×

Limit theorems for some branching measure-valued processes. (English) Zbl 1425.60075

Summary: 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.

MSC:

60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60J75 Jump processes (MSC2010)
60F15 Strong limit theorems
60F05 Central limit and other weak theorems
92D25 Population dynamics (general)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Aïdékon, E., Berestycki, J., Brunet, É. and Shi, Z. (2013). Branching Brownian motion seen from its tip. Prob. Theory Relat. Fields157, 405-451. · Zbl 1284.60154
[2] Athreya, K. B. and Ney, P. E. (2004). Branching Processes. Dover Publications, Mineola, NY. · Zbl 1070.60001
[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. Models6, 219-243. · Zbl 1270.35095
[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. · Zbl 1276.60096
[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. · Zbl 1235.60114
[6] Bertoin, J. (2006). Random Fragmentation and Coagulation Processes (Camb. Stud. Adv. Math. 102). Cambridge University Press. · Zbl 1107.60002
[7] Billingsley, P. (1999). Convergence of Probability Measures, 2nd edn. John Wiley, New York. · Zbl 0944.60003
[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. · Zbl 1196.68028
[9] Chen, M.-F. (2004). From Markov chains to Non-Equilibrium Particle Systems, 2nd edn. World Scientific, River Edge, NJ. · Zbl 1078.60003
[10] Dellacherie, C. and Meyer, P.-A. (1980). Probabilités et Potentiel. Chapitres V à VIII (Current Sci. Indust. Topics 1385). Hermann, Paris.
[11] Delmas, J.-F. and Marsalle, L. (2010). Detection of cellular aging in a Galton-Watson process. Stochastic Process. Appl.120, 2495-2519. · Zbl 1206.60077
[12] Doumic Jauffret, M. and Gabriel, P. (2010). Eigenelements of a general aggregation-fragmentation model. Math. Models Methods Appl. Sci.20, 757-783. · Zbl 1201.35086
[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.
[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. · Zbl 1317.92063
[15] Eberle, A. (1999). Uniqueness and Non-Uniqueness of Semigroups Generated by Singular Diffusion Operators (Lecture Notes Math. 1718). Springer, Berlin. · Zbl 0957.60002
[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. · Zbl 1093.60058
[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. · Zbl 1196.60139
[18] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes. John Wiley, New York. · Zbl 0592.60049
[19] Folland, G. B. (1999). Real Analysis, 2nd edn. John Wiley, New York. · Zbl 0924.28001
[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. · Zbl 1060.92055
[21] Grigorescu, I. and Kang, M. (2010). Steady state and scaling limit for a traffic congestion model. ESAIM Prob. Statist.14, 271-285. · Zbl 1227.60108
[22] Guillemin, F., Robert, P. and Zwart, B. (2004). AIMD algorithms and exponential functionals. Ann. Appl. Prob.14, 90-117. · Zbl 1041.60072
[23] Harris, S. C. and Williams, D. (1996). Large deviations and martingales for a typed branching diffusion. I. Astérisque236, 133-154. · Zbl 0857.60088
[24] Hislop, P. D. and Sigal, I. M. (1996). Introduction to Spectral Theory (Appl. Math. Sci. 113). Springer, New York. · Zbl 0855.47002
[25] Ikeda, N. and Watanabe, S. (1989). Stochastic Differential Equations and Diffusion Processes (North-Holland Math. Library 24), 2nd edn. North-Holland, Amsterdam. · Zbl 0684.60040
[26] Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes (Fundamental Principles Math. Sci. 288), 2nd edn. Springer, Berlin. · Zbl 1018.60002
[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. · Zbl 0595.60008
[28] Kallenberg, O. (2002). Foundations of Modern Probability, 2nd edn. Springer, New York. · Zbl 0996.60001
[29] Kolokoltsov, V. N. (2011). Markov Processes, Semigroups and Generators (De Gruyter Stud. Math. 38). Walter de Gruyter, Berlin. · Zbl 1220.60003
[30] Kubitschek, H. E. (1969). Growth during the bacterial cell cycle: analysis of cell size distribution. Biophys. J.9, 792-809.
[31] Laurençot, P. and Perthame, B. (2009). Exponential decay for the growth-fragmentation/cell-division equation. Commun. Math. Sci.7, 503-510. · Zbl 1183.35038
[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. · Zbl 1142.60049
[33] Méléard, S. (1998). Convergence of the fluctuations for interacting diffusions with jumps associated with Boltzmann equations. Stoch. Stoch. Reports63, 195-225. · Zbl 0905.60073
[34] Méléard, S. and Roelly, S. (1993). Sur les convergences étroite ou vague de processus à valeurs mesures. C. R. Acad. Sci. Paris317, 785-788. · Zbl 0781.60071
[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. · Zbl 1243.60067
[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. · Zbl 0781.60053
[37] Michel, P. (2006). Existence of a solution to the cell division eigenproblem. Math. Models Methods Appl. Sci.16, 1125-1153. · Zbl 1094.92023
[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/.
[39] Perthame, B. (2007). Transport Equations in Biology. Birkhäuser, Basel. · Zbl 1185.92006
[40] Perthame, B. and Ryzhik, L. (2005). Exponential decay for the fragmentation or cell-division equation. J. Differential Equat.210, 155-177. · Zbl 1072.35195
[41] Pinsky, R. G. (1995). Positive Harmonic Functions and Diffusion (Camb. Stud. Adv. Math. 45). Cambridge University Press. · Zbl 0858.31001
[42] Reed, M. and Simon, B. (1978). Methods of Modern Mathematical Physics. IV. Analysis of Operators. Academic Press, New York. · Zbl 0401.47001
[43] Roelly-Coppoletta, S. (1986). A criterion of convergence of measure-valued processes: application to measure branching processes. Stochastics17, 43-65. · Zbl 0598.60088
[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.
[45] Villani, C. (2003). Topics in Optimal Transportation (Grad. Stud. Math. 58). American Mathematical Society, Providence, RI.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.