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 $$ \dd X(t) + \left (\int_0^t b(t-s) A X(s) \, \dd s \right) \, \dd t = \dd W^{_Q}(t), t\in (0,T]; ~ X(0) =X_0\in H, $$ \noindent 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 $A^{(\nu- 1/\rho)/2} Q^{1/2}$ is a Hilbert-Schmidt operator on $H$ for some $\nu \in [0,1/\rho]$. 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 $\varphi : H\rightarrow \R$ is such that $D^2\varphi$ 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 $\varphi : H\rightarrow \R$ twice continuously differentiable test function with bounded second derivative, $$ | \E \varphi(X^N_h) - \E \varphi(X(T)) | \leq C \ln \left(\frac{T}{h^{2/\rho} + \Delta t} \right) (\Delta t^{\rho \nu} + h^{2\nu}), $$ \noindent for any $0\leq \nu \leq 1/\rho$. This is essentially twice the rate of strong convergence under the same regularity assumption on the noise.