Stationary flows and uniqueness of invariant measures

Abstract

In this short paper, we consider a quadruple $(\Omega, \AA, \theta, \mu)$,where $\AA$ is a $\sigma$-algebra of subsets of $\Omega$, and $\theta$ is a measurable bijection from $\Omega$ into itself that preserves the measure $\mu$. For each $B \in \AA$, we consider the measure $\mu_B$ obtained by taking cycles (excursions) of iterates of $\theta$ from $B$. We then derive a relation for $\mu_B$ that involves the forward and backward hitting times of $B$ by the trajectory $(\theta^n \omega, n \in \Z)$ at a point $\omega \in \Omega$. Although classical in appearance, its use in obtaining uniqueness of invariant measures of various stochastic models seems to be new. We apply the concept to countable Markov chains and Harris processes.