The fractional stochastic heat equation on the circle: Time regularity and potential theory

TL;DR

This paper investigates the time regularity and potential theory of solutions to a system of fractional stochastic heat equations driven by fractional Brownian noise on the circle, providing sharp regularity results and bounds on hitting probabilities.

Contribution

It offers new sharp results on the time regularity of solutions and bounds on hitting probabilities for fractional stochastic heat equations on the circle.

Findings

Sharp H"older continuity in time of solutions

Upper and lower bounds on hitting probabilities

Connections to Hausdorff measure and Newtonian capacity

Abstract

We consider a system of $d$ linear stochastic heat equations driven by an additive infinite-dimensional fractional Brownian noise on the unit circle $S^1$. We obtain sharp results on the H\"older continuity in time of the paths of the solution $u=\{u(t, x)\}_{t \in \mathbb{R}_+, x \in S^1}$. We then establish upper and lower bounds on hitting probabilities of $u$, in terms of respectively Hausdorff measure and Newtonian capacity.