CONTINUOUS-TIME VERTEX REINFORCED JUMP PROCESSES ON GALTON-WATSON TREES

• 成果类型: Article
• 署名作者: Basdevant, Anne-Laure; Singh, Arvind
• 署名单位: Universite Paris Saclay; Universite Paris Saclay
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/11-AAP811
• 发表日期: 2012
• 页码: 1728-1743
• 关键词: random environment

摘要

We consider a continuous-time vertex reinforced jump process on a supercritical Galton-Watson tree. This process takes values in the set of vertices of the tree and jumps to a neighboring vertex with rate proportional to the local time at that vertex plus a constant c. The walk is either transient or recurrent depending on this parameter c. In this paper, we complete results previously obtained by Davis and Volkov [Probab. Theory Related Fields 123 (2002) 281-300, Probab. Theory Related Fields 128 (2004) 42-62] and Collevecchio [Ann. Probab. 34 (2006) 870-878, Electron. J. Probab. 14 (2009) 1936-1962] by proving that there is a unique (explicit) positive c(crit) such that the walk is recurrent for c <= c(crit) and transient for c > c(crit).