  Stochastic partial differential equations driven by Poisson random measures
(PRM) have been proposed as models for many different physical systems, where
they are viewed as a refinement of a corresponding noiseless partial
differential equations (PDE). A systematic framework for the study of
probabilities of deviations of the stochastic PDE from the deterministic PDE is
through the theory of large deviations. The goal of this work is to develop the
large deviation theory for small Poisson noise perturbations of a general class
of deterministic infinite dimensional models. Although the analogous questions
for finite dimensional systems have been well studied, there are currently no
general results in the infinite dimensional setting. This is in part due to the
fact that in this setting solutions may have little spatial regularity, and
thus classical approximation methods for large deviation analysis become
intractable. The approach taken here, which is based on a variational
representation for nonnegative functionals of general PRM, reduces the proof of
the large deviation principle to establishing basic qualitative properties for
controlled analogues of the underlying stochastic system. As an illustration of
the general theory, we consider a particular system that models the spread of a
pollutant in a waterway.
