Convergence to global equilibrium for Fokker–Planck equations on a graph and Talagrand-type inequalities

In 2012, Chow, Huang, Li and Zhou [7] proposed the Fokker–Planck equations for the free energy on a finite graph, in which they showed that the corresponding Fokker–Planck equation is a nonlinear ODE defined on a Riemannian manifold of probability distributions. Different choices for inner products result in different Fokker–Planck equations. The unique global equilibrium of each equation is a Gibbs distribution. In this paper we proved that the exponential rate of convergence towards the global equilibrium of these Fokker–Planck equations. The rate is measured by both the decay of the L2L2 norm and that of the (relative) entropy. With the convergence result, we also prove two Talagrand-type inequalities relating relative entropy and Wasserstein metric, based on two different metrics introduced in [7]. The first one is a local inequality, while the second is a global inequality with respect to the “lower bound metric” from [7].