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特征值,不等式和遍历理论(英文版)
Thisbooksurveys,inapopularway,themainprogressmadeinthefieldbyourgroup.Itconsistsoftenchaptersplustwoappendixes.Thefirstchapterisanoverviewofthesecondtotheeighthones.Mainly,westudyseveraldifferentinequalitiesordifferenttypesofconvergencebyusingthreemathematicaltools:aprobabilistictool,thecouplingmethods(Chapters2and3);ageneralizedCheeger'smethodoriginatinginRiemanniangeometry(Chapter4);andanapproachcomingfrompotentialtheoryandharmonicanalysis(Chapters6and7).Theexplicitcriteriafordifferenttypesofconvergenceandtheexplicitestimatesoftheconvergencerates(ortheoptimalconstantsintheinequalities)indimensiononearegiveninChapters5and6;somegeneralizationsaregiveninChapter7.Theproofsofadiagramofninetypesofergodicity(Theorem1.9)arepresentedinChapter8.
PrefaceAcknowledgmentsChapter1AnOverviewoftheBook1.1Introduction1.2Newvariationalformulaforthefirsteigenvalue1.3BasicinequalitiesandnewformsofCheeger'sconstants1.4AnewpictureofergodictheoryandexplicitcriteriaChapter2OptimalMarkovianCouplings2.1CouplingsandMarkoviancouplings2.2Optimalitywithrespecttodistances2.3Optimalitywithrespecttoclosedfunctions2.4ApplicationsofcouplingmethodsChapter3NewVariationalFormulasfortheFirstEigenvalue3.1Background3.2Partialproofinthediscretecase3.3Thethreestepsoftheproofinthegeometriccase3.4Twodifficulties3.5Thefinalstepoftheproofoftheformula3.6Commentsondifferentmethods3.7Proofinthediscretecase(continued)3.8ThefirstDirichleteigenvalueChapter4GeneralizedCheeger'sMethod4.1Cheeger'smethod4.2Ageneralization4.3Newresults4.4Splittingtechniqueandexistencecriterion4.5ProofofTheorem4.44.6LogarithmicSobolevinequality4.7Upperbounds4.8Nashinequality4.9Birth-deathprocessesChapter5TenExplicitCriteriainDimensionOne5.1Threetraditionaltypesofergodicity5.2Thefirst(nontrivial)eigenvalue(spectralgap)5.3Thefirsteigenvaluesandexponentiallyergodicrate5.4Explicitcriteria.5.5Exponentialergodicityforsinglebirthprocesses5.6StrongergodicityChapter6Poincare-TypeInequalitiesinDimensionOne6.1Introduction6.2OrdinaryPoincareinequalities6.3Extension:normedlinearspaces6.4Neumanncase:Orliczspaces6.5NashinequalityandSobolev-typeinequality6.6LogarithmicSobolevinequality6.7PartialproofsofTheorem6.1Chapter7FunctionalInequalities7.1Statementofresults7.2Sketchoftheproofs7.3ComparisonwithCheeger'smethod7.4Generalconvergencespeed7.5Twofunctionalinequalities7.6Algebraicconvergence7.7General(irreversible)caseChapter8ADiagramofNineTypesofErgodicity8.1Statementsofresults8.2Applicationsandcomments8.3ProofofTheorem1.9Chapter9Reaction-DiffusionProcesses9.1Themodels9.2Finite-dimensionalcase9.3Constructionoftheprocesses9.4Ergodicityandphasetransitions9.5HydrodynamiclimitsChapter10StochasticModelsofEconomicOptimization10.1Input-outputmethod10.2L.K.Hua'sfundamentaltheorem10.3Stochasticmodelwithoutconsumption10.4Stochasticmodelwithconsumption10.5ProofofTheorem10.4AppendixASomeElementaryLemmasAppendixBExamplesoftheIsingModelonTwotoFourSitesReferencesAuthorIndexSubjectIndex
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