Stochastic partial differential equations have been intensely studied in the semigroup framework. Due to the semigroup property of the evolution operator the solutions are Markovian and this plays a central role in the classical weak error analysis of numerical schemes. Stochastic Volterra equations are analyzed in the same theoretical framework but for this type of equations the evolution family is not a semigroup and the solution does not satisfy the Markov property.
In this talk I show strong and weak convergence for semilinear stochastic Volterra equations driven by additive noise. We consider approximation by means of a standard finite elements in space, and in time by the backward Euler method combined with a certain convolution quadrature. We do not solely consider weak convergence of the approximate solutions at fixed instances of time but for sums of the solution evaluated at different time points or more generally of integrals of the entire trajectory with respect to Borel measures. This is joint work with Mihály Kovács (University of Otago) and Stig Larsson (Chalmers). The theoretical backbone on which the proof is based is the theory of duality in refined Sobolev-Watanabe spaces that in turn is a joint work with Raphael Kruse (TU, Berlin) and Stig Larsson.