Electronic Journal of Probability

Subcritical contact process seen from the edge: convergence to quasi-equilibrium

Enrique Andjel, François Ezanno, Pablo Groisman, and Leonardo Rolla

Full-text: Open access

Abstract

The subcritical contact process seen from the rightmost infected site has no invariant measures. We prove that nevertheless it converges in distribution to a quasi-stationary measure supported on finite configurations.

Article information

Source
Electron. J. Probab., Volume 20 (2015), paper no. 32, 16 pp.

Dates
Accepted: 28 March 2015
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465067138

Digital Object Identifier
doi:10.1214/EJP.v20-3881

Mathematical Reviews number (MathSciNet)
MR3335823

Zentralblatt MATH identifier
1321.60200

Subjects
Primary: 65K35

Keywords
contact process

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Andjel, Enrique; Ezanno, François; Groisman, Pablo; Rolla, Leonardo. Subcritical contact process seen from the edge: convergence to quasi-equilibrium. Electron. J. Probab. 20 (2015), paper no. 32, 16 pp. doi:10.1214/EJP.v20-3881. https://projecteuclid.org/euclid.ejp/1465067138


Export citation

References

  • Andjel, E. D. Convergence in distribution for subcritical 2D oriented percolation seen from its rightmost point. Ann. Probab. 42 (2014), no. 3, 1285–1296.
  • Andjel, Enrique D.; Schinazi, Rinaldo B.; Schonmann, Roberto H. Edge processes of one-dimensional stochastic growth models. Ann. Inst. H. Poincaré Probab. Statist. 26 (1990), no. 3, 489–506.
  • Bezuidenhout, Carol; Grimmett, Geoffrey. Exponential decay for subcritical contact and percolation processes. Ann. Probab. 19 (1991), no. 3, 984–1009.
  • Collet, Pierre; Martinez, Servet; San Martin, Jaime. Quasi-stationary distributions. Markov chains, diffusions and dynamical systems. Probability and its Applications (New York). Springer, Heidelberg, 2013. xvi+280 pp. ISBN: 978-3-642-33130-5; 978-3-642-33131-2
  • Cox, J. T.; Durrett, R.; Schinazi, R. The critical contact process seen from the right edge. Probab. Theory Related Fields 87 (1991), no. 3, 325–332.
  • Durrett, Richard. Oriented percolation in two dimensions. Ann. Probab. 12 (1984), no. 4, 999–1040.
  • E. Ezanno, Systèmes de particules en interaction et modèles de déposition aléatoire, Ph.D. thesis, Université d'Aix Marseille, 2012.
  • Ferrari, P. A.; Kesten, H.; Martinez, S. $R$-positivity, quasi-stationary distributions and ratio limit theorems for a class of probabilistic automata. Ann. Appl. Probab. 6 (1996), no. 2, 577–616.
  • Galves, Antonio; Presutti, Errico. Edge fluctuations for the one-dimensional supercritical contact process. Ann. Probab. 15 (1987), no. 3, 1131–1145.
  • Kingman, J. F. C. The exponential decay of Markov transition probabilities. Proc. London Math. Soc. (3) 13 1963 337–358.
  • Kuczek, Thomas. The central limit theorem for the right edge of supercritical oriented percolation. Ann. Probab. 17 (1989), no. 4, 1322–1332.
  • Liggett, Thomas M. Interacting particle systems. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 276. Springer-Verlag, New York, 1985. xv+488 pp. ISBN: 0-387-96069-4
  • Méléard, Sylvie; Villemonais, Denis. Quasi-stationary distributions and population processes. Probab. Surv. 9 (2012), 340–410.
  • Schonmann, Roberto Henrique. Absence of a stationary distribution for the edge process of subcritical oriented percolation in two dimensions. Ann. Probab. 15 (1987), no. 3, 1146–1147.
  • Seneta, E.; Vere-Jones, D. On quasi-stationary distributions in discrete-time Markov chains with a denumerable infinity of states. J. Appl. Probability 3 1966 403–434.
  • van Doorn, Erik A.; Pollett, Philip K. Quasi-stationary distributions for discrete-state models. European J. Oper. Res. 230 (2013), no. 1, 1–14.
  • Vere-Jones, D. Some limit theorems for evanescent processes. Austral. J. Statist. 11 1969 67–78.