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

作者：Erbar Matthias^*

来源：Annales de l Institut Henri Poincare-Probabilites et Statistiques, 2010, 46(1): 1-23.

DOI：10.1214/08-AIHP306

• Gradient flow
• Wasserstein metric
• Relative entropy
• Heat equation

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.

• 出版日期2010-2