Preprint:
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.
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.
Pre/E-Prints. This paper is available in
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
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Eulalia Nualart
Institut Galilée
Université Paris 13
93430 Villetaneuse, France
nualart@math.univ-paris13.fr
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Updates: February 15, 2007
© 2007 - Robert C. Dalang, Davar Khoshnevisan, and Eulalia Nualart