Stochastic wave equation with L\'evy white noise

TL;DR

This paper investigates the existence and regularity of solutions to the stochastic wave equation driven by Le9vy white noise with potentially infinite variance in one and two spatial dimensions.

Contribution

It establishes existence results and regularity properties for solutions to the stochastic wave equation with Le9vy noise, including cases with infinite variance.

Findings

Existence of solutions under general Le9vy measure conditions

Solutions have ce0dle0g modifications in fractional Sobolev spaces

Regularity depends on spatial dimension, with different Sobolev order bounds

Abstract

In this article, we study the stochastic wave equation on the entire space $\mathbb{R}^d$, driven by a space-time L\'evy white noise with possibly infinite variance (such as the $\alpha$-stable L\'evy noise). In this equation, the noise is multiplied by a Lipschitz function $\sigma(u)$ of the solution. We assume that the spatial dimension is $d=1$ or $d=2$. Under general conditions on the L\'evy measure of the noise, we prove the existence of the solution, and we show that, as a function-valued process, the solution has a c\`adl\`ag modification in the local fractional Sobolev space of order $r<1/4$ if $d=1$, respectively $r<-1$ if $d=2$.