The Annals of Probability

A phase transition in excursions from infinity of the “fast” fragmentation-coalescence process

Abstract

An important property of Kingman’s coalescent is that, starting from a state with an infinite number of blocks, over any positive time horizon, it transitions into an almost surely finite number of blocks. This is known as “coming down from infinity”. Moreover, of the many different (exchangeable) stochastic coalescent models, Kingman’s coalescent is the “fastest” to come down from infinity. In this article, we study what happens when we counteract this “fastest” coalescent with the action of an extreme form of fragmentation. We augment Kingman’s coalescent, where any two blocks merge at rate $c>0$, with a fragmentation mechanism where each block fragments at constant rate, $\lambda>0$, into its constituent elements. We prove that there exists a phase transition at $\lambda=c/2$, between regimes where the resulting “fast” fragmentation-coalescence process is able to come down from infinity or not. In the case that $\lambda<c/2$, we develop an excursion theory for the fast fragmentation-coalescence process out of which a number of interesting quantities can be computed explicitly.

Article information

Source
Ann. Probab., Volume 45, Number 6A (2017), 3829-3849.

Dates
Revised: September 2016
First available in Project Euclid: 27 November 2017

https://projecteuclid.org/euclid.aop/1511773665

Digital Object Identifier
doi:10.1214/16-AOP1150

Mathematical Reviews number (MathSciNet)
MR3729616

Zentralblatt MATH identifier
06838108

Citation

Kyprianou, Andreas E.; Pagett, Steven W.; Rogers, Tim; Schweinsberg, Jason. A phase transition in excursions from infinity of the “fast” fragmentation-coalescence process. Ann. Probab. 45 (2017), no. 6A, 3829--3849. doi:10.1214/16-AOP1150. https://projecteuclid.org/euclid.aop/1511773665

References

• [1] Aldous, D. and Pitman, J. (1998). The standard additive coalescent. Ann. Probab. 26 1703–1726.
• [2] Aldous, D. J. (1999). Deterministic and stochastic models for coalescence (aggregation and coagulation): A review of the mean-field theory for probabilists. Bernoulli 5 3–48.
• [3] Berestycki, J. (2004). Exchangeable fragmentation-coalescence processes and their equilibrium measures. Electron. J. Probab. 9 770–824.
• [4] Berestycki, J., Berestycki, N. and Limic, V. (2010). The $\Lambda$-coalescent speed of coming down from infinity. Ann. Probab. 38 207–233.
• [5] Bertoin, J. (2000). A fragmentation process connected to Brownian motion. Probab. Theory Related Fields 117 289–301.
• [6] Bertoin, J. (2001). Homogeneous fragmentation processes. Probab. Theory Related Fields 121 301–318.
• [7] Bertoin, J. (2007). Two-parameter Poisson–Dirichlet measures and reversible exchangeable fragmentation-coalescence processes. Combin. Probab. Comput. 17 329–337.
• [8] Dellacherie, C. and Meyer, P.-A. (1987). Probabilités et Potentiel. Chapitres XII–XVI, 2nd ed. Publications de l’Institut de Mathématiques de L’Université de Strasbourg, XIX. Hermann, Paris.
• [9] Foucart, C. (2016). On the coming down from infinity of discrete logistic branching processes. Available at arXiv:1605.07039.
• [10] Horowitz, J. (1972). Semilinear Markov processes, subordinators and renewal theory. Z. Wahrsch. Verw. Gebiete 24 167–193.
• [11] Kingman, J. F. C. (1978). The representation of partition structures. J. Lond. Math. Soc. (2) 18 374–380.
• [12] Kingman, J. F. C. (1982). The coalescent. Stochastic Process. Appl. 13 235–248.
• [13] Möhle, M. and Sagitov, S. (2001). A classification of coalescent processes for haploid exchangeable population models. Ann. Probab. 29 1547–1562.
• [14] Mörters, P. and Peres, Y. (2010). Brownian Motion. Cambridge Series in Statistical and Probabilistic Mathematics 30. Cambridge Univ. Press, Cambridge.
• [15] Pitman, J. (1999). Coalescents with multiple collisions. Ann. Probab. 27 1870–1902.
• [16] Rogers, L. C. G. and Williams, D. (2000). Diffusions, Markov Processes, and Martingales. Vol. 2. Cambridge Mathematical Library. Cambridge Univ. Press, Cambridge. Itô calculus, Reprint of the second (1994) edition.
• [17] Sagitov, S. (1999). The general coalescent with asynchronous mergers of ancestral lines. J. Appl. Probab. 36 1116–1125.
• [18] Schweinsberg, J. (2000). Coalescents with simultaneous multiple collisions. Electron. J. Probab. 5 Paper no. 12, 50 pp. (electronic).