Invariant Measures for Dissipative Systems and Generalised Banach Limits

  • Grzegorz Łukaszewicz
  • José Real
  • James C. Robinson
Open Access


Inspired by a theory due to Foias and coworkers (see, for example, Foias et al. Navier–Stokes equations and turbulence, Cambridge University Press, Cambridge, 2001) and recent work of Wang (Disc Cont Dyn Sys 23:521–540, 2009), we show that the generalised Banach limit can be used to construct invariant measures for continuous dynamical systems on metric spaces that have compact attracting sets, taking limits evaluated along individual trajectories. We also show that if the space is a reflexive separable Banach space, or if the dynamical system has a compact absorbing set, then rather than taking limits evaluated along individual trajectories, we can take an ensemble of initial conditions: the generalised Banach limit can be used to construct an invariant measure based on an arbitrary initial probability measure, and any invariant measure can be obtained in this way. We thus propose an alternative to the classical Krylov–Bogoliubov construction, which we show is also applicable in this situation.


Invariant measure Generalised Banach limit Krylov–Bogoliubov procedure Global attractor 


  1. 1.
    Aliprantis Ch.D., Border K.C.: Infinite Dimensional Analysis. A Hitchhiker’s Guide, 3rd edn. Springer, Berlin (2006)MATHGoogle Scholar
  2. 2.
    Anguiano M., Caraballo T., Real J.: Existence of pullback attractor for a reaction-diffusion equation in some unbounded domains with non-autonomous forcing term in H −1. Int. J. Bifur. Chaos 20, 2645–2656 (2010)MathSciNetMATHGoogle Scholar
  3. 3.
    Babin A.V., Vishik M.I.: Attractors of Evolution Equations. North–Holland, Amsterdam (1992)MATHGoogle Scholar
  4. 4.
    Cholewa J., Dłotko T.: Global Attractors in Abstract Parabolic Problems. Cambridge University Press, Cambridge (2000)MATHCrossRefGoogle Scholar
  5. 5.
    Crauel H.: Random point attractors versus random set attractors. J. Lond. Math. Soc. 63, 413–427 (2001)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Da Prato G., Zabczyk J.: Ergodicity for Infinite Dimensional Systems. Cambridge University Press, Cambridge (1996)MATHCrossRefGoogle Scholar
  7. 7.
    Foias C.: Statistical study of Navier–Stokes equations. I. Rend. Sem. Mat. Univ. Padova 48, 219–348 (1972)MathSciNetGoogle Scholar
  8. 8.
    Foias C.: Statistical study of Navier–Stokes equations. II. Rend. Sem. Mat. Univ. Padova 49, 9–123 (1973)MATHGoogle Scholar
  9. 9.
    Foias C., Manley O., Rosa R., Temam R.: Navier-Stokes Equations and Turbulence. Encyclopedia of Mathematics and its Applications, vol. 83. Cambridge University Press, Cambridge (2001)Google Scholar
  10. 10.
    Hale J.K.: Asymptotic Behavior of Dissipative Systems. AMS, Providence (1988)MATHGoogle Scholar
  11. 11.
    Haraux A.: Systèmes dynamiques dissipatifs et applications. Masson, Paris (1990)Google Scholar
  12. 12.
    Kryloff N., Bogoliouboff N.: La théorie générale de la mesure dans son application à l’étude des systèmes dynamiques de la mécanique non linéaire. Ann. Math. 38, 65–113 (1937)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Langa J.A., Robinson J.C.: Determining asymptotic behavior from the dynamics on attracting sets. J. Dyn. Differ. Equ. 11, 319–331 (1999)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Nemytskii V.V., Stepanov V.V.: Qualitative Theory of Differential Equations. Princeton University Press, Princeton (1960)MATHGoogle Scholar
  15. 15.
    Robinson J.C.: Infinite-Dimensional Dynamical Systems. Cambridge University Press, Cambridge (2001)Google Scholar
  16. 16.
    Rosa R.: The global attractor for the 2D Navier–Stokes flow on some unbounded domains. Nonlinear Anal. 32, 71–85 (1998)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Temam R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer, Berlin (1988)MATHGoogle Scholar
  18. 18.
    van der Vaart A.W., Wellner J.A.: Weak Convergence and Empirical Processes: With Applications to Statistics. Springer, New York (2000)Google Scholar
  19. 19.
    Walters P.: An introduction to Ergodic Theory. Graduate Texts in Mathematics. Springer, New York (1982)Google Scholar
  20. 20.
    Wang X.: Upper semi-continuity of stationary statistical properties of dissipative systems. Disc. Cont. Dyn. Sys. 23, 521–540 (2009)MATHCrossRefGoogle Scholar

Copyright information

© The Author(s) 2011

Authors and Affiliations

  • Grzegorz Łukaszewicz
    • 1
  • José Real
    • 2
  • James C. Robinson
    • 3
  1. 1.Mathematics DepartmentUniversity of WarsawWarsawPoland
  2. 2.Departamento de Ecuaciones Diferenciales y Análisis NuméricoFacultad de MatemáticasSevilleSpain
  3. 3.Mathematics InstituteUniversity of WarwickCoventryUK

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