On the transience of processes defined on Galton-Watson trees

Abstract

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. © Institute of Mathematical Statistics, 2006.