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

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. © Association des Publications de l'Institut Henri Poincaré, 2010.