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

TL;DR

This paper extends the DiPerna-Lions theory to stochastic differential equations with non-smooth drifts on compact Riemannian manifolds, proving the existence and uniqueness of quasi-invariant stochastic flows.

Contribution

It establishes the existence and uniqueness of stochastic flows for SDEs with Sobolev drifts on Riemannian manifolds, generalizing classical results to a geometric stochastic setting.

Findings

Existence of unique stochastic flows almost everywhere.

Flows are quasi-invariant with respect to the Riemannian measure.

Extension of DiPerna-Lions theory to 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.