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In this article we study (possibly degenerate) stochastic differentialequations (SDE) with irregular (or discontiuous) coefficients, and prove thatunder certain conditions on the coefficients, there exists a unique almosteverywhere stochastic (invertible) flow associated with the SDE in the sense ofLebesgue measure. In the case of constant diffusions and BV drifts, we obtainsuch a result by studying the related stochastic transport equation. In thecase of non-constant diffusions and Sobolev drifts, we use a direct method. Inparticular, we extend the recent results on ODEs with non-smooth vector fieldsto SDEs. Moreover, we also give a criterion for the existence of invariantmeasures for the associated transition semigroup.

Stochastic Flows of SDEs with Irregular Coefficients and Stochastic Transport Equations