We analyze the long-time behavior of numerical schemes for a class of monotone SPDEs driven by multiplicative noise. By deriving several time-independent a priori estimates for the numerical solutions, combined with the ergodic theory of Markov processes, we establish the exponential ergodicity of these schemes with a unique invariant measure, respectively. Applying these results to the stochastic Allen--Cahn equation indicates that these 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 these numerical invariant measures are exponentially ergodic and thus give an affirmative answer to a question proposed in (J. Cui, J. Hong, and L. Sun, Stochastic Process. Appl. (2021): 55--93), provided that the interface thickness is not too small.

翻译：摘要：本文分析了一类由乘性噪声驱动的单调SPDEs的数值格式的长时间行为。通过推导数值解的几个与时间无关的先验估计，并结合马尔可夫过程的遍历理论，我们分别建立了这些格式的指数遍历性，并证明了其唯一不变测度的存在性。将这些结果应用于随机Allen–Cahn方程表明，这些格式均至少存在一个不变测度，并且以尖锐的与时间无关的收敛速率强收敛于精确解。我们还证明了这些数值不变测度具有指数遍历性，从而在界面厚度不太小的条件下，对（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。