Stochastic Lagrangian Flows for SDEs with rough coefficients

Abstract

We prove the existence and uniqueness of Stochastic Lagrangian Flows and almost everywhere Stochastic Flows for non-degenearted SDEs with rough coefficients. As an application of our main result, we show that there exists a unique Stochastic Flow corresponding to each Leray-Hopf solution of 3D Navier-Stokes equation in the DiPerna-Lions sense.