Electronic Journal of Probability

Stochastic Order and Attractiveness for Particle Systems with Multiple Births, Deaths and Jumps

Davide Borrello

Full-text: Open access


An approach to analyse the properties of a particle system is to compare it with different processes to understand when one of them is larger than other ones. The main technique for that is coupling, which may not be easy to construct. We give a characterization of stochastic order between different interacting particle systems in a large class of processes with births, deaths and jumps of many particles per time depending on the configuration in a general way: it consists in checking inequalities involving the transition rates. We construct explicitly the coupling that characterizes the stochastic order. As a corollary we get necessary and sufficient conditions for attractiveness. As an application, we first give the conditions on examples including reaction-diffusion processes, multitype contact process and conservative dynamics and then we improve an ergodicity result for an epidemic model.

Article information

Electron. J. Probab., Volume 16 (2011), paper no. 4, 106-151.

Accepted: 9 January 2011
First available in Project Euclid: 1 June 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82C22: Interacting particle systems [See also 60K35]

Stochastic order attractiveness interacting particle systems epidemic model multitype contact process

This work is licensed under aCreative Commons Attribution 3.0 License.


Borrello, Davide. Stochastic Order and Attractiveness for Particle Systems with Multiple Births, Deaths and Jumps. Electron. J. Probab. 16 (2011), paper no. 4, 106--151. doi:10.1214/EJP.v16-852. https://projecteuclid.org/euclid.ejp/1464820173

Export citation


  • W. Allee. Animal aggregation: a study in general sociology. University of Chicago Press (1931). Review number not available.
  • L. Belhadji. Interacting particle systems and epidemic models. Ph.D. Thesis, UniversitÈ de Rouen (2007). Review number not available.
  • D. Borrello. On the role of the Allee effect and mass migration in survival and extinction of a species. Preprint available at ArXiv.
  • M.F. Chen. Ergodic theorems for reaction-diffusion processes. J. Statist. Phys. 58 (1990), no. 5-6, 939-966. 1049053
  • M.F. Chen. From Markov Chains to Non-Equilibrium Particle systems. Second edition. World Scientific Publishing Co., Inc., River Edge, NJ 2091955
  • R. Durrett. Ten lectures on particle systems. Lectures on probability theory (Saint-Flour, 1993) Lecture Notes in Math. 97-201, 1608, Springer, Berlin, 1995. 1383122
  • R. Durrett. Mutual invadability implies coexistence in spatial models. Mem. Amer. Math. Soc. 156 (2002), no. 740. 1879853
  • T. Gobron and E. Saada. Couplings attractiveness and hydrodynamics for conservative particle systems. Ann. I.H.P. 46 (2010), no. 4, 1132-1177. Review number not available.
  • I. Hanski Metapopulation ecology. Oxford University Press, 1999. Review number not available.
  • T.E. Harris. Contact interactions on a lattice. Ann. Probability 2 (1974), 969-988. 0356292
  • T. Liggett. Interacting particle systems. Reprint of the 1985 original. Classics in Mathematics. Springer-Verlag, Berlin, 2005. 2108619
  • T. Liggett. Stochastic Interacting Systems: contact, voter and exclusion processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 324. Springer-Verlag, Berlin, 1999. 1717346
  • C. Neuhauser. An ergodic theorem for Schlogl models with small migration. Probab. Theory Related Fields. 85 (1990), no. 1, 27-32. 1044296
  • C. Neuhauser. Ergodic theorems for the multitype contact process. Probab. Theory Related Fields. 91 (1992), no. 3-4, 467-506. 1151806
  • R.B. Schinazi. Classical and spatial stochastic processes. Birkh?user Boston, Inc., Boston, MA, 1999. 1719718
  • R.B. Schinazi. On the spread of drug resistant diseases. J. Statist. Phys. 97 (1999), no. 1-2, 409-417. 1733477
  • R.B. Schinazi. On the role of social clusters in the transmission of infectious disesases. Theor. Popul. Biol. 61 (2002), 163-169. Review number not available.
  • R.B. Schinazi. Mass extinctions: an alternative to the Allee effect. Ann. Appl. Probab. 15 (2005), no. 1B, 984-991. 2114997
  • P.A. Stephens and W.J. Sutherland. Consequences of the Allee effect for behaviour, ecology and conservation. Trends in Ecology and Evolution. 14 (1999), 401-405.
  • J. Stover. Attractive n-type contact processes. Preprint available at ArXiv.