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

Khoshnevisan, Davar; Nualart, Eulàlia

doi:http://dx.doi.org/10.3150/08-BEJ133

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

Citation

Khoshnevisan D, Nualart E. Level sets of the stochastic wave equation driven by a symmetric Lévy noise. Bernoulli. 2008;14(4):899-925. DOI: 10.3150/08-BEJ133

Abstract

We consider the solution {u(t, x); t≥0, x∈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.