Almost sure CLT for hyperbolic Anderson model with L\'evy colored noise

TL;DR

This paper establishes the Almost Sure Central Limit Theorem for the spatial integral of solutions to the hyperbolic Anderson model driven by Le9vy colored noise, using Malliavin calculus and recent CLT results.

Contribution

It proves the ASCLT for the hyperbolic Anderson model with Le9vy noise, extending previous CLT results to almost sure convergence under specific spatial correlation conditions.

Findings

Proves ASCLT for the model with integrable or Riesz kernel.

Uses Malliavin calculus to estimate derivatives of the solution.

Extends CLT results to almost sure convergence for this stochastic PDE.

Abstract

In this note, we prove the Almost Sure Central Limit Theorem (ASCLT) for the spatial integral of the solution of the hyperbolic Anderson model driven by the L\'evy colored noise introduced in Balan (2015). For this, we use the central limit theorem for the normalized spatial integral, and an estimate for the Malliavin derivative of the solution, both derived in the recent preprint Balan and Stephenson (2026). We assume that the spatial correlation kernel of the noise is either integrable, or it is given by the Riesz kernel.