Long-range exclusion processes, generator and invariant measures

Abstract

We show that if μ is an invariant measure for the long range exclusion process putting no mass on the full configuration, L is the formal generator of that process and f is a cylinder function, then Lf∈L¹(dμ) and ∫Lf dμ=0. This result is then applied to determine (i) the set of invariant and translation-invariant measures of the long range exclusion process on ℤ^d when the underlying random walk is irreducible; (ii) the set of invariant measures of the long range exclusion process on ℤ when the underlying random walk is irreducible and either has zero mean or allows jumps only to the nearest-neighbors.