Ricci curvature of metric spaces

We define a notion of Ricci curvature in metric spaces equipped with a measure or a random walk. For this we use a local contraction coefficient of the random walk acting on the space of probability measures equipped with a transportation distance. This notions allows to generalize several classical theorems associated with positive Ricci curvature, such as a spectral gap bound (Lichnerowicz theorem), Gaussian concentration of measure (Levy-Gromov theorem), logarithmic Sobolev inequalities (a result of Bakry-Emery theory) or the Bonnet-Myers theorem. The definition is compatible with Bakry-Emery theory, and is robust and very easy to implement in concrete examples such as graphs.