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.