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

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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 ○ ϕ −1t that is the image of a certain absolutely continuous measure µ under random mapping ϕt(·). We prove that the restriction of the Hausdorff measure H d−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 H d−1.

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Authors and Affiliations

  1. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev

    A. Yu. Pilipenko

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  1. A. Yu. Pilipenko
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 12, pp. 1663–1673, December, 2006.

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Pilipenko, A.Y. Transfer of absolute continuity by a flow generated by a stochastic equation with reflection. Ukr Math J 58, 1891–1903 (2006). https://doi.org/10.1007/s11253-006-0174-6

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  • DOI: https://doi.org/10.1007/s11253-006-0174-6

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