  We analyze the fluctuation-driven escape of particles from a metastable state
under the influence of a weak periodic force. We develop an asymptotic method
to solve the appropriate Fokker-Planck equation with mixed natural and
absorbing boundary conditions. The approach uses two boundary layers flanking
an interior region; most of the probability is concentrated within the boundary
layer near the metastable point of the potential and particles transit the
interior region before exiting the domain through the other boundary layer,
which is near the unstable maximal point of the potential. The dominant
processes in each region are given by approximate
  time-dependent solutions matched to construct the approximate composite
solution, which gives the rate of escape with weak periodic forcing. Using
reflection we extend the method to a double well potential influenced by white
noise and weak periodic forcing, and thereby derive a two-state stochastic
model--the simplest treatment of stochastic resonance theory--in the
nonadiabatic limit
