The Annals of Probability

Limit Theorems for Nonergodic Set-Valued Markov Processes

David Griffeath

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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

Permanent link to this document
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
links.jstor.org

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
Additive process associate process contact process infinite particle system

Citation

Griffeath, David. Limit Theorems for Nonergodic Set-Valued Markov Processes. Ann. Probab. 6 (1978), no. 3, 379--387. doi:10.1214/aop/1176995524. https://projecteuclid.org/euclid.aop/1176995524


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