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

Quantitative hydrodynamic limits of the Langevin dynamics for gradient interface models