Dirichlet form approach to stochastic thin-film equations

Abstract

The thin-film equation describes the evolution of a thin liquid film driven by capillarity and limited by viscosity. For sufficiently small film heights, thermal fluctuations may not be neglected. Since the thin-film equation as an overdamped evolution equation has a gradient flow structure, the informal implementation of the fluctuation-dissipation principle is straightforward. In this paper, we work out that the concept of Dirichlet forms and the related construction of Markov processes are tailor-made for this type of overdamped continuum evolution with thermal noise. However, the concept of Dirichlet forms relies on the availability of a Gibbsian equilibrium measure, which, in case of the thin-film equation, limits the treatment to the $(1+1)$-dimensional case. By means of Dirichlet form theory, we obtain the existence and uniqueness of the evolution of laws on configuration space. We show that the regularity of the realizations of this Markov process matches the heuristics. In a simplified situation with a Gaussian reference measure and a non-degenerate mobility function, we characterize it as what is called an energy solution. The latter provides a pathwise decomposition of the driving forces into a deterministic and a thermal one, which was the starting point. Applying the same techniques in the spatially discretized setting proposed in [Math. Comp. 92 (2023), 1931--1976], we obtain the same results uniformly in the discretization parameter. In particular, Galerkin as well as spectral approximations of the stochastic thin-film equation are shown to be uniformly tight in optimal topologies. In the simplified setting, we also prove Mosco convergence of the associated Dirichlet forms and thereby obtain convergence of these approximations.