Georgia Institute of Technology College of Sciences

Fokker-Planck equation is a linear parabolic equation which describes the time evolution of of probability distribution of a stochastic process defined on a Euclidean space. Moreover, it is the gradient flow of free energy functional. We will present a Fokker-Planck equation which is a system of ordinary differential equations and describes the time evolution of probability distribution of a stochastic process on a graph with a finite number of vertices. It is shown that there is a strong connection but also substantial differences between the ordinary differential equations and the usual Fokker-Planck equation on Euclidean spaces. Furthermore, the ordinary differential equation is in fact a gradient flow of free energy on a Riemannian manifold whose metric is closely related to certain Wasserstein metrics. Some examples will also be discussed.