Gradient flows with metric and differentiable structures, and applications to the Wasserstein space

In this paper we summarize some of the main results of a orthcoming book on this topic, where we examine in detail the theory of curves of maximal slope in a general metric setting, and study in detail the case of the Wasserstein space of probability measures. In the first part we derive new general conditions ensuring convergence of the implicit time discretization scheme to a curve of maximal slope, the uniqueness, and the error estimates. In the second part we study in detail the differentiable structure of the Wasserstein space, to which the metric theory applies, and use this structure to give also an equivalent concept of gradient flow. Our analysis includes measures in infinite-dimensional Hilbert spaces and it does not require any absolute continuity assumption on the measures involved.