On extremal measures for conservative particle systems

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.