We find the weak rate of convergence of the spatially semidiscrete finite element approximation of the nonlinear stochastic heat equation. Both multiplicative and additive noise is considered under different assumptions. This extends an earlier result of Debussche in which time discretization is considered for the stochastic heat equation perturbed by white noise. It is known that this equation has a solution only in one space dimension. In order to obtain results for higher dimensions, colored noise is considered here, besides white noise in one dimension. Integration by parts in the Malliavin sense is used in the proof. The rate of weak convergence is, as expected, essentially twice the rate of strong convergence.1991 Mathematics Subject Classification. 65M60, 60H15, 60H35, 65C30.
We introduce a new family of refined Sobolev-Malliavin spaces that capture the integrability in time of the Malliavin derivative. We consider duality in these spaces and derive a Burkholder type inequality in a dual norm.The theory we develop allows us to prove weak convergence with essentially optimal rate for numerical approximations in space and time of semilinear parabolic stochastic evolution equations driven by Gaussian additive noise. In particular, we combine a standard Galerkin finite element method with backward Euler timestepping. The method of proof does not rely on the use of the Kolmogorov equation or the Itō formula and is therefore non-Markovian in nature. Test functions satisfying polynomial growth and mild smoothness assumptions are allowed, meaning in particular that we prove convergence of arbitrary moments with essentially optimal rate. 2010 Mathematics Subject Classification. 60H15, 60H07, 65C30, 65M60.The classical Sobolev-Malliavin spaces are obtained for q = 2. We use the refined spaces in a duality argument based on the Gelfand tripleA key ingredient is the following inequality for the H-valued stochastic Itō-integralwhere p ′ , q ′ are the conjugate exponents to p, q ≥ 2. We apply this inequality in situations, where one usually relies on the Burkholder-Davis-Gundy inequality, see Lemma 2.2. There the L 2 (Ω, H)-norm of the stochastic integral is bounded in terms of the L p (Ω, L 2 ([0, T ], L 0 2 ))-norm of Φ, whereas here the dual norm of the integral is bounded by the L p ′ (Ω, L q ′ ([0, T ], L 0 2 ))-norm of Φ. Since q ′ ≤ 2, this allows stronger singularities with respect to t.In defining the spaces M 1,p,q (H) some care needs to be taken. For q ≥ 2 we define the Malliavin derivative on a non-standard core S q (H), see (3.2), (3.3), of smooth and cylindrical random variables, more regular than in the classical theory in which q = 2. By proving that the operator D : S q (H) → L p (Ω, L q ([0, T ], L 0 2 )) is well defined and closable, we show that M 1,p,q (H) are Banach spaces. The proofs are rather elementary and rely to a large extent on existing results for the case q = 2. The spaces are new to the best of our knowledge.The motivation for introducing the spaces described above is found in our aim to develop new methods for the analysis of the weak error of numerical approximations of semilinear parabolic stochastic partial differential equations of the form dX(t) + AX(t) dt = F (X(t)) dt + dW (t), t ∈ (0, T ]; X(0) = X 0 .(1.2) Both space-time white noise and trace class noise are considered and the nonlinearity F is allowed to be a Nemytskii operator. See Assumption 2.3 below for precise conditions on A, F , W , X 0 . We treat discretizations in space and time, allowing for any spatial discretization scheme that satisfies the abstract Assumption 2.4 below. We verify this assumption in Section 5 for piecewise linear finite element approximations of the heat equation. Discretization in time is performed by the semi-implicit backward Euler method. Our main result, weak convergenc...
We prove a weak error estimate for the approximation in space and time of a semilinear stochastic Volterra integro-differential equation driven by additive space-time Gaussian noise. We treat this equation in an abstract framework, in which parabolic stochastic partial differential equations are also included as a special case. The approximation in space is performed by a standard finite element method and in time by an implicit Euler method combined with a convolution quadrature. The weak rate of convergence is proved to be twice the strong rate, as expected. Our convergence result concerns not only functionals of the solution at a fixed time but also more complicated functionals of the entire path and includes convergence of covariances and higher order statistics. The proof does not rely on a Kolmogorov equation. Instead it is based on a duality argument from Malliavin calculus.If (S t ) t∈[0,T ] is the analytic semigroup generated by −A, then (1.1) holds withwhere b : (0, ∞) → R is the Riesz kernel b t = t ρ−2 /Γ(ρ − 1) for some ρ ∈ (1, 2), then (S t ) t∈[0,T ] satisfies (1.1). The latter example is the main motivation of the present paper. In Subsection 5.2 we verify (1.1) for slightly more general kernels b.2010 Mathematics Subject Classification. 60H15, 60H07, 65C30, 65M60. Key words and phrases. Stochastic Volterra equation, finite element method, backward Euler, convolution quadrature, strong and weak convergence, Malliavin calculus, regularity, duality. 1 2 A. ANDERSSON, M. KOVÁCS, AND S. LARSSONThe main object of study in this paper is the stochastic evolution equation
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