We analyze the long-time behavior of numerical schemes, studied by \cite{LQ21} in a finite time horizon, for a class of monotone SPDEs driven by multiplicative noise. We derive several time-independent a priori estimates for both the exact and numerical solutions and establish time-independent strong error estimates between them. These uniform estimates, in combination with ergodic theory of Markov processes, are utilized to establish the exponential ergodicity of these numerical schemes with an invariant measure. Applying these results to the stochastic Allen--Cahn equation indicates that these numerical schemes always have at least one invariant measure, respectively, and converge strongly to the exact solution with sharp time-independent rates. We also show that the invariant measures of these schemes are also exponentially ergodic and thus give an affirmative answer to a question proposed in \cite{CHS21}, provided that the interface thickness is not too small.

翻译：我们分析了由\cite{LQ21}在有限时间区间内研究的数值格式对于一类乘性噪声驱动的单调随机偏微分方程的长时间行为。我们推导了精确解与数值解的几个与时间无关的先验估计，并建立了它们之间的与时间无关的强误差估计。这些一致估计结合马尔可夫过程的遍历理论，用于证明这些数值格式在不变测度下的指数遍历性。将结果应用于随机Allen–Cahn方程表明，这些数值格式分别至少存在一个不变测度，并以与时间无关的强收敛速率收敛到精确解。我们还证明了这些格式的不变测度也是指数遍历的，从而对\cite{CHS21}提出的问题给出了肯定回答，前提是界面厚度非过小。