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Properties of the Flows Generated by Stochastic Equations with Reflection

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Abstract

We consider the properties of a random set ϕ t (ℝ d+ ), where ϕ t (x) is a solution of a stochastic differential equation in ℝ d+ with normal reflection from the boundary that starts from a point x. We characterize inner and boundary points of the set ϕ t (ℝ d+ ) and prove that the Hausdorff dimension of the boundary ∂ϕ t (ℝ d+ ) does not exceed d − 1.

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REFERENCES

  1. I. I. Gikhman and A. V. Skorokhod, Stochastic Differential Equations and Their Applications [in Russian], Naukova Dumka, Kiev (1982).

    Google Scholar 

  2. A. Yu. Pilipenko, “Flows generated by stochastic equations with reflection,” Random Oper. Stochast. Equat., 12, No.4, 389–396 (2004).

    MathSciNet  Google Scholar 

  3. A. P. Calderon, “On the differentiability of absolutely continuous functions,” Riv. Mat. Univ. Parma, 2, 203–213 (1951).

    MathSciNet  MATH  Google Scholar 

  4. N. Bouleau and F. Hirsch, Dirichlet Forms and Analysis on Wiener Space, de Gruyter, Berlin (1991).

    Google Scholar 

  5. P. L. Lions, J.-L. Menaldi, and A.-S. Sznitman, “Construction de processus de diffusion reflechis par penalisation du domaine,” C. R. Acad. Sci. Math., 292, No.11, 559–562 (1981).

    MathSciNet  Google Scholar 

  6. J.-L. Menaldi, “Stochastic variational inequality for reflected diffusion,” Indiana Univ. Math. J., 2, No.5, 733–744 (1983).

    MathSciNet  Google Scholar 

  7. P. Billingsley, Convergence of Probability Measures, Wiley, New York (1968).

    Google Scholar 

  8. O. Kallenberg, Foundations of Modern Probability, Springer, New York (2002).

    Google Scholar 

  9. H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge University Press, Cambridge (1990).

    Google Scholar 

  10. N. Dunford and J. T. Schwartz, Linear Operators. Part 1: General Theory, Interscience, New York (1958).

    Google Scholar 

  11. M. Cranston and Y. Le Jan, “Noncoalescence for the Skorokhod equation in a convex domain of R 2,” Probab. Theory Relat. Fields, 87, No.2, 241–252 (1990).

    Article  Google Scholar 

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Authors and Affiliations

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

    A. Yu. Pilipenko

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  1. A. Yu. Pilipenko
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Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 8, pp. 1069 – 1078, August, 2005.

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Pilipenko, A.Y. Properties of the Flows Generated by Stochastic Equations with Reflection. Ukr Math J 57, 1262–1274 (2005). https://doi.org/10.1007/s11253-005-0260-1

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  • DOI: https://doi.org/10.1007/s11253-005-0260-1

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