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1、Fromconstructivefieldtheorytofractionalstochasticcalculus.(I)TheL´evyareaoffractional11BrownianmotionwithHurstindexα∈(,)84JacquesMagnenandJ´er´emieUnterbergerLetB=(B1(t),...,Bd(t))bead-dimensionalfractionalBrownianmotionwithHurstindexα<1/4.DefiningproperlyiteratedintegralsofBi
2、sadifficulttaskbecauseofthelowH¨olderregularityindexofitspaths.YetroughpaththeoryshowsitisthekeytotheconstructionofastochasticcalculuswithrespecttoB,ortosolvingdifferentialequationsdrivenbyB.Weshowinthispaperhowtoobtainsecond-orderiteratedintegralsasthelimitwhentheultra-violetc
3、ut-offgoestoinfinityofiteratedintegralsofweaklyinteractingfieldsdefinedusingthetoolsofconstructivefieldtheory,inparticular,clusterexpansionandrenormalization.TheconstructionextendstoalargeclassofGaussianfieldswiththesameshort-distancebehaviour,calledmulti-scaleGaussianfields.Previo
4、usconstructions[36,35]wereofalgebraicnatureanddidnotprovidesuchalimitingprocedure.Keywords:fractionalBrownianmotion,stochasticintegrals,roughpaths,constructivefieldtheory,Feynmandiagrams,renormalization,clusterex-pansion.MathematicsSubjectClassification(2000):60F05,60G15,60G18
5、,arXiv:1006.1255v1[math.PR]7Jun201060H05,81T08,81T18.Contents0Introduction21Statementoftheproblemandheuristics101.1AFourieranalysisoftheL´evyarea...............111.2Definitionoftheinteraction...................141.3TowardsaheuristicexpressionfortheL´evyarea.......1512Multisca
6、leGaussianfields192.1Scaledecompositions.......................192.2MultiscaleGaussianfieldsinonedimension..........233Clusterexpansions:anoutline263.1ThegeneralBrydges-Kennedyformula.............273.2Singlescaleclusterexpansion..................293.3Multi-scaleclusterexpansio
7、n..................303.4Mayerexpansion.........................394Power-countingandrenormalization474.1Power-countinganddiverginggraphs..............474.2Dominationproblemandboundarytermintheinteraction.535Definitionofthemodel556Bounds586.1Gaussianbounds......................
8、...586.1.1Wick’sformulaandapplications............586.1.2Gaussianboundsforc