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Journal of Statistical Physics

, Volume 166, Issue 3–4, pp 618–649 | Cite as

On Unique Ergodicity in Nonlinear Stochastic Partial Differential Equations

  • Nathan Glatt-Holtz
  • Jonathan C. Mattingly
  • Geordie Richards
Article

Abstract

We illustrate how the notion of asymptotic coupling provides a flexible and intuitive framework for proving the uniqueness of invariant measures for a variety of stochastic partial differential equations whose deterministic counterpart possesses a finite number of determining modes. Examples exhibiting parabolic and hyperbolic structure are studied in detail. In the later situation we also present a simple framework for establishing the existence of invariant measures when the usual approach relying on the Krylov–Bogolyubov procedure and compactness fails.

Keywords

Ergodicity Coupling SPDEs Stochastic fluid equations 

Notes

Acknowledgments

This work was partially supported by the National Science Foundation under the Grant (NEGH) NSF-DMS-1313272. JCM was partially supported by the Simons Foundation. We would like to thank Peter Constantin, Michele Coti-Zelati, Juraj Földes and Vlad Vicol for helpful feedback. We would also like to express our appreciation to the Mathematical Sciences Research Institute (MSRI) as well as the Duke and Virginia Tech Math Departments where the majority of this work was carried out.

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Tulane UniversityNew OrleansUSA
  2. 2.Duke UniversityDurhamUSA
  3. 3.Utah State UniversityLoganUSA

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