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Length of Galton–Watson trees and blow-up of semilinear systems

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

J. Alfredo López-Mimbela  and
Anton Wakolbinger
J. Alfredo López-Mimbela*
Affiliation:
Centro de Investigación en Matemáticas
Anton Wakolbinger*
Affiliation:
J. W. Goethe-Universität Frankfurt am Main
*
∗Postal address: Apartado Postal 402, Guanajuato 36000, Mexico. Email address: jalfredo@fractal.cimat.mx
∗∗Postal address: FB Mathematik (12), J. W. Goethe Universität, D-60054 Frankfurt am Main, Germany.
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Abstract

By lower estimates of the functionals 𝔼[eStKtNt], where St and Nt denote the total length up to time t and the number of individuals at time t in a Galton-Watson tree, we obtain sufficient criteria for the blow-up of semilinear equations and systems of the type ∂wt/∂t = Awt + Vwtβ. Roughly speaking, the growth of the tree length has to win against the ‘mobility’ of the motion belonging to the generator A, since, in the probabilistic representation of the equations, the latter results in small K(t) as t → ∞. In the single-type situation, this gives a re-interpretation of classical results of Nagasawa and Sirao(1969); in the multitype scenario, part of the results obtained through analytic methods in Escobedo and Herrero (1991) and (1995) are re-proved and extended from the case A = Δ to the case of α-Laplacians.

Keywords

Galton–Watson treesmultitype branching particle systemssemilinear partial differential equations

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60J85: Applications of branching processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 35 , Issue 4 , December 1998 , pp. 802 - 811
Copyright
Copyright © Applied Probability Trust 1998 

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