Contact Interactions on a Lattice

Let $\{\xi_t\}$ be a Markov process whose values are subsets of $Z_d$, the $d$-dimensional integers. Put $\xi_t(x) = 1$ if $x \in \xi_t$ and 0 otherwise. The transition intensity for a change in $\xi_t(x)$ depends on $\{\xi_t(y), y$ a neighbor of $x\}$. The chief concern is with "contact processes," where $\xi_t(x)$ can change from 0 to 1 only if $\xi_t(y) = 1$ for some $y$ neighboring $x$. Let $p_t(\xi) = \operatorname{Prob} \{\xi_t \neq \varnothing \mid \xi_0 = \xi\}$. Under appropriate conditions, $p_t$ is increasing, subadditive, or submodular in $\xi$. In the case of contact processes, conditions are giving implying that $p_\infty(\xi) = 0$ for all finite $\xi$, or that the contrary is true. In other cases conditions for ergodicity are given.