H\"older regularity for a class of nonlinear stochastic heat equations

TL;DR

This paper establishes conditions under which solutions to certain nonlinear stochastic heat equations exhibit local H"older continuity, linking these conditions to previous related results and considering the effects of Lévy process generators.

Contribution

It provides new sufficient conditions for H"older continuity of solutions to stochastic heat equations driven by Lévy noise, extending and connecting prior findings.

Findings

Derived sufficient conditions for H"older continuity

Established equivalence with conditions in prior literature

Analyzed the impact of Lévy process characteristics on regularity

Abstract

We investigate the H\"older continuity of solutions to stochastic partial differential equations of the form $\frac{\partial u}{\partial t}=\mathcal{L}u+\sigma(u)\dot{F}$, subject to a suitable initial condition. The noise term $\dot{F}$ is white in time, colored in space, and $\mathcal{L}$ is the $\mathcal{L}^{2}$-generator of a L\'evy process. Under a growth assumption on the characteristic exponent of the L\'{e}vy process, we derive sufficient conditions for the solution to be locally H\"older continuous. Moreover, we show that these conditions are equivalent to those derived in related papers by Khoshnevisan-Sanz-Sol\'e (2023) and Sanz-Sol\'e-Sarr\'a (2000, 20002).