Skip to main content
SpringerLink
Log in

Multiscale Expansion of Invariant Measures for SPDEs

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Arnold, L.: Random Dynamical Systems. Springer Monographs in Mathematics. Berlin:Springer-Verlag, 1998

  2. Bony, J.-M., Chemin, J.-Y.: Espaces fonctionnels associés au calcul de Weyl-Hörmander. Bull. Soc. Math. France 122(1), 77–118 (1994)

    MATH  Google Scholar 

  3. Berglund, N., Gentz, B.: Geometric singular perturbation theory for stochastic differential equations. J. Differ. Eqs. 191(1), 1–54 (2003)

    Article  MATH  Google Scholar 

  4. Blömker, D.: Amplitude equations for locally cubic non-autonomous nonlinearities. SIAM J. Appl. Dyn. Syst. 3(3), 464–486 (2003)

    Article  Google Scholar 

  5. Blömker, D.: Approximation of the stochastic Rayleigh-Bénard problem near the onset of instability and related problems, 2003. Preprint

  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. Caraballo, T., Crauel, H., Langa, J.A., Robinson, J.C.: Stabilization by additive noise. In preparation

  8. Crauel, H., Flandoli, F.: Additive noise destroys a pitchfork bifurcation. J. Dynam. Differ. Eqs. 10(2), 259–274 (1998)

    Article  MATH  Google Scholar 

  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. 145–154

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

  11. Da Prato, G., Zabczyk, J.: Ergodicity for Infinite Dimensional Systems, Vol. 229 of London Mathematical Society Lecture Note Series. Cambridge: Cambridge University Press, 1996

  12. Eckmann, J.-P., Hairer, M.: Non-equilibrium statistical mechanics of strongly anharmonic chains of oscillators. Commun. Math. Phys. 212(1), 105–164 (2000)

    Article  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  14. Eckmann, J.-P., Hairer, M.: Spectral properties of hypoelliptic operators. Commun. Math. Phys. 235(2), 233–253 (2003)

    MATH  Google Scholar 

  15. Elworthy, K.D., Li, X.-M.: Formulae for the derivatives of heat semigroups. J. Funct. Anal. 125(1), 252–286 (1994)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  18. Hairer, M.: Exponential mixing properties of stochastic PDEs through asymptotic coupling. Probab. Theory Related Fields 124(3), 345–380 (2002)

    Article  MATH  Google Scholar 

  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 Rennes

  20. Hohenberg, P., Swift, J.: Effects of additive noise at the onset of Rayleigh-Bénard convection. Phys. Rev. A 46, 4773–4785 (1992)

    Article  Google Scholar 

  21. Kato, T.: Perturbation Theory for Linear Operators. New York:Springer, 1980

  22. Kuksin, S.B., Shirikyan, A.: Stochastic dissipative PDE’s and Gibbs measures. Commun. Math. Phys. 213, 291–330 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  24. Lai, Z., Das Sarma, S.: Kinetic growth with surface relaxation: Continuum versus atomistic models. Phys. Rev. Lett. 66(18), 2348–2351 (1991)

    Article  Google Scholar 

  25. León, J.A.: Fubini theorem for anticipating stochastic integrals in Hilbert space. Appl. Math. Optim. 27(3), 313–327 (1993)

    Google Scholar 

  26. Lunardi, A.: Analytic Semigroups and Optimal Regularity in Parabolic Problems. Basel:Birkhäuser, 1995

  27. Malliavin, P.: Stochastic Analysis, Vol. 313 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Berlin:Springer-Verlag, 1997

  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)

    Article  MATH  Google Scholar 

  29. Meyn, S.P., Tweedie, R.L.: Markov Chains and Stochastic Stability. New York:Springer, 1994

  30. Nualart, D.: The Malliavin Calculus and Related Topics. Probability and its Applications. New York: Springer-Verlag, 1995

  31. Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. New York: Springer, 1983

  32. Ramachandran, D., Rüschendorf, L.: On the Monge-Kantorovich duality theorem. Teor. Veroyatnost. i Primenen. 45(2), 403–409 (2000)

    MATH  Google Scholar 

  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., 1999

  34. Schneider, G.: The validity of generalized Ginzburg-Landau equations. Math. Methods Appl. Sci. 19(9), 717–736 (1996)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematics Research Centre, University of Warwick, Coventry, CV4 7AL, United Kingdom

    Dirk Blömker & Martin Hairer

  2. Institut für Mathematik, RWTH Aachen , Aachen, Germany

    Dirk Blömker

Authors
  1. Dirk Blömker
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Martin Hairer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dirk Blömker.

Additional information

Communicated by A. Kupiainen

Acknowledgements The work of D.B. was supported by DFG-Forschungsstipendium BL535/5-1. The work of M.H. was supported by the Fonds National Suisse. Both authors would like to thank the MRC at the University of Warwick and especially David Elworthy for their warm hospitality.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Blömker, D., Hairer, M. Multiscale Expansion of Invariant Measures for SPDEs. Commun. Math. Phys. 251, 515–555 (2004). https://doi.org/10.1007/s00220-004-1130-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-004-1130-7

Keywords

Search

Navigation