The aim of this study is the weak convergence rate of a temporal and spatial discretization scheme for stochastic Cahn-Hilliard equation with additive noise, where the spectral Galerkin method is used in space and the backward Euler scheme is used in time. The presence of the unbounded operator in front of the nonlinear term and the lack of the associated Kolmogorov equations make the error analysis much more challenging and demanding. To overcome these difficulties, we further exploit a novel approach proposed in [7] and combine it with Malliavin calculus to obtain an improved weak rate of convergence, in comparison with the corresponding strong convergence rates. The techniques used here are quite general and hence have the potential to be applied to other non-Markovian equations. As a byproduct the rate of the strong error can also be easily obtained.

翻译：这项研究的目的是,在空间使用光谱Galerkin法和后向Euler法的时空分解办法中,与添加性噪声相配的随机卡赫尼-希利亚德方程式的时间和空间分解办法的趋同率较低,这种办法在空间使用光谱Galerkin法,在时间上使用后向的Euler法,在非线性术语之前出现未受约束的操作者,缺乏相关的科尔莫戈罗夫方程式,使得错误分析更具挑战性和难度。为了克服这些困难,我们进一步利用了[7]中提出的新颖办法,并与Malliavin Calculus结合,以获得较弱的趋同率,与相应的强的趋同率相比。这里使用的技术相当笼统,因此有可能适用于其他非马可维恩方程式。作为大误率的一个副产品,也可以轻易获得。