In this paper we summarize some of the main results of a forthcoming book on this topic, where we examine in detail the theory of curves of maximal slope in a general metric setting, following some ideas introduced in [11, 5], and study in detail the case of the Wasserstein space of probability measures. In the first part we derive new general conditions ensuring convergence of the implicit time discretization scheme to a curve of maximal slope, the uniqueness, and the error estimates. In the second part we study in detail the differentiable structure of the Wasserstein space, to which the metric theory applies, and use this structure to give also an equivalent concept of gradient flow. Our analysis includes measures in infinite-dimensional Hilbert spaces and it does not require any absolute continuity assumption on the measures involved.
@article{RLIN_2004_9_15_3-4_327_0,
author = {Luigi Ambrosio and Nicola Gigli and Giuseppe Savar\'e},
title = {Gradient flows with metric and differentiable structures, and applications to the Wasserstein space},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
volume = {15},
year = {2004},
pages = {327-343},
zbl = {1162.35349},
mrnumber = {2148889},
language = {en},
url = {http://dml.mathdoc.fr/item/RLIN_2004_9_15_3-4_327_0}
}
Ambrosio, Luigi; Gigli, Nicola; Savaré, Giuseppe. Gradient flows with metric and differentiable structures, and applications to the Wasserstein space. Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 15 (2004) pp. 327-343. http://gdmltest.u-ga.fr/item/RLIN_2004_9_15_3-4_327_0/
[1] , Existence of solutions to degenerate parabolic equations via the Monge-Kantorovich theory. Arch. Rat. Mech. Anal., to appear (2002). | MR 2703679 | Zbl 1103.35051
[2] , Minimizing movements. Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., (5), 19, 1995, 191-246. | MR 1387558 | Zbl 0957.49029
[3] - - , Gradient flows in metric spaces and in the Wasserstein space of probability measures. Birkhäuser, 2004. | Zbl 1090.35002
[4] - , Selected topics on «analysis in metric spaces». Scuola Normale Superiore, Pisa 2000. | MR 2012736 | Zbl 1084.28500
[5] , Discretization of evolution variational inequalities. In: - - - (eds.), Partial differential equations and the calculus of variations. Vol. I, Birkhäuser, Boston, MA, 1989, 59-92. | MR 1034002 | Zbl 0677.65068
[6] - , A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem. Numer. Math., 84, 2000, n. 3, 375-393. | MR 1738163 | Zbl 0968.76069
[7] , Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland Publishing Co., Amsterdam, 1973, North-Holland Mathematics Studies, No. 5. Notas de Matemática (50). MR 50 #1060 | MR 348562 | Zbl 0252.47055
[8] - , Constrained steepest descent in the -Wasserstein metric. Ann. Math., 157, 2003, n. 3, 807-846. | MR 1983782 | Zbl 1038.49040
[9] - - , Contraction in the -Wasserstein metric length space and thermalization of granular media. To appear. | Zbl 1082.76105
[10] , New problems on minimizing movements. In: - (eds.), Boundary Value Problems for PDE and Applications. Masson, Paris 1993, 81-98. | MR 1260440 | Zbl 0851.35052
[11] - - , Problemi di evoluzione in spazi metrici e curve di massima pendenza. Atti Acc. Lincei Rend. fis., s. 8, v. 68, 1980, 180-187. | MR 636814 | Zbl 0465.47041
[12] - , Measure transport on Wiener space and the Girsanov theorem. C.R. Math. Acad. Sci. Paris, 334, 2002, n. 11, 1025-1028. | MR 1913729 | Zbl 1036.60004
[13] - , The geometry of optimal transportation. Acta Math., 177, 1996, n. 2, 113-161. | MR 1440931 | Zbl 0887.49017
[14] - , Quasiconformal maps in metric spaces with controlled geometry. Acta Math., 181, 1998, n. 1, 1-61. | MR 1654771 | Zbl 0915.30018
[15] - - , The variational formulation of the Fokker-Planck equation. SIAM J. Math. Anal., 29, 1998, n. 1, 1-17 (electronic). | MR 1617171 | Zbl 0915.35120
[16] , Nonpositive curvature: geometric and analytic aspects. Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel1997. | MR 1451625 | Zbl 0896.53002
[17] - - , Curves of maximal slope and parabolic variational inequalities on nonconvex constraints. Ann. Scuola Norm. Sup. Pisa Cl. Sci., 16(4), 1989, n. 2, 281-330. | MR 1041899 | Zbl 0699.49015
[18] , Gradient flows on nonpositively curved metric spaces and harmonic maps. Comm. Anal. Geom., 6, 1998, n. 2, 199-253. | MR 1651416 | Zbl 0914.58008
[19] , A convexity principle for interacting gases. Adv. Math., 128, 1997, n. 1, 153-179. | MR 1451422 | Zbl 0901.49012
[20] - - , A posteriori error estimates for variable time-step discretizations of nonlinear evolution equations. Comm. Pure Appl. Math., 53, 2000, n. 5, 525-589. MR 1 737 503 | MR 1737503 | Zbl 1021.65047
[21] , The geometry of dissipative evolution equations: the porous medium equation. Comm. Partial Differential Equations, 26, 2001, n. 1-2, 101-174. | MR 1842429 | Zbl 0984.35089
[22] , Topics in optimal transportation. Graduate studies in mathematics, 58, AMS, Providence, RI, 2003. | MR 1964483 | Zbl 1106.90001