On Riemannian manifolds satisfying the Transport Continuity Property

Ludovic Rifford

Description

On Riemannian manifolds satisfying the Transport Continuity Property Ludovic Rifford Universite de Nice - Sophia Antipolis (Joint work with A. Figalli and C. Villani) Ludovic Rifford Optimal transportation and applications (Banff 2010)

• riemannian manifold


• optimal transport


• geodesic distance


• transport continuity


• universite de nice


• compact connected

Sujets

• Antipolis
• Riemannian manifold
• Transportation theory
• Distance (graph theory)
• Universidad de Niza Sophia Antipolis

Informations

• Publié par: pefav
• Nombre de lectures: 12
• Langue: English

Extrait

nORiemannianmanifoldssatisfying TransportContinuityProperty LudovicRifford Universite´deNice-SophiaAntipolis (JointworkwithA.FigalliandC.Villani) uLodivciRoffdrpOitamlrtnapsroatitnonadpalpcitaoin teh sB(naff210)0

Statement Ludovic fo Rifford eht Optimal problem transportationdnaapplicationsffnaB()0102

)0102ffnaB(snoitacilppadnanoitatropsnartlamitpOdroffiRcivoduL .)x( 0 µd))x(T,x(c M Z gniziminimdna ,)M ⊂ nailerobB ∀ ,
)B( 1 − T � 0 µ=)B( 1 µ.e.i( 1 µ= 0 µ ] T gniyfsitasM → M:Tpamelbarusem adnfi,Mno 1 µ, 0 µserusaemytilibaborpnaileroBowtneviG .M ∈ y,x ∀ 2 )y,x( g d 2 1 =:)y,x(c yb) ∞ ,0[ → M × M:ctsoccitardauq ehtenfieddnaMnoecnatsidcisedoegeht g dybetoneD manifoldfodimensionn.2 ≥ Let(M,g)beasmoothcompactconnectedRiemannian OptimaltransportonRiemannianmanifolds

)0102ffnaB(snoitacilppadnanoitatropsnartlamitpOdroffiRcivoduL .)x( 0 µd))x(T,x(c M Z gniziminimdna ,)M ⊂ nailero 1 c(x,y):=d g (x,y) 2 2 b ∀ x,y ∈ M. B Denotebyd g thegeodesicdistanceonManddefinethe quadraticcostc:M × M → [0, ∞ )by ∀ Let(M,g)beasmoothcompactconnectedRiemannian manifoldofdimensionn ≥ 2. , OptimaltransportonRiemannianmanifolds
)B( 1 − T � 0 µ=)B( 1 µ.e.i( 1 µ= 0 µ ] T gniyfsitasM → M:Tpamelbarusem adnfi,Mno 1 µ, 0 µserusaemytilibaborpnaileroBowtneviG

OptimaltransportonRiemannianmanifolds Let(M,g)beasmoothcompactconnectedRiemannian manifoldofdimensionn ≥ 2. Denotebyd g thegeodesicdistanceonManddefinethe quadraticcostc:M × M → [0, ∞ )by 1 c(x,y):=d g (x,y) 2 ∀ x,y ∈ M. 2 GiventwoBorelianprobabilitymeasuresµ 0 ,µ 1 onM,finda mesurablemapT:M → Msatisfying
� T ] µ 0 =µ 1 (i.e.µ 1 (B)=µ 0 T − 1 (B), ∀ Bborelian ⊂ M), andminimizing Z M uLd c(x,T(x))dµ 0 (x). vociiRoffdrpOitamlrtnapsroatitnonadpalpcitaoisnB(naff210)0