Average properties of random walks on Galton-Watson trees

Abstract

We study the λ-biased random walk on Galton-Watson trees by the Dirichlet principle and a formula of mean exit time of a Markov chain. We prove that the average of escaping probability and mean exit time are bounded by the counterparts of the corresponding random walks on 0,1,2, ······. In particular we partially verified the recent conjecture of Lyons, Pemantle and Peres on the upper bound of the speed of λ-biased random walk on Galton-Watson trees.