Fix an irrational number $\alpha$ and a smooth, positive, real function $p$ on the circle. If current position is $x$ then in the next step jump to $x+\alpha$ with probability $p(x)$ or to $x-\alpha$ with probability $1-p(x)$. In 1999 Sinai has proven that if $p$ is asymmetric (in certain sense) or $\alpha$ is Diophantine then this Markov process possesses a unique stationary distribution. During the talk I will present the uniqueness of stationary distribution in the case when $\alpha$ is arbitrary irrational and $p$ is symmetric. This answers a recent question posed by D. Dolgopyat, B. Fayad and M. Saprykina.

Meeting ID: 852 4277 3200 Passcode: 103121