On the transience and recurrence of Lamperti's random walk on Galton-Watson trees

• 来源期刊: 中国科学：数学（英文版）
• 年，卷（期）: 2019,（9）
• 页码范围: 1813-1822
• 页数: 10页
• 语种: 英文

摘要

In a Galton-Watson tree generated by a supercritical branching process with offspring N and EN =:m ＞ 1,the conductance assigned to the edge between the vertex x and its parent x* is denoted by C(x) and given by C(x)=(λ+A/|x|a)-|x|,where |x| is the generation of the vertex x.For (Xn)n≥0,a C(x)-biased random walk on the tree,we show that (1) when λ ≠ m,a ＞ 0,(Xn)n≥0 is transient/recurrent according to whether λ ＜ m or λ ＞ m,respectively;(2) when λ =m,0 ＜ a ＜ 1,(Xn)n≥0 is transient/recurrent according to whether A ＜ 0 or A ＞ 0,respectively.In particular,if P(N =1) =1,the C(x)-biased random walk is Lamperti's random walk on the nonnegative integers (see Lamperti (1960)).