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

Davar Khoshnevisan; Eulalia Nualart

doi:10.3150/08-BEJ133

paper

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

arXiv:0709.3165 · doi:10.3150/08-BEJ133

Abstract

We consider the solution ${u (t, x); t \geq 0, x \in R}$ 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.

Published in at http://dx.doi.org/10.3150/08-BEJ133 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)