Random Walks on countable groups

Abstract

We begin by giving a new proof of the equivalence between the Liouville property and vanishing of the drift for symmetric random walks with finite first moments on finitely generated groups; a result which was first established by Kaimanovich-Vershik and Karlsson-Ledrappier. We then proceed to prove that the product of the Poisson boundary of any countable measured group $(G,μ)$ with any ergodic $(G,\checkμ)$-space is still ergodic, which in particular yields a new proof of weak mixing for the double Poisson boundary of $(G,μ)$