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Convergence of path measures arising from a mean field or polaron type interaction. (English) Zbl 0791.60093
We discuss the limiting path measures of Markov processes with either a mean field or a polaron type interaction of the paths. In the polaron type situation the strength is decaying at large distances on the time axis, and so the interaction is of short range in time. In contrast, in the mean field model, the interaction is weak, but of long range in time. M. D. Donsker and S. R. S. Varadhan [Commun. Pure Appl. Math. 36, 505-528 (1983; Zbl 0538.60081)] proved that for the partition functions, there is a transition from the polaron type to the mean field interaction when passing to a limit by letting the strength tend to zero while increasing the range. The discussion of the path measures is more subtle. We treat the mean field case as an example of a differentiable interaction and discuss the transition from the polaron type to the mean field interaction for two instructive examples.
Reviewer: E.Bolthausen

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60F10 Large deviations
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
82C44 Dynamics of disordered systems (random Ising systems, etc.) in time-dependent statistical mechanics
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