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.
Mots clés : return time, random walk, subshift of finite type, recurrence, local limit theorem
@article{AIHPB_2009__45_4_1065_0, author = {P\`ene, Fran\c{c}oise and Saussol, Beno{\^\i}t}, title = {Quantitative recurrence in two-dimensional extended processes}, journal = {Annales de l'I.H.P. Probabilit\'es et statistiques}, pages = {1065--1084}, publisher = {Gauthier-Villars}, volume = {45}, number = {4}, year = {2009}, doi = {10.1214/08-AIHP195}, mrnumber = {2572164}, zbl = {1230.37017}, language = {en}, url = {http://www.numdam.org/articles/10.1214/08-AIHP195/} }
TY - JOUR AU - Pène, Françoise AU - Saussol, Benoît TI - Quantitative recurrence in two-dimensional extended processes JO - Annales de l'I.H.P. Probabilités et statistiques PY - 2009 SP - 1065 EP - 1084 VL - 45 IS - 4 PB - Gauthier-Villars UR - http://www.numdam.org/articles/10.1214/08-AIHP195/ DO - 10.1214/08-AIHP195 LA - en ID - AIHPB_2009__45_4_1065_0 ER -
%0 Journal Article %A Pène, Françoise %A Saussol, Benoît %T Quantitative recurrence in two-dimensional extended processes %J Annales de l'I.H.P. Probabilités et statistiques %D 2009 %P 1065-1084 %V 45 %N 4 %I Gauthier-Villars %U http://www.numdam.org/articles/10.1214/08-AIHP195/ %R 10.1214/08-AIHP195 %G en %F 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/articles/10.1214/08-AIHP195/
[1] Hausdorff dimension of measures via Poincaré recurrence. Commun. Math. Phys. 219 (2001) 443-463. | MR | Zbl
and .[2] Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. Lecture Note in Mathematics 470. Springer, Berlin, 1975. | MR | Zbl
.[3] Probability. Addison-Wesley, Reading, MA, 1968. | MR | Zbl
.[4] Repetition time for gibbsian sources. Nonlinearity 12 (1999) 1225-1237. | MR | Zbl
, and .[5] A note on recurrent random walks. Statist. Probab. Lett. 76 (2006) 1025-1031. | MR | Zbl
.[6] Sur un critère de récurrence en dimension 2 pour les marches stationnaires, applications. Ergodic Theory Dynam. Systems 19 (1999) 1233-1245. | MR | Zbl
.[7] Some problems on random walk in space. In Proc. Berkeley Sympos. Math. Statist. Probab. 353-367. California Univ. Press, Berkeley-Los Angeles, 1951. | MR | Zbl
and .[8] 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 | Zbl
and .[9] Poisson law for Axiom A diffeomorphism. Ergodic Theory Dynam. Systems 13 (1993) 533-556. | MR | Zbl
.[10] Limit Theorems for Markov Chains and Stochastic Properties of Dynamical Systems by Quasi-Compactness. Lecture Notes in Mathematics 1766. Springer, Berlin, 2001. | MR | Zbl
and .[11] Some limit theorems for stationary Markov chains. Theory Probab. Appl. 2 (1957) 378-406. (Translation from Teor. Veroyatn. Primen. 2 (1958) 389-416.) | MR | Zbl
.[12] 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 | Zbl
.[13] Entropy and data compression. IEEE Trans. Inform. Theory 39 (1993) 78-83. | MR | Zbl
and .[14] Recurrence rate in rapidly mixing dynamical systems. Discrete Contin. Dyn. Syst. 15 (2006) 259-267. | MR | Zbl
.[15] Recurrence, dimension and Lyapunov exponents. J. Stat. Phys. 106 (2002) 623-634. | MR | Zbl
, and .[16] On joint recurrence. C. R. Acad. Sci. Paris 327 (1998) 837-842. | MR | Zbl
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