Long-time weak convergence analysis of a semi-discrete scheme for stochastic Maxwell equations

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.