Functional central limit theorems for spatial averages of the parabolic Anderson model with delta initial condition in dimension $d\geq 1$

TL;DR

This paper establishes functional central limit theorems for spatial averages of the parabolic Anderson model with delta initial condition, analyzing the influence of the noise's spatial covariance and dimension.

Contribution

It provides new functional CLTs for the model's spatial averages, explicitly relating the results to the covariance measure and spatial dimension, especially for Riesz kernel noise.

Findings

Functional CLTs depend on the covariance measure and dimension.

Explicit results for Riesz kernel covariance.

Quantitative analysis of noise covariance influences limit behavior.

Abstract

Let $\{u(t,x)\}_{t>0,x\in{{\mathbb R}^{d}}}$ denote the solution to a $d$-dimensional parabolic Anderson model with delta initial condition and driven by a multiplicative noise that is white in time and has a spatially homogeneous covariance given by a nonnegative-definite measure $f$. Let $S_{N,t}:=N^{-d}\int_{{[0,N]}^d}{[U(t,x)-1]}{\rm d}x$ denote the spatial average on ${{\mathbb R}^{d}}$. We obtain various functional central limit theorems (CLTs) for spatial averages based on the quantitative analysis of $f$ and spatial dimension $d$. In particular, when $f$ is given by Riesz kernel, that is, $f({\rm x})={\Vert x \Vert}^{-\beta}{\rm d}x$, $\beta\in(0,2\wedge d)$, the functional CLT is also based on the index $\beta$.