The Annals of Probability

Limit Theorems for Nonergodic Set-Valued Markov Processes

David Griffeath

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.

Article information

Source
Ann. Probab., Volume 6, Number 3 (1978), 379-387.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176995524

Digital Object Identifier
doi:10.1214/aop/1176995524

Mathematical Reviews number (MathSciNet)
MR488378

Zentralblatt MATH identifier
0378.60105

JSTOR