Displacement convexity of generalized relative entropy

The displacement convexity is the convexity of functionals defined on the
space of probability measures equipped with the distance function,
so-called Wasserstein distance function. In this talk, I first introduce  a
class of generalized relative entropies, which stems from the Bregman
divergence, on a  Riemannian manifold with a weighted measure. Then I prove
that the convexity of all the entropies in this class is equivalent to the
combination of the nonnegative weighted Ricci curvature and the convexity
of another function used in the definition of the generalized relative
entropies. From the convexity of the generalized relative entropies, I
derive appropriate variants of the Talagrand and the logarithmic Sobolev
inequalities. I also investigate the gradient flow of the gradient flow of
the generalized relative entropy. This is the joint work with Shin-ichi
Ohta.