Central limit theorems for heat equation with time-independent noise: the regular and rough cases

TL;DR

This paper studies the asymptotic behavior of spatial integrals of solutions to the heat equation with time-independent noise, establishing central limit theorems across various noise regularity cases in different dimensions.

Contribution

It provides new central limit theorems and variance estimates for the heat equation with different types of time-independent noise, including rough fractional noise.

Findings

Variance order of magnitude identified for each noise case

Quantitative CLTs proved with total variation distance estimates

Functional limit theorems established for large domains

Abstract

In this article, we investigate the asymptotic behaviour of the spatial integral of the solution to the parabolic Anderson model with time independent noise in dimension $d\geq 1$, as the domain of the integral becomes large. We consider 3 cases: (a) the case when the noise has an integrable covariance function; (b) the case when the covariance of the noise is given by the Riesz kernel; (c) the case of the rough noise, i.e. fractional noise with index $H \in (\frac{1}{4},\frac{1}{2})$ in dimension $d=1$. In each case, we identify the order of magnitude of the variance of the spatial integral, we prove a quantitative central limit theorem for the normalized spatial integral by estimating its total variation distance to a standard normal distribution, and we give the corresponding functional limit result.