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Random perturbations of dynamical systems and diffusion processes with conservation laws
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  • Published: 02 January 2004

Random perturbations of dynamical systems and diffusion processes with conservation laws

  • Mark Freidlin1 &
  • Matthias Weber2 

Probability Theory and Related Fields volume 128, pages 441–466 (2004)Cite this article

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  • 36 Citations

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An Erratum to this article was published on 27 September 2006

Abstract.

In this paper we consider random perturbations of dynamical systems and diffusion processes with a first integral. We calculate, under some assumptions, the limiting behavior of the slow component of the perturbed system in an appropriate time scale for a general class of perturbations. The phase space of the slow motion is a graph defined by the first integral. This is a natural generalization of the results concerning random perturbations of Hamiltonian systems. Considering diffusion processes as the unperturbed system allows to study the multidimensional case and leads to a new effect: the limiting slow motion can spend non-zero time at some points of the graph. In particular, such delay at the vertices leads to more general gluing conditions. Our approach allows one to obtain new results on singular perturbations of PDE’s.

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References

  1. Bättig, D., Knörrer, H.: Singularitäten. (German), Lectures in Mathematics, ETH Zürich, Birkhäuser Verlag, Basel, 1991

  2. Dynkin, E.B.: Markov Processes. Springer, Berlin, 1965

  3. Feller, W.: Diffusion processes in one dimension. TAMS 97, 1–31 (1954)

    Google Scholar 

  4. Freidlin, M.I.: Functional Integration and Partial Differential Equation. Princeton University Press, Princeton, 1985

  5. Freidlin, M.I., Weber, M.: Random Perturbations of Nonlinear Oscillators. Ann. Probab. 26(3), 1–43 (1998)

    Google Scholar 

  6. Freidlin, M.I., Weber, M.: A Remark on Random Perturbations of Nonlinear Pendulum. Ann. Appl. Probab. 9(3), 611–628 (1999)

    MATH  Google Scholar 

  7. Freidlin, M.I., Weber, M.: On random perturbations of Hamiltonian systems with many degrees of freedom. Stochastic Process. Appl. 94(2), 199–239 (2001)

    Article  Google Scholar 

  8. Freidlin, M.I., Wentzell, A.D.: Diffusion processes on graphs and the averaging principle. Ann. Probab. 21, 2215–2245 (1993)

    MathSciNet  MATH  Google Scholar 

  9. Freidlin, M.I., Wentzell, A.D.: Random perturbations of Hamiltonian Systems. Memoirs of the AMS 109(523), (1994)

  10. Freidlin, M.I., Wentzell, A.D.: Random perturbations of Dynamical Systems. 2nd Edition, Springer, New York, 1998

  11. Freidlin, M.I., Wentzell, A.D.: Diffusion processes on an open book and the averaging principle. submitted to: Stochastic Process. Appl., 2003

  12. Ito, K., McKean, H.P.: Diffusion Processes and their Sample Paths. Springer, Berlin, Heidelberg, New York, 1965

  13. Khas’minskii, R.Z.: Ergodic properties of recurrent diffusion processes and stabilisation of the solution of the Cauchy problem for parabolic equations (Russian). Teor. veriatnost. i primen. 5, 179–196 (1960)

    Google Scholar 

  14. Khas’minskii, R.Z.: Averaging principle for stochastic differential Ito equations (Russian). Kybernetica 4(3), 260–279 (1968)

    Google Scholar 

  15. Krylov, N.V., Safonov, M.V.: On a problem suggested by A. D. Wentzell. In: The Dynkin Festschrift. Markov Processes and their Applications. M. I. Freidlin, (ed.), Birkhäuser, 1994, 209–220

    Google Scholar 

  16. Sowers, R.B.: Stochastic averaging with a flattened Hamiltonian: a Markov process on a stratified space (a whiskered sphere). Trans. Am. Math. Soc. 354(3), 853–900 (2002)

    Article  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Maryland, College Park, Maryland, 20742, U.S.A

    Mark Freidlin

  2. Department of Information Technology and Mathematics, University of Applied Sciences, 01008, Dresden, Post Box 120701, Germany

    Matthias Weber

Authors
  1. Mark Freidlin
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  2. Matthias Weber
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Additional information

Mathematics Subject Classification (2001): 60H10; 34C29; 35B20

An erratum to this article is available at http://dx.doi.org/10.1007/s00440-006-0027-0.

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Cite this article

Freidlin, M., Weber, M. Random perturbations of dynamical systems and diffusion processes with conservation laws. Probab. Theory Relat. Fields 128, 441–466 (2004). https://doi.org/10.1007/s00440-003-0312-0

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  • Received: 18 February 2003

  • Revised: 30 September 2003

  • Published: 02 January 2004

  • Issue Date: March 2004

  • DOI: https://doi.org/10.1007/s00440-003-0312-0

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Keywords

  • Averaging principle
  • Random Perturbations
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