  The large deviation principle is proved for a class of $L^2$-valued processes
that arise from the coarse-graining of a random field. Coarse-grained processes
of this kind form the basis of the analysis of local mean-field models in
statistical mechanics by exploiting the long-range nature of the interaction
function defining such models. In particular, the large deviation principle is
used in a companion paper to derive the variational principles that
characterize equilibrium macrostates in statistical models of two-dimensional
and quasi-geostrophic turbulence. Such macrostates correspond to large-scale,
long-lived flow structures, the description of which is the goal of the
statistical equilibrium theory of turbulence. The large deviation bounds for
the coarse-grained process under consideration are shown to hold with respect
to the strong $L^2$ topology, while the associated rate function is proved to
have compact level sets with respect to the weak topology. This compactness
property is nevertheless sufficient to establish the existence of equilibrium
macrostates for both the microcanonical and canonical ensembles.
