Uniqueness and superposition of the space-distribution dependent Zakai equations

The work concerns the space-distribution dependent Zakai equations from nonlinear filtering problems of McKean-Vlasov stochastic differential equations with correlated noises. First of all, we establish the space-distribution dependent Kushner-Stratonovich equations and the space-distribution dependent Zakai equations. Then, the pathwise uniqueness of their strong solutions is shown. Finally, we prove a superposition principle between the space-distribution dependent Zakai equations and space-distribution dependent Fokker-Planck equations. As a by-product, we give some conditions under which space-distribution dependent Fokker-Planck equations have weak solutions.