Journal of Theoretical Probability

, Volume 14, Issue 2, pp 485–494 | Cite as

Weak Convergence of a Planar Random Evolution to the Wiener Process

  • Alexander D. Kolesnik


The weak convergence of the distributions of a symmetrical random evolution in a plane controlled by a continuous-time homogeneous Markov chain with n, n≥3, states to the distribution of a two-dimensional Brownian motion, as the intensity of transitions tends to infinity, is proved.

weak convergence random evolution random motions 


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  1. 1.
    Griego R., and Hersh R. (1971). Theory of random evolutions with applications to partial differential equations. Trans. Amer. Math. Soc. 156, 405–418.Google Scholar
  2. 2.
    Hersh R. (1974). Random evolutions: a survey of results and problems. Rocky Mount. J. Math. 4, 443–496.Google Scholar
  3. 3.
    Hersh R., and Papanicolaou G. (1972). Non-commuting random evolutions and an operator-valued Feynman–Kac formula. Comm. Pure Appl. Math. 25, 337–367.Google Scholar
  4. 4.
    Hersh R., and Pinsky M. (1972). Random evolutions are asymptotically Gaussian. Comm. Pure Appl. Math. 25, 33–44.Google Scholar
  5. 5.
    Hille E., and Philips R. S. (1957). Functional Analysis and Semigroups, Providence, RI.Google Scholar
  6. 6.
    Kolesnik A. D., and Turbin A. F. (1998). The equation of symmetric Markovian random evolution in a plane. Stoc. Proc. Appl. 75, 67–87.Google Scholar
  7. 7.
    Korolyuk V. S., and Swishchuk A. V. (1994). Semi-Markov Random Evolutions, Kluwer Publ. House, Amsterdam.Google Scholar
  8. 8.
    Kurtz T. (1973). A limit theorem for perturbed operator semigroups with applications to random evolutions. J. Func. Anal. 12, 55–67.Google Scholar
  9. 9.
    Pinsky M. (1968). Differential equations with a small parameter and the central limit theorem for functions defined on a finite Markov chain. Z. Wahrscheinlichkeitstheorie Verw. Geb. 9, 101–111.Google Scholar
  10. 10.
    Pinsky M. (1991). Lectures on Random Evolution, World Scientific.Google Scholar

Copyright information

© Plenum Publishing Corporation 2001

Authors and Affiliations

  • Alexander D. Kolesnik
    • 1
  1. 1.Institute of MathematicsKishinevMoldova

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