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A User’s Guide to Optimal Transport

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Modelling and Optimisation of Flows on Networks

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 2062))

Abstract

This text is an expanded version of the lectures given by the first author in the 2009 CIME summer school of Cetraro. It provides a quick and reasonably account of the classical theory of optimal mass transportation and of its more recent developments, including the metric theory of gradient flows, geometric and functional inequalities related to optimal transportation, the first and second order differential calculus in the Wasserstein space and the synthetic theory of metric measure spaces with Ricci curvature bounded from below.

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Notes

  1. 1.

    Here c − c stands for “convex minus convex” and has nothing to do with the c we used to indicate the cost function.

  2. 2.

    If closed balls in X are compact, the proof greatly simplifies. Indeed in this case the inequality \({R}^{2}\mu (X \setminus {B}_{R}({x}_{0})) \leq \int {d}^{2}(\cdot ,{x}_{0})d\mu \) and the uniform bound on the second moments yields that the sequence \(n\mapsto {\mu }_{n}\) is tight. Thus to prove narrow convergence it is sufficient to check that \(\int fd{\mu }_{n} \rightarrow \int fd\mu \) for every \(f \in {C}_{c}(X)\). Since Lipschitz functions are dense in C c (X) w.r.t. uniform convergence, it is sufficient to check the convergence of the integral only for Lipschitz f’s. This follows from the inequality

    $$\begin{array}{rcl} \left \vert \int fd\mu -\int fd{\mu }_{n}\right \vert & =& \left \vert \int f(x) - f(y)d{\gamma }_{n}(x,y)\right \vert \leq \int \vert f(x) - f(y)\vert d{\gamma }_{n}(x,y) \\ & \leq &\mathrm{Lip}(f) \int d(x,y)d{\gamma }_{n}(x,y) \leq \mathrm{ Lip}(f)\sqrt{\int {d}^{2}(x,y)d{\gamma }_{n}(x,y)} \\ & =& \mathrm{Lip}(f){W}_{2}(\mu ,{\mu }_{n}).\end{array}$$
  3. 3.

    Again, if closed balls in X are compact the argument simplifies. Indeed from the uniform bound on the second moments and the inequality \({R}^{2}\mu (X \setminus {B}_{R}({x}_{0})) \leq \int\limits_{X\setminus {B}_{R}({x}_{0})}{d}^{2}(\cdot ,{x}_{0})d\mu \) we get the tightness of the sequence. Hence up to pass to a subsequence we can assume that (μ n ) narrowly converges to a limit measure μ, and then using the lower semicontinuity of W 2 w.r.t. narrow convergence we can conclude \({\overline{\lim }}_{n}{W}_{2}(\mu ,{\mu }_{n}) \leq {\overline{\lim }}_{n}{\underline{\lim }}_{m}{W}_{2}({\mu }_{m},{\mu }_{n}) = 0\).

  4. 4.

    As for Theorem 3.7 everything is simpler if closed balls in X are compact. Indeed, observe that a geodesic connecting two points in \({B}_{R}({x}_{0})\) lies entirely on the compact set \(\overline{{B}_{2R}({x}_{0})}\), and that the set of geodesics lying on a given compact set is itself compact in \(\mathrm{Geod}(X)\), so that the tightness of (μn) follows directly from the one of \(\{{\mu }_{0},{\mu }_{1}\}\).

  5. 5.

    The assumption \(\lambda \geq 0\) is necessary to have the last inequality in (82). If λ < 0, λ − convexity of \(\mathcal{V}\) along interpolating curves is not anymore true, so that we cannot apply directly the results of Sect. 4.2.4. Yet, adapting the arguments, it possible to show that all the results which we will present hereafter are true for general \(\lambda \in \mathbb{R}\).

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Acknowledgements

Work partially supported by a MIUR PRIN2008 grant.

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Authors and Affiliations

  1. Scuola Normale Superiore, Piazza dei Cavalieri, 7, 56126, Pisa, Italy

    Luigi Ambrosio

  2. Université de Nice, Mathématiques, Parc Valrose, 06108, Nice, France

    Nicola Gigli

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Ambrosio, L., Gigli, N. (2013). A User’s Guide to Optimal Transport. In: Modelling and Optimisation of Flows on Networks. Lecture Notes in Mathematics(), vol 2062. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32160-3_1

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