LA Probability Forum

USC, Kaprelian (KAP) 414

The Poisson boundary of a random walk on a (discrete, infinite) group is an object which serves a dual purpose. It serves as a basis for the space of bounded harmonic functions on a group and it serves to quantify the space of possible asymptotic events. Although apriori the Poisson boundary is quite abstract in a number of cases the Poisson boundary can actually be concretely identified allowing us to classify the space of harmonic functions on the group. Until recently almost all nontrivial classifications have heavily relied on moment conditions. I will discuss some more recent examples which no longer require any control on the moments of the step distribution of the random walk. This is based on joint work with Kunal Chawla, Behrang Forghani, Giullio Tiozzo as well as joint work with Eduardo Silva.