This paper derives an approximation for a generalized Langevin equation driven by a force with random oscillation in time and periodic oscillation in space. By a diffusion approximation and the weak convergence of periodic oscillation function, the solution of the generalized Langevin equation is shown to converge in distribution to the solution of a stochastic partial differential equations (SPDEs) driven by time white noise.

中文翻译：

---

具有高时空振荡的广义朗之万方程的逼近

本文推导了由时间随机振荡和空间周期振荡的力驱动的广义朗之万方程的近似值。通过扩散近似和周期性振荡函数的弱收敛性，证明了广义朗之万方程的解在分布上收敛于由时间白噪声驱动的随机偏微分方程 (SPDE) 的解。