Ukrainian Mathematical Journal

, Volume 57, Issue 7, pp 1035–1054 | Cite as

A Stochastic Analog of Bogolyubov's Second Theorem

  • B. V. Bondarev
  • E. E. Kovtun


We establish an estimate for the rate at which a solution of an ordinary differential equation subject to the action of an ergodic random process converges to a stationary solution of a deterministic averaged system on time intervals of order \(e^{1/\varepsilon ^\rho }\) for some 0 < ρ < 1.


Differential Equation Ordinary Differential Equation Stationary Solution Random Process Stochastic Analog 
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  1. 1.
    A. D. Venttsel' and M. I. Freidlin, Fluctuations in Dynamical Systems under the Action of Small Random Perturbations [in Russian], Nauka, Moscow (1979).Google Scholar
  2. 2.
    R. L. Stratonovich, Conditional Markov Processes and Their Applications in the Theory of Optimal Control [in Russian], Moscow University, Moscow (1966).Google Scholar
  3. 3.
    R. Z. Khas'minskii, “On random processes determined by differential equations with small parameter,” Teor. Ver. Primen., 11, No.2, 240–259 (1966).MathSciNetGoogle Scholar
  4. 4.
    A. N. Borodin, “A limit theorem for solutions of differential equations with random right-hand side,” Teor. Ver. Primen., 22, No.3, 498–511 (1977).zbMATHMathSciNetGoogle Scholar
  5. 5.
    A. V. Skorokhod, Asymptotic Methods in the Theory of Stochastic Differential Equations [in Russian], Naukova Dumka, Kiev (1987).zbMATHGoogle Scholar
  6. 6.
    N. N. Bogolyubov, On Some Statistical Methods in Mathematical Physics [in Russian], Academy of Sciences of Ukr. SSR, Kiev (1945).Google Scholar
  7. 7.
    N. N. Bogolyubov, “Perturbation theory in nonlinear mechanics,” Sb. Tr. Inst. Stroit. Mekh. Akad. Nauk Ukr. SSR, 14, 9–34 (1950).Google Scholar
  8. 8.
    Yu. A. Mitropol'skii, Method of Averaging in Nonlinear Mechanics [in Russian], Naukova Dumka, Kiev (1971).Google Scholar
  9. 9.
    A. Bensoussan, J.-L. Lions, and G. Papanicolau, Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam (1978).zbMATHGoogle Scholar
  10. 10.
    I. A. Ibragimov and Yu. V. Linnik, Independent and Stationary Connected Random Variables [in Russian], Nauka, Moscow (1965).Google Scholar
  11. 11.
    L. Bers, F. John, and M. Schechter, Partial Differential Equations, Interscience, New York (1964).zbMATHGoogle Scholar
  12. 12.
    O. A. Safonova, “On the asymptotic behavior of integral functionals of diffusion processes with periodic coefficients,” Ukr. Mat. Zh., 44, No.2, 245–252 (1992).zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    I. I. Gikhman and A. V. Skorokhod, Stochastic Differential Equations and Their Applications [in Russian], Naukova Dumka, Kiev (1982).zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • B. V. Bondarev
    • 1
  • E. E. Kovtun
    • 1
  1. 1.Donetsk National UniversityDonetskUkraine

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