lectures on finite markov chains

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1、LECTURESONFINITEMARKOVCHAINSLaurentSALOFF-COSTEContentsIntroductionandbackgroundmaterial3041.1Introduction..............................3041.1.1MyownintroductiontofiniteMarkovchains.......3051.1.2Whocares?..........................3071.1.3Asimpleopenproblem...................3091.2Th

2、ePerron-FrobeniusTheorem...................3091.2.1TwoproofsofthePerron-Frobeniustheorem.......3101.2.2CommentsonthePerron-Frobeniustheorem.......3131.2.3Furtherremarksonstrongirreducibility..........3151.3Elementaryfunctionalanalysis...................3161.3.1Operatornorms..........

3、..............3161.3.2Hilbertspacetechniques...................3181.4NotationforfiniteMarkovchains..................3191.4.1Discretetimeversuscontinuoustime............3222Analytictools3262.1Nothingbutthespectralgap3262,1.1TheDirichletform......................3262.1.2Thespectralgap.

4、......................3272.1.3Chernoffboundsandcentrallimittheorems........3312.2Hypercontractivity..........................3322.2.1Thelog-Sobolevconstant..................3322.2.2Hypercontractivity,a,andergodicity............3332.2.3Sometoolsforboundingafrombelow...........3382.3Na

5、shinequalities...........................3442.3.1Nash'sargumentforfiniteMarkovchainsI........3452.3.2Nash'sargumentforfiniteMarkovchainsII........3472.3.3Nashinequalitiesandthelog-Sobolevconstant......3502.3.4AconversetoNash'sargument...............3512.3.5Nashinequalitiesandhigherei

6、genvalues..........3522.3.6NashandSobolevinequalities................3552.4Distances...............................3582.4.1Notationandinequalities..................3582.4.2Thecutoffphenomenonandrelatedquestions.......3613033Geometrictools3673.1Adaptededgesets.........................

7、.3683.2Poincar~inequality..........................3683.3Isoperimetry.............................3793.3.1Isoperimetryandspectralgap................3793.3.2IsoperimetryandNashinequalities.............3863.3.3Isoperimetryandthelog-Sobolevconstant.........3933.4Moderategrowth..........

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