Title: Displacement convexity of generalized relative entropy.
 
Abstract:
The displacement convexity is the convexity of a functional on the space of probability measures equipped with a certain distance function 
called the Wasserstein distance function. We discuss the displacement convexity of a functional consisting of an internal energy and a potential energy, which is a generalization of the 
relative entropy   from the viewpoint of information geometry.
The generalized relative entropies are classified by behavior of internal energies. 
The main theorem of this talk is that, on a weighted Riemannian manifold, the displacement convexity of all the generalized relative 
entropies  in this class is equivalent to the combination of the nonnegative weighted Ricci curvature and the convexity of the potential 
function of the generalized relative entropies. As applications, we analyze the gradient flow of this generalized 
relative entropy. This is a joint work with Shin-ichi Ohta.