This paper focuses on studying the convergence rate of the density function of the Euler--Maruyama (EM) method, when applied to the overdamped generalized Langevin equation with fractional noise which serves as an important model in many fields. Firstly, we give an improved upper bound estimate for the total variation distance between random variables by their Malliavin--Sobolev norms. Secondly, we establish the existence and smoothness of the density function for both the exact solution and the numerical one. Based on the above results, the convergence rate of the density function of the numerical solution is obtained, which relies on the regularity of the noise and kernel. This convergence result provides a powerful support for numerically capturing the statistical information of the exact solution through the EM method.

翻译：本文旨在研究过阻尼广义朗之万方程（以分数噪声为驱动，在多个领域具有重要模型意义）的欧拉-丸山（EM）方法密度函数的收敛速度。首先，我们通过随机变量的Malliavin-Sobolev范数，给出了其全变差距离的改进上界估计。其次，建立了精确解与数值解密度函数的存在性与光滑性。基于上述结果，得到了数值解密度函数的收敛速度，该速度依赖于噪声与核函数的正则性。这一收敛结果为通过EM方法数值捕捉精确解的统计信息提供了有力支持。