On weak convergence, Malliavin calculus and Kolmogorov equations in infinite dimensions
Doctoral thesis, 2014
This thesis is focused around weak convergence analysis of approximations of stochastic evolution equations in Hilbert space. This is a class of problems, which is sufficiently 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 differential equations and to stochastic Volterra integro-differential equations. The solutions to the latter type of equations are not Markov processes, and therefore classical proof techniques do not apply. The final part of the thesis concerns further developments of the Markov theory for stochastic evolution equations with multiplicative non-trace class noise, again motivated by weak convergence analysis. An extension of the transition semigroup is introduced and it is shown to provide a solution operator for the Kolmogorov equation in infinite dimensions. Stochastic evolution equations with irregular initial data are used as a technical tool and existence and uniqueness of such equations are established. Application of this theory to weak convergence analysis is not a part of this thesis, but the tools for it are developed.
Kolmogorov equations in infinite dimensions
backward Euler method
stochastic Volterra equations
finite element method
Stochastic evolution equations