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://www.numdam.org/articles/10.1214/08-AIHP306/

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