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

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.