It is known from the monograph [1, Chapter 5] that the weak convergence analysis of numerical schemes for stochastic Maxwell equations is an unsolved problem. This paper aims to fill the gap by establishing the long-time weak convergence analysis of the semi-implicit Euler scheme for stochastic Maxwell equations. Based on analyzing the regularity of transformed Kolmogorov equation associated to stochastic Maxwell equations and constructing a proper continuous adapted auxiliary process for the semi-implicit scheme, we present the long-time weak convergence analysis for this scheme and prove that the weak convergence order is one, which is twice the strong convergence order. As applications of this result, we obtain the convergence order of the numerical invariant measure, the strong law of large numbers and central limit theorem related to the numerical solution, and the error estimate of the multi-level Monte Carlo estimator. As far as we know, this is the first result on the weak convergence order for stochastic Maxwell equations.

翻译：据专著[1, 第五章]可知，随机麦克斯韦方程数值格式的弱收敛分析仍是一个未解问题。本文旨在通过建立随机麦克斯韦方程半隐式欧拉格式的长时间弱收敛分析来填补这一空白。基于分析随机麦克斯韦方程关联的变换Kolmogorov方程的正则性，并为半隐式格式构造合适的连续适应辅助过程，我们给出了该格式的长时间弱收敛分析，证明了弱收敛阶为1，是强收敛阶的两倍。作为该结果的应用，我们得到了数值不变测度的收敛阶、与数值解相关的强大数定律和中心极限定理，以及多层蒙特卡洛估计量的误差估计。据我们所知，这是关于随机麦克斯韦方程弱收敛阶的首个结果。