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The hyperbolic Anderson model: moment estimates of the Malliavin derivatives and applications
Balan, Raluca M. Nualart, David Quer-Sardanyons, LluÃs Zheng, Guangqu
Balan, Raluca M.
Nualart, David
Quer-Sardanyons, LluÃs
Zheng, Guangqu
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Abstract
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].
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Date
2022-01-18
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Springer
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Keywords
Hyperbolic Anderson model, Wiener chaos expansion, Malliavin calculus, Second-order Poincaré inequality, Quantitative central limit theorem, Riesz kernel, Dalang’s condition
Citation
Balan, R.M., Nualart, D., Quer-Sardanyons, L. et al. The hyperbolic Anderson model: moment estimates of the Malliavin derivatives and applications. Stoch PDE: Anal Comp 10, 757–827 (2022). https://doi.org/10.1007/s40072-021-00227-5