Unique ergodicity of simple symmetric random walks on the circle

TL;DR

This paper provides a new proof demonstrating the uniqueness of the stationary distribution for a class of simple symmetric random walks on the circle with irrational rotation, extending previous results by Sinai and Conze-Guivarc'h.

Contribution

The paper introduces a novel proof of the uniqueness of stationary distribution for simple symmetric random walks on the circle with irrational rotation, generalizing prior work.

Findings

Established the uniqueness of stationary distribution for all irrational angles

Extended previous results by Sinai and Conze-Guivarc'h to a broader setting

Provided a new proof technique for ergodic properties of circle random walks

Abstract

Fix an irrational number $\alpha$ and a smooth, positive, real function $\mathfrak{p}$ on the circle. If current position is $x\in \mathbb R/\mathbb Z$ then in the next step jump to $x+\alpha$ with probability $\mathfrak{p}(x)$ or to $x-\alpha$ with probability $1-\mathfrak{p}(x)$. In 1999 Sinai has proven that if $\mathfrak{p}$ is asymmetric (in certain sense) or $\alpha$ is Diophantine then the Markov process possesses a unique stationary distribution. Next year Conze and Guivarc'h showed the uniqueness of stationary distribution for an arbitrary irrational angle $\alpha$. In this note we present a new proof of latter result.