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

作者：Jingyu Li, Yong Zhang · 发表于：Stochastics · 年份：2022 · DOI：10.1080/17442508.2022.2088236 · 被引用次数：11 · 研究领域：Stochastic processes and statistical mechanics、Probability and Risk Models、Random Matrices and Applications

Consider the parabolic Anderson model of the form ∂tu=12Δu+uη, where u=u(t,x) for t>0 and x∈Rd with u(0)=δ0, 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 pt(x):=(2πt)−d/2exp⁡{−‖x‖2/(2t)} denote the standard Gaussian heat kernel on Rd and set U(t,x):=u(t,x)/pt(x) for all t>0 and x∈Rd. In this paper, we present an almost sure central limit theorem (ASCLT) and a functional ASCLT for spatial averages of the form ∫[0,N]dU(t,x)dx as N→∞ for fixed t>0 based on the quantitative analysis of f. In particular, when f is given by a Riesz kernel, that is, f(dx)=‖x‖−βdx for some β∈(0,d∧2), we can also obtain the ASCLT.