# Transfer of absolute continuity by a flow generated by a stochastic equation with reflection

**Abstract**

Let $\varphi_t(x),\quad x \in \mathbb{R}_+ $, be a value taken at time $t \geq 0$ 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 $\mu$ for random mapping $\varphi_t(\cdot)$. We prove that the contraction of the Hausdorff measure $H^{d-1}$ onto a support of the singular component is $\sigma$-finite. We also present sufficient conditions which guarantee that the singular component is absolutely continuous with respect to $H^{d-1}$.

**English version** (Springer):
Ukrainian Mathematical Journal **58** (2006), no. 12, pp 1891-1903.

**Citation Example:** *Pilipenko A. Yu.* Transfer of absolute continuity by a flow generated by a stochastic equation with reflection // Ukr. Mat. Zh. - 2006. - **58**, № 12. - pp. 1663–1673.

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