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 $d = 1$.