CONVERGENCE TO PERIODIC PROBABILITY SOLUTIONS IN FOKKER-PLANCK EQUATIONS

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.