Stochastic wave equation in a plane driven by spatial stable noise

• Main Authors: Larysa Pryhara, Georgiy Shevchenko
• Format: Article
• Language: English
• Published: VTeX 2016-11-01
• Series: Modern Stochastics: Theory and Applications
• Subjects: Stochastic partial differential equation; wave equation; LePage series; stable random measure; Hölder continuity; generalized solution
• Online Access: https://vmsta.vtex.vmt/doi/10.15559/16-VMSTA62

The main object of this paper is the planar wave equation \[ \bigg(\frac{{\partial }^{2}}{\partial {t}^{2}}-{a}^{2}\varDelta \bigg)U(x,t)=f(x,t),\hspace{1em}t\ge 0,\hspace{2.5pt}x\in {\mathbb{R}}^{2},\] with random source f. The latter is, in certain sense, a symmetric α-stable spatial white noise multiplied by some regular function σ. 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 σ does not vanish. Finally, we prove that U is a generalized solution to the equation.