Additive Set-Valued Markov Processes and Graphical Methods

Let Z be a countable set, Ξ the set of subsets of Z. A Ξ-valued Markov process \ξₜ\ with transition function P(t, ξ, Γ) is called additive if there exists a family \ξAₜ, t 0, A ∈ Ξ\ such that for each A, \ξAₜ\ is Markov with transition function P and ξA₀ = A, and such that ξA ∪ Bₜ = ξAₜ ∪ ξBₜ, A, B ∈ Ξ, t 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 = Zd, 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 $d = 1$.