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 It^o form as $$ dd X(t) + left (int_0^t b(t-s) A X(s) , dd s right) , dd t = dd W^{_Q}(t), tin (0,T]; ~ X(0) =X_0in H, $$ oindent where $W^Q$ is a $Q$-Wiener process on the Hilbert space $H$ and where the time kernel $b$ is the locally integrable potential $t^{rho-2}$, $rho in (1,2)$, 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 $A^{( u- 1/rho)/2} Q^{1/2}$ is a Hilbert-Schmidt operator on $H$ for some $ u in [0,1/rho]$. 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 $Delta t=T/N$). %Let $X_h^N$ be the discrete solution at time $T$. Eventually let $varphi : Hrightarrow R$ is such that $D^2varphi$ is bounded on $H$ but not necessarily bounded and suppose in addition that either its first derivative is bounded on $H$ and $X_0 in L^1(Omega)$ or $varphi = | cdot |^2$ and $X_0 in L^2(Omega)$. We show that for $varphi : Hrightarrow R$ twice continuously differentiable test function with bounded second derivative, $$ | E varphi(X^N_h) - E varphi(X(T)) | leq C ln left(frac{T}{h^{2/rho} + Delta t} right) (Delta t^{rho u} + h^{2 u}), $$ oindent for any $0leq u leq 1/rho$. This is essentially twice the rate of strong convergence under the same regularity assumption on the noise.