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Gradient flows on Wasserstein spaces over compact Alexandrov spaces
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 131, Number 2, April 2009
- pp. 475-516
- 10.1353/ajm.0.0048
- Article
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We establish the existence of Euclidean tangent cones on Wasserstein spaces over compact Alexandrov spaces of curvature bounded below. By using this Riemannian structure, we formulate and construct gradient flows of functions on such spaces. If the underlying space is a Riemannian manifold of nonnegative sectional curvature, then our gradient flow of the free energy produces a solution of the linear Fokker-Planck equation.