Limit theorems for Smoluchowski dynamics associated with critical continuous-state branching processes. (English) Zbl 1312.60100

Summary: We investigate the well-posedness and asymptotic self-similarity of solutions to a generalized Smoluchowski coagulation equation recently introduced by Bertoin and Le Gall in the context of continuous-state branching theory. In particular, this equation governs the evolution of the Lévy measure of a critical continuous-state branching process which becomes extinct (i.e. is absorbed at zero) almost surely. We show that a nondegenerate scaling limit of the Lévy measure (and the process) exists if and only if the branching mechanism is regularly varying at 0. When the branching mechanism is regularly varying, we characterize nondegenerate scaling limits of arbitrary finite-measure solutions in terms of generalized Mittag-Leffler series.


60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60G18 Self-similar stochastic processes
35Q70 PDEs in connection with mechanics of particles and systems of particles
82C28 Dynamic renormalization group methods applied to problems in time-dependent statistical mechanics
Full Text: DOI arXiv Euclid


[1] Bauer, H. (2001). Measure and Integration Theory. de Gruyter Studies in Mathematics 26 . de Gruyter, Berlin. · Zbl 0985.28001
[2] Bertoin, J. (2002). Eternal solutions to Smoluchowski’s coagulation equation with additive kernel and their probabilistic interpretations. Ann. Appl. Probab. 12 547-564. · Zbl 1030.60036
[3] Bertoin, J. and Le Gall, J.-F. (2006). Stochastic flows associated to coalescent processes. III. Limit theorems. Illinois J. Math. 50 147-181 (electronic). · Zbl 1110.60026
[4] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1989). Regular Variation. Encyclopedia of Mathematics and Its Applications 27 . Cambridge Univ. Press, Cambridge. · Zbl 0667.26003
[5] Borovkov, K. A. (1988). A method for the proof of limit theorems for branching processes. Teor. Veroyatn. Primen. 33 115-123. · Zbl 0636.60085
[6] Bricmont, J., Kupiainen, A. and Lin, G. (1994). Renormalization group and asymptotics of solutions of nonlinear parabolic equations. Comm. Pure Appl. Math. 47 893-922. · Zbl 0806.35067
[7] Duquesne, T. and Le Gall, J.-F. (2002). Random trees, Lévy processes and spatial branching processes. Astérisque 281 vi+147. · Zbl 1037.60074
[8] Feller, W. (1971). An Introduction to Probability Theory and Its Applications. Vol. II , 2nd ed. Wiley, New York. · Zbl 0219.60003
[9] Grey, D. R. (1974). Asymptotic behaviour of continuous time, continuous state-space branching processes. J. Appl. Probab. 11 669-677. · Zbl 0301.60060
[10] Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes , 2nd ed. Grundlehren der Mathematischen Wissenschaften 288 . Springer, Berlin. · Zbl 1018.60002
[11] Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications . Springer, Berlin. · Zbl 1104.60001
[12] Kyprianou, A. E. and Pardo, J. C. (2008). Continuous-state branching processes and self-similarity. J. Appl. Probab. 45 1140-1160. · Zbl 1157.60078
[13] Lambert, A. (2007). Quasi-stationary distributions and the continuous-state branching process conditioned to be never extinct. Electron. J. Probab. 12 420-446. · Zbl 1127.60082
[14] Leyvraz, F. (2003). Scaling theory and exactly solved models in the kinetics of irreversible aggregation. Phys. Reports 383 95-212.
[15] Li, Z.-H. (2000). Asymptotic behaviour of continuous time and state branching processes. J. Austral. Math. Soc. Ser. A 68 68-84. · Zbl 0960.60072
[16] Menon, G. and Pego, R. L. (2004). Approach to self-similarity in Smoluchowski’s coagulation equations. Comm. Pure Appl. Math. 57 1197-1232. · Zbl 1049.35048
[17] Menon, G. and Pego, R. L. (2007). Universality classes in Burgers turbulence. Comm. Math. Phys. 273 177-202. · Zbl 1127.76025
[18] Menon, G. and Pego, R. L. (2008). The scaling attractor and ultimate dynamics for Smoluchowski’s coagulation equations. J. Nonlinear Sci. 18 143-190. · Zbl 1142.82012
[19] Norris, J. R. (1999). Smoluchowski’s coagulation equation: Uniqueness, nonuniqueness and a hydrodynamic limit for the stochastic coalescent. Ann. Appl. Probab. 9 78-109. · Zbl 0944.60082
[20] Pakes, A. G. (2010). Critical Markov branching process limit theorems allowing infinite variance. Adv. in Appl. Probab. 42 460-488. · Zbl 1197.60077
[21] Prabhakar, T. R. (1971). A singular integral equation with a generalized Mittag-Leffler function in the kernel. Yokohama Math. J. 19 7-15. · Zbl 0221.45003
[22] Schilling, R. L., Song, R. and Vondraček, Z. (2010). Bernstein Functions : Theory and Applications. de Gruyter Studies in Mathematics 37 . de Gruyter, Berlin.
[23] Slack, R. S. (1972). Further notes on branching processes with mean \(1\). Z. Wahrsch. Verw. Gebiete 25 31-38. · Zbl 0236.60056
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.