Functionals on the space of probabilities

Abstract

We consider in this chapter various classes of functionals on the space \(\mathcal{P}(\varOmega )\), which can be of interest in many variational problems, and are natural in many modeling issues: the potential energy, the interaction energy, the Wasserstein distance to a given measure, the norm in a dual functional space, the integral of a function of the density, and the sum of a function of the masses of the atoms. The scope of the chapter is to study some properties of these functionals. The first questions that we analyze are classical variational issues (semi-continuity, convexity, first variation, etc.). Then, we also introduce and analyze a new notion, the notion of geodesic convexity. In the discussion section, we analyze a typical optimization problem over measures, we present a proof of the Brunn-Minkowski inequality based on geodesic convexity, and we apply the functionals we presented to a model for urban equilibria.