On Weak Convergence, Malliavin Calculus and Kolmogorov Equa- tions in Infinite Dimensions

This thesis is focused around weak convergence analysis of approximations of sto-chastic evolution equations in Hilbert space. This is a class of problems, which is suf-ficiently challenging to motivate new theoretical developments in stochastic analysis. The first paper of the thesis further develops a known approach to weak convergence based on techniques from the Markov theory for the stochastic heat equation, such as the transition semigroup, Kolmogorov’s equation, and also integration by parts from the Malliavin calculus. The thesis then introduces a novel approach to weak convergence analysis, which relies on a duality argument in a Gelfand triple of refined Sobolev-Malliavin spaces. These spaces are introduced and a duality theory is developed for them. The family of refined Sobolev-Malliavin spaces contains the classical Sobolev-Malliavin spaces of Malliavin calculus as a special case. The novel approach is applied to the approximation in space and time of semilinear parabolic stochastic partial differ-ential equations and to stochastic Volterra integro-differential equations. The solutions to the latter type of equations are not Markov processes, and therefore classical proo