Harnack Inequalities for SDEs with Multiplicative Noise and Non-regular Drift

TL;DR

This paper establishes Harnack inequalities for stochastic differential equations with multiplicative noise and irregular drift, extending existing results to non-regular drifts and stable processes using heat kernel estimates.

Contribution

It introduces new Harnack inequalities for SDEs with non-regular, time-dependent drift and multiplicative noise, including stable process-driven equations.

Findings

Harnack inequalities for SDEs with non-regular drift are proved.

Results cover cases with LPS-type integrability and H"older continuous drifts.

Harnack inequality for stable processes with drift is established.

Abstract

The log-Harnack inequality and Harnack inequality with powers for semigroups associated to SDEs with non-degenerate diffusion coefficient and non-regular time-dependent drift coefficient are established, based on the recent papers \cite{Flandoli, Zhang11}. We consider two cases in this work: (1) the drift fulfills the LPS-type integrability, and (2) the drift is uniformly H\"older continuous with respect to the spatial variable. Finally, by using explicit heat kernel estimates for the stable process with drift, the Harnack inequality for the stochastic differential equation driven by symmetric stable process is also proved.