  We study two problems. First, we consider the large deviation behavior of
empirical measures of certain diffusion processes as, simultaneously, the time
horizon becomes large and noise becomes vanishingly small. The law of large
numbers (LLN) of the empirical measure in this asymptotic regime is given by
the unique equilibrium of the noiseless dynamics. Due to degeneracy of the
noise in the limit, the methods of Donsker and Varadhan (1976) are not directly
applicable and new ideas are needed. Second, we study a system of slow-fast
diffusions where both the slow and the fast components have vanishing noise on
their natural time scales. This time the LLN is governed by a degenerate
averaging principle in which local equilibria of the noiseless system obtained
from the fast dynamics describe the asymptotic evolution of the slow component.
We establish a large deviation principle that describes probabilities of
divergence from this behavior. On the one hand our methods require stronger
assumptions than the nondegenerate settings, while on the other hand the rate
functions take simple and explicit forms that have striking differences from
their nondegenerate counterparts.
