Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Stationary flows and uniqueness of invariant measures

Authors: François Baccelli and Takis Konstantopoulos
Journal: Quart. Appl. Math. 68 (2010), 213-228
MSC (2000): Primary 37A05, 60J10; Secondary 37A50, 60G10
DOI: https://doi.org/10.1090/S0033-569X-10-01194-5
Published electronically: February 18, 2010
MathSciNet review: 2662999
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Abstract: We consider a quadruple $ (\Omega, \mathscr{A}, \vartheta, \mu)$, where $ \mathscr{A}$ is a $ \sigma$-algebra of subsets of $ \Omega$, and $ \vartheta$ is a measurable bijection from $ \Omega$ into itself that preserves a finite measure $ \mu$. For each $ B \in \mathscr{A}$, we define and study the measure $ \mu_B$ obtained by integrating on $ B$ the number of visits to a set of the trajectory of a point of $ \Omega$ before returning to $ B$. In particular, we obtain a generalization of Kac's formula and discuss its relation to discrete-time Palm theory. 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 in general state space.

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Additional Information

François Baccelli
Affiliation: Département d’Informatique, École Normale Supérieure, 45 rue d’Ulm, F-75230 Paris Cedex 05, France
Email: Francois.Baccelli@ens.fr

Takis Konstantopoulos
Affiliation: School of Mathematical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, UK
Email: takis@ma.hw.ac.uk

DOI: https://doi.org/10.1090/S0033-569X-10-01194-5
Received by editor(s): February 15, 2008
Published electronically: February 18, 2010
Additional Notes: Research supported by an EPSRC grant
Article copyright: © Copyright 2010 Brown University
The copyright for this article reverts to public domain 28 years after publication.

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