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1、TheCentralLimitTheoremforRandomWalksinMarkovianEnvironmentsDiploma-ThesisbyBenediktJahnelSupervisor:Prof.Dr.JochenBlathTechnischeUniversit•atBerlinInstitutf•urMathematikSeptember27,2010Contents1Introduction22DualityandItsApplications52.1TheVoterModelandDuality-aC
2、lassicResult...............52.2TheCompetingSpeciesModelandRandomWalksinRandomEnvironments...................103TheCentralLimitTheoremforRandomWalksinMarkovianEnvironments173.1TheRandomWalkinaMarkovianEnvironment...............173.2TheCentralLimitTheorem..........
3、................184GibbsMeasuresonShiftspaces194.1GibbsMeasuresinPhysics:TheIsingModel.................204.2ShiftspacesofFiniteTypeandFunctionSpaces...............224.3GibbsMeasuresonShiftspaces.........................264.4Ruelle'sPerron-FrobeniusTheorem........
4、..............304.5TheCentralLimitTheoremforGibbsMeasuresonShiftspaces.......365LocalEnvironmentandProofoftheCentralLimitTheorem405.1TheKey-Lemma................................415.2ProofoftheCentralLimitTheorem......................516MultisiteDependentsandInter
5、actingEnvironments546.1MultisiteDependentsoftheRandomWalk..................546.2TheRandomWalkinanInteractingEnvironment..............57Chapter1IntroductionThestudyofrandomwalksinrandomevolvingenvironmentshasattractedmuchattentionlately.Originally,initsone-dimensi
6、onalversion,thiskindofprocessesappearedasasimplemodelofDNAtranscription.ImagineanRNAstrandasbeingalong(longenoughtobeconsidered'in nite')sequenceoftheletters:A,C,G,orT.ADNAismadeoftwostrandseachofwhichisalsoasequenceoftheletters:A,C,G,orU.TomakeaRNAstrand,anenzym
7、egoesoveroneofthestrandsoftheDNAandcreatesanRNAchainbyputtingaTwhereA'sappear,anAwhereU'sappear,aGwhereC'sappear,andaCwhereG'sappear.ItturnsoutthatwhenreadingtheDNAstrand,theenzymedoesnotmovealonginalinearfashion!Itratherperformsarandomwalkalongthestrand.Imaginet
8、hattheDNAisbeingreadfromlefttoright.Then,atanymomentintime,whatwehaveisanRNAchaintotheleftoftheenzymeandnothingtotherightofit.Atthispoint,theenzymecanchoosetoe
