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On the transience of processes defined on Galton-Watson trees. (English) Zbl 1104.60048

Summary: 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 R. Durrett, H. Kesten and V. Limic [Probab. Theory Relat. Fields 122, No. 4, 567–592 (2002; Zbl 0995.60042)]. 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 [R. Pemantle, Ann. Probab. 16, No. 3, 1229–1241 (1988; Zbl 0648.60077) and R. Lyons, ibid. 18, No. 3, 931–958 (1990; Zbl 0714.60089)] 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. B. Davis and S. Volkov [Probab. Theory Relat. Fields 128, No. 1, 42–62 (2004; Zbl 1048.60062)] proved that a vertex-reinforced jump process defined on the \(b\)-ary tree is transient if \(b\geq 4\) and recurrent if \(b=1\). The case \(b=2\) is still open.

MSC:

60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60G50 Sums of independent random variables; random walks
60J75 Jump processes (MSC2010)
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