Quasi-invariant stochastic flows of SDEs with non-smooth drifts on compact manifolds

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