Limit theorems for locally perturbed planar Lorentz processes

Duke Mathematical Journal

Volumn 148, Issue 3, 2009, Pages 459-499

  Dolgopyat, Dmitry ^a   Szász, Domokos ^a   Varjú, Tamás ^a  

^a NONE

Author keywords

[No Author keywords available]

Indexed keywords

EID: 70449673509     PISSN: 00127094     EISSN: None     Source Type: Journal    
DOI: 10.1215/00127094-2009-031     Document Type: Article

References (30)

1. P. Billingsley, Convergence of Probability Measures, Wiley, New York, 1968. Mathematical Reviews (MathSciNet): MR0233396
2. L. A. Bunimovich and Y. G. Sinai, Statistical properties of Lorentz gas with periodic configuration of scatterers, Comm. Math. Phys. 78 (1980/1981), 479-497. Mathematical Reviews (MathSciNet): MR0606459 Digital Object Identifier: doi:10.1007/BF02046760 Project Euclid: euclid.cmp/1103908786 Zentralblatt MATH: 0459.60099
3. L. A. Bunimovich, Y. G. Sinai, and N. I. Chernov, Statistical properties of two-dimensional hyperbolic billiards, (in Russian), Uspekhi Mat. Nauk 46 (1991), 43-92.; English translation in Russian Math. Surveys 46 (1991), 47-106. Mathematical Reviews (MathSciNet): MR1138952
4. N. I. Chernov, Decay of correlations and dispersing billiards, J. Stat. Phy. 94 (1999), 513-556. Mathematical Reviews (MathSciNet): MR1675363 Digital Object Identifier: doi:10.1023/A:1004581304939 Zentralblatt MATH: 1047.37503
5. N. I. Chernov, Sinai billiards under small external forces, Ann. Henri Poincaré 2 (2001), 197-236. Mathematical Reviews (MathSciNet): MR1832968 Digital Object Identifier: doi:10.1007/PL00001034
6. N. I. Chernov, Advanced statistical properties of dispersing billiards, J. Stat. Phys. 122 (2006), 1061-1094. Mathematical Reviews (MathSciNet): MR2219528 Digital Object Identifier: doi:10.1007/s10955-006-9036-8 Zentralblatt MATH: 1098.82020
7. N. Chernov and D. Dolgopyat, Hyperbolic billiards and statistical physics in Proceedings of the International Congress of Mathematicians, Vol. 2, Eur. Math. Soc., Zürich, 2006, 1679-1704. Mathematical Reviews (MathSciNet): MR2275665 Zentralblatt MATH: 1118.37019
8. N. Chernov and D. Dolgopyat, Particle drift in self-similar billiards, Ergodic Theory Dynam. Systems 28 (2008), 389-403. Mathematical Reviews (MathSciNet): MR2408384 Digital Object Identifier: doi:10.1017/S0143385707000739 Zentralblatt MATH: 1154.37339
9. N. Chernov and D. Dolgopyat, Brownian Brownian Motion-1, Mem. Amer. Math. Soc. 198 (2009), no. 927.
10. N. Chernov and D. Dolgopyat, The Galton board: limit theorems and recurrence, J. Amer. Math. Soc. 22 (2009), 821-858.
11. N. Chernov and R. Markarian, Chaotic Billiards, Math. Surveys and Monog. 127, Amer. Math. Soc., Providence, 2006. Mathematical Reviews (MathSciNet): MR2229799
12. D. Dolgopyat, Fermi acceleration, Contemp. Math. 469 (2008), 149-166. Mathematical Reviews (MathSciNet): MR2478469 Zentralblatt MATH: 05380242
13. D. Dolgopyat, D. SzáSz, and T. Varjú, Recurrence properties of planar Lorentz process, Duke Math J. 142 (2008), 241-281. Mathematical Reviews (MathSciNet): MR2401621 Digital Object Identifier: doi:10.1215/00127094-2008-006 Project Euclid: euclid.dmj/1206642155 Zentralblatt MATH: 1136.37022
14. J.-P. Eckmann and L.-S. Young, Nonequilibrium energy profiles for a class of 1-D models, Comm. Math. Phys. 262 (2006), 237-267. Mathematical Reviews (MathSciNet): MR2200889 Digital Object Identifier: doi:10.1007/s00220-005-1462-y Zentralblatt MATH: 1113.82066
15. M. Freidlin, Markov Processes and Differential Equations: Asymptotic Problems, Lectures in Mathematics ETH Zürich, Birkhäuser, Basel, 1996. Mathematical Reviews (MathSciNet): MR1399081 Zentralblatt MATH: 0863.60049
16. J. M. Harrison and L. A. Shepp, On skew Brownian motion, Ann. Probab. 9 (1981), 309-313. Mathematical Reviews (MathSciNet): MR0606993 Digital Object Identifier: doi:10.1214/aop/1176994472 Project Euclid: euclid.aop/1176994472 Zentralblatt MATH: 0462.60076
17. M. Lenci, Typicality of recurrence for Lorentz guses, Ergodic Theory Dynam. Systems 26 (2006), 799-820. Mathematical Reviews (MathSciNet): MR2237472 Digital Object Identifier: doi:10.1017/S0143385706000022 Zentralblatt MATH: 1098.82028
18. T. Lindvall, Weak convergence of probability measures and random functions in the function space D(0,∞), J. Appl. Probab. 10 (1973), 109-121. Mathematical Reviews (MathSciNet): MR0362429 Digital Object Identifier: doi:10.2307/3212499 JSTOR: links.jstor.org Zentralblatt MATH: 0258.60008
19. R. Markarian, Billiards with polynomial decay of correlations, Ergodic Theory Dynam. Systems 24 (2004), 177-197. Mathematical Reviews (MathSciNet): MR2041267 Digital Object Identifier: doi:10.1017/S0143385703000270 Zentralblatt MATH: 1115.37037
20. I. Melbourne and M. Nicol, A vector-valued almost sure invariance principle for hyperbolic dynamical systems, to appear in Ann. Prob., preprint,\arxivmath/0606535v1[math.DS] R. Pinsky, Normally reflected Brownian motion and spectral properties of the Neumann Laplacian in unbounded domains, preprint,\arxiv0803.3748v1[math.PR]
21. Y. G. Sinai [J. G. Sinaǐ], Dynamical systems with elastic reflections: Ergodic properties of dispersing billiards (in Russian), Uspekhi Mat. Nauk 25 (1970), 141-192.; English translation in Russian Math. Surveys 25 (1970), 137-189.
22. D. W. Stroock and S. R. S. Varadhan, Diffusion processes with boundary conditions, Comm. Pure Appl. Math. 24 (1971), 147-225. Mathematical Reviews (MathSciNet): MR0277037 Digital Object Identifier: doi:10.1002/cpa.3160240206 Zentralblatt MATH: 0227.76131
23. Multidimensional Diffusion Processes, reprint of the 1997 edition, Classics in Math., Springer, Berlin, 2006. Mathematical Reviews (MathSciNet): MR2190038
24. D. SzáSz, Some challenges in the theory of (semi)-dispersing billiards, Nonlinearity 21 (2008), T187-T193. Mathematical Reviews (MathSciNet): MR2439473 Zentralblatt MATH: 1155.37027
25. D. SzáSz and A. Telcs, Random walk in an inhomogeneous medium with local impurities, J. Statist. Phys. 26 (1981), 527-537. Mathematical Reviews (MathSciNet): MR0648199 Digital Object Identifier: doi:10.1007/BF01011433 Zentralblatt MATH: 0509.60071
26. D. SzáSz and T. Varjú, Local limit theorem for the Lorentz process and its recurrence in the plane, Ergodic Theory Dynam. Systems 24 (2004), 257-278. Mathematical Reviews (MathSciNet): MR2041271 Digital Object Identifier: doi:10.1017/S0143385703000439 Zentralblatt MATH: 1115.37009
27. L.-S. Young, Statistical properties of dynamical systems with some hyperbolicity, Ann. of Math. 147 (1998), 585-650. Mathematical Reviews (MathSciNet): MR1637655 Digital Object Identifier: doi:10.2307/120960 JSTOR: links.jstor.org Zentralblatt MATH: 0945.37009
28. Recurrence times and rates of mixing, Israel J. Math. 110 (1999), 153-188. Mathematical Reviews (MathSciNet): MR1750438 Digital Object Identifier: doi:10.1007/BF02808180 Zentralblatt MATH: 0983.37005
29. O. Zeitouni, "Random walks in random environment" in Lectures on Probability Theory and Statistics, Lecture Notes in Math. 1837, Springer, Berlin, 2004, 189-312.
30. Zeitouni. O Random walks in random environments, J. Phys. A 39 (2006), R433-R464. Mathematical Reviews (MathSciNet): MR2261885 Digital Object Identifier: doi:10.1088/0305-4470/39/40/R01 Zentralblatt MATH: 1108.60085