Ricci curvature of Markov chains on Polish spaces revisited

Abstract

Recently, Y. Ollivier defined the Ricci curvature of Markov chains on Polish spaces via the contractivity of transition kernels under the $L^{1}$ Wasserstein metric. In this paper, we will discuss further the spectral gap, entropy decay, and logarithmic Sobolev inequality for the $\lambda$-range gradient operator. As an application, given resistance forms (i.e. symmetric Dirichlet forms with finite effective resistance) on fractals, we can construct Markov chains with positive Ricci curvature, which yields the Gaussian-then-exponential concentration of invariant distributions for Lipschitz test functions.