Transition probabilities, transition functions and an ergodic decomposition

Abstract

Our goal in this paper is to discuss an ergodic decomposition of the state space of a transition probability or transition function. The decomposition allows us to have a better understanding of the structure of the set of invariant probabilities of the transition probability or transition function under consideration, and, as a byproduct, it allows us to obtain various criteria for the existence of invariant probability measures. Also, the decomposition offers a “system of reference” for the invariant ergodic probability measures which is useful in various instances. The present paper is a survey of results that I have obtained during the last ten years and is an extended version of my talk at the 9-eme Colloque Franco-Roumain de Math´ematiques Appliqu´ees,28 aoˆut-2 September 2008, Brasov, Roumanie.