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Self-intersections of trajectories of the Lorentz process
1. | Université de Brest, UMR CNRS 6205, Laboratoire de Mathématique de Bretagne Atlantique, 6 avenue Le Gorgeu, 29238 Brest cedex, France |
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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