Unique ergodicity of simple symmetric random walks on the circle
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arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866917629114974208 |
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| author | Czudek, Klaudiusz |
| author_facet | Czudek, Klaudiusz |
| contents | Fix an irrational number $α$ 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 $x+α$ with probability $\mathfrak{p}(x)$ or to $x-α$ with probability $1-\mathfrak{p}(x)$. In 1999 Sinai has proven that if $\mathfrak{p}$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_01496 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Unique ergodicity of simple symmetric random walks on the circle Czudek, Klaudiusz Dynamical Systems Probability Fix an irrational number $α$ 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 $x+α$ with probability $\mathfrak{p}(x)$ or to $x-α$ with probability $1-\mathfrak{p}(x)$. In 1999 Sinai has proven that if $\mathfrak{p}$ 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. |
| title | Unique ergodicity of simple symmetric random walks on the circle |
| topic | Dynamical Systems Probability |
| url | https://arxiv.org/abs/2301.01496 |