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Large deviations and tunnelling for particle systems with mean field interaction. (English) Zbl 0613.60020

A system of N diffusions on \({\mathbb{R}}^ d\) with mean field interaction is studied. The system is given by the coupled Ito equations \[ dx_ k=[- \nabla U(x_ k)+(\theta /N)\sum^{N}_{j=1}(x_ k-x_ j)dt+\sigma dw_ k,\quad k=1,...,N, \] where \(w_ 1,...,w_ N\) are independent Wiener processes, \(\theta\) and \(\sigma\) are positive constants, and \(U: {\mathbb{R}}^ d\to {\mathbb{R}}\) is a potential satisfying certain conditions; an example for \(d=1\) is \(U(x)=x^ 4/4-x^ 2/2\), which arises in statistical physics. The empirical measure process associated with the system is considered.
Results are presented without proofs on the long time behavior of this process as \(N\to \infty\). These include equilibrium large deviations, an analogue of the results of Freidlin and Wentzell on large deviations from the limiting dynamics (McKean-Vlasov limit), and a result on tunnelling from the domain of attraction of one stable equilibrium to another..
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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