Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

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

Pages: 567 - 587, Volume 24, Issue 2, June 2009      doi:10.3934/dcds.2009.24.567

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Françoise Pène - Université Européenne de Bretagne, Université de Brest, Laboratoire de Mathématiques UMR CNRS 6205, 6 avenue Le Gorgeu, 29285 Brest cedex, France (email)

Abstract: 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, Lorentz gas, range, billiard, limit theorem.
Mathematics Subject Classification:  37D50, 60F99.

Received: March 2008;      Revised: August 2008;      Available Online: March 2009.