Wasserstein metric, gradient flow structure and well-posedness of Fokker-Planck equation on locally finite graphs

Abstract: This paper investigates the gradient flow structure and the well-posedness of the Fokker-Planck equation on locally finite graphs. We first construct a 2-Wasserstein-type metric in the probability density space associated with the underlying graphs that are locally finite. Then, we prove the global existence and asymptotic behavior of the solution to the Fokker-Planck equation using a novel approach that differs significantly from the methods applied in the finite case, as proposed in [S., Chow, W., Huang, Y., Li, H., Zhou, Arch. Rational Mech. Anal., 2012]. This work seems the first result on the study of Wasserstein-type metrics and the Fokker-Planck equation in probability spaces defined on infinite graphs.