From nonlinear Fokker.Planck equations to solutions of distribution dependent SDE

We construct weak solutions to the McKean.Vlasov SDE dX(t)=b(X(t),dLX(t)dx(X(t)))dt+.(X(t),dLX(t)dt(X(t)))dW(t) on Rd for possibly degenerate diffusion matrices . with X(0) having a given law, which has a density with respect to Lebesgue measure, dx. Here, LX(t) denotes the law of X(t). Our approach is to first solve the corresponding nonlinear Fokker.Planck equations and then use the well-known superposition principle to obtain weak solutions of the above SDE.