Central limit theorems for spatial averages of the stochastic heat equation via Malliavin–Stein’s method

Abstract

Suppose that \(\{u(t, x)\}_{t >0, x \in {\mathbb {R}}^d}\) is the solution to a d-dimensional stochastic heat equation driven by a Gaussian noise that is white in time and has a spatially homogeneous covariance that satisfies Dalang’s condition. The purpose of this paper is to establish quantitative central limit theorems for spatial averages of the form \(N^{-d} \int _{[0,N]^d} g(u(t,x))\, \mathrm {d}x\), as \(N\rightarrow \infty \), where g is a Lipschitz-continuous function or belongs to a class of locally-Lipschitz functions, using a combination of the Malliavin calculus and Stein’s method for normal approximations. Our results include a central limit theorem for the Hopf–Cole solution to KPZ equation. We also establish a functional central limit theorem for these spatial averages.