Hitting probabilities for systems of non-linear stochastic heat equations in spatial dimension $k\geq 1$

TL;DR

This paper analyzes hitting probabilities for systems of non-linear stochastic heat equations in various dimensions, providing bounds and conditions for points being polar or not, using Malliavin calculus techniques.

Contribution

It introduces new bounds on the density and hitting probabilities of non-linear stochastic heat systems, linking these to geometric measures and capacity.

Findings

Points are polar when d > (4+2k)/(2-β).

Range Hausdorff dimension is (4+2k)/(2-β) almost surely.

Provides bounds on hitting probabilities using capacity and Hausdorff measures.

Abstract

We consider a system of $d$ non-linear stochastic heat equations in spatial dimension $k \geq 1$, whose solution is an $\R^d$-valued random field $u= \{u(t\,,x),\, (t,x) \in \R_+ \times \R^k\}$. 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 $\R^{2d}$-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 $d < \frac{4+2k}{2-\beta}$. %(if $\frac{4+2k}{2-\beta}$ is an integer, then the case $d=\frac{4+2k}{2-\beta}$ is open). 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.