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Spatial Branching Processes and Subordination

Published online by Cambridge University Press:  20 November 2018

Jean Bertoin ,
Jean-François Le Gall  and
Yves Le Jan
Jean Bertoin
Affiliation:
Laboratoire de Probabilités, Université Paris 6, 4, Place Jussieu, 75252 Paris Cedex 05, France
Jean-François Le Gall
Affiliation:
Laboratoire de Probabilités, Université Paris 6, 4, Place Jussieu, 75252 Paris Cedex 05, France e-mail: gall@ccr.jussieu.fr
Yves Le Jan
Affiliation:
Département de Mathématiques, Université Paris-Sud, Batiment 425, Centre d’Orsay, 91405 Orsay Cedex, France
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Abstract

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We present a subordination theory for spatial branching processes. This theory is developed in three different settings, first for branching Markov processes, then for superprocesses and finally for the path-valued process called the Brownian snake. As a common feature of these three situations, subordination can be used to generate new branching mechanisms. As an application, we investigate the compact support property for superprocesses with a general branching mechanism.

Keywords

60J8060J2560J2760J5560G57
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 49 , Issue 1 , 01 February 1997 , pp. 24 - 54
Copyright
Copyright © Canadian Mathematical Society 1997

References

1. Bertoin, J., Lévy Processes, Cambridge Univ. Press, 1996.Google Scholar
2. Billingsley, P., Convergence of Probability Measures, Wiley, New York, 1968.Google Scholar
3. Chauvin, B., Product martingales and stopping lines for branching Brownian motion, Ann. Probab. 19(1991), 1195.ndash;1205.Google Scholar
4. Dawson, D.A., Iscoe, I. and Perkins, E.A., Super-Brownian motion: Path properties and hitting probabilities, Probab. Theor. Relat. Field. 83(1989), 135–205.Google Scholar
5. Dawson, D.A. and Perkins, E.A., Historical processes, ,Mem. Amer.Math. Soc.. 454(1991).Google Scholar
6. Dawson, D.A. and Vinogradov, V., Almost sure path properties of(2Ò dÒ å) superprocesses, Stochastic Process. Appl. 51(1994), 221–258.Google Scholar
7. Dellacherie, C., Maisonneuve, B. and Meyer, P.A., Probabilités et Potentiel, Processus de Markov (fin), Compléments de Calcul Stochastique, Hermann, Paris, 1992.Google Scholar
8. Delmas, J.F., Path properties of superprocesses with a general branching mechanism, preprint, 1996.Google Scholar
9. Dynkin, E.B., Branching particle systems and superprocesses, Ann. Probab. 19(1991), 1157.ndash;1194.Google Scholar
10. Dynkin, E.B. , A probabilistic approach to one class of nonlinear differential equations, Probab. Theor. Relat. Field. 89(1991), 89–115.Google Scholar
11. Dynkin, E.B., Superdiffusions and parabolic nonlinear differential equations, Ann. Probab. 20(1992), 942– 962.Google Scholar
12. Dynkin, E.B. , Superprocesses and partial differential equations, Ann. Probab. 21(1993), 1185.ndash;1262.Google Scholar
13. Dynkin, E.B. and Kuznetsov, S.E., Markov snakes and superprocesses, Probab. Theor. Relat. Field. 103(1995), 433-473.Google Scholar
14. El Karoui, N. and Roelly, S., Propriétés de martingales, explosion et représentation de Lévy-Khintchine d’une classe de processus de branchement à valeurs mesures, Stochastic Process. Appl. 38(1991), 239–266.Google Scholar
15. Gnedenko, B.V. and Kolmogorov, A.N., Limit Distributions for Sums of Independent Random Variables, Addison-Wesley, Cambridge, 1954.Google Scholar
16. Kaj, I. and Salminen, P., On a first passage problem for branching Brownian motions, Ann. Appl. Probab. 3(1993), 173–185.Google Scholar
17. Le Gall, J.F., A class of path-valued Markov processes and its applications to superprocesses, Probab. Theor. Relat. Field. 95(1993), 25–46.Google Scholar
18. Le Gall, J.F., A path-valued Markov process and its connections with partial differential equations, Proc. 1st European Congress of Math., Vol. II, 185–212, Birkhäuser, Boston, 1994.Google Scholar
19. Le Gall, J.F., The Brownian snake and solutions ofΔu= u2 in a domain, Probab. Theor. Relat. Field. 102(1995), 393–432.Google Scholar
20. Le Gall, J.F. , Brownian snakes, superprocesses and partial differential equations, in preparation.Google Scholar
21. Le Gall, J.F. and Perkins, E.A., TheHausdorff measure of the support of two-dimensional super-Brownian motion, Ann. Probab. 23(1995), 1719.1747.Google Scholar
22. Neveu, J., Vers un problème de Dirichlet pour le processus de branchement brownien, unpublished lecture, Université Paris VI, 1988.Google Scholar
23. Watanabe, S., A limit theorem of branching processes and continuous state branching processes, J.Math. Kyoto Univ. 8(1968), 141–167.Google Scholar