We consider the problem of
characterizing Poisson boundaries of group-invariant time-dependent Markov random
walks on locally compact groups G. We show that such Poisson boundaries, which by
construction are naturally G-spaces, are amenable and approximately transitive (see
Definition 1.1 and Theorem 2.2).
We also establish a relationship between von Neumann algebras and Poisson
boundaries when G = R or Z. More precisely, there is naturally associated to an
eigenvalue list for an ITPFI factor M, a group-invariant time-dependent Markov
random walk on R whose Poisson boundary is the flow of weights for M
(Theorem 3.1).
