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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., Comment: 32Pages, add the existence of invariant measures

We consider a SDE with a smooth multiplicative non-degenerate noise and a possibly unbounded Holder continuous drift term. We prove the existence of a global flow of diffeomorphisms by means of a special transformation of the drift of Ito–Tanaka type. The proof requires non-standard elliptic estimates in Holder spaces. As an application of the stochastic flow, we obtain a Bismut–Elworthy–Li type formula for the first derivatives of the associated diffusion semigroup.

We consider the stochastic flow generated by Stratonovich stochastic differential equations with non-Lipschitz drift coefficients. Based on the author's previous works, we show that if the generalized divergence of the drift is bounded, then the Lebesgue measure on R d is quasi-invariant under the action of the stochastic flow, and the explicit expression of the Radon–Nikodym derivative is also presented. Finally we show in a special case that the unique solution of the corresponding Fokker–Planck equation is given by the density of the stochastic flow.

We study m-dimensional SDE X t = x 0 + ∑ i = 1 ∞ ∫ 0 t σ i ( X s ) d W s i + ∫ 0 t b ( X s ) d s , where { W i } i ⩾ 1 is an infinite sequence of independent standard d-dimensional Brownian motions. The existence and pathwise uniqueness of strong solutions to the SDE was established recently in [Z. Liang, Stochastic differential equations driven by countably many Brownian motions with non-Lipschitzian coefficients, Preprint, 2004]. We will show that the unique strong solution produces a stochastic flow of homeomorphisms if the modulus of continuity of coefficients is less than | x − y | ( log 1 | x − y | ) ϑ , ϑ ∈ [ 0 , 1 ) with ( − 1 ) ϑ = 1 , and the coefficients are compactly supported.

We prove the bicontinuity and homeomorphic property of solutions of stochastic differential equations driven by infinite many Brownian motions and with non-Lipschitz coefficients.