Uniqueness and superposition of the distribution-dependent Zakai equations

The work concerns the Zakai equations from nonlinear filtering problems of McKean-Vlasov stochastic differential equations with correlated noises. First, we establish the Kushner-Stratonovich equations, the Zakai equations and the distribution-dependent Zakai equations. And then, the pathwise uniqueness, uniqueness in joint law and uniqueness in law of weak solutions for the distribution-dependent Zakai equations are shown. Finally, we prove a superposition principle between the distribution-dependent Zakai equations and distribution-dependent Fokker-Planck equations. As a by-product, we give some conditions under which distribution-dependent Fokker-Planck equations have unique weak solutions.