Harnack and Shift Harnack Inequalities for SDEs with Integrable Drifts

TL;DR

This paper develops Harnack and shift Harnack inequalities for certain SDEs with integrable drifts using coupling methods, leading to gradient estimates, heat kernel regularity, and distribution properties.

Contribution

It introduces a coupling by change of measure approach for SDEs with integrable drifts, deriving key inequalities and applications.

Findings

Establishment of Harnack inequalities for the class of SDEs.

Derivation of gradient estimates and heat kernel regularity.

Analysis of the distribution properties of transition probabilities.

Abstract

In this paper, the coupling by change of measure is constructed for a class of SDEs with integrable drift and additive noise, from which the Harnack and shift Harnack inequalities are derived. Finally, as applications, the gradient estimate, the regularity of the heat kernel and the distribution properties of the associated transition probability are also obtained. The important tool is Krylov's estimate.