Abstract
In the present paper, we consider a Fokker-Planck equation associated to periodic stochastic differential equations with irregular coefficients. We define periodic probability solutions to be periodic analogs of stationary measures for stationary Fokker-Planck equations, and study their existence in both non-degenerate and degenerate cases. In the non-degenerate case, a Lyapunov condition is imposed to ensure the existence of periodic probability solutions to the Fokker-Planck equation with Sobolev coefficients. In the degenerate case with slightly more regular coefficients, the existence is established under the same Lyapunov condition. As applications of our results, we construct periodic probability solutions to Fokker-Planck equations associated to stochastic damping Hamiltonian systems and stochastic differential inclusions.
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