Lower bound of coarse Ricci curvature on metric measure spaces and eigenvalues of Laplacian

In this paper,we investigate the coarse Ricci curvature on metric spaces with random walks. There exists no canonical random walk on metric space with a reference measure. However, we prove that a Bishop–Gromov inequality gives a lower bound of coarse Ricci curvature with respect to a random walk called an \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r$$\end{document}-step random walk. The lower bound does not coincide with the constant corresponding to curvature in Bishop–Gromov inequality. As a corollary, we obtain a lower bound of coarse Ricci curvature with respect to an \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r$$\end{document}-step random walk for a metric measure space satisfying the curvature-dimension condition. Moreover we give an important example, Heisenberg group, which does not satisfy the curvature-dimension condition for any constant but has a lower bound of coarse Ricci curvature. We also have an estimate of the eigenvalues of the Laplacian by a lower bound of coarse Ricci curvature.