Stochastic estimates for the thin-film equation with thermal noise

TL;DR

This paper develops uniform stochastic estimates for a class of fourth-order singular SPDEs, notably the stochastic thin-film equation with thermal noise, providing explicit counterterms and advancing understanding of these complex models.

Contribution

It constructs uniform stochastic estimates for a broad class of singular SPDEs, including the stochastic thin-film equation, with explicit counterterm expressions.

Findings

Derived explicit counterterm expressions matching conjectures

Established uniform stochastic estimates in arbitrary dimensions

Analyzed the full subcritical noise regularity regime

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 $d\geq 1$ 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 Remark 9.1 of Math. Comp. 92 (2023), 1931-976.