Ricci curvature and $W_1$-exponential convergence of Markov processes on graphs

In this paper, we show that the Ricci curvature lower bound in Ollivier's Wasserstein metric sense of a continuous time jumping Markov process on a graph can be characterized by some optimal coupling generator and provide the construction of this latter. Some previous results of Ollivier for discrete time Markov chains are generalized to the actual continuous time case. We propose a comparison technique with some death-birth process on $\mathbb N$ to obtain some explicit exponential convergence rate, by modifying the metric. A counterpart of Zhong-Yang's estimate is established in the case where the Ricci curvature with repsect to the graph metric is nonnegative. Moreover we show that the Lyapunov function method for the exponential convergence works with some explicit quantitative estimates, once if the Ricci curvature is bounded from below by a negative constant. Finally we present applications to Glauder dynamics under some dynamical versions of the Dobrushin uniqueness condition or of the Dobrushin-Shlosman analyticity condition.