Unique ergodicity of simple symmetric random walks on the circle

Fix an irrational number $\alpha$ and a smooth, positive, real function $\mathfrak{p}$ on the circle. If current position is $x\in \mathbb{R}\mathrm{/}\mathbb{Z}$ 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.