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.
@article{1176995524,
author = {Griffeath, David},
title = {Limit Theorems for Nonergodic Set-Valued Markov Processes},
journal = {Ann. Probab.},
volume = {6},
number = {6},
year = {1978},
pages = { 379-387},
language = {en},
url = {http://dml.mathdoc.fr/item/1176995524}
}
Griffeath, David. Limit Theorems for Nonergodic Set-Valued Markov Processes. Ann. Probab., Tome 6 (1978) no. 6, pp. 379-387. http://gdmltest.u-ga.fr/item/1176995524/