Stochastic wave equation in a plane driven by spatial stable noise
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 m...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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VTeX
2016-11-01
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| Series: | Modern Stochastics: Theory and Applications |
| Subjects: | |
| Online Access: | https://vmsta.vtex.vmt/doi/10.15559/16-VMSTA62 |
| Summary: | 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. |
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| ISSN: | 2351-6046 2351-6054 |