We prove the existence of non-negative martingale solutions to a class of stochastic degenerate-parabolic fourth-order PDEs arising in surface-tension driven thin-film flow influenced by thermal noise. The construction applies to a range of mobilites including the cubic one which occurs under the assumption of a no-slip condition at the liquid-solid interface. Since their introduction more than 15 years ago, by Davidovitch, Moro, and Stone and by Grün, Mecke, and Rauscher, the existence of solutions to stochastic thin-film equations for cubic mobilities has been an open problem, even in the case of sufficiently regular noise. Our proof of global-in-time solutions relies on a careful combination of entropy and energy estimates in conjunction with a tailor-made approximation procedure to control the formation of shocks caused by the nonlinear stochastic scalar conservation law structure of the noise.

我们证明了在受热噪声影响的表面张力驱动的薄膜流中出现的一类随机简并抛物线四阶偏微分方程的非负鞅解的存在。该构造适用于一系列移动体，包括在假定液固界面无滑移条件下发生的立方体。自从 Davidovitch、Moro 和 Stone 以及 Grün、Mecke 和 Rauscher 于 15 多年前提出以来，三次迁移率的随机薄膜方程的解的存在性一直是一个悬而未决的问题，即使在足够有规律的噪音。