chosen in such a way so that it is not so difficult anymore to estimate E uκ,y N,κ u κ,y N,κ and E ϕy N,κ E ϕp N Y N,κ on theright hand side of 1. More formally, to estimate the term E ϕy N,κ E ϕp N Y N,κ on the right hand side of 1 see Section and Lemma 3.5 in Section 3 we apply the mild tô formula in Corollary in Da Prato et al. 14 to E ϕy N,κ t, t,, andto E ϕp N Y N,κ, t,, andthen estimate the difference of the resulting terms in a straightforward way see the proof of Proposition.1 in Section below for details. his allows us to prove see Proposition.1 below that there exist real numbers C ε 1,, ε,, such that for all ε,κ,, N N it holds that E ϕy N,κ E ϕx N,κ = E ϕy N,κ E ϕp N Y N,κ C 1 ε λ N 1 γ ε. 11 oestimatetheterm E u κ,y N,κ u κ,y N,κ ontherighthandsideof 1weapply the standard tô formula to the stochastic processes u κ t,y N,κ t, κ,, and t, use the fact that the functions u κ, κ,, solve the Kolmogorov backward equation associated to 7 to obtain that for all κ,, N N it holds that E uκ,y N,κ u κ,y N,κ + E x uκ s,ys N,κ B κp N Y N,κ s E x u κ +B κy N,κ s s,y N,κ b, s F κ P N Y N,κ s B κp N Y N,κ s B κy N,κ s 1 F κ Ys N,κ ds b ds 1 where U U is an arbitrary orthonormal basis of U; cf. 174 in Section 3 below. he next key idea in our weak convergence proof is then to again apply the mild tô formula see Da Prato et al. 14 to the terms appearing on the right hand side of 1. After applying the mild tô formula, the resulting terms can be estimated in a straightforward way by using the estimates for the functions u κ, κ,, from Andersson et al. 1. his allows us cf. 16 in Lemma 3.7 and 4 43 in the proof of Proposition 5. to prove that for all ε, there exists a real number C ε, such that for all κ,, N N it holds that E ϕ ˆXξ,κ E ϕy N,κ = E uκ,y N,κ u κ,y N,κ C ε. 13 κ ε λ N 1 γ ε 6

7 Putting 13 and 11 into 1 then proves that for all ε,, κ,, N N it holds that E ϕ ˆXξ,κ E ϕx N,κ C ε κ ε λ N 1 γ ε +C ε 1 λ N 1 γ ε. 14 Estimates 13 and 14 illustrate that we cannot simply let the mollifying parameter κ tend to because the right hand side of 14 diverges as κ tends to. he last key idea in our proof is then to make use of the following somehow nonstandard mollification procedure to overcome this problem. For this mollification procedure we first use wellknown strong convergence analysis to prove cf. Proposition 4.1 and Corollary 4. in Section 4 that for all ε, there exists a real number C ε 3, such that for all κ,, N N it holds that E ϕx E ξ,κ ϕ ˆX + E ϕx N E ϕx N,κ C 3 ε κ ε 1 γ. 15 Combining 15 with 14 then shows that for all ε,, κ,, N N it holds that E ϕx E ϕx N C ε 1 λ N + C 1 γ ε κ ε λ N ε +C3 1 γ ε ε κ 1 γ ε. 16 As the left hand side of 16 is independent of κ,, we can minimize the right hand side of 16 over κ, instead of letting κ tend to and this will allow us to complete the proof of heorem 1.1; see 44 and 46 in the proof of Proposition 5. in Section 5 below for details. 1. Examples n this section we illustrate heorem 1.1 and Corollary 6.1, respectively, by two simple examples. n Subsection 1..1 we apply heorem 1.1 to the continuous version of the onedimensional parabolic Anderson model and in Subsection 1.. we apply heorem 1.1 to a Cahn-Hilliard-Cook type equation Parabolic Anderson model and nonlinear heat-type SPDEs Let H = L,1;R be the R-Hilbert space of equivalence classes of Lebesgue-Borel square integrable functions from,1 to R, let,κ,δ,ν,, ξ H, let Ω,F,P, F t t, be a stochastic basis, let W t t, be an d H -cylindrical Ω,F,P,F t t, - Wiener process, let e n H, n N, be the orthonormal basis of H which satisfies for all n N that e n = sinnπ, let A: DA H H be the linear operator which satisfies DA = {v H: n=1 n4 e n,v H < } and v DA: Av = n=1 νn π e n,v H e n, let H r,, Hr, Hr, r R, be a family of interpolation spaces associated to A, let P N N N LH 1 satisfy for all N N, v H that P N v = N n=1 e n,v H e n, let ψ: H H be a four times continuously Fréchet differentiablefunctionwithgloballyboundedderivatives, andletb: H HSH,H 1/4 δ bethe function which satisfies for all v H and all uniformly continuous functions u:,1 R that Bvu = ψv u. he above assumptions ensure the existence of F t t, -adapted continuous stochastic processes X:, Ω H and X N :, Ω P N H, N N, which satisfy that for all N N, t, it holds P-a.s. that X t = e At ξ + e At s BX s dw s 17 7

8 and X N t = e At P N ξ + eat s P N BX N s dw s. n the case where v H: ψv = 1+ v H 1 v the stochastic process X is a mild solution process of X dx t x = ν t x X x t xdt X ty dy dw tx 18 with X t = X t 1 = and X x = ξx for x,1, t, and the stochastic processes X N, N N, are spatial spectral Galerkin approximations of 18. n the case where v H: ψv = κ v the stochastic process X is a mild solution process of the continuous version of the one-dimensional parabolic Anderson model dx t x = ν x X t xdt+κx t xdw t x 19 with X t = X t 1 = and X x = ξx for x,1, t, cf., e.g., Carmona & Molchanov 1 and the stochastic processes X N, N N, are spatial spectral Galerkin approximations of 19. heorem 1.1 and Corollary 6.1, respectively, apply here with γ = 1/, that is, heorem 1.1 and Corollary 6.1, respectively, ensure that for all ϕ Cb 4H,R, ε, it holds that there exists a real number C R such that for all N N it holds that E ϕx E ϕx N C N 1 ε. heorem 1.1 and Corollary 6.1, respectively, thus demonstrate that the spatial spectral Galerkin approximations X N, N N, of 17, 18, and 19, respectively, converge with rate1 ε tothestochastic process X of 17, 18, and19. o thebest ofour knowledge, heorem 1.1 and Corollary 6.1, respectively, are the first results in the scientific literature which establish essentially sharp weak convergence rates for numerical approximations of 18 and 19, respectively. 1.. A Cahn-Hilliard-Cook type equation Let H = L,1;R be the R-Hilbert space of equivalence classes of Lebesgue-Borel square integrable functions from,1 to R, let,κ,δ,, ξ H, let Ω,F,P, F t t, be a stochastic basis, let W t t, be an d H -cylindrical Ω,F,P,F t t, - Wiener process, let e n H, n N, be the orthonormal basis of H which satisfies for all n N that e = 1 and e n = cosnπ, let A: DA H H be the linear operator which satisfies DA = {v H: n N n8 e n,v H < } and v DA: Av = n= n π n 4 π 4 1 e n,v H e n, let H r,, Hr, Hr, r R, be a family of interpolation spaces associated to A, let P N N N LH 1 satisfy for all N N, v H that P N v = N n= e n,v H e n, and let F: H H 1/4 δ and B: H HSH,H 1/8 δ satisfy forall v H andall uniformly continuous functions u:,1 R that Fv = v and Bvu = κ v u. he above assumptions ensure the existence of F t t, -adapted continuous stochastic processes X:, Ω H and X N :, Ω P N H, N N, which satisfy that for all N N, t, it holds P-a.s. that X t = e At ξ + e At s FX s ds+ e At s BX s dw s 1 and Xt N = e At P N ξ+ eat s P N FXs Nds+ eat s P N BXs NdW s. he stochastic process X is thus a solution process of the Cahn-Hilliard-Cook type equation dx t x = 4 x 4 X t x x X t x dt+κx t xdw t x 8

9 with X t = X t 1 = X3 t = X 3 t 1 = and X x = ξx for x,1, t, and the stochastic processes X N, N N, are spatial spectral Galerkin approximations of. heorem 1.1 and Corollary 6.1, respectively, apply here with γ = 1 /4, that is, heorem 1.1 and Corollary 6.1, respectively, ensure that for all ϕ Cb 4 H,R, ε, it holds that there exists a real number C R such that for all N N it holds that E ϕx E ϕx N C N 3 ε. 3 heorem 1.1 and Corollary 6.1, respectively, thus demonstrate that the spatial spectral Galerkin approximations X N, N N, of converge with rate 3 ε to the solution process X of. o the best of our knowledge, heorem 1.1 and Corollary 6.1, respectively, are the first results in the scientific literature which establish essentially sharp weak convergence rates for numerical approximations of. 1.3 Notation hroughout this article the following notation is used. For every set S we denote by d S : S S the identity mapping on S. For every set S we denote by PS the power set of S. We denote by E r :,,, r,, the functions which satisfy for all r,, x, that E r x = x n Γr n 1/ n= generalized exponential Γnr+1 function; cf., e.g., Exercise 3inChapter 7inHenry 7, 1..1inChapter 1 ingorenflo et al. 3, and 16 in Andersson et al.. For all normed R-vector spaces E 1, E1 and E, E andevery nonnegative integer k N we denoteby Lip k E 1,E, Lip k E 1,E : C k E 1,E, the functions which satisfy for all f C k E 1,E that f Lip k E 1,E = fx fy E x y E1 : k = x,y E1, x y f k x f k y L k E1,E x,y E1, x y x y E1 : k N and f Lip k E 1,E = f E + k l= f Lip l E 1,E and we denote by Lipk E 1,E the set given by Lip k E 1,E = {f C k E 1,E : f Lip k E 1,E < }. For all normed R-vector spaces E 1, E1 and E, E and every natural number k N we denote by C k b E 1,E, C k b E 1,E : Ck E 1,E, the functions which satisfy for all f C k E 1,E that f C k b E 1,E = x E1 f k x L k E 1,E and f C k b E 1,E = f E + k l=1 f Cb le 1,E and we denote by Cb ke 1,E the set given by Cb ke 1,E = {f C k E 1,E : f C k b E 1,E < }. 1.4 Setting hroughout this article the following setting is frequently used. Consider the notation in Section 1.3, let H,, H, H and U,, U, U be separable R-Hilbert spaces, let,, let Ω,F,P,F t t, be a stochastic basis, let W t t, be an d U - cylindrical Ω,F,P,F t t, -Wiener process, let H H be a nonempty orthonormal basis, let λ: H R be a function satisfying b H λ b <, let A: DA H H be a linear operator which satisfies DA = {v H: b H λ b b,v H < } and v DA: Av = b H λ b b,v H b, let H r,, Hr, Hr, r R, be a family of interpolation spaces associated to A, and let P PH LH 1 satisfy for all v H, PH that P v = b b,v H b. 4 9

10 1.5 Auxiliary lemmas hroughout this article we frequently use the following well-known lemmas. Lemma 1.. Assume the setting in Section 1.4. hen it holds for all r,1 that t, ta r e At LH x, x r e x r e r 1. Lemma 1.3 See, e.g., Lemma. in Andersson et al.. Let V k, Vk, k {,1}, be separable R-Banach spaces with V 1 V continuously. hen BV 1 = {B PV 1 : A BV : B = A V 1 } BV. 5 Weak convergence for Galerkin projections of SEEs n this section we establish weak convergence rates for Galerkin projections of SEEs see Proposition.1 below. More specifically, in the framework of Section 1.4 we establish in Proposition.1 below an explicit upper bound for the weak approximation error E ϕx E ϕ P X, 6 where H is a set, where ϕ: H R is a twice continuously Fréchet differentiable function with globally bounded and globally Lipschitz continuous derivatives, and where X:, Ω H is a suitable mild solution process of the SEE 7. n this section the nonlinearities in the SEE 7 are not mollified and may take values in appropriate negative interpolation spaces. Proposition.1, in particular, proves inequality 11 in Section 1.1. n Corollary 3.8 in Section 3 below we will use Proposition.1 to establish weak convergence rates for Galerkin approximations of SEEs with mollified nonlinearities. n particular, in Section 3 we establish upper error bounds for the first summand on the right hand side of 1 see Lemma 3.7 in Subsection 3.3 below and we use these upper error bounds together with Proposition.1 in this section to obtain upper error bounds for the left hand side of 1. Proposition.1 is a slightly modified version of Corollary 8 in Da Prato et al Setting AssumethesettinginSection1.4andletϑ,1,F Lip H,H ϑ,b Lip H,HSU, H ϑ/, ϕ Lip H,R, ξ L 3 P F ;H. heaboveassumptionsensurethatthereexistsanup-to-modificationsuniquef t t, - predictablestochasticprocessx:, Ω H whichsatisfies t, X t L 3 P;H < and which satisfies that for all t, it holds P-a.s. that X t = e At ξ + e At s FX s ds+. A weak convergence result e At s BX s dw s. 7 Proposition.1. Assume the setting in Section.1 and let ρ,1 ϑ, PH. hen E ϕx E ϕ P X ϕ Lip H,R max { 1, t, E } X t 3 H 1 1 ρ ϑ F Lip + H,H ϑ + B Lip H,HSU,H ϑ/ 8 P ρ H\ LH,H ρ. 1 ρ ϑ 1

11 Proof. hroughout this proof let U U be an orthonormal basis of U and let B b CH,H ϑ/, b U, be the functions which satisfy for all b U, v H that B b v = Bvb. Next observe that for all t, it holds P-a.s. that P X t = e At P ξ + t eat s P FX s ds+ eat s P BX s dw s. he mild tô formula in Corollary in Da Prato et al. 14 hence yields that E ϕx E ϕp X = E ϕe A ξ E ϕe A P ξ E ϕ e A t X t e A t FX t E ϕ e A t P X t e A t P FX t dt E ϕ e A t X t e A t B b X t,e A t B b X t dt E ϕ e A t P X t e A t P B b X t,e A t P B b X t dt. Next observe that Lemma 1. implies that 9 E ϕe A ξ E ϕe A P ξ ϕ Lip H,RE ξ H PH\ LH,H ρ ρ. 3 nequality 3 provides us a bound for the first difference on the right hand side of 9. n the next step we bound the second difference on the right hand side of 9. For this observe that for all x H, t, it holds that ϕ e A t x ϕ e A t P x e A t Fx and ϕ Lip 1 H,R P H\ LH,H ρ x H Fx H ϑ t ρ+ϑ 31 ϕ e A t P x d H P e A t Fx ϕ Lip H,R P H\ LH,H ρ Fx H ϑ t ρ+ϑ. Combining 31 and 3 proves that E ϕ e A t X t e A t FX t dt 1 ρ ϑ P H\ LH,H ρ t, E X t H FX t H ϑ ϕ Lip 1H,R + FX t H ϑ ϕ Lip H,R 1 ρ ϑ 3 E ϕ e A t P X t e A t P FX t dt 1 ρ ϑ P H\ LH,H ρ ϕ Lip 1 H,R t, max{e X t H FX t H ϑ,e FX t H ϑ } 1 ρ ϑ 1 ρ ϑ P H\ LH,H ρ ϕ Lip 1 H,R F Lip H,H ϑ max{1, t, E X t H }. 1 ρ ϑ nequality 33providesusaboundfortheseconddifferenceontherighthandsideof 9. Next we bound the third difference on the right hand side of 9. o this end note that for all x H, t, it holds that ϕ e A t x ϕ e A t P x e A t B b x,e A t B b x ϕ Lip H,R Bx HSU,H ϑ/ x 34 H P H\ LH,H ρ t ρ+ϑ 11 33

12 and ϕ e A t P xd H +P e A t B b x,d H P e A t B b x ϕ Lip 1 H,R Bx HSU,H ϑ/ P H\ LH,H ρ t ρ+ϑ. 35 Combining 34 and 35 proves that 1 E ϕ e A t X t e A t B b X t,e A t B b X t dt 1 E ϕ e A t P X t e A t P B b X t,e A t P B b X t dt { } E X t H BX t,e BX HSU,H ϑ/ t HSU,H ϑ/ 1 ρ ϑ P H\ LH,H ρ ϕ Lip H,R t, max 1 ρ ϑ 1 ρ ϑ P H\ LH,H ρ ϕ Lip H,R B max{1, Lip H,HSU,H ϑ/ t,e X t H 3 }. 1 ρ ϑ 36 Combining 9, 3, 33, and 36 finally proves that E ϕx E ϕp X ϕ Lip H,R max { 1, t, E } X t 3 H 1 1 ρ ϑ F Lip + H,H ϑ + B Lip H,HSU,H ϑ/ P ρ H\ LH,H ρ. 1 ρ ϑ 37 his finishes the proof of Proposition.1. 3 Weak convergence for Galerkin approximations of SEEs with mollified nonlinearities n this section we establish weak convergence rates for Galerkin approximations of SEEs with mollified nonlinearities; see Corollary 3.8, Corollary 3.9, and Corollary 3.1 below. Roughly speaking, in the framework of Section 1.4 we establish in Corollary 3.8 below explicit upper bounds for the weak approximation error E ϕx H E ϕx, 38 where H is a set, where ϕ: H R is a four times continuously Fréchet differentiable function with globally bounded derivatives, and where X H :, Ω H and X :, Ω P HareappropriatemildsolutionprocessesoftheSEEsin146. Here, X :, Ω P H is a spectral Galerkin approximation of X H :, Ω H. We prove Corollary 3.8 by using a decomposition of the weak approximation error as in 1 in Section 1.1 above. Corollary 3.8 is then an immediate consequence of the triangle inequality, of Lemma 3.5 below, and of Lemma 3.7 below. n the proof of Corollary 3.9 we further estimate the right hand side of inequality in Corollary 3.8 to obtain a more explicit upper bound for 38 and the right hand side of in Corollary 3.8, respectively. Corollary 3.9, in particular, enables us to prove inequality 14 in the introduction. n Section 5 below we will use Corollary 3.9 to establish weak convergence rates for Galerkin approximations of SEEs with non-mollified nonlinearities. 1

13 3.1 Regularity properties for solutions of infinite dimensional Kolmogorov equations in Hilbert spaces Lemma 3.1. Assume the setting in Section 1.4, let ϕ Cb 4H,R, F C4 b H,H, B Cb 4H,HSU,H, let Xx :, Ω H, x H, be F t t, -predictable stochastic processes which satisfy for all x H that t, E Xt H x 4 < and which satisfy that for all x H, t, it holds P-a.s. that X x t = e At x+ e At s FX x sds+ e At s BX x sdw s, 39 and let φ:, H R be the function which satisfies for all t,, x H that φt,x = EϕX x t. hen i it holds for all t, that H x φt,x R Cb 4 H,R and ii it holds for all k {1,,3,4}, δ 1,...,δ k 1 /, with k i=1 δ i > 1 / that k φt,xv x k 1,...,v k <. 4 t, x H v 1,...,v k H\{} t δ δ k v 1 Hδ1... v k Hδk Proof. Observe that 39 together with items iii & vii of heorem 3.3 in Andersson et al. 1 with =, η =, H = H, U = U, V = R, W = W, A = A, n = 4, ϕ = ϕ, F = F, B = B, k = k, δ 1 = δ 1,...,δ k = δ k, α =, β = for δ 1,...,δ k {x 1,...,x k 1 /, k : k i=1 x i > 1 /}, k {1,,3,4}in the notationof heorem 3.3 in 1 establishes items i ii above. he proof of Lemma 3.1 is thus completed. n the following we add some comments to Lemma 3.1. Lemma 3.1 is used in the proof of Lemma 3.7 below to establish essentially sharp weak convergence rates. As demonstrated above in the proof of Lemma 3.1, Lemma 3.1 is an immediate consequence of heorem 3.3 in Andersson et al. 1. heorem 3.3 in Andersson et al. 1, in particular, establishes a similar result as Lemma 3.1 but under the more general hypothesis that there exists a natural number n N such that F and B are n-times continuously Fréchet differentiable with globally bounded derivatives. However, in the proof of Lemma 3.7 below we merely employ estimates of the form 4 for the first four derivatives of the generalized solution φt,x = EϕXt x, t,x, H, of the Kolmogorovequation associated to 39 and, therefore, we restrict ourselves in Lemma 3.1 above to the case n = 4. Results related to 4 can, e.g., be found in Debussche 19, Lemmas and in Wang & Gan 5, Lemma 3.3. n particular, very roughly speaking, Lemmas in 19 establish 4 for all δ 1,δ k 1 /,, k {1,} without the constraint that δ 1 +δ > 1 / but under the additional assumption. Moreover, very roughly speaking, Lemma 3.3 in 5 establishes 4 for all δ 1,δ k 1,, k {1,} with the constraint that δ 1 +δ > 1 in the case of additive noise. Note that condition is obviously satisfied in the case of additive noise. Next we briefly present the idea of the proof of Lemma 3.1 above and of items iii & vii of heorem 3.3 in Andersson et al. 1, respectively. We first combine Vitali s convergence theorem with repeated applications of the chain rule from calculuscf. Andersson et al. 1, Lemma.1, 77, and 1 to obtain explicit formulas for the higher order space derivatives of φ cf. Andersson et al. 1, tem v of heorem 3.3 in terms of higher order derivatives of the test function ϕ and in terms of higher order derivative processes associated to 39. hereafter, we employ Hölder s inequality and suitable estimates for the higher order derivative processes associated to 39 from Andersson et al. 3, tem ii of heorem.1 cf. Andersson et al. 1, 6, 11, and 13. he next result, Lemma 3. below, is an elementary lemma which provides sufficient conditions for mild solutions of SEEs to be strong solutions. 13

14 Lemma 3.. Considerthenotation in Section1.3, leth,, H, H andu,, U, U be separable R-Hilbert spaces, let,, p,, let Ω,F,P,F t t, be a stochastic basis, let W t t, be an d U -cylindrical Ω,F,P,F t t, -Wiener process, let A: DA H H be a generator of a strongly continuous analytic semigroup with spectruma {z C: Rez < }, let H r,, Hr, Hr, r R, be a family of interpolation spaces associated to A, let ξ L p P;H 1, let X:, Ω H, Y :, Ω H 1, and Z:, Ω HSU,H 1 be F t t, -predictable stochastic processes which satisfy that E Y s p H 1 + Z s p HSU,H 1 ds < and which satisfy that for all t, it holds P-a.s. that eat s Y s H + e At s Z s HSU,H ds < and hen i it holds that X t = e At ξ + e At s Y s ds+ e At s Z s dw s. 41 t, E e At s Y s p H 1 + e At s Z s p HSU,H 1 ds <, 4 ii it holds for all t, that P X t H 1 = 1, 43 iii it holds that E X t ½ H1 X t p H 1 <, 44 t, iv it holds for all t, that lim X s ½ H1 X s X t ½ H1 X t L p P;H 1 =, 45, s t v it holds that P AX s H 1 + AX s ½ H1 X s H + Y s H1 + Z s HSU,H 1 ds < = 1, and vi for all t, it holds P-a.s. that 46 X t = ξ + = ξ + A X s ½ H1 X s +Y s ds+ AX s +Y s ds+ Z s dw s. Z s dw s 47 Proof. hroughout this proof let h N,t,, t,, N N, be the real numbers which satisfy for all N N, t, that h N,t = t N, let N,t: R R, t,, N N, be the functions which satisfy for all N N, t,, s R that s N,t = max,s {, h N,t,h N,t, h N,t,h N,t,...}, 48 let χ, andρ r,, r,1, be the real numbers which satisfy for all r,1 that χ = e At LH and ρ r = ta r e At d H LH 49 t, t, 14

15 cf., e.g., 4, Lemma 11.36, and let X :, Ω H 1 be the F t t, -predictable stochastic process which satisfies for all t, that Observe that for all t, it holds that X t = X t ½ H1 X t. 5 E e At s Y s p H 1 + e At s Z s p t HSU,H 1 ds χ p E Y s p H 1 + Z s p HSU,H 1 ds. 51 herefore, we obtain that E e At s Y s p H 1 + e At s Z s p HSU,H 1 ds t, χ p E 5 Y s p H 1 + Z s p HSU,H 1 ds <. his establishes item i. Moreover, Jensen s inequality and the assumption that p ensure that E 1 Yt Y t H1 dt = E p 1/p H 1 dt 1 E Y t p 1/p H 1 dt = 1 1/p E Y t p 1/p 53 H 1 dt and E 1 Z t HSU,H 1 dt = 1 Zt E p /p HSU,H 1 dt E /p /p Z t p HSU,H 1 dt = 1 E /p Z t p HSU,H 1 dt. 54 Hence, we obtain that his ensures that for all t, it holds that E Y t H1 + Z t HSU,H 1 dt <. 55 E e At s Y s H1 + e At s Z s HSU,H 1 ds χe Y s H1 + χ E 56 Z s HSU,H 1 ds <. his implies that for all t, it holds that P e At s Y s H1 + e At s Z s HSU,H 1 ds < = his, 41, and the assumption that ξ L p P;H 1 prove item ii. tem ii and 41 show that for all t, it holds P-a.s. that X t = X t = e At ξ + e At s Y s ds+ 15 e At s Z s dw s. 58

16 his and the Burkholder-Davis-Gundy type inequality in Lemma 7.7 in Da Prato & Zabczyk 15 imply that for all t, it holds that X t L p P;H 1 e At ξ L p P;H 1 + pp 1 + e At s Y s L p P;H 1 ds e At s Z s L p P;HSU,H 1 ds 1/. 59 Hölder s inequality hence shows that for all t, it holds that X t L p P;H 1 χ ξ L p P;H 1 +t 1 1/p E e At s Y s p H 1 ds 1/p { pp 1t 1 /p + E /p } 1/ e At s Z s p HSU,H 1 ds. 6 his and item i assure that X t L p P;H 1 t, χ ξ L p P;H /p { pp 1 1 /p + t, t, E e At s Y s p H 1 ds 1/p E e At s Z s p HSU,H 1 ds /p } 1/ <. 61 his establishes item iii. Next note that for all s,t, with s t it holds P-a.s. that e At r Y r dr s e As r Y r dr = e At r ½,s re Amax{s r,} Y r dr. 6 his and Hölder s inequality show that for all t,τ, it holds that τ e At s Y s ds e Aτ s Y s ds max{t,τ} 1 1/p L p P;H 1 e Amax{t,τ} s ½,min{t,τ} se Amax{min{t,τ} s,} Y s L p P;H 1 ds max{t,τ} E e Amax{t,τ} s ½,min{t,τ} se Amax{min{t,τ} s,} Y s p = 1 1/p 1/p H 1 ds E ½,max{t,τ} s e Amax{max{t,τ} s,} 63 ½,min{t,τ} se Amax{min{t,τ} s,} Y s p H 1 ds 1/p. Moreover, observe that for all s,t,τ, it holds that ½,max{t,τ} s e Amax{max{t,τ} s,} ½,min{t,τ} se Amax{min{t,τ} s,} Y s H1 χ Y s H

17 Next note that for all t,, v H 1 it holds that lim e At e As v H1 =. 65, s t Combining with Lebesgue s theorem of dominated convergence and the assumption that E Y s p H 1 ds < yields that for all t, it holds that τ lim e At s Y s ds e Aτ s Y s ds =. 66 L p P;H 1, τ t n the next step note that for all s,t, with s t it holds P-a.s. that e At r Z r dw r = s e As r Z r dw r e At r ½,s re Amax{s r,} Z r dw r. his, Hölder s inequality, and the Burkholder-Davis-Gundy type inequality in Lemma 7.7 in Da Prato & Zabczyk 15 show that for all t,τ, it holds that τ L e At s Z s dw s e Aτ s Z s dw s p P;H 1 1/ pp 1 max{t,τ} e Amax{t,τ} s ½,min{t,τ} se Amax{min{t,τ} s,} Z s L p P;HSU,H 1 ds 1/ 67 pp 1 1 /p 1/ max{t,τ} = E e Amax{t,τ} s ½,min{t,τ} se Amax{min{t,τ} s,} Z s p HSU,H 1 pp 1 1 /p 1/ 1/p ds E ½,max{t,τ} s e Amax{max{t,τ} s,} ½,min{t,τ} se Amax{min{t,τ} s,} 1/p Z s p HSU,H 1 ds. 68 Moreover, observe that for all s,t,τ, it holds that ½,max{t,τ} s e Amax{max{t,τ} s,} ½,min{t,τ} se Amax{min{t,τ} s,} Z s HSU,H1 χ Z s HSU,H1. 69 n addition, note that 65 and Lebesgue s theorem of dominated convergence ensure that for all t,, B HSU,H 1 and all orthonormal bases U U of U it holds that lim e At e As B HSU,H 1 = lim e At e As Bu H 1 =. 7, s t, s t u U herefore, we obtain for all t,, B HSU,H 1 that lim e At e As B p HSU,H 1 =. 71, s t 17

18 Combining 68 with 69, Lebesgue s theorem of dominated convergence, and the assumption that E Z s p HSU,H 1 ds < hence yields that for all t, it holds that τ L lim e At s Z s dw s e Aτ s Z s dw s =. 7 p P;H 1, τ t n addition, note that 65 and Lebesgue s theorem of dominated convergence ensure that for all t, it holds that lim e At e As ξ L p P;H 1 =. 73, s t Combining this, 66, and 7 with 58 establishes item iv. Next note that items ii iii imply that E AX s H +E AXs H 1 ds = E X s H1 +E Xs H ds = E X s H1 +E Xs H ds 1+ A 1 LH 1+ A 1 LH E X t H1 <. t, E X s H1 ds 74 Combining this with 55 yields that P AX s H 1 + AX s H + Y s H1 + Z s HSU,H 1 ds < = his proves item v. t thus remains to establish item vi. For this let X:, Ω H be a stochastic process which satisfies that for all t, it holds P-a.s. that X t = X t ξ Y s ds Observe that for all N N, t, it holds P-a.s. that X t = N 1 Xn+1hN,t X nh N,t n= N 1 n+1hn,t = = n= nh N,t X s N,t +h N,t X s N,t N 1 = n= n+1hn,t nh N,t X s N,t +h N,t X s N,t h N,t h N,t ds. ds his implies that for all t, it holds that X t AX s ds L p P;H X liminf s N,t +h N,t X s N,t N N AX s h N,t X liminf s N,t +h N,t X s N,t N N h N,t + liminf AX s N,t X s N N L. ds p P;H 18 Z s dw s. 76 Xn+1hN,t X nhn,t h N,t L p P;Hds AX s N,t L P;Hds p ds 77 78

19 Next note that items iii iv and Lebesgue s theorem of dominated convergence ensure that for all t, it holds that lim AX s N,t X s ds L =. 79 N N p P;H Moreover, observe that item ii and 41 imply that for all s,t, with s t it holds P-a.s. that X t = e At s X s + = e At s X s + s e At r Y r dr + e At r Y r dr+ s s s e At r Z r dw r e At r Z r dw r. 8 his and item ii show that for all N N, s,t, with s < t it holds P-a.s. that X s N,t +h N,t X s N,t h N,t AX s N,t h N,t = eah N,t d H h N,t AX s N,t + h N,t + s N,t +h N,t s N,t +h N,t s N,t e A s N,t+hN,t r d H h N,t Z r dw r. his yields that for all N N, t, it holds that X s N,t +h N,t X s N,t AX s N,t h N,t e Ah N,t d H h N,t AX s N,t + + h N,t s N,t +h N,t s N,t s N,t +h N,t e A s N,t+hN,t r d H s N,t h N,t L p P;Hds L p P;Hds e A s N,t+hN,t r d H Y r LpP;H drds h N,t e A s N,t+hN,t r d H s N,t h N,t Z r dw r L p P;Hds. Next note that for all N N, s,t, with s t it holds that e Ah N,t d H h N,t AX s N,t h N,t L p P;H e Ah N,t d H h N,t AX s N,t X s h N,t L p P;H + e Ah N,t d H h N,t AX s h N,t L p P;H e Ah N,t d H X s N,t X s h N,t + X s N,t X s L p P;H 1 L p P;H + e Ah N,t d H h N,t AX s h N,t L p P;H ρ 1 +1 X s N,t X s L p P;H 1 + e Ah N,t d H h N,t AX s h N,t Y r dr L p P;H

20 n addition, observe that the fact that v H 1 : lim, h e Ah d H hav h H = assures that for all s,t, with s t it holds that lim e Ah N,t d H h N,t AX s N N h N,t =. 84 H Next observe that for all N N, s,t, with s t it holds that e Ah N,t d H h N,t AX s h N,t ρ 1 +1 X s H1. 85 H his, 84, item iii, and Lebesgue s theorem of dominated convergence ensure that for all s,t, with s t it holds that lim e Ah N,t d H h N,t AX s N N h N,t =. 86 L p P;H Combining 83 with 86 and item iv shows that for all s,t, with s t it holds that e lim Ah N,t d H h N,t AX s N,t N N h N,t =. 87 L p P;H his, 85, item iii, and Lebesgue s theorem of dominated convergence yield that for all t, it holds that lim N N e Ah N,t d H h N,t AX s N,t h N,t L p P;Hds Furthermore, observe that for all N N, s,t, with s < t it holds that =. 88 s N,t +h N,t s N,t e A s N,t+hN,t r d H Y r LpP;H h N,t dr s N,t +h N,t s N,t +h N,t r Y r LpP;H1 ρ 1 dr s N,t ρ 1 s N,t +h N,t h N,t s N,t Y r L p P;H 1 dr = ρ 1 ½ s N,t, s N,t +h N,t r Y r L p P;H 1 dr. 89 naddition,notethathölder sinequalityandtheassumptionthat E Y t p H 1 + Z t p HSU,H 1 dt < assure that Y t L p P;H 1 + Z t L p P;HSU,H 1 dt <. 9 his, Lebesgue s theorem of dominated convergence, and 89 imply that for all t, it holds that s N,t +h N,t e A s N,t+hN,t r d H Y r LpP;H lim drds N N s N,t h N,t 91 ρ 1 lim N N ½ s N,t, s N,t +h N,t r Y r L p P;H 1 drds =.

21 Next observe that the Burkholder-Davis-Gundy type inequality in Lemma 7.7 in Da Prato & Zabczyk 15 shows that for all N N, s,t, with s < t it holds that s N,t +h N,t e A s N,t+hN,t r d H Z r dw r s N,t h N,t L p P;H pp 1 s N,t +h N,t ea s N,t+hN,t r d H Z r 1/ L p P;HSU,H dr s N,t h N,t pp 1 ρ1 s N,t +h N,t s N,t +h N,t r Z r 1/ L p P;HSU,H 1 dr 9 s N,t h N,t pp 1 ρ1 s N,t +h N,t 1/ Z r L p P;HSU,H 1 dr s N,t pp 1 ρ1 1/ = ½ s N,t, s N,t +h N,t r Z r L p P;HSU,H 1 dr. Hölder s inequality hence implies that for all N N, t, it holds that s N,t +h N,t e A s N,t+hN,t r d H Z r dw r s N,t h N,t L P;Hds p pp 1 ρ1 1/ 1/ ½ s N,t, s N,t +h N,t r Z r L p P;HSU,H 1 dr ds pp 1 ρ1 t t 1/ ½ s N,t, s N,t +h N,t r Z r L p P;HSU,H 1 drds. his, Lebesgue s theorem of dominated convergence, and 9 ensure that for all t, it holds that s N,t +h N,t lim e A s N,t+hN,t r d H Z r dw r N N s N,t h N,t L P;Hds p pp 1 ρ1 t t 1/ lim ½ s N,t, s N,t +h N,t r Z r L p P;HSU,H 1 drds N N =. 94 Putting 8, 88, 91, and 94 together yields that for all t, it holds that X lim s N,t +h N,t X s N,t AX s N,t N N h N,t =. 95 L P;Hds p Combining this and 79 with 78 shows that for all t, it holds that X t AX s ds =. 96 L p P;H his, 41, and 76 imply that for all t, it holds P-a.s. that X t = ξ + AX s +Y s ds+ 93 Z s dw s. 97 Moreover, items ii & v imply that for all t, it holds P-a.s. that ξ + AX s +Y s ds+ Z s dw s = ξ + AX s +Y s ds+ Z s dw s. 98 his, 5, and 97 establish item vi. he proof of Lemma 3. is thus completed. 1

22 he following result, Lemma 3.3 below, can be shown by employing Lemma 3. above together with the standard tô formula in infinite dimensions cf., e.g., Brzeźniak et al. 9, heorem.4. Lemma 3.3. Consider the notation in Section 1.3, let H,, H, H, U,, U, U, and V,, V, V be separable R-Hilbert spaces, let,, let Ω,F,P, F t t, be a stochastic basis, let W t t, be an d U -cylindrical Ω,F,P,F t t, -Wiener process, let A: DA H H be a generator of a strongly continuous analytic semigroup with spectruma {z C: Rez < }, let H r,, Hr, Hr, r R, be a family of interpolation spaces associated to A, let F Lip H,H 1, B Lip H,HSU,H 1, ϕ Cb H,V, let Xx :, Ω H, x H, be F t t, -predictable stochastic processes which satisfy for all x H that t, E Xt x H < and which satisfy that for all t,, x H it holds P-a.s. that X x t = e At x+ e At s FX x sds+ e At s BX x sdw s, 99 and let u:, H V be the function which satisfies for all t,, x H that ut,x = E ϕx x t. hen i it holds for all x H 1, t, that PX x t H 1 = 1, ii it holds for all p, that x H 1 t, E X x t ½ H1 X x t p H 1 1/p max{1, x H1 } <, 1 iii it holds for all x H 1, t, that lim E Xt x ½ H 1 Xt x Xy s ½ H 1 Xs y H 1 =, 11, H 1 s,y t,x iv it holds for all x H 1 that, t ut,x V C 1,,V, and v it holds that, H 1 t,x t ut,x V C, H 1,V. Proof. hroughout this proof assume w.l.o.g. that H {}, let U U be anorthonormal basis of U, let χ, be the real number given by χ = e At LH, 1 t, let X x :, Ω H 1, x H 1, be the F t t, -predictable stochastic processes which satisfy for all x H 1, t, that X x t = X x t ½ H1 X x t, 13 and let Φ: H 1 V be the function which satisfies for all x H 1 that Φx = ϕ xax+fx+ 1 ϕ xbxb,bxb. 14 Observe that item i of Corollary.1 in Andersson et al. with H = H, U = U, =, η =, α =, β =, W = W, A = A, F = H x Fx H, B = H x U u Bxu H HSU,H in the notation of Corollary.1 in implies that there exist F t t, -predictable stochastic processes X x :, Ω H, x H, which

23 satisfy for all p,, x H that t, E X x t p H < and which satisfy that for all t,, x H it holds P-a.s. that X x t = eat x+ n particular, this implies that for all x H it holds that e At s FX x s ds+ e At s BX x s dw s. 15 E X x t H < 16 t, Combining this and 15 with 99 shows that for all x H, t, it holds P-a.s. that X x t = X x t. 17 cf., e.g., Da Prato & Zabczyk 16, tem i of heorem 7., Da Prato et al. 14, Proposition 3, or Andersson et al., tem i of heorem.9. herefore, we obtain that for all p,, x H, t, it holds that his ensures that for all p,, x H it holds that E X x t p H = E X x t p H. 18 E Xt H x p <. 19 t, his, in turn, demonstrates that for all p,, x H it holds that FX x t L p P;H 1 + BXt x L p P;HSU,H 1 t, F Lip H,H 1 + B Lip H,H 1 max { 1, Xt x L p P;H t, Hence, we obtain that for all p,, x H it holds that } <. 11 E FX x t p H 1 + BX x t p HSU,H 1 dt <. 111 his, 99, and items i, ii, v, and vi of Lemma 3. imply that it holds for all p,, x H 1 that t, E e At s FX x s p H 1 + e At s BX x s p HSU,H 1 ds <, 11 it holds for all x H 1, t, that it holds for all x H 1 that and P X x t H 1 = P X x t = X x t = 1, 113 P AXs x H 1 + FXs x H 1 + BXs x HSU,H 1 ds < = 1, 114 3

24 V for all x H 1, t, it holds P-a.s. that X x t = x+ AXs x +FXx s ds+ BXs x dw s. 115 Observe that item proves item i. n the next step we combine with 99 to obtain that for all x H 1, t, it holds P-a.s. that AX x t = e At Ax+ e At s AFXs x ds+ e At s ABXs x dw s. 116 he Burkholder-Davis-Gundy type inequality in Lemma 7.7 in Da Prato & Zabczyk 15 therefore shows that for all p,, x H 1, t, it holds that AX x t L p P;H e At Ax H + pp 1 + χ Ax H + pp 1 + e At s AFX x s L p P;Hds e At s ABX x s L p P;HSU,Hds e At s FX x s L p P;H 1 ds e At s BX x s L p P;HSU,H 1 ds 1/ 1/. 117 his, Hölder s inequality, and 11 imply that for all p,, x H 1 it holds that herefore, we obtain for all p,, x H that AXt x L p P;H <. 118 t, AX A 1 x t L p P;H <. 119 t, Furthermore, observe that 11 and 113 yield that for all x H 1, t, it holds P-a.s. that and e At s AFX x s ds+ e At s AFXs x H + e At s ABXs x HSU,H ds < 1 e At s ABX x s dw s = e At s AFXs x ds+ e At s ABXs x dw s. 11 his and 116 yield that for all x H 1, t, it holds P-a.s. that AX x t = e At Ax+ = e At Ax+ e At s AFX x s ds+ e At s ABX x s dw s e At s AFA 1 AXs x ds+ e At s ABA 1 AXs x dw s. 1 4