Heat flow on Alexandrov spaces

Nicola Gigli (1) , Kazumasa Kuwada (2) , Ohta Shin Ichi (3)

1 LJAD - Laboratoire Jean Alexandre Dieudonné
2 Ochanomizu University
3 Department of Mathematics, Kyoto University

We prove that on compact Alexandrov spaces with curvature bounded below the gradient flow of the Dirichlet energy in the L2-space produces the same evo- lution as the gradient flow of the relative entropy in the L2-Wasserstein space. This means that the heat flow is well defined by either one of the two gradient flows. Combining properties of these flows, we are able to deduce the Lipschitz continuity of the heat kernel as well as Bakry-Emery gradient estimates and the Γ2-condition. Our identification is established by purely metric means, unlike preceding results relying on PDE techniques. Our approach generalizes to the case of heat flow with drift.

Domaines: Analyse classique [math.CA]

Nicola Gigli, Kazumasa Kuwada, Ohta Shin Ichi. Heat flow on Alexandrov spaces. Communications on Pure and Applied Mathematics, 2012, pp.1-23. ⟨hal-00769390⟩