The asymptotic σ-algebra of a recurrent random walk on a locally compact group

Abstract

Let μ be a probability measure on a locally compact second countable groupG defining a recurrent (but not necessarily Harris) random walk. Denote byG ^∞ the space of paths and byB ^(a)the asymptotic σ-algebra. Let the starting measure be equivalent to the Haar measure and writeQ for the corresponding Markov measure onG ^∞. We prove thatL ^∞(G^∞, B^(a), Q) is in a canonical way isomorphic toL ^∞(G/N) whereN is the smallest closed normal subgroup ofG such that μ(zN)=1 for somez∈G. The groupG/N is either a finite cyclic group with generatorzN or a compact abelian group having the cyclic group$\left\{ {z^n N} \right\}n \in \mathbb{Z}$ as a dense subgroup. As a corollary we obtain that the set of all φ∈L ¹(G) such that$\lim _n \to \infty \left\| {\varphi * \mu ^n } \right\|1 = 0$ coincides with the kernel of the canonical mapping ofL ¹(G)ontoL ¹(G/N). In particular, when μ is aperiodic, i.e.,G=N, then the random walk is mixing:$\lim _n \to \infty \left\| {\varphi * \mu ^n } \right\|1 = 0$ for every φ∈L ¹(G) with ∝ φ=0.