Stationary flows and uniqueness of invariant measures

2007/02/13 by François Baccelli, Baccelli, Francois, Takis Konstantopoulos +1
Computer Science · Mathematics · #28D05 #60G10 (Primary) #60J05 (Secondary) #60J10 #Cellular Automata and Applications #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.math/0702391

openalex publication_date 2007/02/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this short paper, we consider a quadruple (Ω, Å, θ, μ),where Å is a σ-algebra of subsets of Ω, and θ is a measurable bijection from Ω into itself that preserves the measure μ. For each B ∈ Å, we consider the measure μ_B obtained by taking cycles (excursions) of iterates of θ from B. We then derive a relation for μ_B that involves the forward and backward hitting times of B by the trajectory (θⁿ ω, n ∈ \Z) at a point ω∈ Ω. 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.

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