On the Transience of Processes Defined on Galton

We introduce a simple technique for proving the transience of certain processes defined on the random tree G generated by a supercritical branching process. We prove the transience for once-reinforced random walks on 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 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.