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Contraction and regularizing properties of heat flows in metric measure spaces. (English) Zbl 1459.49030

A metric-measure space \((X, d, \mathfrak{m})\) isgiven by a complete and separable metric space \((X, d)\) endowed with a Borel positive measure \(\mathfrak{m}\) with full support satisfying the growth condition \(\exists\; o \in X,\; k>0 : \mathfrak{m}(\{x : d(x, o) < r\})\leq e^{k r^{2}}\). The geometry of metric-measure spaces have had an important role in several classical problems of the calculus of variations, geometric measure theory and mathematical physics. Several authors studied the geometry of metric measure spaces [L. Ambrosio et al., Functions of bounded variation and free discontinuity problems. Oxford: Clarendon Press (2000; Zbl 0957.49001); Nonlinear diffusion equations and curvature conditions in metric measure spaces. Providence, RI: American Mathematical Society (AMS) (2019; Zbl 1477.49003); Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 137, 77–134 (2016; Zbl 1337.49075); Invent. Math. 195, No. 2, 289–391 (2014; Zbl 1312.53056); Ann. Probab. 43, No. 1, 339–404 (2015; Zbl 1307.49044); D. Bakry et al., Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 14, No. 3, 705–727 (2015; Zbl 1331.35151); S. Daneri and G. Savaré, SIAM J. Math. Anal. 40, No. 3, 1104–1122 (2008; Zbl 1166.58011); M. Erbar, Ann. Inst. Henri Poincaré, Probab. Stat. 46, No. 1, 1–23 (2010; Zbl 1215.35016)].
The principal objective in this paper is to 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.

MSC:

49Q20 Variational problems in a geometric measure-theoretic setting
47D07 Markov semigroups and applications to diffusion processes
30L99 Analysis on metric spaces
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References:

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