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

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

Abstract. We consider a system of d non-linear stochastic heat equations in spatial dimension 1 driven by d-dimensional space-time white noise. The non-linearities appear both as additive drift terms and as multipliers of the noise. Using techniques of Malliavin calculus, we establish upper and lower bounds on the one-point density of the solution u(t,x), and upper bounds of Gaussian-type on the two-point density of (u(s,y),u(t,x)). In particular, this estimate quantifies how this density degenerates as (s,y) → (t,x). From these results, we deduce upper and lower bounds on hitting probabilities of the process {u(t,x)}t ∈ R+, x ∈ [0,1], in terms of respectively Hausdorff measure and Newtonian capacity. These estimates make it possible to show that points are polar when d ≥7 and are not polar when d ≤5. We also show that the Hausdorff dimension of the range of the process is 6 when d>6, and give analogous results for the processes t \mapsto u(t,x) and x \mapsto u(t,x). Finally, we obtain the values of the Hausdorff dimensions of the level sets of these processes.

Keywords. Hitting probabilities, stochastic heat equation, space-time white noise, Malliavin calculus.

AMS Classification (2000) Primary: 60H15, 60J45; Secondary: 60H07, 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