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First Variation Formula in Wasserstein Spaces over Compact Alexandrov Spaces

Published online by Cambridge University Press:  20 November 2018

Nicola Gigli  and
Shin-Ichi Ohta
Nicola Gigli
Affiliation:
Institut für Angewandte Mathematik, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germanye-mail: nicolagigli@googlemail.com
Shin-Ichi Ohta
Affiliation:
Department of Mathematics, Kyoto University, Kyoto 606-8502, Japane-mail: sohta@math.kyoto-u.ac.jp
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Abstract

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We extend results proved by the second author (Amer. J. Math., 2009) for nonnegatively curved Alexandrov spaces to general compact Alexandrov spaces $X$ with curvature bounded below. The gradient flow of a geodesically convex functional on the quadratic Wasserstein space $\left( \mathcal{P}\left( X \right),\,{{W}_{2}} \right)$ satisfies the evolution variational inequality. Moreover, the gradient flow enjoys uniqueness and contractivity. These results are obtained by proving a first variation formula for the Wasserstein distance.

Keywords

53C2328A3549Q2058A35Alexandrov spacesWasserstein spacesfirst variation formulagradient flow
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 55 , Issue 4 , 01 December 2012 , pp. 723 - 735
Copyright
Copyright © Canadian Mathematical Society 2012

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