The heat equation on manifolds as a gradient flow in the Wasserstein space

Matthias Erbar

Annales de l'I.H.P. Probabilités et statistiques (2010)

  • Volume: 46, Issue: 1, page 1-23
  • ISSN: 0246-0203

Abstract

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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.

How to cite

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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 46.1 (2010): 1-23. <http://eudml.org/doc/241521>.

@article{Erbar2010,
abstract = {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.},
author = {Erbar, Matthias},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {gradient flow; Wasserstein metric; relative entropy; heat equation; Riemannian manifold},
language = {eng},
number = {1},
pages = {1-23},
publisher = {Gauthier-Villars},
title = {The heat equation on manifolds as a gradient flow in the Wasserstein space},
url = {http://eudml.org/doc/241521},
volume = {46},
year = {2010},
}

TY - JOUR
AU - Erbar, Matthias
TI - The heat equation on manifolds as a gradient flow in the Wasserstein space
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2010
PB - Gauthier-Villars
VL - 46
IS - 1
SP - 1
EP - 23
AB - 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.
LA - eng
KW - gradient flow; Wasserstein metric; relative entropy; heat equation; Riemannian manifold
UR - http://eudml.org/doc/241521
ER -

References

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  11. [11] G. Savaré. Gradient flows and diffusion semigroups in metric spaces under lower curvature bounds. C. R. Acad. Sci. 345 (2007) 151–154. Zbl1125.53064
  12. [12] K.-T. Sturm. Convex functionals of probability measures and nonlinear diffusions on manifolds. J. Math. Pures Appl. 84 (2005) 149–168. Zbl1259.49074
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