## The Annals of Probability

- Ann. Probab.
- Volume 6, Number 3 (1978), 355-378.

### Additive Set-Valued Markov Processes and Graphical Methods

#### Abstract

Let $Z$ be a countable set, $\Xi$ the set of subsets of $Z$. A $\Xi$-valued Markov process $\{\xi_t\}$ with transition function $P(t, \xi, \Gamma)$ is called additive if there exists a family $\{\xi^A_t, t \geqq 0, A \in \Xi\}$ such that for each $A, \{\xi^A_t\}$ is Markov with transition function $P$ and $\xi^A_0 = A$, and such that $\xi^{A \cup B}_t = \xi^A_t \cup \xi^B_t, A, B \in \Xi, t \geqq 0$. Additive processes include symmetric simple exclusion, voter models and all contact processes having associates. The structure of such processes is studied, their construction from sets of independent Poisson flows, and their representations by random graphs. Applications for the case $Z = Z_d$, the $d$-dimensional integers, include individual ergodic theorems for certain cases as well as lower bounds for growth rates, and some results about different kinds of criticality when $d = 1$.

#### Article information

**Source**

Ann. Probab., Volume 6, Number 3 (1978), 355-378.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176995523

**Digital Object Identifier**

doi:10.1214/aop/1176995523

**Mathematical Reviews number (MathSciNet)**

MR488377

**Zentralblatt MATH identifier**

0378.60106

**JSTOR**

links.jstor.org

**Subjects**

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

Secondary: 60G45

**Keywords**

Additive set-valued process interaction percolation contact processes

#### Citation

Harris, T. E. Additive Set-Valued Markov Processes and Graphical Methods. Ann. Probab. 6 (1978), no. 3, 355--378. doi:10.1214/aop/1176995523. https://projecteuclid.org/euclid.aop/1176995523