Communications in Mathematical Physics

, Volume 251, Issue 3, pp 515–555 | Cite as

Multiscale Expansion of Invariant Measures for SPDEs

Article

Abstract

We derive the first two terms in an ɛ-expansion for the invariant measure of a class of semilinear parabolic SPDEs near a change of stability, when the noise strength and the linear instability are of comparable order ɛ2. This result gives insight into the stochastic bifurcation and allows to rigorously approximate correlation functions. The error between the approximate and the true invariant measure is bounded in both the Wasserstein and the total variation distance.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Arnold, L.: Random Dynamical Systems. Springer Monographs in Mathematics. Berlin:Springer-Verlag, 1998Google Scholar
  2. 2.
    Bony, J.-M., Chemin, J.-Y.: Espaces fonctionnels associés au calcul de Weyl-Hörmander. Bull. Soc. Math. France 122(1), 77–118 (1994)MATHGoogle Scholar
  3. 3.
    Berglund, N., Gentz, B.: Geometric singular perturbation theory for stochastic differential equations. J. Differ. Eqs. 191(1), 1–54 (2003)CrossRefMATHGoogle Scholar
  4. 4.
    Blömker, D.: Amplitude equations for locally cubic non-autonomous nonlinearities. SIAM J. Appl. Dyn. Syst. 3(3), 464–486 (2003)CrossRefGoogle Scholar
  5. 5.
    Blömker, D.: Approximation of the stochastic Rayleigh-Bénard problem near the onset of instability and related problems, 2003. PreprintGoogle Scholar
  6. 6.
    Blömker, D., Maier-Paape, S., Schneider, G.: The stochastic Landau equation as an amplitude equation. Discrete and Continuous Dynamical Systems, Series B 1(4), 527–541 (2001)Google Scholar
  7. 7.
    Caraballo, T., Crauel, H., Langa, J.A., Robinson, J.C.: Stabilization by additive noise. In preparationGoogle Scholar
  8. 8.
    Crauel, H., Flandoli, F.: Additive noise destroys a pitchfork bifurcation. J. Dynam. Differ. Eqs. 10(2), 259–274 (1998)CrossRefMATHGoogle Scholar
  9. 9.
    Crauel, H., Imkeller, P., Steinkamp, M.: Bifurcations of one-dimensional stochastic differential equations. In: Stochastic dynamics (Bremen, 1997), 145–154. New York:Springer, 1999, pp. 145–154Google Scholar
  10. 10.
    Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Cambridge: Cambridge University Press, 1992Google Scholar
  11. 11.
    Da Prato, G., Zabczyk, J.: Ergodicity for Infinite Dimensional Systems, Vol. 229 of London Mathematical Society Lecture Note Series. Cambridge: Cambridge University Press, 1996Google Scholar
  12. 12.
    Eckmann, J.-P., Hairer, M.: Non-equilibrium statistical mechanics of strongly anharmonic chains of oscillators. Commun. Math. Phys. 212(1), 105–164 (2000)CrossRefMATHGoogle Scholar
  13. 13.
    Eckmann, J.-P., Hairer, M.: Uniqueness of the invariant measure for a stochastic PDE driven by degenerate noise. Commun. Math. Phys. 219(3), 523–565 (2001)CrossRefMATHGoogle Scholar
  14. 14.
    Eckmann, J.-P., Hairer, M.: Spectral properties of hypoelliptic operators. Commun. Math. Phys. 235(2), 233–253 (2003)MATHGoogle Scholar
  15. 15.
    Elworthy, K.D., Li, X.-M.: Formulae for the derivatives of heat semigroups. J. Funct. Anal. 125(1), 252–286 (1994)CrossRefMATHGoogle Scholar
  16. 16.
    Eckmann, J.-P., Pillet, C.-A., Rey-Bellet, L.: Non-equilibrium statistical mechanics of anharmonic chains coupled to two heat baths at different temperatures. Commun. Math. Phys. 201, 657–697 (1999)CrossRefMathSciNetMATHGoogle Scholar
  17. 17.
    Freidlin, M.I.: Random and deterministic perturbations of nonlinear oscillators. In: Proceedings of the International Congress of Mathematicians, Vol. III (Berlin, 1998), no. Extra Vol. III, 223–235 (electronic) (1998)Google Scholar
  18. 18.
    Hairer, M.: Exponential mixing properties of stochastic PDEs through asymptotic coupling. Probab. Theory Related Fields 124(3), 345–380 (2002)CrossRefMATHGoogle Scholar
  19. 19.
    Helffer, B., Nier, F.: Hypoellipticity and spectral theory for Fokker-Planck operators and Witten Laplacians, 2003. Prépublication 03-25 de l’IRMAR Université de RennesGoogle Scholar
  20. 20.
    Hohenberg, P., Swift, J.: Effects of additive noise at the onset of Rayleigh-Bénard convection. Phys. Rev. A 46, 4773–4785 (1992)CrossRefGoogle Scholar
  21. 21.
    Kato, T.: Perturbation Theory for Linear Operators. New York:Springer, 1980Google Scholar
  22. 22.
    Kuksin, S.B., Shirikyan, A.: Stochastic dissipative PDE’s and Gibbs measures. Commun. Math. Phys. 213, 291–330 (2000)CrossRefMathSciNetMATHGoogle Scholar
  23. 23.
    Kirrmann, P., Schneider, G., Mielke, A.: The validity of modulation equations for extended systems with cubic nonlinearities. Proc. Roy. Soc. Edinburgh Sect. A 122(1-2), 85–91 (1992)Google Scholar
  24. 24.
    Lai, Z., Das Sarma, S.: Kinetic growth with surface relaxation: Continuum versus atomistic models. Phys. Rev. Lett. 66(18), 2348–2351 (1991)CrossRefGoogle Scholar
  25. 25.
    León, J.A.: Fubini theorem for anticipating stochastic integrals in Hilbert space. Appl. Math. Optim. 27(3), 313–327 (1993)Google Scholar
  26. 26.
    Lunardi, A.: Analytic Semigroups and Optimal Regularity in Parabolic Problems. Basel:Birkhäuser, 1995Google Scholar
  27. 27.
    Malliavin, P.: Stochastic Analysis, Vol. 313 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Berlin:Springer-Verlag, 1997Google Scholar
  28. 28.
    Mattingly, J.C.: Exponential convergence for the stochastically forced Navier-Stokes equations and other partially dissipative dynamics. Commun. Math. Phys. 230(3), 421–462 (2002)CrossRefMATHGoogle Scholar
  29. 29.
    Meyn, S.P., Tweedie, R.L.: Markov Chains and Stochastic Stability. New York:Springer, 1994Google Scholar
  30. 30.
    Nualart, D.: The Malliavin Calculus and Related Topics. Probability and its Applications. New York: Springer-Verlag, 1995Google Scholar
  31. 31.
    Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. New York: Springer, 1983Google Scholar
  32. 32.
    Ramachandran, D., Rüschendorf, L.: On the Monge-Kantorovich duality theorem. Teor. Veroyatnost. i Primenen. 45(2), 403–409 (2000)MATHGoogle Scholar
  33. 33.
    Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion, Vol. 293 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Berlin:Springer-Verlag, Third ed., 1999Google Scholar
  34. 34.
    Schneider, G.: The validity of generalized Ginzburg-Landau equations. Math. Methods Appl. Sci. 19(9), 717–736 (1996)CrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Mathematics Research CentreUniversity of WarwickCoventryUnited Kingdom
  2. 2.Institut für Mathematik, RWTH Aachen AachenGermany

Personalised recommendations