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Limit theorems for certain diffusion processes with interaction. (English) Zbl 0552.60051
Stochastic analysis, Proc. Taniguchi Int. Symp., Katata & Kyoto/Jap. 1982, North-Holland Math. Libr. 32, 469-488 (1984).
[For the entire collection see Zbl 0538.00017.]
This is an important paper which develops a method for proving limit theorems for symmetrically interacting diffusions including a central limit theorem and a large deviation theorem. The method consists of two parts. The first is to represent the solution as a functional of a system of independent Brownian motions. The central limit theorem is then obtained by generalizing a method of W. Braun and K. Hepp [The Vlasov dynamics and its fluctuations in the 1/N limit of interacting classical particles. Commun. Math. Phys. 56, 101-113 (1977)]. The large deviation result is obtained by applying a method of Donsker and Varadhan for independent and identically distributed random variables. A large deviation result is also discussed for McKean’s 2-state model of Maxwellian molecules.
Reviewer: D.Dawson

MSC:
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60F05 Central limit and other weak theorems
60F10 Large deviations