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

Let {u( t, x)}(t>0,x epsilon R) denote the solution to the parabolic Anderson model with initial condition delta(0) and driven by space-time white noise on R+ x 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 [6,7] in order to prove that the random field x -> 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 MalliavinStein method introduced in Huang, Nualart, and Viitasaari [11]. (C) 2021 Elsevier Inc. All rights reserved.