Quantitative recurrence in two-dimensional extended processes
Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) no. 4, pp. 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 : 10.1214/08-AIHP195
Classification : 37B20, 37A50, 60Fxx
Mots-clés : return time, random walk, subshift of finite type, recurrence, local limit theorem 
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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://archive.numdam.org/articles/10.1214/08-AIHP195/

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