Poincare inequality, and central limit theorems for parabolic stochastic partial differential equations

Let ${u(t,,x)}_{tge 0, xin mathbb{R}^d}$ denote the solution of a $d$-dimensional nonlinear stochastic heat equation that is driven by a Gaussian noise, white in time with a homogeneous spatial covariance that is a finite Borel measure $f$ and satisfies Dalangs condition. We prove two general functional central limit theorems for occupation fields of the form $N^{-d} int_{mathbb{R}^d} g(u(t,,x)) psi(x/N), mathrm{d} x$ as $Nrightarrow infty$, where $g$ runs over the class of Lipschitz functions on $mathbb{R}^d$ and $psiin L^2(mathbb{R}^d)$. The proof uses Poincare-type inequalities, Malliavin calculus, compactness arguments, and Paul Levys classical characterization of Brownian motion as the only mean zero, continuous Levy process. Our result generalizes central limit theorems of Huang et al cite{HuangNualartViitasaari2018,HuangNualartViitasaariZheng2019} valid when $g(u)=u$ and $psi = mathbf{1}_{[0,1]^d}$.