The heat equation on manifolds as a gradient flow in the Wasserstein space
Annales de l'I.H.P. Probabilités et statistiques, Tome 46 (2010) no. 1, pp. 1-23.

Nous étudions les flux gradients dans l'espace des mesures de probabilité sur une variété riemannienne pas nécessairement compacte. Dans ce but nous munissons l'espace de Wasserstein avec une sorte de structure riemannienne. Si la courbure de Ricci de la variété est bornée inférieurement nous démontrons qu'il existe un flux gradient contractif pour l'entropie relative. Il est construit explicitement en utilisant une approximation variationelle discrète. De plus ses trajectoires Coïncident avec les solutions à l'équation de la chaleur.

We study the gradient flow for the relative entropy functional on probability measures over a riemannian manifold. To this aim we present a notion of a riemannian structure on the Wasserstein space. If the Ricci curvature is bounded below we establish existence and contractivity of the gradient flow using a discrete approximation scheme. Furthermore we show that its trajectories coincide with solutions to the heat equation.

DOI : 10.1214/08-AIHP306
Classification : 35A15, 58J35, 60J60
Mots-clés : gradient flow, Wasserstein metric, relative entropy, heat equation 
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Erbar, Matthias. The heat equation on manifolds as a gradient flow in the Wasserstein space. Annales de l'I.H.P. Probabilités et statistiques, Tome 46 (2010) no. 1, pp. 1-23. doi : 10.1214/08-AIHP306. http://archive.numdam.org/articles/10.1214/08-AIHP306/

[1] L. Ambrosio, N. Gigli and G. Savaré. Gradient Flows in Metric Spaces and in the Space of Probability Measures. Birkhäuser, Basel, 2005. | MR | Zbl

[2] L. Ambrosio and G. Savaré. Gradient Flows of Probability Measures. In Handbook of Differential Equations: Evolution Equations 1-136. Elsevier, 2006. | MR | Zbl

[3] J.-D. Benamou and Y. Brenier. A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem. Numer. Math. 84 (2000) 375-393. | MR | Zbl

[4] P. Bernard. Young measures, superposition and transport. Indiana Univ. Math. J. 57 (2008) 247-276. | MR

[5] I. Chavel. Riemannian Geometry - a Modern Introduction. Cambridge Tracts in Mathematics 108. Cambridge Univ. Press, Cambridge, 1993. | MR | Zbl

[6] J. Dodziuk. Maximum principles for parabolic inequalities and the heat flow on open manifolds. Indiana Univ. Math. J. 32 (1983) 703-716. | MR | Zbl

[7] R. Jordan, D. Kinderlehrer and F. Otto. The variational formulation of the Fokker-Planck equation. SIAM J. Math. Anal. 29 (1998) 1-17. | MR | Zbl

[8] R.J. Mccann. A convexity principle for interacting gases. Adv. Math. 128 (1997) 153-179. | MR | Zbl

[9] R.J. Mccann. Polar factorization of maps on Riemannian manifolds. Geom. Funct. Anal. 11 (2001) 589-608. | MR | Zbl

[10] S.-I. Ohta. Gradient flows on Wasserstein spaces over compact Alexandrov spaces. Amer. J. Math. 2008. To appear. | MR | Zbl

[11] G. Savaré. Gradient flows and diffusion semigroups in metric spaces under lower curvature bounds. C. R. Acad. Sci. 345 (2007) 151-154. | Zbl

[12] K.-T. Sturm. Convex functionals of probability measures and nonlinear diffusions on manifolds. J. Math. Pures Appl. 84 (2005) 149-168.

[13] K.-T. Sturm. On the geometry of metric measure spaces. Acta Math. 196 (2006) 65-131. | MR | Zbl

[14] C. Villani. Optimal Transport, Old and New. Grundlehren der Mathematischen Wissenschaften 338. Springer, Berlin, Heidelberg, 2009. | MR | Zbl

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