分野別セミナー

Abstract:
In recent development on analysis and geometry on metric measure spaces via optimal transport, it is verified that several conditions each of which characterize “Ricci curvature is bounded from below and dimension is bounded from above” on Riemannian manifolds are still equivalent on more singular spaces. Among others, a space-time upper bound of the quadratic Wasserstein distance between two heat distributions is known to be one of the equivalent conditions. In this bound, we have to consider two heat distributions at different time to extract the dimension upper bound.

In this talk, we propose a new bound for two heat distributions at the same time. It involves a control of difference of the relative entropy of heat distributions. We show that this bound is indeed a new characterization of the curvature-dimension condition even in an abstract framework. The talk is based on a joint work with F. Bolley, I. Gentil and A. Guillin.