2000/11/01 by Vadim A. Kaimanovich · 5 citations
Mathematics · #Geometric and Algebraic Topology #Geometry and complex manifolds #Geometric Analysis and Curvature Flows #Mathematics #Poisson distribution #Pure mathematics #Statistics
paper · doi:10.2307/2661351
openalex publication_date 2000/11/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The Poisson boundary of a group G with a probability measure µ 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 µ-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 infinitely many ends, cocompact lattices in Cartan-Hadamard manifolds, discrete subgroups of semisimple Lie groups. The classical Poisson integral representation formula for the harmonic