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1、MetricmeasurespaceswithRiemannianRiccicurvatureboundedfrombelowLuigiAmbrosio∗NicolaGigli†GiuseppeSavar´e‡July30,2011AbstractInthispaperweintroduceasyntheticnotionofRiemannianRicciboundsfrombelowformetricmeasurespaces(X,d,m)whichisstableundermeasuredGromov-Hausdorffconvergencea
2、ndrulesoutFinslergeometries.ItcanbegivenintermsofanenforcementoftheLott,SturmandVillanigeodesicconvexityconditionfortheentropycoupledwiththelinearityoftheheatflow.Besidesstability,itenjoysthesametensorization,global-to-localandlocal-to-globalproperties.Inthesespaces,thatwecall
3、RCD(K,∞)spaces,weprovethattheheatflow(whichcanbeequivalentlycharacterizedeitherastheflowassociatedtotheDirichletform,orastheWassersteingradientflowoftheentropy)satisfiesWassersteincontractionestimatesandseveralregularityproperties,inparticularBakry-EmeryestimatesandtheL∞−LipFelle
4、rregularization.WealsoprovethatthedistanceinducedbytheDirichletformcoincideswithd,thatthelocalenergymeasurehasdensitygivenbythesquareofCheeger’srelaxedslopeand,asaconsequence,thattheunderlyingBrownianmotionhascontinuouspaths.AlltheseresultsareobtainedindependentlyofPoincar´ea
5、nddoublingassumptionsonthemetricmeasurestructureandthereforeapplyalsotospaceswhicharenotlocallycompact,astheinfinite-dimensionalones.Contents1Introduction22Preliminaries62.1Basicnotation,metricandmeasuretheoreticconcepts.............62.2Remindersonoptimaltransport.............
6、............82.3MetricmeasurespacesandSturm’sdistanceD.................112.4Calculusandheatflowinmetricmeasurespaces................132.4.1WeakuppergradientsandgradientflowofCheeger’senergy......132.4.2Convexfunctionals:gradientflows,entropy,andtheCD(K,∞)condition162.4.3Theheatfl
7、owasgradientflowinL2(X,m)andinP2(X).......172.5EVIformulationofgradientflows.........................18∗ScuolaNormaleSuperiore,Pisa.email:l.ambrosio@sns.it†NiceUniversity.email:nicola.gigli@unice.fr‡Universit`adiPavia.email:giuseppe.savare@unipv.it13StrongCD(K,∞)spaces204Keyfor
8、mulas264.1DerivativeofthesquaredWassersteindistance.................
