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Limit theorems for some branching measure-valued processes

Part of: Markov processes Eigenvalue problems Limit theorems

Published online by Cambridge University Press:  26 June 2017

Bertrand Cloez
Bertrand Cloez*
Affiliation:
INRA
*
* Postal address: INRA Montpellier, UMR MISTEA, Bâtiment 29, 2 Place Pierre Viala, 34060 Montpellier Cedex 1, France. Email address: [email protected]
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Abstract

We consider a particle system in continuous time, a discrete population, with spatial motion, and nonlocal branching. The offspring's positions and their number may depend on the mother's position. Our setting captures, for instance, the processes indexed by a Galton–Watson tree. Using a size-biased auxiliary process for the empirical measure, we determine the asymptotic behaviour of the particle system. We also obtain a large population approximation as a weak solution of a growth-fragmentation equation. Several examples illustrate our results. The main one describes the behaviour of a mitosis model; the population is size structured. In this example, the sizes of the cells grow linearly and if a cell dies then it divides into two descendants.

Keywords

Branching Markov processlimit theoremmany-to-one formulasize-biased reproduction distributioneigenproblemdeterministic macroscopic approximationgrowth-fragmentation equation

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60J75: Jump processes 60F15: Strong theorems 60F05: Central limit and other weak theorems 45C05: Eigenvalue problems 92D25: Population dynamics (general)
Type
Research Article
Information
Advances in Applied Probability , Volume 49 , Issue 2 , June 2017 , pp. 549 - 580
Copyright
Copyright © Applied Probability Trust 2017 

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References

[1] Aïdékon, E., Berestycki, J., Brunet, É. and Shi, Z. (2013). Branching Brownian motion seen from its tip. Prob. Theory Relat. Fields 157, 405–451. CrossRefGoogle Scholar
[2] Athreya, K. B. and Ney, P. E. (2004). Branching Processes. Dover Publications, Mineola, NY. Google Scholar
[3] Balagué, D., Cañizo, J. A. and Gabriel, P. (2013). Fine asymptotics of profiles and relaxation to equilibrium for growth-fragmentation equations with variable drift rates. Kinet. Relat. Models 6, 219–243. CrossRefGoogle Scholar
[4] Bansaye, V. and Tran, V. C. (2011). Branching Feller diffusion for cell division with parasite infection. ALEA Lat. Amer. J. Prob. Math. Statist. 8, 95–127. Google Scholar
[5] Bansaye, V., Delmas, J.-F., Marsalle, L. and Tran, V. C. (2011). Limit theorems for Markov processes indexed by continuous time Galton–Watson trees. Ann. App. Prob. 21, 2263–2314. Google Scholar
[6] Bertoin, J. (2006). Random Fragmentation and Coagulation Processes (Camb. Stud. Adv. Math. 102). Cambridge University Press. CrossRefGoogle Scholar
[7] Billingsley, P. (1999). Convergence of Probability Measures, 2nd edn. John Wiley, New York. CrossRefGoogle Scholar
[8] Chafaï, D., Malrieu, F. and Paroux, K. (2010). On the long time behavior of the TCP window size process. Stoch. Process. Appl. 120, 1518–1534. CrossRefGoogle Scholar
[9] Chen, M.-F. (2004). From Markov chains to Non-Equilibrium Particle Systems, 2nd edn. World Scientific, River Edge, NJ. CrossRefGoogle Scholar
[10] Dellacherie, C. and Meyer, P.-A. (1980). Probabilités et Potentiel. Chapitres V à VIII (Current Sci. Indust. Topics 1385). Hermann, Paris. Google Scholar
[11] Delmas, J.-F. and Marsalle, L. (2010). Detection of cellular aging in a Galton–Watson process. Stochastic Process. Appl. 120, 2495–2519. CrossRefGoogle Scholar
[12] Doumic Jauffret, M. and Gabriel, P. (2010). Eigenelements of a general aggregation-fragmentation model. Math. Models Methods Appl. Sci. 20, 757–783. CrossRefGoogle Scholar
[13] Doumic, M., Maia, P. and Zubelli, J. P. (2010). On the calibration of a size-structured population model from experimental data. Acta Biotheoret. 58, 405–413. CrossRefGoogle ScholarPubMed
[14] Doumic, M., Hoffmann, M., Reynaud-Bouret, P. and Rivoirard, V. (2012). Nonparametric estimation of the division rate of a size-structured population. SIAM J. Numer. Anal. 50, 925–950. CrossRefGoogle Scholar
[15] Eberle, A. (1999). Uniqueness and Non-Uniqueness of Semigroups Generated by Singular Diffusion Operators (Lecture Notes Math. 1718). Springer, Berlin. CrossRefGoogle Scholar
[16] Engländer, J. and Winter, A. (2006). Law of large numbers for a class of superdiffusions. Ann. Inst. H. Poincaré Prob. Statist. 42, 171–185. CrossRefGoogle Scholar
[17] Engländer, J., Harris, S. C. and Kyprianou, A. E. (2010). Strong law of large numbers for branching diffusions. Ann. Inst. H. Poincaré Prob. Statist. 46, 279–298. CrossRefGoogle Scholar
[18] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes. John Wiley, New York. CrossRefGoogle Scholar
[19] Folland, G. B. (1999). Real Analysis, 2nd edn. John Wiley, New York. Google Scholar
[20] Fournier, N. and Méléard, S. (2004). A microscopic probabilistic description of a locally regulated population and macroscopic approximations. Ann. Appl. Prob. 14, 1880–1919. CrossRefGoogle Scholar
[21] Grigorescu, I. and Kang, M. (2010). Steady state and scaling limit for a traffic congestion model. ESAIM Prob. Statist. 14, 271–285. CrossRefGoogle Scholar
[22] Guillemin, F., Robert, P. and Zwart, B. (2004). AIMD algorithms and exponential functionals. Ann. Appl. Prob. 14, 90–117. CrossRefGoogle Scholar
[23] Harris, S. C. and Williams, D. (1996). Large deviations and martingales for a typed branching diffusion. I. Astérisque 236, 133–154. Google Scholar
[24] Hislop, P. D. and Sigal, I. M. (1996). Introduction to Spectral Theory (Appl. Math. Sci. 113). Springer, New York. CrossRefGoogle Scholar
[25] Ikeda, N. and Watanabe, S. (1989). Stochastic Differential Equations and Diffusion Processes (North-Holland Math. Library 24), 2nd edn. North-Holland, Amsterdam. Google Scholar
[26] Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes (Fundamental Principles Math. Sci. 288), 2nd edn. Springer, Berlin. CrossRefGoogle Scholar
[27] Joffe, A. and Métivier, M. (1986). Weak convergence of sequences of semimartingales with applications to multitype branching processes. Adv. Appl. Prob. 18, 20–65. CrossRefGoogle Scholar
[28] Kallenberg, O. (2002). Foundations of Modern Probability, 2nd edn. Springer, New York. CrossRefGoogle Scholar
[29] Kolokoltsov, V. N. (2011). Markov Processes, Semigroups and Generators (De Gruyter Stud. Math. 38). Walter de Gruyter, Berlin. Google Scholar
[30] Kubitschek, H. E. (1969). Growth during the bacterial cell cycle: analysis of cell size distribution. Biophys. J. 9, 792–809. CrossRefGoogle ScholarPubMed
[31] Laurençot, P. and Perthame, B. (2009). Exponential decay for the growth-fragmentation/cell-division equation. Commun. Math. Sci. 7, 503–510. CrossRefGoogle Scholar
[32] Löpker, A. H. and van Leeuwaarden, J. S. H. (2008). Transient moments of the TCP window size process. J. Appl. Prob. 45, 163–175. CrossRefGoogle Scholar
[33] Méléard, S. (1998). Convergence of the fluctuations for interacting diffusions with jumps associated with Boltzmann equations. Stoch. Stoch. Reports 63, 195–225. CrossRefGoogle Scholar
[34] Méléard, S. and Roelly, S. (1993). Sur les convergences étroite ou vague de processus à valeurs mesures. C. R. Acad. Sci. Paris 317, 785–788. Google Scholar
[35] Méléard, S. and Tran, V. C. (2012). Slow and fast scales for superprocess limits of age-structured populations. Stoch. Process. Appl. 122, 250–276. CrossRefGoogle Scholar
[36] Meyn, S. P. and Tweedie, R. L. (1993). Stability of Markovian processes. III. Foster-Lyapunov criteria for continuous-time processes. Adv. Appl. Prob. 25, 518–548. CrossRefGoogle Scholar
[37] Michel, P. (2006). Existence of a solution to the cell division eigenproblem. Math. Models Methods Appl. Sci. 16, 1125–1153. CrossRefGoogle Scholar
[38] Ott, T., Kemperman, J. and Mathis, M. (1996). The stationary behavior of ideal TCP congestion avoidance. Unpublished manuscript. Available at http://www.teunisott.com/. Google Scholar
[39] Perthame, B. (2007). Transport Equations in Biology. Birkhäuser, Basel. CrossRefGoogle Scholar
[40] Perthame, B. and Ryzhik, L. (2005). Exponential decay for the fragmentation or cell-division equation. J. Differential Equat. 210, 155–177. CrossRefGoogle Scholar
[41] Pinsky, R. G. (1995). Positive Harmonic Functions and Diffusion (Camb. Stud. Adv. Math. 45). Cambridge University Press. CrossRefGoogle Scholar
[42] Reed, M. and Simon, B. (1978). Methods of Modern Mathematical Physics. IV. Analysis of Operators. Academic Press, New York. Google Scholar
[43] Roelly-Coppoletta, S. (1986). A criterion of convergence of measure-valued processes: application to measure branching processes. Stochastics 17, 43–65. CrossRefGoogle Scholar
[44] Tran, V. C. (2006). Modèles particulaires stochastiques pour des problèmes d'évolution adaptative et pour l'approximation de solutions statistiques. Doctoral thesis. Universités Paris X - Nanterre. Available at http://tel.archives-ouvertes.fr/tel-00125100. Google Scholar
[45] Villani, C. (2003). Topics in Optimal Transportation (Grad. Stud. Math. 58). American Mathematical Society, Providence, RI. CrossRefGoogle Scholar