The Annals of Probability

A continuum-tree-valued Markov process

Romain Abraham and Jean-François Delmas

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Abstract

We present a construction of a Lévy continuum random tree (CRT) associated with a super-critical continuous state branching process using the so-called exploration process and a Girsanov theorem. We also extend the pruning procedure to this super-critical case. Let ψ be a critical branching mechanism. We set ψθ(⋅) = ψ(⋅ + θ) − ψ(θ). Let Θ = (θ, +∞) or Θ = [θ, +∞) be the set of values of θ for which ψθ is a conservative branching mechanism. The pruning procedure allows to construct a decreasing Lévy-CRT-valued Markov process $({\mathcal{T}}_{\theta},\theta\in\Theta)$, such that ${\mathcal{T}}_{\theta}$ has branching mechanism ψθ. It is sub-critical if θ > 0 and super-critical if θ < 0. We then consider the explosion time A of the CRT: the smallest (negative) time θ for which the continuous state branching process (CB) associated with ${\mathcal{T}}_{\theta}$ has finite total mass (i.e., the length of the excursion of the exploration process that codes the CRT is finite). We describe the law of A as well as the distribution of the CRT just after this explosion time. The CRT just after explosion can be seen as a CRT conditioned not to be extinct which is pruned with an independent intensity related to A. We also study the evolution of the CRT-valued process after the explosion time. This extends results from Aldous and Pitman on Galton–Watson trees. For the particular case of the quadratic branching mechanism, we show that after explosion the total mass of the CB behaves like the inverse of a stable subordinator with index 1/2. This result is related to the size of the tagged fragment for the fragmentation of Aldous’s CRT.

Article information

Source
Ann. Probab. Volume 40, Number 3 (2012), 1167-1211.

Dates
First available in Project Euclid: 4 May 2012

Permanent link to this document
https://projecteuclid.org/euclid.aop/1336136062

Digital Object Identifier
doi:10.1214/11-AOP644

Mathematical Reviews number (MathSciNet)
MR2962090

Zentralblatt MATH identifier
1252.60072

Subjects
Primary: 60J25: Continuous-time Markov processes on general state spaces 60G55: Point processes 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Keywords
Continuum random tree explosion time pruning tree-valued Markov process continuous state branching process exploration process

Citation

Abraham, Romain; Delmas, Jean-François. A continuum-tree-valued Markov process. Ann. Probab. 40 (2012), no. 3, 1167--1211. doi:10.1214/11-AOP644. https://projecteuclid.org/euclid.aop/1336136062


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References

  • [1] Abraham, R. and Delmas, J.-F. (2008). Fragmentation associated with Lévy processes using snake. Probab. Theory Related Fields 141 113–154.
  • [2] Abraham, R. and Delmas, J.-F. (2009). Changing the branching mechanism of a continuous state branching process using immigration. Ann. Inst. Henri Poincaré Probab. Stat. 45 226–238.
  • [3] Abraham, R. and Delmas, J.-F. (2009). Williams’ decomposition of the Lévy continuum random tree and simultaneous extinction probability for populations with neutral mutations. Stochastic Process. Appl. 119 1124–1143.
  • [4] Abraham, R., Delmas, J. F. and He, H. (2010). Pruning Galton–Watson trees and tree-valued Markov processes. Preprint. Available at arXiv:1007.0370.
  • [5] Abraham, R., Delmas, J.-F. and Voisin, G. (2010). Pruning a Lévy continuum random tree. Electron. J. Probab. 15 1429–1473.
  • [6] Abraham, R. and Serlet, L. (2002). Poisson snake and fragmentation. Electron. J. Probab. 7 no. 17, 15 pp. (electronic).
  • [7] Aldous, D. (1991). The continuum random tree. II. An overview. In Stochastic Analysis (Durham, 1990). London Mathematical Society Lecture Note Series 167 23–70. Cambridge Univ. Press, Cambridge.
  • [8] Aldous, D. (1991). The continuum random tree. I. Ann. Probab. 19 1–28.
  • [9] Aldous, D. (1993). The continuum random tree. III. Ann. Probab. 21 248–289.
  • [10] Aldous, D. and Pitman, J. (1998). The standard additive coalescent. Ann. Probab. 26 1703–1726.
  • [11] Aldous, D. and Pitman, J. (1998). Tree-valued Markov chains derived from Galton–Watson processes. Ann. Inst. Henri Poincaré Probab. Stat. 34 637–686.
  • [12] Bertoin, J. (1996). Lévy Processes. Cambridge Tracts in Mathematics 121. Cambridge Univ. Press, Cambridge.
  • [13] Bertoin, J. (2002). Self-similar fragmentations. Ann. Inst. Henri Poincaré Probab. Stat. 38 319–340.
  • [14] Bertoin, J. (2006). Random Fragmentation and Coagulation Processes. Cambridge Studies in Advanced Mathematics 102. Cambridge Univ. Press, Cambridge.
  • [15] Delmas, J. F. (2008). Height process for super-critical continuous state branching process. Markov Process. Related Fields 14 309–326.
  • [16] Duquesne, T. and Le Gall, J.-F. (2002). Random trees, Lévy processes and spatial branching processes. Astérisque 281 vi+147.
  • [17] Duquesne, T. and Le Gall, J.-F. (2005). Probabilistic and fractal aspects of Lévy trees. Probab. Theory Related Fields 131 553–603.
  • [18] Duquesne, T. and Winkel, M. (2007). Growth of Lévy trees. Probab. Theory Related Fields 139 313–371.
  • [19] Jiřina, M. (1958). Stochastic branching processes with continuous state space. Czechoslovak Math. J. 8 (83) 292–313.
  • [20] Lamperti, J. (1967). The limit of a sequence of branching processes. Z. Wahrsch. Verw. Gebiete 7 271–288.
  • [21] Le Gall, J.-F. (2006). Random real trees. Ann. Fac. Sci. Toulouse Math. (6) 15 35–62.
  • [22] Le Gall, J.-F. and Le Jan, Y. (1998). Branching processes in Lévy processes: Laplace functionals of snakes and superprocesses. Ann. Probab. 26 1407–1432.
  • [23] Le Gall, J.-F. and Le Jan, Y. (1998). Branching processes in Lévy processes: The exploration process. Ann. Probab. 26 213–252.
  • [24] Uribe Bravo, G. (2009). The falling apart of the tagged fragment and the asymptotic disintegration of the Brownian height fragmentation. Ann. Inst. Henri Poincaré Probab. Stat. 45 1130–1149.
  • [25] Voisin, G. (2012). Dislocation measure of the fragmentation of a general Lévy tree. ESAIM:P&S. To appear.