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

Abstract

Let {u(t,x)}t>0,x∈R denote the solution to the parabolic Anderson model with initial condition δ0 and driven by space-time white noise on R+×R, and let pt(x):=(2πt)−1/2exp⁡{−x2/(2t)} denote the standard Gaussian heat kernel on the line. We use a non-trivial adaptation of the methods in our companion papers [6,7] in order to prove that the random field x↦u(t,x)/pt(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 [11].