Level Sets of the Stochastic Wave Equation Driven by a Symmetric Lévy Noise

D. Khoshnevisan and E. Nualart

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 nonvoid. We also compute the Hausdorff dimension of that zero set, when it is nonempty. These results will follow from more general potential-theoretic theorems on level sets of Lévy sheets.

Keywords. Level sets, stochastic wave equation, Lévy noise, potential theory.

AMS Classification (2000) Primary: 60G60, 60H15; Secondary: 60J45, 60G17.


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Davar Khoshnevisan
Department of Mathematics
University of Utah
155 S, 1400 E JWB 233
Salt Lake City, UT 84112-0090, U.S.A.
Eulalia Nualart
Institut Galilée
Université Paris 13
93430 Villetaneuse, France

Updates: September 20, 2007
© 2007 - Davar Khoshnevisan and Eulalia Nualart