The heat equation on manifolds as a gradient flow in the Wasserstein space
Erbar, Matthias
Ann. Inst. H. Poincaré Probab. Statist., Tome 46 (2010) no. 1, p. 1-23 / Harvested from Project Euclid
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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.
Publié le : 2010-02-15
Classification:  Gradient flow,  Wasserstein metric,  Relative entropy,  Heat equation,  35A15,  58J35,  60J60 
@article{1267454105,
     author = {Erbar, Matthias},
     title = {The heat equation on manifolds as a gradient flow in the Wasserstein space},
     journal = {Ann. Inst. H. Poincar\'e Probab. Statist.},
     volume = {46},
     number = {1},
     year = {2010},
     pages = { 1-23},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1267454105}
}

Erbar, Matthias. The heat equation on manifolds as a gradient flow in the Wasserstein space. Ann. Inst. H. Poincaré Probab. Statist., Tome 46 (2010) no. 1, pp.  1-23. http://gdmltest.u-ga.fr/item/1267454105/