Stochastic domination for a hidden markov chain with applications to the contact process in a randomly evolving environment

The ordinary contact process is used to model the spread of a disease in a population. In this model, each infected individual waits an exponentially distributed time with parameter 1 before becoming healthy. In this paper, we introduce and study the contact process in a randomly evolving environment. Here we associate to every individual an independent two-state, {0, 1}, background process. Given delta(0) < delta(1), if the background process is in state 0, the individual (if infected) becomes healthy at rate delta(0), while if the background process is in state 1, it becomes healthy at rate delta(1). By stochastically comparing the contact process in a randomly evolving environment to the ordinary contact process, we will investigate matters of extinction and that of weak and strong survival. A key step in our analysis is to obtain stochastic domination results between certain point processes. We do this by starting out in a discrete setting and then taking continuous time limits.