# Preprint: Hitting probabilities for systems of non-linear stochastic heat equations in spatial dimension $$k\ge 1$$

## Robert C. Dalang, Davar Khoshnevisan, and Eulalia Nualart

Abstract. We consider a system of d non-linear stochastic heat equations in spatial dimension $$k\ge 1$$, whose solution is an $$\mathbf{R}^d$$-valued random field $$u = \{u(t\,,x),\, (t\,,x)\in\mathbf{R}_+\times\mathbf{R}^k\}.$$ The $$d$$-dimensional driving noise is white in time and with a spatially homogeneous covariance defined as a Riesz kernel with exponent $$\beta$$, where $$0 < \beta < (2\wedge k)$$. The non-linearities appear both as additive drift terms and as multipliers of the noise. Using techniques of Malliavin calculus, we establish an upper bound on the two-point density, with respect to Lebesgue measure, of the $${\bf R}^{2d}$$-valued random vector $$(u(s\,,y)~,~u(t\,,x))$$, that, in particular, quantifies how this density degenerates as $$(s\,, y) \to (t\,, x)$$. From this result, we deduce a lower bound on hitting probabilities of the process $$u$$, in terms of Newtonian capacity. We also establish an upper bound on hitting probabilities of the process in terms of Hausdorff measure. These estimates make it possible to show that points are polar when $$d > \frac{4+2k}{2-\beta}$$ and are not polar when $$d < \frac{4+2k}{2-\beta}$$. In the first case, we also show that the Hausdorff dimension of the range of the process is $$\frac{4+2k}{2-\beta}$$ a.s.

Keywords. Hitting probabilities, systems of non-linear stochastic heat equations, spatially homogeneous Gaussian noise, Malliavin calculus.

AMS Classification (2000) Primary: 60H15, 60J45; Secondary: 60H07, 60G60.

Support. Research supported in part by grants from the Swiss National Science Foundation Fellowship (R.C.D.) and the Unite States' National Science Foundation (D.K.).

Pre/E-Prints. This paper is available in

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

Last Update: March 29, 2012
© 2012 - Robert C. Dalang, Davar Khoshnevisan, and Eulalia Nualart.