### Abstract

It is well known that the exclusion, zero-range and misanthrope particle systems possess families of invariant measures due to the mass conservation property. Although these families have been classified a great deal, a full characterization of their extreme points is not available. In this article, we consider an approach to the study of this classification. One of the results in this note is that the zero-range product invariant measures, ∏_{i∈S}μ_{α(·)}, for an infinite countable set S, under mild conditions, are identified as extremal for α(·)∈H_{ZR} where μ_{α(i)}(k)=Z(α(i))^{-1}α(i) ^{k}/g(1)g(k) with g and Z the rate function and normalization respectively, and H_{ZR} is the set of invariant measures for the transition probability p.

Original language | English (US) |
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Pages (from-to) | 139-154 |

Number of pages | 16 |

Journal | Annales de l'institut Henri Poincare (B) Probability and Statistics |

Volume | 37 |

Issue number | 2 |

DOIs | |

State | Published - Mar 1 2001 |

Externally published | Yes |

### Keywords

- Dirichlet form
- Extreme points
- Invariant measures
- Misanthrope process
- Primary 60K35
- Secondary 60F05
- Simple exclusion process
- Zero-range process

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty