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Clumping in multitype-branching trees

Part of: Markov processes General higher-order equations and systems Partial differential equations Stochastic processes

Published online by Cambridge University Press:  01 July 2016

J. Alfredo López-Mimbela  and
Anton Wakolbinger
J. Alfredo López-Mimbela*
Affiliation:
Centro de Investigación en Matemáticas, Guanajuato
Anton Wakolbinger*
Affiliation:
J. W. Goethe Universität
*
∗ Postal address: Centro de Investigación en Matemáticas, Guanajuato, Mexico.
∗∗ Postal address: Fachbereich Mathematik (12), J.W. Goethe Universität, D-60054 Frankfurt am Main, Germany.
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Abstract

We investigate the ‘clumping versus local finiteness' behavior in the infinite backward tree for a class of branching particle systems in ℝd with symmetric stable migration and critical ‘genuine multitype' branching. Under mild assumptions on the branching we establish, by analysing certain ergodic properties of the individual ancestral process, a critical dimension dc such that the (measure-valued) tree-top is almost surely locally finite if and only if d > dc. This result is used to obtain L1-norm asymptotics of a corresponding class of systems of non-linear partial differential equations.

Keywords

BACKWARD TREEMULTITYPE BRANCHING PARTICLE SYSTEMSANCESTRAL PROCESSNONLINEAR EVOLUTION SYSTEMS

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60G57: Random measures
Secondary: 60J85: Applications of branching processes 35B40: Asymptotic behavior of solutions 35G25: Initial value problems for nonlinear higher-order equations
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 28 , Issue 4 , December 1996 , pp. 1034 - 1050
Copyright
Copyright © Applied Probability Trust 1996 

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References

[1] Aldous, D. (1989) Probability Approximations via the Poisson Clumping Heuristic. Springer, New York.Google Scholar
[2] Chauvin, B., Rouault, A. and Wakolbinger, A. (1991) Growing conditioned trees. Stoch. Proc. Appl. 39, 117–130.Google Scholar
[3] Dawson, D. A. (1977) The critical measure diffusion process. Z. Wahrscheinlichkeitsth. 40, 125–145.Google Scholar
[4] Dawson, D. A. and Fleischmann, K. (1985) Critical dimension for a model of branching in a random medium. Z. Wahrscheinlichkeitsth. 70, 315–334.Google Scholar
[5] Feller, W. (1966) An Introduction to Probability Theory and its Applications. Vol. 2. Wiley, New York.Google Scholar
[6] Gorostiza, L., Roelly, S. and Wakolbinger, A. (1992) Persistence of critical multitype particle and measure branching processes. Prob. Theory Rel. Fields 92, 313–335.CrossRefGoogle Scholar
[7] Gorostiza, L. and Wakolbinger, A. (1991) Persistence criteria for a class of critical branching particle systems in continuous time. Ann. Prob. 19, 266–288.Google Scholar
[8] Gorostiza, L. and Wakolbinger, A. (1993) Asymptotic behavior of a reaction-diffusion system. A probabilistic approach. Rand. Comput. Dynam. 1, 445–463.Google Scholar
[9] Harris, Th. E. (1989) Branching Processes. Dover, New York.Google Scholar
[10] Iscoe, I. (1986) A weighted occupation time for a class of measure-valued branching processes. Prob. Theory Rel. Fields 71, 85–116.CrossRefGoogle Scholar
[11] Kallenberg, O. (1977) Stability of critical cluster fields. Math. Nachr. 77, 7–43.Google Scholar
[12] López-Mimbela, J. A. (1992) Fluctuation limits of multitype branching random fields. J. Multivar. Anal. 40, 56–83.Google Scholar
[13] López-Mimbela, J. A. and Wakolbinger, A. (1994) Which critically branching populations persist? In Classical and Modern Branching Processes (IMA Volumes in Mathematics and its Applications). ed Athreya, K. and Jagers, P. Springer, New York. (in press).Google Scholar
[14] Lyons, R. (1994) Probability and Trees. Lecture notes and forthcoming book.Google Scholar
[15] Lyons, R., Pemantle, R. and Peres, Y. (1995) Conceptual proofs of L log L criteria for mean behavior of branching processes. Ann. Prob. 23, 1125–1138.CrossRefGoogle Scholar
[16] Pazy, A. (1983) Semigroups of Linear Operators with Applications to Partial Differential Equations. Springer, New York.CrossRefGoogle Scholar