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.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 $ \mathrm{RCD}(K, \infty) $ metric measure spaces. As a byproduct, when $ K\ge0 $ we will also find new estimates for the asymptotic decay of the solution.

中文翻译：

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公制度量空间中热流的收缩和正则化特性

我们说明了度量尺度空间中热流的一些新颖的收缩和正则化性质，这些性质强调了Hellinger-Kakutani距离，Kantorovich-Wasserstein距离和Hellinger-Kantorovich距离之间的相互作用。Hellinger-Kakutani距离的收缩性质和广义Csiszár散度在任意度量空间中均成立，并且不需要对流的线性进行假设。当涉及较弱的传输距离时，我们将证明收缩和正则化效应取决于对偶公式距离，并且与$ \ mathrm {RCD}（K，\ infty）$度量度量空间设置中的下Ricci曲率边界严格相关。作为副产品，当$ K \ ge0 $时，我们还将找到溶液渐近衰减的新估计。