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In recent work, Chow, Huang, Li and Zhou introduced the study ofFokker-Planck equations for a free energy function defined on a finite graph.When $N\ge 2$ is the number of vertices of the graph, they show that thecorresponding Fokker-Planck equation is a system of $N$ nonlinear ordinarydifferential equations defined on a Riemannian manifold of probabilitydistributions. The different choices for inner products on the space ofprobability distributions result in different Fokker-Planck equations for thesame process. Each of these Fokker-Planck equations has a unique globalequilibrium, which is a Gibbs distribution. In this paper we study the {\emspeed of convergence} towards global equilibrium for the solution of theseFokker-Planck equations on a graph, and prove that the convergence is indeedexponential. The rate as measured by the decay of the $L_2$ norm can be boundin terms of the spectral gap of the Laplacian of the graph, and as measured bythe decay of (relative) entropy be bound using the modified logarithmic Sobolevconstant of the graph.  With the convergence result, we also prove two Talagrand-type inequalitiesrelating relative entropy and Wasserstein metric, based on two differentmetrics introduced in [CHLZ] The first one is a local inequality, while thesecond is a global inequality with respect to the "lower bound metric" from[CHLZ].

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