# Contraction and regularizing properties of heat flows in metric measure spaces

## TL;DR

This paper explores novel contraction and regularizing effects of heat flows in metric measure spaces, highlighting their relationships with various distances and curvature bounds, without requiring flow linearity.

## Contribution

It demonstrates contraction properties of Hellinger-Kakutani distances and Csiszár divergences in general metric spaces, and links regularization effects to lower Ricci curvature bounds in $RCD(K,
abla)$ spaces.

## Key findings

- Contraction of Hellinger-Kakutani distances holds in arbitrary metric-measure spaces.
- Regularizing effects depend on dual formulations of transport distances.
- Results relate heat flow properties to lower Ricci curvature bounds in $RCD(K,
abla)$ spaces.

## Abstract

We illustrate some novel contraction and regularizing properties of the Heat flow in metric-measure spaces that emphasize an interplay between Hellinger-Kakutani, Kantorovich-Wasserstein and Hellinger-Kantorvich distances. Contraction properties of Hellinger-Kakutani distances and general Csisz\'ar divergences hold in arbitrary metric-measure spaces and do not require assumptions on the linearity of the flow. When weaker transport distances are involved, we will show that contraction and regularizing effects rely on the dual formulations of the distances and are strictly related to lower Ricci curvature bounds in the setting of $RCD(K,\infty)$ metric measure spaces.