| Issue |
ESAIM: COCV
Volume 22, Number 1, January-March 2016
|
|
|---|---|---|
| Page(s) | 169 - 187 | |
| DOI | https://doi.org/10.1051/cocv/2014069 | |
| Published online | 15 January 2016 | |
The exponential formula for the wasserstein metric∗
Dept. of Mathematics, University of
California, Los Angeles, 520
Portola Plaza, Los
Angeles, CA
90095,
USA.
kcraig@math.ucla.edu
Received:
5
June
2014
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.
Mathematics Subject Classification: 47J / 49K / 49J
Key words: Wasserstein metric / gradient flow / exponential formula
© EDP Sciences, SMAI 2016
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