We first establish the unique ergodicity of 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 deriving a minorization condition for the STM. We then generalize the arguments to the 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 Galerkin-based full discretizations are uniquely ergodic for any interface thickness. Numerical experiments verify our theoretical results.

翻译：首先，我们证明了对于由非退化乘性噪声驱动、系数无增长限制的单调随机常微分方程，当θ∈[1/2, 1]时，随机θ方法（STM）具有唯一遍历性。论证的核心在于构造涉及系数、步长和θ的新Lyapunov函数，并为STM推导出一个次要化条件。随后，我们将该论证推广至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的基于Galerkin方法的全离散格式。将这些结果应用于随机Allen-Cahn方程表明，其基于Galerkin方法的全离散格式对于任意界面厚度均具有唯一遍历性。数值实验验证了我们的理论结果。