Stochastic wave equation with L\'evy white noise

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 $r<1\mathrm{/}4$ if $d=1$, respectively $r<-1$ if $d=2$.