Random Walks on Semigroups

Abstract

The term random walk suggests stochastic motion in space, a succession of random steps combined in some way. In Chap. 3, we interpret the term very narrowly: We require the steps to be independent and to have the same probability distribution. The walk is then a succession of products of those steps. Later on, we apply our results to slightly more general situations, e.g., to cases where the steps depend on each other in a Markovian way. Thus, our study of random walks is synonymous with the study of products of independent identically distributed random elements of a semigroup. We study the most basic notions for these processes which are of course discrete-time Markov chains with the semigroups as state spaces. We deal with, for example, communication relations, irreducibility questions, recurrence vs. transience, periodicity and ergodicity. Generally speaking, these probabilistic notions have an algebraic counterpart, in the sense that the probabilistic properties of a random walk cannot be satisfied unless the semigroup supporting the random walk has a certain algebraic structure. The situation is very similar to that in Chap. 2 where we saw, for example, that only completely simple semigroups with a compact group factor support limit points of a tight convolution sequence of measures (see Theorem 2.7).