  We study diffusion processes in $\mathbb{R}^d$ that leave invariant a finite
collection of manifolds (surfaces or points) in $\mathbb{R}^d$ and small
perturbations of such processes. Assuming certain ergodic properties at and
near the invariant surfaces, we describe the rate at which the process gets
attracted to or repelled from the surface, based on the local behavior of the
coefficients. For processes that include, additionally, a small non-degenerate
perturbation, we describe the metastable behavior. Namely, by allowing the time
scale to depend on the size of the perturbation, we observe different
asymptotic distributions of the process at different time scales.
  Stated in PDE terms, the results provide the asymptotics, at different time
scales, for the solution of the parabolic Cauchy problem when the operator that
degenerates on a collection of surfaces is perturbed by a small non-degenerate
term. This asymptotic behavior switches at a finite number of time scales that
are calculated and does not depend on the perturbation.
