Kähler流形上的布朗运动的开题报告IntroductionThestudyofBrownianmotiononRiemannianmanifoldshasbeenanactiveareaofresearchinprobabilityandgeometryinrecentyears.Thetheoryofstochasticprocessesonmanifoldshasawiderangeofapplicationsinmathematicalfinance,physics,andbiology.Inthisreport,wewilldiscussthetopicofBrownianmotiononKählermanifolds,whicharecomplexmanifoldsendowedwithacompatibleRiemannianmetricandasymplecticform.WewillintroducesomebasicconceptsandresultsfromprobabilitytheoryanddifferentialgeometrythatarenecessarytounderstandthetheoryofBrownianmotiononKählermanifolds.WewillalsopresentsomerecentdevelopmentsinthestudyofKählerBrownianmotion.BrownianmotionBrownianmotionisastochasticprocessthatmodelstherandommotionofparticlesinafluid.ItwasfirstdiscoveredbyRobertBrownin1827whenheobservedtherandommotionofpollenparticlesinwater.ThemathematicaltheoryofBrownianmotionwasdevelopedbyAlbertEinsteinin1905.Accordingtothistheory,Brownianmotionisacontinuous-timestochasticprocessthatsatisfiesthefollowingproperties:1.Theprocessstartsatsomeinitialpoint.2.Theprocesshasindependentandidenticallydistributedincrements.3.TheincrementsareGaussiandistributedwithmeanzeroandvarianceproportionaltothetimeinterval.4.Theprocesshascontinuouspathsalmostsurely.Brownianmotionhasanumberofimportantpropertiesthatmakeitusefulformodelingstochasticprocessesinphysics,finance,andotherfields.Forexample,itisthecontinuous-timelimitofarandomwalk,itsatisfiesthecentrallimittheorem,andithasstationaryandMarkovianincrements.StochasticprocessesonmanifoldsStochasticprocessesonmanifoldsareanaturalgeneralizationofBrownianmotioninEuclideanspace.Insteadofconsideringrandommotioninafixedspace,weconsiderrandommotiononamanifold,whichisageometricspacewithmorestructurethanjustasetofpoints.ManifoldscanbeequippedwithaRiemannianmetric,whichallowsustodefinenotionsofdistanceandangle,andavolumeform,whichallowsustointegratefunctions.StochasticprocessesonmanifoldsareusuallydefinedintermsofBrownianmotionwithrespecttoaconnectionoradiffusionprocess