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On extremal measures for conservative particle systems

2001; Institute of Mathematical Statistics; Volume: 37; Issue: 2 Linguagem: Inglês

10.1016/s0246-0203(00)01062-1

1778-7017

Sunder Sethuraman,

Geometry and complex manifolds

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∈Sμα(·), for an infinite countable set S, under mild conditions, are identified as extremal for α(·)∈HZR where μα(i)(k)=Z(α(i))−1α(i)k/g(1)⋯g(k) with g and Z the rate function and normalization respectively, and HZR is the set of invariant measures for the transition probability p. Il est bien connu que le procesus d'exclusion, le processus de zero-range et le processus des misanthropes possdent des familles de mesures invariantes, en raison de la propriete de conservation de la masse. Bien que ces familles aient ete beaucoup etudiees, il n'existe pas de caracterisation complete de leurs points extremaux. Dans cet article, nous considerons une approche de cette clarification. L'un de nos resultats etablit pour le processus de zero-range avec un ensemble denombrable de sites S que les mesures invariantes produits ∏i∈Sμα(·) (ou μα(i)(k)=Z(α(i))−1α(i)k/g(1)⋯g(k), g etant la function de taux et Z la normalisation) sont extremales pour α(·)∈HZR, HZR designant l'ensemble des mesures invariantes pour la probabilite de transition p.