On The Trend To Global Equilibrium For Fokker-Planck Equations On Graph

In [4], Chow, Huang, Li and Zhou consider Fokker-planck equations for a free energy function or Markov process defined on a graph with a finite number of vertices and edges. If N≥2 is the number of vertices of the graph, they shown that the corresponding Fokker-Planck equation is a system of N nonlinear ordinary differential equations defined on a Riemannian manifold of probability distributions. There have 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 unique global equilibrium:Gibbs distribution. In this paper we study the speed of convergence towards global equilibrium for the solution of these Fokker-Planck equations on a graph, and prove the convergence is exponential.