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Averaging method for differential equations perturbed by dynamical systems

Published online by Cambridge University Press:  15 November 2002

Françoise Pène
Françoise Pène*
Affiliation:
UBO, Département de Mathématiques, 29285 Brest Cedex, France; [email protected].
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Abstract

In this paper, we are interested in the asymptotical behaviorof the error between the solution of a differential equation perturbed by a flow (or by a transformation) and the solution of the associated averaged differential equation.The main part of this redaction is devoted to the ascertainment of results of convergence in distribution analogous to those obtained in [10] and [11]. As in [11], we shall use a representation by a suspension flow over a dynamical system. Here, we make an assumption of multiple decorrelation in terms of this dynamical system. We show how this property can be verified for ergodic algebraic toral automorphisms and point out the fact that, fortwo-dimensional dispersive billiards, it is a consequence of the method developed in [18]. Moreover, the singular case of a degenerated limit distribution is also considered.

Keywords

Dynamical systemhyperbolicitybilliardsuspension flowlimit theoremaveraging methodperturbationdifferential equation.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 6: New directions in Time Series Analysis (Guest Editor: Philippe Soulier)New directions in Time Series Analysis (Guest Editor: Philippe Soulier) , 2002 , pp. 33 - 88
Copyright
© EDP Sciences, SMAI, 2002

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References

N. Bary, A treatise on trigonometric series, Vol. 1. Pergamon Press (1984).
P. Billingsley, Convergence of probability measures. J. Wiley and Sons (1968).
S. Le Borgne, Dynamique symbolique et propriétés stochastiques des automorphismes du tore : cas hyperbolique et quasi-hyperbolique, Ph.D. Thesis. University of Rennes I, France (1997).
S. Le Borgne, Un problème de régularité dans l'équation de cobord, in Sémimaires de probabilités de Rennes. Université de Rennes 1 (1998); http://www.maths.univ-rennes1.fr/csp/1998/index.html
Bunimovich, L.A., Chernov, N.I. and Sinai, Y.G., Statistical properties of two-dimensional hyperbolic billiards. Russian Math. Surveys 46 (1991) 47-106. CrossRef
Bunimovich, L.A. and Sinai, Y.G., Statistical properties of Lorentz gaz with periodic configuration of scatterers. Comm. Math. Phys. 78 (1981) 479-497. CrossRef
Chernov, N.I. and Sinai, Y.G., Ergodic properties of certain systems of two-dimensional discs and three-dimensional balls. Russian Math. Surveys 42 (1987) 181-207.
Gordin, M.I., The Central Limit theorem for stationary processes. Soviet Math. Dokl. 10 (1969) 1174-1176.
Katznelson, Y., Ergodic automorphisms of Tn are Bernoulli shifts. Israel J. Math. 10 (1971) 186-195. CrossRef
Khas'minskii, R.Z., On stochastic processes defined by differential equations with a small parameter (translation). Theory Probab. Appl. 11 (1966) 211-228. CrossRef
Kifer, Y., Limit theorem in averaging for dynamical systems. Ergodic Theory Dynam. Systems 15 (1995) 1143-1172. CrossRef
Lind, D.A., Dynamical properties of quasi hyperbolic toral automorphisms. Ergodic Theory Dynam. Systems 2 (1982) 49-68. CrossRef
V.P. Leonov, Quelques applications de la méthode des cumulants à la théorie des processus stochastiques stationnaires (in Russian). Nauka, Moscow (1964).
F. Pène, Applications des propriétés stochastiques des systèmes dynamiques de type hyperbolique : ergodicité du billard dispersif dans le plan, moyennisation d'équations différentielles perturbées par un flot ergodique, Ph.D. Thesis. University of Rennes I, France (2000).
Pène, F., Rates of convergence in the CLT for two-dimensional dispersive billiards. Comm. Math. Phys. 225 (2002) 91-119.
D. Revuz and M. Yor, Continuous martingales and brownian motion. Springer-Verlag (1994).
Sinai, Y.G., Dynamical systems with elastic reflections. Russian Math. Surveys 25 (1970) 137-189. CrossRef
Young, L.-S., Statistical properties of dynamical systems with some hyperbolicity. Ann. of Math. 147 (1998) 585-650. CrossRef