  In this article, we prove that a small random perturbation of dynamical
system with multiple stable equilibria converges to a Markov chain whose states
are neighborhoods of the deepest stable equilibria, under a suitable
time-rescaling, provided that the perturbed dynamics is reversible in time.
Such a result has been anticipated from 1970s, when the foundation of
mathematical treatment for this problem has been established by Freidlin and
Wentzell. We solve this long-standing problem by reducing the entire analysis
to an investigation of the solution of an associated Poisson equation, and
furthermore provide a method to carry out this analysis by using well-known
test functions in a novel manner.
