We prove a weak rate of convergence of a fully discrete scheme for stochastic Cahn--Hilliard equation with additive noise, where the spectral Galerkin method is used in space and the backward Euler method is used in time. Compared with the Allen--Cahn type stochastic partial differential equation, the error analysis here is much more sophisticated due to the presence of the unbounded operator in front of the nonlinear term. To address such issues, a novel and direct approach has been exploited which does not rely on a Kolmogorov equation but on the integration by parts formula from Malliavin calculus. To the best of our knowledge, the rates of weak convergence are revealed in the stochastic Cahn--Hilliard equation setting for the first time.

翻译：我们证明了带加性噪声的随机Cahn--Hilliard方程全离散格式的弱收敛速率，其中空间方向采用谱Galerkin方法，时间方向采用向后欧拉方法。与Allen--Cahn型随机偏微分方程相比，由于非线性项前存在无界算子，这里的误差分析更为复杂。为解决这一问题，我们采用了一种新颖且直接的方法，该方法不依赖于Kolmogorov方程，而是基于Malliavin微积分中的分部积分公式。据我们所知，本文首次揭示了随机Cahn--Hilliard方程框架下的弱收敛速率。