Favorite sites of randomly biased walks on a supercritical Galton-Watson tree

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:128

Article

Chen, Dayue¹  de Raphelis, Loic²  Hu, Yueyun³ 

[1] Peking Univ, Sch Math Sci, Beijing 100871, Peoples R China

[2] ENS Lyon, UMPA, 46 Allee Italie, F-69364 Lyon 07, France

[3] Univ Paris XIII, LAGA, 99 Ave J-B Clement, F-93430 Villetaneuse, France

关键词: Biased random walk on the Galton-Watson tree;    Local times;    Favorite sites;    Multitype Galton-Watson tree;   

DOI  :  10.1016/j.spa.2017.08.002

来源: Elsevier

【 摘 要 】

Erdos and Revesz (1984) initiated the study of favorite sites by considering the one-dimensional simple random walk. We investigate in this paper the same problem for a class of null-recurrent randomly biased walks on a supercritical Galton-Watson tree. We prove that there is some parameter k is an element of (1, infinity] such that the set of the favorite sites of the biased walk is almost surely bounded in the case k is an element of (2, infinity], tight in the case k = 2, and oscillates between a neighborhood of the root and the boundary of the range in the case k is an element of (1, 2). Moreover, our results yield a complete answer to the cardinality of the set of favorite sites in the case k is an element of (2, infinity]. The proof relies on the exploration of the Markov property of the local times process with respect to the space variable and on a precise tail estimate on the maximum of local times, using a change of measure for multi-type Galton-Watson trees. (C) 2017 Elsevier B.V. All rights reserved.

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