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| 內容簡介： |
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Potentialtheoryandcertainaspectsofprobabilitytheoryareintimatelyrelated，perhapsmostobviouslyinthatthetransitionfunctiondeterminingaMarkovprocesscanbeusedtodefinetheGreenfunctionofapotentialtheory.Thusitispossibletodefineanddevelopmanypotentialtheoreticconceptsprobabilistically，aprocedurepotentialtheoristsobservewithjaun-dicedeyesinviewofthefactthatnowasinthepasttheirsubjectprovidesthemotivationformuchofMarkovprocesstheory.Howeverthatmaybeitisclearthatcertainconceptsinpotentialtheorycorrespondcloselytoconceptsinprobabilitytheory，specificallytoconceptsinmartingaletheory.Forexample，superharmonicfunctionscorrespondtosupermartingales.Morespecifically：theFatoutypeboundarylimittheoremsinpotentialtheorycorrespondtosupermartingaleconvergencetheorems;thelimitpropertiesofmonotonesequencesofsuperharmonicfunctionscorrespondsurprisinglycloselytolimitpropertiesofmonotonesequencesofsuper-martingales;certainpositivesuperharmonicfunctions[supermartingales]arecalled"potentials，"haveassociatedmeasuresintheirrespectivetheoriesandaresubjecttodominationprinciples（inequalities）invomngthesupportsofthosemeasures;ineachtheorythereisareductionoperationwhosepropertiesarethesameinthetwotheoriesandthesereductionsinducesweeping（balayage）ofthemeasuresassociatedwithpotentials，and，soon.
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| 目錄：
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Introduction
NotationandConventions
Part1
ClassicalandParabolicPotentialTheory
ChapterI
IntroductiontotheMathematicalBackgroundofClassicalPotentialTheory
1.TheContextofGreen''sIdentity
2.FunctionAverages
3.HarmonicFunctions
4.Maximum-MinimumTheoremforHarmonicFunctions
5.TheFundamentalKernelforRNandItsPotentials
6.GaussIntegralTheorem
7.TheSmoothnessofPotentials;ThePoissonEquation
8.HarmonicMeasureandtheRieszDecomposition
ChapterII
BasicPropertiesofHarmonic,Subharmonic,andSuperharmonicFunctions
1.TheGreenFunctionofaBall;ThePoissonIntegral
2.Hamack''sInequality
3.ConvergenceofDirectedSetsofHarmonicFunctions
4.Harmonic,Subharmonic,andSuperharmorucFunctions
5.MinimumTheoremforSuperharmonicFunctions
6.ApplicationoftheOperationTB
7.CharacterizationofSuperharmonicFunctionsinTermsofHarmonicFunctions
8.DifferentiableSuperharmonicFunctions
9.ApplicationofJensen''sInequality
10.SuperharmonicFunaionsonanAnnulus
II.Examples
12.TheKelvinTransformation
13.GreenianSets
14.TheL1（uB_）andD（uB_）ClassesofHarmonicFunctionsonaBallB;The
Riesz-HerglotzTheorem
15.TheFatouBoundaryLimitTheorem
16.MinimalHarmonicFunctions
ChapterIII
InfimaofFamiliesofSuperharmonicFunctidns
1.LeastSuperharmonicMajorant（LM）andGreatestSubharmonicMinorant（GM）
2.GeneralizationofTheoremI
3.FundamentalConvergenceTheorem（PreliminaryVersion）
4.TheReductionOperation
5.ReductionProperties
6.ASmallnessPropertyofReductionsonCompactSets
7.TheNatural（Pointwise）OrderDecompositionforPositiveSuperharmonk
Functions
Chapter1V
PotentialsonSpecialOpenSets
1.SpecialOpenSets,andPotentialsonThem
2.Examples
3.AFundamentalSmallnessPropertyofPotentials
4.IncreasingSequencesofPotentials
5.SmoothingofaPotential
6.UniquenessoftheMeasureDeterminingaPotential
7.RieszMeasureAssociatedwithaSuperharmonicFunction
8.RieszDecompositionTheorem
9.CounterpartforSuperharmonicFunctionsonR2oftheRiesz
Decomposition
10.AnApproximationTheorem
ChapterV
PolarSetsandTheirApplications
1.Definition
2.SuperharmonicFunctionsAssociatedwithaPolarSet
3.CountableUnionsofPolarSets
4.PropertiesofPolarSets
5.ExtensionofaSuperharmonicFunction
6.GreenianSetsinIR2astheComplementsofNonpolarSets
7.SuperharmonicFunctionMinimumTheorem（ExtensionofTheoremI1.5）
8.Evans-VasilescoTheorem
9.ApproximationofaPotentialbyContinuousPotentials
10.TheDominationPrinciple
I1.TheInfinitySetofaPotentialandtheRieszMeasure
……
Part2
ProbabilisticCountrepartofPart1
Part3
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