Description:
<jats:title>Abstract</jats:title><jats:p>In this article, we study the hyperbolic Anderson model driven by a space-time <jats:italic>colored</jats:italic> Gaussian homogeneous noise with spatial dimension <jats:inline-formula><jats:alternatives><jats:tex-math>$$d=1,2$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mi>d</mml:mi>
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<mml:mn>2</mml:mn>
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</mml:math></jats:alternatives></jats:inline-formula>. Under mild assumptions, we provide <jats:inline-formula><jats:alternatives><jats:tex-math>$$L^p$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
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<mml:mi>L</mml:mi>
<mml:mi>p</mml:mi>
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</mml:math></jats:alternatives></jats:inline-formula>-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 <jats:italic>Wiener chaos expansion</jats:italic> of the solution. Our first application are <jats:italic>quantitative central limit theorems</jats:italic> 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 <jats:italic>novel</jats:italic> ingredient to overcome this difficulty is the <jats:italic>second-order Gaussian Poincaré inequality</jats:italic> coupled with the application of the aforementioned <jats:inline-formula><jats:alternatives><jats:tex-math>$$L^p$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:msup>
<mml:mi>L</mml:mi>
<mml:mi>p</mml:mi>
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</mml:math></jats:alternatives></jats:inline-formula>-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 <jats:inline-formula><jats:alternatives><jats:tex-math>$$L^p$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:msup>
<mml:mi>L</mml:mi>
<mml:mi>p</mml:mi>
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</mml:math></jats:alternatives></jats:inline-formula>-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].</jats:p>