  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
$(G,\mu)$ with any ergodic $(G,\check{\mu})$-space is still ergodic, which in
particular yields a new proof of weak mixing for the double Poisson boundary of
$(G,\mu)$ when $\mu$ is symmetric. Finally, we characterize the failure of
weak-mixing for an ergodic $(G,\mu)$-space as the existence of a non-trivial
measure-preserving isometric factor.
