Unique ergodicity of simple symmetric random walks on the circle

2023/01/04 by Klaudiusz Czudek, Czudek, Klaudiusz
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2301.01496

openalex publication_date 2023/01/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Fix an irrational number α and a smooth, positive, real function \mathfrakp on the circle. If current position is x∈ \mathbb R/\mathbb Z then in the next step jump to x+α with probability \mathfrakp(x) or to x-α with probability 1-\mathfrakp(x). In 1999 Sinai has proven that if \mathfrakp is asymmetric (in certain sense) or α 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 α. In this note we present a new proof of latter result.

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