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1、StochasticAnalysisOnManifoldsEltonP.HsuInMemoryofMyBelovedMotherZhuPei-Ru(1926-1996)Qu¨altmichErinnerung,da?ichver¨ubet,SomancheTat,diedirdasHerzbetr¨ubet?Dassch¨oneHerz,dasmichsosehrgeliebet?............Dochdabistduentgegenmirgekommen,Undach!wasdaindeinemAuggeschwommen,Dasw
2、ardies¨u?e,langgesuchteLiebe.HeinrichHeine,AnMeineMutter.ContentsPrefacexiiiIntroduction1Chapter1.StochasticDi?erentialEquationsandDi?usions5§1.1.SDEoneuclideanspace6§1.2.SDEonmanifolds19§1.3.Di?usionprocesses24Chapter2.BasicStochasticDi?erentialGeometry35§2.1.Framebundleand
3、connection36§2.2.Tensor?elds41§2.3.Horizontalliftandstochasticdevelopment44§2.4.Stochasticlineintegrals51§2.5.Martingalesonmanifolds55§2.6.Martingalesonsubmanifolds60Chapter3.BrownianMotiononManifolds71§3.1.Laplace-Beltramioperator72§3.2.Brownianmotiononmanifolds78§3.3.Examp
4、lesofBrownianmotion82§3.4.Distancefunction88§3.5.Radialprocess92§3.6.Anexittimeestimate98Chapter4.BrownianMotionandHeatSemigroup101ixxContents§4.1.Heatkernelastransitiondensityfunction102§4.2.Stochasticcompleteness107§4.3.C0-propertyoftheheatsemigroup112§4.4.Recurrenceandtra
5、nsience117§4.5.Comparisonofheatkernels127Chapter5.Short-timeAsymptotics129§5.1.Short-timeasymptotics:nearpoints130§5.2.Varadhan’sasymptoticrelation133§5.3.Short-timeasymptotics:distantpoints137§5.4.Brownianbridge142§5.5.Derivativesofthelogarithmicheatkernel147Chapter6.Furthe
6、rApplications157§6.1.Dirichletproblematin?nity158§6.2.Constantupperbound166§6.3.Vanishingupperbound168§6.4.Radiallysymmetricmanifolds172§6.5.CouplingofBrownianmotion179§6.6.Couplingandindexform182§6.7.Eigenvalueestimates186Chapter7.BrownianMotionandIndexTheorems191§7.1.Weitz
7、enb¨ockformula192§7.2.Heatequationondi?erentialforms198§7.3.Gauss-Bonnet-Chernformula201§7.4.Cli?ordalgebraandspingroup208§7.5.SpinbundleandtheDiracoperator213§7.6.Atiyah-Singerindextheorem217§7.7.Brownianholonomy222Chapter8.AnalysisonPathSpaces229§8.1.Quasi-invarianceoftheW
8、ienermeasure230§8.2.Flatpathspace234§8.3.Gradientformulas246§8.4.Integratio
