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),\quad x \in \mathbb{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 $H^{d-1}$.