Optimal regularity of SPDEs with additive noise

TL;DR

This paper establishes optimal regularity conditions for solutions to certain SPDEs driven by Gaussian noise, linking solution smoothness to the properties of the underlying Lévy process and noise covariance.

Contribution

It provides the first precise regularity thresholds for solutions of parabolic and hyperbolic SPDEs with additive noise driven by Lévy processes, generalizing previous results for the Laplacian.

Findings

Derived optimal Hölder regularity conditions for SPDE solutions.

Linked regularity thresholds to the characteristic exponent of the Lévy process.

Extended known results from the Laplacian case to more general operators.

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 $0,\infty)\times\mathbb{R}^d$ of the form $\partial_t u = L u + g(u) + \dot{F} \qquad\text{and}\qquad \partial^2_t u = 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 $L$ of a genuinely $d$-dimensional L\'evy process $X$. We find optimal H\"older 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 $L$ is the Laplace operator on $\mathbb{R}^d$.