We propose a general method to identify nonlinear Fokker–Planck–Kolmogorov equations (FPK equations) as gradient flows on the space \(\mathcal {P}\) of probability measures on \(\mathbb {R}^d\) , based on a classical technique, known as “lifting the geometry,” to define a natural differential structure on \(\mathcal {P}\) (just based on fixing suitable classes of curves in \(\mathcal {P}\) and test functions on \(\mathcal {P}\) ), obtaining a corresponding tangent bundle over \(\mathcal {P}\) and the associated gradient operator. This is achieved without using any metric on \(\mathcal {P}\) , such as the Wasserstein distance. One of the new outcomes of this approach is that the gradient flows are not formulated in some variational or generalized sense, but in a strong sense. We also explicitly identify the associated entropy functions and the corresponding energy functionals E, in particular their domains of definition. Furthermore, the latter functionals are Lyapunov functions for the solutions of the FPK equations. Moreover, we show uniqueness for such gradient flows, and we also prove that the gradient of E at \(\mu \in \mathcal {P}\) 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.
