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

2022/08/24 by Zhang, Wanying, Zhang, Yong, Li, Jingyu
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2208.11323

Abstract

Let \u(t,x)\_t>0,x∈\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∫_[0,N]^d[U(t,x)-1]\rm dx 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^-β\rm dx, β∈(0,2\wedge d), the functional CLT is also based on the index β.

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