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

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.}).

arXiv comment51 pages

References in corpus (8)

• A Central Limit Theorem for the stochastic wave equation with fractional noise
• Gaussian fluctuations for the stochastic heat equation with colored noise
• Central limit theorems for stochastic wave equations in dimensions one and two
• Averaging 2d stochastic wave equation
• Averaging Gaussian functionals
• The hyperbolic Anderson model: Moment estimates of the Malliavin derivatives and applications
• Spatial averages for the Parabolic Anderson model driven by rough noise
• Spatial ergodicity of stochastic wave equations in dimensions 1,2 and 3

Cited by in corpus (3)

• The hyperbolic Anderson model: Moment estimates of the Malliavin derivatives and applications
• Hyperbolic Anderson model with Lévy white noise: spatial ergodicity and fluctuation
• Almost sure central limit theorems for parabolic/hyperbolic Anderson models with Gaussian colored noises