Large deviations from the mckean-vlasov limit for weakly interacting diffusions

A system of N diffusions on R$SUP:D$ESUP:in which the interaction is expressed in terms of the empirical measure is considered. The limiting behavior as N →∞ is described by a McKean_Vlasov equation. The purpose of this paper is to show that the large deviations from the McKean-Vlasov limit can be described by a generalization of the theory of Freidlin and Wentzell and to obtain a characterization of the action functional. In order to obtain this action functional we first obtain results on projective limits of large deviation systems, large deviations on dual vector spaces and a Sanov type theorem for vectors of empirical measures

[1]  R. E. Edwards,et al.  Functional Analysis: Theory and Applications , 1965 .

[2]  O. Ladyženskaja Linear and Quasilinear Equations of Parabolic Type , 1968 .

[3]  S. Varadhan,et al.  Asymptotic evaluation of certain Markov process expectations for large time , 1975 .

[4]  J. Gärtner On Large Deviations from the Invariant Measure , 1977 .

[5]  R. Ellis,et al.  The statistics of Curie-Weiss models , 1978 .

[6]  Robert Zwanzig,et al.  Statistical mechanics of a nonlinear stochastic model , 1978 .

[7]  Piet Groeneboom,et al.  Large Deviation Theorems for Empirical Probability Measures , 1979 .

[8]  J. Jacod Calcul stochastique et problèmes de martingales , 1979 .

[9]  D. W. Stroock,et al.  Multidimensional Diffusion Processes , 1979 .

[10]  Giovanna Jona-Lasinio,et al.  Large fluctution for a non linear heat equation with noise , 1982 .

[11]  H. Bauer,et al.  Probability Theory and Elements of Measure Theory , 1982 .

[12]  L. Rogers Stochastic differential equations and diffusion processes: Nobuyuki Ikeda and Shinzo Watanabe North-Holland, Amsterdam, 1981, xiv + 464 pages, Dfl.175.00 , 1982 .

[13]  D. Dawson Critical dynamics and fluctuations for a mean-field model of cooperative behavior , 1983 .

[14]  Hiroshi Tanaka Limit Theorems for Certain Diffusion Processes with Interaction , 1984 .

[15]  D. Stroock An Introduction to the Theory of Large Deviations , 1984 .

[16]  I. Csiszár Sanov Property, Generalized $I$-Projection and a Conditional Limit Theorem , 1984 .

[17]  Tadahisa Funaki,et al.  A certain class of diffusion processes associated with nonlinear parabolic equations , 1984 .

[18]  Antonio Galves,et al.  Metastable behavior of stochastic dynamics: A pathwise approach , 1984 .

[19]  S. Varadhan Large Deviations and Applications , 1984 .

[20]  M. Freidlin,et al.  Random Perturbations of Dynamical Systems , 1984 .

[21]  R. Ellis,et al.  Entropy, large deviations, and statistical mechanics , 1985 .

[22]  Christian Léonard,et al.  Une loi des grands nombres pour des systèmes de diffusions avec interaction et à coefficients non bornés , 1986 .

[23]  D. Dawson,et al.  Long-time fluctuations of weakly interacting diffusions , 1987 .

[24]  J. Gärtner On the McKean‐Vlasov Limit for Interacting Diffusions , 1988 .