Normal approximation of the solution to the stochastic wave equation with L\'evy noise

TL;DR

This paper establishes Gaussian approximation for solutions to the stochastic wave equation driven by Lévy noise, extending prior results to hyperbolic SPDEs and characterizing path properties crucial for convergence analysis.

Contribution

It provides necessary and sufficient conditions for the Gaussian limit of solutions to hyperbolic SPDEs with Lévy noise, including path regularity results.

Findings

Gaussian approximation of solutions under specific variance conditions

Path properties with càdlàg versions for solutions and derivatives

Extension of previous results to hyperbolic stochastic PDEs

Abstract

For a sequence $\dot{L}^{\varepsilon}$ of L\'evy noises with variance $\sigma^2(\varepsilon)$, we prove the Gaussian approximation of the solution $u^{\varepsilon}$ to the stochastic wave equation driven by $\sigma^{-1}(\varepsilon) \dot{L}^{\varepsilon}$ and thus extend the result of C. Chong and T. Delerue [Stoch. Partial Differ. Equ. Anal. Comput. (2019)] to the class of hyperbolic stochastic PDEs. That is, we find a necessary and sufficient condition in terms of $\sigma^2(\varepsilon)$ for $u^{\varepsilon}$ to converge in law to the solution to the same equation with Gaussian noise. Furthermore, $u^{\varepsilon}$ is shown to have a space-time version with a c\`adl\`ag property determined by the wave kernel, and its derivative $\partial_t u^{\varepsilon}$ a c\`adl\`ag version when viewed as a distribution-valued process. These two path properties are essential to our proof of the normal approximation as the limit is characterized by martingale problems that necessitate both random elements. Our results apply to additive as well as to multiplicative noises.