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 Hölder 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.

Published at http://dx.doi.org/10.15559/16-VMSTA62 in the Modern Stochastics: Theory and Applications (https://www.i-journals.org/vtxpp/VMSTA) by VTeX (http://www.vtex.lt/)

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