Hitting Probabilities for Systems of Non-Linear Stochastic Heat Equations with Additive Noise

TL;DR

This paper investigates hitting probabilities and the geometric properties of solutions to coupled non-linear stochastic heat equations driven by additive noise, providing bounds and dimension results.

Contribution

It establishes bounds on hitting probabilities and determines the Hausdorff dimensions of level sets for the solutions, introducing an anisotropic Kolmogorov continuity theorem.

Findings

Bounds on hitting probabilities in terms of Hausdorff measure and capacity

Hausdorff dimensions of level sets and projections determined

Anisotropic Kolmogorov continuity theorem developed

Abstract

We consider a system of $d$ coupled non-linear stochastic heat equations in spatial dimension 1 driven by $d$-dimensional additive space-time white noise. We establish upper and lower bounds on hitting probabilities of the solution $\{u(t, x)\}_{t \in \mathbb{R}_+, x \in [0, 1]}$, in terms of respectively Hausdorff measure and Newtonian capacity. We also obtain the Hausdorff dimensions of level sets and their projections. A result of independent interest is an anisotropic form of the Kolmogorov continuity theorem.