Stochastic wave equation in a plane driven by spatial stable noise
Volume 3, Issue 3 (2016), pp. 237–248
Pub. online: 8 November 2016
Type: Research Article
Open Access
Open Access
Received
14 October 2016
14 October 2016
Revised
20 October 2016
20 October 2016
Accepted
20 October 2016
20 October 2016
Published
8 November 2016
8 November 2016
Abstract
The main object of this paper is the planar wave equation
\[ \bigg(\frac{{\partial }^{2}}{\partial {t}^{2}}-{a}^{2}\varDelta \bigg)U(x,t)=f(x,t),\hspace{1em}t\ge 0,\hspace{2.5pt}x\in {\mathbb{R}}^{2},\]
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.References
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