N ov 2 00 4 EXACTNESS OF ROKHLIN ENDOMORPHISMS AND WEAK MIXING OF POISSON BOUNDARIES

We give conditions for the exactness of Rokhlin skew products, apply these to random walks on locally compact, second countable topological groups and obtain that the Poisson boundary of a globally supported random walk on such a group is weakly mixing. S is a measurable homomorphism). We give (theorem 2.3) conditions for the exactness of the Rokhlin endomorphism T = These conditions are applied to random walk endomorphisms. Meilijson (in [Me]) gave sufficient conditions for exactness for random walk endomorphisms over G = Z. We clarify Meilijson's theorem, proving a converse (proposition 4.2), extend it to countable Abelian groups (theorem 4.1), characterize the exactness of the Rokhlin endomorphism for a steady random walk (theorem 4.5) and obtain that the right action on Poisson boundary (see §4) of an adapted (i.e. globally supported) random walk is weakly mixing (proposition 4.4). Tools employed include the ergodic theory of " associated actions " (see §1), and the boundary theory of random walks (see §4). §1 Associated actions For an endomorphism R of a measure space (Z, D, ν) set • I(R) := {A ∈ D : R −1 A = A} – the invariant σ-algebra, and • T (R) := ∞ n=0 R −n D – the tail σ-algebra.