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First variation formula in Wasserstein spaces over compact Alexandrov spaces. (English) Zbl 1264.53050

In an earlier paper [Am. J. Math. 131, No. 2, 475–516 (2009; Zbl 1169.53053)], the second author studied the quadratic Wasserstein space \(({\mathcal P}(X),W_2)\) over a compact Alexandrov space \(X\) with curvature bounded below. One of the important results of that earlier paper was the uniqueness and contraction properties of gradient flows of geodesically convex functionals on \(({\mathcal P}(X),W_2)\); the result was obtained only under the more stringent hypothesis that the underlying Alexandrov space has nonnegative curvature. In the paper under review, that more stringent hypothesis is relaxed to allow the Alexandrov space to have a negative lower curvature bound. A key tool in this paper is a first variation formula for the Wasserstein distance from which it follows that the gradient flows satisfy an evolution variational inequality.

MSC:

53C23 Global geometric and topological methods (à 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

Citations:

Zbl 1169.53053
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