Abstract
We apply large-deviation theory to particle systems with a random mean-field interaction in the McKean-Vlasov limit. In particular, we describe large deviations and normal fluctuations around the McKean-Vlasov equation. Due to the randomness in the interaction, the McKean-Vlasov equation is a collection of coupled PDEs indexed by the state space of the single components in the medium. As a result, the study of its solution and of the finite-size fluctuation around this solution requires some new ingredient as compared to existing techniques for nonrandom interaction.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
G. Ben Arous and M. Brunaud, Méthode de Laplace: Etude variationnelle des fluctuations de diffusions de type “champs moyens”,Stoch. Stoch. Rep. 31:79–144 (1990).
E. Bolthausen, Laplace approximations for sums of independent random vectors,Prob. Theory Related Fields 72:305–318 (1986).
H. Brezis,Analyse Fonctionnelle: Théorie et Applications (Masson, Paris, 1983).
M. Brunaud, Finite Kullback information diffusion laws with fixed marginals and associated large deviations functionals,Stoch. Proc. Appl. 44:329–345 (1993).
F. Comets, Nucleation for a long range magnetic model,Ann Inst. H. Poincaré 23: 135–178 (1987).
F. Comets and Th. Eisele, Asymptotic dynamics, non-critical and critical fluctuations for a geometric long-range interacting model,Commun. Math. Phys. 118:531–567 (1988).
D. A. Dawson, Critical dynamics and fluctuations for a mean field model of cooperative behavior.J. Stat. Phys. 31:29–83 (1982).
D. A. Dawson and J. Gärtner, Large deviations from the McKean-Vlasov limit for weakly interacting diffusions,Stochastics 20:247–308 (1987).
J.-D. Deuschel and D. W. Stroock,Large Deviations (Academic Press, Boston, 1989).
S. N. Ethier and T. G. Kurtz,Markov Processes, Characterization and Convergence (Wiley, New York, 1986).
S. Feng, Large deviations for empirical process of mean-field interacting particle systems with unbounded jumps.Ann. Prob. 22:2122–2151 (1994).
H. Föllmer, Random fields and diffusion processes, inÉcole d'Eté de Probabilités de Saint Flour XV-XVII (Springer, Berlin, 1988), pp. 101–204.
I. Karatzas and S. Shreve,Brownian Motion and Stochastic Calculus (Springer, New York, 1988).
S. Kusuoka and Y. Tamura, Gibbs measure for mean field potentials,J. Fac. Sci. Univ. Tokyo I.A. 31:223–245 (1984).
R. S. Lipster and A. N. Shiryaev,Statistics of Random Processes, Vol. 2 (Springer, New York, 1988).
A.-S. Sznitman, Nonlinear reflecting diffusion process, and the propagation of chaos and fluctuations associated.J. Funct. Anal. 56:311–336 (1984).
S. R. S. Varadhan,Large Deviations and Applications (Society for Industrial and Applied Mathematics, Philadelphia, 1984).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Pra, P.D., Hollander, F.d. McKean-Vlasov limit for interacting random processes in random media. J Stat Phys 84, 735–772 (1996). https://doi.org/10.1007/BF02179656
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02179656