SIAM Journal on Mathematical Analysis, Volume 53, Issue 2, Page 1958-1992, January 2021.
The present paper is devoted to the study of convergence of solutions of a Fokker--Planck equation (FPE) associated to a periodic stochastic differential equation with less regular coefficients under various Lyapunov conditions. In the case of nondegenerate noises, we prove two types of convergence of solutions to the unique periodic probability solution, namely, convergence in mean and exponential convergence. In the case of degenerate noises, we show the convergence of solutions in mean to the set of periodic probability solutions. New results on the uniqueness of periodic probability solutions and global probability solutions of the FPE are also obtained. As applications, we study the long-time behaviors of the FPEs associated to stochastic damping Hamiltonian systems and stochastic slow-fast systems, and of weak solutions of periodic stochastic differential equations with less regular coefficients.

中文翻译：

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Fokker-Planck方程的周期概率解的收敛性

SIAM数学分析杂志，第53卷，第2期，第1958-1992页，2021年1月。
本文致力于研究在各种Lyapunov条件下与具有较小规则系数的周期随机微分方程相关的Fokker-Planck方程（FPE）的解的收敛性。在非简并噪声的情况下，我们证明了唯一周期概率解的两种类型的解收敛，即均值收敛和指数收敛。在简并噪声的情况下，我们显示了均值对周期概率解集的解的收敛性。还获得了关于FPE的周期概率解和全局概率解的唯一性的新结果。作为应用，我们研究FPE与随机阻尼汉密尔顿系统和随机慢速快速系统相关的长期行为，