Ballistic regime for random walks in random environment with unbounded jumps and Knudsen billiards
Annales de l'I.H.P. Probabilités et statistiques, Volume 48 (2012) no. 3, p. 721-744

We consider a random walk in a stationary ergodic environment in , with unbounded jumps. In addition to uniform ellipticity and a bound on the tails of the possible jumps, we assume a condition of strong transience to the right which implies that there are no “traps.” We prove the law of large numbers with positive speed, as well as the ergodicity of the environment seen from the particle. Then, we consider Knudsen stochastic billiard with a drift in a random tube in d , d3, which serves as environment. The tube is infinite in the first direction, and is a stationary and ergodic process indexed by the first coordinate. A particle is moving in straight line inside the tube, and has random bounces upon hitting the boundary, according to the following modification of the cosine reflection law: the jumps in the positive direction are always accepted while the jumps in the negative direction may be rejected. Using the results for the random walk in random environment together with an appropriate coupling, we deduce the law of large numbers for the stochastic billiard with a drift.

Nous considérons une marche aléatoire dans un milieu stationnaire ergodique sur , avec des sauts non bornés. En plus de l’uniforme ellipticité et d’une borne uniforme sur la queue de la loi des sauts, nous supposons une condition de transience forte qui garantit l’absence de “pièges.” Nous montrons la loi des grands nombres avec vitesse strictement positive, ainsi que l’ergodicité de l’environnement vu de la particule. Par ailleurs, nous étudions aussi le billard stochastique de Knudsen avec dérive dans un tube aléatoire dans d , d3, qui constitue l’environnement. Le tube est infini dans la première direction, et, vu comme un processus indéxé par la première coordonnée, il est supposé stationnaire ergodique. Une particule se déplace en ligne droite à l’intérieur du tube, avec des rebonds aléatoires sur le bord, selon la modification suivante de la loi de reflexion en cosinus: les sauts dans la direction positive sont toujours acceptés, tandis que ceux dans l’autre direction peuvent être rejetés. En utilisant les résultats pour la marche aléatoire en milieu aléatoire et un couplage approprié, nous obtenons la loi des grands nombres pour le billard stochastique avec dérive.

DOI : https://doi.org/10.1214/11-AIHP439
Classification:  Primary 60K37,  secondary,  37D50,  60J25
Keywords: cosine law, stochastic billiard, Knudsen random walk, random medium, random walk in random environment, unbounded jumps, stationary ergodic environment, regenerative structure, point of view of the particle
@article{AIHPB_2012__48_3_721_0,
     author = {Comets, Francis and Popov, Serguei},
     title = {Ballistic regime for random walks in random environment with unbounded jumps and Knudsen billiards},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {48},
     number = {3},
     year = {2012},
     pages = {721-744},
     doi = {10.1214/11-AIHP439},
     zbl = {1247.60139},
     mrnumber = {2976561},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2012__48_3_721_0}
}
Comets, Francis; Popov, Serguei. Ballistic regime for random walks in random environment with unbounded jumps and Knudsen billiards. Annales de l'I.H.P. Probabilités et statistiques, Volume 48 (2012) no. 3, pp. 721-744. doi : 10.1214/11-AIHP439. http://www.numdam.org/item/AIHPB_2012__48_3_721_0/

[1] S. Alili. Asymptotic behaviour for random walks in random environments. J. Appl. Probab. 36 (1999) 334-349. | Zbl 0946.60046

[2] E. D. Andjel. A zero or one law for one-dimensional random walks in random environments. Ann. Probab. 16 (1988) 722-729. | Zbl 0642.60022

[3] E. Bolthausen and I. Goldsheid. Lingering random walks in random environment on a strip. Commun. Math. Phys. 278 (2008) 253-288. | Zbl 1142.82007

[4] J. Bremont. One-dimensional finite range random walk in random medium and invariant measure equation. Ann. Inst. Henri Poincaré Probab. Stat. 45 (2009) 70-103. | Numdam | Zbl 1171.60395

[5] F. Comets, S. Popov, G. M. Schütz and M. Vachkovskaia. Billiards in a general domain with random reflections. Arch. Ration. Mech. Anal. 191 (2009) 497-537. Erratum: Arch. Ration. Mech. Anal. 193 737-738. | Zbl 1186.37049

[6] F. Comets, S. Popov, G. M. Schütz and M. Vachkovskaia. Quenched invariance principle for Knudsen stochastic billiard in random tube. Ann. Probab. 38 (2010) 1019-1061. | MR 2674993 | Zbl 1200.60091

[7] F. Comets, S. Popov, G. M. Schütz and M. Vachkovskaia. Knudsen gas in a finite random tube: Transport diffusion and first passage properties. J. Statist. Phys. 140 (2010) 948-984. | MR 2673342 | Zbl 1197.82058

[8] R. Feres. Random walks derived from billiards. In Dynamics, Ergodic Theory, and Geometry 179-222. Math. Sci. Res. Inst. Publ. 54, Cambridge Univ. Press, Cambridge, 2007. | MR 2369447 | Zbl 1145.37009

[9] I. Goldsheid. Linear and sub-linear growth and the CLT for hitting times of a random walk in random environment on a strip. Probab. Theory Related Fields 141 (2008) 471-511. | MR 2391162 | Zbl 1141.60070

[10] E. Key. Recurrence and transience criteria for a random walk in a random environment. Ann. Probab. 12 (1984) 529-560. | MR 735852 | Zbl 0545.60066

[11] J. Kemeny and J. L. Snell. Finite Markov Chains. Springer-Verlag, New York, 1976. | MR 410929 | Zbl 0328.60035

[12] J. Ledoux. On weak lumpability of denumerable Markov chains. Statist. Probab. Lett. 25 (1995) 329-339. | MR 1363233 | Zbl 0843.60060

[13] M. Menshikov, M. Vachkovskaia and A. Wade. Asymptotic behaviour of randomly reflecting billiards in unbounded tubular domains. J. Statist. Phys. 132 (2008) 1097-1133. | MR 2430776 | Zbl 1157.82036

[14] L. Shen. Asymptotic properties of certain anisotropic walks in random media. Ann. Appl. Probab. 12 (2002) 477-510. | MR 1910636 | Zbl 1016.60092

[15] F. Solomon. Random walks in a random environment. Ann. Probab. 3 (1975) 1-31. | MR 362503 | Zbl 0305.60029

[16] A.-S. Sznitman. Topics in random walks in random environment. In School and Conference on Probability Theory 203-266 (electronic). ICTP Lect. Notes XVII. Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2004. | MR 2198849 | Zbl 1060.60102

[17] H. Thorisson. Coupling, Stationarity, and Regeneration. Springer-Verlag, New York, 2000. | MR 1741181 | Zbl 0949.60007

[18] O. Zeitouni. Random walks in random environment. In Lectures on Probability Theory and Statistics. Ecole d'Eté de probabilités de Saint-Flour XXXI-2001 191-312. Lecture Notes in Math. 1837. Springer, Berlin, 2000. | MR 2071631 | Zbl 1060.60103