A user’s guide to optimal transport

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1、Auser'sguidetooptimaltransportLuigiAmbrosio∗NicolaGigli†AbstractThistextisanexpandedversionofthelecturesgivenbythefirstauthorinthe2009CIMEsummerschoolofCetraro.Itprovidesaquickandreasonablyaccountoftheclassicaltheoryofoptimalmasstransportationandofitsmorerecentdevelopments,includingthemetrictheor

2、yofgradientflows,geometricandfunctionalinequalitiesrelatedtooptimaltransportation,thefirstandsecondorderdifferentialcalculusintheWassersteinspaceandthesynthetictheoryofmetricmeasurespaceswithRiccicurvatureboundedfrombelow.Contents1Theoptimaltransportproblem41.1MongeandKantorovichformulationsoftheo

3、ptimaltransportproblem........41.2Necessaryandsufficientoptimalityconditions.....................61.3Thedualproblem....................................111.4Existenceofoptimalmaps...............................141.5Bibliographicalnotes..................................222TheWassersteindistanceW224

4、2.1XPolishspace.....................................242.2Xgeodesicspace....................................312.3XRiemannianmanifold................................402.3.1Regularityofinterpolatedpotentialsandconsequences............402.3.2TheweakRiemannianstructureof(P2(M),W2)..............432.4B

5、ibliographicalnotes..................................493Gradientflows493.1Hilbertiantheoryofgradientflows...........................493.2ThetheoryofGradientFlowsinametricsetting....................513.2.1Theframework.................................513.2.2Generall.s.c.functionalsandEDI............

6、...........553.2.3Thegeodesicallyconvexcase:EDEandregularizingeffects.........593.2.4ThecompatibilityofEnergyanddistance:EVIanderrorestimates......633.3ApplicationstotheWassersteincase..........................673.3.1Elementsofsubdifferentialcalculusin(P(Rd),W)............69223.3.2Threeclassical

7、functionals...........................703.4Bibliographicalnotes..................................76∗l.ambrosio@sns.it†nicola.gigli@unice.fr14Geometricandfunctionalinequalities774.1Brunn-Minkowskiinequality...................