Measure concentration, transportation cost, and fu

InstructionalConferenceonCombinatorialAspectsofMathematicalAnalysisEdinburgh,25March-5April2002andSummerSchoolonSingularPhenomenaandScalinginMathematicalModelsBonn,10-13June,2003MEASURECONCENTRATION,TRANSPORTATIONCOST,ANDFUNCTIONALINEQUALITIESM.LedouxUniversityofToulouse,FranceAbstract.—Intheselectures,wepresentatripledescriptionoftheconcentrationofmeasurephenomenon,geometric(throughBrunn-Minkoswkiinequalities),measure-theoretic(throughtransportationcostinequalities)andfunctional(throughlogarithmicSobolevinequalities),andinvestigatetherelationshipsbetweenthesevariousviewpoints.Spe-cialemphasisisputonoptimalmasstransportationandthedualhy-percontractiveboundsonsolutionsofHamilton-Jacobiequationsthatofferaunifiedtreatmentofthesevariousaspects.Thesenotessurveyrecentdevelopmentsaroundtheconcentrationofmeasurephenomenonthroughvariousdescriptions,geometric,measuretheoreticandfunctional.Thesedescriptionsaimtoanalyzemeasureconcentrationforbothproductand(strictly)log-concavemeasures,withaspecialemphasisondimensionfreebounds.Inequalitiesindependentofthenumberofvariablesareindeedakeyinformationinthestudyofanumberofmodelsinprobabilitytheoryandstatisticalmechanics,withaviewtowardsinfinitedimensionalanalysis.Tothistask,wereviewthegeometrictoolofBrunn-Minkowskiinequalities,transportationcostinequalities,andfunctionallogarithmicSobolevinequalitiesandsemigroupmethods.ConnectionsaredevelopedonthebasisofoptimalmasstransportationanddualhypercontractiveboundsonsolutionsofHamilton-Jacobiequations,providingasyntheticviewoftheserecentdevelopments.Resultsandmethodsareonlyoutlinedinthesimplestandbasicsetting.ReferencestorecentPDEextensionsarebrieflydiscussedinthelastpartofthenotes.Thesenotesonlycollectafewbasicresultsonthetopicsoftheselectures,andonlyaimtogiveaflavourofthesubject.Wereferto[Le],[B-G-L],[O-V1],[CE],[CE-G-H],[Vi]...forfurthermaterial,proofsanddetailedreferences.1.TheconcentrationofmeasurephenomenonTheconcentrationofmeasurephenomenonwasputforwardintheearlyseventiesbyV.Milman[Mi1],[Mi3]intheasymptoticgeometryofBanachspacesandtheproofofthefamousDvoretzkytheoremonsphericalsectionsofconvexbodies.Ofisoperimetricinspiration,itisofpowerfulinterestinapplications,invariousareassuchasgeometry,functionalanalysisandinfinitedimensionalintegration,discretemathematicsandcomplexitytheory,andprobabilitytheory.Generalreferences,fromvariousviewpoints,are[Bal],[Grom],[Le],[MD],[Mi2],[Mi4],[M-S],[Sc],[St],[Ta1]...1.1IntroductionTointroducetotheconceptofmeasureconcentration,wefirstbrieflydiscussafewexamples.Afirstillustrationissuggestedbytheexampleofthestandardn-sphereSninRn+1whendimensionnislarge.Byastandardcomputation,uniformmeasureσnonSnisalmostconcentratedwhenthedimensionnislargearoundthe(every)equator.Actually,theisoperimetricinequalityonSnexpressesthatsphericalcaps(geodesicballs)minimizetheboundarymeasureatfixedvolume.Initsintegratedform,givenaBorelsetAonSnwiththesamemeasureasasphericalcapB,thenforeveryr0,σn(Ar)≥σn(Br)whereAr={x∈Sn;d(x,A)r}isthe(open)neighborhoodoforderrforthegeodesicdistanceonSn.Onemainfeatureofconcentrationwithrespecttoisoperimetryistoanalyzethisinequalityforthenon-infinitesimalvaluesofr0.TheexplicitevaluationofthemeasureofsphericalcapsthenimpliesthatgivenanymeasurablesetAwith,say,σn(A)≥12,foreveryr0,σn(Ar)≥1−e−(n−1)r2/2.(1.1)2Therefore,almostallpointsonSnarewithin(geodesic)distance1√nfromAwhichisofparticularinterestwhenthedimensionnislarge.Froma“tomographic”pointofview,thevisualdiameterofSn(forσn)isoftheorderof1√nasn→∞whichisincontrastwiththediameterofSnasmetricspace.Thisexampleisafirst,andmain,instanceoftheconcentrationofmeasurephenomenonforwhichnicepatternsdevelopasthedimensionislarge.Itfurthermoresuggeststheintroductionofaconcentrationfunctioninordertoevaluatethedecayin(1.1).Settingασn(r)=sup1−σn(Ar);A⊂Sn,σn(A)≥12 ,r0,thebound(1.1)amountstosaythatασn(r)≤e−(n−1)r2/2,r0.(1.2)Notethatr0in(1.2)actuallyrangesuptothediameterπofSnandthat(1.2)isthusmainlyofinterestwhennislarge.Byrescalingofthemetric,theprecedingresultsapplysimilarlytouniformmeasureσnRonthen-sphereSnRofradiusR0.Inparticular,ασnR(r)≤e−(n−1)r2/2R2,r0.(1.3)Properlynormalized,uniformmeasuresonhighdimensionalspheresapproximateGaussiandistributions.Moreprecisely,themeasuresσn√nconvergewhenntendstoinfinitytothecanonicalGaussianmeasureonRN.TheisoperimetricinequalityonspheresmaythenbetransferredtoanisoperimetricinequalityforGaussianmeasures.Precisely,ifγ=γkisthecanonicalGaussianmeasureonRkwithdensity(2π)−k/2e−|x|2/2withrespecttoLebesguemeasure,andifAisaBorelsetinRkwithγ(A)=Φ(a)forsomea∈[−∞,+∞]whereΦ(t)=(2π)−1/2Rt−∞e−x2/2dxisthedistributionfunctionofthestandardnormaldistributionontheline,thenforeveryr0,γ(Ar)≥Φ(a+r).HereArdenotesther-neigborhoodofAwithrespecttothestandardEuclideanmetriconRk.Throughoutthesenotes,Rk(orsubsetsofRk)willbeequippedherewiththestandardEuclideanstructureandthemetric|x−y|,x,y∈Rk,inducedbythenorm|x|=Pki=1x2i
1/2,x=(x1,...,xk)∈Rk.Thescalarproductwillbedenotedx·y=Pki=1xiyi,x=(x1,...,xk),y=(y1,...,yk)∈Rk.Definingsimilarlytheconcentrationfunctionforγ