Stochastic Flows of SDEs with Irregular Coefficients and Stochastic Transport Equations

TL;DR

This paper investigates stochastic differential equations with irregular coefficients, establishing the existence and uniqueness of stochastic flows and connecting these results to stochastic transport equations, extending recent work on non-smooth vector fields.

Contribution

It extends the theory of SDEs with irregular coefficients by proving existence and uniqueness of stochastic flows and linking these to stochastic transport equations, including criteria for invariant measures.

Findings

Existence and uniqueness of stochastic flows for irregular SDEs.

Extension of results to SDEs with non-smooth vector fields.

Criteria for invariant measures of the transition semigroup.

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 non-constant 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. Moreover, we also give a criterion for the existence of invariant measures for the associated transition semigroup.