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

Abstract

In this paper, we investigate the asymptotic behavior of the spatial average of the solution to the parabolic Anderson model with time-independent noise in dimension [Formula: see text], as the domain of the integral becomes large. We consider three 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 [Formula: see text] 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.