The present paper is devoted to the study of convergence of solutions of a Fokker-Planck equation (FPE) associated to a periodic stochastic di_erential equation with less regular coe_cients, under various Lyapunov conditions. In the case of non-degenerate 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 di_erential equations with less regular coe_cients.

Publication:

-    SIAM Journal on Mathematical Analysis, (2021)

Authors:

-    Min Ji (Hua Loo-Keng Key Laboratory of Mathematics, AMSS, Chinese Academy of Sciences)

-    Weiwei Qi (University of Chinese Academy of Sciences & Department of Mathematical, and Statistical Sciences, University of Alberta)

-    Zhongwei Shen (Department of Mathematical, and Statistical Sciences, University of Alberta)

-    Yingfei Yi (Department of Mathematical, and Statistical Sciences, University of Alberta)