A Zvonkin's transformation for stochastic differential equations with singular drift and related applications

TL;DR

This paper develops a new Zvonkin-type transformation for stochastic differential equations with singular and Lipschitz drifts, enabling the derivation of Harnack inequalities under minimal regularity conditions.

Contribution

The paper introduces a novel Zvonkin's transformation applicable to SDEs with singular drifts, expanding the analytical tools for such equations.

Findings

Established $L^p$-$L^q$ and Sobolev estimates for PDEs with singular terms

Derived Krylov's estimate for SDEs with irregular drifts

Proved Harnack inequalities for equations with H"older continuous diffusion and singular drift

Abstract

In this paper, by establishing the $L^p$-$L^q$ estimate and Sobolev estimates for parabolic partial differential equations with a singular first order term and a Lipschitz first order term, a new Zvonkin-type transformation is given for stochastic differential equations with singular and Lipschitz drifts. The associated Krylov's estimate is established. As applications, Harnack inequalities are established for stochastic equations with H\"older continuous diffusion coefficient and singular drift term without regularity assumption.