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}.
Categories: math.PR