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

Davide Borrello (Università di Milano and Universié de Rouen)


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.

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Pages: 106-151

Publication Date: January 9, 2011

DOI: 10.1214/EJP.v16-852


  1. W. Allee. Animal aggregation: a study in general sociology. University of Chicago Press (1931). Review number not available.
  2. L. Belhadji. Interacting particle systems and epidemic models. Ph.D. Thesis, UniversitÈ de Rouen (2007). Review number not available.
  3. D. Borrello. On the role of the Allee effect and mass migration in survival and extinction of a species. Preprint available at ArXiv.
  4. M.F. Chen. Ergodic theorems for reaction-diffusion processes. J. Statist. Phys. 58 (1990), no. 5-6, 939-966. 1049053
  5. M.F. Chen. From Markov Chains to Non-Equilibrium Particle systems. Second edition. World Scientific Publishing Co., Inc., River Edge, NJ 2091955
  6. 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
  7. R. Durrett. Mutual invadability implies coexistence in spatial models. Mem. Amer. Math. Soc. 156 (2002), no. 740. 1879853
  8. 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.
  9. I. Hanski Metapopulation ecology. Oxford University Press, 1999. Review number not available.
  10. T.E. Harris. Contact interactions on a lattice. Ann. Probability 2 (1974), 969-988. 0356292
  11. T. Liggett. Interacting particle systems. Reprint of the 1985 original. Classics in Mathematics. Springer-Verlag, Berlin, 2005. 2108619
  12. 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
  13. C. Neuhauser. An ergodic theorem for Schlogl models with small migration. Probab. Theory Related Fields. 85 (1990), no. 1, 27-32. 1044296
  14. C. Neuhauser. Ergodic theorems for the multitype contact process. Probab. Theory Related Fields. 91 (1992), no. 3-4, 467-506. 1151806
  15. R.B. Schinazi. Classical and spatial stochastic processes. Birkh?user Boston, Inc., Boston, MA, 1999. 1719718
  16. R.B. Schinazi. On the spread of drug resistant diseases. J. Statist. Phys. 97 (1999), no. 1-2, 409-417. 1733477
  17. 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.
  18. R.B. Schinazi. Mass extinctions: an alternative to the Allee effect. Ann. Appl. Probab. 15 (2005), no. 1B, 984-991. 2114997
  19. 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.
  20. J. Stover. Attractive n-type contact processes. Preprint available at ArXiv.

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