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

Let ${u(t,, x)}_{t >0, x inmathbb{R}}$ denote the solution to the parabolic Anderson model with initial condition $delta_0$ and driven by space-time white noise on $mathbb{R}_+timesmathbb{R}$, and let $bm{p}_t(x):= (2pi t)^{-1/2}exp{-x^2/(2t)}$ 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 $xmapsto u(t,,x)/bm{p}_t(x)$ 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}.