Schauder estimates for stochastic transport-diffusion equations with L\'{e}vy processes

TL;DR

This paper establishes Schauder estimates for stochastic transport-diffusion equations driven by Lévy processes with Hölder continuous coefficients, demonstrating optimal regularity results in specific cases.

Contribution

It provides new Schauder estimates for solutions of stochastic transport-diffusion equations with Lévy noise, including optimal regularity results when the transport term is absent and p=2.

Findings

Derived Schauder estimates for mild solutions

Proved optimal Hölder regularity in specific cases

Extended regularity theory to Lévy-driven stochastic PDEs

Abstract

We consider a transport-diffusion equation with L\'{e}vy noises and H\"{o}lder continuous coefficients. By using the heat kernel estimates, we derive the Schauder estimates for the mild solutions. Moreover, when the transport term vanishes and $p=2$, we show that the H\"{o}lder index in space variable is optimal.