12 Introduction optimal transportation cost when the cost is a power of a distance. This question is actually much simpler than Question 1, and its answer has been known for some time (see Theorem 7.12). Many many names are associ- ated with these distances (Dall’Aglio, Fr´ echet, Gini, H¨ offding, Hutchinson, Kantorovich, Rubinstein, Tanaka, Wasserstein, and others) and we make no attempt to review the existing literature on the subject. The properties of the Wasserstein distances depend very little on the underlying geometrical structure, and the results in this chapter hold in extreme generality (Polish spaces). A most important particular case is the Wasserstein distance of order 2, or (by abuse of terminology) the quadratic Wasserstein distance, (17) W2(μ, ν) = Td2 (μ, ν). This distance plays a central role among Monge-Kantorovich distances, just as L2 plays a central role in the family of Lp spaces. An application to the study of the Boltzmann equation, due to Tanaka, is expounded (The- orem 7.23) even though it is by now somewhat outdated, it is interesting and not well-known. Chapter 8 is a key chapter: there we introduce and study a differential, dynamical formulation of the mass transportation problem. This idea, which leads to a reformulation from a fluid mechanics point of view, goes back to Benamou and Brenier (see Theorem 8.1), and was exploited by Otto to develop a nice geometric view about optimal transportation. Otto’s work has been very influential over the recent years. As an illustration of the power of this differential point of view, in Chap- ter 9, we explain how it enables one to link mass transportation with several classes of functional inequalities which are useful in many different contexts: logarithmic Sobolev inequalities, entropy-entropy production inequalities, transportation inequalities. These links were first studied systematically by Otto and the author (see the formal Theorem 9.2). Not all chapters are of equal status. Chapters 1, 2, 4 and 7 contain the presentation of the basic theory. These are certainly the chapters which should first be read by a graduate student — particularly Chapter 2. All the proofs of the main results in these chapters are given in detail, except for the last part of Chapter 4. Chapter 5 is crucial for applications, and also deserves careful reading by students the proofs there are rather easy, but the conceptual gain may be very rewarding. Chapter 3 is not diﬃcult, but part of it deals with more original subjects, related to hydrodynamic equations. A reader who would like to know more about the basics of the huge theory of these equations may consult an introductory book on the subject, like Chorin and Marsden [88], intended for mathematicians. The

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