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 $N\to \infty$, 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 {\it Hopf-Cole} solution to KPZ equation. We also establish a functional central limit theorem for these spatial averages.

References in corpus (1)

• Spatial ergodicity for SPDEs via a Poincaré-type inequality