Stoch PDE: Anal Comp https://doi.org/10.1007/s40072-018-0116-y

Exponential moments for numerical approximations of stochastic partial differential equations Arnulf Jentzen1 · Primož Pušnik1

Received: 6 October 2016 © Springer Science+Business Media, LLC, part of Springer Nature 2018

Abstract Stochastic partial differential equations (SPDEs) have become a crucial ingredient in a number of models from economics and the natural sciences. Many SPDEs that appear in such applications include non-globally monotone nonlinearities. Solutions of SPDEs with non-globally monotone nonlinearities are in nearly all cases not known explicitly. Such SPDEs can thus only be solved approximatively and it is an important research problem to construct and analyze discrete numerical approximation schemes which converge with strong convergence rates to the solutions of such SPDEs. In the case of finite dimensional stochastic ordinary differential equations (SODEs) with non-globally monotone nonlinearities it has recently been revealed that exponential integrability properties of the discrete numerical approximation scheme are a key instrument to establish strong convergence rates for the considered approximation scheme. Exponential integrability properties for appropriate approximation schemes have been established in the literature in the case of a large class of finite dimensional SODEs. To the best of our knowledge, there exists no result in the scientific literature which proves exponential integrability properties for a time discrete approximation scheme in the case of a SPDE. In particular, to the best of our knowledge, there exists no result in the scientific literature which establishes strong convergence rates for a time discrete approximation scheme in the case of a SPDE with a non-globally monotone nonlinearity. In this paper we propose a new class of tamed space-time-noise discrete exponential Euler approximation schemes that admit exponential integrability properties in the case of SPDEs. More specifically, the main

B

Primož Pušnik [email protected] Arnulf Jentzen [email protected]

1

Department of Mathematics, Seminar for Applied Mathematics, ETH Zürich, Rämistrasse 101, 8092 Zurich, Switzerland

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result of this article proves that these approximation schemes enjoy exponential integrability properties for a large class of semilinear SPDEs with possibly non-globally monotone nonlinearities. In particular, we establish exponential moment bounds for the proposed approximation schemes in the case of stochastic Burgers equations, stochastic Kuramoto–Sivashinsky equations, and two-dimensional stochastic Navier–Stokes equations. Keywords Stochastic partial differential equation · Numerical analysis · Lyapunov function · Exponential moments · Approximation scheme

1 Introduction Stochastic partial differential equations (SPDEs) have become a crucial ingredient in a number of models from economics and the natural sciences. For example, SPDEs frequently appear in models for the approximative pricing of interest-rate based financial derivatives (cf., e.g., Theorem 3.5 in Harms et al. [26] and (1.3) in Filipovi´c et al. [21]), for the approximative description of random surfaces in surface growth models (cf., e.g., (1) in Blömker and Romito [7] and (1)–(3) in Hairer [24]), for describing the temporal dynamics associated to Euclidean quantum field theories (cf., e.g., (1.1) in Mourrat and Weber [36]), for the approximative description of velocity fields in fully developed turbulent flows (cf., e.g., (7) in Birnir [5] and (1.5) in Birnir [6]), and for the approximative description of the temporal evolution of the concentration of an undesired (chemical or biological) contaminant in water (e.g., in a water basin, the groundwater system, or a river; cf., e.g., (1.1) in Kouritzin and Long [35] and also (2.2) in Kallianpur and Xiong [34]). Many SPDEs that appear in such models include non-globally monotone nonlinearities. Solutions of SPDEs with non-globally monotone nonlinearities are in nearly all cases not known explicitly. Such SPDEs can thus only be solved approximatively and it is an important research problem to construct and analyze discrete numerical approximation schemes which converge with positive strong convergence rates to the solutions of such infinite dimensional SPDEs. In the case of finite dimensional stochastic ordinary differential equations (SODEs) with non-globally monotone nonlinearities it has recently been revealed in the literature that exponential integrability properties of the discrete numerical approximation scheme are a key ingredient to establish positive strong convergence rates for the considered approximation scheme; cf., e.g., Hutzenthaler et al. [31], Hutzenthaler and Jentzen [27], and Cozma and Reisinger [14]. In particular, e.g., Corollary 3.8 in Hutzenthaler et al. [31] and Proposition 3.4, Proposition 3.6, Proposition 3.8, and Proposition 3.9 in Cozma and Reisinger [14] (cf. also Lemma 3.6 in Bou-Rabee and Hairer [9]) establish exponential integrability properties for appropriate stopped/tamed/truncated approximation schemes in the case of a large class of finite dimensional SODEs with non-globally monotone nonlinearities. To the best of our knowledge, there exists no result in the scientific literature which proves exponential integrability properties for a time discrete approximation scheme in the case of an infinite dimensional SPDE. In particular, to the best of our knowledge, there exists no result in the scientific literature which establishes strong convergence rates for a time discrete approximation

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scheme in the case of a SPDE with a non-globally monotone nonlinearity (cf., e.g., Dörsek [19] and Hutzenthaler and Jentzen [27]). In this paper we propose a new class of tamed space-time-noise discrete exponential Euler approximation schemes that admit exponential integrability properties in the case of infinite dimensional SPDEs. More specifically, the main result of this article (see Theorem 3.3 in Sect. 3 below) proves that these approximation schemes enjoy exponential integrability properties for a large class of semilinear SPDEs with possibly non-globally monotone nonlinearities. In particular, we establish exponential moment bounds for the proposed approximation schemes in the case of stochastic Burgers equations (see Corollary 4.11 in Sect. 4.3 below), stochastic Kuramoto–Sivashinsky equations (see Corollary 4.13 in Sect. 4.4 below), and two-dimensional stochastic Navier–Stokes equations (see Corollary 4.15 in Sect. 4.5 below). In this introductory section we now illustrate the proposed approximation schemes and our main result (see Theorem 3.3) in the case of a stochastic Burgers equation (cf., e.g., Da Prato et al. [16, Section 1] and Hairer and Voss [25, Section 2]). Let T ∈ (0, ∞), δ ∈ (0, 1/18), H = L 2 ((0, 1); R), let Q ∈ L 1 (H ) be non-negative and symmetric, let (, F, P) be a probability space, let (Wt )t∈[0,T ] be an Id H -cylindrical P-Wiener process, let A : D(A) ⊆ H → H be the Laplacian with Dirichlet boundary conditions on H , let (en )n∈N ⊆ H , (Pn )n∈N ⊆ L(H ), F : W01,2 ((0, 1), R) → H , ξ ∈ W01,2 ((0, 1), R) satisfy for all n ∈ N, u ∈ H , v ∈ W01,2 ((0, 1), R) that en (·) = √  2 sin(nπ(·)), Pn (u) = nk=1 ek , u H ek , F(v) = −v · v, let W n : [0, T ] ×  → Pn (H ), n ∈ N, be stochastic processes with  t continuous sample paths which satisfy for all n ∈ N, t ∈ [0, T ] that P(Wtn = 0 Pn dWs ) = 1, and let Y N ,M : [0, T ] ×  → PN (H ), N , M ∈ N, be stochastic processes which satisfy for all N , M ∈ N, (m+1)T ] that Y0N ,M = PN (ξ ) and m ∈ {0, 1, . . . , M − 1}, t ∈ [ mT M , M YtN ,M = e(t−

mT/M )A



N ,M YmT /M

+ 1{ (−A)1/2 Y N ,M

2 Mδ δ mT /M H +1≤ /T }

+

1/2 N (WtN −WmT /M ) 1/2 N N 2 1+ PN Q (Wt −WmT /M ) H

Q

   N ,M t− PN F YmT /M

mT M

 .

 (1)

The stochastic processes Y N ,M : [0, T ] ×  → PN (H ), N , M ∈ N, can be viewed as numerical approximations of the up to modification unique mild solution process of the stochastic Burgers partial differential equation d X t (x) =



∂2 ∂x2

X t (x) − X t (x) ·

∂ ∂x

 X t (x) dt + d( QW )t (x)

(2)

with X 0 (x) = ξ(x) and X t (0) = X t (1) = 0 for t ∈ [0, T ], x ∈ (0, 1). The approximation scheme (1) is an extension of the scheme proposed in Hutzenthaler et al. [31, (6)] (cf., e.g., [3,23,28–30,32,38,39] for related approximation schemes). Observe that the scheme (1) differs in two ways from the well-known exponential Euler scheme for SPDEs (see, e.g., Da Prato et al. [17, (130)]). The nonlinear drift coefficient and the diffusion coefficient are truncated by means of the indicator function appearing

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in (1) and, additionally, the diffusion coefficient is tamed in an appropriate way. These modifications of the classical exponential Euler scheme allow us in Corollary 4.11 in Sect. 4.3 below to demonstrate that the approximation scheme (1) enjoys finite exponential moments. More precisely, Corollary 4.11 in Sect. 4.3 proves1 that for all ε ∈ [1, ∞) it holds that sup

 sup E exp

N ,M∈N t∈[0,T ]

ε YtN ,M 2H e2ε trace H (Q)t

 < ∞.

(3)

Corollary 4.11 follows from an application of Corollary 3.4 below (see Sect. 4.3 below for details). Corollary 3.4, in turn, is a direct consequence of Theorem 3.3, which is the main result of this article. Theorem 3.3 establishes exponential integrability properties for a more general class of semilinear SPDEs (such as stochastic Burgers equations with non-additive noise, stochastic Kuramoto–Sivashinsky equations, and two-dimensional stochastic Navier–Stokes equations on a torus) as well as for a more general type of approximation schemes. Exponential integrability properties such as (3) are a key instrument to establish strong convergence rates for SPDEs with non-globally monotone nonlinearities (cf. [27]). In particular, we intend to use (3) and Theorem 3.3, respectively, in succeeding articles to establish strong convergence rates for numerical approximations of stochastic Burgers equations and other SPDEs with non-globally monotone nonlinearities. We also would like to point out that the schemes proposed in this article (cf., e.g., (1) above and (107) in Theorem 3.3 below) employ spatial spectral Galerkin approximations to discretize the infinite dimensional state space H = L 2 ((0, 1); R) of the considered SPDE (2). Spatial spectral Galerkin approximations are, however, only implementable in the case of simple domains where the eigenvectors of the dominant linear operator A are explicitly known. The schemes proposed in this article (cf., e.g., (1) above and (107) in Theorem 3.3 below) are thus only implementable in the case of simple domains where the eigenvectors of the dominant linear operator are explicitly known. It remains the subject of future research to extend the analysis and the schemes in this work to more complicated spatial approximations like finite element methods which are also implementable in the case of more complicated domains where the eigenvectors of the dominant linear operator A are not explicitly known. While polynomial moment bounds for numerical approximations of infinite dimensional SPDEs and exponential moment bounds for numerical approximations of finite dimensional SODEs have been established in the scientific literature, Theorem 3.3 is—to the best of our knowledge—the first result in the literature which establishes exponential moment bounds for time discrete numerical approximations in the case 1 (with d = 1, D = (0, 1), η = 0, γ

√

= 1/2, T = T , ε = ε − 1/ 3, δ = δ, U = H , H = H , H = {en : n ∈ N}, U = {en : n ∈ N}, λe N = −π 2 N 2 , (, F , P, (Ft )t∈[0,T ] ) = (, F , P, (σ ((Ws )s∈[0,t] ))t∈[0,T ] ), W = W , Q = Q, A = A, r = (H1/2 v → √ √ 2ε max{1, trace H (Q)} + 2ε max{1, trace H (Q)} (−A)1/2 v 2H ∈ [0, ∞)), b = ((0, 1) × R (x, y) → √ 1 ∈ R), ϑ = trace H (Q), c = 2ε max{1, trace H (Q)}, R = Id H , F = F, ξ = ( ω → ξ ∈ 1,2 {0,T /M,...,T },{e ,...,e },{e 1 1 ,...,e N } = Y N ,M for N , M ∈ N, ε ∈ [1, ∞) in the notation N W0 ((0, 1), R)), Y of Corollary 4.11).

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of infinite dimensional SPDEs. In particular, Theorem 3.3 and its consequences in Corollaries 3.4, 4.11, 4.13, and 4.15, respectively, are—to the best of our knowledge— the first results in the literature that establish exponential integrability properties for time discrete numerical approximations of stochastic Burgers equations, stochastic Kuramoto Sivashinsky equations, and two-dimensional stochastic Navier Stokes equations.

1.1 Notation Throughout this article the following notation is used. For all sets A and B we denote by M(A, B) the set of all functions from A to B. For every topological space (X, τ ) and every set D ⊆ X we denote by D˚ ⊆ X the interior of D. For every natural number k ∈ N and all normed R-vector spaces (U, · U ) and (V, · V ) we denote by L (k) (U, V ) the set of all continuous k-linear functions from U k to V , we denote by · L (k) (U,V ) : L (k) (U, V ) → [0, ∞) the function which satisfies for all A ∈   1 ,u 2 ,...,u k ) V L (k) (U, V ) that A L (k) (U,V ) = supu 1 ,u 2 ,...,u k ∈U \{0} u A(u , we denote 1 U · u 2 U ··· u k U (0) (0) by L (U, V ) the set given by L (U, V ) = V , and we denote by · L (0) (U,V ) : V → [0, ∞) the function which satisfies for all v ∈ V that v L (0) (U,V ) = v V . For all measurable spaces (1 , F1 ) and (2 , F2 ) we denote by M(F1 , F2 ) the set of all F1 /F2 -measurable functions. For every normed R-vector space (V, · V ), every measure space (, F, μ), every real number p ∈ (0, ∞), and every measurable function f ∈ M(F, B(V )) we denote by f L p (μ;V ) ∈ [0, ∞] and f L∞ (μ;V ) ∈ [0, ∞]  1/p p the extended real numbers given by f L p (μ;V ) =  f (ω) V μ(dω) and

f L∞ (μ;V ) = inf{c ∈ [0, ∞) : μ({v ∈ V : | f (v)| > c}) = 0}. For every topological space (X, τ ) we denote by B(X ) the sigma-algebra of all Borel measurable sets in X . For every natural number d ∈ N and every Borel measurable set A ∈ B(Rd ) we denote by μ A : B(A) → [0, ∞] the Lebesgue-Borel measure on A ⊆ Rd . For all R-Hilbert spaces (H, ·, · H , · H ) and (U, ·, ·U , · U ), every set H ∈ P(H ), and all functions F : H → H and B : H → HS(U, H ) we denote by G F,B : C 2 (H, R) → M(H, R) the function which satisfies for all x ∈ H, φ ∈ C 2 (H, R) that (G F,B φ)(x) = F(x), (∇φ)(x) H +

1 2

  trace H B(x)B(x)∗ (Hess φ)(x) .

(4)

For every set X we denote by P(X ) the power set of X , we denote by # X ∈ N0 ∪ {∞} the number of elements of X , and we denote by P0 (X ) the set given by P0 (X ) = {θ ∈ P(X ) : #θ < ∞}. For all sets  and A ⊆ P() we denote by σ (A) the smallest sigma-algebra on  which contains A. For every normed R-vector space (V, · V ) with #V > 1, all real numbers n ∈ N, c ∈ [1, ∞), every set B ⊆ R, and every open and convex set A ⊆ V we denote by Ccn (A, B) the set given by  Ccn (A,

B) =

f ∈C

n−1

 ∀ x, y ∈ A, i ∈ N0 ∩ [0, n) : f (i) (x) − f (i) (y) L (i) (H,R) (A, B) : ≤ c x − y H (1 + supr ∈[0,1] | f (r x + (1 − r )y)|)1−1/c

(5)

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(cf., e.g., (1.12) in Hutzenthaler and Jentzen [28]). We denote by (·) ∧ (·) : R2 → R the function which satisfies for all x, y ∈ R that x ∧ y = min{x, y}. For every real number T ∈ (0, ∞) we denote by T the set given by T = {θ ⊆ [0, T ] : {0, T } ⊆ θ and #θ < ∞}. For every real number T ∈ (0, ∞) we denote by |·|T : T → [0, T ] the function which satisfies for all θ ∈ T that  

 |θ |T = max x ∈ (0, ∞) : ∃ a, b ∈ θ : x = b − a and θ ∩ (a, b) = ∅ ∈ (0, T ]. (6) For every θ ∈ (∪T ∈(0,∞) T ) we denote by ·θ : [0, ∞) → [0, ∞) and ·θ : [0, ∞) → [0, ∞) the functions which satisfy for all t ∈ (0, ∞) that tθ = max([0, t] ∩ θ ), tθ = max([0, t) ∩ θ ), and 0θ = 0θ = 0. For every measure space (, F, μ), every measurable space (S, S), every set R, and every function f :  → R we denote by [ f ]μ,S the set given by [ f ]μ,S = {g ∈ M(F, S) : (∃ A ∈ F : μ(A) = 0 and {ω ∈  : f (ω) = g(ω)} ⊆ A)}.

2 Exponential moments for time discrete approximation schemes 2.1 From one-step estimates to exponential moments In this subsection we establish in Corollary 2.2 below exponential integral properties for approximation schemes [see (12) in Corollary 2.2] under a general one-step condition on the considered approximation scheme [see (11) in Corollary 2.2 below]. We will verify this one-step condition for a specific class of approximation schemes in Sect. 2.2 below. Corollary 2.2 is an extension of Corollary 2.3 in Hutzenthaler et al. [31]. Our proof of Corollary 2.2 is based on an application of Lemma 2.1 below. Lemma 2.1, in turn, is, e.g., proved as Lemma 2.2 in Hutzenthaler et al. [31]. For completeness the proof of Lemma 2.1 is also provided in this article. Lemma 2.1 Consider the notation in Sect. 1.1, let (, F, P) be a probability space, let (E, E) be a measurable space, let T, ρ ∈ [0, ∞), θ ∈ T , c ∈ R, V, V¯ ∈ M(E, B(R)), A ∈ E, and let Y : [0, T ] ×  → Ebe a product measurable stochastic t process which satisfies for all t ∈ [0, T ] that P( 0 1 A (Yr θ )|V¯ (Yr )| dr < ∞) = 1 and  

  t 1 A (Yr θ )V¯ (Yr )  V (Yt ) dr (Yr )r ∈[0,tθ ] E exp − ct + eρt + eρr 0    tθ 1 A (Yr θ )V¯ (Yr ) V (Ytθ ) dr . ≤ exp − c tθ + eρtθ + eρr

(7)

0

Then it holds for all t ∈ [0, T ] that

  t) E exp Ve(Y + ρt

0

123

t

1 A (Yr θ )V¯ (Yr ) eρr

 dr

 ≤ ect E e V (Y0 ) .

(8)

Stoch PDE: Anal Comp

Proof of Lemma 2.1 Assumption 7 implies for all t ∈ [0, T ] that

   t 1 A (Yr θ )V¯ (Yr ) V (Yt ) E exp − ct + eρt + dr eρr 0 

   t 1 A (Yr θ )V¯ (Yr )  V (Yt ) = E E exp − ct + eρt + dr (Ys )s∈[0,tθ ] eρr 0 

  tθ 1 A (Yr θ )V¯ (Yr ) V (Ytθ ) dr ≤ E exp − c tθ + eρtθ + eρr 0     0) = E e V (Y0 ) . ≤ · · · ≤ E exp Ve(Y ρ·0

(9)  

This completes the proof of Lemma 2.1.

Corollary 2.2 Consider the notation in Sect. 1.1, let (H, ·, · H , · H ) and (U, ·, ·U ,

· U ) be separable R-Hilbert spaces, let T ∈ (0, ∞), θ ∈ T , ρ, c ∈ [0, ∞), V ∈   M B(H ), B([0, ∞)) , V¯ ∈ M B(H ), B(R) ,  ∈ M B(H × [0, T ] × U ), B(H ) , E ∈ B(H ), S ∈ M((0, T ], L(H )), let (, F, P, (Ft )t∈[0,T ] ) be a filtered probability space, let W : [0, T ] ×  → U be an IdU -cylindrical (Ft )t∈[0,T  ] -Wiener process with continuous sample paths, let Y ∈ M B([0, T ]) ⊗ F, B(H ) be an (Ft )t∈[0,T ] adapted stochastic process, assume for all t ∈ (0, T ], x ∈ H that V (St x) ≤ V (x), V¯ (St x) ≤ V¯ (x), and

  Yt = St−tθ 1 H \E (Ytθ ) · Ytθ + 1 E (Ytθ ) ·  Ytθ , t − tθ , Wt − Wtθ , (10) and assume for all x ∈ E, t ∈ (0, |θ |T ] that ∫0T 1 E (Ysθ ) |V¯ (Ys )| ds +  |θ|T |V¯ ((x, s, Ws ))| ds < ∞ and 0

 t t )) +∫ E exp V ((x,t,W eρt

V¯ ((x,s,Ws )) eρs



≤ ect+V (x) .

(11)

   t 1 (Y E sθ ) V¯ (Ys ) V (Yt ) ct V (Y0 ) . E exp eρt + ∫ ds ≤ e E e ρs e

(12)

0

ds

Then it holds for all t ∈ [0, T ] that

0

Proof of Corollary 2.2 We prove Corollary 2.2 through an application of Lemma 2.1. Assumption (11) implies that for all t ∈ (0, |θ |T ], x ∈ H it holds that

  t 1 E (x) V¯ ((x,s,Ws )) t )) ∫ E exp 1 E (x) V ((x,t,W + ds ≤ ect+1 E (x) V (x) . eρs eρt

(13)

0

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Stoch PDE: Anal Comp

Next note that (10) ensures that for all t ∈ (0, T ] it holds that 

  t 1 (Y 1 (Yt ) V (Yt )  t ) V¯ (Ys ) E exp E eρtθ + ∫ E eρsθ ds  (Ys )s∈[0,tθ ] tθ

 1 (Ytθ ) V (St−tθ (Ytθ ,t−tθ ,Wt −Wtθ )) = E exp E eρt   t 1 (Y  tθ ) V¯ (Ss−tθ (Ytθ ,s−tθ ,Ws −Wtθ )) + ∫ E ds (Y )  s s∈[0,tθ ] eρs tθ

  1 (Ytθ ) V ((Ytθ ,t−tθ ,Wt −Wtθ )) ≤ E exp E ρ(t−tθ ) e

+

t−tθ 1 (Y E tθ ) V¯ ((Ytθ ,s,Wtθ +s −Wtθ )) ∫ eρs 0

ds

 exp(−ρtθ )    (Ys )s∈[0,tθ ] . (14)

Jensen’s inequality, (13), and, e.g., Lemma 2.9 in Jentzen and Pušnik [33] hence imply for all t ∈ (0, T ] that

   t 1 (Y 1 (Yt ) V (Yt )  t ) V¯ (Ys ) E exp E eρtθ + ∫ E eρsθ ds  (Ys )s∈[0,tθ ] tθ    1 (Ytθ ) V ((Ytθ ,t−tθ ,Wt −Wtθ )) ≤  E exp E ρ(t−tθ ) e

exp(−ρtθ )     (Ys )s∈[0,tθ ] 

t−tθ 1 (Y E tθ ) V¯ ((Ytθ ,s,Wtθ +s −Wtθ )) + ∫ ds eρs 0 e−ρtθ    c(t−t )+1 (Y ) V (Y ) t t θ E   θ θ ≤ e P,B([0,∞]) 

   1 (Ytθ ) V (Ytθ ) ≤ exp c(t − tθ ) + E eρt θ

P,B([0,∞])

.

(15)

This and (10) show for all t ∈ (0, T ] that

 E exp −ct +

V (Yt ) eρt

t 1 (Y E r θ ) V¯ (Yr ) eρr 0

+∫

   dr  (Yr )r∈[0,tθ ]

 

 t 1 (Yt ) V (Yt ) 1 (Yt ) V (Yt ) 1 E (Ytθ ) V¯ (Yr )  + H \E eρt θ + ∫ dr  (Yr )r∈[0,tθ ] = E exp E eρtθ eρr   tθ 1 (Y E r θ ) V¯ (Yr ) dr · exp −ct + ∫ ρr e

tθ

0

 

 t 1 (Yt ) V (Yt ) 1 (Ytθ ) V (St−tθ Ytθ ) 1 E (Ytθ ) V¯ (Yr )  + H \E + ∫ dr  (Yr )r∈[0,tθ ] = E exp E eρtθ eρr eρt tθ

  tθ 1 (Y E r θ ) V¯ (Yr ) dr · exp −ct + ∫ ρr e 0



 tθ 1 (Y 1 (Ytθ ) V (Ytθ ) 1 (Ytθ ) V (Ytθ ) E r θ ) V¯ (Yr ) + H \E − ct + ∫ dr ≤ exp c(t − tθ ) + E eρt θ eρr eρt

 ≤ exp −ctθ +

123

V (Ytθ ) eρtθ

tθ 1 (Y E r θ ) V¯ (Yr ) + ∫ eρr 0

0

 dr P,B([0,∞])

.

P,B([0,∞])

(16)

Stoch PDE: Anal Comp

Combining Lemma 2.1 and (16) establishes (12). The proof of Corollary 2.2 is thus completed.   Remark 2.3 Let (U, ·, ·U , · U ) be a separable R-Hilbert space, let T ∈ (0, ∞), let (, F, P) be a probability space, and let W : [0, T ] ×  → U be an IdU -cylindrical P-Wiener process. Then dim(U ) < ∞. 2.2 A one-step estimate for exponential moments In this subsection we establish in (39) in Lemma 2.13 below an appropriate exponential one-step estimate for a general class of one-step approximation schemes. This exponential one-step estimate and Corollary 2.2 above (cf. (39) in Lemma 2.13 below with (11) in Corollary 2.2 above) will allow us to establish exponential integrability properties for some tamed approximation schemes in Sect. 2.3 below. Lemma 2.13 below extends Lemma 2.8 in Hutzenthaler et al. [31] from finite dimensional stochastic ordinary differential equations to infinite dimensional stochastic partial differential equations. Our proof of Lemma 2.13 exploits several elementary/well known auxiliary lemmas (see Lemmas 2.4–2.12 below). Lemma 2.4 below is a straightforward extension of Lemma 2.5 in Hutzenthaler et al. [31]. Lemma 2.5 is, e.g., proved as Lemma 2.13 in Jentzen and Pušnik [33]. Lemma 2.7 is well known in the literature (cf., e.g., Bogachev [8, Theorem 5.8.12]). Lemma 2.4 Let (H, ·, · H , · H ) and (U, ·, ·U , · U ) be separable R-Hilbert spaces, let T ∈ (0, ∞), B ∈ HS(U, H ), let (, F, P) be a probability space, and let (Wt )t∈[0,T ] be an IdU -cylindrical P-Wiener process. Then it holds for all t ∈ [0, T ] that   

 E exp ∫t0 B dWs H ≤ 2 exp 2t B 2HS(U,H ) .

(17)

Lemma 2.5 Let (E, d E ), (F, d F ), and (G, dG ) be metric spaces and let f : E → F and g : F → G be locally Lipschitz continuous functions. Then it holds that g ◦ f : E → G is a locally Lipschitz continuous function. Lemma 2.6 Consider the notation in Sect. 1.1, let (V, · V ) be a normed R-vector space with #V > 1, and let c ∈ [1, ∞), n ∈ N0 , U ∈ Ccn+1 (V, R), i ∈ N0 ∩ [0, n]. Then it holds that U (i) is locally Lipschitz continuous. Proof of Lemma 2.6 The fact that U is continuous proves that for every (x, ε) ∈ V × (0, ∞) there exists a real number δx,ε ∈ (0, ∞) such that for all v ∈ V with

x − v V < δx,ε it holds that |U (x) − U (v)| < ε. This and the triangle inequality prove that for all x, x1 , x2 ∈ V with max{ x − x1 V , x − x2 V } < δx,1 it holds that

U (i) (x1 ) − U (i) (x2 ) L (i) (V,R) ≤ c x1 − x2 V (1 + supr ∈[0,1] |U (r x1 + (1 − r )x2 )|) ≤ c x1 − x2 V (1 + |U (x)| + supr ∈[0,1] |U (r x1 + (1 − r )x2 ) − U (x)|) ≤ c x1 − x2 V (2 + |U (x)|).

The proof of Lemma 2.6 is thus completed.

(18)  

123

Stoch PDE: Anal Comp

Lemma 2.7 Consider the notation in Sect. 1.1, let a ∈ R, b ∈ (a, ∞), and let f ∈ C([a, b], R) be a locally Lipschitz continuous function. Then (i) it holds that {s ∈ [a, b] : f is differentiable at s} ∈ B(R), (ii) it holds that μR ([a, b]\{s ∈ [a, b] : f is differentiable at s}) = 0, and (iii) it holds that f is absolutely continuous. Lemma 2.8 Consider the notation in Sect. 1.1, let (H, ·, · H , · H ) be an R-Hilbert space with # H > 1, and let c ∈ [1, ∞), n ∈ N0 , x, y ∈ H , V ∈ Ccn+1 (H, [0, ∞)). Then (i) it holds for all t ∈ {s ∈ [0, 1] : R u → V (x + uy) ∈ R is differentiable at s} that | ∂t∂ V (x + t y)| ≤ c y H (1 + V (x + t y))1−1/c and (ii) it holds for all i ∈ N ∩ [0, n], z 1 , . . . , z i ∈ H , t ∈ {s ∈ [0, 1] : R u → V (i) (x + uy)(z 1 , . . . , z i ) ∈ R is differentiable at s} that     ∂ 1  ∂t V (i) (x + t y)(z 1 , . . . , z i )  ≤ c z 1 H · · · z i H y H (1 + V (x + t y))1− /c .

(19) Proof of Lemma 2.8 First of all, note that the assumption that V ∈ Ccn+1 (H, [0, ∞)) ensures that for all t ∈ [0, 1], h ∈ R it holds that |V (x + t y) − V (x + (t + h)y)| ≤ c|h| y H [1 + supr ∈[0,1] V (x + (t + (1 − r )h)y)]1− /c 1

= c|h| y H [1 + supr ∈[0,1] V (x + (t + r h)y)]

1−1/c

(20)

.

Next observe that Lemma 2.6 ensures for all t ∈ [0, 1] that lim sup(R\{0}) h→0 | supr ∈[0,1] V (x + (t + r h)y) − V (x + t y)| = 0.

(21)

Combining this with (20) proves (i). In the next step observe that for all i ∈ N ∩ [0, n], z 1 , . . . , z i ∈ H \{0}, t ∈ [0, 1], h ∈ R it holds that |V (i) (x+t y)(z 1 ,...,z i )−V (i) (x+(t+h)y)(z 1 ,...,z i )|

z 1 H ··· z i H (i) (i)

≤ V

(x + t y) − V

(x + (t + h)y) L (i) (H,R)

≤ c|h| y H [1 + supr ∈[0,1] V (x + (t + (1 − r )h)y)]1− /c 1

(22)

= c|h| y H [1 + supr ∈[0,1] V (x + (t + r h)y)]1− /c . 1

This and (21) establish (ii). The proof of Lemma 2.8 is thus completed.

123

 

Stoch PDE: Anal Comp

Lemma 2.9 Consider the notation in Sect. 1.1, let (H, ·, · H , · H ) be an R-Hilbert space with # H > 1, and let c ∈ [1, ∞), x, y ∈ H , V ∈ Cc1 (H, [0, ∞)). Then 1 + V (x + y) ≤ 2c−1 (1 + V (x) + y cH ). Proof of Lemma 2.9 Throughout this proof let f : R → R be the function which satisfies for all t ∈ R that f (t) = V (x + t y). Next observe that item (i) of Lemma 2.8 implies that for all t ∈ {s ∈ [0, 1] : R u → f (u) ∈ R is differentiable at s} it holds that | ∂t∂ (1 + f (t))| ≤ c y H (1 + f (t))1− /c . 1

(23)

Lemma 2.11 in Hutzenthaler and Jentzen [28] and Lemmas 2.5, 2.6, and 2.7 hence prove for all t ∈ [0, 1] that

 1 + f (t) ≤ 2c−1 1 + f (0) + t c y cH .

(24)

 This implies that 1 + V (x + y) ≤ 2c−1 1 + V (x) + y cH . The proof of Lemma 2.9 is thus completed.   Lemma 2.10 Consider the notation in Sect. 1.1, let (H, ·, · H , · H ) be an R-Hilbert space with # H > 1, and let c ∈ [1, ∞), n ∈ N0 , x, y ∈ H , V ∈ Ccn+1 (H, [0, ∞)). Then maxi∈{0,1,...,n} V (i) (x) − V (i) (y) L (i) (H,R)   . ≤ c 2c−1 x − y H 1 + V (x) + x − y c−1 H

(25)

Proof of Lemma 2.10 First, note that Lemmas 2.5, 2.6, and 2.7, item (i) of Lemma 2.8, Lemma 2.9, and the fact that ∀ r ∈ [0, 1], a ∈ [1, ∞), b ∈ [0, ∞) : (a + b)r ≤ a + br prove that |V (y) − V (x)|  ≤ ≤ ≤

{s∈[0,1] : R u →V (x+u(y−x))∈R is differentiable at s}  1 1 c x − y H [1 + V (x + r (y − x))]1− /c dr 0   . c 2c−1 x − y H 1 + V (x) + y − x c−1 H

 ∂  V (x + r (y − x)) dr ∂r (26)

Moreover, Lemmas 2.5, 2.6, and 2.7, item (ii) of Lemma 2.8, Lemma 2.9, and the fact that ∀ r ∈ [0, 1], a ∈ [1, ∞), b ∈ [0, ∞) : (a + b)r ≤ a + br ensure that for all i ∈ N ∩ [0, n], z 1 , . . . , z i ∈ H \{0} it holds that |(V (i) (y)−V (i) (x))(z 1 ,...,z i )|

z 1 H ··· z i H



·

≤

1

z 1 H ··· z i H

{s∈[0,1] : R u →V (i) (x+u(y−x))(z 1 ,...,z i )∈R is differentiable at s}

123

Stoch PDE: Anal Comp

 ∂  (i)   V (x + r (y − x))(z 1 , . . . , z i )  dr ∂r  1 1 ≤ c x − y H [1 + V (x + r (y − x))]1− /c dr 0   ≤ c 2c−1 x − y H 1 + V (x) + y − x c−1 . H

(27)

Combining this with (26) completes the proof of Lemma 2.10.

 

Lemma 2.11 Consider the notation in Sect. 1.1, let (H, ·, · H , · H ) be an R-Hilbert space with # H > 1, and let c ∈ [1, ∞), n ∈ N0 , x ∈ H , V ∈ Ccn+1 (H, [0, ∞)). Then maxi∈{0,1,...,n} V (i) (x) L (i) (H,R) ≤ c (1 + V (x)). Proof of Lemma 2.11 Lemma 2.10 proves for all y ∈ H , t ∈ [0, 1] that   . (28) |V (t x + (1 − t)y) − V (x)| ≤ c 2c−1 x − y H 1 + V (x) + x − y c−1 H This implies for all ε ∈ (0, ∞), y ∈ H with x − y H < ε that   | supr ∈[0,1] V (r x + (1 − r )y) − V (x)| ≤ c 2c−1 ε 1 + V (x) + εc−1 .

(29)

Hence, we obtain that   lim sup y→x supr ∈[0,1] V (r x + (1 − r )y) − V (x) = 0.

(30)

Moreover, the assumption that V ∈ Ccn+1 (H, [0, ∞)) assures for all i ∈ N0 ∩ [0, n], y ∈ H that

V (i) (x) − V (i) (y) L (i) (H,R) ≤ c x − y H (1 + supr ∈[0,1] V (r x + (1 − r )y))1− /c . (31) 1

 

Combining this with (30) completes the proof of Lemma 2.11.

Lemma 2.12 Consider the notation in Sect. 1.1, let (V, · V ) be a normed R-vector space, let (, F, μ) be a finite measure space, and let X ∈ M(F, B(V )). Then lim inf p→∞ X L p (μ;V ) = lim sup p→∞ X L p (μ;V ) = X L∞ (μ;V ) . Proof of Lemma 2.12 Throughout this proof assume w.l.o.g. that X L∞ (μ;V ) > 0 and let Aδ ⊆ , δ ∈ (0, ∞), be the sets with the property that for all δ ∈ (0, ∞) it holds that Aδ = {ω ∈  : X (ω) V ≥ δ}. Next observe that for all p ∈ (0, ∞), δ ∈ (0, X L∞ (μ;V ) ) it holds that

X L p (μ;V ) ≥ X 1 Aδ L p (μ;V ) ≥ δ 1 Aδ L p (μ;R) = δ [μ(Aδ )] /p . 1

(32)

Hence, we obtain for all δ ∈ (0, X L∞ (μ;V ) ) that lim inf p→∞ X L p (μ;V ) ≥ δ. This shows that lim inf X L p (μ;V ) ≥ X L∞ (μ;V ) . p→∞

123

(33)

Stoch PDE: Anal Comp

Moreover, note that for all p ∈ (0, ∞), q ∈ (0, p) it holds that 

X L p (μ;V ) =

q



p−q

X (ω) V X (ω) V

1 p ( p−q)/ p q/ p μ(dω) ≤ X L∞ (μ;V ) X Lq (μ;V ) . (34)

This implies that lim sup p→∞ X L p (μ;V ) ≤ X L∞ (μ;V ) . The proof of Lemma 2.12 is thus completed.   Lemma 2.13 Consider the notation in Sect. 1.1, let (H, ·, · H , · H ) and  (U, ·, ·U ,

· U ) be separable R-Hilbert spaces > 1 < # , let F ∈ M B(H ), B(H ) , with # H U  B ∈ M B(H ), B(HS(U, H )) , ς, h ∈ (0, ∞), c, γ0 , γ1 ∈ [1, ∞), ρ, δ ∈ [0, ∞), γ2 ∈ [0, 1/2], x ∈ H , V¯ ∈ C(H, R), V ∈ Cc3 (H, [0, ∞)),  ∈ C 1,2 ([0, h] × U, H ), let (, F, P) be a probability space, let W : [0, h] ×  → U be an IdU -cylindrical PWiener process with continuous sample paths, assume for all r ∈ [1, ∞), s ∈ (0, h], y, z ∈ H that |V¯ (y) − V¯ (z)| ≤ c (1 + |V (y)|γ0 + |V (z)|γ0 ) y − z H , |V¯ (y)| ≤ c (1 + |V (y)|γ1 ), max{ F(x) H , B(x) HS(U,H ) } ≤ ch −δ , V (x) ≤ ch −ς , (0, 0) = x, and 2  (G F,B V )(x) + 21  B(x)∗ (∇V )(x)U + V¯ (x) ≤ ρV (x), (35)   ∂    ∂   max  ∂s  (s, Ws ) − F(x)L4 (P;H ) ,  ∂ y  (s, Ws ) − B(x)L8 (P;HS(U,H ))

≤ cs γ2 , (36)     ∂ 2  γ   2  (s, Ws ) (u, u) L4 (P;H ) ≤ cs , (37) u∈U ∂ y2         (s, Ws ) − x  r L (P;H ) ≤ c min 1, F(x) H s + B(x)Ws H Lr (P;R) . (38) Then it holds for all t ∈ (0, h] that

  t ¯ V ((s,Ws )) t )) ∫ E exp V ((t,W + ds ρs ρt e e 0      t [2c](2c+13)γ0 [max{s, ρ, 1}]4 [2c](2c+6)γ1 s V (x) ≤e 1+ exp ds . min{s 2δ+max{2,γ1 }ς , 1} [min{s, 1}]ς +ς γ0 +4δ−γ2 0

(39) Proof of Lemma 2.13 Throughout this proof let U ∈ P(U ) be an orthonormal basis of U , let Y : [0, h] ×  → H be the function which satisfies for all s ∈ [0, h] that Ys = (s, Ws ), and let τn :  → [0, h], n ∈ N, be the functions which satisfy for all n ∈ N that τn = inf({s ∈ [0, h] : Ws U > n} ∪ {h}). Next observe that P(Y0 = x) = P((0, W0 ) = x) = P((0, 0) = x) = 1. Itô’s formula hence implies for all t ∈ [0, h] that

123

Stoch PDE: Anal Comp

    t exp e−ρt V (Yt ) + e−ρr V¯ (Yr ) dr − e V (x) =

 t

0

  exp e−ρs V (Ys ) +

0

s

e

0



  exp e−ρs V (Ys ) +

t

+ 0

∂   (s, Ws ) + + V (Ys ) ∂s

+

−ρr

 

1  2eρs



∂ ∂y 

s



   V¯ (Yr ) dr e−ρs V (Ys ) ∂∂y  (s, Ws ) dWs

 e−ρr V¯ (Yr ) dr e−ρs V¯ (Ys ) − ρV (Ys )

0 1 2

P,B(R)

 

traceU

 ∗   (s, Ws ) (Hess V )(Ys ) ∂∂y  (s, Ws )    2    V (Ys ) ∂∂y 2  (s, Ws ) (u, u) ds



∂ ∂y 

2 ∗ (s, Ws ) (∇V )(Ys )U +

1 2

u∈U

P,B(R)

.

(40) Therefore, we obtain for all t ∈ [0, h], n ∈ N that

  −ρ(t∧τn ) E exp e V (Yt∧τn ) +

 0

+ V (Ys ) +



 

1  2eρs

e

∂ ∂s 

∂ ∂y 

s

(s, Ws ) +

1 2

V¯ (Yr ) dr



− e V (x)

  e−ρr V¯ (Yr ) dr e−ρs V¯ (Ys ) − ρV (Ys )

0





−ρr

0

  exp e−ρs V (Ys ) +

t∧τn

=E

t∧τn

traceU

 

∂ ∂y 

2 ∗ (s, Ws ) (∇V )(Ys )U +

1 2



 ∗   (s, Ws ) (Hess V )(Ys ) ∂∂y  (s, Ws )    2   V (Ys ) ∂∂y 2  (s, Ws ) (u, u) ds .

u∈U

This implies for all t ∈ [0, h], n ∈ N that

(41)



  t∧τn e−ρr V¯ (Yr ) dr − e V (x) E exp e−ρ(t∧τn ) V (Yt∧τn ) + 0 

 t∧τn   s −ρs −ρr ¯ −ρs (G F,B V )(x) =E exp e V (Ys ) + ∫ e V (Yr ) dr e 0

0

   B(x)∗ (∇V )(x)2 + V¯ (x) − ρV (x) + V¯ (Ys ) − V¯ (x) + U     ∗ ∂  1 ∂ − ρ(V (Ys ) − V (x)) + 2 traceU  (s, W ) (Hess V )(Y )  (s, W ) s s s ∂y ∂y  ∂   ∗ 1 + V (Ys ) ∂s  (s, Ws ) − V (x)F(x) − 2 traceU B(x) (Hess V )(x)B(x) 2 2     ∗ + 2e1ρs  ∂∂y  (s, Ws ) (∇V )(Ys )U − 2e1ρs  B(x)∗ (∇V )(x)U    2   + 21 V (Ys ) ∂∂y 2  (s, Ws ) (u, u) ds . (42) 1 2eρs

u∈U

Assumption (35) hence proves for all t ∈ [0, h], n ∈ N that

   t∧τn −ρ(t∧τn ) −ρr ¯ E exp e V (Yt∧τn ) + e − e V (x) V (Yr ) dr 0

 t∧τn   s ≤E exp e−ρs V (Ys ) + ∫ e−ρr V¯ (Yr ) dr 0

123

0

Stoch PDE: Anal Comp



      V¯ (Ys ) − V¯ (x) + ρ|V (Ys ) − V (x)| + V (Ys ) ∂  (s, Ws ) − V (x)F(x) ∂s       ∗ ∂  1 ∂ + 2  traceU  (s, W ) (Hess V )(Y )  (s, W ) s s s ∂y ∂y       2     V (Ys ) ∂∂y 2  (s, Ws ) (u, u) − traceU B(x)∗ (Hess V )(x)B(x)  + 21  u∈U     2  2   ∗  + 2e1ρs  ∂∂y  (s, Ws ) (∇V )(Ys )U −  B(x)∗ (∇V )(x)U  ds . (43)

·

Fatou’s lemma therefore shows for all t ∈ [0, h] that

   t −ρt −ρr ¯ E exp e V (Yt ) + e − e V (x) V (Yr ) dr 0

   t∧τn −ρ(t∧τn ) −ρr ¯ ≤ lim inf E exp e V (Yt∧τn ) + e − e V (x) V (Yr ) dr n→∞ 0

 t   s   −ρr ¯ V¯ (Ys ) − V¯ (x) + ρ|V (Ys ) − V (x)| ≤E exp V (Ys ) + ∫ e V (Yr ) dr 0 0    1  ∂     + V (Ys ) ∂s  (s, Ws ) − V (x)F(x) + 2 traceU B(x)∗ (Hess V )(x)B(x)     ∗ ∂  ∂  − traceU ∂ y  (s, Ws ) (Hess V )(Ys ) ∂ y  (s, Ws )    2  2   ∗  + 2e1ρs  ∂∂y  (s, Ws ) (∇V )(Ys )U −  B(x)∗ (∇V )(x)U      ∂ 2    1 (44) + 2 V (Ys ) ∂ y 2  (s, Ws ) (u, u) ds . u∈U

Tonelli’s theorem and Hölder’s inequality hence imply for all t ∈ [0, h] that

   t −ρt −ρr ¯ E exp e V (Yt ) + e − e V (x) V (Yr ) dr 0   t   s    −ρr ¯   exp V (Y ρ V (Ys ) − V (x) L2 (P;R) ≤ ) + e ) dr V (Y s r   0

+ + − + + +

0

L2 (P;R)

    V (Ys ) ∂  (s, Ws ) − V (x)F(x) 2 ∂s L (P;R)    ∗ 1 2 traceU B(x) (Hess V )(x)B(x)     ∗ ∂   ∂ traceU ∂ y  (s, Ws ) (Hess V )(Ys ) ∂ y  (s, Ws ) L2 (P;R)   2  2   ∗ ∗ 1  ∂     B(x)  (s, W ) (∇V )(Y ) − (∇V )(x)  s s U 2eρs ∂y U L2 (P;R)    2    1 V (Ys ) ∂∂y 2  (s, Ws ) (u, u) 2 2 u∈U L (P;R)    V¯ (Ys ) − V¯ (x) 2 L (P;R) ds.

(45)

123

Stoch PDE: Anal Comp

Next we estimate the L2 (P; R)-semi-norms on the right-hand side of (45) separately. Lemma 2.10 implies for all y, z ∈ H , i ∈ {0, 1, 2} that   . (46)

V (i) (y) − V (i) (z) L (i) (H,R) ≤ c 2c−1 y − z H 1 + V (y) + y − z c−1 H The assumption that ∀ y ∈ H : |V¯ (y)| ≤ c (1 + |V (y)|γ1 ) and Hölder’s inequality hence prove for all s ∈ (0, h] that 

  s e−ρr V¯ (Yr ) dr − 2V (x) E exp 2V (Ys ) + 2 0 

 s   ≤ E exp 2 |V (Ys ) − V (x)| + 2 ∫ V¯ (Yr ) dr 0

  s 

  γ1 ∫ ≤ E exp c 2c 1+V (x) + Ys −x c−1

Y 1 + |V (Y dr − x

+ 2c )| s H r H 0   

  c−1  c ≤ exp c 2 1 + V (x) + Ys − x H Ys − x H  1 L (P;R)   s     ·exp 2c ∫ 1 + |V (Yr )|γ1 dr  ∞ L (P;R) 0   



Y ≤ E exp c 2c 1 + V (x) + Ys − x c−1 − x

s H H   s

 γ (47) exp 2c ∫ 1 + V (Yr ) L1∞ (P;R) dr . 0

Next we estimate the two factors on the right-hand side of (47) separately. Observe that (38) ensures that for all r ∈ [1, ∞), s ∈ (0, h] it holds that (s, Ws ) − x Lr (P;H ) ≤ c. Combining this with Lemma 2.12 establishes that for all s ∈ (0, h] it holds that (s, Ws ) − x L∞ (P;H ) ≤ c. Hölder’s inequality, Tonelli’s theorem, and (38) therefore show that for all s ∈ (0, h] it holds that   



Y E exp c 2c 1 + V (x) + Ys − x c−1 − x

s H H ∞  (2c c)n

c−1 n n

(s, Ws ) − x H =E 1 + V (x) + (s, Ws ) − x H n! n=0 ∞

= n=0 ∞

≤ n=0 ∞

≤ n=0

123

 c n

 n  (2 c) 

(s, Ws ) − x nH   n! 1 + V (x) + (s, Ws ) − x c−1 H

L1 (P;R)

(2c c)n

1+ n!

V (x) + (s, Ws ) − x c−1 L∞ (P;H )

(2c c)n

1+ n!

V (x) + cc−1

n

n

(s, Ws ) − x nLn (P;H )

(s, Ws ) − x nLn (P;H )

Stoch PDE: Anal Comp ∞

≤

(2c c)n n!

  n

 c 1 + V (x) + cc E ( F(x) H s + B(x)Ws H )n

n=0



  = E exp c 2c c(1 + V (x)) + cc ( F(x) H s + B(x)Ws H ) .

(48)

The assumption that ∀ s ∈ (0, h] : max{ F(x) H , B(x) HS(U,H ) } ≤ ch −δ ≤ cs −δ , the fact that ∀ a, b ∈ [0, ∞) : (a + b)c ≤ 2c−1 (a c + bc ), and Lemma 2.4 hence show that for all s ∈ (0, h] it holds that   



Ys − x H E exp c 2c 1 + V (x) + Ys − x c−1 H 

  ≤ E exp c2 2c 1 + V (x) + cc−1 ( F(x) H s + B(x)Ws H ) 

  ≤ E exp c2 2c 1 + c[min{s, 1}]−ς + cc−1 ( F(x) H s + B(x)Ws H )    ≤ E exp 3 · 2c cc+2 [min{s, 1}]−ς ( F(x) H s + B(x)Ws H )   = exp 3 · 2c cc+2 [min{s, 1}]−ς F(x) H s    ·E exp 3 · 2c cc+2 [min{s, 1}]−ς B(x)Ws H   2c 2c+4   9·2 c

B(x) 2HS(U,H ) s ≤ exp 3 · 2c cc+2 [min{s, 1}]−ς F(x) H s 2 exp 2 [min{s,1}]2ς    ≤ 2 exp 9 · 22c−1 c2c+4 s[min{s, 1}]−2ς F(x) H + B(x) 2HS(U,H )    ≤ 2 exp 9 · 22c−1 c2c+4 s[min{s, 1}]−2ς cs −δ + c2s −2δ   ≤ 2 exp 9 · 22c c2c+6 s[min{s, 1}]−2δ−2ς . (49) In the next step we combine (38), (46), and Lemma 2.12 to obtain for all s ∈ (0, h] that

V (Ys ) L∞ (P;R) ≤ V (x) + V (Ys ) − V (x) L∞ (P;R)     

Y ≤ V (x) + c 2c−1  1 + V (x) + Ys − x c−1 − x

s H ∞ H L (P;R)  c−1 2 −ς c c + c [min{s, 1}] + c ≤ V (x) + c 2  ≤ cs −ς + c 2c−1 2c2 [min{s, 1}]−ς + cc ≤ 2c+1 cc+2 [min{s, 1}]−ς .

(50)

Therefore, we obtain for all s ∈ (0, h] that s

  γ γ 2c ∫ 1 + V (Yr ) L1∞ (P;R) dr ≤ 2cs + 2cs 2c+1 cc+2 [min{s, 1}]−ς 1 0

≤2

2+(c+1)γ1 1+(c+2)γ1

c

[min{s, 1}]

−γ1 ς

(51)

s.

123

Stoch PDE: Anal Comp

Combining this with (47) and (49) ensures that for all s ∈ (0, h] it holds that

  s −ρr ¯ E exp 2V (Ys ) + 2 ∫ e V (Yr ) dr − 2V (x) 0   ≤ 2 exp 9 · 22c c2c+6 s[min{s, 1}]−2δ−2ς   · exp 22+(c+1)γ1 c1+(c+2)γ1 [min{s, 1}]−γ1 ς s   ≤ 2 exp 9 · 22c c2c+6 + 22+(c+1)γ1 c1+(c+2)γ1 s[min{s, 1}]−2δ−max{2,γ1 }ς   ≤ 2 exp 22cγ1 +4 c2cγ1 +γ1 +5 s[min{s, 1}]−2δ−max{2,γ1 }ς . (52) Hence, we obtain for all s ∈ (0, h] that   s  exp V (Ys ) + ∫  0

V¯ (Yr ) eρr

  dr  

L2 (P;R)

√ ≤ 2 exp



22cγ1 +3 c2cγ1 +γ1 +5 s [min{s, 1}]2δ+max{2,γ1 }ς

 e V (x) . (53)

Moreover, (38), the assumption that ∀ s ∈ (0, h] : max{ F(x) H , B(x) HS(U,H ) } ≤ ch −δ ≤ cs −δ , the triangle inequality, and the Burkholder-Davis-Gundy type inequality in Lemma 7.7 in Da Prato and Zabczyk [18] assure that for all r ∈ [2, ∞), s ∈ (0, h] it holds that  

Ys −x Lr (P;H ) = (s, Ws ) − x Lr (P;H ) ≤ c F(x) H s + B(x)Ws H Lr (P;R)   ≤ c F(x) H s + sr (r − 1)/2 B(x) HS(U,H )   1 ≤ c cs 1−δ + c r (r − 1)/2 s /2−δ √ 1 1 ≤ c2 r s /2−δ max{ s, 1} ≤ c2 r [min{s, 1}] /2−δ max{s, 1}. (54) Combining this with (38), (46), and Hölder’s inequality implies that for all r ∈ [2, ∞), i ∈ {0, 1, 2}, s ∈ (0, h] it holds that   (i) V (Ys ) − V (i) (x) r L (P;L (i) (H,R))       c−1

Y 1 + V (x) + Ys − x c−1 ≤ c 2 − x

 r s H H L (P;R)   ≤ c 2c−1 Ys − x Lr (P;H ) + V (x) Ys − x Lr (P;H ) + Ys − x cLr c (P;H )   ≤ c 2c−1 1 + cs −ς + Ys − x c−1 r c L (P;H ) Ys − x Lr c (P;H )  1 ≤ c4 2c−1 2c[min{s, 1}]−ς + cc−1 r [min{s, 1}] /2−δ max{s, 1} ≤ r cc+4 2c+1 [min{s, 1}] /2−δ−ς max{s, 1}. 1

123

(55)

Stoch PDE: Anal Comp

Hölder’s inequality, the assumption that ∀ s ∈ (0, h] : max{ F(x) H , B(x) HS(U,H ) } ≤ ch −δ ≤ cs −δ , (36), and Lemma 2.11 hence show for all s ∈ (0, h] that     V (Ys ) ∂  (s, Ws ) − V (x)F(x) 2 ∂s L (P;R)   ≤ V (Ys ) − V (x)L2 (P;L(H,R)) F(x) H      ∂    + V (Ys ) L(H,R)  ∂s  (s, Ws ) − F(x) H  2 L (P;R)     ∂      ≤ V (x) L(H,R) ∂s  (s, Ws ) − F(x) L2 (P;H )  

+ V (Ys ) − V (x)L4 (P;L(H,R)) F(x) H  ∂   +  ∂s  (s, Ws ) − F(x)L4 (P;H )

 1 ≤ c2 [1 + V (x)] s γ2 + 2c+3 cc+4 [min{s, 1}] /2−δ−ς cs −δ + cs γ2 max{s, 1}

 1 ≤ c2 1 + cs −ς s γ2 + 2c+4 cc+5 [min{s, 1}] /2−2δ−ς [max{s, 1}]1+γ2 ≤ 2c+5 cc+5 [min{s, 1}]γ2 −2δ−ς [max{s, 1}]1+γ2 .

(56)

In addition, observe that (55), the assumption that ∀ s ∈ (0, h] : max{ F(x) H ,

B(x) HS(U,H ) } ≤ ch −δ ≤ cs −δ , Hölder’s inequality, (36), and Lemma 2.11 ensure that for all s ∈ (0, h] it holds that     V (Ys ) ∂  (s, Ws ) − V (x)B(x) 4 ∂y L (P;L(U,R))      ∂      ≤  V (Ys ) L(H,R) ∂ y  (s, Ws ) − B(x)HS(U,H )  4 L (P;R)   + V (Ys ) − V (x)L4 (P;L(H,R)) B(x) HS(U,H )      ≤ V (x) L(H,R)  ∂∂y  (s, Ws ) − B(x)L4 (P;HS(U,H ))    + V (Ys ) − V (x)L8 (P;L(H,R)) B(x) HS(U,H )    +  ∂  (s, Ws ) − B(x) 8 L (P;HS(U,H ))

∂y

γ2

≤ c [1 + V (x)] cs + 2 c [min{s, 1}] /2−δ−ς [cs −δ + cs γ2 ] max{s, 1}

 1 ≤ c2 s γ2 1 + cs −ς + 2c+5 cc+5 [min{s, 1}] /2−2δ−ς [max{s, 1}]1+γ2 c+4 c+4

1

≤ 2c+6 cc+5 [min{s, 1}]γ2 −2δ−ς [max{s, 1}]1+γ2 .

(57)

Moreover, note that for all A1 , A2 ∈ HS(U, H ), B1 , B2 ∈ L(H ) it holds that      ∗    ∗ ∗ ∗  | traceU A1 B1 A1 − A2 B2 A2 | =   A1 B1 A1 − A2 B2 A2 u, uU  u∈U

  = 

  B2 A2 u, A2 u H 

B1 A1 u, A1 u H − u∈U

u∈U

= |A1 , B1 A1 HS(U,H ) − A2 , B2 A2 HS(U,H ) | = |A1 − A2 , B1 A1 HS(U,H ) + A2 , B1 (A1 − A2 )HS(U,H )

123

Stoch PDE: Anal Comp

+ A2 , (B1 − B2 )A2 HS(U,H ) | ≤ A1 − A2 HS(U,H ) B1 A1 HS(U,H ) + A2 HS(U,H ) B1 (A1 − A2 ) HS(U,H ) + A2 HS(U,H ) (B1 − B2 )A2 HS(U,H )

 ≤ A1 − A2 HS(U,H ) B1 L(H ) A1 HS(U,H ) + A2 HS(U,H ) + B1 − B2 L(H ) A2 2HS(U,H )  ≤ A1 − A2 2HS(U,H ) + 2 A1 − A2 HS(U,H ) A2 HS(U,H )

 · B1 − B2 L(H ) + B2 L(H ) + B1 − B2 L(H ) A2 2HS(U,H )

(58)

(cf., e.g., (78) in Hutzenthaler et al. [31]). Next we apply (58) (with A1 = ( ∂∂y )(s, Ws ), A2 = B(x), B1 = (Hess V )(Ys ), and B2 = (Hess V )(x) for s ∈ [0, h] in the notation of (58)), we apply Hölder’s inequality, we use the assumption that ∀ s ∈ (0, h] : max{ F(x) H , B(x) HS(U,H ) } ≤ ch −δ ≤ cs −δ , we apply Lemma 2.11, and we apply (36) and (55) to obtain that for all s ∈ (0, h] it holds that     ∗    traceU ∂∂y  (s, Ws ) (Hess V )(Ys ) ∂∂y  (s, Ws )   − B(x)∗ (Hess V )(x)B(x)  2 L (P;R)  2    ∂   ≤  ∂ y  (s, Ws ) − B(x) HS(U,H ) + 2 ∂∂y  (s, Ws )    − B(x)HS(U,H ) B(x) HS(U,H )  4 L (P;R) 

· (Hess V )(x) L(H ) + (Hess V )(Ys ) − (Hess V )(x) L4 (P;L(H )) + (Hess V )(Ys ) − (Hess V )(x) L2 (P;L(H )) B(x) 2HS(U,H ) 

 1 ≤ c2 s 2γ2 + 2cs γ2 cs −δ c(1 + V (x)) + 2c+3 cc+4 [min{s, 1}] /2−δ−ς max{s, 1} + 2c+2 cc+4 [min{s, 1}] /2−δ−ς max{s, 1}c2 s −2δ

≤ 3c2 [min{s, 1}]γ2 −δ [max{s, 1}]2γ2 2c2 [min{s, 1}]−ς  1 + 2c+3 cc+4 [min{s, 1}] /2−δ−ς max{s, 1} 1

+ 2c+2 cc+6 [min{s, 1}] /2−3δ−ς max{s, 1}

≤ 3c2 [max{s, 1}]1+2γ2 2c2 [min{s, 1}]γ2 −δ−ς  1 + 2c+3 cc+4 [min{s, 1}] /2+γ2 −2δ−ς 1

+ 2c+2 cc+6 [min{s, 1}] /2−3δ−ς max{s, 1} 1

≤ 2c+5 cc+6 [max{s, 1}]1+2γ2 [min{s, 1}]γ2 −3δ−ς .

(59)

 2 − b 2 | ≤ a − b 2 b

Furthermore,  the fact that ∀ a, b ∈ U : | a U U U U + a − b U , Hölder’s inequality, (57), Lemma 2.11, and the assumption that ∀ s ∈ (0, h] : max{ F(x) H , B(x) HS(U,H ) } ≤ ch −δ ≤ cs −δ show that for all s ∈ (0, h] it holds that

123

Stoch PDE: Anal Comp

  2  2  ∗  ∂    ∂ y  (s, Ws ) (∇V )(Ys )U −  B(x)∗ (∇V )(x)U  2 L (P;R)   ∂   ∗ ∗   ≤ ∂ y  (s, Ws ) (∇V )(Ys ) − B(x) (∇V )(x) L4 (P;U )      · 2 B(x)∗ HS(H,U ) (∇V )(x) H      ∗  +  ∂∂y  (s, Ws ) (∇V )(Ys ) − B(x)∗ (∇V )(x)U  4

L (P;R)

γ2 −2δ−ς

≤ 2 c [min{s, 1}] [max{s, 1}] 

2 −δ −ς c+6 c+5 · 2c s [1 + cs ] + 2 c [min{s, 1}]γ2 −2δ−ς [max{s, 1}]1+γ2 c+6 c+5

1+γ2

≤ 22c+13 c2c+10 [min{s, 1}]γ2 −4δ−2ς [max{s, 1}]2+2γ2 .

(60)

In addition, note that Hölder’s inequality, Lemma 2.11, (37), and (55) imply for all s ∈ (0, h] that    2      u∈U V (Ys ) ∂∂y 2  (s, Ws ) (u, u) 2 L (P;R)    2       ∂ ≤ V (Ys )L4 (P;L(H,R))  u∈U ∂ y 2  (s, Ws ) (u, u) 4 L (P;H )          ≤ V (Ys ) − V (x) L4 (P;L(H,R)) + V (x) L(H,R)      2     ∂  (s, W ) (u, u) ·  s ∂ y2   u∈U L4 (P;H )   1 ≤ 2c+3 cc+4 [min{s, 1}] /2−δ−ς max{s, 1} + c[1 + V (x)] cs γ2   1 ≤ 2c+3 cc+4 [min{s, 1}] /2−δ−ς max{s, 1} + 2c2 [min{s, 1}]−ς cs γ2 ≤ 2c+4 cc+5 [min{s, 1}]γ2 −δ−ς [max{s, 1}]1+γ2 .

(61)

 Moreover, the assumption that ∀ y, z ∈ H : |V¯ (y) − V¯ (z)| ≤ c 1 + |V (y)|γ0 + |V (z)|γ0 y − z H , (50), and (54) show for all s ∈ (0, h] that       γ0 γ0 V¯ (Ys ) − V¯ (x) 2 

Ys − x H L2 (P;R) L (P;R) ≤ c 1 + |V (x)| + |V (Ys )|   γ ≤ c 1 + |V (x)|γ0 + V (Ys ) L0∞ (P;R) Ys − x L2 (P;H ) 

γ  1 ≤ c 1 + |V (x)|γ0 + 2c+1 cc+2 [min{s, 1}]−ς 0 2c2 [min{s, 1}] /2−δ max{s, 1}  1 ≤ 2c3 [min{s, 1}] /2−δ max{s, 1} 1 + cγ0 s −ςγ0 + 2γ0 (c+1) cγ0 (c+2) [min{s, 1}]−ςγ0 ≤ 2γ0 (c+1)+2 cγ0 (c+2)+3 max{s, 1}[min{s, 1}] /2−δ−ςγ0 . 1

(62)

In the next step we insert (53), (55), (56), (59), (60), (61), and (62) into (45) to obtain for all t ∈ (0, h] that

123

Stoch PDE: Anal Comp

  t E exp e−ρt V (Yt ) + ∫ e−ρr V¯ (Yr ) dr − e V (x) 0

 t√  (2c+3)γ (2c+6)γ  1c 1s 2 e V (x) ≤ 2 exp [min{s,1}] 2δ+max{2,γ1 }ς 0

1 · ρ 2c+2 cc+4 [min{s, 1}] /2−δ−ς max{s, 1} + 2c+5 cc+5 [min{s, 1}]γ2 −2δ−ς [max{s, 1}]1+γ2 + 21 · 2c+5 cc+6 [max{s, 1}]1+2γ2 [min{s, 1}]γ2 −3δ−ς + 21 · 22c+13 c2c+10 [min{s, 1}]γ2 −4δ−2ς [max{s, 1}]2+2γ2 + 21 · 2c+4 cc+5 [min{s, 1}]γ2 −δ−ς [max{s, 1}]1+γ2

 1 + 2γ0 (c+1)+2 cγ0 (c+2)+3 max{s, 1} [min{s, 1}] /2−δ−ςγ0 ds  t√  (2c+3)γ (2c+6)γ  25 1c 1s 2 ≤ e V (x) max{ρ, 1} c(2c+10)γ0 2(2c+ /2)γ0 2 exp [min{s,1}] 2δ+max{2,γ1 }ς 0

· [max{s, 1}]2+2γ2 [min{s, 1}]γ2 −4δ−ς−ςγ0 ds.

(63)

This implies for all t ∈ (0, h] that

 t t) E exp Ve(Y ρt + ∫ 0

V¯ (Yr ) eρr



≤ e V (x)

dr

 (2c+13)γ0 · 1 + max{ρ, 1} 2 0

 t exp



2(2c+3)γ1 c(2c+6)γ1 s c(2c+10)γ0 [max{s,1}]2+2γ2 [min{s,1}]2δ+max{2,γ1 }ς [min{s,1}]ς+ςγ0 +4δ−γ2

 ds . (64)

The proof of Lemma 2.13 is thus completed.

 

2.3 Exponential moments for tamed approximation schemes In this subsection we apply Lemmas 2.2 and 2.13 above to establish in Proposition 2.14 below exponential moment bounds for an appropriate tamed exponential Euler-type approximation scheme (cf., e.g., [28,29,31,38,39] for related schemes in the case of finite dimensional SODEs and, e.g., [3,23,30,32] for related schemes in the case of infinite dimensional SPDEs). Proposition 2.14 Consider the notation in Sect. 1.1, let (H, ·, · H , · H ) and (U, ·, ·U , · U ) be separable R-Hilbert spaces with # H >  1 < #U , let T ∈ (0, ∞),  , F ∈ M B(H ), B(H ) , ρ ∈ [0, ∞), δ ∈ [0, 1/14), c, γ ∈ [1, ∞), ς ∈ 0, 1−14δ 2+2γ   B ∈ M B(H ), B(HS(U, H )) , V ∈ Cc3 (H, [0, ∞)), V¯ ∈ C(H, R), let S : (0, T ] → L(H ) and D : (0, T ] → B(H ) be functions, let (, F, P, (Ft )t∈[0,T ] ) be a filtered probability space, let W : [0, T ] ×  → U be an IdU -cylindrical (Ft )t∈[0,T ] -Wiener

123

Stoch PDE: Anal Comp

process with continuous sample paths, assume for all h ∈ (0, T ], x, y ∈ H that   |V¯ (x) − V¯ (y)| ≤ c 1 + |V (x)|γ + |V (y)|γ x − y H ,   |V¯ (x)| ≤ c 1 + |V (x)|γ , V¯ (Sh x) ≤ V¯ (x), and V (Sh x) ≤ V (x), −ς Dh ⊆ {v ∈ H : V (v) ≤ ch },

(65) (66)

assume for all h ∈ (0, T ], x ∈ Dh that max{ F(x) H , B(x) HS(U,H ) } ≤ ch −δ and (G F,B V )(x) +

1 2

2

B(x)∗ (∇V )(x) U + V¯ (x) ≤ ρV (x),

(67)

and let Y θ : [0, T ] ×  → H , θ ∈ T , be (Ft )t∈[0,T ] -adapted stochastic processes with continuous sample paths which satisfy for all θ ∈ T , t ∈ (0, T ] that 

θ θ θ Ytθ = St−tθ Yt + 1 D|θ|T (Yt ) F(Yt ) (t − tθ ) θ θ θ  θ B(Ytθ )(Wt −Wtθ ) + . θ 2

(68)

1+ B(Ytθ )(Wt −Wtθ ) H

Then (i) it holds that

  θ ) V¯ (Y θ ) t 1 D (Ys

θ  s V (Y θ ) |θ|T θ ds ≤ lim sup E e V (Y0 ) lim sup sup E exp eρtt + ∫ eρs |θ|T 0 t∈[0,T ]

|θ|T 0

0

(69) and (ii) it holds for all θ ∈ T that    θ ) V¯ (Y θ ) t 1 D (Ys s V (Y θ ) |θ|T θ ds sup E exp eρtt + ∫ ρs e

t∈[0,T ]

≤ exp



0   3 exp 2[720 max{T,ρ,1}c3 ](720c max{T,1}+7)γ [min{|θ|T ,1}]ς+ςγ +7δ−1/2

θ  E e V (Y0 ) .

(70)

Proof of Proposition 2.14 Throughout this proof let cˆ ∈ [1, ∞) and h ∈ (0, ∞), h ∈ (0, T ], be the real numbers which satisfy for all s ∈ (0, T ] that cˆ = 360c3 max{T, 1} and   ˆ ˆ [2c] ˆ (2c+13)γ [max{s,ρ,1}]4 c] ˆ (2c+6)γ s , s = exp min{s[22δ+max{2,γ (71) }ς ,1} ς+ςγ +7δ−1/2 [min{s,1}]

let ψ : H → H be the function which satisfies for all x ∈ H that ψ(x) = and let  : H × [0, T ] × U → H and

hx :

x 1+ x 2H

,

[0, h] × U → H , (x, h) ∈ H × (0, T ],

123

Stoch PDE: Anal Comp

be the functions which satisfy for all h ∈ (0, T ], x ∈ H , s ∈ [0, h], y ∈ U that (x, s, y) = x + F(x)s +

B(x)y 1+ B(x)y 2H

and

hx (s, y) = (x, s, y). (72)

We now verify step by step the assumptions of Lemma 2.13. First, note that for all h ∈ (0, T ], x ∈ H it holds that hx (0, 0) = x.

(73)

Furthermore, observe that for all h ∈ (0, T ], x ∈ H , s ∈ (0, h], y ∈ U it holds that ∂

x ∂s h

 (s, y) = F(x).

(74)

Next we note that ψ ∈ C 2 (H, H ) and we observe that for all z, u, v ∈ H it holds that ψ (z)u =

u 1+ z 2H

−

2zz,u H (1+ z 2H )2

(75)

and +vz,u H +zu,v H ] ψ (z)(u, v) = − 2[uz,v H(1+ z

+ 2 )2 H

8zz,u H z,v H (1+ z 2H )3

.

(76)

Moreover, note that (75) ensures for all h ∈ (0, T ], x ∈ H , s ∈ (0, h], y, u ∈ U that 

x ∂ ∂ y h

 (s, y)u =

B(x)u 1+ B(x)y 2H

−

2B(x)yB(x)y,B(x)u H (1+ B(x)y 2H )2

.

(77)

The Cauchy–Schwarz inequality hence implies for all h ∈ (0, T ], x ∈ H , s ∈ (0, h], y ∈ U that   (s, y) − B(x)HS(U,H )     B(x)  ≤  1+ B(x)y 2 − B(x) 

 

x ∂ ∂ y h

HS(U,H )

H

   2B(x)yB(x)y,B(x)(·) H    +  (1+ B(x)y 2H )2

HS(U,H )

    

2B(x)y H B(x)y H B(x) HS(U,H ) 1  ≤ − 1 B(x) +  1+ B(x)y 2H  (1+ B(x)y 2H )2 HS(U,H )     2 1  ≤  1+ B(x)y

(78) 2 − 1 B(x) HS(U,H ) + 2 B(x)y H B(x) HS(U,H ) . H

This ensures for all h ∈ (0, T ], x ∈ H , s ∈ (0, h], y ∈ U that  

 (s, y) − B(x)HS(U,H )     2 1  ≤ 1 − 1+ B(x)y

2  B(x) HS(U,H ) + 2 B(x)y H B(x) HS(U,H ) x ∂ ∂ y h



H

123

Stoch PDE: Anal Comp

=

B(x)y 2H B(x) HS(U,H ) 1+ B(x)y 2H

+ 2 B(x)y 2H B(x) HS(U,H )

≤ 3 B(x)y 2H B(x) HS(U,H ) .

(79)

The Burkholder–Davis–Gundy type inequality in Lemma 7.7 in Da Prato and Zabczyk [18] therefore proves that for all h ∈ (0, T ], x ∈ Dh , s ∈ (0, h] it holds that  

x ∂ ∂ y h

  (s, Ws ) − B(x)L8 (P;HS(U,H )) ≤ 3 B(x) HS(U,H )

B(x)Ws 2H L8 (P;R)

= 3 B(x) HS(U,H ) B(x)Ws 2L16 (P;H )  

B(x) 2HS(U,H ) s ≤ 3 B(x) HS(U,H ) 16·15 2 = 360 B(x) 3HS(U,H ) s ≤ 360(ch −δ )3 s ≤ 360c3 s 1−3δ .

(80)

This and (74) show that for all h ∈ (0, T ], x ∈ Dh , s ∈ (0, h] it holds that        ∂ max  ∂s hx (s, Ws ) − F(x)L4 (P;H ) ,  ∂∂y hx (s, Ws ) − B(x)L8 (P;HS(U,H )) ˆ /2−3δ . ≤ 360c3 s 1−3δ ≤ cs 1

(81) Next observe that (76) implies that for all z, u ∈ H it holds that ψ (z)(u, u) =

8z|z,u H |2  3 1+ z 2H

−



2 2uz,u H +z u 2H .   2 1+ z 2H

(82)

Therefore, we obtain that for all x ∈ H , y, u ∈ U it holds that 

∂2 ψ(B(x)y) ∂ y2



   (u, u) = ψ B(x)y B(x)u, B(x)u =

8B(x)y|B(x)y,B(x)u H |2  3 1+ B(x)y 2H



−

2 2B(x)uB(x)y,B(x)u H +B(x)y B(x)u 2H  2 1+ B(x)y 2H

 .

(83)

The Cauchy–Schwarz inequality hence shows for all x ∈ H , y, u ∈ U that  2    ∂   ∂ y 2 ψ(B(x)y) (u, u)

H

≤

8 B(x)y 3H B(x)u 2H  3 1+ B(x)y 2H

=

+

6 B(x)y H B(x)u 2H  2 1+ B(x)y 2H

8 B(x)y 2H +6(1+ B(x)y 2H )  3 1+ B(x)y 2H

≤ 6 B(x)y H B(x)u 2H .



B(x)y H B(x)u 2H (84)

This, the triangle inequality, and the Burkholder–Davis–Gundy type inequality in Lemma 7.7 in Da Prato and Zabczyk [18] imply that for all h ∈ (0, T ], x ∈ Dh , s ∈ (0, h] it holds that

123

Stoch PDE: Anal Comp

  

 u∈U

∂2 x ∂ y2 h

    (s, Ws ) (u, u)

L4 (P;H )

B(x)u 2H u∈U

≤ 6 B(x)Ws L4 (P;H ) √ √ 1 1 ≤ 6 B(x) 3HS(U,H ) 6s ≤ 6 6c3 s /2−3δ ≤ cs ˆ /2−3δ .

(85)

Next observe that for all h ∈ (0, T ], x ∈ Dh , s ∈ (0, h], r ∈ [1, ∞) it holds that    

B(x)Ws H   h Lr (P;H ) ≤  F(x) H s + 1+ B(x)Ws 2H  r L (P;R)      1   ≤ min F(x) H s + 2 Lr (P;R) , F(x) H s + B(x)Ws H Lr (P;R)     ≤ min 21 + ch −δ s,  F(x) H s + B(x)Ws H Lr (P;R)     ≤ min 21 + cT 1−δ ,  F(x) H s + B(x)Ws H Lr (P;R)     ≤ cˆ min 1,  F(x) H s + B(x)Ws H Lr (P;R) . (86)

  x  (s, Ws ) − x 

Moreover, note that the fact that ψ ∈ C 2 (H, H ) implies that for all h ∈ (0, T ], x ∈ H it holds that hx ∈ C 1,2 ([0, h] × U, H ). Combining this, (73), (81), (85), and (86) allows us to apply Lemma 2.13 (with ς = ς , h = h, c = c, ˆ γ0 = γ , γ1 = γ , ρ = ρ, δ = δ, γ2 = 1/2 − 3δ, x = x, F = F, B = B, V¯ = V¯ , V = V ,  = hx for x ∈ Dh , h ∈ (0, T ] in the notation of Lemma 2.13) to obtain that for all h ∈ (0, T ], x ∈ Dh , t ∈ (0, h] it holds that

 t V (hx (t,Wt )) +∫ E exp eρt 0

V¯ (hx (s,Ws )) eρs

 ds

  ≤ 1 + ∫t0 s ds e V (x) .

(87)

Next note that the estimates 1 − 2δ − max{2, γ }ς ≥ 0 and 1/2 − ς − ς γ − 7δ > 0 ensure that the function (0, T ] h → h ∈ (0, ∞) is non-decreasing and that lim suph0 h = 0. Combining this with (87) implies that for all h ∈ (0, T ], x ∈ Dh , t ∈ (0, h] it holds that

 t V (hx (t,Wt )) +∫ E exp eρt 0

≤ (1 + h t) e

V (x)

V¯ (hx (s,Ws )) eρs

 ds

  ≤ 1 + ∫t0 s ds e V (x)

(88)

.

This ensures for all θ ∈ T , x ∈ D|θ|T , t ∈ (0, |θ |T ] that

 x t V (|θ| (t,Wt )) T E exp +∫ eρt 0

x (s,Ws )) V¯ (|θ| T eρs

 ds

≤ e|θ|T t+V (x) .

(89)

Hence, we obtain for all θ ∈ T , x ∈ D|θ|T , t ∈ (0, |θ |T ] that

 t t )) E exp V ((x,t,W +∫ eρt 0

123

V¯ ((x,s,Ws )) eρs

 ds

≤ e|θ|T t+V (x) .

(90)

Stoch PDE: Anal Comp

Corollary 2.2 (with T = T , θ = θ , ρ = ρ, c = |θ|T , V = V , V¯ = V¯ ,  = , E = Dh , S = S, W = W , Y = Y θ for θ ∈ T , h ∈ (0, T ] in the notation of Corollary 2.2) therefore yields that for all θ ∈ T , t ∈ [0, T ] it holds that 

  θ ) V¯ (Y θ ) t 1D (Ys s V (Y θ ) |θ|T |θ|T t V (Y0θ ) θ . ds ≤ e E e E exp eρtt + ∫ ρs e

(91)

0

This assures that for all θ ∈ T it holds that

   θ ) V¯ (Y θ ) t 1D (Ys s V (Y θ ) |θ|T |θ|T T V (Y0θ ) θ sup E exp eρtt + ∫ . ds ≤ e E e ρs e

t∈[0,T ]

(92)

0

This and the fact that lim suph0 h = 0 establish (69). It thus remains to prove (70).   For this observe that the fact that ∀ x ∈ [220 , ∞) : x ≤ exp x 1/4 and the fact that ∀ θ ∈ T : (|θ |T )1−2δ−max{2,γ }ς ≤ max{1, T } show that for all θ ∈ T it holds that  |θ|T T = exp

3

[720c3 max{T,1}](720c max{T,1}+6)γ |θ|T [min{|θ|T ,1}]2δ+max{2,γ }ς



(720c3 max{T,1}+13)γ

[max{|θ|T ,ρ,1}] T · [720c max{T,1}] [min{|θ|T ,1}]ς+ςγ +7δ−1/2   3 (720c3 max{T,1}+18)γ 3 ≤ exp [720c3 max{T, 1}](720c max{T,1}+7)γ [720c max{T,ρ,1}]ς+ςγ +7δ−1/2 [min{|θ|T ,1}]   3 1 ≤ exp 2[720c3 max{T, ρ, 1}](720c max{T,1}+7)γ . ς+ςγ +7δ−1/2 3

4

[min{|θ|T ,1}]

(93) Combining (92) with (93) establishes (70). The proof of Proposition 2.14 is thus completed.   Let us add a few words about the functions V : H → [0, ∞) and V¯ : H → R appearing in Proposition 2.14. The functions V and V¯ are used as ingredients of a Lyapunov-type function for the evolution of the numerical approximation processes (Ytθ )t∈[0,T ] , θ ∈ T , in (68) above. To be more specific, it holds, roughly speaking, that the functions 

 t 1 D (ys ) V¯ (ys ) θ |θ|T V (yt ) ds ∈ [0, ∞) [0, T ] × C([0, T ], H ) (t, y) → E exp eρt + ∫ eρs 0

(94) for θ ∈ T are suitable Lyapunov-type functions for the numerical approximation processes (Ytθ )t∈[0,T ] , θ ∈ T , (cf., e.g., (69), Gyöngy and Krylov [22, Section 2], (1.2) in [28], Corollary 2.4 in [13], and (11) in [31]). Moreover, note that the hypotheses in (65), roughly speaking, ensure that the function V¯ does neither grow nor change too rapidly.

123

Stoch PDE: Anal Comp

3 Exponential moments for space-time-noise discrete approximation schemes In Proposition 2.14 in Sect. 2 above we established exponential moment bounds for a class of time discrete approximation schemes. In this section we extend this result in Theorem 3.3 and Corollary 3.4 below to obtain exponential moments for a class of space-time-noise discrete approximation schemes. Theorem 3.3 below proves exponential moment bounds for numerical approximations of SPDEs whose coefficients satisfy a general Lyapunov-type condition. Corollary 3.4 below specialises Theorem 3.3 to the case where the considered Lyapunov-type function is an affine linear transformation of the squared Hilbert space norm. Our proof of Theorem 3.3 uses two well-known auxiliary lemmas (see Lemmas 3.1 and 3.2 below).

3.1 Setting Consider the notation in Sect. 1.1, let (H, ·, · H , · H ) and (U, ·, ·U , · U ) be separable R-Hilbert spaces, let H ⊆ H be a non-empty orthonormal basis of H , let U ⊆ U be a non-empty orthonormal basis of U , let λ : H → R be a function which satisfies sup(im(λ)) <  0, let A :  D(A) ⊆ H → H be the  lin2 |λ | h, v < ∞ and ear operator which satisfies D(A) = v ∈ H : H h∈H h 

·

λ h, v h, let (H , ·, · , ), r ∈ R, be a ∀ v ∈ D(A) : Av = H r Hr Hr h∈H h family of interpolation spaces associated to −A (cf., e.g., [40, Section 3.7]), let T ∈ (0, ∞), γ ∈ [0, ∞), δ ∈ [0, 1/14), let (, F, P, (Ft )t∈[0,T ] ) be a filtered probability space, let (Wt )t∈[0,T ] be an IdU -cylindrical (Ft )t∈[0,T ] -Wiener process, let ξ ∈ M(F0 , B(Hγ )), F ∈ M(B(Hγ ), B(H )), B ∈ M(B(Hγ ), B(HS(U, H ))), let DhI ∈ B(Hγ ), (I, h) ∈ P(H) × (0, T ], be a family of sets, and let PI ∈ L(H ), I ∈ P(H), and Pˆ J ∈ L(U ), J ∈ P(U), be the linear operators  which satisfy for all I ∈ P(H), J ∈ P(U), x ∈ H , y ∈ U that PI (x) = h∈I h, x H h and  Pˆ J (y) = u∈J u, yU u.

3.2 Exponential moments for tamed approximation schemes Lemma 3.1 (cf., e.g., Lemma 1 in Da Prato et al. [17]) Consider the notation in Sect. 1.1, let (, F, μ) be a sigma-finite measure space, and let T ∈ (0, ∞), Y, Z ∈ T M(F ⊗B([0, T ]), B(R)) satisfy for all t ∈ [0, T ] that μ(Yt = Z t ) = μ 0 |Ys | ds = T T    ∞ = 0. Then μ \ ω ∈  : 0 |Ys (ω)| + |Z s (ω)| ds < ∞ and 0 Ys (ω) ds = T  = 0. 0 Z s (ω) ds Proof of Lemma 3.1 First, note that the Tonelli theorem implies that    T ∫ |Ys − Z s | ds dμ = 

123

0

T 0



 ∫ |Ys − Z s | dμ ds = 0.



(95)

Stoch PDE: Anal Comp

T  T This shows that μ ∫ |Ys − Z s | ds > 0 = 0. Therefore, we obtain that μ 0 |Ys − 0  T  Z s | ds = ∞ = 0. This and the assumption that μ 0 |Ys | ds = ∞ = 0 proves that T  T μ ∫ |Ys | ds + ∫ |Ys − Z s | ds = ∞ 0

0

 T   T = μ ∫ |Ys | ds = ∞ ∪ ∫ |Ys − Z s | ds = ∞ 0

(96)

0 T

  T ≤ μ ∫ |Ys | ds = ∞ + μ ∫ |Ys − Z s | ds = ∞ = 0. 0

0

The triangle inequality hence proves that    T μ ∫ |Z s | ds = ∞ ≤ μ 0

T

 |Z s − Ys | ds +

0

T

 |Ys | ds = ∞ = 0.

(97)

0

Next note that (95) ensures that    T  T   ∫ 1  T  ∫ dμ ≤ (Y − Z ) ds |Y − Z | ds dμ = 0. (98) s s s  { 0 |Yu −Z u | du<∞} s   0



0

Hence, we obtain that  T μ ∫ 1{ T |Y 0

0

u −Z u

 (Y − Z ) ds  = 0 = 0. s | du<∞} s

(99)

This shows that  μ

T

∫ 1{ T |Y |+|Z | du<∞} (Ys u u 0 0

 − Z s ) ds = 0 = 0.

Combining (97) and (100) completes the proof of Lemma 3.1.

(100)  

Lemma 3.2 Consider the notation in Sect. 1.1, let (H, ·, · H , · H ) and (U, ·, ·U ,

· U ) be separable R-Hilbert spaces, let T ∈ (0, ∞), let Q ∈ L(U ) be a non-negative and symmetric linear operator, let (, F, P, (Ft )t∈[0,T ] ) be a filtered probability space, let (Wt )t∈[0,T ] be a Q-cylindrical (Ft )t∈[0,T ] -Wiener process, let Gt ⊆ F, t ∈ [0, T ], satisfy for all t ∈ [0, T ] that Gt = σ (Ft ∪ {C ∈ F : P(C) = 0}), let R ∈ HS(Q 1/2 (U ), H ), and let W˜ : [0, T ] ×  → H be a stochastic processwith cont tinuous sample paths which satisfies for all t ∈ [0, T ] that [W˜ t ]P,B(H ) = 0 R dWs . Then it holds that W˜ is an R R ∗ -standard (Gt+ )t∈[0,T ] -Wiener process. Proof of Lemma 3.2 Throughout this proof let U0 ⊆ U be an orthonormal basis of Kern(Q 1/2 ) and let U1 ⊆ U be an orthonormal basis of Kern(Q 1/2 )⊥ . Next note that for all v, w ∈ H , s ∈ [0, T ), t ∈ (s, T ] it holds that

123

Stoch PDE: Anal Comp

" ! " 

! t t E[v, W˜ t − W˜ s  H w, W˜ t − W˜ s  H ] = E v, ∫ R dWr w, ∫ R dWr s s H H

   t t ∫w, R dWr  H = E ∫v, R dWr  H s s  

 t t ∫R ∗ w, dWr  Q 1/2 (U ) . (101) = E ∫R ∗ v, dWr  Q 1/2 (U ) s

s

Itô’s isometry hence shows for all v, w ∈ H , s ∈ [0, T ), t ∈ (s, T ] that 1 (t−s)

#

E[v, W˜ t − W˜ s  H w, W˜ t − W˜ s  H ]

= (Q /2 (U ) z → R ∗ v, z Q 1/2 (U ) ∈ R), $ 1 (Q /2 (U ) z → R ∗ w, z Q 1/2 (U ) ∈ R) HS(Q 1/2 (U ),R) # 1 = (U z → R ∗ v, Q /2 z Q 1/2 (U ) ∈ R), $ 1 (U z → R ∗ w, Q /2 z Q 1/2 (U ) ∈ R) HS(U,R) =

1

R ∗ v, Q /2 u Q 1/2 (U ) R ∗ w, Q /2 u Q 1/2 (U ) 1

u∈U0 ∪U1

=

1

R ∗ v, Q /2 u Q 1/2 (U ) R ∗ w, Q /2 u Q 1/2 (U ) 1

u∈U1

=

1

Q − /2 (R ∗ v), Q − /2 (Q /2 u)U Q − /2 (R ∗ w), Q − /2 (Q /2 u)U 1

u∈U1

=

1

=

1

1

1

Q − /2 (R ∗ v), uU Q − /2 (R ∗ w), uU 1

= Q

1

Q − /2 (R ∗ v), uU Q − /2 (R ∗ w), uU 1

u∈U1

1

u∈U0 ∪U1 −1/2 ∗

1

(R v), Q − /2 (R ∗ w)U = R ∗ v, R ∗ w Q 1/2 (U ) = v, R R ∗ w H . 1

(102) Next observe that the assumption that W is a Q-cylindrical (Ft )t∈[0,T ] -Wiener process ensures that (, F, P, (Gt+ )t∈[0,T ] ) is a stochastic basis and that W is a Q-cylindrical (Gt+ )t∈[0,T ] -Wiener process. This implies that for all s ∈ [0, T ), t ∈ (s, T ] it holds that σ (W˜ t − W˜ s ) and Gs+ are P-independent and that W˜ is (Gr+ )r ∈[0,T ] -adapted. Combining this with (102) completes the proof of Lemma 3.2.   Theorem the setting in Sect. 3.1, let ρ ∈ [0, ∞), c, ι ∈ [1, ∞), ς ∈  1−14δ 3.3 Assume 0, 2+2ι , V ∈ Cc3 (H, [0, ∞)), V¯ ∈ C(H, R), assume for all h ∈ (0, T ], x, y ∈ H that   |V¯ (x) − V¯ (y)| ≤ c 1 + |V (x)|ι + |V (y)|ι x − y H ,   |V¯ (x)| ≤ c 1 + |V (x)|ι , V¯ (eh A x) ≤ V¯ (x), and V (eh A x) ≤ V (x),

V (P ξ )  I sup I ∈P0 (H) E e < ∞,

123

(103) (104)

Stoch PDE: Anal Comp

assume for all I ∈ P0 (H), J ∈ P0 (U), h ∈ (0, T ], x ∈ DhI that DhI ⊆ {v ∈ H : V (v) ≤ ch −ς }, max{ PI F(x) H , PI B(x) Pˆ J HS(U,H ) } ≤ ch −δ , (105) ∗ 2 1 ˆ ¯ and (G V )(x) +

(P B(x) P ) (∇V )(x)

+ V (x) ≤ ρV (x), (106) I J ˆ U 2

PI F,PI B PJ

and let Y θ,I,J : [0, T ] ×  → PI (Hγ ), θ ∈ T , I ∈ P0 (H), J ∈ P0 (U), be (Ft )t∈[0,T ] -adapted stochastic processes with continuous sample paths which satisfy for all θ ∈ T , I ∈ P0 (H), J ∈ P0 (U), t ∈ (0, T ] that Y0θ,I,J = PI (ξ ) and [Ytθ,I,J ]P,B(PI (Hγ ))

= e

(t−tθ )A t

+

tθ



θ,I,J Yt θ

e(t−tθ )A 1+

t

+ 1DI

|θ|T

θ,I,J θ,I,J (Yt )PI F(Yt ) (t θ θ

θ,I,J θ,I,J ˆ 1 D I (Yt ) PI B(Yt ) P J d Ws θ θ

tθ

|θ|T

θ,I,J ˆ PI B(Yt ) PJ d Ws 2H θ

 − tθ )

 P,B(PI (Hγ ))

.

(107)

Then it holds that lim sup sup

sup

 t V (Ytθ,I,J ) θ,I,J sup E exp + ∫ 1 D I (Ys ) eρt θ

|θ|T 0 I ∈P0 (H) J ∈P0 (U) t∈[0,T ]

≤

|θ|T

0

 sup E e V (PI ξ )

V¯ (Ysθ,I,J ) eρs

 ds

I ∈P0 (H)

≤ sup

sup

sup

sup

θ∈T I ∈P0 (H) J ∈P0 (U) t∈[0,T ]

 t V (Ytθ,I,J ) θ,I,J E exp + ∫ 1 D I (Ys ) eρt θ 0

|θ|T

V¯ (Ysθ,I,J ) eρs

 ds

< ∞.

(108)

Proof of Theorem 3.3 Throughout this proof let Gt ⊆ F, t ∈ [0, T ], be the sets which satisfy for all t ∈ [0, T ] that   Gt = σ Ft ∪ {C ∈ F : P(C) = 0} ,

(109)

let u 0 ∈ U, let W J : [0, T ] ×  → Pˆ J ∪{u 0 } (U ), J ∈ P0 (U), be stochastic processes with continuous  t sample paths which satisfy for all J ∈ P0 (U), t ∈ [0, T ] that [WtJ ]P,B(U ) = 0 Pˆ J ∪{u 0 } dWs and W0J = 0, let F˜ I : H → H , I ∈ P0 (H), and B˜ I,J : H → HS( Pˆ J ∪{u 0 } (U ), H ), I ∈ P0 (H), J ∈ P0 (U), be the functions which satisfy for all I ∈ P0 (H), J ∈ P0 (U), x ∈ H , u ∈ Pˆ J ∪{u 0 } (U ) ⊆ U that  F˜ I (x) = 

PI (F(x)) : x ∈ Hγ 0 : x ∈ H \Hγ

and

PI (B(x) Pˆ J u) : x ∈ Hγ B˜ I,J (x)u = , 0 : x ∈ H \Hγ

(110)

123

Stoch PDE: Anal Comp

and let Y˜ θ,I,J : [0, T ] ×  → H , θ ∈ T , I ∈ P0 (H), J ∈ P0 (U), be the functions which satisfy for all θ ∈ T , I ∈ P0 (H), J ∈ P0 (U), t ∈ [0, T ] that Y˜0θ,I,J = PI (ξ ) and 

θ,I,J θ,I,J θ,I,J ˜ Y˜tθ,I,J = e(t−tθ )A Y˜t + 1 ( Y ) ) (t − tθ ) F˜ I (Y˜t I tθ D|θ| θ θ T  θ,I,J J )(WtJ −Wt ) B˜ I,J (Y˜t θ θ . + J J 2 ˜ ˜ θ,I,J 1+ B I,J (Ytθ )(Wt −Wtθ ) H

(111)

In the next step observe that, e.g., Andersson et al. [1, Lemma 2.2] (cf., e.g., Parthasarathy [37, Theorem 2.4 in Chapter V]) ensures that B(Hγ ) ⊆ B(H ). This implies that for all I ∈ P0 (H), J ∈ P0 (U), h ∈ (0, T ] it holds that DhI ∈ B(H ), F˜ I ∈ M(B(H ), B(H )), and B˜ I,J ∈ M(B(H ), B(HS( Pˆ J ∪{u 0 } (U ), H ))).

(112)

In addition, note that (Gt+ )t∈[0,T ] is a normal filtration on (, F, P) and that W is an IdU -cylindrical (Gt+ )t∈[0,T ] -Wiener process. Lemma 3.2 (with H = Pˆ J ∪{u 0 } (U ), U = U , R = (U u → Pˆ J ∪{u 0 } (u) ∈ Pˆ J ∪{u 0 } (U )), Q = IdU , (Ft )t∈[0,T ] = (Ft )t∈[0,T ] , W = W , W˜ = W J for J ∈ P0 (U) in the notation of Lemma 3.2) hence assures that for all J ∈ P0 (U) it holds that W J is an ((U u → Pˆ J ∪{u 0 } (u) ∈ Pˆ J ∪{u 0 } (U ))(U u → Pˆ J ∪{u 0 } (u) ∈ Pˆ J ∪{u 0 } (U ))∗ )-standard (Gt+ )t∈[0,T ] -Wiener process. This shows that for all J ∈ P0 (U) it holds that W J is an Id PˆJ ∪{u } (U ) -standard 0

(Gt+ )t∈[0,T ] -Wiener process with continuous sample paths. Combining the fact that for all θ ∈ T , I ∈ P0 (H), J ∈ P0 (U) it holds that Y˜ θ,I,J is a (Gt+ )t∈[0,T ] -adapted stochastic process with continuous sample paths, the fact that ∀ I ∈ P0 (H), h ∈ (0, T ] : DhI ⊆ {v ∈ H : V (v) ≤ ch −ς }, (112), and item (ii) of Proposition 2.14 (with H = H , U = Pˆ J ∪{u 0 } (U ), T = T , ρ = ρ, δ = δ, c = c, γ = ι, ς = ς , F = F˜ I , B = B˜ I,J , V = V , V¯ = V¯ , S = ((0, T ] t → (H x → et A x ∈ H ) ∈ L(H )), Dh = DhI , (Ft )t∈[0,T ] = (Gt+ )t∈[0,T ] , W = W J , Y θ = Y˜ θ,I,J for h ∈ (0, T ], θ ∈ T , I ∈ P0 (H), J ∈ P0 (U) in the notation of Proposition 2.14) hence proves that for all θ ∈ T , I ∈ P0 (H), J ∈ P0 (U) it holds that     ˜ θ,I,J t V ( Yt ) θ,I,J ˜ ∫ Y + 1 sup E exp I D sθ eρt

t∈[0,T ]

≤ exp



V¯ (Y˜sθ,I,J ) eρs

|θ|T 0   3 exp 2[720 max{T,ρ,1}c3 ](720c max{T,1}+7)ι

[min{|θ|T

,1}]ς+ςι+7δ−1/2

 ds

θ,I,J  E e V (Y0 ) .

(113)

This, the fact that 1/2−ς −ς ι−7δ > 0, and the assumption that sup I ∈P0 (H) E[e V (PI (ξ )) ] < ∞ imply that

123

Stoch PDE: Anal Comp

lim sup sup

sup

sup

|θ|T 0 I ∈P0 (H) J ∈P0 (U) t∈[0,T ]

 t V (Y˜tθ,I,J ) θ,I,J ∫ E exp + 1 D I (Y˜s ) ρt e θ |θ| T 0  ≤ sup E e V (PI ξ )

V¯ (Y˜sθ,I,J ) eρs

 ds

I ∈P0 (H)

≤ sup

sup

sup

sup

θ∈T I ∈P0 (H) J ∈P0 (U) t∈[0,T ]

 t V (Y˜tθ,I,J ) θ,I,J E exp + ∫ 1 D I (Y˜s ) eρt θ 0

|θ|T

V¯ (Y˜sθ,I,J ) eρs

 < ∞.

ds

(114)

Furthermore, note that (107) and (111) ensure that for all θ ∈ T , I ∈ P0 (H), J ∈ P0 (U), t ∈ [0, T ] it holds that [Ytθ,I,J ]P,B(PI (Hγ )) = [Y˜tθ,I,J ]P,B(PI (Hγ )) . Combining this, Lemma 3.1, and (114) establishes (108). The proof of Theorem 3.3 is thus completed.   Corollary 3.4 Assume the setting in Sect. 3.1, let ϑ ∈ [supx∈Hγ B(x) 2HS(U,H ) , ∞]∩ √   , c ∈ [2 max{1, εb1 , ε ϑ, ε}, ∞), R, b1 , b2 ∈ [0, ∞), ε ∈ (0, ∞), ς ∈ 0, 1−14δ 4

assume that E[eε ξ H ] < ∞, assume for all h ∈ (0, T ], I ∈ P0 (H), J ∈ P0 (U), x ∈ DhI that 2

max{ PI F(x) H , PI B(x) Pˆ J HS(U,H ) } ≤ ch −δ , √ DhI ⊆ {v ∈ H : ϑ + ε v 2H ≤ ch −ς }, and

(115)

x, PI F(x) H ≤ b1 + b2 x 2H ,

(116)

and let Y θ,I,J : [0, T ] ×  → PI (Hγ ), θ ∈ T , I ∈ P0 (H), J ∈ P0 (U), be (Ft )t∈[0,T ] -adapted stochastic processes with continuous sample paths which satisfy for all θ ∈ T , I ∈ P0 (H), J ∈ P0 (U), t ∈ (0, T ] that Y0θ,I,J = PI (ξ ) and

  θ,I,J  θ,I,J θ,I,J [Ytθ,I,J ]P,B(PI (Hγ )) = e(t−tθ )A Yt + 1 (Y )P F(Y ) (t − t ) I I θ tθ tθ D θ |θ|T

t

+

tθ

θ,I,J θ,I,J ˆ 1 D I (Yt ) PI B(Yt ) PJ θ θ |θ|T t θ,I,J ˆ 2 1+ t PI B(Yt ) P d W

s H J θ θ

e(t−tθ )A

d Ws

P,B(PI (Hγ ))

.

(117)

Then it holds that lim sup sup

sup

√ t ϑ+ε Ytθ,I,J 2H θ,I,J sup E exp − ∫ 1 D I (Ys ) θ e2(b2 +εϑ)t

|θ |T 0 I ∈P0 (H) J ∈P0 (U) t∈[0,T ]

√ϑ+ε ξ 2  H

≤E e

≤ sup

sup

sup

√ t ϑ+ε Ytθ,I,J 2H θ,I,J sup E exp − ∫ 1 D I (Ys ) θ e2(b2 +εϑ)t

θ ∈T I ∈P0 (H) J ∈P0 (U) t∈[0,T ] √ ϑ

≤e

sup

sup

|θ|T

0

ε(2b1 +ϑ) e2(b2 +εϑ)s

sup

0

|θ|T

 θ,I,J  ε Y

2 < ∞. sup E exp e2(bt 2 +εϑ)tH

θ ∈T I ∈P0 (H) J ∈P0 (U) t∈[0,T ]

 ds

ε(2b1 +ϑ) e2(b2 +εϑ)s

 ds

(118)

123

Stoch PDE: Anal Comp

Proof of Corollary 3.4 Throughout this proof let V : H → [0, ∞) and V¯√: H → R be the functions with the property that for all x ∈ H it holds that V (x) = ϑ + ε x 2H and V¯ (x) = −2εb1 √ − εϑ. First of all, observe that for all x, y ∈ H it holds that |V (x) − V (y)| ≤ 2 ε x − y H (1 + supr ∈[0,1] |V (r x + (1 − r )y)|)1/2 , V (x) − V (y) L (1) (H,R) ≤ 2ε x − y H , and |V (x) − V (y) L (2) (H,R) = 0. Hence, we obtain that V ∈ C23 max{1,ε} (H, [0, ∞)).

(119)

Next note that the assumption that ∀ h ∈ (0, T ], I ∈ P0 (H), J ∈ P0 (U), x ∈ DhI : x, PI F(x) H ≤ b1 + b2 x 2H shows that for all h ∈ (0, T ], I ∈ P0 (H), J ∈ P0 (U), x ∈ DhI it holds that 2  (G PI F,PI B PˆJ V )(x) + 21 (PI B(x) Pˆ J )∗ (∇V )(x)U + V¯ (x) = 2εx, PI F(x) H + ε

u∈U

PI B(x) Pˆ J u, PI B(x) Pˆ J uU

2 + 2ε2 (PI B(x) Pˆ J )∗ x U − 2εb1 − εϑ = 2εx, PI F(x) H + ε PI B(x) Pˆ J 2

HS(U,H )

2 + 2ε2 (PI B(x) Pˆ J )∗ x U − 2εb1 − εϑ

≤ 2ε(b2 + εϑ) x 2H ≤ 2(b2 + εϑ)V (x).

(120) √

Combining this, (119), the fact that sup I ∈P0 (H) E[e V (PI ξ ) ] ≤ e ϑ E[eε ξ H ], the 2 assumption that E[eε ξ H ] < ∞, the fact that ∀ x ∈ H : |V¯ (x)| ≤ c(1 + |V (x)|), and Theorem 3.3 (with ρ = 2(b2 + εϑ), c = c, ι = 1, δ = δ, ς = ς , V = V , V¯ = V¯ , Y θ,I,J = Y θ,I,J for θ ∈ T , I ∈ P0 (H), J ∈ P0 (U) in the notation of Theorem 3.3) establishes (118). The proof of Corollary 3.4 is thus completed.   2

4 Examples In this section we illustrate Corollary 3.4 by some examples. In particular, we prove in the case of a class of stochastic Burgers equations (see Sect. 4.3), stochastic Kuramoto– Sivashinsky equations (see Sect. 4.4), and two-dimensional stochastic Navier–Stokes equations (see Sect. 4.5) that a certain tamed and space-time-noise discrete approximation scheme [see (125) below] has bounded exponential moments. 4.1 Setting Consider the notation in Sect. 1.1, let d ∈ N, D = (0, 1)d , η, γ ∈ [0, ∞), T, ε ∈ (0, ∞), δ ∈ (0, 1/18), (U, ·, ·U , · U ) = (L 2 (μD ; Rd ), ·, · L 2 (μD ;Rd ) ,

· L 2 (μD ;Rd ) ), let U ⊆ U be an orthonormal basis of U , let H ⊆ U be a closed subvector space of U , let H ⊆ H be a non-empty orthonormal basis of H , let λ : H → R be a function which satisfies sup(im(λ)) < 0, let (, F, P, (Ft )t∈[0,T ] ) be a filtered

123

Stoch PDE: Anal Comp

probability space, let (Wt )t∈[0,T ] be an IdU -cylindrical (Ft )t∈[0,T ] -Wiener process, let Q ∈ L(U ) be a non-negative symmetric trace class  operator, let A : D(A) ⊆ 2H → H be the linear operator which satisfies D(A) = v ∈ H : h∈H |λh h, v H | < ∞ λ h, v h, let (H , ·, · and ∀ v ∈ D(A) : Av = h H r Hr , · Hr ), r ∈ R, be h∈H a family of interpolation spaces associated to −A (cf., e.g., [40, Section 3.7]), let r ∈ M(B(Hγ ), B([0, ∞))), b ∈ M(B(D × Rd ), B(Rd×d )) satisfy  supx∈D,y∈Rd ,z∈Rd \{y} b(x, y) Rd×d +

b(x,y)−b(x,z) Rd×d

y−z Rd



< ∞,

(121)

√ let ϑ = traceU (Q)(supx∈D,y∈Rd b(x, y) 2Rd×d ), c ∈ [2 max{1, ε ϑ, ε}, ∞), let PI ∈ L(H ), I ∈ P(H), and Pˆ J ∈ L(U ), J ∈ P(U), be the linear operators which satisfy for all I ∈ P(H), J ∈ P(U), v ∈ H , w ∈ U that PI (v) =

h∈I

h, v H h

Pˆ J (w) =

and

u∈U

u, wU u,

(122)

for every I ∈ P0 (H), h ∈ (0, T ] let DhI ∈ P(Hγ ) be the set given by DhI = {x ∈ PI (Hγ ) : r (x) ≤ ch −δ }, let R ∈ L(U ) be the orthogonal projection of U on H , for every n ∈ N, v ∈ W 1,2 (D, Rn ) let ∂v = (∂1 v, . . . , ∂d v) ∈ ∞ (D, Rn ) L 2 (μD ; Rn×d ) be the vector which satisfies for all i ∈ {1, . . . , d}, φ ∈ Ccpt ∂ that ∂i v, [φ]μD ,B(Rn )  L 2 (μD ;Rn ) = −v, [ ∂ xi φ]μD ,B(Rn )  L 2 (μD ;Rn ) , let ξ ∈ M(F0 ,

2  B(Hγ )) satisfy E eε ξ H < ∞, let F : Hγ → H and B : Hγ → HS(U, H ) be functions which satisfy for all u ∈ U , v ∈ M(B(D), B(Rd )), w ∈ [Hγ ∩ W 1,2 (D, Rd ) ∩ L ∞ (μD ; Rd )] with [v]μD ,B(Rd ) ∈ Hγ that   B([v]μD ,B(Rd ) )u = R [{b(x, v(x))}x∈D ]μD ,B(Rd×d ) ( Qu)   d  wi ∂i w , and F(w) = R ηw −

(123) (124)

i=1

and let Y θ,I,J : [0, T ]× → PI (H ), θ ∈ T , I ∈ P0 (H), J ∈ P0 (U), be (Ft )t∈[0,T ] adapted stochastic processes with continuous sample paths which satisfy for all t ∈ (0, T ], θ ∈ T , I ∈ P0 (H), J ∈ P0 (U) that Y0θ,I,J = PI (ξ ) and   θ,I,J [Ytθ,I,J ]P,B(PI (Hγ )) = e(t−tθ )A Yt θ + 1{r (Y θ,I,J )≤c[|θ| tθ

t

+

tθ

T

θ,I,J P F(Yt ) (t − tθ ) θ ]−δ } I

 P,B(PI (Hγ ))

θ,I,J ˆ e(t−tθ )A 1 PI B(Yt ) P J d Ws θ,I,J θ {r (Yt )≤c[|θ|T ]−δ } θ t . θ,I,J ˆ 1+ t PI B(Ytθ ) PJ d Ws 2H

(125)

θ

123

Stoch PDE: Anal Comp

4.2 Properties of the nonlinearities In this subsection we establish a few elementary properties for the nonlinearities F and B in Sect. 4.1 (see Lemmas 4.1, 4.2, 4.8, 4.9, and Corollary 4.10 below). To do so, we also recall in this subsection some well-known properties of the involved Sobolev and interpolation spaces (see Lemmas 4.3–4.8 below). Lemma 4.1 Assume the setting in Sect. 4.1 and let v, w ∈ [Hγ ∩ W 1,2 (D, Rd ) ∩ L ∞ (μD ; Rd )]. Then it holds that

F(v) L 2 (μD ;Rd ) ≤ η v L 2 (μD ;Rd ) + d v L ∞ (μD ;Rd ) ∂v L 2 (μD ;Rd×d ) < ∞ (126) and

F(v) − F(w) L 2 (μD ;Rd ) ≤ η v − w L 2 (μD ;Rd )  + d ∂v L 2 (μD ;Rd×d ) v − w L ∞ (μD ;Rd )  + w L ∞ (μD ;Rd ) ∂(v − w) L 2 (μD ;Rd×d ) < ∞.

(127)

Proof of Lemma 4.1 Note that the triangle inequality and Hölder’s inequality imply that  

F(v) L 2 (μD ;Rd ) ≤ η v L 2 (μD ;Rd ) +  dj=1 v j ∂ j v  L 2 (μ ;Rd ) D  ≤ η v L 2 (μD ;Rd ) + dj=1 v j L ∞ (μD ;R) ∂ j v L 2 (μD ;Rd ) % % d d 2 2 ≤ η v L 2 (μD ;Rd ) +

v

∞ j j=1 j=1 ∂ j v L 2 (μ L (μ ;R) D

≤ η v L 2 (μD ;Rd ) + d v L ∞ (μD ;Rd ) ∂v L 2 (μD ;Rd×d ) .

d D ;R )

(128)

In addition, observe that



F(v) − F(w) L 2 (μD ;Rd ) − η v − w L 2 (μD ;Rd ) ≤  dj=1 (∂ j v)v j   − dj=1 (∂ j w)w j  L 2 (μ ;Rd ) D     ≤  dj=1 (∂ j v)(v j − w j ) L 2 (μ ;Rd ) +  dj=1 (∂ j v − ∂ j w)w j  L 2 (μ ;Rd ) D D d d ≤ j=1 (∂ j v)(v j − w j ) L 2 (μD ;Rd ) + j=1 (∂ j v − ∂ j w)w j L 2 (μD ;Rd )   ≤ dj=1 ∂ j v L 2 (μD ;Rd ) v j − w j L ∞ (μD ;R) + dj=1 ∂ j v − ∂ j w L 2 (μD ;Rd ) w j L ∞ (μD ;R) .

(129)

Hölder’s inequality hence proves that % % d d 2 2

F(v) − F(w) L 2 (μD ;Rd ) ≤

∂ v

j j=1 j=1 v j − w j L ∞ (μD ;R) L 2 (μD ;Rd ) % % d d 2 2 + j=1 ∂ j v − ∂ j w L 2 (μ ;Rd ) j=1 w j L ∞ (μD ;R) + η v − w L 2 (μD ;Rd ) D

123

Stoch PDE: Anal Comp

  ≤ d ∂v L 2 (μD ;Rd×d ) v − w L ∞ (μD ;Rd ) + w L ∞ (μD ;Rd ) ∂(v − w) L 2 (μD ;Rd×d ) + η v − w L 2 (μD ;Rd ) .

(130)  

The proof of Lemma 4.1 is thus completed. Lemma 4.2 Assume the setting in Sect. 4.1. Then it holds for all v, w ∈ Hγ that

√  

B(v) HS(U,H ) ≤ supx∈D,y∈Rd b(x, y) Rd×d traceU (Q) = ϑ < ∞ (131) and

B(v) − B(w) HS(U,H )  ≤ supx∈D,y∈Rd ,z∈Rd \{y}

b(x,y)−b(x,z) Rd×d

y−z Rd



v − w L ∞ (μD ;Rd ) traceU (Q). (132)

Proof of Lemma 4.2 First of all, note for all v ∈ M(B(D), B(Rd )) with [v]μD ,B(Rd ) ∈ Hγ that

B([v]μD ,B(Rd ) ) 2HS(U,H ) =

B([v]μD ,B(Rd ) )u 2H u∈U 2  1/2 [{b(x, v(x))}x∈D ] ≤ u)U d×d ) (Q μ , B (R D   u∈U 1 2 ≤ supx∈D,y∈Rd b(x, y) 2Rd×d

Q /2 u U u∈U   = supx∈D,y∈Rd b(x, y) 2Rd×d traceU (Q).

(133)

Next observe for all v, w ∈ M(B(D), B(Rd )) with [v]μD ,B(Rd ) , [w]μD ,B(Rd ) ∈ Hγ that

B([v]μD ,B(Rd ) ) − B([w]μD ,B(Rd ) ) 2HS(U,H ) 2  1/2  [{b(x, v(x)) − b(x, w(x))}x∈D ] ≤ μD ,B(Rd×d ) (Q u) U u∈U

(134) 1 2 ≤ {b(x, v(x)) − b(x, w(x))}x∈D 2L∞ (μ ;Rd×d )

Q /2 u U D u∈U 

b(x,y)−b(x,z) Rd×d 2 ≤ supx∈D,y∈Rd ,z∈Rd \{y}

v − w 2L∞ (μ ;Rd ) traceU (Q).

y−z d D

R

 

Combining (133) and (134) completes the proof of Lemma 4.2. Lemma 4.3 Assume the setting in Sect. 4.1 and let ρ ∈ [0, ∞), v ∈ Hρ . Then 

v L ∞ (μD ;Rd ) ≤ v Hρ



|λh |−2ρ

sup h L ∞ (μD ;Rd )

h∈H

1/2 .

(135)

h∈H

123

Stoch PDE: Anal Comp

Proof of Lemma 4.3 Note that Hölder’s inequality proves that 



h, v H h L ∞ (μD ;Rd ) ≤ h∈H

sup h L ∞ (μD ;Rd )



h∈H

|h, v H | 

h∈H

|λh |ρ |h, v H | |λh |−ρ

sup h L ∞ (μD ;Rd )

≤

h∈H

h∈H

 ≤ v Hρ



sup h L ∞ (μD ;Rd )

h∈H

|λh |

−2ρ

1/2 .

h∈H

(136)  

This completes the proof of Lemma 4.3.

Lemma 4.4 Assume the setting in Sect. 4.1, let ρ ∈ [0, ∞),  and assume for all j ∈ {1, . . . , d}, v, w ∈ H that H ⊆ W 1,2 (D, Rd ), suph∈H ∂ j h U |λh |−ρ < ∞, ∂ j v, ∂ j wU 1H\{v} (w) = 0. Then (i) it holds that Hρ ⊆ W 1,2 (D, Rd ), (ii) it holds for all u ∈ Hρ , j ∈ {1, . . . , d} that

∂ j u U =

 h∈H

2

h, u H ∂ j h U

1/2

 ≤ suph∈H

∂ j h U |λh |ρ



u Hρ < ∞ (137)

 and ∂ j u = h∈H h, u H ∂ j h, and (iii) it holds for all u ∈ Hρ that

d 

∂u L 2 (μD ;Rd×d ) ≤ ≤

j=1

√



∂ j u 2L 2 (μ ;Rd ) D

d sup

sup

h∈H j∈{1,...,d}

1/2

∂ j h U |λh |ρ



u Hρ < ∞.

(138)

Proof of Lemma 4.4 Note that for all u ∈ Hρ , j ∈ {1, . . . , d} it holds that

h∈H

2

h, u H ∂ j h U ≤

 sup 

h∈H

2

∂ j h U |λh |2ρ

= suph∈H

123



2

∂ j h U |λh |2ρ

h∈H

|λh |2ρ |h, u H |2



u 2Hρ < ∞.

(139)

Stoch PDE: Anal Comp

The fact that for all j ∈ {1, . . . , d}, v, w ∈ H with v = w it holds that ∂ j v, ∂ j wU = 0 ∞ (D, Rd ), j ∈ {1, . . . , d} it holds that hence shows that for all u ∈ Hρ , φ ∈ Ccpt & u, [ ∂ ∂x j

φ]μD ,B(Rd ) U = !

=

h, uU h, h∈H

=−



∂ ∂x j

' h, uU h, [ ∂∂x j h∈H

φ

φ]μD ,B(Rd ) U

"



μD ,B(Rd ) U

(140) &

'

h, uU ∂ j h, [φ]μD ,B(Rd ) U = − h∈H

h, uU ∂ j h, [φ]μD ,B(Rd ) h∈H

. U

 

This and (139) complete the proof of Lemma 4.4.

Lemma 4.5 (Weak product rule (cf., e.g., Proposition 7.1.11 in Atkinson and Han [2])) Let d ∈ N, u, v ∈ [W 1,2 ((0, 1)d , R) ∩ L ∞ (μ(0,1)d ; R)], j ∈ {1, . . . , d}. Then it holds that u · v ∈ [W 1,2 ((0, 1)d , R) ∩ L ∞ (μ(0,1)d ; R)] and ∂ j (uv) = u ∂ j v + v ∂ j u. In the next two well-known auxiliary results, Lemmas 4.6 and 4.7 below, we recall some basic properties of certain Sobolev spaces with periodic boundary conditions. For the formulation of Lemmas 4.6 and 4.7 below we recall that for all d ∈ N it holds that d C∞ P ((0, 1) , R)   d = f ∈ C((0, 1) , R) :

∃ ϕ ∈ C ∞ (Rd , R) : ( f = ϕ|(0,1)d ) and (∀ i ∈ [1, d] ∩ N, x ∈ Rd : ϕ(x + ei ) = ϕ(x))

 (141)

and d W P1,2 ((0, 1)d , R) = C ∞ P ((0, 1) , R)

W 1,2 ((0,1)d ,R)

.

(142)

Lemma 4.6 (Weak integration by parts) Let d ∈ N, u, v ∈ W P1,2 ((0, 1)d , R), j ∈ {1, . . . , d}. Then it holds that ∂ j u, vL 2 (μ

(0,1)d ;R)

= −u, ∂ j vL 2 (μ

(0,1)d ;R)

.

(143)

Lemma 4.7 (Weak integration by parts revisited) Let d ∈ N, u, v, w ∈ [W P1,2 ((0, 1)d , R) ∩ L ∞ (μ(0,1)d ; R)], j ∈ {1, . . . , d}. Then it holds that u · v ∈ [W 1,2 ((0, 1)d , R) ∩ L ∞ (μ(0,1)d ; R)] and ∂ j (uv), wL 2 (μ

(0,1)d ;R)

= −uv, ∂ j w L 2 (μ

(0,1)d ;R)

.

(144)

123

Stoch PDE: Anal Comp d Proof of Lemma 4.7 Throughout this proof let u˜ n , v˜n , w˜ n ∈ C ∞ P ([0, 1] , R), n ∈ N, 1,2 and let u n , vn , wn ∈ W P ((0, 1)d , R), n ∈ N, satisfy for all n ∈ N that u n = [u˜ n |(0,1)d ]μ(0,1)d ,B(R) , vn = [v˜n |(0,1)d ]μ(0,1)d ,B(R) , wn = [w˜ n |(0,1)d ]μ(0,1)d ,B(R) , and   lim supm→∞ u−u m W 1,2 ((0,1)d ,R) + v−vm W 1,2 ((0,1)d ,R) + w−wm W 1,2 ((0,1)d ,R) = 0. Observe that Lemma 4.5 (with d = d, u = u, v = v, j = j in the notation of Lemma 4.5) and the product rule for differentiation prove that u · v ∈ [W 1,2 ((0, 1)d , R) ∩ L ∞ (μ(0,1)d ; R)] and

∂ j (uv), w L 2 (μ d ;R) = u ∂ j v + v ∂ j u, w L 2 (μ d ;R) (0,1) (0,1)   = lim u l ∂ j v, w L 2 (μ d ;R) + v ∂ j u l , w L 2 (μ d ;R) (0,1) (0,1) l→∞    = lim lim u l ∂ j v, wn  L 2 (μ d ;R) + v ∂ j u l , wn  L 2 (μ d ;R) (0,1) (0,1) l→∞ n→∞     = lim lim lim u l ∂ j vm , wn  L 2 (μ d ;R) + vm ∂ j u l , wn  L 2 (μ d ;R) (0,1) (0,1) l→∞ n→∞ m→∞    = lim lim lim u l ∂ j vm + vm ∂ j u l , wn  L 2 (μ d ;R) (0,1) l→∞ n→∞ m→∞    = lim lim lim ∂ j (u l vm ), wn  L 2 (μ d ;R) . (145) l→∞

n→∞

(0,1)

m→∞

d Integration by parts and the fact that ∀ n ∈ N : u˜ n , v˜n , w˜ n ∈ C ∞ P ([0, 1] , R) hence show that

 ∂ j (uv), w L 2 (μ

(0,1)d ;R)

 = − lim

l→∞



lim



= lim

n→∞

n→∞

l→∞



lim



 lim

m→∞ (0,1)d





lim

m→∞ (0,1)d

u˜ l (x) v˜m (x)



∂ ∂x j

lim u l vm , ∂ j wn  L 2 (μ d ;R) (0,1) l→∞ n→∞ m→∞   = − lim lim u l v, ∂ j wn  L 2 (μ d ;R) = − lim

l→∞

∂ ∂x j

 (u˜ l (x) · v˜m (x)) w˜ n (x) d x

 w˜ n (x) d x







lim

(0,1)

n→∞

= − lim u l v, ∂ j w L 2 (μ

(0,1)d ;R)

l→∞

= −uv, ∂ j w L 2 (μ

(0,1)d ;R)

.

(146)  

The proof of Lemma 4.7 is thus completed.

Lemma 4.8 Assume the setting in Sect. 4.1, let ρ ∈ [γ , ∞), u ∈ Hρ , and assume for all . . , d}, v, w ∈ H that H ⊆ W 1,2 (D, Rd ), ∂ j v,∂ j wU 1H\{v} (w) = 0,  j ∈ {1, .−2ρ + suph∈H h L ∞ (μD ;Rd ) + ∂ j h U |λh |−ρ < ∞. h∈H |λh | Then it holds that u ∈ [W 1,2 (D, Rd ) ∩ L ∞ (μD ; Rd )] and

∂u L 2 (μD ;Rd×d ) ≤

√ ≤ d sup

d  j=1

sup

h∈H j∈{1,...,d}

123

∂ j u 2L 2 (μ

D ;R

1/2 d)



∂ j h U

u Hρ < ∞, |λh |ρ

(147)

Stoch PDE: Anal Comp



u L ∞ (μD ;Rd ) ≤ u Hρ

 sup h L ∞ (μD ;Rd )

h∈H

|λh |

1/2 < ∞, (148)

h∈H

√

F(u) H ≤ η u H + d d u 2Hρ

 

∂ j h U · sup sup

h

sup ∞ d L (μD ;R ) |λh |ρ h∈H j∈{1,...,d}

−2ρ

h∈H

|λh |−2ρ

1/2

(149) < ∞.

h∈H

Proof of Lemma 4.8 First, note that Lemma 4.3 (with ρ = ρ, v = u in the notation of Lemma 4.3) proves (148). Moreover, observe that Lemma 4.4 (with ρ = ρ in the notation of Lemma 4.4) establishes that u ∈ W 1,2 (D, Rd ) and (147). This and (148) ensure that u ∈ [W 1,2 (D, Rd ) ∩ L ∞ (μ D ; Rd )]. Combining Lemma 4.1 (with v = u, w = u in the notation of Lemma 4.1), (147), and (148) hence proves (149). The proof of Lemma 4.8 is thus completed.   Lemma 4.9 Assume the setting in Sect. 4.1, let ρ ∈ [γ , ∞), u = (u 1 , . . . , u d ) ∈ for all j  ∈ {1, . . . , d}, v, w ∈ H that H ⊆ W P1,2 (D, Rd ), ρ , and assume  H −2ρ + sup −ρ < ∞, ∂ v, ∂ w j j U h∈H h L ∞ (μD ;Rd ) + ∂ j h U |λh | h∈H |λh | 1,2 d ∞ 1H\{v} (w) = 0. Then it holds that u ∈ [W P (D, R ) ∩ L (μD ; Rd )] and 2u, F(u) H = 2η u 2H + =

2η u 2H

+

d

j=1 u, u ∂ j u j U $ d 2 i=1 (u i ) , j=1 ∂ j u j L 2 (μD ;R) .

# d

(150)

Proof of Lemma 4.9 First, note that Lemma 4.8 (with ρ = ρ, u = u in the notation of Lemma 4.8) ensures that u ∈ [W 1,2 (D, Rd ) ∩ L ∞ (μD ; Rd )].

(151)

Moreover, observe that lim supP0 (H) I →H u −



h∈I h, u H h L 2 (μD ;Rd )

= 0.

(152)

In addition, note that item (ii) of Lemma 4.4 (with ρ = ρ, u = u, j = j for j ∈ {1, . . . , d} in the notation of Lemma 4.4) proves that for all j ∈ {1, . . . , d} it holds that lim supP0 (H) I →H ∂ j u −



h∈I h, u H ∂ j h L 2 (μD ;Rd )

= 0.

(153)

Combining the fact that ∀ v ∈ W 1,2 (D, Rd ) : v 2W 1,2 (D,Rd ) = v 2L 2 (μ ;Rd ) + D d 2 j=1 ∂ j v L 2 (μ ;Rd ) with (151)–(153) proves that D

lim supP0 (H) I →H u −



h∈I h, u H h W 1,2 (D ;Rd )

= 0.

(154)

123

Stoch PDE: Anal Comp

 1,2 d d The fact that ∀ I ∈ P0 (H) : h∈I h, u H h ∈ W P ((0, 1) , R ), the fact that 1,2 W P ((0, 1)d , Rd ) is a closed subspace of W 1,2 ((0, 1)d , Rd ), and (151) hence show that u ∈ [W P1,2 (D, Rd ) ∩ L ∞ (μD ; Rd )].

(155)

This and Lemma 4.5 (with d = d, u = u i , v = u j , j = j for i, j ∈ {1, . . . , d} in the notation of Lemma 4.5) prove that for all i, j ∈ {1, . . . , d} it holds that u i u j ∈ W 1,2 (D, R) and ∂ j (u i u j ) = u i ∂ j u j + u j ∂ j u i . Combining this and the fact that ∀ i ∈ {1, . . . , d} : u i ∈ [W P1,2 (D, R) ∩ L ∞ (μD ; R) ∩ H ] with Lemma 4.7 (with d = d, u = u i , v = u j , w = u i , j = j for i, j ∈ {1, . . . , d} in the notation of Lemma 4.7) ensures that   u, F(u) H = u, R(ηu − dj=1 u j ∂ j u) H = η u 2H − dj=1 Ru, u j ∂ j uU d   = η u 2H − dj=1 u, u j ∂ j uU = η u 2H − dj=1 i=1 u i , u j ∂ j u i  L 2 (μD ;R)   d = η u 2H − dj=1 i=1 u i u j , ∂ j u i  L 2 (μD ;R) = η u 2H + = η u 2H +

d

d

d

j=1

j=1

d

i=1

∂ j (u i u j ), u i  L 2 (μD ;R)

i=1 u i

∂ j u j + u j ∂ j u i , u i  L 2 (μD ;R) .

(156)

Hence, we obtain that  d u i , u i ∂ j u j + u j ∂ j u i  L 2 (μD ;R) u, F(u)U = η u 2H + dj=1 i=1 d d 2 = η u H + j=1 u, u ∂ j u j U + j=1 u, u j ∂ j uU  

 = 2η u 2H − η u 2H − Ru, dj=1 u j ∂ j uU + dj=1 u, u ∂ j u j U  

d = 2η u 2H − u, R(ηu) H − u, R( i=1 u i ∂i u) H + dj=1 u, u ∂ j u j U  = 2η u 2H − u, F(u) H + dj=1 u, u ∂ j u j U . (157) In addition, note that d d j=1 u, u ∂ j u j U = j=1 i=1 u i , u i d d 2 = j=1 i=1 (u i ) , ∂ j u j  L 2 (μD ;R) $ # d d 2 = i=1 (u i ) , j=1 ∂ j u j L 2 (μD ;R) .

d

∂ j u j  L 2 (μD ;R)

Combining this, (155), and (157) completes the proof of Lemma 4.9.

(158)

 

Corollary 4.10 Assume the setting in Sect. 4.1 and assume for all . . ,d},  j ∈ {1, .−2γ v, w ∈ H that H ⊆ W 1,2 (D, Rd ), ∂ j v, ∂ j wU 1H\{v} (w) = 0, + h∈H |λh | suph∈H h L ∞ (μD ;Rd ) + ∂ j h U |λh |−γ < ∞. Then it holds that F ∈ C(Hγ , H ) and B ∈ C(Hγ , HS(U, H )).

123

Stoch PDE: Anal Comp

Proof of Corollary 4.10 First of all, note that Lemma 4.8 (with ρ = γ in the notation of Lemma 4.8) assures that Hγ ⊆ W 1,2 (D, Rd ) continuously

Hγ ⊆ L ∞ (μD ; Rd ) continuously. (159)

and

This and Lemma 4.1 (with v = v, w = w for v, w ∈ Hγ in the notation of Lemma 4.1) show that for all v, w ∈ Hγ it holds that

F(v) − F(w) L 2 (μD ;Rd ) ≤ η v − w L 2 (μD ;Rd )  + d ∂v L 2 (μD ;Rd×d ) v − w L ∞ (μD ;Rd )  + w L ∞ (μD ;Rd ) ∂(v − w) L 2 (μD ;Rd×d ) < ∞.

(160)

In addition, Lemma 4.2 (with v = v, w = w for v, w ∈ Hγ in the notation of Lemma 4.2) proves that for all v, w ∈ Hγ it holds that

B(v) − B(w) HS(U,H )  ≤ supx∈D,y∈Rd ,z∈Rd \{y}

b(x,y)−b(x,z) Rd×d

y−z Rd



v − w L ∞ (μD ;Rd ) traceU (Q). (161)

Combining this and (160) with (159) completes the proof of Corollary 4.10.

 

4.3 Stochastic Burgers equations Corollary 4.11 Assume the setting in Sect. 4.1, let (en )n∈N √ ⊆ H, and assume for all n ∈ N, v ∈ Hγ that η = 0, d = 1, γ ≥ 1/2, en = [{ 2 sin(nπ x)}x∈D ]μD ,B(R) ,  √ λen = −π 2 n 2 , r (v) ≥ max ϑ + ε v 2H , √1 v 2Hγ . Then 3

sup

sup

sup

 θ,I,J  ε Yt

2H < ∞. sup E exp e2εϑt

θ∈T I ∈P0 (H) J ∈P0 (U) t∈[0,T ]

(162)

Proof of Corollary 4.11 First of all, note that the fact that {en : n ∈ N} ⊆ H ⊆ H ⊆ U shows that {en : n ∈ N} = H

and

H = U.

(163)

In the next step we observe that

h∈H

|λh |−2γ = =π

n∈N −4γ

|π 2 n 2 |−2γ n∈N

n −4γ ≤ π −2

n∈N

n −2 < ∞.

(164)

123

Stoch PDE: Anal Comp

Moreover, note that for all n ∈ N it holds that √

∂en U |λen |−γ = [{π n 2 cos(nπ x)}x∈D ]μD ,B(R) U |π 2 n 2 |−γ = π n|π 2 n 2 |−γ =

1 (π n)2γ −1

≤ 1.

(165)

√ Combining (163)–(165), the fact that suph∈H h L ∞ (μD ;R) = 2, and Lemma 4.8 (with ρ = γ , u = v for v ∈ Hγ in the notation of Lemma 4.8) proves that for all v ∈ Hγ it holds that Hγ ⊆ [W 1,2 (D, R) ∩ L ∞ (μD ; R)] and

F(v) H ≤

√

2 π



n∈N n

 −2 1/2 v 2 Hγ

=

√1 v 2 Hγ 3

< ∞.

(166)

Next note that ∞ (D, R) H ⊆ D(A) = [W 2,2 (D, R) ∩ W01,2 (D, R)] ⊆ W01,2 (D, R) = Ccpt

⊆

W C∞ P (D, R)

1,2 (D ,R)

=

W 1,2 (D ,R)

(167)

W P1,2 (D, R).

√ This, (163)–(165), the fact that suph∈H h L ∞ (μD ;R) = 2, and Lemma 4.9 (with ρ = γ , u = x for x ∈ Hγ in the notation of Lemma 4.9) ensure that for all x ∈ Hγ it holds that Hγ ⊆ [W P1,2 (D, R) ∩ L ∞ (μD ; R)] and 2x, F(x) H = 2η x 2H + x, x ∂ xU = 2η x 2H + x, R(x ∂ x) H = 2η x 2H − x, F(x) H = −x, F(x) H .

(168)

Hence, we obtain that for all I ∈ P0 (H), x ∈ PI (H ) it holds that x, PI F(x) H = PI x, F(x) H = x, F(x) H = 0.

(169)

√ In the next step we observe that (163)–(165), the fact that suph∈H h L ∞ (μD ;R) = 2, and Corollary 4.10 assure that F ∈ C(Hγ , H ) and B ∈ C(Hγ , HS(U, H )). This proves that F ∈ M(B(Hγ ), B(H ))

and

B ∈ M(B(Hγ ), B(HS(U, H ))).

(170)

Moreover, (166) and Lemma 4.2 (with v = x, w = x for x ∈ ∪h∈(0,T ] ∪ I ∈P0 (H) DhI in the notation of Lemma 4.2) imply for all h ∈ (0, T ], I ∈ P0 (H), J ∈ P0 (U), x ∈ DhI that     max PI F(x) H , PI B(x) Pˆ J HS(U,H ) ≤ max F(x) H , B(x) HS(U,H ) √   ≤ max √1 x 2Hγ , ϑ ≤ r (x) ≤ ch −δ . 3

(171)

123

Stoch PDE: Anal Comp

Furthermore, we observe that the fact that ∀ v ∈ Hγ : for all I ∈ P0 (H), h ∈ (0, T ] it holds that

√ θ + ε v 2H ≤ r (v) shows that

√ DhI = {x ∈ PI (H ) : r (x) ≤ ch −δ } ⊆ {x ∈ PI (H ) : ϑ + ε x 2H ≤ ch −δ } √ ⊆ {v ∈ H : ϑ + ε v 2H ≤ ch −δ }. (172) In addition, we note that Lemma 4.2 ensures that supx∈Hγ B(x) 2HS(U,H ) ≤ ϑ < ∞. Combining (169)–(172) and Corollary 3.4 (with H = H , U = U , H = H, U = U, T = T , γ = γ , δ = δ, λ = λ, A = A, ξ = ξ , F = F, B = B, DhI = DhI , ϑ = ϑ, b1 = 0, b2 = 0, ε = ε, ς = δ, c = c, Y θ,I,J = Y θ,I,J for h ∈ (0, T ], θ ∈ T , I ∈ P0 (H), J ∈ P0 (U) in the notation of Corollary 3.4) hence completes the proof of Corollary 4.11.   Remark 4.12 Consider the setting of Corollary 4.11. Then the stochastic processes Y θ,I,J : [0, T ] ×  → PI (H ), θ ∈ T , I ∈ P0 (H), J ∈ P0 (U), are space-timenoise discrete numerical approximation processes for the stochastic Burgers equation d X t (x) =



∂2 ∂x2

X t (x) − X t (x) ·

∂ ∂x

 X t (x) dt + b(x, X t (x)) d( QW )t (x) (173)

with X 0 (x) = ξ(x) and X t (0) = X t (1) = 0 for t ∈ [0, T ], x ∈ (0, 1) (cf., e.g., Section 1 in Da Prato et al. [16] and Section 2 in Hairer and Voss [25]). 4.4 Stochastic Kuramoto–Sivashinsky equations Corollary 4.13 Assume the setting in Sect. 4.1, let (ek )k∈Z ⊆ H, and assume for all n ∈ N, k ∈ Z, v ∈ Hγ that η ∈ (0, ∞), d = 1, γ ≥ 1/4, e0 = [{1}x∈D ]μD ,B(R) , √ √ en = [{ 2 cos(2π nx)}x∈D ]μD ,B(R) , e−n = [{ 2 sin(2π nx)}x∈D ]μD ,B(R) , r (v) ≥ √  max ϑ + ε v 2H , η v H +5 max{1, η−γ } v 2Hγ , λek = 4k 2 π 2 − 16k 4 π 4 − η. Then

 θ,I,J  ε Yt

2H < ∞. sup sup sup sup E exp e2(η+εϑ)t (174) θ∈T I ∈P0 (H) J ∈P0 (U) t∈[0,T ]

Proof of Corollary 4.13 First of all, note that the fact that {el : l ∈ Z} ⊆ H ⊆ H ⊆ U shows that {el : l ∈ Z} = H

and

H = U.

(175)

Hence, we obtain that 

h∈H |λh |

−2γ

= =





−2γ = 4 4 2 2 −2γ k∈Z |λek | k∈Z |16k π − 4k π + η|  η−2γ + 2 ∞ |16k 4 π 4 − 4k 2 π 2 + η|−2γ ≤ η−2γ ∞ k=1 4 4 + 2 k=1 |12k π + η|−2γ

123

Stoch PDE: Anal Comp

≤ η−2γ + 2

∞

k=1 |12k

≤ η−2γ +

2 (12π 4 )1/2

= η−2γ +

π2 6

4 π 4 |−2γ

∞

1 k=1 k 2

= η−2γ + |12π24 |2γ  1 ≤ η−2γ + ∞ k=1 k 2

∞

1 k=1 k 8γ

< ∞.

(176)

Moreover, note that for all n ∈ N it holds that

∂en U |λen |−γ =

√

[{2π n 2 sin(2π nx)}x∈(0,1) ]μ(0,1) ,B(R) U |16n 4 π 4 −4n 2 π 2 +η|γ

≤

2π n |12n 4 π 4 +η|γ

≤

2π n |12n 4 π 2 |γ

≤

=

2π n |16n 4 π 4 −4n 2 π 2 +η|γ

(177) 2π n √ n(12)1/4 π

=

√

2 π (12)1/4

≤ 2.

This shows that for all n ∈ N it holds that

∂e−n U |λe−n |−γ = =

√

[{2π n 2 cos(2π nx)}x∈(0,1) ]μ(0,1) ,B(R) U |16n 4 π 4 −4n 2 π 2 +η|γ 2π n ≤ 2. |16n 4 π 4 −4n 2 π 2 +η|γ

(178)

√ Combining (175)–(178), the fact that suph∈H h L ∞ (μD ;R) = 2, and Lemma 4.8 (with ρ = γ , u = v for v ∈ Hγ in the notation of Lemma 4.8) proves that for all v ∈ Hγ it holds that Hγ ⊆ [W 1,2 (D, R) ∩ L ∞ (μD ; R)] and √  2 1/2

F(v) H ≤ η v H + 2 2 η−2γ + π6

v 2Hγ  2 1/2 = η v H + 8η−2γ + 4π3

v 2Hγ  2 1/2 ≤ η v H + max{1, η−γ } 8 + 4π3

v 2Hγ

(179)

≤ η v H + 5 max{1, η−γ } v 2Hγ . Next note that H ⊆ C∞ P (D, R)

W 1,2 (D ,R)

= W P1,2 (D, R).

(180)

√ This, (175)–(178), the fact that suph∈H h L ∞ (μD ;R) = 2, and Lemma 4.9 (with ρ = γ , u = x for x ∈ Hγ in the notation of Lemma 4.9) ensure that for all x ∈ Hγ it holds that Hγ ⊆ [W P1,2 (D, R) ∩ L ∞ (μD ; R)] and 2x, F(x) H = 2η x 2H + x, x ∂ xU = 2η x 2H + x, R(x ∂ x) H

(181) = 3η x 2H − [x, R(ηx) H − x, R(x ∂ x) H ] = 3η x 2H − x, F(x) H .

Hence, we obtain that for all I ∈ P0 (H), x ∈ PI (H ) it holds that x, PI F(x) H = PI x, F(x) H = x, F(x) H = η x 2H .

123

(182)

Stoch PDE: Anal Comp

√ In the next step we observe that (175)–(178), the fact that suph∈H h L ∞ (μD ;R) = 2, and Corollary 4.10 assure that F ∈ C(Hγ , H ) and B ∈ C(Hγ , HS(U, H )). This proves that F ∈ M(B(Hγ ), B(H ))

B ∈ M(B(Hγ ), B(HS(U, H ))).

and

(183)

Moreover, (179) and Lemma 4.2 (with v = x, w = x for x ∈ ∪h∈(0,T ] ∪ I ∈P0 (H) DhI in the notation of Lemma 4.2) imply that for all h ∈ (0, T ], I ∈ P0 (H), J ∈ P0 (U), x ∈ DhI it holds that max{ PI F(x) H , PI B(x) Pˆ J HS(U,H ) } ≤ max{ F(x) H , B(x) HS(U,H ) } √   ≤ max η x H + 5 max{1, η−γ } x 2Hγ , ϑ ≤ r (x) ≤ ch −δ . (184) Furthermore, we observe that the fact that ∀ v ∈ Hγ : that for all I ∈ P0 (H), h ∈ (0, T ] it holds that

√ θ + ε v 2H ≤ r (v) implies

√ DhI = {x ∈ PI (H ) : r (x) ≤ ch −δ } ⊆ {x ∈ PI (H ) : ϑ + ε x 2H ≤ ch −δ } √ ⊆ {v ∈ H : ϑ + ε v 2H ≤ ch −δ }. (185) In addition, we note that Lemma 4.2 ensures that supx∈Hγ B(x) 2HS(U,H ) ≤ ϑ < ∞. Combining (182)–(185) and Corollary 3.4 (with H = H , U = U , H = H, U = U, T = T , γ = γ , δ = δ, λ = λ, A = A, ξ = ξ , F = F, B = B, DhI = DhI , ϑ = ϑ, b1 = 0, b2 = η, ε = ε, ς = δ, c = c, Y θ,I,J = Y θ,I,J for h ∈ (0, T ], θ ∈ T , I ∈ P0 (H), J ∈ P0 (U) in the notation of Corollary 3.4) hence completes the proof of Corollary 4.13.   Remark 4.14 Consider the setting of Corollary 4.13. Then the stochastic processes Y θ,I,J : [0, T ]× → PI (H ), θ ∈ T , I ∈ P0 (H), J ∈ P0 (U), are space-time-noise discrete numerical approximation processes for the stochastic Kuramoto–Sivashinsky equation

d X t (x) = −

∂4 ∂x4

X t (x) −

∂2 ∂x2

X t (x) − X t (x) ·

+ b(x, X t (x)) d( QW )t (x)

∂ ∂x

 X t (x) dt (186)

(3)

(3)

with X t (0) = X t (1), X t (0) = X t (1), X t (0) = X t (1), X t (0) = X t (1), and X 0 (x) = ξ(x) for t ∈ [0, T ], x ∈ (0, 1) (cf., e.g, Duan and Ervin [20] and Section 1 in Hutzenthaler et al. [30]). 4.5 Two-dimensional stochastic Navier–Stokes equations Corollary 4.15 Assume the setting in Sect. 4.1, assume that η > 0, d = 2, γ > 1/2, let (ϕk )k∈Z ⊆ C((0, 1), R) satisfy for all n ∈ N, x ∈ (0, 1) that √ √ ϕ0 (x) = 1, ϕn (x) = 2 cos(2nπ x), ϕ−n (x) = 2 sin(2nπ x), (187)

123

Stoch PDE: Anal Comp

let (φk,l )k,l∈Z ⊆ C(D, R), (ei, j,0 )i, j∈Z ⊆ U , e0,0,1 ∈ U , assume for all (k, l) ∈ Z2 \{(0, 0)}, x, y ∈ (0, 1) that H = {e0,0,1 } ∪ {ei, j,0 : i, j ∈ Z}, φk,l (x, y) = ϕk (x)ϕl (y), e0,0,0 = [{(1, 0)}(x,y)∈D ]μD ,B(R2 ) , e0,0,1 = [{(0, 1)}(x,y)∈D ]μD ,B(R2 ) , 

√ ek,l,0 = {1/ k 2 +l 2 (lφk,l (x, y), kφ−k,−l (x, y))}(x,y)∈D μ ,B(R2 ) , λek,l,0 = −η − D 4π 2 (k 2 + l 2 ), λe0,0,0 = λe0,0,1 = −η, and assume for all v ∈ Hγ that √

r (v) ≥ max ϑ + ε v 2H , η v H + 6 η−2γ 1/2   + i, j∈Z (η + 4π 2 (i 2 + j 2 ))−2γ v 2Hγ .

(188)

Then sup

sup

 θ,I,J  ε Yt

2H < ∞. sup E exp e2(η+εϑ)t

sup

(189)

θ∈T I ∈P0 (H) J ∈P0 (U) t∈[0,T ]

Proof of Corollary 4.15 Observe that 

|λh |−2γ = η−2γ +

h∈H

= η−2γ + η−2γ + 2

 (k,l)∈Z2 ∞ 

(η + 4π 2 (k 2 + l 2 ))−2γ

(η + 4π 2 l 2 )−2γ + 2

l=1

+4

∞ ∞  

∞ 

(η + 4π 2 k 2 )−2γ

k=1

(η + 4π 2 (k 2 + l 2 ))−2γ

k=1 l=1

≤ 2η−2γ + 4

∞ 

(4π 2 k 2 )−2γ + 4

k=1

∞  k=1

k −4γ +

k −4γ + 41−2γ π −4γ

k=1 ∞ 

k −4γ = 1 +

k=1

∞  k=2

∞ ∞  

(k 2 + l 2 )−2γ

k=1 l=1

(k 2 + l 2 )−2γ .

k,l=1

Next note that the fact that ∀ k ∈ N : k −4γ ≤ ∞ 

(190)

(4π 2 (k 2 + l 2 ))−2γ

k=1 l=1 ∞ 

= 2η−2γ + 41−2γ π −4γ ≤ 2η−2γ +

∞ ∞  

k −4γ ≤ 1 +

∞ 

k

k k−1

x −4γ d x proves that ∞

∫ x −4γ d x = 1 + ∫ x −4γ d x = 1 +

k=2 k−1

1

1 4γ −1 .

(191) k l In addition, we observe that the fact that ∀ k, l ∈ N : (k 2 + l 2 )−2γ = k−1 l−1 (k 2 + k l k l l 2 )−2γ d x d y ≤ k−1 l−1 (y 2 + x 2 )−2γ d x d y = k−1 l−1 (x 2 + y 2 )−2γ d x d y proves that

123

Stoch PDE: Anal Comp ∞ 

∞ 

(k 2 + l 2 )−2γ =

k,l=1

≤2 =2

∞  k=1 ∞ 

(k 2 + 1)−2γ +

k=1 ∞ ∞  

k −4γ +

k

l 2 k−1 l−1 (x

∞∞ 1

k=1

1

1−4γ = 2π ∫∞ ds + 2 1 s

(1 + l 2 )−2γ +

l=2

k=2 l=2

k −4γ +

∞ 

∞ ∞  

+ y 2 )−2γ d x d y

(x 2 + y 2 )−2γ d x d y = 2

∞ 

k −4γ +

 2π  ∞ 0

1

π 2γ −1

+2

k=1 ∞ 

k −4γ =

k=1

(k 2 + l 2 )−2γ

k=2 l=2

2π 4γ −2

∞ 

+2

k −4γ =

k=1

s 1−4γ ds du

∞ 

k −4γ .

k=1

(192) Combining (190)–(192) proves that 

|λh |−2γ < ∞.

h∈H

(193)

Moreover, note that for all (k, l) ∈ Z2 \{(0, 0)} it holds that

∂1 ek,l,0 U |λek,l,0 |−γ = [{(k 2 + l 2 )− /2 (l 1

∂ ∂ x φk,l (x,

y), k

−γ ∂ ∂ x φ−k,−l (x, y))}(x,y)∈D ]μD ,B(R2 ) U |λek,l,0 | y), 2πk 2 φk,−l (x, y))}(x,y)∈D ]μD ,B(R2 ) U |λek,l,0 |−γ

= [{(k 2 + l 2 )− /2 (−2πklφ−k,l (x, 1 = (k 2 + l 2 )− /2 2πk k 2 + l 2 |λek,l,0 |−γ = 2πk|4π 2 (k 2 + l 2 ) + η|−γ 1

≤

2π k |4π 2 (k 2 +l 2 )+η|1/2

≤

2π k 2π(k 2 +l 2 )1/2

≤1

(194)

and

∂2 ek,l,0 U |λek,l,0 |−γ = [{(k 2 + l 2 )− /2 (l

−γ ∂ ∂ ∂ y φk,l (x, y), k ∂ y φ−k,−l (x, y))}(x,y)∈D ]μD ,B(R2 ) U |λek,l,0 | 1

[{(k 2 + l 2 )− /2 (−2πl 2 φk,−l (x, y), 2πklφ−k,l (x, y))}(x,y)∈D ]μD ,B(R2 ) U |λek,l,0 |−γ 1

=

1 = (k 2 + l 2 )− /2 2πl k 2 + l 2 |λek,l,0 |−γ = 2πl|4π 2 (k 2 + l 2 ) + η|−γ ≤

2πl |4π 2 (k 2 +l 2 )+η|1/2

≤

2πl 2π(k 2 +l 2 )1/2

≤ 1.

(195)

Furthermore, observe that for all (k, l) ∈ Z2 \{(0, 0)} it holds that

ek,l,0 L ∞ (μD ;R2 ) = =

√ 1

(lφk,l , kφ−k,−l ) L ∞ (μD ;R2 ) k 2 +l 2 1/2 √ 1

l 2 |φk,l (·)|2 + k 2 |φ−k,−l (·)|2 L ∞ (μD ;R) k 2 +l 2

=

√ 1 sup k 2 +l 2 x ,x ∈(0,1) 1 2

≤

√ 1 k 2 +l 2

% l 2 |ϕk (x1 )|2 |ϕl (x2 )|2 + k 2 |ϕ−k (x1 )|2 |ϕ−l (x2 )|2

l24 + k24 =

√ 2√ l 2 +k 2 k 2 +l 2

= 2.

(196)

123

Stoch PDE: Anal Comp

Hence, we obtain that suph∈H h L ∞ (μD ;R2 )  = max e0,0,0 L ∞ (μD ;R2 ) , e0,0,1 L ∞ (μD ;R2 ) ,  sup(k,l)∈Z2 \{(0,0)} ek,l,0 L ∞ (μD ;R2 )

(197)

≤ max{1, 1, 2} = 2. Combining (190), (193), (194), (195), (197), and Lemma 4.8 (with ρ = γ , u = v for v ∈ Hγ in the notation of Lemma 4.8) proves that for all v ∈ Hγ it holds that Hγ ⊆ [W 1,2 (D, R2 ) ∩ L ∞ (μD ; R2 )] and 1/2



F(v) H ≤ η v H + 6 η−2γ + (k,l)∈Z2 (η + 4π 2 (k 2 + l 2 ))−2γ v 2Hγ . (198) Next note that 2 H ⊆ C∞ P (D; R )

W 1,2 (D ,R2 )

= W P1,2 (D, R).

(199)

This, (193), (194), (195), (197), and Lemma 4.9 (with ρ = γ , u = u for u = (u 1 , u 2 ) ∈ Hγ in the notation of Lemma 4.9) ensure that for all u = (u 1 , u 2 ) ∈ Hγ it holds that Hγ ⊆ [W P1,2 (D, R2 ) ∩ L ∞ (μD ; R2 )] and 2u, F(u) H = 2η u 2H + (u 1 )2 + (u 2 )2 , ∂1 u 1 + ∂2 u 2  L 2 (μD ;R) .

(200)

In addition, note that for all (k, l) ∈ Z2 , x, y ∈ (0, 1) it holds that     ∂ l φk,l (x, y) + k ∂ φ−k,−l (x, y) =  − 2π klφ−k,l (x, y) + 2π klφ−k,l (x, y) = 0. ∂x ∂y (201) This assures that for all h = (h 1 , h 2 ) ∈ H it holds that ∂1 h 1 + ∂2 h 2 = [{0}x∈D ]μD ,B(R) .

(202)

Moreover, note that (194), (195), and item (ii) of Lemma 4.4 (with ρ = γ , u = u, j = j for u ∈ Hγ , j ∈ {1, 2} in the notation of Lemma 4.4) prove that for all u ∈ Hγ , j ∈ {1, 2} it holds that lim supP0 (H) I →H ∂ j u −



h∈I h, u H ∂ j h L 2 (μD ;R2 )

= 0.

(203)

This implies that for all u = (u 1 , u 2 ) ∈ Hγ , j ∈ {1, 2} it holds that lim supP0 (H) I →H ∂ j u j −



h=(h 1 ,h 2 )∈I h, u H ∂ j h j L 2 (μD ;R)

= 0.

(204)

Next note that (202) ensures that for all u = (u 1 , u 2 ) ∈ W 1,2 (D, R2 ), I ∈ P0 (H) it holds that

123

Stoch PDE: Anal Comp

∂1 u 1 + ∂2 u 2 L 2 (μ D ;R)  = ∂1 u 1 + ∂2 u 2 − h=(h 1 ,h 2 )∈I h, u H (∂1 h 1 + ∂2 h 2 ) L 2 (μD ;R)  ≤ ∂1 u 1 − h=(h 1 ,h 2 )∈I h, u H ∂1 h 1 L 2 (μD ;R)  + ∂2 u 2 − h=(h 1 ,h 2 )∈I h, u H ∂2 h 2 L 2 (μD ;R) .

(205)

Combining (204) with the fact that Hγ ⊆ W 1,2 (D, R2 ) hence shows that for all u = (u 1 , u 2 ) ∈ Hγ it holds that

∂1 u 1 + ∂2 u 2 L 2 (μ D ;R)

 = lim supP0 (H) I →H ∂1 u 1 + ∂2 u 2 − h=(h 1 ,h 2 )∈I h, u H (∂1 h 1 + ∂2 h 2 ) L 2 (μD ;R)  ≤ lim supP0 (H) I →H ∂1 u 1 − h=(h 1 ,h 2 )∈I h, u H ∂1 h 1 L 2 (μD ;R)  + lim supP0 (H) I →H ∂2 u 2 − h=(h 1 ,h 2 )∈I h, u H ∂2 h 2 L 2 (μD ;R) = 0. (206)

This assures that for all u = (u 1 , u 2 ) ∈ Hγ it holds that ∂1 u 1 + ∂2 u 2 = [{0}x∈D ]μD ,B(R) .

(207)

Equation (200) therefore proves that for all I ∈ P0 (H), x ∈ PI (H ) it holds that x, PI F(x) H = PI x, F(x) H = x, F(x) H = η x 2H .

(208)

In the next step we observe that (193), (194), (195), (197), and Corollary 4.10 assure that F ∈ C(Hγ , H ) and B ∈ C(Hγ , HS(U, H )). This proves that F ∈ M(B(Hγ ), B(H ))

and

B ∈ M(B(Hγ ), B(HS(U, H ))).

(209)

Moreover, note that (198) and Lemma 4.2 imply that for all h ∈ (0, T ], I ∈ P0 (H), J ∈ P0 (U), x ∈ DhI it holds that     max PI F(x) H , PI B(x) Pˆ J HS(U,H ) ≤ max F(x) H , B(x) HS(U,H ) √  1/2

  ≤ max η x H + 6 η−2γ + (k,l)∈Z2 (η + 4π 2 (k 2 + l 2 ))−2γ x 2Hγ , ϑ ≤ r (x) ≤ ch −δ . Furthermore, we observe that the fact that ∀ v ∈ Hγ : that for all I ∈ P0 (H), h ∈ (0, T ] it holds that

(210) √ θ + ε v 2H ≤ r (v) implies

√ DhI = {x ∈ PI (H ) : r (x) ≤ ch −δ } ⊆ {x ∈ PI (H ) : ϑ + ε x 2H ≤ ch −δ } √ ⊆ {v ∈ H : ϑ + ε v 2H ≤ ch −δ }. (211) In addition, we note that Lemma 4.2 ensures that supx∈Hγ B(x) 2HS(U,H ) ≤ ϑ < ∞. Combining (208)–(211) and Corollary 3.4 (with H = H , U = U , H = H, U = U, T = T , γ = γ , δ = δ, λ = λ, A = A, ξ = ξ , F = F, B = B, DhI = DhI , ϑ = ϑ,

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b1 = 0, b2 = η, ε = ε, ς = δ, c = c, Y θ,I,J = Y θ,I,J for h ∈ (0, T ], θ ∈ T , I ∈ P0 (H), J ∈ P0 (U) in the notation of Corollary 3.4) hence completes the proof of Corollary 4.15.   Remark 4.16 Consider the setting of Corollary 4.15. Then the stochastic processes Y θ,I,J : [0, T ] ×  → PI (H ), θ ∈ T , I ∈ P0 (H), J ∈ P0 (U), are space-timenoise discrete numerical approximation processes for the two-dimensional stochastic Navier–Stokes equations

2 d X t (x) = ( ∂∂x 2 +

∂2 )X t (x) + (R(( ∂∂x ∂ x22

+ b(x, X t (x)) d( QW )t (x) 1

 X t ) · X t ))(x) dt (212)

with periodic boundary conditions, (div X t )(x) = 0, and X 0 (x) = ξ(x) for t ∈ [0, T ], x = (x1 , x2 ) ∈ (0, 1)2 (cf., e.g., Da Prato and Debussche [15, Section 2], Carelli and Prohl [12], Carelli et al. [11], Brze´zniak et al. [10], and Bessaih et al. [4]). Acknowledgements We gratefully acknowledge Zdzisław Brze´zniak for several useful comments that helped to improve the presentation of the results. This project has been supported through the SNSFResearch Project 200021_156603 “Numerical approximations of nonlinear stochastic ordinary and partial differential equations”.

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