Harmonic measures and Poisson boundaries

ESB 4127

The Poisson boundary of a random walk on a group is a probability space used to study the long-term behavior of the random walk. Because the group naturally acts on the Poisson boundary, various questions regarding the structure of this action can be studied. In this talk, I will show that the set of probability measures with equivalent harmonic measures may not necessarily form a convex algebra (a convex algebra is closed under convex combinations and convolutions). To provide such examples, I will compute the harmonic measures for some finitely supported probability measures on the modular group. This talk is based on joint work with Vadim Kaimanovich.