  The goal of this paper is to study the Moderate Deviation Principle (MDP) for
a system of stochastic reaction-diffusion equations with a time-scale
separation in slow and fast components and small noise in the slow component.
Based on weak convergence methods in infinite dimensions and related stochastic
control arguments, we obtain an exact form for the moderate deviations rate
function in different regimes as the small noise and time-scale separation
parameters vanish. Many issues that come up due to the infinite dimensionality
of the problem are completely absent in their finite-dimensional counterpart.
In comparison to corresponding Large Deviation Principles, the moderate
deviation scaling necessitates a more delicate approach to establishing
tightness and properly identifying the limiting behavior of the underlying
controlled problem. The latter involves regularity properties of a solution of
an associated elliptic Kolmogorov equation on Hilbert space along with a
finite-dimensional approximation argument.
