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

Abstract

We consider the solution $\{u(t, x); t≥0,x∈\mathbf R\}$ 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.