Skip to main content
SpringerLink
Log in

The Conjugacy of Stochastic and Random Differential Equations and the Existence of Global Attractors

  • Published:
Journal of Dynamics and Differential Equations Aims and scope Submit manuscript

Abstract

We consider stochastic differential equations in d-dimensional Euclidean space driven by an m-dimensional Wiener process, determined by the drift vector field f0 and the diffusion vector fields f1,...,fm, and investigate the existence of global random attractors for the associated flows φ. For this purpose φ is decomposed into a stationary diffeomorphism Φ given by the stochastic differential equation on the space of smooth flows on Rd driven by m independent stationary Ornstein Uhlenbeck processes z1,...,zm and the vector fields f1,...,fm, and a flow χ generated by the nonautonomous ordinary differential equation given by the vector field (∂Φt/∂x)−1[f0(Φt)+∑ 1i=1 fi(Φt)z it ]. In this setting, attractors of χ are canonically related with attractors of φ. For χ, the problem of existence of attractors is then considered as a perturbation problem. Conditions on the vector fields are derived under which a Lyapunov function for the deterministic differential equation determined by the vector field f0 is still a Lyapunov function for χ, yielding an attractor this way. The criterion is finally tested in various prominent examples.

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. (1998). Random Dynamical Systems, Springer, Berlin.

    Google Scholar 

  2. Arnold, L., and Scheutzow, M. (1995). Perfect cocycles through stochastic differential equations. Probab. Th. Rel. Fields 101, 65-88.

    Google Scholar 

  3. Bismut, J. M., and Michel, D. (1981). Diffusions conditionnelles. I. J. Funct. Anal. 445, 174-211.

    Google Scholar 

  4. Bismut, J. M., and Michel, D. (1982). Diffusions conditionnelles. II. J. Funct. Anal. 45, 274-292.

    Google Scholar 

  5. Crauel, H. (1993). Markov measures for random dynamical systems. Stochast. Stochast. Rep. 37, 13-28.

    Google Scholar 

  6. Crauel, H., Debussche, A., and Flandoli, F. (1997). Random attractors. J. Dynam. Diff. Eqs. 9(2), 307-341.

    Google Scholar 

  7. Crauel, H., and Flandoli, F. (1994). Attractors for random dynamical systems. Probab. Th. Rel. Fields 100, 365-393.

    Google Scholar 

  8. Flandoli, F., and Schmalfuss, B. (1996). Random attractors for the 3D Navier Stokes equation with multiplicative white noise. Stochast. Stochast. Rep. 59, 21-45.

    Google Scholar 

  9. Kunita, H. (1990). Stochastic Flows and Stochastic Differential Equations, Cambridge University Press, Cambridge.

    Google Scholar 

  10. Keller, H., and Ochs, G. (1998). Numerical Approximation of Random Attractors, Report No. 431, Universität Bremen.

  11. Keller, H., and Schmalfuss, B. (1998). Attractors for stochastic differential equations with non-trivial noise. Izvest. Akad. Nauk RM 26, 43-54.

    Google Scholar 

  12. Ledrappier, F. (1986). Positivity of the exponent for stationary sequences. In Arnold, L., and Wihstutz, V. (eds.), Lyapunov Exponents, LNM 1186, Springer, Berlin.

    Google Scholar 

  13. Le Jan, Y. (1986). Equilibre statistique pour les produits de difféomorphismes aléatories indépendants. CRAS 302, 351-354.

    Google Scholar 

  14. Leonov, G. A., and Boichenko, V. A. (1992). Lyapunov's direct method in the estimation of the Hausdorff dimension of attractors. Acta Appl. Math. 26, 1-60.

    Google Scholar 

  15. Lyons, T. (1998). Differential equations driven by rough signals. Preprint, Imperial College, London.

  16. Ocone, D., and Pardoux, E. (1989). A generalized Itô-Ventzell formula. Application to a class of anticipating stochastic differential equations. Ann. Inst. H. Poincaré 25, 39-71.

    Google Scholar 

  17. Schenk-Hoppé, K. R. (1996). Bifurcation scenarios of the noisy Duffing-van der Pol oscillator. Nonlin. Dynam. 11, 255-274.

    Google Scholar 

  18. Schenk-Hoppé, K. R. (1998). Random attractors: General properties, existence and applications to stochastic bifurcation theory. Discrete Cont. Dynam. Syst. 4, 99-130.

    Google Scholar 

  19. Schmalfuss, R. (1997). The random attractor of the stochastic Lorenz system. ZAMP 48, 951-974.

    Google Scholar 

  20. Schmalfuss, B. (1998). A measurable fixed point theorem and the measurable graph trans-formation. JMAA 225, 91-113.

    Google Scholar 

  21. Spivak, M. (1979). A Comprehensive Introduction to Differential Geometry , Vol. 1, 2nd ed., Pub. Berkeley.

  22. Sznitman, A. S. (1982). Martingales dépendant d'un paramétre: Une formule d'Itô. Z. Wahr-scheinlichkeitstheorie verw. Geb. 60, 41-70.

    Google Scholar 

  23. Ventzell, A. D. (1965). On the equation of the theory of conditional Markov processes. Theory Probab. Appl. 10, 357-361.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099, Berlin, Germany

    Peter Imkeller

  2. FB Angewandte Naturwissenschaften, FH Merseburg, Geusaer Strasse, 06217, Merseburg, Germany

    Björn Schmalfuss

Authors
  1. Peter Imkeller
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Björn Schmalfuss
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Imkeller, P., Schmalfuss, B. The Conjugacy of Stochastic and Random Differential Equations and the Existence of Global Attractors. Journal of Dynamics and Differential Equations 13, 215–249 (2001). https://doi.org/10.1023/A:1016673307045

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1016673307045

Search

Navigation