In this talk, we generalize the Ricci curvature lower bound in Ollivier’s Wasserstein metric sense from discrete time to continuous time jumping Markov processes. This becomes more flexible when the generator equals to J −I for a transition probability kernel J than the Ollivier’s one. We show that this lower bound of Ricci curvature can be characterized by some optimal coupling generator and we provide the construction of this latter, by introducing the “false” jumps. Some previous results of Ollivier for discrete time Markov chains are generalized to the actual continuous time case. But our main attention will be concentrated on new phenomena appeared in the continuous time case. The main results of this paper are
(1) we propose a new comparison condition with some death-birth process on N to obtain some explicit exponential convergence rate, by modifying the metric, even if the Ricci curvature with respect to the graph metric is not positive;
(2) a counterpart of Zhong-Yang’s estimate is established in the case where the Ricci curvature with respect to the graph metric is nonnegative;
(3) we use the Lyapunov function condition for the exponential convergence in Wasserstein metric with explicit quantitative estimate, even if the Ricci curvature is bounded from below by a negative constant. That generalizes the result of Hairer-Mattingly from Markov chains to continuous time Markov processes.
Moreover we present applications of the previous general results to Glauder dynamics under some dynamical versions of the Dobrushin uniqueness condition or of the Dobrushin-Shlosman analyticity condition, and a number of examples for illustration.