Skip to main content
Log in

Central limit theorems for the ℤ2-periodic Lorentz gas

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

This paper is devoted to the stochastic properties of dynamical systems preserving an infinite measure. More precisely we prove central limit theorems for Birkhoff sums of observables of ℤ2-extensions of dynamical systems (satisfying some nice spectral properties). The motivation of our paper is the ℤ2-periodic Lorentz process for which we establish a functional central limit theorem for Hölder continuous observables (in discrete time as well as in continuous time).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. J. Aaronson, An Introduction to Infinite Ergodic Theory, Mathematical Surveys and Monographs, Vol. 50, American Mathematical Society, Providence, RI, 1997.

    Book  Google Scholar 

  2. R. Bowen, Symbolic dynamics for hyperbolic flows, American Journal of Mathematics 95 (1973), 429–460.

    Article  MathSciNet  Google Scholar 

  3. R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics, Vol. 470, Springer, Berlin-New York, 1975.

    Book  Google Scholar 

  4. L. A. Bunimovich, N. I. Chernov and Ya. G. Sinai, Statistical properties of two dimensional hyperbolic billiards, Uspekhi Matematicheskikh Nauk 46 (1991), 47–106.

    MathSciNet  MATH  Google Scholar 

  5. L. A. Bunimovich and Ya. G. Sinai, Statistical properties of Lorentz gas with periodic configuration of scatterers, Communications in Mathematical Physics 78 (1981), 479–497.

    Article  MathSciNet  Google Scholar 

  6. N. Chernov and H.-K. Zhang, Billiards with polynomial mixing rates, Nonlinearity 18 (2005), 1527–1553.

    Article  MathSciNet  Google Scholar 

  7. J.-P. Conze, Sur un critère de récurrence en dimension 2 pour les marches stationnaires, applications, Ergodic Theory and Dynamical Systems 19 (1999), 1233–1245.

    Article  MathSciNet  Google Scholar 

  8. M. Demers, F. Pène and H.-K. Zhang, Local limit theorem for randomly deforming billiards, Communications in Mathematical Physics 375 (2020), 2281–2334.

    Article  MathSciNet  Google Scholar 

  9. M. Demers and H.-K. Zhang, Spectral analysis of hyperbolic systems with singularities, Nonlinearity 27 (2014), 379–433.

    Article  MathSciNet  Google Scholar 

  10. R. L. Dobrushin, Two limit theorems for the simplest random walk on a line, Uspekhi Matematicheskikh Nauk 10 (1955), 139–146.

    MathSciNet  Google Scholar 

  11. D. Dolgopyat, D. Szász and T. Varjú, Recurrence properties of Lorentz gas, Duke Mathematical Journal 142 (2008), 241–281

    Article  MathSciNet  Google Scholar 

  12. W. Feller, An Introduction to Probability Theory and its Applications. Vol. II, John Wiley & Sons, New York-London-Sydney, 1966.

    MATH  Google Scholar 

  13. Y. Guivarc’h and J. Hardy, Théorèmes limites pour une classe de chaînes de Markov et applications aux difféomorphismes d’Anosov, Annales de l’Institut Henri Poincaré 24 (1988), 73–98.

    MathSciNet  MATH  Google Scholar 

  14. H. Hennion and L. Hervé, Limit Theorems for Markov Chains and Stochastic Properties of Dynamical Systems by Quasi-Compactness, Lecture Notes in Mathematics, Vol. 1766, Springer, Berlin, 2001.

    Book  Google Scholar 

  15. E. Hopf, Ergodentheorie, Springer, Berlin, 1937.

    Book  Google Scholar 

  16. Y. Kasahara, Two limit theorems for occupation times of Markov processes, Japanese Journal of Mathematics 7 (1981), 291–300.

    Article  MathSciNet  Google Scholar 

  17. Y. Kasahara, A limit theorem for sums of random number of i.i.d. random variables and its application to occupation times of Markov chains, Journal of the Mathematical Society of Japan 37 (1985), 197–205.

    Article  MathSciNet  Google Scholar 

  18. G. Keller and C. Liverani, Stability of the spectrum for transfer operators, Annali della Scuola Normale Superiore di Pisa. Classe di Scienze 28 (1999), 141–152.

    MathSciNet  MATH  Google Scholar 

  19. H. Kesten, Occupation times for Markov and semi-Markov chains, Transactions of the American Mathematical Society 103 (1962), 82–112.

    Article  MathSciNet  Google Scholar 

  20. S. V. Nagaev, Some limit theorems for stationary Markov chains, Theory of probability and its applications, 11 (1957), 378–406.

    Article  Google Scholar 

  21. S. V. Nagaev, More exact statements of limit theorems for homogeneous Markov chains, Teoriya Veroyatnostei i ee Primeneniya 6 (1961), 67–87.

    Google Scholar 

  22. F. Pène and D. Thomine, Potential kernel, hitting probabilities and distributional asymptotics, Ergodic Theory and Dynamical Systems 40 (2020), 1894–1967.

    Article  MathSciNet  Google Scholar 

  23. M. Ratner, The central limit theorem for geodesic flows on n-dimensional manifolds of negative curvature, Israel Journal of Mathematics 16 (1973), 181–197.

    Article  MathSciNet  Google Scholar 

  24. N. Simányi, Towards a proof of recurrence for the Lorentz process. in Dynamical Systems and Ergodic Theory (Warsaw, 1986), Banach Center Publications, Vol. 23, PWN, Warsaw, 1989, pp. 265–276.

    Google Scholar 

  25. Ja. G. Sinaĭ, Dynamical systems with elastic reflections. Ergodic properties of dispersing billiards, Uspehi Matematiceskih Nauk 25 (1970), 141–192.

    MathSciNet  MATH  Google Scholar 

  26. D. Szász and T. Varjú, Limit laws and recurrence for the planar Lorentz process with infinite horizon, Journal of Statistical Physics 129 (2007), 59–80.

    Article  MathSciNet  Google Scholar 

  27. D. Thomine, Théorèmes limites pour les sommes de Birkhoff de fonctions d’intégrale nulle en théorie ergodique en mesure infinie, PhD thesis, Université de Rennes 1, 2013 version.

  28. D. Thomine, A generalized central limit theorem in infinite ergodic theory, Probability Theory and Related Fields 158 (2014), 597–636.

    Article  MathSciNet  Google Scholar 

  29. D. Thomine, Variations on a central limit theorem in infinite ergodic theory, Ergodic Theory and Dynamical Systems 35 (2015), 1610–1657.

    Article  MathSciNet  Google Scholar 

  30. L.-S. Young, Statistical properties of dynamical systems with some hyperbolicity, Annals of Mathematics 147 (1998), 585–650.

    Article  MathSciNet  Google Scholar 

  31. R. Zweimüller, Mixing limit theorems for ergodic transformations, Journal of Theoretical Probability 20 (2007), 1059–1071.

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

This research has been done mostly in the Mathematical Departments of Brest and Orsay Universities, and also at the Institut Henri Poincaré that we thank for their hospitalities. FP is grateful to the Institut Universitaire de France (IUF) for its important support.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Françoise Pène.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pène, F., Thomine, D. Central limit theorems for the ℤ2-periodic Lorentz gas. Isr. J. Math. 241, 539–582 (2021). https://doi.org/10.1007/s11856-021-2106-4

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-021-2106-4

Navigation