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# First Variation Formula in Wasserstein Spaces over Compact Alexandrov Spaces

Published:2011-06-08
Printed: Dec 2012
• Nicola Gigli,
Institut für Angewandte Mathematik, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany
• Shin-Ichi Ohta,
Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan
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## Abstract

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 $(\mathcal P(X),W_2)$ 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: Alexandrov spaces, Wasserstein spaces, first variation formula, gradient flow
 MSC Classifications: 53C23 - Global geometric and topological methods (a la Gromov); differential geometric analysis on metric spaces 28A35 - Measures and integrals in product spaces 49Q20 - Variational problems in a geometric measure-theoretic setting 58A35 - Stratified sets [See also 32S60]

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