American Institute of Mathematical Sciences

Advanced
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

Françoise Pène 1,

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

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
[1]

Mark F. Demers, Hong-Kun Zhang. Spectral analysis of the transfer operator for the Lorentz gas. Journal of Modern Dynamics, 2011, 5 (4) : 665-709. doi: 10.3934/jmd.2011.5.665
[2]

Françoise Pène. Self-intersections of trajectories of the Lorentz process. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4781-4806. doi: 10.3934/dcds.2014.34.4781
[3]

Paolo Maremonti. A remark on the Stokes problem in Lorentz spaces. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1323-1342. doi: 10.3934/dcdss.2013.6.1323
[4]

Manuel Gutiérrez. Lorentz geometry technique in nonimaging optics. Conference Publications, 2003, 2003 (Special) : 386-392. doi: 10.3934/proc.2003.2003.386
[5]

Neal Bez, Sanghyuk Lee, Shohei Nakamura, Yoshihiro Sawano. Sharpness of the Brascamp–Lieb inequality in Lorentz spaces. Electronic Research Announcements, 2017, 24: 53-63. doi: 10.3934/era.2017.24.006
[6]

David Cowan. A billiard model for a gas of particles with rotation. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 101-109. doi: 10.3934/dcds.2008.22.101
[7]

Junjie Zhang, Shenzhou Zheng. Weighted lorentz estimates for nondivergence linear elliptic equations with partially BMO coefficients. Communications on Pure & Applied Analysis, 2017, 16 (3) : 899-914. doi: 10.3934/cpaa.2017043
[8]

A. Carati. On the existence of scattering solutions for the Abraham-Lorentz-Dirac equation. Discrete & Continuous Dynamical Systems - B, 2006, 6 (3) : 471-480. doi: 10.3934/dcdsb.2006.6.471
[9]

Shuang Liang, Shenzhou Zheng. Variable lorentz estimate for stationary stokes system with partially BMO coefficients. Communications on Pure & Applied Analysis, 2019, 18 (6) : 2879-2903. doi: 10.3934/cpaa.2019129
[10]

Lucas C. F. Ferreira, Jhean E. Pérez-López, Élder J. Villamizar-Roa. On the product in Besov-Lorentz-Morrey spaces and existence of solutions for the stationary Boussinesq equations. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2423-2439. doi: 10.3934/cpaa.2018115
[11]

Sun-Sig Byun, Yumi Cho. Lorentz-Morrey regularity for nonlinear elliptic problems with irregular obstacles over Reifenberg flat domains. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 4791-4804. doi: 10.3934/dcds.2015.35.4791
[12]

Stéphane Brull, Pierre Degond, Fabrice Deluzet, Alexandre Mouton. Asymptotic-preserving scheme for a bi-fluid Euler-Lorentz model. Kinetic & Related Models, 2011, 4 (4) : 991-1023. doi: 10.3934/krm.2011.4.991
[13]

Alessio Pomponio. Oscillating solutions for prescribed mean curvature equations: euclidean and lorentz-minkowski cases. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3899-3911. doi: 10.3934/dcds.2018169
[14]

Mathias Staudigl. A limit theorem for Markov decision processes. Journal of Dynamics & Games, 2014, 1 (4) : 639-659. doi: 10.3934/jdg.2014.1.639
[15]

Jean-Pierre Conze, Stéphane Le Borgne, Mikaël Roger. Central limit theorem for stationary products of toral automorphisms. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1597-1626. doi: 10.3934/dcds.2012.32.1597
[16]

James Nolen. A central limit theorem for pulled fronts in a random medium. Networks & Heterogeneous Media, 2011, 6 (2) : 167-194. doi: 10.3934/nhm.2011.6.167
[17]

Micol Amar, Daniele Andreucci, Paolo Bisegna, Roberto Gianni. Homogenization limit and asymptotic decay for electrical conduction in biological tissues in the high radiofrequency range. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1131-1160. doi: 10.3934/cpaa.2010.9.1131
[18]

Arnaud Debussche, Sylvain De Moor, Julien Vovelle. Diffusion limit for the radiative transfer equation perturbed by a Wiener process. Kinetic & Related Models, 2015, 8 (3) : 467-492. doi: 10.3934/krm.2015.8.467
[19]

Manuel Torrilhon. H-Theorem for nonlinear regularized 13-moment equations in kinetic gas theory. Kinetic & Related Models, 2012, 5 (1) : 185-201. doi: 10.3934/krm.2012.5.185
[20]

Oliver Díaz-Espinosa, Rafael de la Llave. Renormalization and central limit theorem for critical dynamical systems with weak external noise. Journal of Modern Dynamics, 2007, 1 (3) : 477-543. doi: 10.3934/jmd.2007.1.477

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (17)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences