  We study the Langevin dynamics corresponding to the $\nabla\phi$ (or
Ginzburg-Landau) interface model with a uniformly convex interaction potential.
We interpret these Langevin dynamics as a nonlinear parabolic equation forced
by white noise, which turns the problem into a nonlinear homogenization
problem. Using quantitative homogenization methods, we prove a quantitative
hydrodynamic limit, obtain the $C^2$ regularity of the surface tension, prove a
large-scale Lipschitz-type estimate for the trajectories of the dynamics, and
show that the fluctuation-dissipation relation can be seen as a commutativity
of homogenization and linearization. Finally, we explain why we believe our
techniques can be adapted to the setting of degenerate (non-uniformly) convex
interaction potentials.
