Stochastic Homeomorphism Flows of SDEs with Singular Drifts and Sobolev Diffusion Coefficients

TL;DR

This paper establishes the stochastic homeomorphism flow and strong Feller properties for SDEs with singular, time-dependent drifts and Sobolev diffusion coefficients, extending prior results to non-constant diffusions.

Contribution

It extends Krylov and R"ockner's results to SDEs with non-constant diffusion coefficients, proving key properties under less restrictive conditions.

Findings

Proves stochastic homeomorphism flow property for SDEs with singular drifts.

Establishes strong Feller property for these SDEs.

Demonstrates local well-posedness under local assumptions.

Abstract

In this paper we prove the stochastic homeomorphism flow property and the strong Feller property for stochastic differential equations with sigular time dependent drifts and Sobolev diffusion coefficients. Moreover, the local well posedness under local assumptions are also obtained. In particular, we extend Krylov and R\"ockner's results in \cite{Kr-Ro} to the case of non-constant diffusion coefficients.