The hyperbolic Anderson model: Moment estimates of the Malliavin derivatives and applications

TL;DR

This paper develops $L^p$-estimates for Malliavin derivatives of solutions to the hyperbolic Anderson model driven by colored Gaussian noise, enabling new quantitative CLTs and absolute continuity results for the model's law.

Contribution

It introduces novel $L^p$-estimates of Malliavin derivatives using Wiener chaos expansion, leading to quantitative CLTs and absolute continuity results for the hyperbolic Anderson model.

Findings

Quantitative central limit theorems with convergence rates

Absolute continuity of the solution's law established

Simplified approach for the one-dimensional case

Abstract

In this article, we study the hyperbolic Anderson model driven by a space-time \emph{colored} Gaussian homogeneous noise with spatial dimension $d=1,2$. Under mild assumptions, we provide $L^p$-estimates of the iterated Malliavin derivative of the solution in terms of the fundamental solution of the wave solution. To achieve this goal, we rely heavily on the \emph{Wiener chaos expansion} of the solution. Our first application are \emph{quantitative central limit theorems} for spatial averages of the solution to the hyperbolic Anderson model, where the rates of convergence are described by the total variation distance. These quantitative results have been elusive so far due to the temporal correlation of the noise blocking us from using the It\^o calculus. A \emph{novel} ingredient to overcome this difficulty is the \emph{second-order Gaussian Poincar\'e inequality} coupled with the application of the aforementioned $L^p$-estimates of the first two Malliavin derivatives. Besides, we provide the corresponding functional central limit theorems. As a second application, we establish the absolute continuity of the law for the hyperbolic Anderson model. The $L^p$-estimates of Malliavin derivatives are crucial ingredients to verify a local version of Bouleau-Hirsch criterion for absolute continuity. Our approach substantially simplifies the arguments for the one-dimensional case, which has been studied in the recent work by Balan, Quer-Sardanyons and Song (2019).