Stochastic analysis on Wiener space and applications to distributional asymptotics

Abstract

In this thesis we study Malliavin calculus on infinite dimensional Wiener space and study properties of Malliavin operators. We then see how these along with what is known as Stein’s method for distributional approximation is used to obtain quantitative limit theorems inside a fixed Wiener chaos and also sometimes more generally. In a joint work with David Nualart which is the content of chapter 4, we apply these results to prove an invariance principle for functionals of Gaussian random vector fields on Euclidean space for a large class of covariances. This is an extension of the original famous result by Breuer and Major and recent functional convergence results by Nualart et. al. to the case of vector valued fields. We then briefly also look into further applications in the area of geometry of random fields.