Stochastic Estimates for the Thin-Film Equation with Thermal Noise

Abstract

We construct and derive uniform stochastic estimates on the renormalised model for a class of fourth-order conservative quasilinear singular SPDEs in arbitrary dimension \(d\ge 1 \) and in the full subcritical regime of noise regularity. The prototype of the class of equations we study is the so-called thin-film equation with thermal noise, also commonly referred to in the literature as the stochastic thin-film equation. We derive an explicit expression for the form of the counterterm as a function of the film mobility which is in surprising agreement with the form conjectured in [GGKO23, Remark 9.1].