Second order analysis on (p 2(M), W 2)

Memoirs of the American Mathematical Society

Volumn 216, Issue 1018, 2012, Pages 1-173

  Gigli, Nicola ^a  

^a NONE

Author keywords

Weak Riemannian structure; Wesserstein distance

Indexed keywords

EID: 84860630927     PISSN: 00659266     EISSN: 19476221     Source Type: Journal    
DOI: 10.1090/S0065-9266-2011-00619-2     Document Type: Article

References (27)

1. L. AMBROSIO, N. GIGLI, Construction of parallel transport in the Wasserstein space, Methods and Applications of Analysis, 15(2008), vol. 1, pp. 1-30. MR2482206
2. L. AMBROSIO, N. GIGLI, AND G. SAVARÉ, Gradient flows in metric spaces and in spaces of probability measures, Birkḧauser, 2005. MR2129498 (2006k:49001)
3. J.-D. BENAMOU AND Y. BRENIER, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem, Numer. Math., 84(2000), pp. 375-393. MR1738163 (2000m:65111)
4. X. CABRÉ, Nondivergent elliptic equations on manifolds with nonegative curvature, Comm. on Pure and Applied Math., 50(1998), 623-665. MR1447056 (98b:58157)
5. J. A. CARRILLO, R. J. MCCANN, AND C. VILLANI, Contractions in the 2-Wasserstein length space and thermalization of granular media, Arch. Ration. Mech. Anal., 179(2006), pp. 217-263. MR2209130 (2006j:76121) (Pubitemid 43237778)
6. M. P. DO CARMO, Riemannian geometry, Mathematics: Theory & Applications, Birkhäuser Boston Inc., Boston, MA, 1992. Translated from the second Portuguese edition by Francis Flaherty. MR1138207 (92i:53001)
7. L. C. EVANS AND W. GANGBO, Differential equations methods for the Monge-Kantorovich mass transfer problem, Mem. Amer. Math. Soc., 137(1999), pp. viii+66. MR1464149 (99g:35132)
8. A. FATHI, A. FIGALLI, Optimal transportation on non-compact manifolds, Israel J. Math. 175(2010), pp. 1-59. MR2607536 (2011c:49091)
9. W. GANGBO, T. NGUYEN, A. TUDORASCU, Hamilton-Jacobi equations in the Wasserstein space, Methods and Applications of Analysis, 15(2008), vol. 2, pp. 155-184. MR2481677
10. d endowed with the quadratic optimal transportation distance, Ph. D. thesis, Scuola Normale Superiore, Pisa, 2008. Available at: http://cvgmt.sns.it
11. N. GIGLI, On Hölder continuity in time of the optimal transport map towards measures along a curve, Proceedings of the Edinburgh Math. Soc., to appear.
12. 2), Method and App. of Anal., to appear.
13. R. JORDAN, D. KINDERLEHRER, AND F. OTTO, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29(1998), pp. 1-17 (electronic). MR1617171 (2000b:35258)
14. J. LOTT, Some geometric calculations in the Wasserstein space, Comm. Math. Phys. 277(2007), pp. 423-437.
15. J. LOTT AND C. VILLANI, Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. (2) 169(2009), pp. 903-991. MR2480619 (2010i:53068) MR2480619
16. X.-N. MA, N. S. TRUDINGER, AND X.-J. WANG, Regularity of potential functions of the optimal transportation problem. Arch. Ration. Mech. Anal. 177, no. 2(2005), pp. 151-183. MR2188047 (2006m:35105) (Pubitemid 41053542)
17. R. J. MCCANN, A convexity principle for interacting gases, Adv. Math., 128(1997), pp. 153-179. MR1451422 (98e:82003)
18. R. J. MCCANN, Polar factorization of maps on Riemannian manifolds, Geom. Funct. Anal. 11, no. 3(2001), pp. 589-608. MR1844080 (2002g:58017)
19. L. NATILE AND G. SAVARÉ, A Wasserstein approach to the one-dimensional sticky particle system, Siam J. Math. Anal. 41, no. 4(2009), pp. 1340-1365. MR2540269
20. S.-i. OHTA, Gradient flows on Wasserstein spaces over compact Alexandrov spaces, tech. rep., Universität Bonn, 2007.
21. F. OTTO, The geometry of dissipative evolution equations: the porous medium equation, Comm. Partial Differential Equations, 26(2001), pp. 101-174. MR1842429 (2002j:35180) (Pubitemid 33731270)
22. G. SAVARÉ, Gradient flows and diffusion semigroups in metric spaces under lower curvature bounds, C. R. Math. Acad. Sci. Paris, 345(2007), pp. 151-154. MR2344814 (2008j:49113)
23. K.-T. STURM, M. K. VON RENESSE, Entropic Measure and Wasserstein Diffusion, Ann. Probab. 37(2009), pp. 1114-1191. MR2537551 (2010k:60185)
24. K.-T. STURM, Entropic measure on multidimensional spaces, Stochastic analysis, random fields and applications VI. Progress in Probability, Birkhauser, 2010.
25. K.-T. STURM, On the geometry of metric measure spaces. I-II, Acta Math. 196, 1(2006), 65-177. MR2237206 (2007k:53051a)
26. C. VILLANI, Optimal transport, old and new, Springer-Verlag, 2008. MR2459454
27. L. ZAJÍČEK, On the differentiability of convex functions in finite and infinite dimensional spaces, Czechoslovak Math. J. 29(104) (1979), pp. 340-348. MR536060 (80h:46063)