The problem of recurrence for the planar Lorentz gas
2003, Volume 8, Number 4, pp. 395-411
Author(s): Kramli A.
Author(s): Kramli A.
This paper is a brief survey of solving the problem of the recurrence for planar Lorentz process. There are two different ways to do this.
1. Using Lai-Sang Young's construction [27] one proves the local central limit theorem from which Pólya's theorem is then deduced — this is the method of D.Szász and T.Varjú [25].
2. Klaus Schmidt [21] and J.-P.Conze [8] proved that the recurrence of the planar Lorentz process follows from the global central limit theorem, established by Bunimovich and Sinai [7].
The history of the problem and the main ingredients of the proofs are given. The details of K.Schmidt's method are analysed in the Appendix written by V.Bognár.
1. Using Lai-Sang Young's construction [27] one proves the local central limit theorem from which Pólya's theorem is then deduced — this is the method of D.Szász and T.Varjú [25].
2. Klaus Schmidt [21] and J.-P.Conze [8] proved that the recurrence of the planar Lorentz process follows from the global central limit theorem, established by Bunimovich and Sinai [7].
The history of the problem and the main ingredients of the proofs are given. The details of K.Schmidt's method are analysed in the Appendix written by V.Bognár.
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