Geodesic
First Variation Formula in Wasserstein Spaces over Compact Alexandrov Spaces
Canadian mathematical bulletin, Tome 55 (2012) no. 4, pp. 723-735

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DOI

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 $\left( \mathcal{P}\left( X \right),\,{{W}_{2}} \right)$ 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.
DOI : 10.4153/CMB-2011-110-3
Mots-clés : 53C23, 28A35, 49Q20, 58A35, Alexandrov spaces, Wasserstein spaces, first variation formula, gradient flow 
Gigli, Nicola; Ohta, Shin-Ichi. First Variation Formula in Wasserstein Spaces over Compact Alexandrov Spaces. Canadian mathematical bulletin, Tome 55 (2012) no. 4, pp. 723-735. doi: 10.4153/CMB-2011-110-3
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     title = {First {Variation} {Formula} in {Wasserstein} {Spaces} over {Compact} {Alexandrov} {Spaces}},
     journal = {Canadian mathematical bulletin},
     pages = {723--735},
     year = {2012},
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