  We investigate the behaviour of a family of entropy production functionals
associated to stochastic differential equations of the form $\mathrm{d} X_s =
-\nabla V(X_s) \, \mathrm{d} s + b(X_s) \, \mathrm{d} s + \sqrt{2\epsilon} \,
\mathrm{d} W_s $, where $b$ is a globally Lipschitz nonconservative vector
field keeping the system out of equilibrium, with emphasis on the large-time
limit and then the vanishing-noise limit. Different members of the family
correspond to different choices of boundary terms. Our analysis yields a law of
large numbers and a local large deviation principle which does not depend on
the choice of boundary terms and which exhibits a Gallavotti--Cohen symmetry.
We use techniques from the theory of semigroups and from semiclassical analysis
to reduce the description of the asymptotic behaviour of the functional to the
study of the leading eigenvalue of a quadratic approximation of a deformation
of the infinitesimal generator near critical points of $V$.
