Quantitative recurrence in two-dimensional extended processes
Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) no. 4, p. 1065-1084
Sous certaines conditions, une marche aléatoire dans le plan est récurrente. En particulier, chaque trajectoire est dense, et il est naturel d'estimer le temps nécessaire pour revenir dans un petit voisinage de l'origine. Nous nous intéressons à cette question dans le cas de systèmes dynamiques étendus similaires à des marches aléatoires planaires, notamment celui des ℤ2-extension de sous-shifts de type fini mélangeants. Nous déterminons une vitesse de convergence ponctuelle que nous relions à la dimension du processus et nous établissons un résultat de convergence en loi du temps de retour à l'origine, correctement normalisé.
Under some mild condition, a random walk in the plane is recurrent. In particular each trajectory is dense, and a natural question is how much time one needs to approach a given small neighbourhood of the origin. We address this question in the case of some extended dynamical systems similar to planar random walks, including ℤ2-extension of mixing subshifts of finite type. We define a pointwise recurrence rate and relate it to the dimension of the process, and establish a result of convergence in distribution of the rescaled return time near the origin.
DOI : https://doi.org/10.1214/08-AIHP195
Classification:  37B20,  37A50,  60Fxx
@article{AIHPB_2009__45_4_1065_0,
     author = {P\`ene, Fran\c coise and Saussol, Beno\^\i t},
     title = {Quantitative recurrence in two-dimensional extended processes},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {45},
     number = {4},
     year = {2009},
     pages = {1065-1084},
     doi = {10.1214/08-AIHP195},
     zbl = {1230.37017},
     mrnumber = {2572164},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2009__45_4_1065_0}
}
Pène, Françoise; Saussol, Benoît. Quantitative recurrence in two-dimensional extended processes. Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) no. 4, pp. 1065-1084. doi : 10.1214/08-AIHP195. http://www.numdam.org/item/AIHPB_2009__45_4_1065_0/

[1] L. Barreira and B. Saussol. Hausdorff dimension of measures via Poincaré recurrence. Commun. Math. Phys. 219 (2001) 443-463. | MR 1833809 | Zbl 1007.37012

[2] R. Bowen. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. Lecture Note in Mathematics 470. Springer, Berlin, 1975. | MR 2423393 | Zbl 0308.28010

[3] L. Breiman. Probability. Addison-Wesley, Reading, MA, 1968. | MR 229267 | Zbl 0174.48801

[4] P. Collet, A. Galves and B. Schmitt. Repetition time for gibbsian sources. Nonlinearity 12 (1999) 1225-1237. | MR 1709841 | Zbl 0945.60017

[5] D. Cheliotis. A note on recurrent random walks. Statist. Probab. Lett. 76 (2006) 1025-1031. | MR 2269338 | Zbl 1090.60041

[6] J.-P. Conze. Sur un critère de récurrence en dimension 2 pour les marches stationnaires, applications. Ergodic Theory Dynam. Systems 19 (1999) 1233-1245. | MR 1721618 | Zbl 0973.37007

[7] A. Dvoretzky and P. Erdös. Some problems on random walk in space. In Proc. Berkeley Sympos. Math. Statist. Probab. 353-367. California Univ. Press, Berkeley-Los Angeles, 1951. | MR 47272 | Zbl 0044.14001

[8] Y. Guivarc'H and J. Hardy. Théorèmes limites pour une classe de chaînes de Markov et applications aux difféomorphismes d'Anosov. Ann. Inst. H. Poincaré, Probab. Statist. 24 (1988) 73-98. | Numdam | MR 937957 | Zbl 0649.60041

[9] M. Hirata. Poisson law for Axiom A diffeomorphism. Ergodic Theory Dynam. Systems 13 (1993) 533-556. | MR 1245828 | Zbl 0828.58026

[10] H. Hennion and L. Hervé. Limit Theorems for Markov Chains and Stochastic Properties of Dynamical Systems by Quasi-Compactness. Lecture Notes in Mathematics 1766. Springer, Berlin, 2001. | MR 1862393 | Zbl 0983.60005

[11] S. V. Nagaev. Some limit theorems for stationary Markov chains. Theory Probab. Appl. 2 (1957) 378-406. (Translation from Teor. Veroyatn. Primen. 2 (1958) 389-416.) | MR 94846 | Zbl 0078.31804

[12] S. V. Nagaev. More exact statement of limit theorems for homogeneous Markov chains. Theory Probab. Appl. 6 (1961) 62-81. (Translation from Teor. Veroyatn. Primen 6 (1961) 67-86.) | MR 131291 | Zbl 0116.10602

[13] D. Ornstein and B. Weiss. Entropy and data compression. IEEE Trans. Inform. Theory 39 (1993) 78-83. | MR 1211492 | Zbl 0764.94003

[14] B. Saussol. Recurrence rate in rapidly mixing dynamical systems. Discrete Contin. Dyn. Syst. 15 (2006) 259-267. | MR 2191396 | Zbl 1175.37006

[15] B. Saussol, S. Troubetzkoy and S. Vaienti. Recurrence, dimension and Lyapunov exponents. J. Stat. Phys. 106 (2002) 623-634. | MR 1884547 | Zbl 1138.37300

[16] K. Schmidt. On joint recurrence. C. R. Acad. Sci. Paris 327 (1998) 837-842. | MR 1663750 | Zbl 0923.60090