Contraction and regularizing properties of heat flows in metric measure spaces

Giulia Luise; Giuseppe Savaré

doi:10.3934/dcdss.2020327

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2021, Volume 14, Issue 1: 273-297. Doi: 10.3934/dcdss.2020327

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Contraction and regularizing properties of heat flows in metric measure spaces

Giulia Luise^1, and
Giuseppe Savaré^2, ,

1.
Department of Computer Science, University College London, Gower Street, London WC1E 6BT, UK
2.
Dipartimento di Matematica 'Felice Casorati', University of Pavia, Via Ferrata 1, 27100 Pavia, Italy

^* Corresponding author: Giuseppe Savaré
^* Corresponding author: Giuseppe Savaré

Dedicated to Alexander Mielke on the occasion of his 60th birthday

Received: March 31, 2019

Revised: September 30, 2019

Early access: April 2020

Published: January 2021

The second author is partially supported by PRIN2015 grant from MIUR for the project Calculus of Variations and by IMATI-CNR

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Abstract

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.

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Mathematics Subject Classification: Primary: 49Q20, 47D07; Secondary: 30L99.

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