Level sets of the stochastic wave equation driven by a symmetric Lévy noise

Davar Khoshnevisan; Eulalia Nualart

doi:10.3150/08-bej133

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arXiv · 0709.3165

Level sets of the stochastic wave equation driven by a symmetric Lévy noise

Davar Khoshnevisan·Eulalia Nualart

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Abstract

We consider the solution $\{u(t,x);t\geq0,x\in\mathbf{R}\}$ of a system of $d$ linear stochastic wave equations driven by a $d$-dimensional symmetric space-time Lévy noise. We provide a necessary and sufficient condition on the characteristic exponent of the Lévy noise, which describes exactly when the zero set of $u$ is non-void. We also compute the Hausdorff dimension of that zero set when it is non-empty. These results will follow from more general potential-theoretic theorems on the level sets of Lévy sheets.

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Davar Khoshnevisan, Eulalia Nualart. 2008-11-14. Level sets of the stochastic wave equation driven by a symmetric Lévy noise. https://doi.org/10.3150/08-bej133

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