Asymptotic of the number of obstacles visited by the planar Lorentz process

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June 2009, 24(2): 567-587. doi: 10.3934/dcds.2009.24.567

Asymptotic of the number of obstacles visited by the planar Lorentz process

1.	Université Européenne de Bretagne, Université de Brest, Laboratoire de Mathématiques UMR CNRS 6205, 6 avenue Le Gorgeu, 29285 Brest cedex, France

Received March 2008 Revised August 2008 Published March 2009

Abstract
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We are interested in the planar Lorentz process with a periodic configuration of strictly convex obstacles and with finite horizon. Its recurrence comes from a criteria of Conze in [8] or of Schmidt in [15] and from the central limit theorem for the billiard in the torus ([2,4,19]) Another way to prove recurrence is given by Szász and Varjú in [18]. Total ergodicity follows from these results (see [16] and [12]). In this paper we answer a question of Szász about the asymptotic behaviour of the number of visited cells when the time goes to infinity. It is not more difficult to study the asymptotic of the number of obstacles hit by the particle when the time goes to infinity. We give an estimate for the expectation and a result of almost sure convergence. For the simple random walk in Z², this question has been studied by Dvoretzky and Erdös in [10]. We adapt the proof of Dvoretzky and Erdös. The lack of independence is compensated by a strong decorrelation result due to Chernov ([6])and by some refinement (got in [14])of the local limit theorem proved by Szász and Varjú in [18].

Keywords: Lorentz process, limit theorem., Lorentz gas, billiard, range.

Mathematics Subject Classification: 37D50, 60F9.

Citation: Françoise Pène. Asymptotic of the number of obstacles visited by the planar Lorentz process. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 567-587. doi: 10.3934/dcds.2009.24.567

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