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

Abstract

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

$$C\left( x \right) = {\left( {\lambda + \frac{A}{{{{\left| x \right|}^\alpha }}}} \right)^{ - \left| x \right|}}$$

where |x| is the generation of the vertex x. For (X_n)_n⩾0, a C(x)-biased random walk on the tree, we show that (1) when λ ≠ m, α > 0, (X_n)_n⩾0 is transient/recurrent according to whether λ < m or λ > m, respectively; (2) when λ ≠ m, 0 < λ < 1, (X_n)_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)).