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Liouville property and Poisson boundary of random walks with infinite entropy: what's amiss?

Vadim Kaimanovich

Abstract

We discuss the qualitatively new properties of random walks on groups that arise in the situation when the entropy of the step distribution is infinite.

Liouville property and Poisson boundary of random walks with infinite entropy: what's amiss?

Abstract

We discuss the qualitatively new properties of random walks on groups that arise in the situation when the entropy of the step distribution is infinite.

Paper Structure

This paper contains 6 sections, 7 theorems, 78 equations.

Table of Contents

  1. Poisson boundary and Liouville property
  2. Entropy of random walks
  3. Wreath products, dynamical configurations, etc.
  4. Random walks on the group $\mathcal{L}$
  5. Infinite entropy
  6. Appendix. "Liouville's theorem" and "Shannon's theorem"

Key Result

Theorem 14

A measure $\mu$ with finite entropy $H(\mu)$ is Liouville if and only if $h(\mu)=0$.

Theorems & Definitions (24)

  • Remark 11
  • Claim 12
  • Remark 13
  • Theorem 14: Vershik -- Kaimanovich Vershik-Kaimanovich79rKaimanovich-Vershik83, Derriennic Derriennic80
  • Proposition 16: Kaimanovich83arKaimanovich-Vershik83
  • Corollary 17: Kaimanovich83arKaimanovich-Vershik83
  • Remark 18
  • Proposition 21
  • Remark 22
  • Remark 23
  • ...and 14 more