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

Suppose that {u(t, x))(t>0,x is an element of 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) integral[0,N](d) g(u(t, x)) dx, as N -> infinity, 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.