Probability Theory and Related Fields

, Volume 144, Issue 1–2, pp 137–177 | Cite as

Averaging principle for a class of stochastic reaction–diffusion equations

  • Sandra CerraiEmail author
  • Mark Freidlin


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.


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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  1. 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. 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. 3.
    Brin, M., Freidlin, M.I.: On stochastic behavior of perturbed Hamiltonian systems. Ergod. Theory Dyn. Syst. 20, 55–76 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 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)Google Scholar
  5. 5.
    Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (1992)zbMATHGoogle Scholar
  6. 6.
    Freidlin, M.I.: On stable oscillations and equilibrium induced by small noise. J. Stat. Phys. 103, 283–300 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Freidlin, M.I., Wentzell, A.D.: Random Perturbations of Dynamical Systems, 2nd edn. Springer, Heidelberg (1998)zbMATHGoogle Scholar
  8. 8.
    Freidlin, M.I., Wentzell, A.D.: Long-time behavior of weakly coupled oscillators. J. Stat. Phys. 123, 1311–1337 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 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)CrossRefGoogle Scholar
  10. 10.
    Khasminskii, R.Z.: On the principle of averaging the Itô’s stochastic differential equations (Russian). Kibernetika 4, 260–279 (1968)Google Scholar
  11. 11.
    Kifer, Y.: Some recent advances in averaging. In: Modern Dynamical Systems and Applications, pp. 385–403. Cambridge University Press, Cambridge (2004)Google Scholar
  12. 12.
    Kifer, Y.: Diffusion approximation for slow motion in fully coupled averaging. Probab Theory Relat Fields 129, 157–181 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Kifer, Y.: Averaging and climate models. In: Stochastic Climate Models (Chorin, 1999). Progress in Probability, vol. 49, pp. 171–188. Birkhuser, Basel (2001)Google Scholar
  14. 14.
    Kifer, Y.: Stochastic versions of Anosov’s and Neistadt’s theorems on averaging. Stochast. Dyn. 1, 1–21 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Kuksin, S.B., Piatnitski, A.L.: Khasminski–Whitman averaging for randonly perturbed KdV equations. J Math Pures Appl (2008, in press)Google Scholar
  16. 16.
    Maslowski, B., Seidler, J., Vrkoč, I.: An averaging principle for stochastic evolution equations. II. Mathematica Bohemica 116, 191–224 (1991)zbMATHMathSciNetGoogle Scholar
  17. 17.
    Neishtadt, A.: Averaging in multyfrequency systems. Sov. Phys. Doktagy 21, 80–82 (1976)Google Scholar
  18. 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)Google Scholar
  19. 19.
    Seidler, J., Vrkoč, I.: An averaging principle for stochastic evolution equations. Časopis Pěst. Mat. 115, 240–263 (1990)zbMATHGoogle Scholar
  20. 20.
    Veretennikov, A.Y.: On the averaging principle for systems of stochastic differential equation. Math. USSR-Sbornik 69, 271–284 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Volosov, V.M.: Averaging in systems of ordinary differential equations. Russ. Math. Surv. 17, 1–126 (1962)zbMATHCrossRefMathSciNetGoogle Scholar

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© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Dipartimento di Matematica per le DecisioniUniversità di FirenzeFirenzeItaly
  2. 2.Department of MathematicsUniversity of MarylandCollege ParkUSA

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