Nonlinear Fokker-Planck equations and their asymptotic properties

We study the nonlinear Fokker-Planck equation on graphs, which is the gradient flow in the space of probability measures supported on the nodes with respect to the discrete Wasserstein metric. The energy functional driving the gradient flow consists of a Boltzmann entropy, a linear potential and a quadratic interaction energy. We show that the solution converges to the Gibbs measures exponentially fast with a rate that can be given analytically. The continuous analog of this asymptotic rate is related to the Yano's formula.