Ukrainian Mathematical Journal

, Volume 52, Issue 9, pp 1457–1469 | Cite as

A Probabilistic Representation for the Solution of One Problem of Mathematical Physics

  • N. I. Portenko
Article

Abstract

We consider a multidimensional Wiener process with a semipermeable membrane located on a given hyperplane. The paths of this process are the solutions of a stochastic differential equation, which can be regarded as a generalization of the well-known Skorokhod equation for a diffusion process in a bounded domain with boundary conditions on the boundary. We randomly change the time in this process by using an additive functional of the local-time type. As a result, we obtain a probabilistic representation for solutions of one problem of mathematical physics.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

REFERENCES

  1. 1.
    E. B. Dynkin, Markov Processes, Vols. I, II, Academic Press–Springer, New York–Berlin (1965).Google Scholar
  2. 2.
    N. I. Portenko, “Diffusion processes with generalized drift coefficient,” Teor. Ver. Primen., 24, 62–77 (1979).Google Scholar
  3. 3.
    N. I. Portenko, “Stochastic differential equations with generalized drift vector,” Teor. Ver. Primen., 24, 332–347 (1979).Google Scholar
  4. 4.
    N. I. Portenko, Generalized Diffusion Processes, Amer. Math. Soc. (1990).Google Scholar
  5. 5.
    I. I. Gikhman and A. V. Skorokhod, The Theory of Stochastic Processes, Vol. 2, Springer, Berlin (1975).Google Scholar
  6. 6.
    B. I. Kopytko and Zh. J. Tsapovska, “Diffusion processes with discontinuous local characteristics of the movement,” Theory Stochast. Process, 4(20), No. 1–2, 139–146 (1998).Google Scholar
  7. 7.
    S. V. Anulova, “Diffusion processes with singular characteristics,” Int. Symp. Stochast. Different. Equat., AbstractsVilnius (1978), pp. 7–11.Google Scholar
  8. 8.
    I. I. Gikhman and A. V. Skorokhod, Stochastic Differential Equations, Springer, Berlin (1972).Google Scholar
  9. 9.
    D. W. Stroock and S. R. S. Varadhan, Multidimensional Diffusion Processes, Springer, Berlin (1979).Google Scholar
  10. 10.
    K. Itô and H. McKean, Diffusion Processes and Their Sample Paths, Academic Press– Springer, New York–Berlin (1965).Google Scholar
  11. 11.
    N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland (1981).Google Scholar
  12. 12.
    J. M. Harrison and L. A. Shepp, “On skew Brownian motion,” Ann. Probab., 9, 309–313 (1981).Google Scholar
  13. 13.
    N. I. Portenko, “Diffusion processes with irregular drift,” Proc School and Sem. Theory of Random Process. (Druskininkai, 1974), Part II, Institute of Physics and Mathematics, Lithuanian Academy of Sciences, Vilnius (1975), pp. 127–146.Google Scholar
  14. 14.
    W. Rosenkrantz, “Limit theorems for solutions to a class of stochastic differential equations,” Indiana Univ. Math. J., 24, 613–625 (1975).Google Scholar
  15. 15.
    J. B. Walsh, “A diffusion with discontinuous local time,” Asterisque, 52–53, 37–45 (1978).Google Scholar
  16. 16.
    N. I. Portenko, “On multidimensional skew Brownian motion and the Feynman–Kac formula,” Theory Stochast. Process, 4(20), No. 1–2, 60–70 (1998).Google Scholar
  17. 17.
    V. G. Papanicolau, “The probabilistic solution of the third boundary-value problem for second-order elliptic equations,” Probab. Theory Related Fields, 87, 27–77 (1990).Google Scholar
  18. 18.
    A. K. Durdiev and N. I. Portenko, “On the number of crossings by a multidimensional Brownian motion of a membrane on a hyperplane with killing,” Teor. Ver. Mat. Statist., No. 56, 67–80 (1997).Google Scholar

Copyright information

© Plenum Publishing Corporation 2000

Authors and Affiliations

  • N. I. Portenko
    • 1
  1. 1.Institute of MathematicsUkrainian Academy of SciencesKiev

Personalised recommendations