  The aim of this paper is to develop tractable large deviation approximations
for the empirical measure of a small noise diffusion. The starting point is the
Freidlin-Wentzell theory, which shows how to approximate via a large deviation
principle the invariant distribution of such a diffusion. The rate function of
the invariant measure is formulated in terms of quasipotentials, quantities
that measure the difficulty of a transition from the neighborhood of one
metastable set to another. The theory provides an intuitive and useful
approximation for the invariant measure, and along the way many useful related
results (e.g., transition rates between metastable states) are also developed.
With the specific goal of design of Monte Carlo schemes in mind, we prove large
deviation limits for integrals with respect to the empirical measure, where the
process is considered over a time interval whose length grows as the noise
decreases to zero. In particular, we show how the first and second moments of
these integrals can be expressed in terms of quasipotentials. When the dynamics
of the process depend on parameters, these approximations can be used for
algorithm design, and applications of this sort will appear elsewhere. The use
of a small noise limit is well motivated, since in this limit good sampling of
the state space becomes most challenging. The proof exploits a regenerative
structure, and a number of new techniques are needed to turn large deviation
estimates over a regenerative cycle into estimates for the empirical measure
and its moments.
