Contraction and regularizing properties of heat flows in metric measure spaces

Abstract

<p style='text-indent:20px;'>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-Kantorovich distances. Contraction properties of Hellinger-Kakutani distances and general Csiszár divergences hold in arbitrary metric-measure spaces and do not require assumptions on the linearity of the flow. <p style='text-indent:20px;'>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 <inline-formula><tex-math id="M1">\begin{document}$ \mathrm{RCD}(K, \infty) $\end{document}</tex-math></inline-formula> metric measure spaces. As a byproduct, when <inline-formula><tex-math id="M2">\begin{document}$ K\ge0 $\end{document}</tex-math></inline-formula> we will also find new estimates for the asymptotic decay of the solution.