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Mixing of a generic simple symmetric random walk on the circle

Klaudiusz Czudek

Abstract

Fix an irrational number $α$ and a real function $\mathfrak{p}$ on the circle with $0<\mathfrak{p}<1$. If a particle is placed at a point $x\in \mathbb R/\mathbb Z$, then in the next step it jumps to $x+α$ with probability $\mathfrak{p}(x)$ and to $x-α$ with probability $1-\mathfrak{p}(x)$. Sinai and Kaloshin proved that if $\mathfrak{p}$ is smooth then the random walk is uniquely ergodic and mixing, unless $α$ is Liouville and $\mathfrak{p}$ is symmetric. Unique ergodicity in the general case has been obtained by Conze and Guivarc'h. Here we give an alternative proof of the latter as well as some generic result about mixing, which partially solves a recent open problem.

Mixing of a generic simple symmetric random walk on the circle

Abstract

Fix an irrational number and a real function on the circle with . If a particle is placed at a point , then in the next step it jumps to with probability and to with probability . Sinai and Kaloshin proved that if is smooth then the random walk is uniquely ergodic and mixing, unless is Liouville and is symmetric. Unique ergodicity in the general case has been obtained by Conze and Guivarc'h. Here we give an alternative proof of the latter as well as some generic result about mixing, which partially solves a recent open problem.
Paper Structure (10 sections, 11 theorems, 61 equations)

This paper contains 10 sections, 11 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Definition of the process
  3. Random walks in quasiperiodic environment
  4. Motivation and results
  5. Notation
  6. Acknowledgements
  7. Basic facts about symmetric random walks on $\mathbb{Z}$
  8. The proof of Theorem \ref{['T:1']}
  9. The proof of Theorem \ref{['T:2']}
  10. Proof of Theorem \ref{['T:3']}

Key Result

Theorem 1

If $\alpha\not\in\mathbb{Q}$, $\mathfrak{p}$ is symmetric, $\log\frac{\mathfrak{p}(x)}{\mathfrak{q}(x)}$ is continuous of bounded variation then the Markov process (E:1.1) is uniquely ergodic.

Theorems & Definitions (19)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 4
  • Corollary 1
  • Lemma 3
  • ...and 9 more