Stochastic Flows of Sdes with Irregular Coefficients and Stochastic Transport Equations

Abstract. In this article we study (possibly degenerate) stochastic differential equations (SDE) with irregular (or discontiuous) coefficients, and prove that under certain conditions on the coefficients, there exists a unique almost everywhere stochastic (invertible) flow associated with the SDE in the sense of Lebesgue measure. In the case of constant diffusions and BV drifts, we obtain such a result by studying the related stochastic transport equation. In the case of nonconstant diffusions and Sobolev drifts, we use a direct method. In particular, we extend the recent results on ODEs with non-smooth vector fields to SDEs. As an application, we obtain the well posedness of an SDE with singular drift and discontinuous, degenerate diffusion coefficients that appears in the study of polymeric fluids.