## The Annals of Probability

- Ann. Probab.
- Volume 3, Number 4 (1975), 643-663.

### Ergodic Theorems for Weakly Interacting Infinite Systems and the Voter Model

Richard A. Holley and Thomas M. Liggett

#### Abstract

A theorem exhibiting the duality between certain infinite systems of interacting stochastic processes and a type of branching process is proved. This duality is then used to study the ergodic properties of the infinite system. In the case of the vector model a complete understanding of the ergodic behavior is obtained.

#### Article information

**Source**

Ann. Probab. Volume 3, Number 4 (1975), 643-663.

**Dates**

First available: 19 April 2007

**Permanent link to this document**

http://projecteuclid.org/euclid.aop/1176996306

**JSTOR**

links.jstor.org

**Digital Object Identifier**

doi:10.1214/aop/1176996306

**Mathematical Reviews number (MathSciNet)**

MR402985

**Zentralblatt MATH identifier**

0367.60115

**Subjects**

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

Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

**Keywords**

Infinite particle system ergodic theorem branching process with interference Markov chain harmonic function

#### Citation

Holley, Richard A.; Liggett, Thomas M. Ergodic Theorems for Weakly Interacting Infinite Systems and the Voter Model. The Annals of Probability 3 (1975), no. 4, 643--663. doi:10.1214/aop/1176996306. http://projecteuclid.org/euclid.aop/1176996306.