Stochastic wave equation in a plane driven by spatial stable noise

Volume 3, Issue 3 (2016), pp. 237–248

Larysa Pryhara

^{ }Georgiy Shevchenko^{ }
Pub. online: 8 November 2016
Type: Research Article
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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