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Fluctuations of spatial branching processes with mean-field interaction

Published online by Cambridge University Press:  01 July 2016

B. Chauvin ,
P. Olivares-Rieumont  and
A. Rouault
B. Chauvin*
Affiliation:
Université Paris VI
P. Olivares-Rieumont*
Affiliation:
Universidad La Habana
A. Rouault*
Affiliation:
Université Paris XI
*
∗Postal address: Université Paris VI, Laboratoire de Probabilités, 4, Place Jussieu, tour 56, 3 ème étage, 75230 Paris Cedex 05, France.
∗∗Postal address: Universidad La Habana, Departamento de Cybernetica y Matematica, San Lazaro y L (Vedado), La Habana, Cuba.
∗∗∗Postal address: UA-CNRS 743, Statistique Appliquée, Université Paris Sud, Mathématiques (Bat. 425) 91405 Orsay Cedex, France.
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Abstract

We consider a branching Brownian motion on starting with n particles of mass 1/n, with interactive branching dynamics. The parameters are unsealed, but depend on the present state of the measure-valued process. For this mean-field model, which is a generalization of Chauvin and Rouault (1990) and Nappo and Orlandi (1988), we prove a propagation of chaos and a fluctuation theorem in ([0, T]; W–5).

Keywords

BRANCHING BROWNIAN MOTIONPARTICLE SYSTEMSFLUCTUATIONSMEAN FIELDSOBOLEV SPACES
Type
Research Article
Information
Advances in Applied Probability , Volume 23 , Issue 4 , December 1991 , pp. 716 - 732
Copyright
Copyright © Applied Probability Trust 1991 

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References

Adams, R. (1975) Sobolev Spaces. Academic Press, New York.Google Scholar
Borde, A. M. (1990) Stochastic demographic models. Age of a population. Stoch. Proc. Appl. Google Scholar
Chauvin, B. and Rouault, A. (1990) A stochastic simulation for solving scalar reaction–diffusion equations. Adv. Appl. Prob. 22, 88–100.CrossRefGoogle Scholar
Comets, F. and Eisele, Th. (1988) Asymptotic dynamics, non-critical and critical fluctuations for a geometric long-range interaction model. Commun. Math. Phys. 118, 531–567.CrossRefGoogle Scholar
Dawson, D. and Gartner, J. (1989) Large deviations, free energy functional and quasi-potential for a mean field model of interacting diffusions. Mem. Amer. Math. Soc. 78, N° 398.Google Scholar
Dittrich, P. (1988) A stochastic model of a chemical reaction with diffusion. Prob. Theory Rel. Fields 79, 115–128.CrossRefGoogle Scholar
Dittrich, P. (1988) A stochastic particle system: fluctuations around a nonlinear reaction–diffusion equation. Stoch. Proc. Appl. 30, 149–164.CrossRefGoogle Scholar
Ethier, S. and Kurtz, T. (1986) Markov Processes. Characterization and Convergence. Wiley, New York.CrossRefGoogle Scholar
Joffe, A. and Metivier, M. (1986) Weak convergence of sequences of semimartingales with applications to multitype branching processes. Adv. Appl. Prob. 18, 20–65.CrossRefGoogle Scholar
Kotelenez, P. (1989) A stochastic reaction diffusion model. Lecture Notes in Mathematics 1390, 132–137.CrossRefGoogle Scholar
Leonard, C. (1987) Large deviations and law of large numbers for a mean field type interacting particle system. Stoch. Proc. Appl. 25, 215–235.CrossRefGoogle Scholar
Metivier, M. (1984) Quelques problèmes liés aux systèmes infinis de particules et leurs limites. Lecture Notes in Mathematics 1204, 426–446.CrossRefGoogle Scholar
Metivier, M. (1985) Weak convergence of measure valued processes using Sobolev imbedding techniques. Lecture Notes in Mathematics 1236, 172–183.CrossRefGoogle Scholar
Nappo, N. and Orlandi, E. (1988) Limit laws for a coagulation model of interacting random particles. Ann. Inst. Henri Poincaré 24, 319–344.Google Scholar
Roelly-Coppoletta, S. (1986) A criterion of convergence of measure valued processes: application to measure branching processes. Stochastics 17, 43–65.CrossRefGoogle Scholar
Roelly, S. and Rouault, A. (1990) Construction et propriétés des processus de branchement à valeurs mesures. Internat. Statist. Rev. CrossRefGoogle Scholar
Sznitman, A. S. (1984) Nonlinear reflecting diffusion processes and the propagation of chaos and fluctuations associated. J. Funct. Anal. 56, 311–386.CrossRefGoogle Scholar