### Abstract

We consider a quadruple (O, A, ?, µ), where A is a s-algebra of subsets of O, and ? is a measurable bijection from O into itself that preserves a finite measure µ.For each B ? A, we define and study the measure µB obtained by integrating on B the number of visits to a set of the trajectory of a point of O 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. © 2010 Brown University.

Original language | English |
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Pages (from-to) | 213-228 |

Number of pages | 16 |

Journal | Quarterly of Applied Mathematics |

Volume | 68 |

Issue number | 2 |

Publication status | Published - 2010 |

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*Quarterly of Applied Mathematics*,

*68*(2), 213-228.

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*Quarterly of Applied Mathematics*, vol. 68, no. 2, pp. 213-228.

**Stationary flows and uniqueness of invariant measures.** / Baccelli, François; Konstantopoulos, Panagiotis Takis.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Stationary flows and uniqueness of invariant measures

AU - Baccelli, François

AU - Konstantopoulos, Panagiotis Takis

PY - 2010

Y1 - 2010

N2 - We consider a quadruple (O, A, ?, µ), where A is a s-algebra of subsets of O, and ? is a measurable bijection from O into itself that preserves a finite measure µ.For each B ? A, we define and study the measure µB obtained by integrating on B the number of visits to a set of the trajectory of a point of O 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. © 2010 Brown University.

AB - We consider a quadruple (O, A, ?, µ), where A is a s-algebra of subsets of O, and ? is a measurable bijection from O into itself that preserves a finite measure µ.For each B ? A, we define and study the measure µB obtained by integrating on B the number of visits to a set of the trajectory of a point of O 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. © 2010 Brown University.

UR - http://www.scopus.com/inward/record.url?scp=77957367970&partnerID=8YFLogxK

M3 - Article

VL - 68

SP - 213

EP - 228

JO - Quarterly of Applied Mathematics

JF - Quarterly of Applied Mathematics

SN - 0033-569X

IS - 2

ER -