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