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arXiv · 2109.03875

Quantitative central limit theorems for the parabolic Anderson model driven by colored noises

David Nualart·Panqiu Xia·Guangqu Zheng
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Abstract

In this paper, we study the spatial averages of the solution to the parabolic Anderson model driven by a space-time Gaussian homogeneous noise that is colored in time and space. We establish quantitative central limit theorems (CLT) of this spatial statistics under some mild assumptions, by using the Malliavin-Stein approach. The highlight of this paper is the obtention of rate of convergence in the colored-in-time setting, where one can not use Itô's calculus due to the lack of martingale structure. In particular, modulo highly technical computations, we apply a modified version of second-order Gaussian Poincaré inequality to overcome this lack of martingale structure and our work improves the results by Nualart-Zheng (2020 \emph{Electron. J. Probab.}) and Nualart-Song-Zheng (2021 \emph{ALEA, Lat. Am. J. Probab. Math. Stat.}).

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David Nualart, Panqiu Xia, Guangqu Zheng. 2021-09-08. Quantitative central limit theorems for the parabolic Anderson model driven by colored noises. https://doi.org/10.1214/22-ejp847

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