Stochastic wave equation with heavy-tailed noise: Uniqueness of solutions and past light-cone property

In this article, we study the stochastic wave equation in spatial dimensions $d\le 2$ with multiplicative Lévy noise that can have infinite $p$th moments. Using the past light-cone property of the wave equation, we prove the existence and uniqueness of a solution, considering only the $p$-integrability of the Lévy measure $\nu$ for the region corresponding to the small jumps of the noise. For $d=1$, there are no restrictions on $\nu$. For $d=2$, we assume that there exists a value $p\in (0,2)$ for which ${\int }_{\{\left|z\right|\le 1\}}{\left|z\right|}^p\nu \left(dz\right)<+\infty$.