Lower bounds for weak approximation errors for spatial spectral Galerkin approximations of stochastic wave equations
LLower bounds for weak approximation errors for spatial spectral Galerkin approximationsof stochastic wave equations
Ladislas Jacobe de Naurois, Arnulf Jentzen, and Timo Welti
ETH Zürich, Switzerland
October 9, 2018
Abstract
Although for a number of semilinear stochastic wave equations existence and uniquenessresults for corresponding solution processes are known from the literature, these solutionprocesses are typically not explicitly known and numerical approximation methods are neededin order for mathematical modelling with stochastic wave equations to become relevant forreal world applications. This, in turn, requires the numerical analysis of convergence rates forsuch numerical approximation processes. A recent article by the authors proves upper boundsfor weak errors for spatial spectral Galerkin approximations of a class of semilinear stochasticwave equations. The findings there are complemented by the main result of this work, thatprovides lower bounds for weak errors which show that in the general framework consideredthe established upper bounds can essentially not be improved.
In this work we consider numerical approximation processes of solution processes of stochasticwave equations and examine corresponding weak convergence properties. As opposed to strongconvergence, weak convergence even in the case of stochastic evolution equations with regularnonlinearities are still only poorly understood (see, e.g., [3, 6, 7, 8, 12] for several weak convergenceresults for stochastic wave equations and, e.g., the references in Section 1 in [4] for further resultson weak convergence in the literature). Therefore, equations available to current numerical analysisare limited to model problems such as the ones considered in the present article that cannot takeinto account the full complexity of models for evolutionary processes under influence of randomnessappearing in real world applications (see, e.g., the references in Section 1 in [4]). The recentarticle [4] by the authors provides upper bounds for weak errors for spatial spectral Galerkinapproximations of a class of semilinear stochastic wave equations, including equations driven bymultiplicative noise and, in particular, the hyperbolic Anderson model. The purpose of this work isto show that the weak convergence rates for stochastic wave equations established in Theorem 1.1in [4] can in the general setting there essentially not be improved . This is achieved by provinglower bounds for weak errors in the case of concrete examples of stochastic wave equations withadditive noise and without drift nonlinearity (see Corollary 2.10 below). We argue similarly to thereasoning in Section 6 in Conus et al. [1] and Section 9 in Jentzen & Kurniawan [5]. First resultson lower bounds for strong errors for two examples of stochastic heat equations were achieved inDavie & Gaines [2]. Furthermore, lower bounds for strong errors for examples and whole classesof stochastic heat equations have been established in Müller-Gronbach et al. [10] (see also thereferences therein) and in Müller-Gronbach & Ritter [9], respectively. Results on lower bounds for1 a r X i v : . [ m a t h . P R ] J a n eak errors in in the case of a few specific examples of stochastic heat equations can be found inConus et al. [1] and in Jentzen & Kurniawan [5]. Theorem 1.1.
For all real numbers η, T ∈ (0 , ∞ ) , every R -Hilbert space ( H, (cid:104)· , ·(cid:105) H , (cid:107)·(cid:107) H ) , everyprobability space (Ω , F , P ) with a normal filtration ( F t ) t ∈ [0 ,T ] , every id H -cylindrical (Ω , F , P , ( F t ) t ∈ [0 ,T ] ) -Wiener process ( W t ) t ∈ [0 ,T ] , and every orthonormal basis ( e n ) n ∈ N : N → H of H there exist an in-creasing sequence ( λ n ) n ∈ N : N → (0 , ∞ ) , a linear operator A : D ( A ) ⊆ H → H with D ( A ) = (cid:8) v ∈ H : (cid:80) n ∈ N | λ n (cid:104) e n , v (cid:105) H | < ∞ (cid:9) and ∀ v ∈ D ( A ) : Av = (cid:80) n ∈ N − λ n (cid:104) e n , v (cid:105) H e n , a family ofinterpolation spaces ( H r , (cid:104)· , ·(cid:105) H r , (cid:107)·(cid:107) H r ) , r ∈ R , associated to − A (cf., e.g., [11, Section 3.7]),a family of R -Hilbert spaces ( H r , (cid:104)· , ·(cid:105) H r , (cid:107)·(cid:107) H r ) , r ∈ R , with ∀ r ∈ R : ( H r , (cid:104)· , ·(cid:105) H r , (cid:107)·(cid:107) H r ) = (cid:0) H r / × H r / − / , (cid:104)· , ·(cid:105) H r/ × H r/ − / , (cid:107)·(cid:107) H r/ × H r/ − / (cid:1) , families of functions P N : (cid:83) r ∈ R H r → (cid:83) r ∈ R H r , N ∈ N ∪ {∞} , and P N : (cid:83) r ∈ R H r → (cid:83) r ∈ R H r , N ∈ N ∪ {∞} , with ∀ N ∈ N ∪ {∞} , r ∈ R , u ∈ H r , ( v, w ) ∈ H r : (cid:0) P N ( u ) = (cid:80) Nn =1 (cid:104) ( λ n ) − r e n , u (cid:105) H r ( λ n ) − r e n and P N ( v, w ) = ( P N ( v ) , P N ( w )) (cid:1) , a lin-ear operator A : D ( A ) ⊆ H → H with D ( A ) = H and ∀ ( v, w ) ∈ H : A ( v, w ) = ( w, Av ) ,real numbers γ, c ∈ (0 , ∞ ) , and functions ξ ∈ L ( P | F ; H γ ) , ϕ ∈ C ( H , R ) , F ∈ C ( H , H ) , B ∈ C ( H , HS( H, H )) and ( C ε ) ε ∈ (0 , ∞ ) : (0 , ∞ ) → [0 , ∞ ) with ∀ β ∈ ( γ / , γ ] : ( − A ) − β / ∈ HS( H ) , F ∈ C ( H , H γ ) , and B ∈ C ( H , L ( H, H γ )) such that(i) it holds that there exist up to modifications unique ( F t ) t ∈ [0 ,T ] -predictable stochastic processes X N : [0 , T ] × Ω → P N ( H ) , N ∈ N ∪ {∞} , which satisfy for all N ∈ N ∪ {∞} , t ∈ [0 , T ] that sup s ∈ [0 ,T ] (cid:107) X Ns (cid:107) L ( P ; H ) < ∞ and P -a.s. that X Nt = e t A P N ξ + (cid:90) t e ( t − s ) A P N F ( X Ns ) d s + (cid:90) t e ( t − s ) A P N B ( X Ns ) d W s (1.1) (ii) and it holds for all ε ∈ (0 , ∞ ) , N ∈ N that c · ( λ N ) − η ≤ (cid:12)(cid:12) E (cid:2) ϕ (cid:0) X ∞ T (cid:1)(cid:3) − E (cid:2) ϕ (cid:0) X NT (cid:1)(cid:3)(cid:12)(cid:12) ≤ C ε · ( λ N ) ε − η . (1.2)Here and below we denote for every non-trivial R -Hilbert space ( V, (cid:104)· , ·(cid:105) V , (cid:107)·(cid:107) V ) and every R -Hilbert space ( W, (cid:104)· , ·(cid:105) W , (cid:107)·(cid:107) W ) by C ( V, W ) the set of all globally bounded twice Fréchet dif-ferentiable functions from V to W with globally bounded derivatives. Theorem 1.1 is a directconsequence of Theorem 1.1 in [4] (with γ = 2 η , β = min { η + ε, η } , ρ = 0 in the notation ofTheorem 1.1 in [4]) and Corollary 2.10 below (with p = / η , δ = / − η in the notation of Corol-lary 2.10 below). Inequality (1.2) reveals that the weak convergence rates in Theorem 1.1 in [4]are essentially sharp. More details and further lower bounds for weak approximation errors forstochastic wave equations can be found in Corollary 2.10 below. Let ( H, (cid:104)· , ·(cid:105) H , (cid:107)·(cid:107) H ) be a separable R -Hilbert space, for every set A let P ( A ) be the power setof A , let T ∈ (0 , ∞ ) , let (Ω , F , P ) be a probability space with a normal filtration ( F t ) t ∈ [0 ,T ] ,let ( W t ) t ∈ [0 ,T ] be an id H -cylindrical (Ω , F , P , ( F t ) t ∈ [0 ,T ] ) -Wiener process, let H ⊆ H be a non-empty orthonormal basis of H , let λ : H → R be a function with sup h ∈ H λ h < , let A : D ( A ) ⊆ H → H be the linear operator which satisfies D ( A ) = (cid:8) v ∈ H : (cid:80) h ∈ H | λ h (cid:104) h, v (cid:105) H | < ∞ (cid:9) and ∀ v ∈ D ( A ) : Av = (cid:80) h ∈ H λ h (cid:104) h, v (cid:105) H h , let ( H r , (cid:104)· , ·(cid:105) H r , (cid:107)·(cid:107) H r ) , r ∈ R , be a family of interpolationspaces associated to − A , let ( H r , (cid:104)· , ·(cid:105) H r , (cid:107)·(cid:107) H r ) , r ∈ R , be the family of R -Hilbert spaces whichsatisfies for all r ∈ R that ( H r , (cid:104)· , ·(cid:105) H r , (cid:107)·(cid:107) H r ) = (cid:0) H r / × H r / − / , (cid:104)· , ·(cid:105) H r/ × H r/ − / , (cid:107)·(cid:107) H r/ × H r/ − / (cid:1) ,let P I : (cid:83) r ∈ R H r → (cid:83) r ∈ R H r , I ∈ P ( H ) , and P I : (cid:83) r ∈ R H r → (cid:83) r ∈ R H r , I ∈ P ( H ) , be the functions2hich satisfy for all I ∈ P ( H ) , r ∈ R , u ∈ H r , ( v, w ) ∈ H r that P I ( u ) = (cid:80) h ∈ I (cid:104)| λ h | − r h, u (cid:105) H r | λ h | − r h and P I ( v, w ) = (cid:0) P I ( v ) , P I ( w ) (cid:1) , let A : D ( A ) ⊆ H → H be the linear operator which satisfies D ( A ) = H and ∀ ( v, w ) ∈ H : A ( v, w ) = ( w, Av ) , let µ : H → R be a function which satisfies (cid:80) h ∈ H | µ h | | λ h | < ∞ , let B ∈ HS( H, H ) be the linear operator which satisfies for all v ∈ H that B v = (cid:0) , (cid:80) h ∈ H µ h (cid:104) h, v (cid:105) H h (cid:1) , and let X I = ( X I, , X I, ) : Ω → P I ( H ) , I ∈ P ( H ) , be randomvariables which satisfy for all I ∈ P ( H ) that it holds P -a.s. that X I = (cid:82) T e ( T − s ) A P I B d W s . Lemma 2.1.
Assume the setting in Section 2.1. Then for all I ∈ P ( H ) it holds P -a.s. that X I = P I X H = (cid:18) X I, X I, (cid:19) = (cid:80) h ∈ I (cid:16) µ h | λ h | / (cid:82) T sin (cid:0) | λ h | / ( T − s ) (cid:1) d (cid:104) h, W s (cid:105) H (cid:17) h (cid:80) h ∈ I (cid:16) µ h | λ h | / (cid:82) T cos (cid:0) | λ h | / ( T − s ) (cid:1) d (cid:104) h, W s (cid:105) H (cid:17) | λ h | / h . (2.1) Proof of Lemma 2.1.
Lemma 2.5 in [4] proves that it holds P -a.s. that X H = (cid:90) T e ( T − s ) A B d W s = (cid:88) h ∈ H (cid:90) T e ( T − s ) A B h d (cid:104) h, W s (cid:105) H = (cid:88) h ∈ H (cid:32) µ h (cid:82) T ( − A ) − / sin (cid:0) ( − A ) / ( T − s ) (cid:1) h d (cid:104) h, W s (cid:105) H µ h (cid:82) T cos (cid:0) ( − A ) / ( T − s ) (cid:1) h d (cid:104) h, W s (cid:105) H (cid:33) = (cid:88) h ∈ H (cid:32) µ h | λ h | / (cid:82) T sin (cid:0) | λ h | / ( T − s ) (cid:1) h d (cid:104) h, W s (cid:105) H µ h (cid:82) T cos (cid:0) | λ h | / ( T − s ) (cid:1) h d (cid:104) h, W s (cid:105) H (cid:33) = (cid:80) h ∈ H (cid:16) µ h | λ h | / (cid:82) T sin (cid:0) | λ h | / ( T − s ) (cid:1) d (cid:104) h, W s (cid:105) H (cid:17) h (cid:80) h ∈ H (cid:16) µ h | λ h | / (cid:82) T cos (cid:0) | λ h | / ( T − s ) (cid:1) d (cid:104) h, W s (cid:105) H (cid:17) | λ h | / h . (2.2)Furthermore, Lemma 2.7 in [4] shows for all I ∈ P ( H ) that it holds P -a.s. that P I X H = (cid:90) T P I e ( T − s ) A B d W s = (cid:90) T e ( T − s ) A P I B d W s = X I . (2.3)This and (2.2) complete the proof of Lemma 2.1. Lemma 2.2.
Assume the setting in Section 2.1 and let I ∈ P ( H ) . Then(i) it holds that (cid:104) h, X I, (cid:105) H , h ∈ H , is a family of independent centred Gaussian random variables,(ii) it holds that (cid:10) | λ h | / h, X I, (cid:11) H − / , h ∈ H , is a family of independent centred Gaussian randomvariables, and(iii) it holds for all h ∈ H that Var (cid:0) (cid:104) h, X I, (cid:105) H (cid:1) = I ( h ) | µ h | | λ h | (cid:18) T − sin (cid:0) | λ h | / T (cid:1) | λ h | / (cid:19) , (2.4) Var (cid:16)(cid:10) | λ h | / h, X I, (cid:11) H − / (cid:17) = I ( h ) | µ h | | λ h | (cid:18) T + sin (cid:0) | λ h | / T (cid:1) | λ h | / (cid:19) , (2.5) Cov (cid:16) (cid:104) h, X I, (cid:105) H , (cid:10) | λ h | / h, X I, (cid:11) H − / (cid:17) = I ( h ) | µ h | | λ h | (cid:18) − cos (cid:0) | λ h | / T (cid:1) | λ h | / (cid:19) . (2.6)3 roof of Lemma 2.2. Observe that Lemma 2.1 implies (i) and (ii). It thus remains to prove (iii).Lemma 2.1 implies for all h ∈ H that it holds P -a.s. that (cid:104) h, X I, (cid:105) H = I ( h ) µ h | λ h | / (cid:90) T sin (cid:0) | λ h | / ( T − s ) (cid:1) d (cid:104) h, W s (cid:105) H , (2.7) (cid:10) | λ h | / h, X I, (cid:11) H − / = I ( h ) µ h | λ h | / (cid:90) T cos (cid:0) | λ h | / ( T − s ) (cid:1) d (cid:104) h, W s (cid:105) H . (2.8)Itô’s isometry hence shows for all h ∈ H that Var (cid:0) (cid:104) h, X I, (cid:105) H (cid:1) = E (cid:2) |(cid:104) h, X I, (cid:105) H | (cid:3) = I ( h ) | µ h | | λ h | (cid:90) T (cid:12)(cid:12) sin (cid:0) | λ h | / ( T − s ) (cid:1)(cid:12)(cid:12) d s = I ( h ) | µ h | | λ h | (cid:18) T − sin (cid:0) | λ h | / T (cid:1) | λ h | / (cid:19) , (2.9) Var (cid:16)(cid:10) | λ h | / h, X I, (cid:11) H − / (cid:17) = E (cid:104)(cid:12)(cid:12)(cid:12)(cid:10) | λ h | / h, X I, (cid:11) H − / (cid:12)(cid:12)(cid:12) (cid:105) = I ( h ) | µ h | | λ h | (cid:90) T (cid:12)(cid:12) cos (cid:0) | λ h | / ( T − s ) (cid:1)(cid:12)(cid:12) d s = I ( h ) | µ h | | λ h | (cid:18) T + sin (cid:0) | λ h | / T (cid:1) | λ h | / (cid:19) . (2.10)Furthermore, observe for all h ∈ H that Cov (cid:16) (cid:104) h, X I, (cid:105) H , (cid:10) | λ h | / h, X I, (cid:11) H − / (cid:17) = E (cid:104) (cid:104) h, X I, (cid:105) H (cid:10) | λ h | / h, X I, (cid:11) H − / (cid:105) = I ( h ) | µ h | | λ h | (cid:90) T sin (cid:0) | λ h | / ( T − s ) (cid:1) cos (cid:0) | λ h | / ( T − s ) (cid:1) d s = I ( h ) | µ h | | λ h | (cid:18) (cid:12)(cid:12) sin (cid:0) | λ h | / T (cid:1)(cid:12)(cid:12) | λ h | / (cid:19) = I ( h ) | µ h | | λ h | (cid:18) − cos (cid:0) | λ h | / T (cid:1) | λ h | / (cid:19) . (2.11)The proof of Lemma 2.2 is thus completed. Corollary 2.3.
Assume the setting in Section 2.1 and let I ∈ P ( H ) . Then it holds for all ( v, w ) ∈ P I ( H ) that CovOp( X I )( v, w ) = 12 | µ h | | λ h | (cid:88) h ∈ I (cid:20)(cid:18) T − sin (cid:0) | λ h | / T (cid:1) | λ h | / (cid:19) (cid:104) h, v (cid:105) H (cid:18) h (cid:19) + (cid:18) − cos (cid:0) | λ h | / T (cid:1) | λ h | / (cid:19)(cid:10) | λ h | / h, w (cid:11) H − / (cid:18) h (cid:19) (2.12) + (cid:18) − cos (cid:0) | λ h | / T (cid:1) | λ h | / (cid:19) (cid:104) h, v (cid:105) H (cid:18) | λ h | / h (cid:19) + (cid:18) T + sin (cid:0) | λ h | / T (cid:1) | λ h | / (cid:19)(cid:10) | λ h | / h, w (cid:11) H − / (cid:18) | λ h | / h (cid:19)(cid:21) ∈ P I ( H ) . roof of Corollary 2.3. Lemma 2.2 and Lemma 2.1 prove for all x = ( v , w ) , x = ( v , w ) ∈ P I ( H ) that (cid:104) x , CovOp( X I ) x (cid:105) H = Cov (cid:0) (cid:104) x , X I (cid:105) H , (cid:104) x , X I (cid:105) H (cid:1) = E (cid:2) (cid:104) x , X I (cid:105) H (cid:104) x , X I (cid:105) H (cid:3) = E (cid:2)(cid:0) (cid:104) v , X I, (cid:105) H + (cid:104) w , X I, (cid:105) H − / (cid:1)(cid:0) (cid:104) v , X I, (cid:105) H + (cid:104) w , X I, (cid:105) H − / (cid:1)(cid:3) = (cid:88) h ∈ H Var (cid:0) (cid:104) h, X I, (cid:105) H (cid:1) (cid:104) h, v (cid:105) H (cid:104) h, v (cid:105) H + (cid:88) h ∈ H Cov (cid:16) (cid:104) h, X I, (cid:105) H , (cid:10) | λ h | / h, X I, (cid:11) H − / (cid:17) (cid:104) h, v (cid:105) H (cid:10) | λ h | / h, w (cid:11) H − / + (cid:88) h ∈ H Cov (cid:16) (cid:104) h, X I, (cid:105) H , (cid:10) | λ h | / h, X I, (cid:11) H − / (cid:17) (cid:104) h, v (cid:105) H (cid:10) | λ h | / h, w (cid:11) H − / + (cid:88) h ∈ H Var (cid:16)(cid:10) | λ h | / h, X I, (cid:11) H − / (cid:17)(cid:10) | λ h | / h, w (cid:11) H − / (cid:10) | λ h | / h, w (cid:11) H − / = (cid:28) v , (cid:88) h ∈ H (cid:104) Var (cid:0) (cid:104) h, X I, (cid:105) H (cid:1) (cid:104) h, v (cid:105) H + Cov (cid:16) (cid:104) h, X I, (cid:105) H , (cid:10) | λ h | / h, X I, (cid:11) H − / (cid:17)(cid:10) | λ h | / h, w (cid:11) H − / (cid:105) h (cid:29) H + (cid:28) w , (cid:88) h ∈ H (cid:104) Cov (cid:16) (cid:104) h, X I, (cid:105) H , (cid:10) | λ h | / h, X I, (cid:11) H − / (cid:17) (cid:104) h, v (cid:105) H + Var (cid:16)(cid:10) | λ h | / h, X I, (cid:11) H − / (cid:17)(cid:10) | λ h | / h, w (cid:11) H − / (cid:105) | λ h | / h (cid:29) H − / = (cid:28) x , (cid:88) h ∈ H (cid:104) Var (cid:0) (cid:104) h, X I, (cid:105) H (cid:1) (cid:104) h, v (cid:105) H + Cov (cid:16) (cid:104) h, X I, (cid:105) H , (cid:10) | λ h | / h, X I, (cid:11) H − / (cid:17)(cid:10) | λ h | / h, w (cid:11) H − / (cid:105) (cid:18) h (cid:19)(cid:29) H + (cid:28) x , (cid:88) h ∈ H (cid:104) Cov (cid:16) (cid:104) h, X I, (cid:105) H , (cid:10) | λ h | / h, X I, (cid:11) H − / (cid:17) (cid:104) h, v (cid:105) H + Var (cid:16)(cid:10) | λ h | / h, X I, (cid:11) H − / (cid:17)(cid:10) | λ h | / h, w (cid:11) H − / (cid:105) (cid:18) | λ h | / h (cid:19)(cid:29) H . (2.13)This and again Lemma 2.2 complete the proof of Corollary 2.3. Lemma 2.4.
Assume the setting in Section 2.1 and let I ∈ P ( H ) . Then it holds for all i ∈ { , } that X I ∈ L ( P ; H ) and E (cid:2) (cid:107) X I (cid:107) H (cid:3) = T (cid:88) h ∈ I | µ h | | λ h | < ∞ , (2.14) E (cid:104) (cid:107) X I,i (cid:107) H / − i/ (cid:105) = 12 (cid:88) h ∈ I | µ h | | λ h | (cid:18) T + sin (cid:0) | λ h | / T (cid:1) ( − i | λ h | / (cid:19) < ∞ . (2.15) Proof of Lemma 2.4.
Itô’s isometry and Lemma 2.6 in [4] imply that E (cid:2) (cid:107) X I (cid:107) H (cid:3) = E (cid:20)(cid:13)(cid:13)(cid:13)(cid:13)(cid:90) T e ( T − s ) A P I B d W s (cid:13)(cid:13)(cid:13)(cid:13) H (cid:21) = T (cid:107) P I B (cid:107) H, H ) = T (cid:88) h ∈ I | µ h | | λ h | < ∞ . (2.16)5n addition, Lemma 2.2 shows for all i ∈ { , } that E (cid:104) (cid:107) X I,i (cid:107) H / − i/ (cid:105) = (cid:88) h ∈ H E (cid:104)(cid:12)(cid:12)(cid:12)(cid:10) | λ h | i / − / h, X I,i (cid:11) H / − i/ (cid:12)(cid:12)(cid:12) (cid:105) = 12 (cid:88) h ∈ I | µ h | | λ h | (cid:18) T + sin (cid:0) | λ h | / T (cid:1) ( − i | λ h | / (cid:19) < ∞ . (2.17)The proof of Lemma 2.4 is thus completed. Proposition 2.5.
Assume the setting in Section 2.1 and assume inf h ∈ H | µ h | > . Then it holdsfor all I ∈ P ( H ) \ { H } that E (cid:2) (cid:107) X H (cid:107) H (cid:3) − E (cid:2) (cid:107) X I (cid:107) H (cid:3) = E (cid:2) (cid:107) X H \ I (cid:107) H (cid:3) ≥ T inf h ∈ H | µ h | (cid:88) h ∈ H \ I | λ h | > . (2.18) Proof of Proposition 2.5.
Orthogonality and Lemma 2.1 imply for all I ∈ P ( H ) \ { H } that E (cid:2) (cid:107) X I (cid:107) H (cid:3) + E (cid:2) (cid:107) X H \ I (cid:107) H (cid:3) = E (cid:2) (cid:107) P I X H (cid:107) H (cid:3) + E (cid:2) (cid:107) P H \ I X H (cid:107) H (cid:3) = E (cid:2) (cid:107) ( P I + P H \ I ) X H (cid:107) H (cid:3) = E (cid:2) (cid:107) X H (cid:107) H (cid:3) . (2.19)This and Lemma 2.4 show for all I ∈ P ( H ) \ { H } that E (cid:2) (cid:107) X H (cid:107) H (cid:3) − E (cid:2) (cid:107) X I (cid:107) H (cid:3) = E (cid:2) (cid:107) X H \ I (cid:107) H (cid:3) = T (cid:88) h ∈ H \ I | µ h | | λ h | ≥ T inf h ∈ H | µ h | (cid:88) h ∈ H \ I | λ h | > . (2.20)The proof of Proposition 2.5 is thus completed.In Corollary 2.7 and Corollary 2.8 below lower bounds on the weak approximation error withthe squared norm as test function are presented. Our proofs of Corollary 2.7 and Corollary 2.8 usethe following elementary and well-known lemma (cf., e.g., Proposition 6.4 in Conus et al. [1]). Lemma 2.6.
Let p ∈ (0 , ∞ ) , δ ∈ ( −∞ , / − / (2 p ) ) . Then it holds for all N ∈ N that ∞ (cid:88) n = N +1 n p (2 δ − ≥ N p (2 δ − [ p (1 − δ ) − p (1 − δ ) − . (2.21) Proof of Lemma 2.6.
Observe that the assumption that δ ∈ ( −∞ , / − / (2 p ) ) ensures that p (2 δ − ∈ ( −∞ , − . This implies for all N ∈ N that ∞ (cid:88) n = N +1 n p (2 δ − = ∞ (cid:88) n = N +1 (cid:90) n +1 n n p (2 δ − d x ≥ ∞ (cid:88) n = N +1 (cid:90) n +1 n x p (2 δ − d x = (cid:90) ∞ N +1 x p (2 δ − d x = − ( N + 1) p (2 δ − p (2 δ −
1) + 1 ≥ N p (2 δ − [ p (1 − δ ) − p (1 − δ ) − . (2.22)This completes the proof of Lemma 2.6. Corollary 2.7.
Assume the setting in Section 2.1, let c ∈ (0 , ∞ ) , p ∈ (1 , ∞ ) , let e : N → H be abijection which satisfies for all n ∈ N that λ e n = − cn p , and let I N ∈ P ( H ) , N ∈ N , be the setswhich satisfy for all N ∈ N that I N = { e , e , . . . , e N } ⊆ H . Then it holds for all N ∈ N that E (cid:2) (cid:107) X H (cid:107) H (cid:3) − E (cid:2) (cid:107) X I N (cid:107) H (cid:3) ≥ T inf h ∈ H | µ h | N − p c ( p − p − . (2.23)6 roof of Corollary 2.7. Proposition 2.5 and Lemma 2.6 prove for all N ∈ N that E (cid:2) (cid:107) X H (cid:107) H (cid:3) − E (cid:2) (cid:107) X I N (cid:107) H (cid:3) ≥ T inf h ∈ H | µ h | (cid:88) h ∈ H \ I N | λ h | = c − T inf h ∈ H | µ h | ∞ (cid:88) n = N +1 n p ≥ T inf h ∈ H | µ h | N − p c ( p − p − . (2.24)The proof of Corollary 2.7 is thus completed. Corollary 2.8.
Assume the setting in Section 2.1, let c, p ∈ (0 , ∞ ) , δ ∈ ( −∞ , / − / (2 p ) ) , let e : N → H be a bijection which satisfies for all n ∈ N that λ e n = − cn p , let I N ∈ P ( H ) , N ∈ N , bethe sets which satisfy for all N ∈ N that I N = { e , e , . . . , e N } ⊆ H , and assume for all h ∈ H that | µ h | = | λ h | δ . Then it holds for all N ∈ N that E (cid:2) (cid:107) X H (cid:107) H (cid:3) − E (cid:2) (cid:107) X I N (cid:107) H (cid:3) ≥ T c δ − N p (2 δ − [ p (1 − δ ) − p (1 − δ ) − . (2.25) Proof of Corollary 2.8.
Proposition 2.5, Lemma 2.4, and Lemma 2.6 show for all N ∈ N that E (cid:2) (cid:107) X H (cid:107) H (cid:3) − E (cid:2) (cid:107) X I N (cid:107) H (cid:3) = T (cid:88) h ∈ H \ I N | µ h | | λ h | = T (cid:88) h ∈ H \ I N | λ h | δ − = T c δ − ∞ (cid:88) n = N +1 n p (2 δ − ≥ T c δ − N p (2 δ − [ p (1 − δ ) − p (1 − δ ) − . (2.26)This completes the proof of Corollary 2.8. The next proposition, Proposition 2.9 below, follows directly from Lemma 2.2 above and Lemma 9.5in Jentzen & Kurniawan [5].
Proposition 2.9.
Assume the setting in Section 2.1 and let ϕ i : H → R , i ∈ { , } , be thefunctions which satisfy for all i ∈ { , } , ( v , v ) ∈ H that ϕ i ( v , v ) = exp (cid:0) −(cid:107) v i (cid:107) H / − i/ (cid:1) . Thenit holds for all i ∈ { , } , I ∈ P ( H ) that ϕ i ∈ C ( H , R ) and E [ ϕ i ( X I )] − E [ ϕ i ( X H )] ≥ E (cid:104) (cid:107) X H ,i (cid:107) H / − i/ (cid:105) − E (cid:104) (cid:107) X I,i (cid:107) H / − i/ (cid:105) exp (cid:16) E (cid:104) (cid:107) X H ,i (cid:107) H / − i/ (cid:105)(cid:17) . (2.27) Corollary 2.10.
Assume the setting in Section 2.1, let c, p ∈ (0 , ∞ ) , δ ∈ ( −∞ , / − / (2 p ) ) , let e : N → H be a bijection which satisfies for all n ∈ N that λ e n = − cn p , let I N ∈ P ( H ) , N ∈ N ,be the sets which satisfy for all N ∈ N that I N = { e , e , . . . , e N } ⊆ H , assume for all h ∈ H that | µ h | = | λ h | δ , and let ϕ i : H → R , i ∈ { , } , be the functions which satisfy for all i ∈ { , } , ( v , v ) ∈ H that ϕ i ( v , v ) = exp (cid:0) −(cid:107) v i (cid:107) H / − i/ (cid:1) . Then it holds for all i ∈ { , } , N ∈ N that ϕ i ∈ C ( H , R ) and E [ ϕ i ( X I N )] − E [ ϕ i ( X H )] ≥ (cid:20) x ∈ [2 c / T, ∞ ) sin( x )( − i x (cid:21) T c δ − p (2 δ − N p (2 δ − [ p (1 − δ ) −
1] exp (cid:0) p (2 δ − T c δ − p (2 δ − (cid:1) > . (2.28)7 roof of Corollary 2.10. Lemma 2.4, Lemma 2.6, and the fact that ∀ x ∈ (0 , ∞ ) : (cid:12)(cid:12) sin( x ) x (cid:12)(cid:12) < provefor all i ∈ { , } , N ∈ N that E (cid:104) (cid:107) X H ,i (cid:107) H / − i/ (cid:105) − E (cid:104) (cid:107) X I,i (cid:107) H / − i/ (cid:105) = 12 (cid:88) h ∈ H \ I N | µ h | | λ h | (cid:18) T + sin (cid:0) | λ h | / T (cid:1) ( − i | λ h | / (cid:19) ≥ (cid:18) h ∈ H sin (cid:0) | λ h | / T (cid:1) ( − i | λ h | / T (cid:19) T (cid:88) h ∈ H \ I N | λ h | δ − ≥ (cid:18) x ∈ [2 c / T, ∞ ) sin( x )( − i x (cid:19) T c δ − ∞ (cid:88) n = N +1 n p (2 δ − ≥ (cid:18) x ∈ [2 c / T, ∞ ) sin( x )( − i x (cid:19) T c δ − p (2 δ − N p (2 δ − [ p (1 − δ ) − > . (2.29)Furthermore, the assumption that δ ∈ ( −∞ , / − / (2 p ) ) ensures that p (2 δ − ∈ ( −∞ , − . Hence,we obtain that ∞ (cid:88) n =1 n p (2 δ − ≤ ∞ (cid:88) n =1 (cid:90) n +1 n x p (2 δ − d x = 1 + (cid:90) ∞ x p (2 δ − d x = 1 − p (2 δ −
1) + 1 = p (2 δ − p (2 δ −
1) + 1 . (2.30)Lemma 2.4 and (2.30) imply for all i ∈ { , } that exp (cid:16) − E (cid:104) (cid:107) X H ,i (cid:107) H / − i/ (cid:105)(cid:17) ≥ exp (cid:0) − E (cid:2) (cid:107) X H (cid:107) H (cid:3)(cid:1) = exp (cid:18) − T c δ − ∞ (cid:88) n =1 n p (2 δ − (cid:19) ≥ exp (cid:18) − p (2 δ − T c δ − p (2 δ −
1) + 1 (cid:19) > . (2.31)Combining this and (2.29) with Proposition 2.9 concludes the proof of Corollary 2.10.Corollary 2.11 below specifies Corollary 2.10 to the case where the linear operator A : D ( A ) ⊆ H → H in the setting in Section 2.1 is the Laplacian with Dirichlet boundary conditions on H . Itis an immediate consequence of Corollary 2.10. Corollary 2.11.
Assume the setting in Section 2.1, let δ ∈ ( −∞ , / ) , let e : N → H be a bijectionwhich satisfies for all n ∈ N that λ e n = − π n , let I N ∈ P ( H ) , N ∈ N , be the sets which satisfyfor all N ∈ N that I N = { e , e , . . . , e N } ⊆ H , assume for all h ∈ H that | µ h | = | λ h | δ , andlet ϕ i : H → R , i ∈ { , } , be the functions which satisfy for all i ∈ { , } , ( v , v ) ∈ H that ϕ i ( v , v ) = exp (cid:0) −(cid:107) v i (cid:107) H / − i/ (cid:1) . Then it holds for all i ∈ { , } , N ∈ N that ϕ i ∈ C ( H , R ) and E [ ϕ i ( X I N )] − E [ ϕ i ( X H )] ≥ (cid:20) x ∈ [2 πT, ∞ ) sin( x )( − i x (cid:21) T (4 π ) δ − N δ − [1 − δ ] exp (cid:0) δ − T π δ − δ − (cid:1) > . (2.32) Acknowledgements
This project has been partially supported through the ETH Research Grant ETH-47 15-2 “Mildstochastic calculus and numerical approximations for nonlinear stochastic evolution equations withLévy noise”. 8 eferences [1]
Conus , D.,
Jentzen , A., and Kurniawan , R. Weak convergence rates of spectral Galerkinapproximations for SPDEs with nonlinear diffusion coefficients.
ArXiv e-prints (Aug. 2014).arXiv: . Minor revision requested from Ann. Appl. Probab. [2]
Davie , A. M., and Gaines , J. G. Convergence of numerical schemes for the solution ofparabolic stochastic partial differential equations.
Math. Comp.
70, 233 (2001), 121–134. issn : 0025-5718. doi : . url : http://dx.doi.org/10.1090/S0025-5718-00-01224-2 .[3] Hausenblas , E. Weak approximation of the stochastic wave equation.
J. Comput. Appl.Math. issn : 0377-0427. doi :
10 . 1016 / j . cam . 2010 . 03 . 026 . url : http://dx.doi.org/10.1016/j.cam.2010.03.026 .[4] Jacobe de Naurois , L.,
Jentzen , A., and Welti , T. Weak convergence rates for spatialspectral Galerkin approximations of semilinear stochastic wave equations with multiplicativenoise.
ArXiv e-prints (Aug. 2015). arXiv: .[5]
Jentzen , A., and Kurniawan , R. Weak convergence rates for Euler-type approximationsof semilinear stochastic evolution equations with nonlinear diffusion coefficients.
ArXiv e-prints (Jan. 2015). arXiv: .[6]
Kovács , M.,
Lindner , F., and Schilling , R. L. Weak convergence of finite elementapproximations of linear stochastic evolution equations with additive Lévy noise.
ArXiv e-prints (Nov. 2014). arXiv: .[7]
Kovács , M.,
Larsson , S., and Lindgren , F. Weak convergence of finite element ap-proximations of linear stochastic evolution equations with additive noise.
BIT
52, 1 (2012),85–108. issn : 0006-3835. doi : . url : http://dx.doi.org/10.1007/s10543-011-0344-2 .[8] Kovács , M.,
Larsson , S., and Lindgren , F. Weak convergence of finite element ap-proximations of linear stochastic evolution equations with additive noise II. Fully discreteschemes.
BIT
53, 2 (2013), 497–525. issn : 0006-3835.[9]
Müller-Gronbach , T., and Ritter , K. Lower bounds and nonuniform time discretiza-tion for approximation of stochastic heat equations.
Found. Comput. Math.
7, 2 (2007), 135–181. issn : 1615-3375. doi : . url : http://dx.doi.org/10.1007/s10208-005-0166-6 .[10] Müller-Gronbach , T.,
Ritter , K., and Wagner , T. Optimal pointwise approximationof infinite-dimensional Ornstein-Uhlenbeck processes.
Stoch. Dyn.
8, 3 (2008), 519–541. issn :0219-4937. doi :
10 . 1142 / S0219493708002433 . url : http : / / dx . doi . org / 10 . 1142 /S0219493708002433 .[11] Sell , G. R., and You , Y.
Dynamics of evolutionary equations . Vol. 143. Applied Math-ematical Sciences. Springer-Verlag, New York, 2002, xiv+670. isbn : 0-387-98347-3. doi : . url : http://dx.doi.org/10.1007/978-1-4757-5037-9 .[12] Wang , X. An exponential integrator scheme for time discretization of nonlinear stochasticwave equation.
J. Sci. Comput.
64, 1 (2015), 234–263. issn : 0885-7474. doi :
10 . 1007 /s10915-014-9931-0 . url : http://dx.doi.org/10.1007/s10915-014-9931-0http://dx.doi.org/10.1007/s10915-014-9931-0