Continuous-time vertex reinforced jump processes on galton-watson trees

Abstract

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 ccrit such that the walk is recurrent for c > ccrit and transient for c > ccrit. © 2012 Institute of Mathematical Statistics.