An almost sure central limit theorem for the parabolic Anderson model with delta initial condition

Abstract

Consider the parabolic Anderson model of the form , where for t>0 and with , and η is a centered Gaussian noise that is white in time and has a spatially homogeneous covariance given by a nonnegative-definite measure f that satisfies Dalang's condition. Let denote the standard Gaussian heat kernel on and set for all t>0 and . In this paper, we present an almost sure central limit theorem (ASCLT) and a functional ASCLT for spatial averages of the form as for fixed t>0 based on the quantitative analysis of f. In particular, when f is given by a Riesz kernel, that is, for some , we can also obtain the ASCLT.