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

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 $\boldsymbol{p}_t(x):=(2\pi t)^{-d/2}\exp{-|x|^2/(2t)}$ denote the standard Gaussian heat kernel on $\mathbb{R}^d$. We prove that for all $t>0$, the process $U(t):={u(t\,, x)/\boldsymbol{p}_t(x): x\in \mathbb{R}^d}$ 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., $f(\mathrm{d} x) = |x|^{-\beta}\mathrm{d} x$, we obtain a multiple phase transition for the CLT for $U(t)$ from $\beta\in(0\,,1)$ to $\beta=1$ to $\beta\in(1\,,d\wedge 2)$.