An Approach to the Existence of Unique Invariant Probabilities for Markov Processes

A notion of localized splitting is introduced as a further extension of the splitting notions for iterated monotone maps introduced earlier by Dubins and Freedman (1966) and more generally by Bhattacharya and Majumdar (1999). We will see that under quite general conditions, localized splitting theory is a natural extension of the Doeblin (1937) minorization theory, Harris (1956) recurrence theory, splitting theory of Nummelin (1978) and regeneration theory of Athreya and Ney (1978), under which we can prove the existence of a unique invariant probability. By way of introduction we also provide a natural coupling proof of the ergodic problem for Markov processes on general state spaces under Doeblin’s minorization condition which seems to have heretofore gone unnoticed. The paper is concluded with some new applications of splitting theory to random iterated quadratic maps.