## 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)

### Citations:

Zbl 0995.60042; Zbl 0648.60077; Zbl 0714.60089; Zbl 1048.60062
Full Text:

### References:

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