Poisson representation and Furstenberg entropy of hypergroups

Abstract

We extend the theory of Poisson boundary, tail boundary and the associated entropy theory of groups to the class of discrete hypergroups. We establish a zero-entropy criterion for the Liouville property of random walks on discrete hypergroups and provide various other characterizations for it. Finally, we will solve the identification problem for the Poisson boundary of finite range random walks on permutation hypergroups associated with affine groups of homogenous trees. As a byproduct, we obtain the first examples of random walks on these hypergroups that have a countable infinite Poisson boundary.