Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-26T01:10:17.817Z Has data issue: false hasContentIssue false
  • English
  • Français

Limit theorems in averaging for dynamical systems

Published online by Cambridge University Press:  14 October 2010

Yuri Kifer
Yuri Kifer
Affiliation:
Institute of Mathematics, Hebrew University, Jerusalem, Givat Ram 91904, Israel
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper yields diffusion and moderate deviation type asymptotics for solutions of differential equations of the form dZε(t)/dt = εB(Zε(t), fty) where ft is a suspension flow (in particular, a hyperbolic flow) over a sufficiently fast mixing transformation. Such problems emerge in the study of perturbed Hamiltonian systems. These exhibit a new class of limit theorems for dynamical systems and extend a number of previously known results.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 15 , Issue 6 , December 1995 , pp. 1143 - 1172
Copyright
Copyright © Cambridge University Press 1995

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[A]Arnold, V. I.. Geometric Methods in the Theory of Ordinary Differential Equations. Springer: Berlin, 1983.CrossRefGoogle Scholar
[B]Bowen, R.. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. (Lecture Notes in Mathematics, 470) Springer: Berlin, 1975.CrossRefGoogle Scholar
[Bi]Billingsley, P.. Convergence of Probability Measures. Wiley: New York, 1968.Google Scholar
[Br]Bryc, W.. A remark on the connection between the large deviation principle and the central limit theorem. Statist. Probab. Lett. 18 (1993), 253–256.CrossRefGoogle Scholar
[BF]Borodin, A. N. and Freidlin, M. I.. Fast oscillating random perturbations of dynamical systems with conservation laws. Ann. de L'I. H. P. (probab. et stat.) 31 (1995), 485–525.Google Scholar
[BR]Bowen, R. and Ruelle, D.. The ergodic theory of Axiom A flows. Invent. Math. 29 (1975), 181–202.CrossRefGoogle Scholar
[CP]Coelho, Z. and Parry, W.. Central limit asymptotics for shifts of finite type. Isr. J. Math. 69 (1990), 235–249.CrossRefGoogle Scholar
[CE]Cogburn, R. and Ellison, J. A.. A stochastic theory of adiabatic invariance. Comm. Math. Phys. 149 (1992) 97–126.CrossRefGoogle Scholar
[D]Denker, M.. The central limit theorem for dynamical systems. Dynamical Systems and Ergodic Theory, Banach Center Publications 23 PWN-Polish Sci. Publ.: Warszawa, 1989.Google Scholar
[DP]Denker, M. and Philipp, W.. Approximation by Brownian motion for Gibbs measures and flows under a function. Ergod. Th. & Dynam. Sys. 4 (1984), 541–552.CrossRefGoogle Scholar
[Fr]Franko, E.. Flows with unique equilibrium state. Am. J. Math. 99 (1977), 486–514.CrossRefGoogle Scholar
[Fre]Freidlin, M. I.. The averaging principle and theorems on large deviations. Russ. Math. Surv. 33 (5) (1978), 107–160.CrossRefGoogle Scholar
[FW]Freidlin, M. I. and Wentzel, A. D.. Random Perturbations of Dynamical Systems. Springer: Berlin, 1984.CrossRefGoogle Scholar
[G]Gartner, J.. On large deviations from the invariant measure. Theory Probab. Appl. 22 (1977), 24–39.CrossRefGoogle Scholar
[I]Ibragimov, I. A.. Some limit theorems for stochastic processes. Theory Probab. Appl. 7 (1962), 349–382.CrossRefGoogle Scholar
[Kh]Khasminskii, R. Z.. On stochastic processes defined by differential equations with a small parameter. Theory Probab. Appl. 11 (1966), 211–228.CrossRefGoogle Scholar
[K]Kato, T.. Perturbation Theory for Linear Operators. Springer: Berlin, 1966.Google Scholar
[Kl]Kifer, Y.. Large deviations in dynamical systems and stochastic processes. Trans. Am. Math. Soc. 321 (1990), 505–524.CrossRefGoogle Scholar
[K2]Kifer, Y.. Averaging in dynamical systems and large deviations. Invent. Math. 110 (1992), 337–370.CrossRefGoogle Scholar
[K3]Kifer, Y.. Large deviations, averaging, and periodic orbits of dynamical systems. Comm. Math. Phys. 162 (1994), 33–46.CrossRefGoogle Scholar
[L]Lalley, S. P.. Distribution of periodic orbits of symbolic and Axiom A flows. Adv. Appl. Math. 8 (1987), 154–193.CrossRefGoogle Scholar
[P]Pollicott, M.. Large deviations, Gibbs measures, and closed orbits for hyperbolic flows. Preprint 1993.Google Scholar
[PP]Parry, W. and Pollicott, M.. Zeta functions and the periodic orbit structure of hyperbolic dynamics. Asterisque 187–188 (1990).Google Scholar
[R]Ratner, M.. The central limit theorem for geodesic flows on n-dimensional manifolds of negative curvature. Isr. J. Math. 16 (1973), 181–197.CrossRefGoogle Scholar
[Ru]Ruelle, D.. Thermodynamic Formalism. Addison-Wesley: Reading, MA, 1978.Google Scholar
[RV]Rozanov, Yu. A. and Volkonskii, A.. Some limit theorems for random functions I. Theory Probab. Appl. 4 (1959), 178–197.Google Scholar
[SV]Sanders, J. A. and Verhulst, F.. Averaging Methods in Nonlinear Dynamical Systems. Springer: Berlin, 1985.CrossRefGoogle Scholar
[W]Waddington, S.. Large deviation asymptotics for Anosov flows. Preprint 1993.Google Scholar