Hölder regularity for a class of nonlinear stochastic heat equations

We investigate the Hölder continuity of solutions to stochastic partial differential equations of the form $\frac{\mathrm{\partial }u}{\mathrm{\partial }t}=\mathcal{L}u+\sigma (u)\dot{F}$, 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 Lévy process. Under a growth assumption on the characteristic exponent of the Lévy process, we derive sufficient conditions for the solution to be locally Hölder continuous. Moreover, we show that these conditions are equivalent to those derived in related papers by Khoshnevisan-Sanz-Solé (2023) and Sanz-Solé-Sarrá (2000, 20002).