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

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}

Gaussian fluctuation for Gaussian Wishart matrices of overall correlation

In this note, we study the Gaussian fluctuations for the Wishart matrices $d^{-1}\mathcal{X}_{n, d}\mathcal{X}^{T}_{n, d}$, where $\mathcal{X}_{n, d}$ is a $n\times d$ random matrix whose entries are jointly Gaussian and correlated with row and column covariance functions given by $r$ and $s$ respectively such that $r(0)=s(0)=1$. Under the assumptions $s\in \ell^{4/3}(\mathbb{Z})$ and $\|r\|_{\ell^1(\mathbb{Z})}< \sqrt{6}/2$, we establish the $\sqrt{n^3/d}$ convergence rate for the Wasserstein distance between a normalization of $d^{-1}\mathcal{X}_{n, d}\mathcal{X}^{T}_{n, d}$ and the corresponding Gaussian ensemble. This rate is the same as the optimal one computed in \cite{JL15,BG16,BDER16} for the total variation distance, in the particular case where the Gaussian entries of $\mathcal{X}_{n, d}$ are independent. Similarly, we obtain the $\sqrt{n^{2p-1}/d}$ convergence rate for the Wasserstein distance in the setting of random $p$-tensors of overall correlation. Our analysis is based on the Malliavin-Stein approach