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-timecoloredGaussian homogeneous noise with spatial dimension$$d=1,2$$$d=1,2$. Under mild assumptions, we provide$$L^p$$${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 theWiener chaos expansionof the solution. Our first application arequantitative central limit theoremsfor 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. Anovelingredient to overcome this difficulty is thesecond-order Gaussian Poincaré inequalitycoupled with the application of the aforementioned$$L^p$$${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$$L^p$$${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 [2].

Authors:
; ; ;
Publication Date:
NSF-PAR ID:
10372299
Journal Name:
Stochastics and Partial Differential Equations: Analysis and Computations
Volume:
10
Issue:
3
Page Range or eLocation-ID:
p. 757-827
ISSN:
2194-0401
Publisher:
National Science Foundation
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