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Probabilistic aspects of critical growth-fragmentation equations

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

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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References

[1] Bertoin, J. (1996).Lévy Processes (Camb. Tracts Math. 121).Cambridge University Press.Google Scholar
[2] Bertoin, J. (2006).Random Fragmentation and Coagulation Processes (Camb. Stud. Adv. Math. 102).Cambridge University Press.CrossRefGoogle Scholar
[3] Bertoin, J. (2014).Compensated fragmentation processes and limits of dilated fragmentations. To appear in Ann. Prob. Preprint available at https://hal.archives-ouvertes.fr/hal-00966190v2.Google Scholar
[4] Bertoin, J. (2015).Markovian growth-fragmentation processes. To appear in Bernoulli. Preprint available at https://hal.archives-ouvertes.fr/hal-01152370v1.Google Scholar
[5] Bertoin, J. and Yor, M. (2002).The entrance laws of self-similar Markov processes and exponential functionals of Lévy processes.Potential Anal. 17,389–400.CrossRefGoogle Scholar
[6] Bertoin, J. and Yor, M. (2002).On the entire moments of self-similar Markov processes and exponential functionals of Lévy processes.Ann. Fac. Sci. Toulouse Math. (6) 11,33–45.CrossRefGoogle Scholar
[7] Biggins, J. D. (1977).Chernoff's theorem in the branching random walk.J. Appl. Prob. 14,630–636.CrossRefGoogle Scholar
[8] Blumenthal, R. M. and Getoor, R. K. (1968).Markov Processes and Potential Theory (Pure Appl. Math. 29).Academic Press,New York.Google Scholar
[9] Chaumont, L. and Rivero, V. (2007).On some transformations between positive self-similar Markov processes.Stoch. Process. Appl. 117,1889–1909.CrossRefGoogle Scholar
[10] Davies, B. (2002).Integral Transforms and Their Applications (Texts Appl. Math. 41),3rd edn.Springer,New York.CrossRefGoogle Scholar
[11] Doumic, M. and Escobedo, M. (2015).Time asymptotics for a critical case in fragmentation and growth-fragmentation equations. Preprint. Available at https://hal.archives-ouvertes.fr/hal-01080361v2.Google Scholar
[12] Ethier, S. N. and Kurtz, T. G. (1986).Markov Processes: Characterization and Convergence.John Wiley,New York.CrossRefGoogle Scholar
[13] Haas, B. (2003).Loss of mass in deterministic and random fragmentations.Stoch. Process. Appl. 106,245–277.CrossRefGoogle Scholar
[14] Haas, B. and Rivero, V. (2012).Quasi-stationary distributions and Yaglom limits of self-similar Markov processes.Stoch. Process. Appl. 122,4054–4095.CrossRefGoogle Scholar
[15] Hardy, R. and Harris, S. C. (2009).A spine approach to branching diffusions with applications to ℒ p -convergence of martingales. In Séminaire de Probabilités XLII (Lecture Notes Math. 1979),Springer,Berlin, pp. 281–330.CrossRefGoogle Scholar
[16] Kyprianou, A. E. (2014).Fluctuations of Lévy Processes with Applications,2nd edn.Springer,Berlin.CrossRefGoogle Scholar
[17] Lamperti, J. (1972).Semi-stable Markov processes, I.Z. Wahrscheinlichkeitsth. 22,205–225.CrossRefGoogle Scholar
[18] Protter, P. E. (2004).Stochastic Integration and Differential Equations (Appl. Math. 21),2nd edn.Springer,Berlin.Google Scholar
[19] Rivero, V. (2005).Recurrent extensions of self-similar Markov processes and Cramér's condition.Bernoulli 11,471–509.CrossRefGoogle Scholar
[20] Rivero, V. (2012).Tail asymptotics for exponential functionals of Lévy processes: the convolution equivalent case.Ann. Inst. H. Poincaré Prob. Statist. 48,1081–1102.CrossRefGoogle Scholar
[21] Sato, K. (1999).Lévy Processes and Infinitely Divisible Distributions (Camb. Stud. Adv. Math. 68).Cambridge University Press.Google Scholar
[22] Vuolle-Apiala, J. (1994).Itô excursion theory for self-similar Markov processes.Ann. Prob. 22,546–565.CrossRefGoogle Scholar