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June, 1978 Additive Set-Valued Markov Processes and Graphical Methods
T. E. Harris
Ann. Probab. 6(3): 355-378 (June, 1978). DOI: 10.1214/aop/1176995523


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 $d = 1$.


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T. E. Harris. "Additive Set-Valued Markov Processes and Graphical Methods." Ann. Probab. 6 (3) 355 - 378, June, 1978.


Published: June, 1978
First available in Project Euclid: 19 April 2007

zbMATH: 0378.60106
MathSciNet: MR488377
Digital Object Identifier: 10.1214/aop/1176995523

Primary: 60K35
Secondary: 60G45

Keywords: Additive set-valued process , contact processes , interaction percolation

Rights: Copyright © 1978 Institute of Mathematical Statistics

Vol.6 • No. 3 • June, 1978
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