Introduction

These notes are based on a series of lectures given by the second author for the Summer School “Optimal Transportation: Theory and Applications” in Grenoble during the week of June 22–26, 2009.

We try to summarize some of the main results concerning gradient flows of geodesically λ-convex functionals in metric spaces and applications to diffusion partial differential equations (PDEs) in the Wasserstein space of probability measures. Due to obvious space constraints, the theory and the references presented here are largely incomplete and should be intended as an oversimplified presentation of a quickly evolving subject. We refer to the books [3, 68] for a detailed account of the large literature available on these topics.

In the Section 6.2 we collect some elementary and well-known results concerning gradient flows of smooth convex functions in ℝd. We selected just a few topics, which are well suited for a “metric” formulation and provide a useful guide for the more abstract developments. In the Section 6.3 we present the main (and strongest) notion of gradient flow in metric spaces characterized by the solution of a metric evolution variational inequality: the aim here is to show the consequence of this definition, without any assumptions on the space and on the functional (except completeness and lower semicontinuity); we shall see that solutions to evolution variational inequalities enjoy nice stability, asymptotic, and regularization properties.