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

Abstract

In this paper, by establishing the localized 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, dimension-free Harnack inequalities are established for stochastic equations with Hölder continuous diffusion coefficient and singular drift term without regularity assumption.