Central limit theorems for spatial averages of the stochastic heat equation via Malliavin-Steins method

Suppose that ${u(t,, x)}_{t >0, x inmathbb{R}^d}$ 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 Dalangs 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 $Nrightarrowinfty$, where $g$ is a Lipschitz-continuous function or belongs to a class of locally-Lipschitz functions, using a combination of the Malliavin calculus and Steins 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.