Ollivier curvature and continuous time

We provide a generalization of Ollivier curvature to weighted graphs with potentially unbounded Laplacian. An immediate consequence of this is a discrete version of the Laplacian comparison principle. We give a continuous time heat semigroup characterization of lower bounds on the Ollivier curvature via the perpetual cutoff method. Using this, we show that a lower bound on the Ollivier curvature implies stochastic completeness.