Transfer of absolute continuity by a flow generated by a stochastic equation with reflection - Pilipenko, A. Yu. (Ukrains’kyi Matematychnyi Zhurnal, 2006)
Let φ_t(x),x R_+ , be a value taken at time t ≥ 0 by a solution of stochastic equation with normal reflection from the hyperplane starting at initial time from x. We characterize an absolutely continuous (with respect to the Lebesgue measure) component and a singular component of the stochastic measure-valued process µ_t = µ ○ ϕ_t^−1, which is an image of some absolutely continuous measure μ for random mapping φ_t(·). We prove that the contraction of the Hausdorff measure H^d-1 onto a support of the singular component is σ-finite. We also present sufficient conditions which guarantee that the singular component is absolutely continuous with respect to H^d-1.