Hitting probabilities for systems of non-linear stochastic heat equations with multiplicative noise

TL;DR

This paper analyzes the hitting probabilities and fractal properties of solutions to a system of non-linear stochastic heat equations with multiplicative noise, using Malliavin calculus to derive bounds and dimensional results.

Contribution

It provides new bounds on the densities and hitting probabilities of the solutions, and determines the Hausdorff dimensions of the process's range and level sets, extending understanding of non-linear SPDEs.

Findings

Points are polar when d > 6.

Hausdorff dimension of the range is 6 when d > 6.

Derived bounds on densities and hitting probabilities.

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) converges to (t,x). From these results, we deduce upper and lower bounds on hitting probabilities of the process {u(t,x)}_{t \in \mathbb{R}_+, x \in [0,1]}, in terms of respectively Hausdorff measure and Newtonian capacity. These estimates make it possible to show that points are polar when d >6 and are not polar when d<6. 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.