In this work, we establish the Freidlin--Wentzell large deviations principle (LDP) of the stochastic Cahn--Hilliard equation with small noise, which implies the one-point LDP. Further, we give the one-point LDP of the spatial finite difference method (FDM) for the stochastic Cahn--Hilliard equation. Our main result is the convergence of the one-point large deviations rate function (LDRF) of the spatial FDM, which is about the asymptotical limit of a parametric variational problem. The main idea for proving the convergence of the LDRF of the spatial FDM is via the $\Gamma$-convergence of objective functions, which relies on the qualitative analysis of skeleton equations of the original equation and the numerical method. In order to overcome the difficulty that the drift coefficient is not one-side Lipschitz, we use the equivalent characterization of the skeleton equation of the spatial FDM and the discrete interpolation inequality to obtain the uniform boundedness of the solution to the underlying skeleton equation. This plays an important role in deriving the $\Gamma$-convergence of objective functions.

翻译：本文建立了小噪声随机Cahn--Hilliard方程的Freidlin--Wentzell大偏差原理（LDP），由此可推导出单点LDP。进一步地，我们给出了随机Cahn--Hilliard方程空间有限差分方法（FDM）的单点LDP。主要结果是空间FDM的单点大偏差率函数（LDRF）的收敛性，该收敛性涉及参数变分问题的渐近极限。证明空间FDM的LDRF收敛性的核心思想是通过目标函数的$\Gamma$-收敛，这依赖于原始方程及数值方法的骨架方程的定性分析。为克服漂移系数非单侧Lipschitz的困难，我们利用空间FDM骨架方程的等价表征及离散插值不等式，获得底层骨架方程解的一致有界性。这一结果在推导目标函数的$\Gamma$-收敛中起到关键作用。