Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-23T06:03:19.525Z Has data issue: false hasContentIssue false
  • English
  • Français

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: bertrand.cloez@supagro.inra.fr
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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