Advertisement

Ukrainian Mathematical Journal

, Volume 57, Issue 8, pp 1262–1274 | Cite as

Properties of the Flows Generated by Stochastic Equations with Reflection

  • A. Yu. Pilipenko
Article

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.

Keywords

Differential Equation Boundary Point Stochastic Differential Equation Hausdorff Dimension Stochastic Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

REFERENCES

  1. 1.
    I. I. Gikhman and A. V. Skorokhod, Stochastic Differential Equations and Their Applications [in Russian], Naukova Dumka, Kiev (1982).Google Scholar
  2. 2.
    A. Yu. Pilipenko, “Flows generated by stochastic equations with reflection,” Random Oper. Stochast. Equat., 12, No.4, 389–396 (2004).MathSciNetGoogle Scholar
  3. 3.
    A. P. Calderon, “On the differentiability of absolutely continuous functions,” Riv. Mat. Univ. Parma, 2, 203–213 (1951).MathSciNetzbMATHGoogle Scholar
  4. 4.
    N. Bouleau and F. Hirsch, Dirichlet Forms and Analysis on Wiener Space, de Gruyter, Berlin (1991).Google Scholar
  5. 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).MathSciNetGoogle Scholar
  6. 6.
    J.-L. Menaldi, “Stochastic variational inequality for reflected diffusion,” Indiana Univ. Math. J., 2, No.5, 733–744 (1983).MathSciNetGoogle Scholar
  7. 7.
    P. Billingsley, Convergence of Probability Measures, Wiley, New York (1968).Google Scholar
  8. 8.
    O. Kallenberg, Foundations of Modern Probability, Springer, New York (2002).Google Scholar
  9. 9.
    H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge University Press, Cambridge (1990).Google Scholar
  10. 10.
    N. Dunford and J. T. Schwartz, Linear Operators. Part 1: General Theory, Interscience, New York (1958).Google Scholar
  11. 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).CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • A. Yu. Pilipenko
    • 1
  1. 1.Institute of MathematicsUkrainian Academy of SciencesKievUkraine

Personalised recommendations