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

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