Self-intersections of trajectories of the Lorentz process

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November 2014, 34(11): 4781-4806. doi: 10.3934/dcds.2014.34.4781

Self-intersections of trajectories of the Lorentz process

Françoise Pène ^1,

1.	Université de Brest, UMR CNRS 6205, Laboratoire de Mathématique de Bretagne Atlantique, 6 avenue Le Gorgeu, 29238 Brest cedex, France

Received April 2013 Revised February 2014 Published May 2014

Abstract
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We are interested in the asymptotic behaviour of the number of self-intersections of a trajectory of a Lorentz process in a $\mathbb Z^2$-periodic planar domain with strictly convex obstacles and with finite horizon. We give precise estimates for its expectation and its variance. As a consequence, we establish the almost sure convergence of the self-intersections with a suitable normalization.

Keywords: Lorentz process, Sinai billiard, self intersection..

Mathematics Subject Classification: Primary: 60F99, 37D5.

Citation: Françoise Pène. Self-intersections of trajectories of the Lorentz process. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4781-4806. doi: 10.3934/dcds.2014.34.4781

References:

[1]	E. Bolthausen, A central limit theorem for two-dimensional random walks in random sceneries, Ann. Probab., 17 (1989), 108-115. doi: 10.1214/aop/1176991497.
[2]	L. A. Bunimovich and Ya. G. Sinai, Markov partitions for dispersed billiards,, Comm. Math. Phys., 78 (): 247. doi: 10.1007/BF01942372.
[3]	L. A. Bunimovich and Ya. G. Sinai, Statistical properties of Lorentz gas with periodic configuration of scatterers,, Comm. Math. Phys., 78 (): 479.
[4]	L. A. Bunimovich, Ya. G. Sinai and N. I. Chernov, Markov partitions for two-dimensional hyperbolic billiards, Russian Math. Surveys, 45 (1990), 105-152 (Translation from Uspekhi Mat. Nauk 45 (1990), 97-134). doi: 10.1070/RM1990v045n03ABEH002355.
[5]	L. A. Bunimovich, Ya. G. Sinai and N. I. Chernov, Statistical properties of two-dimensional hyperbolic billiards, Russian Math. Surveys, 46 (1991), 47-106 (Translation from Usp. Mat. Nauk 46 (1991) 43-92). doi: 10.1070/RM1991v046n04ABEH002827.
[6]	X Chen, Random Walk Intersections. Large Deviations and Related Topics, Math. Surv. and Monog., 157. Amer. Math. Soc., Providence, RI, 2010. doi: 10.1090/surv/157.
[7]	N. Chernov and R. Markarian, Chaotic Billiards, Math. Surv. and Monog., 127. Amer. Math. Soc., Providence, RI, 2006. doi: 10.1090/surv/127.
[8]	J.-P. Conze, Sur un critère de récurrence en dimension 2 pour les marches stationnaires, applications, Erg. Th. & Dynam. Syst., 19 (1999), 1233-1245. doi: 10.1017/S0143385799141701.
[9]	G. Deligiannidis and S. Utev, Asymptotic variance of the self-intersections of stable random walks, Sib. Math. J., 52 (2011), 639-650. doi: 10.1134/S0037446611040082.
[10]	D. Dolgopyat, D. Szász and T. Varjú, Recurrence properties of planar Lorentz gas, Duke Math. J., 142 (2008), 241-281. doi: 10.1215/00127094-2008-006.
[11]	A. Dvoretzky and P. Erdös, Some problems on random walk in space, Proc. Berkeley Sympos. math. Statist. Probab., 1950 (1951), 353-367.
[12]	Y. Guivarc'h and J. Hardy, Théorèmes limites pour une classe de chaînes de Markov et applications aux difféomorphismes d'Anosov, Ann. Inst. H. Poincaré (B), Probab. Stat., 24 (1988), 73-98.
[13]	S. V. Nagaev, Some limit theorems for stationary Markov chains, Theor. Probab. Appl., 2 (1957), 378-406; translation from Teor. Veroyatn. Primen., 2 (1958), 389-416.
[14]	S. V. Nagaev, More exact statement of limit theorems for homogeneous Markov chains, Theor. Probab. Appl., 6 (1961), 62-81; translation from Teor. Veroyatn. Primen, 6 (1961), 67-86.
[15]	F. Pène, Applications des propriétés stochastiques de billards dispersifs, C. R. Acad. des Sci., 330 (2000), 1103-1106. doi: 10.1016/S0764-4442(00)00318-9.
[16]	F. Pène, Rates of convergence in the CLT for two-dimensional dispersive billiards, Comm. Math. Phys., 225 (2002), 91-119. doi: 10.1007/s002201000573.
[17]	F. Pène, Planar Lorentz process in a random scenery, Ann. Inst. Henri Poincaré, Probab. Stat., 45 (2009), 818-839. doi: 10.1214/08-AIHP191.
[18]	F. Pène and B. Saussol, Back to balls in billiards, Comm. Math. Phys., 293 (2010), 837-866. doi: 10.1007/s00220-009-0911-4.
[19]	Ya. G. Sinai, Dynamical systems with elastic reflections, Russian Math. Surveys, 25 (1970), 141-192. doi: 10.1070/RM1970v025n02ABEH003794.
[20]	D. Szász and T. Varjú, Local limit theorem for the Lorentz process and its recurrence in the plane, Erg. Th. & Dynam. Syst., 24 (2004), 257-278. doi: 10.1017/S0143385703000439.
[21]	L. -S. Young, Statistical properties of dynamical systems with some hyperbolicity, Ann. Math., 147 (1998), 585-650. doi: 10.2307/120960.

show all references

References:

[1]	E. Bolthausen, A central limit theorem for two-dimensional random walks in random sceneries, Ann. Probab., 17 (1989), 108-115. doi: 10.1214/aop/1176991497.
[2]	L. A. Bunimovich and Ya. G. Sinai, Markov partitions for dispersed billiards,, Comm. Math. Phys., 78 (): 247. doi: 10.1007/BF01942372.
[3]	L. A. Bunimovich and Ya. G. Sinai, Statistical properties of Lorentz gas with periodic configuration of scatterers,, Comm. Math. Phys., 78 (): 479.
[4]	L. A. Bunimovich, Ya. G. Sinai and N. I. Chernov, Markov partitions for two-dimensional hyperbolic billiards, Russian Math. Surveys, 45 (1990), 105-152 (Translation from Uspekhi Mat. Nauk 45 (1990), 97-134). doi: 10.1070/RM1990v045n03ABEH002355.
[5]	L. A. Bunimovich, Ya. G. Sinai and N. I. Chernov, Statistical properties of two-dimensional hyperbolic billiards, Russian Math. Surveys, 46 (1991), 47-106 (Translation from Usp. Mat. Nauk 46 (1991) 43-92). doi: 10.1070/RM1991v046n04ABEH002827.
[6]	X Chen, Random Walk Intersections. Large Deviations and Related Topics, Math. Surv. and Monog., 157. Amer. Math. Soc., Providence, RI, 2010. doi: 10.1090/surv/157.
[7]	N. Chernov and R. Markarian, Chaotic Billiards, Math. Surv. and Monog., 127. Amer. Math. Soc., Providence, RI, 2006. doi: 10.1090/surv/127.
[8]	J.-P. Conze, Sur un critère de récurrence en dimension 2 pour les marches stationnaires, applications, Erg. Th. & Dynam. Syst., 19 (1999), 1233-1245. doi: 10.1017/S0143385799141701.
[9]	G. Deligiannidis and S. Utev, Asymptotic variance of the self-intersections of stable random walks, Sib. Math. J., 52 (2011), 639-650. doi: 10.1134/S0037446611040082.
[10]	D. Dolgopyat, D. Szász and T. Varjú, Recurrence properties of planar Lorentz gas, Duke Math. J., 142 (2008), 241-281. doi: 10.1215/00127094-2008-006.
[11]	A. Dvoretzky and P. Erdös, Some problems on random walk in space, Proc. Berkeley Sympos. math. Statist. Probab., 1950 (1951), 353-367.
[12]	Y. Guivarc'h and J. Hardy, Théorèmes limites pour une classe de chaînes de Markov et applications aux difféomorphismes d'Anosov, Ann. Inst. H. Poincaré (B), Probab. Stat., 24 (1988), 73-98.
[13]	S. V. Nagaev, Some limit theorems for stationary Markov chains, Theor. Probab. Appl., 2 (1957), 378-406; translation from Teor. Veroyatn. Primen., 2 (1958), 389-416.
[14]	S. V. Nagaev, More exact statement of limit theorems for homogeneous Markov chains, Theor. Probab. Appl., 6 (1961), 62-81; translation from Teor. Veroyatn. Primen, 6 (1961), 67-86.
[15]	F. Pène, Applications des propriétés stochastiques de billards dispersifs, C. R. Acad. des Sci., 330 (2000), 1103-1106. doi: 10.1016/S0764-4442(00)00318-9.
[16]	F. Pène, Rates of convergence in the CLT for two-dimensional dispersive billiards, Comm. Math. Phys., 225 (2002), 91-119. doi: 10.1007/s002201000573.
[17]	F. Pène, Planar Lorentz process in a random scenery, Ann. Inst. Henri Poincaré, Probab. Stat., 45 (2009), 818-839. doi: 10.1214/08-AIHP191.
[18]	F. Pène and B. Saussol, Back to balls in billiards, Comm. Math. Phys., 293 (2010), 837-866. doi: 10.1007/s00220-009-0911-4.
[19]	Ya. G. Sinai, Dynamical systems with elastic reflections, Russian Math. Surveys, 25 (1970), 141-192. doi: 10.1070/RM1970v025n02ABEH003794.
[20]	D. Szász and T. Varjú, Local limit theorem for the Lorentz process and its recurrence in the plane, Erg. Th. & Dynam. Syst., 24 (2004), 257-278. doi: 10.1017/S0143385703000439.
[21]	L. -S. Young, Statistical properties of dynamical systems with some hyperbolicity, Ann. Math., 147 (1998), 585-650. doi: 10.2307/120960.

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