Advertisement

Local Limit Theorem for Randomly Deforming Billiards

  • Mark F. DemersEmail author
  • Françoise Pène
  • Hong-Kun Zhang
Article
  • 21 Downloads

Abstract

We study limit theorems in the context of random perturbations of dispersing billiards in finite and infinite measure. In the context of a planar periodic Lorentz gas with finite horizon, we consider random perturbations in the form of movements and deformations of scatterers. We prove a central limit theorem for the cell index of planar motion, as well as a mixing local limit theorem for piecewise Hölder continuous observables. In the context of the infinite measure random system, we prove limit theorems regarding visits to new obstacles and self-intersections, as well as decorrelation estimates. The main tool we use is the adaptation of anisotropic Banach spaces to the random setting.

Notes

Acknowledgements

This work was begun at the AIM workshop Stochastic Methods for Non-Equilibrium Dynamical Systems, in June 2015. Part of this work was carried out during visits by the authors to ESI, Vienna in 2016, to CIRM, Luminy in 2017 and 2018, and to BIRS, Canada in 2018, and by a visit of FP to the University of Massachusetts at Amherst in 2018. MD was supported in part by NSF Grant DMS 1800321. FP is grateful to the IUF for its important support.

References

  1. 1.
    Aaronson, J., Denker, M.: Local limit theorems for partial sums of stationary sequences generated by Gibbs–Markov maps. Stoch. Dyn. 1(2), 193–237 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Aimino, R., Nicol, M., Vaienti, S.: Annealed and quenched limit theorems for random expanding dynamical systems. Probab. Theory Relat. Fields 162(1), 233–274 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Bálint, P., Tóth, P.: Correlation decay in certain soft billiards. Commun. Math. Phys. 243, 55–91 (2003)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Billingsley, P.: Convergence of Probability Measures, 2nd edn. Wiley, New York (1999)zbMATHCrossRefGoogle Scholar
  5. 5.
    Bolthausen, E.: A central limit theorem for two-dimensional random walks in random sceneries. Ann. Probab. 17, 108–115 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Borodin, A.N.: A limit theorem for sums of independent random variables defined on a recurrent random walk. (Russian) Dokl Akad. Nauk SSSR 246(4), 786–787 (1979)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Borodin, A.N.: Limit theorems for sums of independent random variables defined on a transient random walk. Investigations in the theory of probability distributions, IV. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 85, 17–29, 237, 244 (1979)Google Scholar
  8. 8.
    Bunimovich, L.A., Sinai, Y.G., Chernov, N.I.: Markov partitions for two-dimensional billiards. Russ. Math. Surv. 45, 105–152 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Bunimovich, L.A., Sinai, Y.G., Chernov, N.I.: Statistical properties of two-dimensional hyperbolic billiards. Russ. Math. Surv. 46, 47–106 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Castell, F., Guillotin-Plantard, N., Pène, F.: Limit theorems for one and two-dimensional random walks in random scenery. Ann. Inst. Henri Poincaré 49, 506–528 (2013)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Chernov, N.: Sinai billiards under small external forces. Ann. Henri Poincaré 2(2), 197–236 (2001)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Chernov, N., Markarian, R.: Chaotic Billiards. Math. Surveys and Monographs, vol. 127. AMS, Providence, RI (2006)zbMATHCrossRefGoogle Scholar
  13. 13.
    Cohen, G., Conze, J.-P.: On the quenched functional CLT in 2d random sceneries, examples. Preprint arXiv:1908.03777
  14. 14.
    Deligiannidis, G., Utev, S.: An asymptotic variance of the self-intersections of random walks. Sib. Math. J. 52, 639–650 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Demers, M.F., Zhang, H.-K.: Spectral analysis of the transfer operator for the Lorentz gas. J. Mod. Dyn. 5(4), 665–709 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Demers, M.F., Zhang, H.-K.: A functional analytic approach to perturbations of the Lorentz gas. Commun. Math. Phys. 324(3), 767–830 (2013)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Demers, M.F., Zhang, H.-K.: Spectral analysis of hyperbolic systems with singularities. Nonlinearity 27, 379–433 (2014)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Dolgopyat, D., Szász, D., Varjú, T.: Recurrence properties of planar Lorentz process. Duke Math. J. 142, 241–281 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Dunford, N., Schwartz, J.T.: Linear Operators. Part I: General Theory. Pure and Applied Mathematics, vol. VII. Wiley, New York (1964)zbMATHGoogle Scholar
  20. 20.
    Dvoretzky, A., Erdös, P.: Some problems on random walk in space. In: Proc. Berkeley Sympos. Math. Statist. Probab., pp. 353–367 (1955)Google Scholar
  21. 21.
    Guillotin-Plantard, N., Dos Santos, R.S., Poisat, J.: A quenched central limit theorem for planar random walks in random sceneries. Electron. Commun. Probab. 19(3), 1–9 (2014)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Guivarc’h, Y., Hardy, J.: Théorèmes limites pour une classe de chaînes de Markov et applications aux difféomorphismes d’Anosov. Ann. Inst. Henri Poincaré 24(1), 73–98 (1988)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Hennion, H., Hervé, L.: Stable laws and products of positive random matrices. J. Theor. Probab. 21(4), 966–981 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Kalikow, S.A.: \(T, T^{-1}\) transformation is not loosely Bernoulli. Ann. Math. Second Ser. 115(2), 393–409 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Keller, G., Liverani, C.: Stability of the spectrum for transfer operators. Annali della Scuola Normale Superiore di Pisa, Scienze Fisiche e Matematiche, (4) XXVIII, 141–152 (1999)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Kesten, H., Spitzer, F.: A limit theorem related to an new class of self similar processes. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 50, 5–25 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Nagaev, S.V.: Some limit theorems for stationary Markov chains. Theory Probab. Appl. 11(4), 378–406 (1957)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Pène, F.: Applications des propriétés stochastiques de billards dispersifs. C. R. Acad. Sci. 330(I), 1103–1106 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Pène, F.: Asymptotic of the number of obstacles visited by the planar Lorentz process. Discrete Contin. Dyn. Syst. Ser. A 24(2), 567–588 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Pène, F.: Planar Lorentz process in a random scenery. Ann. l’Inst. Henri Poincaré, Probab. Stat. 45(3), 818–839 (2009)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Pène, F.: An asymptotic estimate of the variance of the self-intersections of a planar periodic Lorentz process. arXiv:1303.3034
  32. 32.
    Pène, F.: Mixing and decorrelation in infinite measure: the case of the periodic sinai billiard. Ann. Institut Henri Poincaré 55(1), 378–411 (2019)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Pène, F., Saussol, B.: Back to balls in billiards. Commun. Math. Phys. 293, 837–866 (2010)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Pène, F., Thomine, D.: Potential kernel, hitting probabilities and distributional asymptotics. Ergodic Theory Dyn. Syst.  https://doi.org/10.1017/etds.2018.136
  35. 35.
    Szász, D., Varjú, T.: Local limit theorem for the Lorentz process and its recurrence in the plane. Ergodic Theory Dyn. Syst. 24(1), 257–278 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Serfling, R.J.: Moment inequalities for the maximum cumulative sum. Ann. Math. Stat. 41, 1227–1234 (1970)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Sinai, Y.G.: Dynamical systems with elastic reflections. Ergodic properties of dispersing billiards. Uspehi Mat. Nauk 25, 141–192 (1970)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Young, L.-S.: Statistical properties of dynamical systems with some hyperbolicity. Ann. Math. Second Ser. 147(3), 585–650 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Weiss, B.: The isomorphism problem in ergodic theory. Bull. A.M.S. 78, 668–684 (1972)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2020

Authors and Affiliations

  1. 1.Department of MathematicsFairfield UniversityFairfieldUSA
  2. 2.Laboratoire de Mathématique de Bretagne Atlantique, LMBA, UMR CNRS 6205, Institut Universitaire de France, IUFUniv Brest, Université de BrestBrest CedexFrance
  3. 3.Department of Mathematics and StatisticsUniversity of Massachusetts AmherstAmherstUSA

Personalised recommendations