First Variation Formula in Wasserstein Spaces over Compact Alexandrov Spaces

Canadian mathematical bulletin

Shin-Ichi Ohta¹  Nicola Gigli² 

[1] Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan;Institut für Angewandte Mathematik, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany

关键词: Alexandrov spaces;    Wasserstein spaces;    first variation formula;    gradient flow;   

DOI  :  10.4153/CMB-2011-110-3

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

We extend results proved by the second author (Amer. J. Math., 2009)for nonnegatively curved Alexandrov spacesto general compact Alexandrov spaces $X$ with curvature boundedbelow. The gradient flow of a geodesically convex functional on the quadratic Wassersteinspace $(mathcal P(X),W_2)$ satisfies the evolution variational inequality.Moreover, the gradient flow enjoys uniqueness and contractivity.These results are obtained by proving a first variation formula forthe Wasserstein distance.

【 授权许可】

Unknown