The stochastic thin film equation via Langevin dynamics

It is known that the thin film equation is formally a gradient flow with respect to the Dirichlet energy and the Wasserstein metric. Then it is natural to consider the formal Fokker-Planck equation to make sure that the associated stochastic equation satisfies the detailed balance condition which means that the invariant measure is the Gibbs measure with respect to the energy. We make this argument rigorous by discretizing in space via a Galerkin scheme. This leads to a high dimensional stochastic differential equation in Ito form which corresponds to the discretized stochastic thin film equation. Contrary to the literature our ansatz yields an additional Ito correction term.

This is ongoing work with Benjamin Gess and Felix Otto.