Evolution equations as Wasserstein gradient flows

In this talk, I study a certain evolution equation on a weighted Riemannian manifold as the gradient flow of a generalized relative entropy with respect to the Wasserstein metric. The Wasserstein metric is a metric on the space of probability measures. For example, it is known that the porous medium equation is the Wasserstein gradient flow of the Tsallis entropy. I give a criterion for generalized relative entropies to be convex. From the convexity of generalized relative entropies, I derive appropriate variants of the Poincaré and the logarithmic Sobolev inequalities, also the contraction propety of its gradient flow. This is a joint work with Shin-ichi Ohta.