Random walk on the Poincaré disk induced by a group of Möbius transformations.

We consider a discrete-time random motion, Markov chain on the Poincaré disk. In the basic variant of the model a particle moves along certain circular arcs within the disk, its location is determined by a composition of random Möbius transformations. We exploit an isomorphism between the underlying group of Möbius transformations and to study the random motion through its relation to a one-dimensional random walk.