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.