Optimal Transport: Old and New by Cédric Villani

By Cédric Villani

At the shut of the Nineteen Eighties, the self sustaining contributions of Yann Brenier, Mike Cullen and John Mather introduced a revolution within the venerable box of optimum delivery based via G. Monge within the 18th century, which has made breathtaking forays into quite a few different domain names of arithmetic ever for the reason that. the writer offers a huge evaluate of this sector, delivering whole and self-contained proofs of all of the basic result of the speculation of optimum shipping on the applicable point of generality. hence, the booklet encompasses the extensive spectrum starting from simple thought to the newest examine effects.

PhD scholars or researchers can learn the whole publication with none earlier wisdom of the sector. A complete bibliography with notes that widely speak about the present literature underlines the book’s price as a so much welcome reference textual content in this topic.

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M The formula of conservation of mass is an Eulerian description of the physical world, which means that the unknowns are ﬁelds. The next theorem links it with the Lagrangian description, in which everything is expressed in terms of particle trajectories, that are integral curves of the velocity ﬁeld: d Tt (x). 6) ξ t, Tt (x) = dt If ξ is (locally) Lipschitz continuous, then the Cauchy–Lipschitz theorem guarantees the existence of a ﬂow Tt locally deﬁned on a maximal time interval, and itself locally Lipschitz in both arguments t and x.

3). e. π = π . 8 (Convexity of the optimal cost). Let X and Y be two Polish spaces, let c : X ×Y → R∪{+∞} be a lower semicontinuous function, and let C be the associated optimal transport cost functional on P (X ) × P (Y). Let (Θ, λ) be a probability space, and let µθ , νθ be two measurable functions deﬁned on Θ, with values in P (X ) and P (Y) respectively. Assume that c(x, y) ≥ a(x) + b(y), where a ∈ L1 (dµθ dλ(θ)), b ∈ L1 (dνθ dλ(θ)). Then µθ λ(dθ), C Θ νθ λ(dθ) ≤ C(µθ , νθ ) λ(dθ) . 8. First notice that a ∈ L1 (µθ ), b ∈ L1 (νθ ) for λalmost all values of θ.

The modern formulation seems to have emerged around 1980, independently by Berkes and Philipp [101], Kallenberg, Thorisson, and maybe others. g. 1]; see also the bibliographic comments in [317, p. 20]. 6]. A comment about terminology: I like the word “gluing” which gives a good indication of the construction, but many authors just talk about “composition” of plans. g. Evans and Gariepy [331, Chapter 3]. 5]. Such a generality is interesting in the context of optimal transportation, where changes of variables are often very rough (say BV , which means of bounded variation).

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