The hyperbolic Anderson model : moment estimates of the Malliavin derivatives and applications
Balan, Raluca M. (University of Ottawa. Department of Mathematics and Statistics)
Nualart, David (University of Kansas. Department of Mathematics)
Quer i Sardanyons, Lluís (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Zheng, Guangqu (The University of Edinburg. School of Mathematics)

Data:	2022
Resum:	In this article, we study the hyperbolic Anderson model driven by a space-time colored Gaussian homogeneous noise with spatial dimension d= 1 , 2. Under mild assumptions, we provide Lp-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 Wiener chaos expansion of the solution. Our first application are 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 novel ingredient to overcome this difficulty is the second-order Gaussian Poincaré inequality coupled with the application of the aforementioned Lp-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 Lp-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].
Ajuts:	Agencia Estatal de Investigación PGC2018-097848-B-I00
Drets:	Aquest document està subjecte a una llicència d'ús Creative Commons. Es permet la reproducció total o parcial, la distribució, la comunicació pública de l'obra i la creació d'obres derivades, fins i tot amb finalitats comercials, sempre i quan es reconegui l'autoria de l'obra original.
Llengua:	Anglès
Document:	Article ; recerca ; Versió publicada
Matèria:	Dalang's condition ; Hyperbolic Anderson model ; Malliavin calculus ; Quantitative central limit theorem ; Riesz kernel ; Second-order Poincaré inequality ; Wiener chaos expansion
Publicat a:	Stochastics and Partial Differential Equations: Analysis and Computations, Vol. 10, Issue 3 (January 2022) , p. 757-827, ISSN 2194-041X

DOI: 10.1007/s40072-021-00227-5
PMID: 36196215

71 p, 954.7 KB

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