Malliavin regularity and weak approximation of semilinear SPDE with Lévy noise
aa r X i v : . [ m a t h . P R ] A ug MALLIAVIN REGULARITY AND WEAK APPROXIMATION OFSEMILINEAR SPDE WITH L´EVY NOISE
ADAM ANDERSSON AND FELIX LINDNER
Abstract.
We investigate the weak order of convergence for space-time dis-crete approximations of semilinear parabolic stochastic evolution equationsdriven by additive square-integrable L´evy noise. To this end, the Malliavinregularity of the solution is analyzed and recent results on refined Malliavin-Sobolev spaces from the Gaussian setting are extended to a Poissonian setting.For a class of path-dependent test functions, we obtain that the weak rate ofconvergence is twice the strong rate. Introduction
Stochastic partial differential equations (SPDE) with L´evy noise occur in variousapplications, ranging from environmental pollution models [19] to the statisticaltheory of turbulence [6], to mention only two examples. In the context of thenumerical approximation of the solution processes of such equations, the quantityof interest is typically the expected value of some functional of the solution and oneis thus interested in the weak convergence rate of the considered numerical scheme.While the weak convergence analysis for numerical approximations of SPDE withGaussian noise is meanwhile relatively far developed, see, e.g., [1, 2, 3, 7, 8, 9, 10,11, 12, 14, 15, 16, 17, 18, 23, 30], available results for non-Gaussian L´evy noise havebeen restricted to linear equations so far [4, 5, 21, 25]. In this article, we analyzefor the first time the weak convergence rate of numerical approximations for a classof semi-linear SPDE with non-Gaussian L´evy noise.We consider equations of the typed X ( t ) + AX ( t ) d t = F ( X ( t )) d t + d L ( t ) , t ∈ [0 , T ] , X (0) = X , (1)where X takes values in a separable real Hilbert space H and A : D ( A ) ⊂ H → H is an unbounded linear operator such that − A generates an analytic semigroup( S ( t )) t > ⊂ L ( H ). By ˙ H ρ , ρ ∈ R , we denote the smoothness spaces associatedto A via ˙ H ρ = D ( A ρ ), see Subsection 2.1 for details. The driving L´evy process L = ( L ( t )) t ∈ [0 ,T ] is assumed to be ˙ H β − -valued for some regularity parameter β ∈ (0 , F : H → ˙ H β − is supposed to satisfy suitable Lipschitz conditions. The preciseassumptions are stated in Subsection 2.3 and 3.1. We remark that for a strongconvergence analysis one could allow F to be only ˙ H β − -valued, but to obtaina weak convergence rate which is twice the strong rate we need to assume morethan that. Our main example for the abstract equation (1) is the semilinear heat Mathematics Subject Classification.
Key words and phrases.
Malliavin calculus, Poisson random measure, L´evy process, stochasticpartial differential equation, numerical approximation, weak convergence. equation(2) ˙ u ( t, ξ ) − ∆ ξ u ( t, ξ ) = f ( u ( t, ξ )) + ˙ η ( t, ξ ) , ( t, ξ ) ∈ [0 , T ] × O ,u ( t, ξ ) = 0 , ( t, ξ ) ∈ [0 , T ] × ∂ O ,u (0 , ξ ) = u ( ξ ) , ξ ∈ O . Here
O ⊂ R d is an open, bounded, convex, polygonal/polyhedral domain, d ∈{ , , } , f : R → R is twice continuously differentiable with bounded derivatives,and ˙ η is an impulsive space-time noise, cf. Example 3.2. The discretization in spaceis performed by a standard finite element method and in time by an implicit Eulermethod, cf. Subsection 4.1.Several approaches to analyzing the weak error of numerical approximations ofSPDE can be found in the literature. We follow the the approach from [1, 2, 4, 5, 22],which is based on duality principles in Malliavin calculus. We remark that Malliavincalculus for Poisson or L´evy noise is fundamentally different from that for Gaussiannoise. Our analysis heavily relies on the results on Hilbert space-valued PoissonMalliavin calculus from [4]. Following the ideas in [24, 28], the Malliavin derivativein [4] is in fact a finite difference operator D : L (Ω; H ) → L (Ω × [0 , T ] × U ; H ) , (3)where (Ω , F , P ) is the underlying probability space and U = ˙ H β − is the statespace of the L´evy process L , endowed with the Borel- σ -algebra B ( U ) and the L´evymeasure ν of L . Starting with the operator (3), one can in a second step defineMalliavin-Sobolev-type spaces as classes of H -valued random variables satisfyingcertain integrability properties together with their Malliavin derivatives, cf. Sub-sections 2.3 and 3.2.In this article, we extend the strategy for semilinear SPDE from [1, 2] to Poissonnoise and analyze the weak approximation error in a framework of Gelfand triplesof refined Malliavin-Sobolev spaces M ,p,q ( H ) ⊂ L (Ω; H ) ⊂ ( M ,p,q ( H )) ∗ , seeSubsection 3.2 for the definition of these spaces. We first investigate in Section 3the Malliavin regularity of the mild solution X = ( X ( t )) t > to Eq. (1). We startby proving in Proposition 3.3 that the Malliavin derivative DX ( t ) of X ( t ) satisfiesfor all t ∈ [0 , T ] the equality(4) D s,x X ( t ) = s t · Z ts S ( t − r ) (cid:2) F (cid:0) X ( r ) + D s,x X ( r ) (cid:1) − F (cid:0) X ( r ) (cid:1)(cid:3) d r + s t · S ( t − s ) x P ⊗ d s ⊗ ν (d x )-almost everywhere on Ω × [0 , T ] × U . The terms on the righthand side are understood to be zero for s > t . Based on this equality we derivein Proposition 3.5 and 3.7 suitable integrability and time regularity properties of DX ( t ) by using Gronwall-type arguments. The regularity results from Section 3are then used in Section 4 for the analysis of the weak error E [ f ( ˜ X h,k ) − f ( X )],where X h,k = ( ˜ X h,k ( t )) t ∈ [0 ,T ] , h, k ∈ (0 , X . We use a standard finite element method with maximal meshsize h for the discretization in space and an implicit Euler method with step size k for the discretization in time. For finite Borel measures µ , . . . , µ n on [0 , T ],we consider path-dependent functionals f : L ([0 , T ] , P ni =1 µ i ; H ) → R of the form f ( x ) = ϕ (cid:0) R [0 ,T ] x ( t ) µ (d t ) , . . . , R [0 ,T ] x ( t ) µ n (d t ) (cid:1) , where ϕ : L ni =1 H → R is as-sumed to be Fr´echet differentiable with globally Lipschitz continuous derivative ALLIAVIN REGULARITY AND WEAK APPROXIMATION 3 mapping ϕ ′ : L ni =1 H → L (cid:0) L ni =1 H, R (cid:1) . Our main result, Theorem 4.5, states thatfor all γ ∈ [0 , β ) there exists a finite constant C such that | E [ f ( ˜ X h,k ) − f ( X )] | C ( h γ + k γ ) , h, k ∈ (0 , . (5)For the considered class of test functions, the weak rate of convergence is thus twicethe strong rate. The idea of the proof is to exploit the Malliavin regularity of X and˜ X h,k in order to estimate the weak error | E [ f ( ˜ X h,k ) − f ( X )] | in terms of the normof the error ˜ X h,k ( t ) − X ( t ) in the dual space ( M ,p,q ( H )) ∗ , for suitable exponents p, q ∈ [2 , ∞ ). As an exemplary application, we consider in Corollary 4.6 the ap-proximation of covariances Cov( h X ( t ) , ψ i , h X ( t ) , ψ i ), t , t ∈ [0 , T ], ψ , ψ ∈ H of the solution process.We remark that weak error estimates for SPDE involving path-dependent func-tionals have been derived so far only in [1, 4, 10]. Our setting allows for integral-typefunctionals as well as for functionals of the form f ( x ) = ϕ ( x ( t ) , . . . , x ( t n )), where x = ( x ( t )) t ∈ [0 ,T ] is an H -valued path, 0 t . . . t n T , and ϕ : L nj =1 H → R .The paper is organized as follows: In Section 2 we collect some general notation(Subsection 2.1), introduce the precise assumptions on the L´evy process L (Subsec-tion 2.2), and review fundamental concepts and results from Hilbert space-valuedPoisson Malliavin calculus (Subsection 2.3). Section 3 is concerned with the Malli-avin regularity of the mild solution X to Eq. (1). Here we first describe in detail ourassumptions on the considered equation (Subsection 3.1) before we analyse the reg-ularity of X (Subsection 3.2) and derive some auxiliary results concerning refinedMalliavin-Sobolev spaces (Subsection 3.3). The weak convergence analysis is foundin Section 4, where we present the numerical scheme and our main result (Sub-section 4.1), analyze the regularity of the approximation process (Subsection 4.2)as well as convergence in negative order Malliavin-Sobolev spaces (Subsection 4.3),and finally prove the main result by combining the results previously collected(Subsection 4.4.) 2. Preliminaries
General notation.
If ( U, k · k U , h· , ·i U ) and ( V, k · k V , h· , ·i V ) are separablereal Hilbert spaces, we denote by L ( U, V ) and L ( U, V ) ⊂ L ( U, V ) the spaces ofbounded linear operators and Hilbert-Schmidt operators from U to V , respectively.By C ( U, V ) we denote the space of Fr´echet differentiable functions f : U → V withcontinuous derivative f ′ : U → L ( U, V ). In the special case V = R we identify L ( U, R ) with U via the Riesz isomorphism and consider f ′ as a U -valued mapping.The Lipschitz spacesLip ( U, V ) := { f ∈ C ( U, V ) : | f | Lip ( U,V ) < ∞} , Lip ( U, V ) := { f ∈ C ( U, V ) : | f | Lip ( U,V ) + | f | Lip ( U,V ) < ∞} , are defined in terms of the semi-norms | f | Lip ( U,V ) := sup (cid:16)n k f ( x ) − f ( y ) k V k x − y k U : x, y ∈ U, x = y o ∪ { } (cid:17) , | f | Lip ( U,V ) := sup (cid:16)n k f ′ ( x ) − f ′ ( y ) k L ( U,V ) k x − y k U : x, y ∈ U, x = y o ∪ { } (cid:17) , compare, e.g., [11, Sec. 1.2]. We also use the norm k f k Lip ( U,V ) := k f (0) k V + | f | Lip ( U,V ) . If ( S, S , m ) is a σ -finite measure space and ( X, k · k X ) is a Ba-nach space, we denote by L ( S ; X ) := L ( S, S , m ; X ) the space of (equivalence A. ANDERSSON AND F. LINDNER classes of) strongly S -measurable functions f : S → X . As usual, we identifyfunctions which coincide m -almost everywhere. The space L ( S ; X ) is endowedwith the topology of local convergence in measure. For p ∈ [1 , ∞ ], we denote by L p ( S ; X ) := L p ( S, S , m ; X ) the subspace of L ( S ; X ) consisting of all (equivalenceclasses of) strongly S -measurable mappings f : S → X such that k f k L p ( S ; X ) := (cid:0) R S k f ( s ) k pX m (d s ) (cid:1) /p < ∞ if p ∈ [1 , ∞ ) and k f k L ∞ ( S ; X ) := ess sup s ∈ S k f ( s ) k X < ∞ if p = ∞ . By λ we denote one-dimensional Lebesgue measure and we sometimesalso write λ (d t ), d t , λ (d s ), d s etc. in place of λ to improve readability.2.2. L´evy processes and Poisson random measures.
Here we describe in de-tail the setting concerning the driving process L in Eq. (1). Our standard referencefor Hilbert space-valued L´evy processes is [27]. Assumption 2.1.
The following setting is considered throughout the article. • (Ω , F , P ) is a complete probability space. The σ -algebra F coincides with the P -completion of the σ -algebra σ ( L ( t ) : t ∈ [0 , T ]) generated by the L´evy process L introduced below. • L = ( L ( t )) t ∈ [0 ,T ] is a L´evy process defined on (Ω , F , P ) , taking values in aseparable real Hilbert space ( U, k · k U , h· , ·i U ) . Here T ∈ (0 , ∞ ) is fixed. Weassume that L is square-integrable with mean zero, i.e., L ( t ) ∈ L (Ω; U ) and E ( L ( t )) = 0 , and that the Gaussian part of L is zero. • ( H, k · k , h· , ·i ) is a further separable real Hilbert space. The jump intensity measure (L´evy measure) ν : B ( U ) → [0 , ∞ ] of a general U -valued L´evy process L satisfies ν ( { } ) = 0 and R U min( k x k U , ν (d x ) < ∞ , cf. [27,Section 4]. Due to our square integrability assumption on L we additionally have | ν | := (cid:16) Z U k y k U ν (d y ) (cid:17) < ∞ , (6)see, e.g., [27, Theorem 4,47]. As a further consequence of our assumptions on L ,the characteristic function of L ( t ) is of given by E e i h x,L ( t ) i U = exp (cid:16) − t Z U (cid:0) − e i h x,y i U + i h x, y i U (cid:1) ν (d y ) (cid:17) , x ∈ U, (7)cf. [27, Theorem 4.27]. Conversely, every U -valued L´evy process L satisfying (6)and (7) is square-integrable with mean zero and vanishing Gaussian part.We always consider a fixed c`adl`ag (right continuous with left limits) modificationof L . The jumps of L determine a Poisson random measure on B ([0 , T ] × U ) asfollows: For ( ω, t ) ∈ Ω × (0 , T ] we denote by ∆ L ( t )( ω ) := L ( t )( ω ) − lim s ր t L ( s )( ω ) ∈ U the jump of a trajectory of L at time t . Then(8) N ( ω ) := X t ∈ (0 ,T ]:∆ L ( t )( ω ) =0 δ ( t, ∆ L ( t )( ω )) , ω ∈ Ω , defines a Poisson random measure N on B ([0 , T ] × U ) with intensity measure λ ⊗ ν ,where δ ( t,x ) denotes Dirac measure at ( t, x ) ∈ [0 , T ] × U and ν is the L´evy measureof L . This follows, e.g., from Theorem 6.5 in [27] together with Theorems 4.9, 4.15,4.23 and Lemma 4.25 therein. It the context of Poisson Malliavin calculus it is usefulto consider N as a random variable with values in the space N = N ([0 , T ] × U ) ofall σ -finite N ∪ { + ∞} -valued measures on B ([0 , T ] × U ). It is endowed with the σ -algebra N = N ([0 , T ] × U ) generated by the mappings N ∋ µ µ ( B ) ∈ N ∪{ + ∞} , B ∈ B ([0 , T ] × U ). ALLIAVIN REGULARITY AND WEAK APPROXIMATION 5
We now list some important notation used in the present context.
Notation 2.2.
The following notation is used throughout the article. • ν : B ( U ) → [0 , ∞ ] and ( U , k · k U , h· , ·i U ) are the L´evy measure and the re-producing kernel Hilbert space of L , respectively; cf. [27, Definition 4.28 and7.2] . • N : Ω → N is the Poisson random measure (Poisson point process) on [0 , T ] × U determined by the jumps of L as specified in Eq. (8) above. The compensatedPoisson random measure is denoted by ˜ N := N − λ ⊗ ν , i.e., ˜ N ( B ) = N ( B ) − ( λ ⊗ ν )( B ) for all B ∈ B ([0 , T ] × U ) with ( λ ⊗ ν )( B ) < ∞• ( F t ) t ∈ [0 ,T ] is the filtration given by F t := T u ∈ ( t,T ] ˜ F u , where ˜ F u is the P -completion of σ ( L ( s ) : s ∈ [0 , u ]) . • For p ∈ { }∪ [1 , ∞ ] set L p (Ω; H ) := L p (Ω , F , P ; H ) and L p (Ω × [0 , T ] × U ; H ) := L p (Ω × [0 , T ] × U, F ⊗B ([0 , T ] × U ) , P ⊗ λ ⊗ ν ; H ) . Moreover, P T ⊂ F ⊗B ([0 , T ]) denotes the σ -algebra of predictable sets w.r.t. to ( F t ) t ∈ [0 ,T ] and we further set L (Ω × [0 , T ] × U ; H ) := L (cid:0) Ω × [0 , T ] × U, P T ⊗ B ( U ) , P ⊗ λ ⊗ ν ; H (cid:1) . We end this section by recalling some basics on stochastic integration w.r.t. L and ˜ N , cf. [27]. The H -valued L stochastic integral R T Φ( s ) d L ( s ) w.r.t. L is de-fined for all Φ ∈ L (Ω × [0 , T ]; L ( U , H )) := L (Ω × [0 , T ] , P T , P ⊗ λ ; L ( U , H )),and we have the Itˆo isometry E (cid:13)(cid:13) R T Φ( s ) d L ( s ) (cid:13)(cid:13) = R T E k Φ( s ) k L ( U ,H ) d s . The H -valued L stochastic integral R T R U Φ( s, x ) ˜ N (d s, d x ) w.r.t. ˜ N is defined for allΦ ∈ L (Ω × [0 , T ] × U ; H ), and here it holds that E (cid:13)(cid:13) R T R U Φ( s, x ) ˜ N (d s, d x ) (cid:13)(cid:13) = R T R U E k Φ( s, x ) k ν (d x )d s . As usual, we set R t Φ( s ) d L ( s ) := R T (0 ,t ] ( s )Φ( s ) d L ( s )and R t R U Φ( s, x ) ˜ N (d s, d x ) := R T R U (0 ,t ] ( s )Φ( s, x ) ˜ N (d s, d x ), t ∈ [0 , T ]. A use-ful property shown in [21, Lemma 3.1] is the following: There exists an isomet-ric embedding κ : L (Ω × [0 , T ]; L ( U , H )) → L (Ω × [0 , T ] × U ; H ) such that R T Φ( s ) d L ( s ) = R T R U Φ( s ) x ˜ N (d s, d x ) for Φ ∈ L (Ω × [0 , T ]; L ( U , H )), wherewe set Φ( s ) x := κ (Φ)( s, x ) to simplify notation.2.3. Poisson-Malliavin calculus in Hilbert space.
In this subsection we collectsome concepts and results from Hilbert space-valued Poisson Malliavin calculus. Werefer to [4] and the references therein for a more detailed exposition.While in the Gaussian case the Malliavin derivative is a differential operator, onepossible analogue in the Poisson case is a finite difference operator D : L (Ω; H ) → L (Ω × [0 , T ] × U ; H ) defined as follows. Recall that F is the P -completion of the σ -algebra generated by the L´evy process L , which coincides with the P -completion ofthe σ -algebra generated by the Poisson random measure N . This and the factoriza-tion theorem from measure theory imply that for every random variable F : Ω → H there exists a N - B ( H )-measurable function f : N → H , called a representative of F ,such that F = f ( N ) P -almost surely. In this situation we set ε + t,x F := f ( N + δ ( t,x ) ),where δ ( t,x ) denotes Dirac measure at ( t, x ) ∈ [0 , T ] × U . As a consequence ofMecke’s formula, this definition is P ⊗ d t ⊗ ν (d x )-almost everywhere independent ofthe choice of the representative f , so that F (cid:0) ε + t,x F (cid:1) is well-defined as a mappingfrom L (Ω; H ) to L (Ω × [0 , T ] × U ; H ), cf. [4, Lemma 2.5]. The difference operator D : L (Ω; H ) → L (Ω × [0 , T ] × U ; H ) , F DF = (cid:0) D t,x F (cid:1) is then defined by D t,x F := ε + t,x F − F, ( t, x ) ∈ [0 , T ] × U. (9) A. ANDERSSON AND F. LINDNER
The Malliavin-Sobolev space D , ( H ) consists of all F ∈ L (Ω; H ) satisfying DF ∈ L (Ω × [0 , T ] × U ; H ). In Subsection 3.2 we introduce refined Malliavin-Sobolevspaces M ,p,q ( H ), p, q ∈ (1 , ∞ ].The following basic lemmata are taken from [4, Lemma 3.2 and Corollary 4.2]. Lemma 2.3.
Let F ∈ L (Ω; H ) and h be a measurable mapping from H to anotherseparable real Hilbert space V . Then it holds that Dh ( F ) = h ( F + DF ) − h ( F ) . Lemma 2.4.
Let t ∈ [0 , T ] and F : Ω → H be F t - B ( H ) -measurable. Then theequality D s,x F = 0 holds P ⊗ d s ⊗ ν (d x ) -almost everywhere on Ω × ( t, T ] × U . The next result is a special case of the general duality formula in [4, Propo-sition 4.9]. It is crucial for our approach to weak error analysis for L´evy drivenSPDE.
Proposition 2.5 (Duality formula) . For all F ∈ D , ( H ) and Φ ∈ L (Ω × [0 , T ] × U ; H ) we have E D F, Z T Z U Φ( t, x ) ˜ N (d t, d x ) E = E Z T Z U (cid:10) D t,x F, Φ( t, x ) (cid:11) ν (d x ) d t. Before we proceed with two further important results, we need to discuss theapplication of D on stochastic processes. Remark 2.6 (Difference operator for stochastic processes) . One can define in aanalogous way as above for stochastic processes a further difference operator D mapping X ∈ L (Ω × [0 , T ]; H ) to DX = (cid:0) D s,x X ( t ) (cid:1) t ∈ [0 ,T ] , ( s,x ) ∈ [0 ,T ] × U ∈ L (cid:0) Ω × [0 , T ] × [0 , T ] × U ; H (cid:1) , see [4, Remark 3.10]. Then it holds for λ -almost all t ∈ [0 , T ]that D s,x X ( t ) = D s,x ( X ( t )) P ⊗ d s ⊗ ν (d x )-a.e. , (10)where D ( X ( t )) = (cid:0) D s,x ( X ( t )) (cid:1) ( s,x ) ∈ [0 ,T ] × U ∈ L (Ω × [0 , T ] × U ; H ) is for fixed t the Malliavin derivative of the random variable F = X ( t ) as introduced above.We will, however, typically encounter the situation where X = ( X ( t )) t ∈ [0 ,T ] isnot given as an equivalence class of stochastic processes but as a single stochas-tic process with X ( t ) being specifically defined for every t ∈ [0 , T ]. If X isnot only F ⊗ B ([0 , T ])-measurable but also stochastically continuous or piece-wise stochastically continuous, then there exists a P ⊗ d t ⊗ d s ⊗ ν (d x )-versionof DX = (cid:0) D s,x X ( t ) (cid:1) t ∈ [0 ,T ] , ( s,x ) ∈ [0 ,T ] × U such that (10) holds for every t ∈ [0 , T ],cf. [4, Lemma 4.3]. We also use a further analogously defined difference operator D mapping Φ ∈ L (Ω × [0 , T ] × U ; H ) to D Φ = (cid:0) D s,x Φ( t, y ) (cid:1) ( t,y ) , ( s,x ) ∈ [0 ,T ] × U ∈ L (Ω × ([0 , T ] × U ) ; H ) in such a way that for λ ⊗ ν -almost all ( t, y ) ∈ [0 , T ] × U we have D s,x Φ( t, y ) = D s,x (Φ( t, y )) P ⊗ d s ⊗ ν (d x )-a.e., cf. [4, Remark 3.10].In the regularity analysis of SPDEs it is important to know how D acts onLebesgue integrals and stochastic integrals. For this purpose we recall the followingresults. The first one is taken from [4, Proposition 4.5], the second is a special caseof [4, Proposition 4.13] combined with [4, Lemma 4.11]. Proposition 2.7 (Malliavin derivative of time integrals) . Let X : Ω × [0 , T ] → H bea stochastic process which is F ⊗ B ([0 , T ]) -measurable and piecewise stochasticallycontinuous, let µ be a σ -finite Borel-measure on [0 , T ] , and assume that X belongsto L ([0 , T ] , µ ; L p (Ω; H )) for some p > . Consider a fixed version of DX = ALLIAVIN REGULARITY AND WEAK APPROXIMATION 7 ( D s,x X ( t )) t ∈ [0 ,T ] , ( s,x ) ∈ [0 ,T ] × U such that for all t ∈ [0 , T ] the identity D s,x X ( t ) = D s,x ( X ( t )) holds P ⊗ d s ⊗ ν (d x ) -almost everywhere, cf. Remark 2.6. Then, for all B ∈ B ( U ) with ν ( B ) < ∞ we have E h Z [0 ,T ] Z B Z [0 ,T ] k D s,x X ( t ) k µ (d t ) ν (d x ) d s i < ∞ , so that the integral R [0 ,T ] D s,x X ( t ) µ (d t ) is defined P ⊗ d s ⊗ ν (d x ) -almost everywhereon Ω × [0 , T ] × U as an H -valued Bochner integral. Moreover, the equality D s,x Z [0 ,T ] X ( t ) µ (d t ) = Z [0 ,T ] D s,x X ( t ) µ (d t ) holds P ⊗ d s ⊗ ν (d x ) -almost everywhere on Ω × [0 , T ] × U . Proposition 2.8 (Malliavin derivative of stochastic integrals) . Let Φ ∈ L (Ω × [0 , T ] × U ; H ) . Then the derivative D Φ ∈ L (cid:0) Ω × ([0 , T ] × U ) ; H (cid:1) has a P T ⊗B ( U ) ⊗ B ([0 , T ] × U ) -measurable version, i.e., the mapping D Φ : Ω × ([0 , T ] × U ) → H, ( ω, t, y, s, x ) D s,x Φ( ω, t, y ) has a P ⊗ ( λ ⊗ ν ) ⊗ -version which is P T ⊗ B ( U ) ⊗ B ([0 , T ] × U ) -measurable. Ifmoreover E R T R U k D s,x Φ( t, y ) k ν (d y ) d t < ∞ for λ ⊗ ν -almost all ( s, x ) ∈ [0 , T ] × U , then the equality D s,x Z T Z U Φ( t, y ) ˜ N (d t, d y ) = Z T Z U D s,x Φ( t, y ) ˜ N (d t, d y ) + Φ( s, x )(11) holds P ⊗ d s ⊗ ν (d x ) -almost everywhere on Ω × [0 , T ] × U . Malliavin regularity for a class of semilinear SPDE
Assumptions on the considered equation.
We next state the precise as-sumptions on the operator A , the driving noise L , the nonlinearity F , and the initialvalue X in Eq. (1). Assumption 3.1.
In addition to Assumption 2.1, suppose that the following holds:(i) The operator A : D ( A ) ⊂ H → H is densely defined, linear, self-adjoint,positive definite and has a compact inverse. In particular, − A is the generatorof an analytic semigroup of contractions, which we denote by ( S ( t )) t > ⊂L ( H ) . The spaces ˙ H ρ , ρ ∈ R , are defined for ρ > as ˙ H ρ := D ( A ρ ) withnorm k ·k ˙ H ρ := k A ρ ·k and for ρ < as the closure of H w.r.t. the analogouslydefined k · k ˙ H ρ -norm.(ii) For some β ∈ (0 , , the state space U of the L´evy process L = ( L ( t )) t ∈ [0 ,T ] inAssumption 2.1 is given by U = ˙ H β − .(iii) For some δ ∈ [1 − β, , the drift function F : H → ˙ H β − belongs to the class Lip ( H, ˙ H β − ) ∩ Lip ( H, ˙ H − δ ) .(iv) The initial value X is an element of the space ˙ H β . It is well known that, under Assumption 3.1(i), there exist constants C ρ ∈ [0 , ∞ )(independent of t ) such that (cid:13)(cid:13) A ρ S ( t ) (cid:13)(cid:13) L ( H ) C ρ t − ρ , t > , ρ > , (12) (cid:13)(cid:13) A − ρ ( S ( t ) − id H ) (cid:13)(cid:13) L ( H ) C ρ t ρ , t > , ρ ∈ (0 , , (13)see, e.g., [26, Section 2.6]. Concerning Assumption 3.1(iii), let us remark thatLipschitz continuity of the derivative F ′ of F is needed for the weak convergence A. ANDERSSON AND F. LINDNER analysis in Section 4. Assuming F ∈ Lip ( H, H ) is sufficient for the analysis,compare, e.g., [2]. In applications to SPDE this assumption is not satisfactory asthe most important type of nonlinear drift, the Nemytskii type drift, typically doesnot satisfy the assumption. By assuming that F ′ is Lipschitz continuous only as amapping into the larger space ˙ H − δ , for suitable δ , the Sobolev embedding theoremcan be used to prove that Nemytskii type nonlinearities are in fact included in d ∈ { , , } space dimensions. More precisely this holds for δ > d , compare [29,Example 3.2]. Example 3.2.
For d ∈ { , , } let O ⊂ R d be an open, bounded, convex, poly-gonal/polyhedral domain and set H := L ( O ). Our standard example for A isa second order elliptic partial differential operator with zero Dirichlet boundarycondition of the form Au := −∇ · ( a ∇ u ) + cu, u ∈ D ( A ) := H ( O ) ∩ H ( O ) , with bounded and sufficiently smooth coefficients a, c : O → R such that a ( ξ ) > θ > c ( ξ ) > ξ ∈ O . Here H ( O ) and H ( O ) are the classical L -Sobolev spaces of order one with zero Dirichlet boundary condition and of order two,respectively. As an example for the drift function F we consider the Nemytskii typenonlinearity given by ( F ( x ))( ξ ) = f ( x ( ξ )), x ∈ L ( O ), ξ ∈ O , where f : R → R istwice continuously differentiable with bounded first and second derivative. In thissituation, Assumption 3.1(iii) is fullfilled for δ > d , compare [29, Example 3.2].Concrete examples for the L´evy process L can be found in [21, Subsection 2.1].By a mild solution to Eq. (1) we mean an ( F t ) t ∈ [0 ,T ] -predictable stochastic pro-cess X : Ω × [0 , T ] → H such thatsup t ∈ [0 ,T ] k X ( t ) k L (Ω; H ) < ∞ , (14)and such that for all t ∈ [0 , T ] it holds P -almost surely that X ( t ) = S ( t ) X + Z t S ( t − s ) F ( X ( s )) d s + Z t S ( t − s ) d L ( s ) . (15)Under Assumption 3.1 there exists a unique (up to modification) mild solution X to Eq. (1). This follows, e.g., from a straightforward modification of the proof of[27, Theorem 9.29], where slightly different assumptions are used. Moreover, thissolution is mean-square continuous, i.e., X ∈ C ([0 , T ] , L (Ω; H )), which can be seenby using standard arguments analogous to those used in the Gaussian case.3.2. Regularity results for the solution process.
We are now ready to analyzethe Malliavin regularity of the mild solution to Eq. (1).
Proposition 3.3.
Let Assumption 3.1 hold, let X = ( X ( t )) t ∈ [0 ,T ] be the mild solu-tion to Eq. (1) , and consider a fixed version of DX = ( D s,x X ( t )) t ∈ [0 ,T ] , ( s,x ) ∈ [0 ,T ] × U such that for all t ∈ [0 , T ] the identity D s,x X ( t ) = D s,x ( X ( t )) holds P ⊗ d s ⊗ ν (d x ) -almost everywhere, cf. Remark 2.6. Then for all t ∈ [0 , T ] and all B ∈ B ( U ) with ν ( B ) < ∞ we have E Z T Z B Z t (cid:13)(cid:13)(cid:13) S ( t − r ) h F (cid:0) X ( r ) + D s,x X ( r ) (cid:1) − F (cid:0) X ( r ) (cid:1)i(cid:13)(cid:13)(cid:13) d r ν (d x ) d s < ∞ , (16) so that for all t ∈ [0 , T ] the integral R t S ( t − r ) (cid:2) F (cid:0) X ( r )+ D s,x X ( r ) (cid:1) − F (cid:0) X ( r ) (cid:1)(cid:3) d r isdefined P ⊗ d s ⊗ ν (d x ) -almost everywhere on Ω × [0 , T ] × U as an H -valued Bochnerintegral. Moreover, for all t ∈ [0 , T ] the equality (4) holds P ⊗ d s ⊗ ν (d x ) -almosteverywhere on Ω × [0 , T ] × U . ALLIAVIN REGULARITY AND WEAK APPROXIMATION 9
Proof.
We fix t ∈ [0 , T ] and apply the difference operator D : L (Ω; H ) → L (Ω × [0 , T ] × U ; H ) to the single terms in (15). As the initial value X is deterministic, itis clear that D s,x ( S ( t ) X ) = 0 P ⊗ d s ⊗ ν (d x )-almost everywhere on Ω × [0 , T ] × U .Next, observe that by (12), the linear growth of F and (14) we have(17) Z t (cid:13)(cid:13) S ( t − r ) F ( X ( r )) (cid:13)(cid:13) L (Ω; H ) d r C − β k F k Lip( H, ˙ H β − ) Z t ( t − r ) β − (cid:0) k X ( r ) k L (Ω; H ) (cid:1) d r C − β k F k Lip( H, ˙ H β − ) β + 1 T β +12 (cid:0) t ∈ [0 ,T ] k X ( t ) k L (Ω; H ) (cid:1) < ∞ . Proposition 2.7 thus implies (16) and that the equality D s,x R t S ( t − r ) F ( X ( r )) d r = R t D s,x (cid:0) S ( t − r ) F ( X ( r )) (cid:1) d r holds P ⊗ d s ⊗ ν (d x )-a.e. on Ω × [0 , T ] × U . Herebywe consider a version of (cid:0) D s,x (cid:0) S ( t − r ) F ( X ( r )) (cid:1)(cid:1) r ∈ [0 ,t ) , ( s,x ) ∈ [0 ,T ] × U which is F ⊗B ([0 , t )) ⊗ B ([0 , T ] × U )-measurable, cf. Remark 2.6. Using also Lemma 2.3 andLemma 2.4, we obtain D s,x Z t S ( t − r ) F ( X ( r )) d r = s t · Z ts S ( t − r ) (cid:2) F (cid:0) X ( r ) + D s,x X ( r ) (cid:1) − F (cid:0) X ( r ) (cid:1)(cid:3) d r P ⊗ d s ⊗ ν (d x )-a.e. on Ω × [0 , T ] × U . Finally, the identity R t S ( t − r ) d L ( r ) = R t R U S ( t − r ) x ˜ N (d r, d x ) and the commutation relation in Proposition 2.8 yield D s,x Z t S ( t − r ) d L ( r ) = s t · S ( t − s ) x P ⊗ d s ⊗ ν (d x )-a.e. on Ω × [0 , T ] × U . Summing up, we have shown that (4) holdsfor every fixed t ∈ [0 , T ] as an equality in L (Ω × [0 , T ] × U ; H ). (cid:3) The refined Malliavin-Sobolev spaces introduced next and the subsequent regu-larity results have Gaussian counterparts in [1, 2].
Definition 3.4 (Refined Sobolev-Malliavin spaces) . Consider the setting describedin Subsections 2.2 and 2.3. For p, q ∈ (1 , ∞ ] we define M ,p,q ( H ) as the spaceconsisting of all F ∈ L p (Ω; H ) such that DF ∈ L p (Ω; L q ([0 , T ]; L ( U ; H ))) . It isequipped with the seminorm | F | M ,p,q ( H ) := k DF k L p (Ω; L q ([0 ,T ]; L ( U ; H ))) and norm k F k M ,p,q ( H ) := (cid:16) k F k pL p (Ω; H ) + | F | p M ,p,q ( H ) (cid:17) p . For p, p ′ , q, q ′ ∈ (1 , ∞ ) such that p + p ′ = q + q ′ = 1 , the space M − ,p ′ ,q ′ ( H ) isdefined as the (topological) dual space of M ,p,q ( H ) . Arguing as in [4, Proposition 3.7] one finds that M ,p,q ( H ) is a Banach spacefor all p, q ∈ (1 , ∞ ). If additionally p ∈ [2 , ∞ ), then M ,p,q ( H ) is continuouslyembedded in L (Ω; H ). This embedding is dense according to [4, Lemma 3.8]. Inthis situation we will use the Gelfand triple M ,p,q ( H ) ⊂ L (Ω; H ) ⊂ M − ,p ′ ,q ′ ( H ). Proposition 3.5 (Regularity I) . Let Assumption 3.1 hold. Depending on the valueof β ∈ (0 , , we assume either that q ∈ (1 , − β ) if β ∈ (0 , or q = ∞ if β = 1 . Then it holds that sup t ∈ [0 ,T ] | X ( t ) | M , ∞ ,q ( H ) < ∞ . (18) As a consequence, we also have sup t ∈ [0 ,T ] k X ( t ) k M , ,q ( H ) < ∞ .Proof. We consider a fixed version of DX = ( D s,x X ( t )) t ∈ [0 ,T ] , ( s,x ) ∈ [0 ,T ] × U such thatfor all t ∈ [0 , T ] the identity D s,x X ( t ) = D s,x ( X ( t )) holds P ⊗ d s ⊗ ν (d x )-almosteverywhere, cf. Remark 2.6. As a consequence of Proposition 3.3, the smoothingproperty (12), the fact that U = ˙ H β − and the Lipschitz continuity of F , we knowthat for all t ∈ [0 , T ] the estimate(19) k D s,x X ( t ) k s t · C − β | F | Lip ( H, ˙ H β − ) Z ts ( t − r ) β − k D s,x X ( r ) k d r + s t · C − β k x k U ( t − s ) β − holds P ⊗ d s ⊗ ν (d x )-almost everywhere on Ω × [0 , T ] × U . Moreover, Proposition 2.7and (14) imply that Z T k D s,x X ( t ) k d t < ∞ (20) P ⊗ d s ⊗ ν (d x )-almost everywhere on Ω × [0 , T ] × U .In order to be able to apply the generalized Gronwall Lemma A.1, we constructa new version of ( D s,x X ( t )) t ∈ [0 ,T ] , ( s,x ) ∈ [0 ,T ] × U such that the estimates (19) and(20) hold everywhere on Ω × [0 , T ] × [0 , T ] × U and Ω × [0 , T ] × U , respectively.For this purpose, let A ∈ F ⊗ B ([0 , T ]) ⊗ B ([0 , T ] × U ) be the set consisting of all( ω, t, s, x ) ∈ Ω × [0 , T ] × [0 , T ] × U for which (19) holds. Let B ∈ F ⊗ B ([0 , T ] × U )be the set consisting of all ( ω, s, x ) ∈ Ω × [0 , T ] × U for which (19) holds d t -almosteverywhere on [0 , T ]. Finally, let C ∈ F ⊗ B ([0 , T ] × U ) be the set consisting of all( ω, s, x ) ∈ Ω × [0 , T ] × U for which (20) holds. Let Γ : Ω × [0 , T ] × [0 , T ] × U → H bedefined by Γ := A ∩ π − ( B ∩ C ) DX , where π : Ω × [0 , T ] × [0 , T ] × U → Ω × [0 , T ] × U is the coordinate projection given by π ( ω, t, s, x ) := ( ω, s, x ). Note that for all t ∈ [0 , T ] the identity Γ( · , t, · , · ) = DX ( t ) holds P ⊗ λ ⊗ ν -almost everywhere onΩ × [0 , T ] × U . We choose Γ as our new version of DX and henceforth write DX =( D s,x X ( t )) t ∈ [0 ,T ] , ( s,x ) ∈ [0 ,T ] × U instead of Γ to simplify notation. Observe that forthis new version the estimates (19) and (20) hold indeed everywhere on Ω × [0 , T ] × [0 , T ] × U and Ω × [0 , T ] × U , respectively. The generalized Gronwall Lemma A.1thus implies that there exists a constant C = C (cid:0) C − β | F | Lip ( H, ˙ H β − ) , T, β (cid:1) ∈ [0 , ∞ )such that the estimate k D s,x X ( t ) k s t C C − β k x k U ( t − s ) β − (21)holds everywhere on Ω × [0 , T ] × [0 , T ] × U .Assume the case where β ∈ (0 , q ∈ (1 , − β ) and consider the version of DX = ( D s,x X ( t )) t ∈ [0 ,T ] , ( s,x ) ∈ [0 ,T ] × U constructed above. Integration of (21) yieldssup t ∈ [0 ,T ] h Z T (cid:16) Z U k D s,x X ( t ) k ν (d x ) (cid:17) q d s i q C C − β | ν | sup t ∈ [0 ,T ] h Z t ( t − s ) q · β − d s i q C C − β | ν | q · β − + 1) q T β − + q , which implies (18). The case where β = 1 and q = ∞ is treated similarly. Finally,the second assertion in Proposition 3.5 follows from (18) and (14). (cid:3) ALLIAVIN REGULARITY AND WEAK APPROXIMATION 11
Proposition 3.6 (Negative norm inequality) . Consider the setting described inSubsections 2.2 and 2.3. Let p ′ , q ′ ∈ (1 , . For predictable integrands Φ ∈ L (Ω × [0 , T ] × U ; H ) it holds that (cid:13)(cid:13)(cid:13) Z T Z U Φ( t, y ) ˜ N (d t, d y ) (cid:13)(cid:13)(cid:13) M − ,p ′ ,q ′ ( H ) k Φ k L p ′ (Ω; L q ′ ([0 ,T ]; L ( U ; H ))) . Proof.
Let p, q ∈ [2 , ∞ ) satisfy p + p ′ = q + q ′ = 1. By the duality formula fromProposition 2.5, duality in the Gelfand triple M ,p,q ( H ) ⊂ L (Ω; H ) ⊂ M − ,p ′ ,q ′ ( H ),and by the H¨older inequality it holds that (cid:13)(cid:13)(cid:13) Z T Z U Φ( t, y ) ˜ N (d t, d y ) (cid:13)(cid:13)(cid:13) M − ,p ′ ,q ′ ( H ) = sup Z ∈ M ,p,q ( H ) \{ } (cid:10) Z, R T R U Φ( t, y ) ˜ N (d t, d y ) (cid:11) L (Ω; H ) k Z k M ,p,q ( H ) = sup Z ∈ M ,p,q ( H ) \{ } h DZ, Φ i L (Ω × [0 ,T ] × U ; H ) k Z k M ,p,q ( H ) sup Z ∈ M ,p,q ( H ) \{ } k DZ k L p (Ω; L q ([0 ,T ]; L ( U ; H ))) k Φ k L p ′ (Ω; L q ′ ([0 ,T ]; L ( U ; H ))) k Z k M ,p,q ( H ) k Φ k L p ′ (Ω; L q ′ ([0 ,T ]; L ( U ; H ))) . (cid:3) Proposition 3.7 (Regularity II) . Let Assumption 3.1 hold and X = ( X ( t )) t ∈ [0 ,T ] be the mild solution to Eq. (1) . For all γ ∈ [0 , β ) and q ′ = γ there exist a constant C ∈ [0 , ∞ ) such that k X ( t ) − X ( t ) k M − , ,q ′ ( H ) C | t − t | γ , t , t ∈ [0 , T ] . Proof.
Let 0 t t T . Representing the increment X ( t ) − X ( t ) via (15),taking norms and using the continuous embedding L (Ω; H ) ⊂ M − , ,q ′ ( H ), weobtain k X ( t ) − X ( t ) k M − , ,q ′ ( H ) k ( S ( t − t ) − id H ) A − γ S ( t ) A γ X k + (cid:13)(cid:13)(cid:13) Z t ( S ( t − t ) − id H ) A − γ A γ S ( t − s ) A − β A β − F ( X ( s )) d s (cid:13)(cid:13)(cid:13) L (Ω; H ) + (cid:13)(cid:13)(cid:13) Z t t S ( t − s )) A − β A β − F ( X ( s )) d s (cid:13)(cid:13)(cid:13) L (Ω; H ) + (cid:13)(cid:13)(cid:13) Z t Z ˙ H β − ( S ( t − t ) − id H ) A − γ A γ S ( t − s ) x ˜ N (d s, d x ) (cid:13)(cid:13)(cid:13) M − , ,q ′ ( H ) + (cid:13)(cid:13)(cid:13) Z t t Z ˙ H β − S ( t − s ) A − β A β − x ˜ N (d s, d x ) (cid:13)(cid:13)(cid:13) M − , ,q ′ ( H ) . Further, from (12), (13), (14), the linear growth of F and the negative norm in-equality in Proposition 3.6 we obtain k X ( t ) − X ( t ) k M − , ,q ′ ( H ) C C γ k X k ˙ H γ ( t − t ) γ + C γ +1 − β Z t ( t − s ) − γ − − β d s C γ ( t − t ) γ · k F k Lip ( H, ˙ H β − ) (cid:0) t ∈ [0 ,T ] k X ( t ) k L (Ω; H ) (cid:1) + C − β Z t t ( t − s ) − − β d s k F k Lip ( H, ˙ H β − ) (cid:0) t ∈ [0 ,T ] k X ( t ) k L (Ω; H ) (cid:1) + h Z t (cid:16) Z ˙ H β − (cid:13)(cid:13) ( S ( t − t ) − id H ) A − γ A γ S ( t − s ) A − β A β − x (cid:13)(cid:13) ν (d x ) (cid:17) q ′ d s i q ′ + h Z t t (cid:16) Z ˙ H β − (cid:13)(cid:13) S ( t − s ) A − β A β − x (cid:13)(cid:13) ν (d x ) (cid:17) q ′ d s i q ′ . (22)Note that γ + − β < − γ + β ) − T − γ + β . The integral in the third termon the right hand side of (22) satisfies R t t ( t − s ) − − β d s = β ( t − t ) β . ( t − t ) γ . The fourth term on the right hand side of (22) can be estimated by C γ ( t − t ) γ C γ +(1 − β ) (cid:0) R t ( t − s ) − γ γ +1 − β d s (cid:1) γ | ν | . ( t − t ) γ . The latterintegral is bounded by R T s − γ γ +1 − β d s , which is finite since21 + γ γ + 1 − β γ − ( β − γ )1 + γ < . (23)Finally, the the last term (22) is bounded by C − β (cid:0) R t t ( t − s ) γ β − d s (cid:1) γ | ν | = (cid:0) β + γ γ (cid:1) − γ C − β | ν | ( t − t ) β + γ . ( t − t ) γ . This completes the proof. (cid:3)
Auxiliary results on refined Malliavin Sobolev spaces.
In the sequel,we consider the setting described in Subsection 2.2 and 2.3.
Lemma 3.8.
Let p, q ∈ (1 , ∞ ) , let V , V be separable real Hilbert spaces, and let ϕ : H → L ( V , V ) be a bounded function belonging to the class Lip ( H, L ( V , V )) .For all Y ∈ L (Ω; H ) satisfying DY ∈ L ∞ (Ω; L q ([0 , T ]; L ( U ; H ))) and all Z ∈ M ,p,q ( V ) it holds that ϕ ( Y ) Z ∈ M ,p,q ( V ) and k ϕ ( Y ) Z k M ,p,q ( V ) p (cid:16) x ∈ H k ϕ ( x ) k L ( V ,V ) + | ϕ | Lip ( H, L ( V ,V )) | Y | M , ∞ ,q ( H ) (cid:17) · k Z k M ,p,q ( V ) . Proof.
Take Y and Z as in the statement and observe that(24) k ϕ ( Y ) Z k L p (Ω; V ) sup x ∈ H k ϕ ( x ) k L ( V ,V ) k Z k L p (Ω; V ) . Next, due to the definition of the difference operator D in Subsection 2.3 and theidentities D t,y Y = ε + t,y Y − Y and ε + t,y Y = Y + D t,y Y it holds P ⊗ d t ⊗ ν (d y )–almost everywhere on Ω × [0 , T ] × U that D t,y ( ϕ ( Y ) Z ) = ϕ ( ε + t,y Y ) ε + t,y Z − ϕ ( Y ) Z = ALLIAVIN REGULARITY AND WEAK APPROXIMATION 13 ϕ ( Y + D t,y Y ) D t,y Z + ( ϕ ( Y + D t,y Y ) − ϕ ( Y )) Z. As a consequence, we obtain(25) (cid:13)(cid:13) D (cid:0) ϕ ( Y ) Z (cid:1)(cid:13)(cid:13) L p (Ω; L q ([0 ,T ]; L ( U ; V ))) (cid:16) sup x ∈ H k ϕ ( x ) k L ( V ,V ) + | ϕ | Lip ( H, L ( V ,V )) k DY k L ∞ (Ω; L q ([0 ,T ]; L ( U ; H ))) (cid:17) k Z k M ,p,q ( V ) . Combining (24) and (25) finishes the proof. (cid:3)
Proposition 3.9 (Local Lipschitz bound) . Let p ′ , q ′ ∈ (1 , , q ∈ [2 , ∞ ) be suchthat q + q ′ = 1 , let V be a separable real Hilbert space, and ψ ∈ Lip ( H, V ) .Then there exists C ∈ [0 , ∞ ) such that for all Y , Y ∈ L (Ω; H ) with DY , DY ∈ L ∞ (Ω; L q ([0 , T ]; L ( U ; H ))) it holds that k ψ ( Y ) − ψ ( Y ) k M − ,p ′ ,q ′ ( V ) (cid:16) | ψ | Lip ( H,V ) + | ψ | Lip ( H,V ) 2 X i =1 | Y i | M , ∞ ,q ( H ) (cid:17) k Y − Y k M − ,p ′ ,q ′ ( H ) . Proof.
Let p = p ′ / ( p ′ − M ,p,q ( V ) ⊂ L (Ω; V ) ⊂ M − ,p ′ ,q ′ ( V ), it holds that k ψ ( Y ) − ψ ( Y ) k M − ,p ′ ,q ′ ( V ) = (cid:13)(cid:13)(cid:13) Z ψ ′ (cid:0) Y + λ ( Y − Y ) (cid:1) ( Y − Y ) d λ (cid:13)(cid:13)(cid:13) M − ,p ′ ,q ′ ( V ) = sup Z ∈ M ,p,q ( V ) k Z k M ,p,q ( V ) =1 D Z, Z ψ ′ (cid:0) Y + λ ( Y − Y ) (cid:1) ( Y − Y ) d λ E L (Ω; V ) sup Z ∈ M ,p,q ( V ) k Z k M ,p,q ( V ) =1 Z (cid:13)(cid:13)(cid:2) ψ ′ (cid:0) Y + λ ( Y − Y ) (cid:1)(cid:3) ∗ Z (cid:13)(cid:13) M ,p,q ( H ) d λ k Y − Y k M − ,p ′ ,q ′ ( H ) , where for x ∈ H we denote by [ ψ ′ ( x )] ∗ ∈ L ( V, H ) the Hilbert space adjoint of ψ ′ ( x ) ∈ L ( H, V ). Note that the mapping ϕ : H → L ( V, H ) defined by ϕ ( x ) :=[ ψ ′ ( x )] ∗ , x ∈ H , is bounded and belongs to the class Lip ( H, L ( V, H )). We havesup x ∈ H k ϕ ( x ) k L ( V,H ) | ψ | Lip ( H,V ) and | ϕ | Lip ( H, L ( V,H )) | ψ | Lip ( H,V ) . An appli-cation of Lemma 3.8 with V = V , V = H thus yields the assertion. (cid:3) Lemma 3.10.
Let p ′ , q ′ ∈ (1 , , F ∈ L (Ω; H ) and S ∈ L ( H ) . It holds that k SF k M − ,p ′ ,q ′ ( H ) k S k L ( H ) k F k M − ,p ′ ,q ′ ( H ) .Proof. Let p, q ∈ [2 , ∞ ) satisfy p + p ′ = q + q ′ = 1. For notational convenience let B = M ,p,q ( H ) and hence B ∗ = M − ,p ′ ,q ′ ( H ). Assuming without loss of generalitythat k S k L ( H ) >
0, it holds that k SF k B ∗ = sup k Z k B =1 h SF, Z i L (Ω; H ) = k S ∗ k L ( H ) sup k Z k B =1 D F, S ∗ Z k S ∗ k L ( H ) E L (Ω; H ) k S ∗ k L ( H ) sup k Z k B =1 h F, Z i L (Ω; H ) = k S k L ( H ) k F k B ∗ (cid:3) Weak approximation for a class of semilinear SPDE
The main result and an application.
Here we describe the numericalspace-time discretization scheme for Eq. (1) and formulate our main result onweak convergence in Theorem 4.5. For the sake of comparability, we also statea corresponding strong convergence result in Proposition 4.3. An application of
Theorem 4.5 to covariance convergence in presented in Corollary 4.6, see [20] forrelated results.
Assumption 4.1 (Discretization) . For the spatial discretization we use a family ( V h ) h ∈ (0 , of finite dimensional subspaces of H and linear operators A h : V h → V h that serve as discretizations of A . By P h : H → V h we denote the orthogonalprojectors w.r.t. the inner product in H . For the discretization in time we use alinearly implicit Euler scheme with uniform grid t m = km , m ∈ { , . . . , M } , where k ∈ (0 , is the stepsize and M = M k ∈ N is determined by t M T < t M + k .The operators S h,k := (id V h + kA h ) − P h thus serve as discretizations of S ( k ) , and E mh,k := S mh,k − S ( t m ) are the corresponding error operators. We assume that thereare constants D ρ , D ρ,σ ∈ [0 , ∞ ) (independent of h , k , m ) such that, k A ρ h S mh,k k L ( H ) + k S mh,k A min( ρ, k L ( H ) D ρ t − ρ m , ρ > , (26) k E mh,k A ρ k L ( H ) D ρ,σ t − ρ + σ m (cid:0) h σ + k σ (cid:1) , σ ∈ [0 , , ρ ∈ [ − σ, min(1 , − σ )] , (27) for all h, k ∈ (0 , and m ∈ { , . . . , M } . Example 4.2.
In the situation of Example 3.2, the spaces V h can be chosen asstandard finite element spaces consisting of continuous, piecewise linear functionsw.r.t. regular triangulations of O , with maximal mesh size bounded by h . See, e.g.,[2, Section 5] for a proof of the estimates (26), (27) in this case.For h, k ∈ (0 ,
1) and M = M k ∈ N the approximation ( X mh,k ) m ∈{ ,...,M } of themild solution ( X ( t )) t ∈ [0 ,T ] to Eq. (1) is defined recursively by X h,k = P h X and X mh,k = S mh,k X + k m − X j =0 S m − jh,k F ( X jh,k ) + m − X j =0 S m − jh,k ( L ( t j +1 ) − L ( t j )) , (28) m ∈ { , . . . , M } . By ( ˜ X h,k ( t )) t ∈ [0 ,T ] we denote the piecewise constant interpolationof ( X mh,k ) m ∈{ ,...,M } which is defined as˜ X h,k ( t ) = M − X m =0 [ t m ,t m +1 ) ( t ) X mh,k + [ t M ,T ] ( t ) X Mh,k . (29)The following strong convergence result can be proven analogously to the Gauss-ian case, cf. [1, Theorem 4.2]. Proposition 4.3 (Strong convergence) . Let Assumption 3.1 hold, let ( X ( t )) t ∈ [0 ,T ] be the mild solution to Eq. (1) and ( ˜ X h,k ( t )) t ∈ [0 ,T ] be its discretization given by (28) , (29) . Then, for every γ ∈ [0 , β ) there exists a constant C ∈ [0 , ∞ ) , which doesnot depend on h, k , such that sup t ∈ [0 ,T ] k X ( t ) − ˜ X h,k ( t ) k L (Ω; H ) C ( h γ + k γ ) , h, k ∈ (0 , . For the weak convergence we consider path dependent functionals as speci-fied by the next assumption. In the related work [1] functionals of the form f ( x ) = Q ni =1 ϕ i (cid:0) R [0 ,T ] x ( t ) µ i (d t ) (cid:1) , with ϕ , . . . , ϕ n being twice differentiable withpolynomially growing derivatives of some fixed but arbitrary degree, and µ , . . . , µ n being finite Borel measures on [0 , T ], were considered for equations with Gaussiannoise. Here we generalize by removing the product structure, but we only allow forquadratically growing test functions. The reason for the latter restrition is that the ALLIAVIN REGULARITY AND WEAK APPROXIMATION 15 solution to our equation has in general only finite moments up to order two whilesolutions to equations with Gaussian noise have all moments finite.
Assumption 4.4 (Test function f ) . Let n ∈ N and ϕ : L ni =1 H → R be Fr´echetdifferentiable with globally Lipschitz continuous derivative mapping ϕ ′ : L ni =1 H →L (cid:0) L ni =1 H, R (cid:1) . Let µ , . . . , µ n be finite Borel-measures on [0 , T ] . The functional f : L ([0 , T ] , P ni =1 µ i ; H ) → R is given by f ( x ) := ϕ (cid:16) Z [0 ,T ] x ( t ) µ (d t ) , . . . , Z [0 ,T ] x ( t ) µ n (d t ) (cid:17) . Observe that X, ˜ X h,k ∈ L (Ω; L ([0 , T ] , P ni =1 µ i ; H )) due to (14) and, e.g., theestimate (32) below. In particular, the random variables f ( X ), f ( ˜ X h,k ) are definedand integrable.We next state our main result on weak convergence. The proof is postponed toSubsections 3.2–4.4. Note that the obtained weak rate of convergence is twice thestrong rate from Propostion 4.3. Theorem 4.5 (Weak convergence) . Let Assumption 3.1, 4.1 and 4.4 hold. Let X = ( X ( t )) t ∈ [0 ,T ] be the mild solution to Eq. (1) and ˜ X h,k = ( ˜ X h,k ( t )) t ∈ [0 ,T ] be itsdiscretization given by (28) , (29) . Then, for every γ ∈ [0 , β ) there exists a constant C ∈ [0 , ∞ ) , which does not depend on h, k , such that the weak error estimate (5) holds. Corollary 4.6 (Covariance convergence) . Consider the setting of Theorem 4.5.For all γ ∈ [0 , β ) , t , t ∈ (0 , T ] and φ , φ ∈ H there exists a constant C ∈ [0 , ∞ ) ,which does not depend on h, k , such that (cid:12)(cid:12) Cov (cid:0)(cid:10) X ( t ) , φ (cid:11) , (cid:10) X ( t ) , φ (cid:11)(cid:1) − Cov (cid:0)(cid:10) ˜ X h,k ( t ) , φ (cid:11) , (cid:10) ˜ X h,k ( t ) , φ (cid:11)(cid:1)(cid:12)(cid:12) C ( h γ + k γ ) , h, k ∈ (0 , . Proof of Corollary 4.6.
For random variables Y , Y , Z , Z ∈ L (Ω; H ) and vectors φ , φ ∈ H it holds thatCov (cid:0) h Y , φ i , h Y , φ i (cid:1) − Cov (cid:0) h Z , φ i , h Z , φ i (cid:1) = E (cid:2) h Y , φ ih Y , φ i − h Z , φ ih Z , φ i (cid:3) − E (cid:2) h Y , φ i − h Z , φ i (cid:3) E h Y , φ i− E h Z , φ i E (cid:2) h Y , φ i − h Z , φ i (cid:3) (30)We consider the Borel measure µ := δ t + δ t on [0 , T ] as well as the functionals f i : L ([0 , T ] , µ ; H ) → R , i ∈ { , , } , given by f ( x ) := h x ( t ) , φ ih x ( t ) , φ i , f ( x ) := h x ( t ) , φ i , f ( x ) := h x ( t ) , φ i . These functionals satisfy Assumption 4.4.From (30) with Y = X ( t ), Y = X ( t ), Z = ˜ X h,k ( t ) and Z = ˜ X h,k ( t ) weobtain (cid:12)(cid:12) Cov (cid:0) h X ( t ) , φ i , h X ( t ) , φ i (cid:1) − Cov (cid:0) h ˜ X h,k ( t ) , φ i , h ˜ X h,k ( t ) , φ i (cid:1)(cid:12)(cid:12) (cid:12)(cid:12) E (cid:2) f ( X ) − f ( ˜ X h,k ) (cid:3)(cid:12)(cid:12) + k φ k sup t ∈ [0 ,T ] k X ( t ) k L (Ω; H ) (cid:12)(cid:12) E (cid:2) f ( X ) − f ( ˜ X h,k ) (cid:3)(cid:12)(cid:12) + k φ k sup h,k ∈ (0 , k ˜ X h,k ( t ) k L (Ω; H ) (cid:12)(cid:12) E (cid:2) f ( X ) − f ( ˜ X h,k ) (cid:3)(cid:12)(cid:12) Three applications of Theorem 4.5 together with (14) and the estimate (32) belowcomplete the proof. (cid:3)
A regularity result for the discrete solution.
Here we prove an analogueof Proposition 3.5 for the discrete solution. It has Gaussian counterparts in [1,Proposition 4.3] and [2, Proposition 3.17].
Proposition 4.7.
Let Assumption 3.1 and 4.1 hold. Depending on the value of β ∈ (0 , , we assume either that q ∈ (1 , − β ) if β ∈ (0 , or q = ∞ if β = 1 .Then, sup h,k ∈ (0 , sup m ∈{ ,...,M k } k X mh,k k M , ,q ( H ) < ∞ . (31) Proof.
By a classical Gronwall argument based on Lemma A.2, it holds thatsup h,k ∈ (0 , sup m ∈{ ,...,M k } k X mh,k k L (Ω; H ) < ∞ . (32)Up to some straightforward modifications, the proof of (32) is analogous to thatof [2, Proposition 3.16] in the Gaussian case and is therefore omitted. Next, werewrite the scheme (28) in the form X mh,k = S mh,k X + k m − X j =0 S m − jh,k F ( X jh,k ) + m − X j =0 Z T Z U ( t j ,t j +1 ] ( s ) S m − jh,k x ˜ N (d s, d x ) ,m ∈ { , . . . , M } . Applying the difference operator D on the single terms in thisequation and taking into account Lemma 2.3, Lemma 2.4 and Proposition 2.8, weobtain(33) D s,x X mh,k = k m − X j = ⌈ s ⌉ k S m − jh,k (cid:2) F ( X jh,k + D s,x X jh,k ) − F ( X jh,k ) (cid:3) + m − X j =0 ( t j ,t j +1 ] ( s ) S m − jh,k x holding P ⊗ d s ⊗ ν (d x )-almost everywhere on Ω × [0 , T ] × U . Here we denote for s ∈ [0 , T ] by ⌈ s ⌉ k is the smallest number i ∈ N such that ik > s . According toLemma 2.4, the identity D s,x X jh,k = 0 holds P ⊗ d s ⊗ ν (d x )-almost everywhere onΩ × ( t j , T ] × U . Taking norms in (33) yields(34) k D s,x X mh,k k L ∞ (Ω; L q ([0 ,T ]; L ( U ; H ))) k m − X j =0 (cid:13)(cid:13)(cid:13) S m − jh,k (cid:2) F ( X jh,k + DX jh,k ) − F ( X jh,k ) (cid:3)(cid:13)(cid:13)(cid:13) L ∞ (Ω; L q ([0 ,T ]; L ( U ; H ))) + (cid:13)(cid:13)(cid:13) ( s, x ) m − X j =0 ( t j ,t j +1 ] ( s ) S m − jh,k x (cid:13)(cid:13)(cid:13) L q ([0 ,T ]; L ( U ; H )) . Using the estimate (26) and the Lipschitz assumption on F , we obtain(35) (cid:13)(cid:13)(cid:13) S m − jh,k (cid:2) F ( X jh,k + DX jh,k ) − F ( X jh,k ) (cid:3)(cid:13)(cid:13)(cid:13) L ∞ (Ω; L q ([0 ,T ]; L ( U ; H ))) D − β t β − m − j | F | Lip( H, ˙ H β − ) k DX jh,k k L ∞ (Ω; L q ([0 ,T ]; L ( U,H ))) . ALLIAVIN REGULARITY AND WEAK APPROXIMATION 17
Concerning the second term in (34) we apply the estimate (26) together with theidentity U = ˙ H β − and observe that(36) (cid:13)(cid:13)(cid:13) ( s, x ) m − X j =0 ( t j ,t j +1 ] ( s ) S m − jh,k x (cid:13)(cid:13)(cid:13) L q ([0 ,T ]; L ( U ; H )) = (cid:16) Z T (cid:16) Z U (cid:13)(cid:13)(cid:13) m − X j =0 ( t j ,t j +1 ] ( s ) S m − jh,k x (cid:13)(cid:13)(cid:13) ν (d x ) (cid:17) q d s (cid:17) q D − β | ν | (cid:16) k m − X j =0 t q ( β − m − j (cid:17) q D − β | ν | (cid:16) Z T ( T − r ) q ( β − d r (cid:17) q < ∞ . The penultimate inequality follows by approximating the sum by a Riemann integraland observing that the singularity is integrable. From (34), (35) and (36) weconclude that for all m ∈ { , . . . M k } , uniformly in h, k ∈ (0 , k DX mh,k k L ∞ (Ω; L q ([0 ,T ]; L ( U,H ))) . k m − X j =0 t β − m − j k DX jh,k k L ∞ (Ω; L q ([0 ,T ]; L ( U,H ))) . By induction we obtain that DX mh,k ∈ L ∞ (Ω; L q ([0 , T ]; L ( U ; H ))) for all m , sothat (32) and an application of the discrete Gronwall Lemma A.2 yield the uniformbound (31). (cid:3) Convergence in negative order spaces.
The following crucial result hasGaussian counterparts in [1, Lemma 4.6] and [2, Lemma 4.6].
Lemma 4.8.
Let Assumption 3.1 and 4.1 hold, let ( X ( t )) t ∈ [0 ,T ] be the mild solutionto Eq. (1) and ( ˜ X h,k ( t )) t ∈ [0 ,T ] be its discretization given by (28) , (29) . Then, forevery γ ∈ [0 , β ) and q ′ = γ there exists a constant C ∈ [0 , ∞ ) , which does notdepend on h, k , such that sup t ∈ [0 ,T ] k ˜ X h,k ( t ) − X ( t ) k M − , ,q ′ ( H ) C (cid:0) h γ + k γ (cid:1) , h, k ∈ (0 , . Proof.
For notational convenience we introduce the piecewise continuous error map-ping ˜ E h,k : [0 , T ) → L ( H ) given by ˜ E h,k ( t ) := S mh,k − S ( t ) for t ∈ [ t m − , t m ), so that X mh,k − X ( t m ) = E mh,k X + Z t m ˜ E h,k ( t m − s ) F ( ˜ X h,k ( s )) d s + Z t m S ( t m − s ) (cid:0) F ( ˜ X h,k ( s )) − F ( X ( s )) (cid:1) d s + Z t m Z ˙ H β − ˜ E h,k ( t m − s ) x ˜ N (d s, d x ) . Taking norms and using the continuous embedding L (Ω; H ) ⊂ M − , ,q ′ ( H ) as wellas Minkowski’s integral inequality yields(37) (cid:13)(cid:13) X mh,k − X ( t m ) (cid:13)(cid:13) M − , ,q ′ ( H ) k E mh,k X k + Z t m (cid:13)(cid:13) ˜ E h,k ( t m − s ) F ( ˜ X h,k ( s )) (cid:13)(cid:13) L (Ω; H ) d s + Z t m (cid:13)(cid:13) S ( t m − s ) (cid:0) F ( ˜ X h,k ( s )) − F ( X ( s )) (cid:1)(cid:13)(cid:13) M − , ,q ′ ( H ) d s + (cid:13)(cid:13)(cid:13) Z t m Z ˙ H β − ˜ E h,k ( t m − s ) x ˜ N (d s, d x ) (cid:13)(cid:13)(cid:13) M − , ,q ′ ( H ) . We estimate the terms on the right hand side separately. To this end, notethat the error estimate (27) extends to the piecewise continuous error mapping˜ E h,k . Indeed, as a consequence of the identity ˜ E h,k ( t ) = E mh,k + ( S ( t m ) − S ( t )), t ∈ [ t m − , t m ), and the estimates (12), (13), (27), we have (cid:13)(cid:13) ˜ E h,k ( t ) A ρ (cid:13)(cid:13) L ( H ) ( D ρ,σ + C σ C σ + ρ ) t − ρ + σ (cid:0) h σ + k σ (cid:1) , (38)holding for σ ∈ [0 , ρ ∈ [ − σ, min(1 , − σ )] and h, k ∈ (0 , t ∈ (0 , T ].Concerning the first two terms on the right hand side of (37) we observe that(13), (38), and the linear growth of F yield(39) k E mh,k X k + Z t m (cid:13)(cid:13) ˜ E h,k ( t m − s ) F ( ˜ X h,k ( s )) (cid:13)(cid:13) L (Ω; H ) d s D − γ, γ k X k ˙ H γ (cid:0) h γ + k γ (cid:1) + ( D − β, γ + C γ C γ +1 − β ) T β − γ (1 + β ) / − γ · k F k Lip ( H, ˙ H β − ) (cid:0) t ∈ [0 ,T ] k X ( t ) k L (Ω; H ) (cid:1)(cid:0) h γ + k γ (cid:1) . Next, we use Lemma 3.10, (12) and Proposition 3.9 to estimate the third termon the right hand side of (37) from above by(40) Z t m (cid:13)(cid:13) S ( t m − s ) A δ (cid:13)(cid:13) L ( H ) (cid:13)(cid:13) A − δ (cid:0) F ( ˜ X h,k ( s )) − F ( X ( s )) (cid:1)(cid:13)(cid:13) M − , ,q ′ ( H ) d s C δ K m − X i =0 Z t i +1 t i ( t m − s ) − δ (cid:13)(cid:13) X ih,k − X ( t i ) (cid:13)(cid:13) M − , ,q ′ ( H ) d s + C δ K m − X i =0 Z t i +1 t i ( t m − s ) − δ (cid:13)(cid:13) X ( t i ) − X ( s ) (cid:13)(cid:13) M − , ,q ′ ( H ) d s, where K is, by Proposition 3.5, Proposition 3.9 and Lemma 4.7, the finite constant K = 4 (cid:16) | F | Lip ( H, ˙ H − δ ) + | F | Lip ( H, ˙ H − δ ) sup s ∈ [0 ,T ] | X ( s ) | M , ∞ ,q ( H ) + | F | Lip ( H, ˙ H − δ ) sup h,k ∈ (0 , sup m ∈{ ,...,M k } | X mh,k | M , ∞ ,q ( H ) (cid:17) < ∞ . The terms on the right hand side of (40) can be estimated as follows: We have m − X i =0 Z t i +1 t i ( t m − s ) − δ (cid:13)(cid:13) X ih,k − X ( t i ) (cid:13)(cid:13) M − , ,q ′ ( H ) d s k m − X i =0 t − δ m − i − (cid:13)(cid:13) X ih,k − X ( t i ) (cid:13)(cid:13) M − , ,q ′ ( H ) + k − δ − δ (cid:13)(cid:13) X m − h,k − X ( t m − ) (cid:13)(cid:13) M − , ,q ′ ( H ) . ALLIAVIN REGULARITY AND WEAK APPROXIMATION 19
Since for all m ∈ { , , . . . } it holds that max i ∈{ , ,...,m − } (cid:0) t − δ m − i − · t δ m − i (cid:1) =max i ∈{ , ,...,m − } (cid:0) ( m − i − − δ · ( m − i ) δ (cid:1) = 2 δ , we obtain for all m ∈ N = { , , . . . } (41) m − X i =0 Z t i +1 t i ( t m − s ) − δ (cid:13)(cid:13) X ih,k − X ( t i ) (cid:13)(cid:13) M − , ,q ′ ( H ) d s δ k m − X i =0 t − δ m − i (cid:13)(cid:13) X ih,k − X ( t i ) (cid:13)(cid:13) M − , ,q ′ ( H ) + k − δ − δ (cid:13)(cid:13) X m − h,k − X ( t m − ) (cid:13)(cid:13) M − , ,q ′ ( H ) δ k − δ m − X i =0 t − δ m − i (cid:13)(cid:13) X ih,k − X ( t i ) (cid:13)(cid:13) M − , ,q ′ ( H ) . Moreover, by the H¨older continuity of Proposition 3.7 it holds(42) m − X i =0 Z t i +1 t i ( t m − s ) − δ (cid:13)(cid:13) X ( t i ) − X ( s ) (cid:13)(cid:13) M − , ,q ′ ( H ) d s Ck γ Z t m ( t m − s ) − δ d s . k γ . Concerning the fourth term on the right hand side of (37), note that the negativenorm inequality in Proposition 3.6 yields(43) (cid:13)(cid:13)(cid:13) Z t m Z ˙ H β − ˜ E h,k ( t m − s ) x ˜ N (d s, d x ) (cid:13)(cid:13)(cid:13) M − , ,q ′ ( H ) h Z t m (cid:16) Z ˙ H β − k ˜ E h,k ( t m − s ) x k ν (d x ) (cid:17) q ′ d s i q ′ D − β, γ | ν | (cid:0) h γ + k γ (cid:1)(cid:16) Z t m ( t m − s ) γ β − − γ d s (cid:17) γ . h γ + k γ , where the last integral is finite due to (23).Combining the estimates (37)–(43) yields (cid:13)(cid:13) X mh,k − X ( t m ) (cid:13)(cid:13) M − , ,q ′ ( H ) . h γ + k γ + k m − X i =0 t − δ m − i (cid:13)(cid:13) X ih,k − X ( t i ) (cid:13)(cid:13) M − , ,q ′ ( H ) . The discrete Gronwall Lemma A.2 thus implies that there exists C ′ ∈ [0 , ∞ ), whichdoes not depend on h, k , such that sup m ∈{ ,...,M k } k X mh,k − X ( t m ) k M − , ,q ′ ( H ) C ′ (cid:0) h γ + k γ (cid:1) for all h, k ∈ (0 , . This and Proposition 3.7 imply the claimedassertion. (cid:3)
Proof of the main result.
We are finally prepared to prove the weak con-vergence result in Theorem 4.5. Recall from Subsection 4.1 that the processes X =( X ( t )) t ∈ [0 ,T ] and ˜ X h,k = ( ˜ X h,k ( t )) t ∈ [0 ,T ] belong to L (Ω; L ([0 , T ] , P ni =1 µ i ; H )).To simplify notation, we introduce the ( L ni =1 H )-valued random variables Y =( Y (1) , . . . , Y ( n ) ), ˜ Y h,k = ( ˜ Y (1) h,k , . . . , ˜ Y ( n ) h,k ) and Φ h,k = (Φ (1) h,k , . . . , Φ ( n ) h,k ) defined by(44) Y ( i ) := Z [0 ,T ] X ( t ) µ i (d t ) , ˜ Y ( i ) h,k := Z [0 ,T ] ˜ X h,k ( t ) µ i (d t ) , Φ ( i ) h,k := Z ∂ i ϕ (cid:0) (1 − θ ) Y + θ ˜ Y h,k (cid:1) d θ. Here we denote for x = ( x (1) , . . . , x ( n ) ) ∈ L nj =1 H by ∂ i ϕ ( x ) = ∂∂x ( i ) ϕ ( x ) the Fr´echetderivative of ϕ w.r.t. the i -th coordinate of x , considered as an element of H viathe Riesz isomorphism L ( H, R ) ≡ H . Moreover, we set set q := − γ and q ′ := γ . Using the notation above, the fundamental theorem of calculus, and duality inthe Gelfand triple M , ,q ( H ) ⊂ L (Ω; H ) ⊂ M − , ,q ′ ( H ), we represent and estimatethe weak error as follows:(45) (cid:12)(cid:12) E (cid:2) f ( ˜ X h,k ) − f ( X ) (cid:3)(cid:12)(cid:12) = (cid:12)(cid:12) E (cid:2) ϕ ( ˜ Y h,k ) − ϕ ( Y ) (cid:3)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) E n X i =1 (cid:10) Φ ( i ) h,k , ˜ Y ( i ) h,k − Y ( i ) (cid:11)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) n X i =1 Z [0 ,T ] E (cid:10) Φ ( i ) h,k , ˜ X h,k ( t ) − X ( t ) (cid:11) µ i (d t ) (cid:12)(cid:12)(cid:12) n X i =1 µ i ([0 , T ]) (cid:13)(cid:13) Φ ( i ) h,k (cid:13)(cid:13) M , ,q ( H ) sup t ∈ [0 ,T ] (cid:13)(cid:13) ˜ X h,k ( t ) − X ( t ) (cid:13)(cid:13) M − , ,q ′ ( H ) The assertion of Theorem 4.5 now follows from (45) together with Lemma 4.8 andLemma 4.9 below.
Lemma 4.9.
Let Assumption 3.1, 4.1 and 4.4 hold. Let ( X ( t )) t ∈ [0 ,T ] be the mildsolution to Eq. (1) , ( ˜ X h,k ( t )) t ∈ [0 ,T ] be its discretization given by (28) , (29) , and let Φ ( i ) h,k , i ∈ { , . . . , n } , h, k ∈ (0 , be the H -valued random variables defined by (44) .For all γ ∈ [0 , β ) and q = − γ it holds that max i ∈{ ,...,n } sup h,k ∈ (0 , (cid:13)(cid:13) Φ ( i ) h,k (cid:13)(cid:13) M , ,q ( H ) < ∞ . Proof.
First note that the linear growth of ∂ i ϕ : L nj =1 H → H , the estimates(14), (32), and the fact that µ i ([0 , T ]) < ∞ imply for all i ∈ { , . . . , n } thatsup h,k ∈ (0 , (cid:13)(cid:13) Φ ( i ) h,k (cid:13)(cid:13) L (Ω; H ) < ∞ . It remains to check that sup h,k ∈ (0 , (cid:12)(cid:12) Φ ( i ) h,k (cid:12)(cid:12) M , ,q ( H ) is finite. The chain rule from Lemma 2.3, applied to the function h : ( L nj =1 H ) ⊕ ( L nj =1 H ) → H, ( y, ˜ y ) R ∂ i ϕ (cid:0) (1 − θ ) y + θ ˜ y (cid:1) d θ , yields for all i ∈ { , . . . , n } D s,x Φ ( i ) h,k = D s,x Z ∂ i ϕ (cid:0) (1 − θ ) Y + θ ˜ Y h,k (cid:1) d θ = Z h ∂ i ϕ (cid:16) (1 − θ ) (cid:0) Y + D s,x Y (cid:1) + θ (cid:0) ˜ Y h,k + D s,x ˜ Y h,k (cid:1)(cid:17) − ∂ i ϕ (cid:0) (1 − θ ) Y + θ ˜ Y h,k (cid:1)i d θ P ⊗ d s ⊗ ν (d x )-almost everywhere on Ω × [0 , T ] × U . This, the global Lipschitzcontinuity of ∂ i ϕ : L nj =1 H → H , and Proposition 2.7 imply (cid:13)(cid:13) D s,x Φ ( i ) h,k (cid:13)(cid:13) | ϕ | Lip ( L nj =1 H ; R ) (cid:16)(cid:13)(cid:13) D s,x Y ( i ) (cid:13)(cid:13) + (cid:13)(cid:13) D s,x ˜ Y ( i ) h,k (cid:13)(cid:13)(cid:17) | ϕ | Lip ( L nj =1 H ; R ) Z [0 ,T ] (cid:16)(cid:13)(cid:13) D s,x X ( t ) (cid:13)(cid:13) + (cid:13)(cid:13) D s,x ˜ X h,k ( t ) (cid:13)(cid:13)(cid:17) µ i (d t )Iterated integration w.r.t. ν (d x ), d s , P , and three applications of Minkowski’s in-tegral inequality lead to (cid:12)(cid:12) Φ ( i ) h,k (cid:12)(cid:12) M , ,q ( H ) | ϕ | Lip ( L nj =1 H ; R ) Z [0 ,T ] (cid:16) | X ( t ) | M , ,q ( H ) + (cid:12)(cid:12) ˜ X h,k ( t ) (cid:12)(cid:12) M , ,q ( H ) (cid:17) µ i (d t )The estimates (18), (31) and the assumption that µ i ([0 , T ]) < ∞ thus imply for all i ∈ { , . . . , n } the finiteness of sup h,k ∈ (0 , (cid:12)(cid:12) Φ ( i ) h,k (cid:12)(cid:12) M , ,q ( H ) . (cid:3) Acknowledgement
Kristin Kirchner, Raphael Kruse, Annika Lang and Stig Larsson are gratefullyacknowledged for participating in early discussion regarding this work and [4].
ALLIAVIN REGULARITY AND WEAK APPROXIMATION 21
Appendix A. Gronwall Lemmata
In this section we state two versions of Gronwall’s lemma. The first one followsfrom the arguments in the proof of [13, Lemma 6.3] together with the standardversion of Gronwall’s lemma for measurable functions. The second one is a slightmodification of [22, Lemma A.4], compare also [13, Lemma 7.1].
Lemma A.1 (Generalized Gronwall lemma) . Let T ∈ (0 , ∞ ) and φ : { ( t, s ) : 0 s t T } → [0 , ∞ ) be a Borel measurable function satisfying R Ts φ ( r, s ) d r < ∞ for all s ∈ [0 , T ] . If φ ( t, s ) A ( t − s ) − α + B Z ts ( t − r ) − β φ ( r, s ) d r, s t T, for some constants A, B ∈ [0 , ∞ ) , α, β ∈ (0 , ∞ ) , then there exists a constant C = C ( B, T, α, β ) ∈ [0 , ∞ ) such that φ ( t, s ) C A ( t − s ) − α , s t T. Lemma A.2 (Discrete Gronwall lemma) . Let T ∈ (0 , ∞ ) , k ∈ (0 , and M = M k ∈ N be such that M k T < ( M + 1) k , and set t m := mk , m ∈ { , . . . , M } .Let ( φ i ) Mi =0 be a sequence of nonnegative real numbers. If φ m A + B k m − X i =0 t − βm − i φ i , m ∈ { , . . . , M } , for some constants A, B ∈ [0 , ∞ ) , β ∈ (0 , , then there exists a constant C = C ( B, T, β ) ∈ [0 , ∞ ) such that φ m C A , m ∈ { , . . . , M } . References [1] A. Andersson, M. Kov´acs, and S. Larsson. Weak and strong error analysis for semilinearstochastic Volterra equations.
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