The law of the iterated logarithm (LIL) for the time-homogeneous Markov process with a unique invariant measure characterizes the almost sure maximum possible fluctuation of time averages around the ergodic limit. Whether a numerical approximation can preserve this asymptotic pathwise behavior remains an open problem. In this work, we give a positive answer to this question and establish the LIL for the numerical approximation of such a process under verifiable assumptions. The Markov process is discretized by a decreasing time-step strategy, which yields the non-homogeneous numerical approximation but facilitates a martingale-based analysis. The key ingredient in proving the LIL for such numerical approximation lies in extracting a quasi-uniform time-grid subsequence from the original non-uniform time grids and establishing the LIL for a predominant martingale along it, while the remainder terms converge to zero. Finally, we illustrate that our results can be flexibly applied to numerical approximations of a broad class of stochastic systems, including SODEs and SPDEs.

翻译：具有唯一不变测度的时间齐次马尔可夫过程的重对数律（LIL）刻画了时间平均围绕遍历极限的几乎必然最大可能波动。数值逼近能否保持这种渐近路径行为仍是一个开放问题。本文对该问题给出了肯定回答，并在可验证的假设下为此类过程的数值逼近建立了重对数律。马尔可夫过程通过递减时间步长策略进行离散化，这产生了非齐次数值逼近，但便于基于鞅的分析。证明此类数值逼近重对数律的关键在于从原始非均匀时间网格中提取拟均匀时间网格子序列，并沿该子序列为主鞅建立重对数律，同时余项收敛至零。最后，我们阐明结果可灵活应用于包括随机常微分方程（SODE）和随机偏微分方程（SPDE）在内的广泛随机系统的数值逼近。