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

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 Itô calculus. A \emph{novel} ingredient to overcome this difficulty is the \emph{second-order Gaussian Poincaré 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).

Version 2: updated references, fixed typos. Page numbers are different from published version

References in corpus (9)

• Central limit theorems for sequences of multiple stochastic integrals
• Central limit theorems for stochastic wave equations in dimensions one and two
• Spatial ergodicity for SPDEs via a Poincaré-type inequality
• Spatial averages for the Parabolic Anderson model driven by rough noise
• Quantitative central limit theorems for the parabolic Anderson model driven by colored noises
• Central limit theorems for spatial averages of the stochastic heat equation via Malliavin-Stein's method
• Central limit theorems for parabolic stochastic partial differential equations
• Gaussian fluctuation for spatial average of parabolic Anderson model with Neumann/Dirichlet/periodic boundary conditions
• Limit theorems for time-dependent averages of nonlinear stochastic heat equations

Cited by in corpus (4)

• Quantitative central limit theorems for the parabolic Anderson model driven by colored noises
• Hyperbolic Anderson model with Lévy white noise: spatial ergodicity and fluctuation
• Almost sure central limit theorem for the hyperbolic Anderson model with Lévy white noise
• Evolving disorder and chaos enhances the wave speed of elastic waves