A Uniqueness Theorem for Stationary Measures of Ergodic Markov Processes

Abstract

If $\beta_1$ and $\beta_2$ are not identically zero $\sigma$-finite invariant measures for a measurable invertible ergodic transformation $S$ on a measure space, and $\beta_1(E) > 0$ implies $\beta_2(E) > 0$ for measurable sets $E$, then $\beta_2 = c\beta_1$ for some constant $c \neq 0$ ([4], p. 35). In this paper a corresponding result will be proved for stationary measures of a Markov process (Theorem 1). Theorem 1 is a generalization of the corollary of [6], p. 863. In that paper, the authors impose conditions ensuring that the shift transformation has no wandering sets of positive measure, and then they use Hopf's theorem. In Section 3, some new and known results are seen to follow readily from Theorem 1. The recurrence condition introduced by Harris [5] is discussed, and Theorem 1 is used to give a new proof of the uniqueness theorem of [5] independent of the existence of stationary measures, and generalizing the theorem to $\sigma$-fields which are not necessarily separable.