Level sets of the stochastic wave equation driven by a symmetric L\'{e}vy noise

TL;DR

This paper investigates the conditions under which the solution to a stochastic wave equation driven by symmetric Lévy noise has a non-empty zero set and determines its Hausdorff dimension, extending potential-theoretic results.

Contribution

It provides a necessary and sufficient condition on the Lévy noise's characteristic exponent for the zero set to be non-empty and computes its Hausdorff dimension, advancing understanding of Lévy-driven wave equations.

Findings

Characterization of when the zero set is non-empty based on Lévy noise

Explicit computation of the Hausdorff dimension of the zero set

Extension of potential-theoretic theorems to Lévy sheets

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 L\'{e}vy noise. We provide a necessary and sufficient condition on the characteristic exponent of the L\'{e}vy 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 L\'{e}vy sheets.