Superposition principles for the Zakai equations and the Fokker-Planck equations on measure spaces

The work concerns the superposition between the Zakai equations and the Fokker-Planck equations on measure spaces. First, we prove a superposition principle for the Fokker-Planck equations on $mR^mN$ under the integrable condition. And then by means of it, we show two superposition principles for the weak solutions of the Zakai equations from the nonlinear filtering problems and the weak solutions of the Fokker-Planck equations on measure spaces. As a by-product, we give some weak conditions under which the Fokker-Planck equations can be solved in the weak sense.