In recent work, Chow, Huang, Li and Zhou introduced the study of
Fokker-Planck equations for a free energy function defined on a finite graph.
When N≥2 is the number of vertices of the graph, they show that the
corresponding Fokker-Planck equation is a system of N nonlinear ordinary
differential equations defined on a Riemannian manifold of probability
distributions. The different choices for inner products on the space of
probability distributions result in different Fokker-Planck equations for the
same process. Each of these Fokker-Planck equations has a unique global
equilibrium, which is a Gibbs distribution. In this paper we study the {\em
speed of convergence} towards global equilibrium for the solution of these
Fokker-Planck equations on a graph, and prove that the convergence is indeed
exponential. The rate as measured by the decay of the L2 norm can be bound
in terms of the spectral gap of the Laplacian of the graph, and as measured by
the decay of (relative) entropy be bound using the modified logarithmic Sobolev
constant of the graph.
With the convergence result, we also prove two Talagrand-type inequalities
relating relative entropy and Wasserstein metric, based on two different
metrics introduced in [CHLZ] The first one is a local inequality, while the
second is a global inequality with respect to the "lower bound metric" from
[CHLZ]