The Annals of Probability

Branching processes in Lévy processes: Laplace functionals of snakes and superprocesses

Jean-François Le Gall and Yves Le Jan

Full-text: Open access

Abstract

We use the exploration process introduced in a previous work to develop a new construction of superprocesses with a general branching mechanism. This construction depends on a path-valued process called the Lévy snake, which is of independent interest. Our method of proof involves a calculation of the Laplace functional of the occupation field of the Lévy snake. This calculation relies on an evaluation of the corresponding moment functionals, which requires precise information about the underlying genealogical structure.

Article information

Source
Ann. Probab., Volume 26, Number 4 (1998), 1407-1432.

Dates
First available in Project Euclid: 31 May 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1022855868

Digital Object Identifier
doi:10.1214/aop/1022855868

Mathematical Reviews number (MathSciNet)
MR1675019

Zentralblatt MATH identifier
0945.60090

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60J30 60G57: Random measures

Keywords
Branching process Lévy process Lévy snake superprocess exploration process Laplace functionals moment functionals

Citation

Le Gall, Jean-François; Le Jan, Yves. Branching processes in Lévy processes: Laplace functionals of snakes and superprocesses. Ann. Probab. 26 (1998), no. 4, 1407--1432. doi:10.1214/aop/1022855868. https://projecteuclid.org/euclid.aop/1022855868


Export citation

References

  • [1] Bertoin, J. (1996). L´evy processes. Cambridge Univ. Press.
  • [2] Bingham, N. (1975). Fluctuation theory in continuous time. Adv. in Appl. Probab. 7 705-766.
  • [3] Dawson, D. A. and Perkins, E. A. (1991). Historical processes. Mem. Amer. Math. Soc. 454 1-179.
  • [4] Dellacherie, C., Maisonneuve, B. and Meyer, P. A. (1992). Probabilit´es et Potentiel, Chapitres 17 a 24. Hermann, Paris.
  • [5] Dynkin, E. B. (1991). Branching particle systems and superprocesses. Ann. Probab. 19 1157-1194.
  • [6] Dynkin, E. B. (1993). Superprocesses and partial differential equations. Ann. Probab. 21 1185-1262.
  • [7] Dynkin, E. B. (1994). An Introduction to Branching Measure-Valued Processes. Amer. Math. Soc., Providence.
  • [8] Le Gall, J. F. (1993). A class of path-valued Markov processes and its applications to superprocesses. Probab. Theory Related Fields 95 25-46.
  • [9] Le Gall, J. F. (1993). The uniform random tree in a Brownian excursion. Probab. Theory Related Fields 96 369-383.
  • [10] Le Gall, J. F. (1995). The Brownian snake and solutions of u = u2 in a domain. Probab. Theory Related Fields 102 393-432.
  • [11] Le Gall, J. F. (1998). Branching processes in L´evy processes: The L´evy snake, local times and superprocesses. Unpublished manuscript.
  • [12] 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.
  • [13] Le Gall, J. F. and Le Jan, Y. (1998). Branching processes in L´evy processes: The exploration process. Ann. Probab. 26 213-252.
  • [14] Le Jan, Y. (1991). Superprocesses and projective limits of branching particle systems. Ann. Inst. H. Poincar´e Probab. Statist. 27 91-106.
  • [15] Liemant, A., Matthes, K. and Wakolbinger, A. (1988). Equilibrium Distributions of Branching Processes. Kluwer, Dordrecht.
  • [16] Neveu, J. (1964). Bases Math´ematiques du Calcul des Probabilit´es. Masson, Paris.