Contraction and regularizing properties of heat flows in metric measure spaces

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.