In this paper, we study the diffusive limit of solutions to the generalized Langevin equation (GLE) in a periodic potential. Under the assumption of quasi-Markovianity, we obtain sharp longtime equilibration estimates for the GLE using techniques from the theory of hypocoercivity. We then show asymptotic results for the effective diffusion coefficient in the small correlation time regime, as well as in the overdamped and underdamped limits. Finally, we employ a recently developed numerical method (Roussel and Stoltz in ESAIM Math Model Numer Anal 52(3):1051–1083, 2018) to calculate the effective diffusion coefficient for a wide range of (effective) friction coefficients, confirming our asymptotic results.

在本文中，我们研究了周期势中广义Langevin方程（GLE）解的扩散极限。在准马尔可夫性的假设下，我们使用低矫顽力理论的技术对GLE进行了长期的精确平衡估计。然后，我们在较小的相关时间范围内以及过阻尼和欠阻尼极限中显示了有效扩散系数的渐近结果。最后，我们采用一种最新开发的数值方法（ESAIM数学模型Numer Anal 52（3）：1051-1083，2018中的Roussel和Stoltz）来计算各种（有效）摩擦系数的有效扩散系数，从而确认了我们的渐近线结果。