  The aim of this paper is twofold. The first is to study the asymptotics of a
parabolically scaled, continuous and space-time stationary in time version of
the well-known Funaki-Spohn model in Statistical Physics. After a change of
unknowns requiring the existence of a space-time stationary eternal solution of
a stochastically perturbed heat equation, the problem transforms to the
qualitative homogenization of a uniformly elliptic, space-time stationary,
divergence form, nonlinear partial differential equation, the study of which is
the second aim of the paper. An important step is the construction of
correctors with the appropriate behavior at infinity.
