Preprint:
Hitting Probabilities for Systems of NonLinear
Stochastic Heat Equations with Multiplicative Noise
R. C. Dalang, D. Khoshnevisan, and E. Nualart
Abstract.
We consider a system of d nonlinear stochastic heat equations in spatial
dimension 1 driven by ddimensional spacetime white noise. The nonlinearities 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 onepoint density of the
solution u(t,x), and upper bounds of Gaussiantype on the twopoint 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, spacetime 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/EPrints. This paper is available in
Robert C. Dalang
Institut de Mathématiques, Ecole Polytechnique
Fédérale de Lausanne
Station 8, CH1015
Lausanne, Switzerland
robert.dalang@epfl.ch

Davar Khoshnevisan
Department of Mathematics
University of Utah
155 S, 1400 E JWB 233
Salt Lake City, UT 841120090, U.S.A.
davar@math.utah.edu

Eulalia Nualart
Institut Galilée
Université Paris 13
93430 Villetaneuse, France
nualart@math.univparis13.fr

Updates: February 15, 2007
© 2007  Robert C. Dalang, Davar Khoshnevisan, and Eulalia Nualart