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Functionals on the space of probabilities

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Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE,volume 87)

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.

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Notes

  1. 1.

    This is an approach used by P. L. Lions in his lectures at Collège de France; see [211].

  2. 2.

    For the case of the entropy, the lower semi-continuity on the whole \(\mathbb{R}^{d}\) is false, but it is true under stronger convergence assumptions; see Ex (45). This is the usual strategy to study on the whole space the variational problems that we will present in bounded domains in Section 8.3 for the Fokker-Planck equation.

  3. 3.

    Actually, it is also known that \(D^{2}\varphi\) exists a.e.: indeed, convex functions are twice differentiable a.e. (see, for instance, [160], the original result being stated in [6]).

  4. 4.

    By the way, this functional can even be proven to be somehow geodesically concave, as it is shown in [15], Theorem 7.3.2.

  5. 5.

    Thanks to an observation by R. McCann himself, this corresponds to the fact that the utility is a concave function not only of land “volume” but also of land “linear size,” which seems reasonable since it has to be compared to “linear” quantities such as distances in the interaction term.

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Santambrogio, F. (2015). Functionals on the space of probabilities. In: Optimal Transport for Applied Mathematicians. Progress in Nonlinear Differential Equations and Their Applications, vol 87. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-20828-2_7

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