Abstract
Let ϕt(x), x ∈ ℝ+ be a value taken at time t ≥ 0 by a solution of a stochastic equation with normal reflection from a hyperplane starting at initial time from x. We characterize the absolutely continuous (with respect to Lebesgue measure) component and the singular component of a stochastic measure-valued process µt = µ ○ ϕt−1 that is the image of a certain absolutely continuous measure µ under random mapping ϕt(·). We prove that the restriction of the Hausdorff measure Hd−1 to the support of the singular component is σ-finite and give sufficient conditions guaranteeing that the singular component is absolutely continuous with respect to Hd−1.
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