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When the Ricci curvature of a Riemannian manifold is not lower bounded by aconstant, but lower bounded by a continuous function, we give a newcharacterization of this lower bound through the convexity of relative entropyon the probability space over the Riemannian manifold. Hence, we generalizeK.T. Sturm and von Renesse's result (Comm. Pure Appl. Math. 2005) to the casewith non-constant lower bound of Ricci curvature.

Characterization of non-constant lower bound of Ricci curvature via entropy inequality on Wasserstein space