Stochastic wave equation in a plane driven by spatial stable noise

Abstract

The main object of this paper is the planar wave equation\[\bigg(\frac{\partial^{2}}{\partial t^2}- a^2 \varDelta\bigg)U(x,t)=f(x,t), \quad t\ge0,~x\in\mathbb{R}^2,\]with random source $f$. The latter is, in certain sense, a symmetric $\alpha$-stable spatial white noise multiplied by some regular function $\sigma$. We define a candidate solution $U$ to the equation via Poisson's formula and prove that the corresponding expression is well defined at each point almost surely, although the exceptional set may depend on the particular point $(x,t)$. We further show that $U$ is H\"older continuous in time but with probability 1 is unbounded in any neighborhood of each point where $\sigma$ does not vanish. Finally, we prove that $U$ is a generalized solution to the equation.