Open Access
May 2006 On the transience of processes defined on Galton–Watson trees
Andrea Collevecchio
Ann. Probab. 34(3): 870-878 (May 2006). DOI: 10.1214/009117905000000837


We introduce a simple technique for proving the transience of certain processes defined on the random tree $\mathcal{G}$ generated by a supercritical branching process. We prove the transience for once-reinforced random walks on $\mathcal{G}$, that is, a generalization of a result of Durrett, Kesten and Limic [Probab. Theory Related Fields 122 (2002) 567–592]. Moreover, we give a new proof for the transience of a family of biased random walks defined on $\mathcal{G}$. Other proofs of this fact can be found in [Ann. Probab. 16 (1988) 1229–1241] and [Ann. Probab. 18 (1990) 931–958] as part of more general results. A similar technique is applied to a vertex-reinforced jump process. A by-product of our result is that this process is transient on the 3-ary tree. Davis and Volkov [Probab. Theory Related Fields 128 (2004) 42–62] proved that a vertex-reinforced jump process defined on the b-ary tree is transient if b≥4 and recurrent if b=1. The case b=2 is still open.


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Andrea Collevecchio. "On the transience of processes defined on Galton–Watson trees." Ann. Probab. 34 (3) 870 - 878, May 2006.


Published: May 2006
First available in Project Euclid: 27 June 2006

zbMATH: 1104.60048
MathSciNet: MR2243872
Digital Object Identifier: 10.1214/009117905000000837

Primary: 60G50 , 60J80
Secondary: 60J75

Keywords: branching processes , random walk on trees , Reinforced random walk

Rights: Copyright © 2006 Institute of Mathematical Statistics

Vol.34 • No. 3 • May 2006
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