Titlebook: ;

Definitive

The Wasserstein Distance and its Behaviour along GeodesicsIn this chapter we will introduce the .-th Wasserstein distance .(.) between two measures . ∈ .(.). The first section is devoted to its preliminary properties, in connection with the optimal transportation problems studied in the previous chapter and with narrow convergence: the main topological results are valid in general metric spaces.

invulnerable

AppendixIn this section we recall some standard facts about integrands depending on two variables, measurable w.r.t. the first one, and more regular w.r.t. the second one.

飞来飞去真休

宽容

Distribution of Media and Informationderivative of an absolutely continuous curve with values in . and the upper gradients of a functional defined in .. The related definitions are presented in the next two sections (a more detailed treatment of this topic can be found for instance in [20]); the last one deals with curves of maximal slope.

indoctrinate

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Media and Global Climate Knowledgegnificant result in the quite general framework of . in view of possible applications to infinite dimensional Hilbert (or Banach) spaces, thus avoiding any local compactness assumption (we refer to the treatises [126, 71, 72, 136, 67] for comprehensive presentations of this subject).

尽责

debacle

Nick Couldry,Sonia Livingstone,Tim Markhamed in a Hilbert space ., the . : . → 2. of . is a multivalued operator defined as . which we will also write in the equivalent form for . ∈ .(.) . As usual in multivalued analysis, the proper domain .(.) ⊂ .(.) is defined as the set of all . ∈ . such that .(.) ≠ φ; we will use this convention for all the multivalued operators we will introduce.

gene-therapy

Global Justice and Global Mediat flows . generated by a proper, l.s.c. functional . in ., . being a separable Hilbert space. Taking into account the first part of this book and the (sub)differential theory developed in the previous chapter, there are at least four possible approaches to gradient flows which can be adapted to the framework of Wasserstein spaces:

解脱

Introductions made of two parts, the first one concerning gradient flows in metric spaces and the second one devoted to gradient flows in the .-Wasserstein space of probability measures on a separable Hilbert space . endowed with the Wasserstein . metric (we consider the .-Wasserstein distance, . ∈ (1, ∞), as well).