Exchangeable Fragmentation-Coalescence Processes and their Equilibrium Measures

Julien Berestycki (Université de Provence, France)


We define and study a family of Markov processes with state space the compact set of all partitions of $N$ that we call exchangeable fragmentation-coalescence processes. They can be viewed as a combination of homogeneous fragmentation as defined by Bertoin and of homogenous coalescence as defined by Pitman and Schweinsberg or Möhle and Sagitov. We show that they admit a unique invariant probability measure and we study some properties of their paths and of their equilibrium measure.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 770-824

Publication Date: November 17, 2004

DOI: 10.1214/EJP.v9-227


  1. D.J. Aldous, Exchangeability and related topics. École d'été de probabilités de Saint-Flour, XIII---1983, 1--198, Lecture Notes in Math., 1117, Springer, Berlin, 1985. MR0883646
  2. D.J. Aldous, Deterministic and stochastic models for coalescence (aggregation and coagulation): a review of the mean-field theory for probabilists. Bernoulli 5 (1999), no. 1, 3--48. MR1673235
  3. J. Berestycki,Ranked fragmentations. ESAIM Probab. Statist. 6 (2002), 157--175 (electronic). MR1943145
  4. J. Berestycki, Multifractal spectra of fragmentation processes. J. Statist. Phys. 113 (2003), no. 3-4, 411--430. MR2013691
  5. J. Bertoin, Random fragmentation and coagulation. In preparation.
  6. J. Bertoin, Homogeneous fragmentation processes. Probab. Theory Related Fields 121 (2001), no. 3, 301--318. MR1867425
  7. J. Bertoin, Self-similar fragmentations. Ann. Inst. H. Poincaré Probab. Statist. 38 (2002), no. 3, 319--340. MR1899456
  8. J. Bertoin. The asymptotic behavior of fragmentation processes. J. Eur. Math. Soc. (JEMS) 5 (2003), no. 4, 395--416. MR2017852
  9. D. Beysens, X. Campi and E. Peffekorn (Editors), Proceedings of the workshop : Fragmentation phenomena, Les Houches Series. World Scientific, 1995. Math. Review number not available.
  10. E. Bolthausen, A.-S. Sznitman, On Ruelle's probability cascades and an abstract cavity method. Comm. Math. Phys. 197 (1998), no. 2, 247--276. MR1652734
  11. P. Diaconis; E. Mayer-Wolf; O. Zeitouniand M.P. Zerner, The Poisson-Dirichlet law is the unique invariant distribution for uniform split-merge transformations. Ann. Probab. 32 (2004), no. 1B, 915--938. MR2044670
  12. R. Durrett and V. Limic, A surprising Poisson process arising from a species competition model. Stochastic Process. Appl. 102 (2002), no. 2, 301--309. MR1935129
  13. E.B. Dynkin, Markov processes. Vols. I, II. Translated with the authorization and assistance of the author by J. Fabius, V. Greenberg, A. Maitra, G. Majone. Die Grundlehren der Mathematischen Wi ssenschaften, Bände 121, 122 Academic Press Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg 1965 Vol. I: xii+365 pp.; Vol. II: viii+274 pp. MR0193671
  14. S.N. Ethier and T.G. Kurtz, Thomas G. Markov processes. Characterization and convergence. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York, 1986. x+534 pp. ISBN: 0-471-08186-8 MR0838085
  15. J. Jacod and A.N. Shiryaev, Albert N. Limit theorems for stochastic processes. Second edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 288. Springer-Verlag, Berlin, 2003. xx+661 pp. ISBN: 3-540-43932-3 MR1943877
  16. J. F. C. Kingman, On the genealogy of large populations. Essays in statistical science. J. Appl. Probab. 1982, Special Vol. 19A, 27--43. MR0633178
  17. J. F. C. Kingman, The representation of partition structures. J. London Math. Soc. (2) 18 (1978), no. 2, 374--380. MR0509954
  18. A. Lambert, The branching process with logistic growth. To appear in Ann. Appl. Prob.(2004).
  19. M. Möhle and S. Sagitov, A classification of coalescent processes for haploid exchangeable population models. Ann. Probab. 29 (2001), no. 4, 1547--1562. MR1880231
  20. J. Pitman, Coalescents with multiple collisions. Ann. Probab. 27 (1999), no. 4, 1870--1902. MR1742892
  21. J. Pitman, Poisson-Dirichlet and GEM invariant distributions for split-and-merge transformation of an interval partition. Combin. Probab. Comput. 11 (2002), no. 5, 501--514. MR1930355
  22. L. C. G. Rogers and D. Williams, Diffusions, Markov processes, and martingales. Vol. 1. Foundations. Reprint of the second (1994) edition. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 2000. xx+386 pp. ISBN: 0-521-77594-9 MR1796539
  23. J. Schweinsberg, Coalescents with simultaneous multiple collisions. Electron. J. Probab. 5 (2000), Paper no. 12, 50 pp. (electronic). MR1781024
  24. J. Schweinsberg, A necessary and sufficient condition for the $Lambda$-coalescent to come down from infinity. Electron. Comm. Probab. 5 (2000), 1--11 (electronic). MR1736720 (2001g:60025)

Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.