Optimal H\"older Continuity and Hitting Probabilities for SPDEs with Rough Fractional Noises

Abstract

We investigate the optimal H\"older continuity and hitting probabilities for systems of stochastic heat equations and stochastic wave equations driven by an additive fractional Brownian sheet with temporal index $1/2$ and spatial index $H\le1/2$. Using stochastic calculus for fractional Brownian motion, we prove that these systems are well-posed and the solutions are H\"older continuous. Furthermore, the optimal H\"older exponents are obtained, which is the first result, as far as we knew, on the optimal H\"older continuity of SHEs and SWEs driven by fractional Brownian sheet that is rough in space. Based on this sharp regularity, we obtain lower and upper bounds of hitting probabilities of the solutions in terms of Bessel--Riesz capacity and Hausdorff measure, respectively.