We consider the periodic planar Lorentz process with convex obstacles (and with finite horizon). In this model, a point particle moves freely with elastic reflection at the fixed convex obstacles. The random scenery is given by a sequence of independent, identically distributed, centered random variables with finite and non-null variance. To each obstacle, we associate one of these random variables. We suppose that each time the particle hits an obstacle, it wins the amount given by the random variable associated to the obstacle. We prove a convergence in distribution to a Wiener process for the total amount won by the particle (normalized by $\sqrt{n\log(n)}$ ) when the time n goes to infinity. Such a result has been established by Bolthausen [Ann. Probab. 17 (1989) 108–115)] in the case of random walks in ℤ2 given by sums of independent identically distributed random variables. We follow the scheme of his proof. The lack of independence will be compensated by some extensions of the local limit theorem proved by Szász and Varjú in [Ergodic Theory Dynam. Systems 24 (2004) 257–278]. This paper answers a question of Szász about the asymptotic behaviour of ∑k=0n−1ζSk where (ζℓ)ℓ is a sequence of i.i.d. centered random variables (with finite and non-null variance) and where Sk is the number of the cell at the kth reflection.
Publié le : 2009-08-15
Classification:
Lorentz process,
Finite horizon,
Random scenery,
Limit theorem,
Billiard,
Infinite measure,
37D50,
60F05
@article{1249391387,
author = {P\`ene, Fran\c coise},
title = {Planar Lorentz process in a random scenery},
journal = {Ann. Inst. H. Poincar\'e Probab. Statist.},
volume = {45},
number = {1},
year = {2009},
pages = { 818-839},
language = {en},
url = {http://dml.mathdoc.fr/item/1249391387}
}
Pène, Françoise. Planar Lorentz process in a random scenery. Ann. Inst. H. Poincaré Probab. Statist., Tome 45 (2009) no. 1, pp. 818-839. http://gdmltest.u-ga.fr/item/1249391387/