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 $c_{\mathrm{crit}}$ such that the walk is recurrent for $c\le c_{\mathrm{crit}}$ and transient for $c>c_{\mathrm{crit}}$.

Published in at http://dx.doi.org/10.1214/11-AAP811 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (1)

• On the transience of processes defined on Galton--Watson trees

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