Optimal regularity of SPDEs with additive noise

Abstract

The sample-function regularity of the random-field solution to a stochastic partial differential equation (SPDE) depends naturally on the roughness of the external noise, as well as on the properties of the underlying integro-differential operator that is used to define the equation. In this paper, we consider parabolic and hyperbolic SPDEs on $(0,\mathrm{\infty })\times {\mathbb{R}}^d$ of the form

$${\partial }_{t}u=\mathcal{L}u+g(u)+\dot{F}\text{and}{\partial }_t^{2}u=\mathcal{L}u+c+\dot{F},$$

with suitable initial data, forced with a space-time homogeneous Gaussian noise $\dot{F}$ that is white in its time variable and correlated in its space variable, and driven by the generator $\mathcal{L}$ of a genuinely d-dimensional Lévy process X. We find optimal Hölder conditions for the respective random-field solutions to these SPDEs. Our conditions are stated in terms of indices that describe thresholds on the integrability of some functionals of the characteristic exponent of the process X with respect to the spectral measure of the spatial covariance of $\dot{F}$. Those indices are suggested by references [45, 46] on the particular case that $\mathcal{L}$ is the Laplace operator on ${\mathbb{R}}^d$.