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Large deviations from the McKean-Vlasov limit for weakly interacting diffusions. (English) Zbl 0613.60021
A system of N interacting diffusions on \({\mathbb{R}}^ d\) is considered. The system is given by the N coupled Ito equations \[ dx_ k=\sigma (x_ k)dw_ k+b(x_ k,\epsilon_ x)dt,\quad k=1,...,N, \] where \(w_ 1,...,w_ N\) are independent Wiener processes and \(\epsilon_ x=N^{- 1}\sum^{N}_{k=1}\delta_{x_ k}\) is the empirical measure of the particle configuration \(x=(x_ 1,...,x_ N)\). The limit as \(N\to \infty\) of the empirical measure process of the system is a McKean-Vlasov limit.
The purpose of the paper is to build a framework in which to study large deviations from the McKean-Vlasov limit. The main result for the above system is a large deviation theorem for the empirical measure process under certain conditions in the coefficients \(\sigma\) and b. The idea of the proof rests on the observation that locally, along a fixed path \({\bar \mu}\)(.), the N-particle system behaves for large N nearly as if it were a superposition of N independent copies of a diffusion process with diffusion matrix \(\sigma (x)\sigma^*(x)\) and ”frozen” drift vector \(\bar b(x,t)=b(x,{\bar \mu}(t))\). This allows to convert the large deviation result for independent diffusions into a local large deviation result for interacting diffusions.
The methods of proof involve an infinite dimensional generalization of the theory of Freidlin and Wentzell. In order to obtain the action functional results are obtained on projective limits of large deviation systems, large deviations on dual vector spaces and a Sanov type theorem for vectors of empirical measures.
Reviewer: L.G.Gorostiza

MSC:
60F10 Large deviations
60J60 Diffusion processes
60K35 Interacting random processes; statistical mechanics type models; percolation theory
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