Gradient flows in general metric and Wasserstein spaces

Abstract

In this thesis we approach gradient flows in metric spaces which are not necessarily endowed with a natural linear structure. The first part is an introduction to the theory in abstract metric spaces: We present three possible generalizations of gradient flows to the metric setting and investigate uniqueness and contraction properties of the various notions. However, an existence theory may not easily be archived at this level of generality. Therefore, we content ourselves by presenting only the principal definitions of the so-called minimizing movements scheme which provides a variational interpolation technique for gradient flows in the metric setting. In the second part of this thesis we deal with gradient flows in Wasserstein spaces. We introduce the theory of optimal transportation. We focus on the Kantorovich problem which is a relaxation of the Monge problem. Specifically, we show that the Kantorovich problem admits an optimal solution under very general assumptions. Moreover, one may utilize the theory of optimal transportation to define a family of metrics on a certain subset in the class of probability measures on a metric space. The resulting family of metric spaces, called Wasserstein spaces, inherit to a great extend the topological and geometrical structure of the underlying metric space. We give proof to an existence result of gradient flows in the quadratic Wasserstein space, employing the aforementioned minimizing movements scheme. This result allows comprehensive application of gradient flows in the field of partial differential equations. In a concluding example we show the main ideas applied to the heat equation.