Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Mixing and decorrelation in infinite measure: The case of the periodic Sinai billiard

Françoise Pène

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Abstract

We investigate the question of the rate of mixing for observables of a $\mathbb{Z}^{d}$-extension of a probability preserving dynamical system with good spectral properties. We state general mixing results, including expansions of every order. The main motivation of this article is the study of mixing rates for smooth observables of the $\mathbb{Z}^{2}$-periodic Sinai billiard, with different kinds of results depending on whether the horizon is finite or infinite. We establish a first order mixing result when the horizon is infinite. In the finite horizon case, we establish an asymptotic expansion of every order, enabling the study of the mixing rate even for observables with null integrals. This result is related to an Edgeworth expansion in the local limit theorem.

Résumé

Cet article est une contribution à l’étude du mélange d’observables de systèmes dynamiques préservant une mesure infinie. Nous étudions le cas de $\mathbb{Z}^{d}$-extensions de systèmes dynamiques probabilisés ayant de bonnes propriétés spectrales. Nous établissons des résultats généraux et les illustrons par plusieurs exemples. Notre motivation principale est l’étude de la vitesse de mélange pour des observables régulières du billard de Sinai $\mathbb{Z}^{2}$-périodique, pour lequel nous obtenons des résultats de types différents selon que l’horizon soit fini ou infini. Nous établissons un résultat de mélange du premier ordre lorsque l’horizon est infini. Dans le cas où l’horizon est fini, nous établissons un développement asymptotique de tout ordre, permettant l’étude de la vitesse de mélange pour des observables d’intégrale nulle. Ce dernier résultat est relié à un développement de Edgeworth dans le théorème limite local.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 55, Number 1 (2019), 378-411.

Dates
Received: 17 June 2017
Revised: 16 January 2018
Accepted: 17 January 2018
First available in Project Euclid: 18 January 2019

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1547802404

Digital Object Identifier
doi:10.1214/18-AIHP885

Mathematical Reviews number (MathSciNet)
MR3901650

Zentralblatt MATH identifier
07039774

Subjects
Primary: 37A25: Ergodicity, mixing, rates of mixing

Keywords
Sinai Billiard Lorentz process Young tower Local limit theorem Decorrelation Mixing Infinite measure Edgeworth expansion

Citation

Pène, Françoise. Mixing and decorrelation in infinite measure: The case of the periodic Sinai billiard. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 1, 378--411. doi:10.1214/18-AIHP885. https://projecteuclid.org/euclid.aihp/1547802404


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