We discuss various characterizations of synthetic upper Ricci bounds for metric measure spaces in terms of heat flow, entropy and optimal transport. In particular, we present a characterization in terms of semiconcavity of the entropy along certain Wasserstein geodesics which is stable under convergence of mm-spaces. And we prove that a related characterization is equivalent to an asymptotic lower bound on the growth of the Wasseretein distance between heat flows. For weighted Riemannian manifolds, the crucial result will be a precise uniform two-sided bound for \begin{ eqnarray*}\frac{ d}{ dt}\Big|_{ t=0}W\big(\hat P_t\delta_x,\hat P_t\delta_y\big)\end{ eqnarray*} in terms of the mean value of the Bakry-Emery Ricci tensor $\mathrm{ Ric}+\mathrm{ Hess}\, f$ along the minimizing geodesic from $x$ to $y$ and an explicit correction term depending on the bound for the curvature along this curve.
Categories: math.DG math.AP math.FA