Central limit theorems for stochastic heat equation driven by Gaussian noise with heat-kernel covariance

Abstract

In this article, we study the spatial fluctuations of the Skorohod solution to a stochastic heat equation on $\mathbb R^d$ for $d<4$, driven by multiplicative Gaussian noise with the non-separable covariance kernel $p_{|t-s|}(x-y)$. We prove that the solution is strictly stationary and spatially ergodic at every fixed time. For the centered spatial integral $F_R(t)=\int_{\{|x|<R\}}(u(t,x)-1)dx,$ we show that $\mathbb E[F_R(t)F_R(s)]\sim K(t,s)R^d$ as $R\to\infty$. Using moment estimates for the first two Malliavin derivatives and a second-order Gaussian Poincaré inequality, we establish a quantitative central limit theorem in total variation distance with rate $R^{-d/2}$. We also prove a functional central limit theorem for the process $\{R^{-d/2}F_R(t)\}_{t\geq0}$.