The exponential formula for the wasserstein metric

Craig, Katy

ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 1, pp. 169-187.

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.

Classification : 47J, 49K, 49J
Mots clés : Wasserstein metric, gradient flow, exponential formula

L. Ambrosio, N. Gigli, and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lect. Math. ETH Zürich 2nd edition. Birkhäuser Verlag, Basel (2008). | MR | Zbl

H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland Mathematics Studies, No. 5. Notas de Matemática (50). North-Holland Publishing Co., Amsterdam (1973). | MR | Zbl

H. Brézis and A. Pazy, Convergence and approximation of semigroups of nonlinear operators in Banach spaces. J. Functional Analysis 9 (1972) 63–74. | DOI | MR | Zbl

E.A. Carlen and K. Craig, Contraction of the proximal map and generalized convexity of the Moreau−Yosida regularization in the 2-Wasserstein metric. Math. Mech. Complex Systems 1 (2013) 33–65. | DOI | Zbl

J.A. Carrillo, M. Di Francesco, A. Figalli, T. Laurent, and D. Slepčev, Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations. Duke Math. J. 156 (2011) 229–271. | DOI | MR | Zbl

J. A. Carrillo, M. Di Francesco, A. Figalli, T. Laurent and D. Slepčev, Confinement in nonlocal interaction equations. Nonlin. Anal. 75 (2012) 550–558. | DOI | MR | Zbl

Ph. Clément and W. Desch, A Crandall-Liggett approach to gradient flows in metric spaces. J. Abstr. Differ. Equ. Appl. 1 (2010) 46–60. | MR | Zbl

M.G. Crandall, Semigroups of nonlinear transformations in Banach spaces. In Contributions to nonlinear functional analysis Proc. of Sympos., Math. Res. Center, Univ. Wisconsin, Madison. Publ. Math. Res. Center Univ. Wisconsin, No. 27. Academic Press, New York (1971) 157–179. | MR | Zbl

M.G. Crandall and T.M. Liggett, Generation of semi-groups of nonlinear transformations on general Banach spaces. Amer. J. Math. 93 (1971) 265–298. | DOI | MR | Zbl

N. Gigli, On the inverse implication of Brenier-McCann theorems and the structure of $(P_2\left(M\right),W_2)$. Methods Appl. Anal. 18 (2011) 127–158. | DOI | MR | Zbl

J. Jost, Convex functionals and generalized harmonic maps into spaces of nonpositive curvature. Comment. Math. Helv. 70 (1995) 659–673. | DOI | MR | Zbl

U.F. Mayer, Gradient flows on nonpositively curved metric spaces and harmonic maps. Comm. Anal. Geom. 6 (1998) 199–253. | DOI | MR | Zbl

R.J. Mccann, Existence and uniqueness of monotone measure-preserving maps. Duke Math. J. 80 (1995) 309–323. | DOI | MR | Zbl

F. Otto, Doubly degenerate diffusion equations as steepest descent, manuscript (1996).

F. Otto, The geometry of dissipative evolution equations: the porous medium equation. Commun. Partial Differ. Equ. 26 (2001) 101–174. | DOI | MR | Zbl

S. Rasmussen, Non-linear Semi-Groups, Evolution Equations and Product Integral Representations. Aarhus Universitet (1971). | Zbl

J. Rulla, Error analysis for implicit approximations to solutions to Cauchy problems. SIAM J. Numer. Anal. 33 (1996) 68–87. | DOI | MR | Zbl

K. Yosida, Functional Analysis. Classics in Mathematics. Springer-Verlag, Berlin (1995). Reprint of the sixth edition (1980). | MR