The Annals of Probability

Averaging of Hamiltonian flows with an ergodic component

Dmitry Dolgopyat and Leonid Koralov

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Abstract

We consider a process on $\mathbb{T}^{2}$, which consists of fast motion along the stream lines of an incompressible periodic vector field perturbed by white noise. It gives rise to a process on the graph naturally associated to the structure of the stream lines of the unperturbed flow. It has been shown by Freidlin and Wentzell [Random Perturbations of Dynamical Systems, 2nd ed. Springer, New York (1998)] and [Mem. Amer. Math. Soc. 109 (1994)] that if the stream function of the flow is periodic, then the corresponding process on the graph weakly converges to a Markov process. We consider the situation where the stream function is not periodic, and the flow (when considered on the torus) has an ergodic component of positive measure. We show that if the rotation number is Diophantine, then the process on the graph still converges to a Markov process, which spends a positive proportion of time in the vertex corresponding to the ergodic component of the flow.

Article information

Source
Ann. Probab. Volume 36, Number 6 (2008), 1999-2049.

Dates
First available in Project Euclid: 19 December 2008

Permanent link to this document
http://projecteuclid.org/euclid.aop/1229696595

Digital Object Identifier
doi:10.1214/07-AOP372

Mathematical Reviews number (MathSciNet)
MR2478675

Zentralblatt MATH identifier
1156.60038

Subjects
Primary: 60J60: Diffusion processes [See also 58J65] 34E10: Perturbations, asymptotics

Keywords
Averaging Markov process Diophantine condition Hamiltonian flow gluing conditions diffusion on a graph

Citation

Dolgopyat, Dmitry; Koralov, Leonid. Averaging of Hamiltonian flows with an ergodic component. Ann. Probab. 36 (2008), no. 6, 1999--2049. doi:10.1214/07-AOP372. http://projecteuclid.org/euclid.aop/1229696595.


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References

  • [1] Arnold, V. I. (1991). Topological and ergodic properties of closed 1-forms with incommensurable periods. Funct. Anal. Appl. 25 81–90.
  • [2] Cornfeld, I. P., Fomin, S. V. and Sinai, Ya. G. (1982). Ergodic Theory. Springer, New York.
  • [3] Doob, J. L. (1953). Stochastic Processes. Wiley, New York.
  • [4] Fannjiang, A. and Papanicolaou, G. (1994). Convection enhanced diffusion for periodic flows. SIAM J. Appl. Math. 54 333–408.
  • [5] Freidlin, M. I. (1996). Markov Processes and Differential Equations: Asymptotic Problems. Birkhauser, Basel.
  • [6] Freidlin, M. I. and Wentzell, A. D. (1998). Random Perturbations of Dynamical Systems, 2nd ed. Springer, New York.
  • [7] Freidlin, M. I. and Wentzell, A. D. (1994). Random perturbations of Hamiltonian systems. Mem. Amer. Math. Soc. 109.
  • [8] Khanin, K. M. and Sinai, Ya. G. (1992). Mixing of some classes of special flows over rotations of the circle. Funct. Anal. Appl. 26 155–169.
  • [9] Khinchin, A. Ya. (1997). Continued Fractions. Dover, Mineola, NY.
  • [10] Koralov, L. (2004). Random perturbations of 2-dimensional Hamiltonian flows. Probab. Theory Related Fields 129 37–62.
  • [11] Mandl, P. (1968). Analytical Treatment of One-Dimensional Markov Provesses. Springer, Berlin.
  • [12] Miranda, C. (1955). Partial Differential Equations of Elliptic Type, 1st ed. Springer, New York.
  • [13] Novikov, A., Papanicolaou, G. and Ryzhik, L. (2005). Boundary layers for cellular flows at high Peclet numbers. Comm. Pure Appl. Math. 58 867–922.
  • [14] Sowers, R. (2006). Random perturbations of two-dimensional pseudoperiodic flows. Illinois J. Math. 50 853–959.
  • [15] Sowers, R. (2005). A boundary-layer theory for diffusively perturbed transport around a heteroclinic cycle. Comm. Pure Appl. Math. 58 30–84.