# On the speed in Trasportation costs of heat distributions

### (Ochanomizu University)

**Event:** ERC Workshop on Optimal Transportation and Applications

**Date:** Oct 27, 2014,
**time:** 14:30

**Abstract.** By regarding heat distributions as a curve in the space of probability measures,

we can consider its speed measured by some transportation costs. Our main concern in

this talk is heat distributions on (backward) Ricci ﬂow. The speed can be expressed

explicitly when heat distribution is identiﬁed with a gradient ﬂow of the relative entropy.

On Ricci ﬂow, this interpretation does not seem to work well. Nevertheless we can show

some results for a suitably chosen transportation costs. By combining this result with

monotonicity of those transportation costs, we can show the monotonicity of Perelmans

F-functional as well as W-entropy. Indeed it extends some known results on Ricci ﬂow to

noncompact case.