We 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 the construction of new Lyapunov functions, involving the coefficients, the stepsize, and $\theta$, and the irreducibility and the strong Feller property for the STM. We also generalize the arguments to the STM 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 drift-implicit Euler 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：本文针对由非退化乘性噪声驱动的单调随机微分方程（SODEs），在系数无增长限制的条件下，建立了当参数θ∈[1/2,1]时随机θ方法（STM）生成马尔可夫链的唯一遍历性。论证的核心在于构建涉及系数、步长及θ的新Lyapunov函数，并证明STM的不可约性与强Feller性。我们将该论证推广至无穷维非退化乘性迹类噪声驱动的一类单调随机偏微分方程（SPDEs）的STM及其基于Galerkin的全离散格式。将结果应用于随机Allen-Cahn方程表明，其漂移隐式Euler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）提出的问题提供了肯定答案。数值实验验证了我们的理论结果。