Synthetic versus distributional lower Ricci curvature bounds

We compare two standard approaches to defining lower Ricci curvature bounds for Riemannian metrics of regularity below C2. These are, on the one hand, the synthetic definition via weak displacement convexity of entropy functionals in the framework of optimal transport, and the distributional one based on non-negativity of the Ricci-tensor in the sense of Schwartz. It turns out that distributional bounds imply entropy bounds for metrics of class C1 and that the converse holds for C1,1-metrics under an additional convergence condition on regularisations of the metric.