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

Abstract

We construct weak solutions to the McKean–Vlasov SDE \begin{equation*}dX(t)=b\biggl(X(t),\frac{d{\mathcal{L}}_{X(t)}}{dx}\bigl(X(t)\bigr)\biggr)\,dt+\sigma\biggl(X(t),\frac{d{\mathcal{L}}_{X(t)}}{dt}\bigl(X(t)\bigr)\biggr)\,dW(t)\end{equation*} on ${\mathbb{R}}^{d}$ for possibly degenerate diffusion matrices $\sigma$ 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.