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

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).