Quantitative hydrodynamic limits of the Langevin dynamics for gradient interface models

We study the Langevin dynamics corresponding to the $\nablaφ$ (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 traj...