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
Full-text PDF Free Access
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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
MR Author ID:
28845
Email:
Francois.Baccelli@ens.fr
Takis Konstantopoulos
Affiliation:
School of Mathematical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, UK
MR Author ID:
251724
Email:
takis@ma.hw.ac.uk
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.