## The Annals of Probability

- Ann. Probab.
- Volume 4, Number 3 (1976), 339-356.

### Coupling the Simple Exclusion Process

#### Abstract

Consider the infinite particle system on the countable set $S$ with the simple exclusion interaction and one-particle motion determined by the stochastic transition matrix $p(x, y)$. In the past, the ergodic theory of this process has been treated successfully only when $p(x, y)$ is symmetric, in which case great simplifications occur. In this paper, coupling techniques are used to give a complete description of the set of invariant measures for the system in the following three cases: (a) $p(x, y)$ is translation invariant on the integers and has mean zero, (b) $p(x, y)$ corresponds to a birth and death chain on the nonnegative integers, and (c) $p(x, y)$ corresponds to the asymmetric simple random walk on the integers.

#### Article information

**Source**

Ann. Probab. Volume 4, Number 3 (1976), 339-356.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176996084

**Digital Object Identifier**

doi:10.1214/aop/1176996084

**Mathematical Reviews number (MathSciNet)**

MR418291

**Zentralblatt MATH identifier**

0339.60091

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

**Keywords**

Infinite particle system invariant measures simple exclusion process coupling

#### Citation

Liggett, Thomas M. Coupling the Simple Exclusion Process. Ann. Probab. 4 (1976), no. 3, 339--356. doi:10.1214/aop/1176996084. https://projecteuclid.org/euclid.aop/1176996084.