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07 February 2025
 
  » arxiv » 2301.01496

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Unique ergodicity of simple symmetric random walks on the circle
Klaudiusz Czudek ;
Date 4 Jan 2023
AbstractFix an irrational number $alpha$ and a smooth, positive, real function $mathfrak{p}$ on the circle. If current position is $xin mathbb R/mathbb Z$ 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.
Source arXiv, 2301.01496
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