## The Annals of Probability

- Ann. Probab.
- Volume 23, Number 1 (1995), 141-156.

### A Limit Theorem for a Class of Interacting Particle Systems

#### Abstract

Let $S$ be a countable set and $\Lambda$ the collection of all subsets of $S$. We consider interacting particle systems (IPS) $\{\eta_k\}$ on $\Lambda$, with duals $\{\tilde\eta_t\}$ and duality equation $P\lbrack |\eta^\zeta_t \cap A| \operatorname{odd} = \tilde{P}\lbrack |\tilde\eta_t^A \cap \zeta| \operatorname{odd} \rbrack, \zeta, A \subset S, A$ finite Under certain conditions we find all the extreme invariant distributions that arise as limits of translation invariant initial configurations. Specific systems will be considered. A new property of the annihilating particle model is then used to prove a limiting relation between the annihilating and coalescing particle models.

#### Article information

**Source**

Ann. Probab., Volume 23, Number 1 (1995), 141-156.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176988380

**Digital Object Identifier**

doi:10.1214/aop/1176988380

**Mathematical Reviews number (MathSciNet)**

MR1330764

**Zentralblatt MATH identifier**

0833.60094

**JSTOR**

links.jstor.org

**Subjects**

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

Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

**Keywords**

Cancellative systems duality equation annihilation

#### Citation

Simonelli, Italo. A Limit Theorem for a Class of Interacting Particle Systems. Ann. Probab. 23 (1995), no. 1, 141--156. doi:10.1214/aop/1176988380. https://projecteuclid.org/euclid.aop/1176988380