Quasi Invariant Stochastic Flows of SDEs with Non-smooth Drifts on Riemannian Manifolds$^*$

Abstract: In this article we prove that stochastic differential equation (SDE) with Sobolev drift on compact Riemannian manifold admits a unique $\nu$-almost everywhere stochastic invertible flow, where $\nu$ is the Riemannian measure, which is quasi-invariant with respect to $\nu$. In particular, we extend the well known DiPerna-Lions flows of ODEs to SDEs on Riemannian manifold.