Malliavin derivatives of hyperbolic Anderson model with applications to its absolute continuity and spatial averages.

In this talk, we present recent work on the hyperbolic Anderson model driven by a space-time colored Gaussian homogeneous noise with spatial dimension one and two. Under mild assumptions, we provide Lp-estimates of the iterated Malliavin derivatives of the solution in terms of the fundamental solution of the wave solution. We present two applications:
(1) We present 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.
(2) 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 Balan, Quer-Sardanyons and Song (2019).
This talk is based on the joint work (arXiv:2101.10957) with R. Balan, D. Nualart and L. Quer-Sardanyons (2021).