Hitting with probability one for stochastic heat equations with additive noise

TL;DR

This paper investigates the conditions under which solutions to stochastic heat equations with additive noise almost surely hit certain sets, linking hitting probabilities to the capacity of those sets.

Contribution

It establishes that solutions to the stochastic heat equations with additive noise hit any bounded Borel set with positive $d-6$-dimensional capacity almost surely.

Findings

Solutions hit sets with positive $d-6$-capacity almost surely

Hitting probabilities are characterized by set capacity

Results extend understanding of stochastic heat equation behavior

Abstract

We study the hitting probabilities of the solution to a system of $d$ stochastic heat equations with additive noise subject to Dirichlet boundary conditions. We show that for any bounded Borel set with positive $d-6$-dimensional capacity, the solution visits this set almost surely.