Hölder regularity for a class of nonlinear stochastic heat equations
Sudheesh Surendranath
TL;DR
The paper addresses Hölder regularity of the random-field solution to a nonlinear stochastic heat equation driven by a Lévy-generator-based operator $\mathcal{L}$ and multiplicative noise. It employs a Kolmogorov continuity framework and relies on new $L^1$ bounds for transition densities derived from growth conditions on the characteristic exponent $\Psi$ to establish local Hölder continuity. A central result is that the fractal indices satisfy $\mathsf{IND}_{\ell}=\mathsf{IND}_{m}=\mathsf{IND}_{u}$ under the assumptions, and Hölder continuity holds whenever $\int_{\mathbb{R}^n} \frac{\mu(d\xi)}{(1+\mathrm{Re}\Psi(\xi))^{1-\eta}}<\infty$ for some $\eta\in(0,1)$; the corresponding space and time Hölder exponents are determined by these indices and the growth of $\Psi$. The work connects to and shows equivalence with criteria in Khoshnevisan–Sanz-Solé (2023) and Sanz–Solé–Sarrá (2000, 2002), extending Hölder regularity results to nonlinear $\sigma(u)$ and general Lévy generators, while noting that the obtained exponents may not be optimal and that the method does not directly extend to wave equations.
Abstract
We investigate the Hölder continuity of solutions to stochastic partial differential equations of the form $\frac{\partial u}{\partial t}=\mathcal{L}u+σ(u)\dot{F}$, 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).