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

We construct weak solutions to a class of distribution dependent SDE, of type $dX(t)=b\left( X(t), \displaystyle\frac{d$\mathcal{L}${X(t)}}{dx}(X(t))\right) dt+\sigma\left( X(t),\displaystyle\frac{d$\mathcal{L}${X(t)}}{dt}(X(t))\right) dW(t)$forpossiblydegeneratediffusionmatrices$\sigma$with$X(0)$ having a given law, which has a density with respect to Lebesgue measure, $dx$. Here ${\mathcal{L}}_{X(t)}$ 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.