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Averaging principle for a class of stochastic reaction–diffusion equations
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  • Published: 04 March 2008

Averaging principle for a class of stochastic reaction–diffusion equations

  • Sandra Cerrai1 &
  • Mark Freidlin2 

Probability Theory and Related Fields volume 144, pages 137–177 (2009)Cite this article

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Abstract

We consider the averaging principle for stochastic reaction–diffusion equations. Under some assumptions providing existence of a unique invariant measure of the fast motion with the frozen slow component, we calculate limiting slow motion. The study of solvability of Kolmogorov equations in Hilbert spaces and the analysis of regularity properties of solutions, allow to generalize the classical approach to finite-dimensional problems of this type in the case of SPDE’s.

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References

  1. Arnold, V.I., Kozlov, V.V., Neishtadt, A.I.: Mathematical aspects of classical and celestial mechanics. Dynamical systems. III. In: Encyclopaedia of Mathematical Sciences, 3rd edn. Springer, Berlin (2006)

    Google Scholar 

  2. Bogoliubov, N.N., Mitropolsky, Y.A.: Asymptotic Methods in the Theory of Non-linear Oscillations. Gordon and Breach Science Publishers, New York (1961)

    Google Scholar 

  3. Brin, M., Freidlin, M.I.: On stochastic behavior of perturbed Hamiltonian systems. Ergod. Theory Dyn. Syst. 20, 55–76 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  4. Cerrai, S.: Second order PDE’s in finite and infinite dimension. A probabilistic approach. In: Lecture Notes in Mathematics, vol. 1762, x+330 pp. Springer, Heidelberg (2001)

  5. Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (1992)

    MATH  Google Scholar 

  6. Freidlin, M.I.: On stable oscillations and equilibrium induced by small noise. J. Stat. Phys. 103, 283–300 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  7. Freidlin, M.I., Wentzell, A.D.: Random Perturbations of Dynamical Systems, 2nd edn. Springer, Heidelberg (1998)

    MATH  Google Scholar 

  8. Freidlin, M.I., Wentzell, A.D.: Long-time behavior of weakly coupled oscillators. J. Stat. Phys. 123, 1311–1337 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  9. Gyöngy, I., Krylov, N.V.: Existence of strong solutions for Itô’s stochastic equations via approximations. Probab. Theory Relat. Fields 103, 143–158 (1996)

    Article  Google Scholar 

  10. Khasminskii, R.Z.: On the principle of averaging the Itô’s stochastic differential equations (Russian). Kibernetika 4, 260–279 (1968)

    Google Scholar 

  11. Kifer, Y.: Some recent advances in averaging. In: Modern Dynamical Systems and Applications, pp. 385–403. Cambridge University Press, Cambridge (2004)

  12. Kifer, Y.: Diffusion approximation for slow motion in fully coupled averaging. Probab Theory Relat Fields 129, 157–181 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  13. Kifer, Y.: Averaging and climate models. In: Stochastic Climate Models (Chorin, 1999). Progress in Probability, vol. 49, pp. 171–188. Birkhuser, Basel (2001)

  14. Kifer, Y.: Stochastic versions of Anosov’s and Neistadt’s theorems on averaging. Stochast. Dyn. 1, 1–21 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  15. Kuksin, S.B., Piatnitski, A.L.: Khasminski–Whitman averaging for randonly perturbed KdV equations. J Math Pures Appl (2008, in press)

  16. Maslowski, B., Seidler, J., Vrkoč, I.: An averaging principle for stochastic evolution equations. II. Mathematica Bohemica 116, 191–224 (1991)

    MATH  MathSciNet  Google Scholar 

  17. Neishtadt, A.: Averaging in multyfrequency systems. Sov. Phys. Doktagy 21, 80–82 (1976)

    Google Scholar 

  18. Papanicolaou, G.C., Stroock, D., Varadhan, S.R.S.: Martingale approach to some limit theorems. In: Papers from the Duke Turbolence Conference (Duke Univ. Duhram, N.C. 1976), Paper 6, ii+120 pp. Duke Univ. Math. Ser., vol. III. Duke Univ. Duhram, N.C. (1977)

  19. Seidler, J., Vrkoč, I.: An averaging principle for stochastic evolution equations. Časopis Pěst. Mat. 115, 240–263 (1990)

    MATH  Google Scholar 

  20. Veretennikov, A.Y.: On the averaging principle for systems of stochastic differential equation. Math. USSR-Sbornik 69, 271–284 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  21. Volosov, V.M.: Averaging in systems of ordinary differential equations. Russ. Math. Surv. 17, 1–126 (1962)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Matematica per le Decisioni, Università di Firenze, via Cesare Lombroso 6/17, 50134, Firenze, Italy

    Sandra Cerrai

  2. Department of Mathematics, University of Maryland, College Park, MD, USA

    Mark Freidlin

Authors
  1. Sandra Cerrai
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  2. Mark Freidlin
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Correspondence to Sandra Cerrai.

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Cerrai, S., Freidlin, M. Averaging principle for a class of stochastic reaction–diffusion equations. Probab. Theory Relat. Fields 144, 137–177 (2009). https://doi.org/10.1007/s00440-008-0144-z

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  • Received: 17 November 2006

  • Revised: 24 January 2008

  • Published: 04 March 2008

  • Issue Date: May 2009

  • DOI: https://doi.org/10.1007/s00440-008-0144-z

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Keywords

  • Stochastic reaction–diffusion equations
  • Invariant measures and ergodicity
  • Averaging principle
  • Kolmogorov equations in Hilbert spaces

Mathematical Subject Classification (2000).

  • 60F99
  • 60H15
  • 70K65
  • 70K70
  • 37A25
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