Limit Theorems for Nonergodic Set-Valued Markov Processes

Abstract

Certain Markov processes on the state space of subsets of the integers have $\varnothing$ as a trap, but have an equilibrium $\nu \neq \delta_\varnothing$. In this paper we prove weak convergence to a mixture of $\delta_\varnothing$ and $\nu$ from any initial state for some of these processes. In particular, we prove that the basic symmetric one-dimensional contact process of Harris has only $\delta_\varnothing$ and $\nu$ as extreme equilibria when the infection rate is large enough in comparison to the recovery rate.