On the McKean‐Vlasov Limit for Interacting Diffusions

For a system of diffusions in a domain of Rd with long-range weak interaction the behavior of the associated empirical process is studied. Under mild growth and smoothness assumptions on the drift and diffusion coefficients such as coercivity and monotonicity conditions the law of large numbers and the propagation of chaos are proved. Existence and uniqueness of the weak solution to the McKean - Vlasov equation and the associated non-linear martingale problem are investigated.

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

[2]  P. Billingsley,et al.  Convergence of Probability Measures , 1969 .

[3]  H. McKean Fluctuations in the kinetic theory of gases , 1975 .

[4]  N. Krylov Sequences of convex functions and estimates of the maximum of the solution of a parabolic equation , 1976 .

[5]  W. Braun,et al.  The Vlasov dynamics and its fluctuations in the 1/N limit of interacting classical particles , 1977 .

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

[7]  M. Nagasawa Segregation of a population in an environment , 1980 .

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

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

[10]  A. Sznitman Nonlinear reflecting diffusion process, and the propagation of chaos and fluctuations associated , 1984 .

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

[12]  A. Sznitman Équations de type de Boltzmann, spatialement homogènes , 1984 .

[13]  K. Oelschlager A Martingale Approach to the Law of Large Numbers for Weakly Interacting Stochastic Processes , 1984 .

[14]  Hiroshi Tanaka,et al.  Central limit theorem for a system of Markovian particles with mean field interactions , 1985 .

[15]  Hiroshi Tanaka,et al.  A diffusion process in a singular mean-drift-field , 1985 .

[16]  M. Scheutzow Periodic behavior of the stochastic Brusselator in the mean-field limit , 1986 .

[17]  Hiroshi Tanaka,et al.  Diffusion with interactions and collisions between coloured particles and the propagation of chaos , 1987 .

[18]  J. Gärtner,et al.  Large deviations from the mckean-vlasov limit for weakly interacting diffusions , 1987 .