The Annals of Probability

Additive Set-Valued Markov Processes and Graphical Methods

T. E. Harris

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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

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

First available in Project Euclid: 19 April 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60G45

Additive set-valued process interaction percolation contact processes


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

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