  We consider a diffusion given by a small noise perturbation of a dynamical
system driven by a potential function with a finite number of local minima. The
classical results of Freidlin and Wentzell show that the time this diffusion
spends in the domain of attraction of one of these local minima is
approximately exponentially distributed and hence the diffusion should behave
approximately like a Markov chain on the local minima. By the work of Bovier
and collaborators, the local minima can be associated with the small
eigenvalues of the diffusion generator. In Part I of this work, by applying a
Markov mapping theorem, we used the eigenfunctions of the generator to couple
this diffusion to a Markov chain whose generator has eigenvalues equal to the
eigenvalues of the diffusion generator that are associated with the local
minima and established explicit formulas for conditional probabilities
associated with this coupling. The fundamental question now becomes to relate
the coupled Markov chain to the approximate Markov chain suggested by the
results of Freidlin and Wentzel. In this paper, we take up this question and
provide a complete analysis of this relationship in the special case of a
double-well potential in one dimension.
