SPDEs with rough noise in space: H\"older continuity of the solution

TL;DR

This paper proves that solutions to certain stochastic wave and heat equations with rough spatial noise exhibit specific H"older continuity properties in both space and time, extending understanding of their regularity.

Contribution

The authors demonstrate spatial and temporal H"older continuity of solutions to SPDEs driven by fractional-like noise, building on previous existence and uniqueness results.

Findings

Solutions are H"older continuous in space with order less than H

Solutions are H"older continuous in time with order less than H for wave, H/2 for heat

Regularity results apply to equations with Gaussian noise rough in space

Abstract

We consider the stochastic wave and heat equations with affine multiplicative Gaussian noise which is white in time and behaves in space like the fractional Brownian motion with index $H \in (\frac14,\frac12)$. The existence and uniqueness of the solution to these equations has been proved recently by the authors. In the present note we show that these solutions have modifications which are H\"older continuous in space of order smaller than $H$, and H\"older continuous in time of order smaller than $\gamma$, where $\gamma=H$ for the wave equation and $\gamma=H/2$ for the heat equation.