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Criteria for Strong and Weak Random Attractors

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Abstract

The theory of random attractors has different notions of attraction, amongst them pullback attraction and weak attraction. We investigate necessary and sufficient conditions for the existence of pullback attractors as well as of weak attractors.

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References

  1. Arnold L.: Random Dynamical Systems. Springer, New York (1998)

    MATH  Google Scholar 

  2. Arnold L., Scheutzow M.: Perfect cocycles through stochastic differential equations. Probab. Theory Relat. Fields 101, 65–88 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  3. Arnold L., Schmalfuß B.: Lyapunov’s second method for random dynamical systems. J. Differ. Equ. 177, 235–265 (2001)

    Article  MATH  Google Scholar 

  4. Ashwin P., Ochs G.: Convergence to local random attractors. Dyn. Sys. 18, 139–158 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  5. Chueshov I., Scheutzow M.: On the structure of attractors and invariant measures for a class of monotone random systems. Dyn. Sys. 19, 127–144 (2004)

    MATH  MathSciNet  Google Scholar 

  6. Crauel H.: Global random attractors are uniquely determined by attracting deterministic compact sets. Ann. Mat. Pura Appl., IV. Ser. CLXXVI, 57–72 (1999)

    Article  MathSciNet  Google Scholar 

  7. Crauel H.: Random point attractors versus random set attractors. J. London Math. Soc., II. Ser. 63, 413–427 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  8. Crauel H.: A uniformly exponential random forward attractor which is not a pullback attractor. Archiv der Mathematik 78, 329–336 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  9. Crauel H.: Random Probability Measures on Polish Spaces, Series Stochastics Monographs, vol. 11. Taylor & Francis, London and New York (2002)

    Google Scholar 

  10. Crauel H., Debussche A., Flandoli F.: Random attractors. J. Dyn. Diff. Eqn. 9, 307–341 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  11. Crauel H., Siegmund S., Siegmund S.: Towards a Morse theory for random dynamical systems. Stochastics Dyn. 4, 277–296 (2004)

    Article  MATH  Google Scholar 

  12. Crauel H., Flandoli F.: Attractors for random dynamical systems. Probab. Theory Relat. Fields 100, 365–393 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  13. Dimitroff, G.: Some properties of isotropic Brownian and Ornstein-Uhlenbeck flows. Dissertation, Technische Universität Berlin (2006)

  14. Dimitroff, G., Scheutzow, M.: Attractors and expansion for Brownian flows. (submitted for publication) (2009)

  15. Shouchuan Hu., Papageorgiou N.S.: Handbook of Multivalued Analysis. Vol. I: Theory. Kluwer, Dordrecht (1997)

    Google Scholar 

  16. Ochs, G.: Weak random attractors, Report 449. Institut für Dynamische Systeme. Universität Bremen (1999)

  17. Scheutzow M.: Comparison of various concepts of a random attractor: a case study. Archiv der Mathematik 78, 233–240 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  18. Scheutzow M.: Attractors for ergodic and monotone random dynamical systems. In: : Dalang, R., Dozzi, M., Russo, F. (eds) Seminar on Stochastic Analysis, Random Fields and Applications V, pp. 331–344. Birkhäuser Verlag, USA (2008)

    Chapter  Google Scholar 

  19. Scheutzow M.: Chaining techniques and their application to stochastic flows, to appear. In: Blath, J., Mörters, P., Scheutzow, M. (eds) Trends in Stochastic Analysis. London Mathematical Society Lecture Note Series No. 353, pp. 35–63. Cambridge University Press, Cambridge (2009)

    Google Scholar 

  20. Schmalfuß, B.: Backward cocycles and attractors of stochastic differential equations. In: Reitmann, V., Riedrich, T., Koksch, N. (eds.) International Seminar on Applied Mathematics – Nonlinear Dynamics: Attractor Approximation and Global Behaviour. International Seminar on Applied Mathematics, pp. 185–192. Technische Universität Dresden (1992)

  21. Schmalfuß B.: Stochastische Attraktoren des stochastischen Lorenz-Systems. Z. Angew. Math. Mech. 74(6), T627–T628 (1994)

    Google Scholar 

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Authors and Affiliations

  1. Fachbereich Mathematik, Johann Wolfgang Goethe-Universität Frankfurt, Robert-Mayer-Straße 10, 60325, Frankfurt, Germany

    Hans Crauel

  2. Institut Techno- und Wirtschaftsmathematik, Fraunhofer-Platz 1, 67663, Kaiserslautern, Germany

    Georgi Dimitroff

  3. Institut für Mathematik, MA 7-5, Fakultät II, Technische Universität Berlin, Straße des 17, Juni 136, 10623, Berlin, Germany

    Michael Scheutzow

Authors
  1. Hans Crauel
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  2. Georgi Dimitroff
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  3. Michael Scheutzow
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Correspondence to Hans Crauel.

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Crauel, H., Dimitroff, G. & Scheutzow, M. Criteria for Strong and Weak Random Attractors. J Dyn Diff Equat 21, 233–247 (2009). https://doi.org/10.1007/s10884-009-9135-8

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  • DOI: https://doi.org/10.1007/s10884-009-9135-8

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