2004/03/16 by Vadim A. Kaimanovich, Yuri Kifer, Kaimanovich, Vadim A. +3
Mathematics · #37A30 #60B99 #60J50 #Advanced Operator Algebra Research #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Probability (math.PR) #math.DS #math.PR #msc:37A30 #msc:60B99 #msc:60J50
paper · pdf · doi:10.48550/arxiv.math/0403255
arxiv created 2004/03/16 · openalex publication_date 2004/03/16 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The usual random walk on a group (homogeneous both in time and in space) is determined by a probability measure on the group. In a random walk with random transition probabilities this single measure is replaced with a stationary sequence of measures, so that the resulting (random) Markov chains are still space homogeneous, but no longer time homogeneous. We study various notions of measure theoretical boundaries associated with this model and establish an analogue of the Poisson formula for (random) bounded harmonic functions. Under natural conditions on transition probabilities we identify these boundaries for several classes of groups with hyperbolic properties and prove the boundary triviality (i.e., the absence of non-constant random bounded harmonic functions) for groups of subexponential growth, in particular, for nilpotent groups.