Regularity of Ornstein-Uhlenbeck processes driven by a L{\'e}vy white noise

TL;DR

This paper investigates the regularity properties of solutions to linear stochastic evolution equations driven by Lévy white noise, providing conditions for spatial continuity and analyzing the existence of càdlàg modifications, with applications to fractional Laplacian and Burgers equations.

Contribution

It offers new sufficient conditions for spatial continuity of solutions and demonstrates that solutions generally lack càdlàg modifications, extending understanding of Lévy-driven stochastic equations.

Findings

Solutions can be spatially continuous under certain conditions.

Generally, solutions do not admit càdlàg modifications.

Applications include equations with fractional Laplacian and Burgers equations.

Abstract

The paper is concerned with spatial and time regularity of solutions to linear stochastic evolution equation perturbed by L\'evy white noise "obtained by subordination of a Gaussian white noise". Sufficient conditions for spatial continuity are derived. It is also shown that solutions do not have in general \cadlag modifications. General results are applied to equations with fractional Laplacian. Applications to Burgers stochastic equations are considered as well.