Skip to main content
SpringerLink
Log in

On Stochastic Perturbations of Dynamical Systems with a “Rough” Symmetry. Hierarchy of Markov Chains.

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

We consider long-time behavior of dynamical systems perturbed by a small noise. Under certain conditions, a slow component of such a motion, which is most important for long-time evolution, can be described as a motion on the cone of invariant measures of the non-perturbed system. The case of a finite number of extreme points of the cone is considered in this paper. As is known, in the generic case, the long-time evolution can be described by a hierarchy of cycles defined by the action functional for corresponding stochastic processes. This, in particular, allows to study metastable distributions and such effects as stochastic resonance. If the system has some symmetry in the logarithmic asymptotics of transition probabilities (rough symmetry),the hierarchy of cycles should be replaced by a hierarchy of Markov chains and their invariant measures.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  1. Brin, M., Freidlin, M.: On stochastic behavior of perturbed Hamiltonian systems. Ergod. Theory Dyn. Syst. 20(1), 55–76 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  2. Doob, J.L.: Stochastic Processes. Wiley, New York (1953)

    MATH  Google Scholar 

  3. Freidlin, M.: Sublimiting distributions and stabilization of solutions of parabolic equations with a small parameter. Sov. Math. Dokl. 18(4), 1114–1118 (1977)

    MATH  Google Scholar 

  4. Freidlin, M.: Quasi-deterministic approximation, metastability and stochastic resonance. Phys. D 137, 333–352 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  5. Freidlin, M., Wentzell, A.: Random Perturbations of Dynamical Systems, 3rd edn. Springer, Heidelberg (2012)

    Book  MATH  Google Scholar 

  6. Krein, M., Milman, D.: On extreme points of regular convex sets. Studia Math. 9, 133–138 (1940)

    MathSciNet  Google Scholar 

Download references

Acknowledgments

The work was supported in part by the National Science Foundation, Grant DMS 1411866.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Maryland, College Park, MD, 20742-4015, USA

    M. Freidlin

Authors
  1. M. Freidlin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Freidlin.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Freidlin, M. On Stochastic Perturbations of Dynamical Systems with a “Rough” Symmetry. Hierarchy of Markov Chains.. J Stat Phys 157, 1031–1045 (2014). https://doi.org/10.1007/s10955-014-1110-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-014-1110-z

Keywords

Search

Navigation