The Annals of Probability

Branching processes in Lévy processes: the exploration process

Jean-Francois Le Gall and Yves Le Jan

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Abstract

The main idea of the present work is to associate with a general continuous branching process an exploration process that contains the desirable information about the genealogical structure. The exploration process appears as a simple local time functional of a Lévy process with no negative jumps, whose Laplace exponent coincides with the branching mechanism function. This new relation between spectrally positive Lévy processes and continuous branching processes provides a unified perspective on both theories. In particular, we derive the adequate formulation of the classical Ray–Knight theorem for such Lévy processes. As a consequence of this theorem, we show that the path continuity of the exploration process is equivalent to the almost sure extinction of the branching process.

Article information

Source
Ann. Probab., Volume 26, Number 1 (1998), 213-252.

Dates
First available in Project Euclid: 31 May 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1022855417

Digital Object Identifier
doi:10.1214/aop/1022855417

Mathematical Reviews number (MathSciNet)
MR1617047

Zentralblatt MATH identifier
0948.60071

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60J30
Secondary: 60J25: Continuous-time Markov processes on general state spaces 60J55: Local time and additive functionals

Keywords
Branching processes Lévy processes genealogy local time exploration process random tree jump processes

Citation

Le Gall, Jean-Francois; Le Jan, Yves. Branching processes in Lévy processes: the exploration process. Ann. Probab. 26 (1998), no. 1, 213--252. doi:10.1214/aop/1022855417. https://projecteuclid.org/euclid.aop/1022855417


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References

  • [1] Adhikari, A. (1986). Skip free processes. Ph.D. dissertation, Univ. California, Berkeley.
  • [2] Aldous, D. J. (1993). The continuum random tree III. Ann. Probab. 21 248-289.
  • [3] Athreya, K. B. and Ney, P. (1973). Branching Processes. Springer, Berlin.
  • [4] Bennies, J. and Kersting, G. (1995). A random walk approach to Galton-Watson trees. Preprint.
  • [5] Bertoin, J. (1992). An extension of Pitman's theorem for spectrally positive L´evy processes. Ann. Probab. 20 1464-1483.
  • [6] Bertoin, J. (1996). L´evy Processes. Cambridge Univ. Press.
  • [7] Bingham, N. (1975). Fluctuation theory in continuous time. Adv. in Appl. Probab. 7 705-766.
  • [8] Bingham, N. (1976). Continuous branching processes and spectral positivity. Stochastic Process. Appl. 4 217-242.
  • [9] Borovkov, A. A. (1976). Stochastic Processes in Queueing Theory. Springer, Berlin.
  • [10] Borovkov, K. A. and Vatutin, V. A. (1996). On distribution tails and expectations of maxima in critical branching processes. J. Appl. Probab. 33 614-622.
  • [11] Dawson, D. A. and Perkins, E. A. (1991). Historical processes. Mem. Amer. Math. Soc. 454 1-179.
  • [12] Duquesne, T. (1997). Th´eor emes limites pour le processus d'exploration d'arbres de Galton- Watson. Unpublished manuscript.
  • [13] Durrett, R. T. (1984). Oriented percolation in two dimensions. Ann. Probab. 12 999-1040.
  • [14] Dwass, M. (1969). The total progeny in a branching process. J. Appl. Probab. 6 682-686.
  • [15] Dynkin, E. B. (1993). Superprocesses and partial differential equations. Ann. Probab. 21 1185-1262.
  • [16] Dynkin, E. B. (1994). An Introduction to Branching Measure-Valued Processes. Amer. Math. Soc., Providence, RI.
  • [17] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.
  • [18] Feller, W. (1971). An Introduction to Probability Theory and Its Applications 2. Wiley, New York.
  • [19] Geiger, J. (1995). Contour processes of random trees. In Stochastic Partial Differential Equations (A. Etheridge, ed.) 72-96. Cambridge Univ. Press.
  • [20] Geiger, J. (1996). Size-biased and conditioned random splitting trees. Stochastic Process. Appl. 65 187-207.
  • [21] Green, P. J. (1996). The maximum and time to absorption of a left-continuous random walk. J. Appl. Probab. 13 444-454.
  • [22] Grimvall, A. (1974). On the convergence of a sequence of branching processes. Ann. Probab. 2 1027-1045.
  • [23] Harris, T. E. (1974). First passage and recurrence distributions. Trans. Amer. Math. Soc. 73 471-486.
  • [24] Helland, I. S. (1978). Continuity of a class of random time transformations. Stochastic Process. Appl. 7 79-99.
  • [25] Jacod, J. and Shiryaev, A. N. (1987). Limit Theorems for Stochastic Processes. Springer, Berlin.
  • [26] Kendall, D. G. (1951). Some problems in the theory of queues. J. Roy. Statist. Soc. Ser. B 13 151-185.
  • [27] Knight, F. B. (1963). Random walks and a sojourn density of Brownian motion. Trans. Amer. Math. Soc. 109 56-86.
  • [28] Lamperti, J. (1967). The limit of a sequence of branching processes.Wahrsch. Verw. Gebiete 7 271-288.
  • [29] Lamperti, J. (1967). Continuous-state branching processes. Bull. Amer. Math. Soc. 73 382- 386.
  • [30] Le Gall, J. F. (1991). Brownian excursions, trees and measure-valued branching processes. Ann. Probab. 19 1399-1439.
  • [31] Le Gall, J. F. (1993). The uniform random tree in a Brownian excursion. Probab. Theory Related Fields 96 369-383.
  • [32] Le Gall, J. F. (1993). A class of path-valued Markov processes and its applications to superprocesses. Probab. Theory Related Fields 95 25-46.
  • [33] Le Gall, J. F. (1995). The Brownian snake and solutions of u = u2 in a domain. Probab. Theory Related Fields 102 393-432.
  • [34] Le Gall, J. F. and Le Jan, Y. (1995). Arbres al´eatoires et processus de L´evy. C.R. Acad. Sci. Paris S´er. I Math. 321 1241-1244.
  • [35] Le Gall, J. F. and Le Jan, Y. (1997). Branching processes in L´evy processes: Laplace functionals of snakes and superprocesses. Preprint.
  • [36] Lindvall, T. (1976). On the maximum of a branching process. Scand. J. Statist. 3 209-214.
  • [37] Neveu, J. (1986). Arbres et processus de Galton-Watson. Ann. Inst. H. Poincar´e S´er. B 22 199-207.
  • [38] Neveu, J. and Pitman, J. (1989). The branching process in a Brownian excursion. S´eminaire de Probabilit´es XXIII. Lecture Notes in Math. 1372 248-257. Springer, Berlin.
  • [39] Otter, R. (1949). The multiplicative process. Ann. Math. Statist. 20 206-224.
  • [40] Pitman, J. (1997). Enumerations of trees and forests related to branching processes and random walks. Preprint.
  • [41] Rogers, L. C. G. (1984). A new identity for real L´evy processes. Ann. Inst. H. Poincar´e Probab. Statist. 20 21-34.
  • [42] Spitzer, F. (1964). Principles of Random Walk. Van Nostrand, Princeton.
  • [43] Tak´acs, L. (1993). Limit distributions for queues and random rooted trees. J. Appl. Math. Stochastic Anal. 6 189-216.