Strong order of convergence of a fully discrete approximation of a linear stochastic Volterra type evolution equation

In this paper we investigate a discrete approximation in time and in space of a Hilbert space valued stochastic process $\{u(t)\}_{t\in [0,T]}$ satisfying a stochastic linear evolution equation with a positive-type memory term driven by an additive Gaussian noise. The equation can be written in an abstract form as $$\dd u + (\int_0^t b(t-s) Au(s) \, \dd s)\, \dd t = \dd W^{_Q}, t\in (0,T]; \quad u(0)=u_0 \in H,$$ where $W^{_Q}$ is a $Q$-Wiener process on $H=L^2({\mathcal D})$ and where the main example of $b$ we consider is given by $$b(t) = t^{\beta-1}/\Gamma(\beta), \quad 0 < \beta <1.$$ We let $A$ be an unbounded linear self-adjoint positive operator on $H$ and we further assume that there exist $\alpha >0$ such that $A^{-\alpha}$ has finite trace and that $Q$ is bounded from $H$ into $D(A^\kappa)$ for some real $\kappa$ with $\alpha-\frac{1}{\beta+1}<\kappa \leq \alpha$. The discretization is achieved via an implicit Euler scheme and a Laplace transform convolution quadrature in time (parameter $\Delta t =T/n$), and a standard continuous finite element method in space (parameter $h$). Let $u_{n,h}$ be the discrete solution at $T=n\Delta t$. We show that $$(\E \| u_{n,h} - u(T)\|^2)^{1/2}={\mathcal O}(h^{\nu} + \Delta t^\gamma),$$ for any $\gamma< (1 - (\beta+1)(\alpha - \kappa))/2$ and $\nu \leq \frac{1}{\beta+1}-\alpha+\kappa$.