Couplings and attractiveness for general exclusion processes

TL;DR

This paper investigates the conditions for attractiveness in general exclusion processes where jump rates depend on the entire configuration, extending classical results and analyzing invariant measures.

Contribution

It provides necessary and sufficient conditions for attractiveness in complex exclusion processes and introduces new coupled processes that differ from basic coupling.

Findings

Attractiveness conditions are characterized for these processes.

Basic coupling is not attractive except in simple exclusion.

Invariant measures are explicitly determined for studied examples.

Abstract

Attractiveness is a fundamental tool to study interacting particle systems and the basic coupling construction is a usual route to prove this property, as for instance in the simple exclusion process. We consider here general exclusion processes where jump rates from an occupied site to an empty one depend not only on the location of the jump but also possibly on the whole configuration. These processes include in particular exclusion processes with speed change introduced by F. Spitzer in [18]. For such processes we derive necessary and sufficient conditions for attractiveness, through the construction of a coupled process under which discrepancies do not increase. We emphasize the fact that basic coupling is never attractive for this class of processes, except in the case of simple exclusion, and that the coupled processes presented here necessarily differ from it. We study various examples, for which we determine the set of extremal translation invariant and invariant probability measures.