  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. Applying a Markov mapping theorem, we
use 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 establish explicit
formulas for conditional probabilities associated with this coupling. The
fundamental question then becomes to relate the coupled Markov chain to the
approximate Markov chain suggested by the results of Freidlin and Wentzel. In
Part II of this work, we provide a complete analysis of this relationship in
the special case of a double-well potential in one dimension. More generally,
the coupling can be constructed for a general class of Markov processes and any
finite set of eigenvalues of the generator.
