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Probabilistic aspects of critical growth-fragmentation equations

Part of: Equations of mathematical physics and other areas of application Stochastic processes

Published online by Cambridge University Press:  25 July 2016

Jean Bertoin  and
Alexander R. Watson
Jean Bertoin*
Affiliation:
University of Zurich
Alexander R. Watson*
Affiliation:
University of Manchester
*
Institute of Mathematics, University of Zurich, Winterthurerstrasse 190, 8057 Zürich, Switzerland. Email address: [email protected]
School of Mathematics, University of Manchester, Manchester, M13 9PL, UK. Email address: [email protected]
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Abstract

The self-similar growth-fragmentation equation describes the evolution of a medium in which particles grow and divide as time proceeds, with the growth and splitting of each particle depending only upon its size. The critical case of the equation, in which the growth and division rates balance one another, was considered in Doumic and Escobedo (2015) for the homogeneous case where the rates do not depend on the particle size. Here, we study the general self-similar case, using a probabilistic approach based on Lévy processes and positive self-similar Markov processes which also permits us to analyse quite general splitting rates. Whereas existence and uniqueness of the solution are rather easy to establish in the homogeneous case, the equation in the nonhomogeneous case has some surprising features. In particular, using the fact that certain self-similar Markov processes can enter (0,∞) continuously from either 0 or ∞, we exhibit unexpected spontaneous generation of mass in the solutions.

Keywords

Growth-fragmentation equationself-similarityself-similar Markov processbranching process

MSC classification

Primary: 35Q92: PDEs in connection with biology and other natural sciences
Secondary: 45K05: Integro-partial differential equations 60G18: Self-similar processes 60G51: Processes with independent increments; L'evy processes
Type
Research Article
Information
Advances in Applied Probability , Volume 48 , Issue A: Probability, Analysis and Number Theory , July 2016 , pp. 37 - 61
Copyright
Copyright © Applied Probability Trust 2016 

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