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The central limit theorem for the Smoluchovski coagulation model

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Kolokoltsov, V. N. (Vasiliĭ Nikitich). (2010) The central limit theorem for the Smoluchovski coagulation model. Probability Theory and Related Fields, Vol.146 (No.1-2). pp. 87-153. ISSN 0178-8051

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Official URL: http://dx.doi.org/10.1007/s00440-008-0186-2

Abstract

The general model of coagulation is considered. For basic classes of unbounded coagulation kernels the central limit theorem (CLT) is obtained for the fluctuations around the dynamic law of large numbers (LLN) described by the Smoluchovski equation. A rather precise rate of convergence is given both for LLN and CLT.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Statistics
Library of Congress Subject Headings (LCSH):	Coagulation, Central limit theorem, Limit theorems (Probability theory), Jump processes, Markov processes
Journal or Publication Title:	Probability Theory and Related Fields
Publisher:	Springer
ISSN:	0178-8051
Date:	January 2010
Volume:	Vol.146
Number:	No.1-2
Page Range:	pp. 87-153
Identification Number:	10.1007/s00440-008-0186-2
Status:	Peer Reviewed
Access rights to Published version:	Restricted or Subscription Access
References:	[1] D.J. Aldous. Deterministic and stochastic models for coalescence (aggregation and coagulation): a review of the mean-field theory for probabilists. Bernoulli 5:1 (1999), 3-48. [2] V.P. Belavkin. A Quantum Nonadapted Ito Formula and Stochastic Analysis in Fock Scale. J. Funct. Anal. 102:2 (1991), 414-447. [3] V. Belavkin, V. Kolokoltsov. On general kinetic equation for many particle systems with interaction, fragmentation and coagulation. Proc. R. Soc. Lond. A 459 (2002), 1-22. [4] P. Bremaud. Point Processes and Queues, Springer, 1981. [5] Mu Fa Chen. From Markov Chains to Non-Equilibrium Particle Systems. World Scientific, 1992. [6] D. Dawson. Critical dynamics and fluctuations for a mean field model of cooperative behavior. J. Stat. Phys. 31 (1983), 29-85. [7] D. Dawson et al. Generalized Mehler Semigroups and Catalytic Branching Processes with Immigration. Potential Anal. 21:1 (2004), 75-97. [8] M. Deaconu, N. Fournier, E. Tanre. Rate of convergence of a stochastic particle system for the Smoluchowski coagulation equation. Methodol. Comput. Appl. Probab. 5:2 (2003), 131-158. [9] A.M. Etheridge. An Introduction to Superprocesses. University Lecture Series, v. 20. AMS Providence, 2000. [10] A. Joffe, M. Metivier. Weak convergence of sequence of semimatingales with applications to multitype branching processes. Adv. Appl. Probab. 18 (1986), 20-65. [11] S.N. Ethier, T.G. Kurtz. Markov Processes. Characterization and convergence. John Wiley Sons 1986. [12] M. Freidlin. Functional Integration and Partial Differential Equations. Princeton Univ. Press, Princeton, NY 1985. [13] E. Gine, J.A. Wellner. Uniform convergence in some limit theorem for multiple particle systems. Stochastic Processes and their Applications 72 (1997), 47-72. [14] O. Kallenberg. Foundations of Modern Probability. Second ed., Springer 2002. [15] A. Kolodko, K. Sabelfeld, W. Wagner. A stochastic Method for Solving Smoluchowski's coagulation equation. Math. Comput. Simulation 49 (1999), 57-79. [16] V. N. Kolokoltsov. On Extension of Molli¯ed Boltzmann and Smoluchovski Equations to Particle Systems with a k-ary Interaction. Russian Journal of Math.Phys. 10:3 (2003), 268-295. [17] V. N. Kolokoltsov. Hydrodynamic Limit of Coagulation-Fragmentation Type Models of k-nary Interacting Particles. Journal of Statistical Physics 115, 5/6 (2004), 1621-1653. [18] V. N. Kolokoltsov. Kinetic equations for the pure jump models of k-nary interacting particle systems. Markov Processes and Related Fields 12 (2006), 95-138. [19] V. N. Kolokoltsov. On the regularity of solutions to the spatially homogeneous Boltzmann equation with polynomially growing collision kernel. Preprint Universidad Autonoma Metropolitana, 04.0402.1.I.003.2005, Mexico. Published in Adv.Stud.Cont.Math. 12:1 (2006), 9-38. [20] V. N. Kolokoltsov. Symmetric Stable Laws and Stable-Like Jump-Diffusions. Proc. London Math. Soc. 3:80 (2000), 725-768. [21] V. N. Kolokoltsov. Nonlinear Markov Semigroups and Interacting Levy Type Processes. Journ. Stat. Physics 126:3 (2007), 585-642. [22] M. Lachowicz, Ph. Laurencot, D. Wrzosek. On the Oort-Hulst-Savronov coagulation equation and its relation to the Smoluchowski equation. SIAM J. Math. Anal. 34 (2003), 1399-1421. [23] P. Lescot, M. Roeckner. Perturbations of Generalized Mehler Semigroups and Applications to Stochastic Heat Equation with Lèvy Noise and Singular Drift. Potential Anal. 20:4 (2004), 317-344. [24] F. Leyvraz. Scaling theory and exactly solved models in the kinetics of irreversible aggregation. Physics Reports 383, 2-3 (2003), 95-212. [25] R.H. Martin. Nonlinear operators and differential equations in Banach spaces. New York, 1976. [26] V.P. Maslov. Perturbation Theory and Asymptotical Methods. Moscow State University Press, 1965 (in Russian). French Transl. Dunod, Paris, 1972. [27] V.P. Maslov. Mèthodes Opèratorielles. Moscow, Nauka 1974 (in Russian). French transl. Moscow, Mir, 1987. [28] V.P. Maslov. Complex Markov Chains and Functional Feynman Integral. Moscow, Nauka, 1976 (in Russian). [29] S. Mèlèard. Convergence of the fluctuations for interacting diffusions with jumps associated with Boltzmann equations. Stochastics Stochastics Rep. 63: 3-4 (1998), 195-225. [30] M. Mètivier.Weak convergence of mesaure-valued processes using Sobolev imbedding techniques. Proceedings 'Stochastic Partial Differential Equations', Trento 1985, Springer LNM 1236, 172-183. [31] I. Mitoma. Tightness of probabilities on C([0; 1]; S0) and D([0; 1]; S0). Ann. Probab. 11:4 (1983), 989{999. [32] I. Mitoma. An 1-dimensional inhomogeneous Langevin's equation. J. Functional Analysis 61 (1985), 342-359. [33] J.M. van Neerven. Continuity and Representation of Gaussian Mehler Semigroups. Potential Anal. 13:3 (2000), 199-211. [34] J. Norris. Cluster Coagulation. Comm. Math. Phys. 209(2000), 407-435. [35] A. Ja. Povzner. The Boltzmann equation in the kinetic theory of gases. Mat. Sbornik 58 (1962), 65-86. [36] R. Rebolledo. Sur l'existence de solution a certain problèmes de semimartingales, C.R. Acad. Sci. Paris, Ser. A-B 290:18 (1980), 843-846.
URI:	http://wrap.warwick.ac.uk/id/eprint/2496