Vadim A. Kaimanovich

The Poisson formula for groups with hyperbolic properties

The Poisson boundary of a group G with a probability measure „ is the space of ergodic components of the time shift in the path space of the associated random walk. Via a generalization of the classical Poisson formula it gives an integral representation of bounded „-harmonic functions on G. In this paper we develop a new method of identifying the Poisson boundary based on entropy estimates for conditional random walks. It leads to simple purely geometric criteria of boundary maximality which bear hyperbolic nature and allow us to identify the Poisson boundary with natural topological boundaries for several classes of groups: word hyperbolic groups and discontinuous groups of isometries of Gromov hyperbolic spaces, groups with inflnitely many ends, cocompact lattices in Cartan-Hadamard manifolds, discrete subgroups of semisimple Lie groups.

Poisson Boundaries of Random Walks on Discrete Solvable Groups

Let G be a topological group, and μ — a probability measure on G. A function f on G is called harmonic if it satisfies the mean value property

The Poisson boundary of covering Markov operators

f(g) = \int {f(gx)d\mu (x)}

Random Walks on Discrete Groups: Boundary and Entropy

for all g ∈ G. It is well known that under natural assumptions on the measure μ there exists a measure G-space Γ with a quasi-invariant measure v such that the Poisson formula

The Poisson boundary of hyperbolic groups

f(g) =

Ergodic properties of boundary actions and the Nielsen-Schreier theory

states an isometric isomorphism between the Banach space H ∞(G, μ) of bounded harmonic functions with sup-norm and the space X∞(Γ, μ). The space (Γ, v) is called the Poisson boundary of the pair (G, μ). Thus triviality of the Poisson boundary is equivalent to absence of non-constant bounded harmonic functions for the pair (G, μ) (the Liouville property).