Long-range exclusion processes, generator and invariant measures

Abstract

We show that if $\mu$ 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 $Lf\in {\mathbf{L}}^1(d\mu )$ and $\int Lfd\mu =0$. This result is then applied to determine (i) the set of invariant and translation-invariant measures of the long range exclusion process on ${\mathbb{Z}}^d$ when the underlying random walk is irreducible; (ii) the set of invariant measures of the long range exclusion process on $\mathbb{Z}$ when the underlying random walk is irreducible and either has zero mean or allows jumps only to the nearest-neighbors.

Published at http://dx.doi.org/10.1214/009117905000000486 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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