Stochastic wave equation in a plane driven by spatial stable noise

TL;DR

This paper studies the stochastic wave equation in a plane driven by spatial stable noise, establishing existence, regularity, and generalized solution properties of the model.

Contribution

It introduces a well-defined solution via Poisson's formula for the stochastic wave equation with stable noise and analyzes its regularity and boundedness properties.

Findings

Solution is well-defined almost surely at each point.

The solution is Hölder continuous in time.

The solution is unbounded near points where the noise coefficient does not vanish.

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\"{o}lder 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.