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

TL;DR

This paper establishes the optimal H"older continuity and hitting probability bounds for stochastic heat and wave equations driven by rough fractional Brownian sheets, advancing understanding of their regularity and probabilistic properties.

Contribution

It provides the first results on optimal H"older continuity for SHEs and SWEs driven by rough fractional Brownian sheets, including precise regularity exponents and hitting probability bounds.

Findings

Solutions are well-posed and H"older continuous.

Derived sharp H"older exponents for the solutions.

Established bounds on hitting probabilities using capacity and measure.

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.