Applications of Malliavin-Stein Method: Spatial averages of solution to stochastic heat equation and Breuer-Major theorem

Abstract

This thesis includes four main parts. The first part is an exposition about Malliavin calculus, Malliavin-Stein method, Walsh stochastic integral and existence and regularity of mild solution to stochastic heat equation. In the second part, we study Malliavin differentibility of the solution of stochastic heat equation and establishing Lp-bounds for Malliavin derivatives. One way to obtain such results is through Feynman-Kac formula which is studied in the second part as well. Last two parts are devoted to quantitative rates of convergences corresponding to some central limit theorems: We start with studying such problem for spatial averages of the solution to the stochastic heat equation. Then, we establish rate of convergence results in total variation as well as in Wasserstein distances for the Breuer-Major theorem.