# Stationary flows and uniqueness of invariant measures

Baccelli, Francois and Konstantopoulos, Takis
(2009) Stationary flows and uniqueness of invariant measures.

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## 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.

Item Type: Article math.PR, math.PR, 28D05; 60G10 (Primary); 60J10; 60J05 (Secondary) Symplectic Admin 27 Nov 2018 08:45 03 Mar 2021 10:43 Author https://livrepository.liverpool.ac.uk/id/eprint/3029072