**Abstract:**
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. Donsker and Varadhan 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.

**Keywords:**
maximum entropy principle,
large deviations,
weak convergence,
polaron problem,
mean field interaction,
interacting Markov processes

**2010
Mathematics Subject Classification:**

- 60F05 Central limit and other weak theorems
- 60F10 Large deviations
- 60G15 Gaussian processes
- 60J25 Markov processes with continuous parameter

**Reference:**
Probability
Theory and Related Fields, Vol. 95 (1993) 283-310.

DOI: 10.1007/BF01192166

The paper (28 pages) is available in:

- PDF / portable document format (273kB)

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