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 ${\rho }_t-\Delta \beta (\rho )+\mathrm{div}(Db(\rho )\rho )=0$ 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 $D:{\mathbb{R}}^d\to {\mathbb{R}}^d$, $d\ge 1$, 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.