Mass Transportation on the Earth

Ludovic Rifford

Description

Mass Transportation on the Earth Ludovic Rifford Universite de Nice - Sophia Antipolis & Institut Universitaire de France UPV/EHU Ludovic Rifford Mass Transportation on the Earth

• distance between


• quadratic monge's


• µ0


• any measurable map


• mass transportation


• transport map


• universite de nice

Sujets

• Institut universitaire de France
• Antipolis
• Public transport
• Universidad de Niza Sophia Antipolis

Informations

• Publié par: pefav
• Nombre de lectures: 11
• Langue: English

Extrait

Mass

Transportation on the

Ludovic Rifford

Universit´edeNice-SophiaAntipolis & Institut Universitaire de France

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UPV/EHU

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Earth

heEarth

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LetMbe asmooth connected compact surfaceinR n . For anyx,y∈M, we define the geodesic distance betweenxand y, denoted byd(x,ythe minimum of the lengths of the), as curves (drawn onM) joiningxtoy.

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∀Bmeasurable

Letµ 0 andµ 1 beprobability measuresonM. We call transport mapfromµ 0 toµ 1 any measurable map T:M→Msuch thatT ] µ 0 =µ 1 , that is µ 1 (B) =µ 0  T −1 (B)  ,∀Bmeas

⊂M.

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c(x,T(x))dµ 0 (x). M

Given two probabilities measuresµ 0 , µ 1 surM, find a transport mapT:M→Mfromµ 0 toµ 1 which minimizes the quadratic cost (c=d 2 /2) Z

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c(x,T(x))dµ 0 (x). M

Given two probabilities measuresµ 0 , µ 1 surM, find a transport mapT:M→Mfromµ 0 toµ 1 which minimizes the quadratic cost (c=d 2 /2) Z

Existence ?

Regularity ?

Uniqueness ?

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