Existence of periodic probability measure solutions to Fokker-Planck equations for SDEs with random switching

This paper is concerned with the existence of periodic probability measure solutions to the Fokker-Planck equation for a periodic stochastic differential equation driving by a continuous-time, discrete-state jump process. The jump rates of this jump process can also be time-periodic and dependent on the state variables of the system. We prove the existence and smoothness of principal eigenfunctions for a cooperative weakly coupled periodic-parabolic system of partial differential equations (PDEs), in which the boundary operator is time-dependent and its zero-order coefficients may be negative. In addition, the results on Hölder estimates and Harnack inequality for a cooperative weakly coupled parabolic PDE system are extended. Based on these results, we establish a Lyapunov function criterion for the existence of periodic probability measure solutions to the Fokker-Planck equation in both non-degenerate and degenerate cases.