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The propagation of chaos of multitype mean field interacting particle systems

Part of: Special processes Markov processes Limit theorems

Published online by Cambridge University Press:  14 July 2016

Shui Feng
Shui Feng*
Affiliation:
McMaster University
*
∗Postal address: Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario L8S 4K1, Canada.
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Abstract

A result for the propagation of chaos is obtained for a class of pure jump particle systems of two species with mean field interaction. This result leads to the corresponding result for particle systems with one species and the argument used is valid for particle systems with more than two species. The model is motivated by the study of the phenomenon of self-organization in biology, chemistry and physics, and the technical difficulty is the unboundedness of the jump rates.

Keywords

PURE JUMP MARKOV PROCESSNONLINEAR MASTER EQUATIONMEAN FIELD INTERACTIONPROPAGATION OF CHAOSEMPIRICAL DISTRIBUTION

MSC classification

Primary: 60F05: Central limit and other weak theorems 60J75: Jump processes 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Type
Research Article
Information
Journal of Applied Probability , Volume 34 , Issue 2 , June 1997 , pp. 346 - 362
Copyright
Copyright © Applied Probability Trust 1997 

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Footnotes

Research supported by the SERB grant at McMaster University and by the Natural Sciences and Engineering Research Council of Canada.

References

[1] Chen, M. F. (1992) From Markov Chains to Non-Equilibrium Particle System. World Scientific, Singapore.Google Scholar
[2] Dawson, D. A. (1983) Critical dynamics and fluctuations for a mean-field model of cooperative behaviour. J. Statist. Phys. 31, 29–85.Google Scholar
[3] Dawson, D. A. and Zheng, X. (1991) Law of large numbers and a central limit theorem for unbounded jump mean-field models. Adv. Appl. Math. 12, 293–326.Google Scholar
[4] Feng, S. (1995) Nonlinear master equation of multitype particle systems. Stoch. Proc. Appl. to appear.Google Scholar
[5] Feng, S. and Zheng, X. (1992) Solutions of a class of nonlinear master equations. Stoch. Proc. Appl. 43, 65–84.Google Scholar
[6] Ferland, R. and Giroux, G. (1992) An unbounded mean field intensity model: propagation of the convergence of the empirical laws and compactness of the fluctuations. Preprint. Google Scholar
[7] Lotka, A. J. (1920) Undamped oscillations derived from the law of mass action. J. Amer. Chem. Soc. 42, 1595–1599.Google Scholar
[8] Scheutzow, M. (1985) Some examples of nonlinear diffusions having a time-periodic law. Ann. Prob. 13, 379–384.Google Scholar
[9] Scheutzow, M. (1986) Periodic behavior of the stochastic Brusselator in the mean-field limit. Prob. Theory Rel. Fields 72, 425–462.Google Scholar
[10] Shiga, T. and Tanaka, H. (1985) Central limit theorem for a system of Markovian particles with mean-field interactions. Z. Wahrscheinlichkeitsth. 69, 439–459.Google Scholar
[11] Sznitman, S. A. (1984) Nonlinear reflecting diffusion process, and the propagation of chaos and fluctuations associated. J. Funct. Anal. 56, 311–336.Google Scholar
[12] Sznitman, S. A. (1991) Topics in propagation of chaos. In Lecture Notes in Mathematics 1464. Springer, Berlin, pp. 167–251.Google Scholar
[13] Tanaka, H. (1984) Limit theorems for certain diffusion processes with interactions. In Taniguchi Symp., Katata. Kinokuniya, Tokyo, pp. 469–488.Google Scholar
[14] Volterra, V. (1926) Variazioni fluttuazioni del numero d'individui in specie animali conviventi. Mem. Acad. Lincei. 2, 31–113.Google Scholar
[15] Zheng, J. L. and Zheng, X. G. (1987) A martingale approach to q-process. Chinese Kexue Tongbao 22, 1457–1459.Google Scholar