Spatial ergodicity and central limit theorems for parabolic Anderson model with delta initial condition

Abstract: Let ${u(t\,, x)}{t >0, x \in\mathbb{R}}$ denote the solution to the parabolic Anderson model with initial condition $\delta_0$ and driven by space-time white noise on $\mathbb{R}+\times\mathbb{R}$, and let $p_t(x):= (2\pi t)^{-1/2}\exp{-x^2/(2t)}$ denote the standard Gaussian heat kernel on the line. We use a non-trivial adaptation of the methods in our companion papers \cite{CKNP,CKNP_b} in order to prove that the random field $x\mapsto u(t\,,x)/p_t(x)$ is ergodic for every $t >0$. And we establish an associated quantitative central limit theorem following the approach based on the Malliavin-Stein method introduced in Huang, Nualart, and Viitasaari \cite{HNV2018}.