Contact Interactions on a Lattice

Let \ₜ\ be a Markov process whose values are subsets of Zd, the d-dimensional integers. Put ₜ (x) = 1 if x ₜ and 0 otherwise. The transition intensity for a change in ₜ (x) depends on \ₜ (y), y a neighbor of x\. The chief concern is with "contact processes, " where ₜ (x) can change from 0 to 1 only if ₜ (y) = 1 for some y neighboring x. Let pₜ () = Prob \ₜ ₀ = \. Under appropriate conditions, pₜ is increasing, subadditive, or submodular in. In the case of contact processes, conditions are giving implying that p_ () = 0 for all finite, or that the contrary is true. In other cases conditions for ergodicity are given.