Boundaries and harmonic functions for random walks with random transition probabilities

V. A. Kaimanovich, Y. Kifer, B. Z. Rubshtein

Research output: Contribution to journalArticlepeer-review

6 Scopus citations

Abstract

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.

Original languageEnglish
Pages (from-to)605-646
Number of pages42
JournalJournal of Theoretical Probability
Volume17
Issue number3
DOIs
StatePublished - Jul 2004

Keywords

  • harmonic function
  • Poisson boundary
  • random transition probability
  • Random walk

Fingerprint

Dive into the research topics of 'Boundaries and harmonic functions for random walks with random transition probabilities'. Together they form a unique fingerprint.

Cite this