Hitting Probabilities for Systems of Non-Linear Stochastic Heat Equations with Additive Noise

R. C. Dalang, D. Khoshnevisan, and E. Nualart

Abstract. We consider a system of d coupled non-linear stochastic heat equations in spatial dimension 1 driven by d-dimensional additive space-time white noise. We establish upper and lower bounds on hitting probabilities of the solution {u(t,x); tR+, x ∈[0,1]}, in terms of respectively Hausdorff measure and Newtonian capacity. We also obtain the Hausdorff dimensions of level sets and their projections. A result of independent interest is an anisotropic form of the Kolmogorov continuity theorem.

Keywords. Hitting probabilities, systems of non-linear stochastic heat equations, space-time white noise, capacity, Hausdorff measure, anisotropic Kolmogorov continuity theorem

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


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Robert C. Dalang
Institut de Mathématiques, Ecole Polytechnique
Fédérale de Lausanne
Station 8, CH-1015
Lausanne, Switzerland
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: February 15, 2007
© 2007 - Robert C. Dalang, Davar Khoshnevisan, and Eulalia Nualart