Thermodynamically consistent discretization of the stochastic thin film equation

We begin by informally demonstrating how the gradient flow structure of the deterministic thin film equation, which encodes the balance between driving capillary forces and limiting viscous forces, can be used as a foundation for the thermodynamically consistent introduction of fluctuations. This approach is then pursued at the level of a spatial discretization of the gradient flow structure, specifically, by discretizing the energy functional and the dissipative metric tensor. This leads us to derive a stochastic differential equation that can be interpreted as an approximation of the stochastic thin film equation. Subsequently, we show that a suitable choice of the discrete mobility function results in solutions that strictly preserve positivity. The talk closes by discussing the probability of solutions to the stochastic thin film equation to approach zero.