Spatial stationarity, ergodicity and CLT for parabolic Anderson model with delta initial condition in dimension $d\ge 1$

Abstract

Suppose that $\{u(t,x){\}}_{t>0,x\in {\mathbb{R}}^d}$ is the solution to a $d$-dimensional parabolic Anderson model with delta initial condition and driven by a Gaussian noise that is white in time and has a spatially homogeneous covariance given by a nonnegative-definite measure $f$ which satisfies Dalang's condition. Let ${\mathbf{p}}_t(x):=(2\pi t{)}^{-d/2}\exp \{-\Vert x{\Vert }^2/(2t)\}$ denote the standard Gaussian heat kernel on ${\mathbb{R}}^d$. We prove that for all $t>0$, the process $U(t):=\{u(t,x)/{\mathbf{p}}_t(x):x\in {\mathbb{R}}^d\}$ is stationary using Feynman-Kac's formula, and is ergodic under the additional condition $\hat{f}\{0\}=0$, where $\hat{f}$ is the Fourier transform of $f$. Moreover, using Malliavin-Stein method, we investigate various central limit theorems for $U(t)$ based on the quantitative analysis of $f$. In particular, when $f$ is given by Riesz kernel, i.e., $f(\mathrm{d}x)=\Vert x{\Vert }^{-\beta }\mathrm{d}x$, we obtain a multiple phase transition for the CLT for $U(t)$ from $\beta \in (0,1)$ to $\beta =1$ to $\beta \in (1,d\wedge 2)$.