Uniqueness for nonlinear Fokker-Planck equations and for McKean-Vlasov SDEs: The degenerate case

Abstract

This work is concerned with the existence and uniqueness of generalized (mild or distributional) solutions to (possibly degenerate) Fokker-Planck equations $\rho_t-\Delta\beta(\rho)+{\rm div}(Db(\rho)\rho)=0$ in $(0,\infty)\times\mathbb{R}^d,$ $\rho(0,x) \equiv \rho_0(x)$. Under suitable assumptions on $\beta:\mathbb{R}\to\mathbb{R},\,b:\mathbb{R}\to\mathbb{R}$ and $D:\mathbb{R}^d\to\mathbb{R}^d$, $d\ge1$, this equation generates a unique flow $\rho(t)=S(t)\rho_0:[0,\infty)\to L^1(\mathbb{R}^d)$ as a mild solution in the sense of nonlinear semigroup theory. This flow is also unique in the class of $L^\infty((0,T)\times\mathbb{R}^d)\cap L^1((0,T)\times\mathbb{R}^d),$ $\forall T>0$, Schwartz distributional solutions on $(0,\infty)\times\mathbb{R}^d$. Moreover, for $\rho_0\in L^1(\mathbb{R}^d)\cap H^{-1}(\mathbb{R}^d)$, $t\to S(t)\rho_0$ is differentiable from the right on $[0,\infty)$ in $H^{-1}(\mathbb{R}^d)$-norm. As a main application, the weak uniqueness of the corresponding McKean-Vlasov SDEs is proven.