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Exponential mixing for stochastic PDEs: the non-additive case
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  • Published: 14 February 2007

Exponential mixing for stochastic PDEs: the non-additive case

  • Cyril Odasso1 

Probability Theory and Related Fields volume 140, pages 41–82 (2008)Cite this article

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Abstract

We establish a general criterion which ensures exponential mixing of parabolic stochastic partial differential equations (SPDE) driven by a non additive noise which is white in time and smooth in space. We apply this criterion on two representative examples: 2D Navier–Stokes (NS) equations and Complex Ginzburg–Landau (CGL) equation with a locally Lipschitz noise. Due to the possible degeneracy of the noise, Doob theorem cannot be applied. Hence, a coupling method is used in the spirit of Kuksin and Shirikyan (J. Math. Pures Appl. 1:567–602, 2002) and Mattingly (Commun. Math. Phys. 230:421–462, 2002).

Previous results require assumptions on the covariance of the noise which might seem restrictive and artificial. For instance, for NS and CGL, the covariance operator is supposed to be diagonal in the eigenbasis of the Laplacian and not depending on the high modes of the solutions. The method developed in the present paper gets rid of such assumptions and only requires that the range of the covariance operator contains the low modes.

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Authors and Affiliations

  1. Université Paris 12, Val-de-Marne, UFR Des Sciences et Technologie, Bâtiment, P3, 4ème étage, Bureau 432, 61, Avenue du Général de Gaulle, 94010, Créteil Cedex, France

    Cyril Odasso

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  1. Cyril Odasso
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Odasso, C. Exponential mixing for stochastic PDEs: the non-additive case. Probab. Theory Relat. Fields 140, 41–82 (2008). https://doi.org/10.1007/s00440-007-0057-2

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  • Received: 24 May 2005

  • Revised: 11 January 2007

  • Published: 14 February 2007

  • Issue Date: January 2008

  • DOI: https://doi.org/10.1007/s00440-007-0057-2

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Keywords

  • Two-dimensional Navier–Stokes equations
  • Complex Ginzburg–Landau equations
  • Markov transition semi-group
  • Invariant measure
  • Ergodicity
  • Coupling method
  • Girsanov formula
  • Expectational Foias–Prodi estimate
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