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We consider a diffusion given by a small noise perturbation of a dynamicalsystem driven by a potential function with a finite number of local minima. Theclassical results of Freidlin and Wentzell show that the time this diffusionspends in the domain of attraction of one of these local minima isapproximately exponentially distributed and hence the diffusion should behaveapproximately like a Markov chain on the local minima. By the work of Bovierand collaborators, the local minima can be associated with the smalleigenvalues of the diffusion generator. In Part I of this work, by applying aMarkov mapping theorem, we used the eigenfunctions of the generator to couplethis diffusion to a Markov chain whose generator has eigenvalues equal to theeigenvalues of the diffusion generator that are associated with the localminima and established explicit formulas for conditional probabilitiesassociated with this coupling. The fundamental question now becomes to relatethe coupled Markov chain to the approximate Markov chain suggested by theresults of Freidlin and Wentzel. In this paper, we take up this question andprovide a complete analysis of this relationship in the special case of adouble-well potential in one dimension.

Finite Markov chains coupled to general Markov processes and an application to metastability II