Geodesic
The exponential formula for the wasserstein metric
Craig, Katy1
1 Dept. of Mathematics, University of California, Los Angeles, 520 Portola Plaza, Los Angeles, CA 90095, USA.
ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 1, pp. 169-187

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A recurring obstacle in the study of Wasserstein gradient flow is the lack of convexity of the square Wasserstein metric. In this paper, we develop a class of transport metrics that have better convexity properties and use these metrics to prove an Euler−Lagrange equation characterizing Wasserstein discrete gradient flow. We then apply these results to give a new proof of the exponential formula for the Wasserstein metric, mirroring Crandall and Liggett’s proof of the corresponding Banach space result [M.G. Crandall and T.M. Liggett, Amer. J. Math. 93 (1971) 265–298]. We conclude by using our approach to give simple proofs of properties of the gradient flow, including the contracting semigroup property and energy dissipation inequality.

Reçu le :
DOI : 10.1051/cocv/2014069
Classification : 47J, 49K, 49J
Keywords: Wasserstein metric, gradient flow, exponential formula

Craig, Katy 1

1 Dept. of Mathematics, University of California, Los Angeles, 520 Portola Plaza, Los Angeles, CA 90095, USA. 
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     title = {The exponential formula for the wasserstein metric},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {169--187},
     publisher = {EDP-Sciences},
     volume = {22},
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Craig, Katy. The exponential formula for the wasserstein metric. ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 1, pp. 169-187. doi: 10.1051/cocv/2014069

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