Electronic Journal of Probability

Branching processes seen from their extinction time via path decompositions of reflected Lévy processes

Miraine Dávila Felipe and Amaury Lambert

Full-text: Open access

Abstract

We consider a spectrally positive Lévy process $X$ that does not drift to $+\infty $, viewed as coding for the genealogical structure of a (sub)critical branching process, in the sense of a contour or exploration process [34, 29]. We denote by $I$ the past infimum process defined for each $t\geq 0$ by $I_t:= \inf _{[0,t]} X$ and we let $\gamma $ be the unique time at which the excursion of the reflected process $X-I$ away from 0 attains its supremum. We prove that the pre-$\gamma $ and the post-$\gamma $ subpaths of this excursion are invariant under space-time reversal, which has several other consequences in terms of duality for excursions of Lévy processes. It implies in particular that the local time process of this excursion is also invariant when seen backward from its height. As a corollary, we obtain that some (sub)critical branching processes such as the binary, homogeneous (sub)critical Crump-Mode-Jagers (CMJ) processes and the excursion away from 0 of the critical Feller diffusion, which is the width process of the continuum random tree, are invariant under time reversal from their extinction time.

Article information

Source
Electron. J. Probab., Volume 23 (2018), paper no. 98, 30 pp.

Dates
Received: 24 August 2017
Accepted: 4 September 2018
First available in Project Euclid: 25 September 2018

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1537841130

Digital Object Identifier
doi:10.1214/18-EJP221

Zentralblatt MATH identifier
06964792

Subjects
Primary: 60G51: Processes with independent increments; Lévy processes
Secondary: 60G17: Sample path properties 60J55: Local time and additive functionals 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Keywords
space-time reversal excursion measure duality Ray-Knight theorem Williams decomposition Lévy process conditioned to stay positive Esty transform

Rights
Creative Commons Attribution 4.0 International License.

Citation

Dávila Felipe, Miraine; Lambert, Amaury. Branching processes seen from their extinction time via path decompositions of reflected Lévy processes. Electron. J. Probab. 23 (2018), paper no. 98, 30 pp. doi:10.1214/18-EJP221. https://projecteuclid.org/euclid.ejp/1537841130


Export citation

References

  • [1] Romain Abraham and Jean-François Delmas. Williams’ decomposition of the Lévy continuum random tree and simultaneous extinction probability for populations with neutral mutations. Stochastic Processes and their Applications, 119(4):1124 – 1143, 2009.
  • [2] David Aldous. The continuum random tree. I. Ann. Probab., 19(1):1–28, 1991.
  • [3] David Aldous. The continuum random tree. III. Ann. Probab., 21(1):248–289, 1993.
  • [4] David Aldous and Lea Popovic. A critical branching process model for biodiversity. Adv. in Appl. Probab., 37(4):1094–1115, 2005.
  • [5] Gerold Alsmeyer and Uwe Rösler. Asexual versus promiscuous bisexual Galton-Watson processes: the extinction probability ratio. Ann. Appl. Probab., 12(1):125–142, 2002.
  • [6] Krishna B. Athreya and Peter E. Ney. Branching processes. Springer-Verlag, New York-Heidelberg, 1972. Die Grundlehren der mathematischen Wissenschaften, Band 196.
  • [7] Jean Bertoin. Splitting at the infimum and excursions in half-lines for random walks and Lévy processes. Stochastic Process. Appl., 47(1):17–35, 1993.
  • [8] Jean Bertoin. Lévy processes, volume 121 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1996.
  • [9] Gabriel Berzunza and Juan Carlos Pardo. Asymptotic behaviour near extinction of continuous-state branching processes. J. Appl. Probab., 53(2):381–391, 2016.
  • [10] Hongwei Bi and Jean-François Delmas. Total length of the genealogical tree for quadratic stationary continuous-state branching processes. Ann. Inst. Henri Poincaré Probab. Stat., 52(3):1321–1350, 2016.
  • [11] Ma. Emilia Caballero, Amaury Lambert, and Gerónimo Uribe Bravo. Proof(s) of the Lamperti representation of continuous-state branching processes. Probab. Surveys, 6(0):62–89, 2009.
  • [12] Loïc Chaumont. Sur certains processus de Lévy conditionnés à rester positifs. Stochastics Stochastics Rep., 47(1-2):1–20, 1994.
  • [13] Loïc Chaumont. Conditionings and path decompositions for Lévy processes. Stochastic Process. Appl., 64(1):39–54, 1996.
  • [14] Loïc Chaumont. On the law of the supremum of Lévy processes. Ann. Probab., 41(3A):1191–1217, 2013.
  • [15] Loïc Chaumont and Ronald A. Doney. On Lévy processes conditioned to stay positive. Electron. J. Probab., 10:no. 28, 948–961, 2005.
  • [16] Miraine Dávila Felipe and Amaury Lambert. Time reversal dualities for some random forests. ALEA Lat. Am. J. Probab. Math. Stat., 12(1):399–426, 2015.
  • [17] Jean-François Delmas and Olivier Hénard. A Williams decomposition for spatially dependent superprocesses. Electron. J. Probab., 18(0), 2013.
  • [18] Ronald A. Doney. Fluctuation theory for Lévy processes, volume 1897 of Lecture Notes in Mathematics. Springer, Berlin, 2007.
  • [19] Thomas Duquesne. Path decompositions for real Levy processes. Ann. Inst. H. Poincaré Probab. Statist., 39(2):339–370, 2003.
  • [20] Thomas Duquesne and Jean-François Le Gall. Random trees, Lévy processes and spatial branching processes. Astérisque, (281):vi+147, 2002.
  • [21] Warren W. Esty. The reverse Galton-Watson process. J. Appl. Probability, 12(3):574–580, 1975.
  • [22] Steven N. Evans. Probability and Real Trees: École d’Été de Probabilités de Saint-Flour XXXV-2005. Lecture Notes in Mathematics. Springer Berlin Heidelberg, 2007.
  • [23] Priscilla Greenwood and Jim Pitman. Fluctuation identities for Lévy processes and splitting at the maximum. Adv. in Appl. Probab., 12(4):893–902, 1980.
  • [24] Jean Jacod and Albert N. Shiryaev. Limit theorems for stochastic processes, volume 288 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, second edition, 2003.
  • [25] Peter Jagers. Branching processes with biological applications. Wiley-Interscience [John Wiley & Sons], London-New York-Sydney, 1975. Wiley Series in Probability and Mathematical Statistics—Applied Probability and Statistics.
  • [26] Fima C. Klebaner, Uwe Rösler, and Serik Sagitov. Transformations of galton-watson processes and linear fractional reproduction. Advances in Applied Probability, 39(4):1036–1053, 2007.
  • [27] Andreas E. Kyprianou. Introductory lectures on fluctuations of Lévy processes with applications. Universitext. Springer-Verlag, Berlin, 2006.
  • [28] Andreas E. Kyprianou and Juan Carlos Pardo. Continuous-state branching processes and self-similarity. J. Appl. Probab., 45(4):1140–1160, 2008.
  • [29] Amaury Lambert. The contour of splitting trees is a Lévy process. Ann. Probab., 38(1):348–395, 2010.
  • [30] Amaury Lambert, Florian Simatos, and Bert Zwart. Scaling limits via excursion theory: interplay between Crump-Mode-Jagers branching processes and processor-sharing queues. Ann. Appl. Probab., 23(6):2357–2381, 2013.
  • [31] Amaury Lambert and Gerónimo Uribe Bravo. Totally ordered measured trees and splitting trees with infinite variation, 2016.
  • [32] John Lamperti. Continuous state branching processes. Bull. Amer. Math. Soc., 73:382–386, 1967.
  • [33] Jean-François Le Gall. Random trees and applications. Probab. Surv., 2:245–311, 2005.
  • [34] Jean-Francois Le Gall and Yves Le Jan. Branching processes in Lévy processes: the exploration process. Ann. Probab., 26(1):213–252, 1998.
  • [35] Grégory Miermont. Ordered additive coalescent and fragmentations associated to Levy processes with no positive jumps. Electron. J. Probab., 6:no. 14, 33 pp. (electronic), 2001.
  • [36] Pressley Warwick Millar. Exit properties of stochastic processes with stationary independent increments. Trans. Amer. Math. Soc., 178:459–479, 1973.
  • [37] Pressley Warwick Millar. Random times and decomposition theorems. In Probability (Proc. Sympos. Pure Math., Vol. XXXI, Univ. Illinois, Urbana, Ill., 1976), pages 91–103. Amer. Math. Soc., Providence, R. I., 1977.
  • [38] Pressley Warwick Millar. Zero-one laws and the minimum of a Markov process. Trans. Amer. Math. Soc., 226:365–391, 1977.
  • [39] Etienne Pardoux and Anton Wakolbinger. From Brownian motion with a local time drift to Feller’s branching diffusion with logistic growth. Electronic Communications in Probability, 16(0):720–731, 2011.
  • [40] Daniel Revuz and Marc Yor. Continuous martingales and Brownian motion, volume 293 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1991.
  • [41] Boris Alekseevich Rogozin. On distributions of functionals related to boundary problems for processes with independent increments. Theory of Probability & Its Applications, 11(4):580–591, jan 1966.
  • [42] David Williams. Path decomposition and continuity of local time for one-dimensional diffusions. I. Proc. London Math. Soc. (3), 28:738–768, 1974.
  • [43] Ahmed I. Zayed. Handbook of function and generalized function transformations. Mathematical Sciences Reference Series. CRC Press, Boca Raton, FL, 1996.