We first establish the unique ergodicity of the Markov chain generated by the stochastic theta method (STM) with $\theta \in [1/2, 1]$ for monotone SODEs, without growth restriction on the coefficients, driven by nondegenerate multiplicative noise. The main ingredient of the arguments lies in constructing new Lyapunov functions involving the coefficients, the stepsize, and $\theta$, and the irreducibility and the strong Feller property for the STM. We then generalize the arguments to the temporal drift-implicit Euler (DIE) method and its Galerkin-based full discretizations for a class of monotone SPDEs driven by infinite-dimensional nondegenerate multiplicative trace-class noise. Applying these results to the stochastic Allen--Cahn equation indicates that its DIE scheme is uniquely ergodic for any interface thickness, which gives an affirmative answer to a question proposed in (J. Cui, J. Hong, and L. Sun, Stochastic Process. Appl. (2021): 55--93). Numerical experiments verify our theoretical results.

翻译：首先，针对系数无增长限制、由非退化乘性噪声驱动的单调随机常微分方程，我们建立了当$\theta \in [1/2, 1]$时随机θ方法（STM）所生成马尔可夫链的唯一遍历性。论证的核心在于构造了包含系数、步长及$\theta$的新李雅普诺夫函数，并结合STM的不可约性与强Feller性质。随后，我们将该论证推广至时间漂移隐式欧拉（DIE）方法及其基于伽辽金法的全离散格式，用于处理一类由无限维非退化乘性迹类噪声驱动的单调随机偏微分方程。将这些结果应用于随机Allen--Cahn方程表明，其DIE格式对任意界面厚度均具有唯一遍历性，这为（J. Cui, J. Hong, and L. Sun, Stochastic Process. Appl. (2021): 55--93）中提出的问题给出了肯定回答。数值实验验证了我们的理论结果。