Stochastic flows with reflection

Flows generated by SDEs in Euclidean space is a well-studied topic nowadays. It is well known for example (cf. [1] and ref. therein) that if the coefficients of SDE are Liphitzian then the SDE generates a flow of homeomorphisms, if coefficients are of the class C then SDE generates C-flow of diffeomorphisms, equations for derivatives are obtained by formal differentiation of the SDE etc. Note that the similar questions for SDEs with reflection is much harder to answer. Even the problems about coalescence of two reflecting Brownian motions [2, 3, 4, 5] or differentiability of the Brownian reflecting flow (σ(x) = const) [6, 7] need accurate and non-trivial considerations. The article below was published in Reports of Ukrainian Nat.Acad. of Sci. [8] (2005). Only some new references or minor remarks are added. Assume that functions ak : R d + → R d satisfy the Lipschitz condition. Here