Limit distributions for Skorohod integrals and spatial averages of the stochastic wave and heat equation

Abstract

In this dissertation, we study some problems related to the convergence in distribution of functionals of Gaussian processes. The approach used to address the problems presented in this thesis is based on Malliavin calculus techniques. In Chapter 1, we prove the convergence in distribution of sequences of It\^o and Skorohod integrals with integrands having singular asymptotic behavior. These sequences include stochastic convolutions and generalize the example $\sqrt n\int _0^1 t^n B_tdB_t$ first studied by Peccati and Yor in 2004. In Chapter 2, we prove a functional central limit theorem for the spatial average of the mild solution to the 2D stochastic wave equation driven by a Gaussian noise, which is temporally white and spatially colored described by the Riesz kernel. We also establish a quantitative central limit theorem for the marginal and the rate of convergence is described by the total-variation distance. A fundamental ingredient in our proofs is the pointwise $L^p$-estimate of the Malliavin derivative, which is of independent interest. In Chapter 3, we prove a quantitative central limit theorem for the spatial average of the mild solution to the 1D stochastic heat equation driven by space-time white noise with an initial condition given by an independent white noise. As part of this chapter, we also prove the existence, uniqueness, stationarity, and differentiability (in the Malliavin calculus sense) of the mild solution.