Averaging method for differential equations perturbed by dynamical systems
ESAIM: Probability and Statistics (2010)
- Volume: 6, page 33-88
- ISSN: 1292-8100
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topPène, Françoise. "Averaging method for differential equations perturbed by dynamical systems." ESAIM: Probability and Statistics 6 (2010): 33-88. <http://eudml.org/doc/104296>.
@article{Pène2010,
abstract = {
In this paper, we are interested in the asymptotical behavior
of 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, for
two-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.
},
author = {Pène, Françoise},
journal = {ESAIM: Probability and Statistics},
keywords = {Dynamical system; hyperbolicity; billiard; suspension flow;
limit theorem; averaging method; perturbation; differential equation.; exponential decay of correlations; limit theorem; toral automorphism; Sinai billiards},
language = {eng},
month = {3},
pages = {33-88},
publisher = {EDP Sciences},
title = {Averaging method for differential equations perturbed by dynamical systems},
url = {http://eudml.org/doc/104296},
volume = {6},
year = {2010},
}
TY - JOUR
AU - Pène, Françoise
TI - Averaging method for differential equations perturbed by dynamical systems
JO - ESAIM: Probability and Statistics
DA - 2010/3//
PB - EDP Sciences
VL - 6
SP - 33
EP - 88
AB -
In this paper, we are interested in the asymptotical behavior
of 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, for
two-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.
LA - eng
KW - Dynamical system; hyperbolicity; billiard; suspension flow;
limit theorem; averaging method; perturbation; differential equation.; exponential decay of correlations; limit theorem; toral automorphism; Sinai billiards
UR - http://eudml.org/doc/104296
ER -
References
top- 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
- L.A. Bunimovich, N.I. Chernov and Y.G. Sinai, Statistical properties of two-dimensional hyperbolic billiards. Russian Math. Surveys46 (1991) 47-106.
- L.A. Bunimovich and Y.G. Sinai, Statistical properties of Lorentz gaz with periodic configuration of scatterers. Comm. Math. Phys.78 (1981) 479-497.
- N.I. Chernov and Y.G. Sinai, Ergodic properties of certain systems of two-dimensional discs and three-dimensional balls. Russian Math. Surveys42 (1987) 181-207.
- M.I. Gordin, The Central Limit theorem for stationary processes. Soviet Math. Dokl.10 (1969) 1174-1176.
- Y. Katznelson, Ergodic automorphisms of Tn are Bernoulli shifts. Israel J. Math.10 (1971) 186-195.
- R.Z. Khas'minskii, On stochastic processes defined by differential equations with a small parameter (translation). Theory Probab. Appl.11 (1966) 211-228.
- Y. Kifer, Limit theorem in averaging for dynamical systems. Ergodic Theory Dynam. Systems15 (1995) 1143-1172.
- D.A. Lind, Dynamical properties of quasi hyperbolic toral automorphisms. Ergodic Theory Dynam. Systems2 (1982) 49-68.
- 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).
- F. Pène, 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).
- Y.G. Sinai, Dynamical systems with elastic reflections. Russian Math. Surveys25 (1970) 137-189.
- L.-S. Young, Statistical properties of dynamical systems with some hyperbolicity. Ann. of Math.147 (1998) 585-650.
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