By Cédric Villani
On the shut of the Eighties, the self sufficient contributions of Yann Brenier, Mike Cullen and John Mather introduced a revolution within the venerable box of optimum delivery based by means of G. Monge within the 18th century, which has made breathtaking forays into numerous different domain names of arithmetic ever considering. the writer provides a vast assessment of this region, providing whole and self-contained proofs of the entire primary result of the idea of optimum shipping on the acceptable point of generality. hence, the ebook encompasses the vast spectrum starting from easy thought to the newest learn effects. PhD scholars or researchers can learn the complete booklet with none previous wisdom of the sphere. A complete bibliography with notes that generally talk about the present literature underlines the book’s worth as a so much welcome reference textual content in this topic.
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Additional resources for Optimal Transport: Old and New (Grundlehren der mathematischen Wissenschaften)
The modern formulation seems to have emerged around 1980, independently by Berkes and Philipp , 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).
An inﬁnite-dimensional generalization was studied by Bogachev, Kolesnikov and Medvedev [134, 135]. FKG inequalities were introduced in , and have since then played a crucial role in statistical mechanics. Holley’s proof by coupling appears in . Recently, Caﬀarelli  has revisited the subject in connection with optimal transport. It was in 1965 that Moser proved his coupling theorem, for smooth compact manifolds without boundaries ; noncompact manifolds were later considered by Greene and Shiohama .
The situation above appears in a number of problems in statistical mechanics, in connection with the so-called FKG (Fortuin–Kasteleyn–Ginibre) inequalities. 2) intuitively says that ν puts more mass on large values than µ. 6. Probabilistic representation formulas for solutions of partial diﬀerential equations. There are hundreds of them (if not thousands), representing solutions of diﬀusion, transport or jump processes as the laws of various deterministic or stochastic processes. Some of them are recalled later in this chapter.
Optimal Transport: Old and New (Grundlehren der mathematischen Wissenschaften) by Cédric Villani