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Nonexplosion of a class of semilinear equations via branching particle representations

Part of: Markov processes

Published online by Cambridge University Press:  01 July 2016

Santanu Chakraborty  and
Jose Alfredo López-Mimbela
Santanu Chakraborty*
Affiliation:
University of Texas - Pan American
Jose Alfredo López-Mimbela*
Affiliation:
CIMAT
*
∗ Postal address: Department of Mathematics, University of Texas - Pan American, 1201 West University Drive, Edinburg, TX 78539-2999, USA. Email address: schakraborty@utpa.edu
∗∗ Postal address: CIMAT, Apartado Postal 402, Guanajuato 36000, Mexico.
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Abstract

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We consider a branching particle system where an individual particle gives birth to a random number of offspring at the place where it dies. The probability distribution of the number of offspring is given by pk, k = 2, 3, …. The corresponding branching process is related to the semilinear partial differential equation for x ∈ ℝd, where A is the infinitesimal generator of a multiplicative semigroup and the pks, k = 2, 3, …, are nonnegative functions such that We obtain sufficient conditions for the existence of global positive solutions to semilinear equations of this form. Our results extend previous work by Nagasawa and Sirao (1969) and others.

Keywords

Branching particle systemsemilinear partial differential equationglobal solution

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60J85: Applications of branching processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 40 , Issue 1 , March 2008 , pp. 250 - 272
Copyright
Copyright © Applied Probability Trust 2008 

References

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