"Preprint by Dalang, Khoshnevisan, and Nualart"

Preprint:
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); t ∈ R₊, 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

Support.

The research of R.C.D. was supported in part by a grant from the Swiss National Foundation for Scientific Research.
The research of D.K. was supported in part by a grant from the United States National Science Foundation.

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Robert C. Dalang
Institut de Mathématiques, Ecole Polytechnique
Fédérale de Lausanne
Station 8, CH-1015
Lausanne, Switzerland
robert.dalang@epfl.ch Davar Khoshnevisan
Department of Mathematics
University of Utah
155 S, 1400 E JWB 233
Salt Lake City, UT 84112-0090, U.S.A.
davar@math.utah.edu Eulalia Nualart
Institut Galilée
Université Paris 13
93430 Villetaneuse, France
nualart@math.univ-paris13.fr

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

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

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