Weak convergence of a fully discrete approximation of a linear stochastic evolution equation with a positive-type memory term

In this paper we are interested in the numerical approximation of the marginal distributions of the Hilbert space valued solution of a stochastic Volterra equation driven by an additive Gaussian noise. This equation can be written in the abstract Itô form as d X ( t ) + ( ∫ 0 t b ( t − s ) A X ( s ) d s ) d t = d W Q ( t ) , t ∈ ( 0 , T ] ; X ( 0 ) = X 0 ∈ H , where W Q is a Q-Wiener process on the Hilbert space H and where the time kernel b is the locally integrable potential t ρ − 2 , ρ ∈ ( 1 , 2 ) , or slightly more general. The operator A is unbounded, linear, self-adjoint, and positive on H. Our main assumption concerning the noise term is that A ( ν − 1 / ρ ) / 2 Q 1 / 2 is a Hilbert–Schmidt operator on H for some ν ∈ [ 0 , 1 / ρ ] . The numerical approximation is achieved via a standard continuous finite element method in space (parameter h) and an implicit Euler scheme and a Laplace convolution quadrature in time (parameter Δ t = T / N ). We show that for φ : H → R twice continuously differentiable test function with bounded second derivative, | E φ ( X h N ) − E φ ( X ( T ) ) | ⩽ C ln ( T h 2 / ρ + Δ t ) ( Δ t ρ ν + h 2 ν ) , for any 0 ⩽ ν ⩽ 1 / ρ . This is essentially twice the rate of strong convergence under the same regularity assumption on the noise.