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 Hellinger-Kantorovich distances. Contraction Hellinger-Kakutani distances general Csiszár divergences hold arbitrary do not require assumptions on linearity flow. style='text-indent:20px;'>When weaker transport are involved, we will show effects rely dual formulations strictly related to lower Ricci curvature bounds setting \begin{document}$ \mathrm{RCD}(K, \infty) $\end{document} metric measure spaces. As a byproduct, when id="M2">\begin{document}$ K\ge0 also find new estimates for asymptotic decay solution.