We introduce a new method for proving weak convergence for stochastic
evolution problems. The proof is based on refined Sobolev-Malliavin
spaces from the Malliavin calculus. It does not rely on the use of the
Kolmogorov equation or the Ito formula and is therefore applicable also to
non-Markovian equations, where these are not available. We use it to
prove weak convergence of fully discrete approximations of the solution of
the semilinear stochastic parabolic evolution equation with additive noise
as well as a semilinear stochastic Volterra integro-differential equation.
This is joint work with Adam Andersson, Mihaly Kovacs, and Raphael Kruse.