Nonlinear Fokker--Planck--Kolmogorov equations as gradient flows on the space of probability measures

Abstract

We propose a general method to identify nonlinear Fokker--Planck--Kolmogorov equations (FPK equations) as gradient flows on the space of probability measures on $\mathbb{R}^d$ with a natural differential geometry. Our notion of gradient flow does not depend on any underlying metric structure such as the Wasserstein distance, but is derived from purely differential geometric considerations. We explicitly identify the associated entropy functions, which are not necessarily convex, and also the corresponding energy functionals, in particular their domains of definition. Furthermore, the latter functions are Lyapunov functions for the solutions of the FPK equations. Moreover, we show uniqueness results for such gradient flows, and we also prove that the gradient of $E$ is a gradient field on $\mathbb{R}^d$, which can be approximated by smooth gradient fields. These results cover classical and generalized porous media equations, where the latter have a generalized diffusivity function and a nonlinear transport-type first-order perturbation.