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1、Branchingprocessesinrandomenvironment–aviewoncriticalandsubcriticalcasesM.Birkner1,J.Geiger2,andG.Kersting31Weierstraß-Institutf¨urAngewandteAnalysisundStochastik,Mohrenstr.39,D-10117Berlin,Germanybirkner@wias-berlin.de2FachbereichMathematik,TechnischeUniversit¨atKaiserslautern,Postfach3049,D
2、-67653Kaiserslautern,Germanyjgeiger@mathematik.uni-kl.de3FachbereichMathematik,Universit¨atFrankfurt,Fach187,D-60054FrankfurtamMain,Germanykersting@math.uni-frankfurt.deSummary.Branchingprocessesexhibitaparticularlyrichlongtimebehaviourwhenevolvinginarandomenvironment.Thenthetransitionfromsub
3、criticalitytosupercriticalityproceedsinseveralsteps,andthereoccursasecond‘transition’inthesubcriticalphase(besidesthephase-transitionfrom(sub)criticalitytosu-percriticality).Herewepresentanddiscusslimitlawsforbranchingprocessesincriticalandsubcriticali.i.d.environment.Theresultsrelyonastimula
4、tinginter-playbetweenbranchingprocesstheoryandrandomwalktheory.Wealsoconsideraspatialversionofbranchingprocessesinrandomenvironmentforwhichwederiveextinctionandultimatesurvivalcriteria.MSC2000subjectclassifications.Primary60J80,Secondary60G50,60F17Keywordsandphrases.Branchingprocess,randomenvi
5、ronment,randomwalk,conditionedrandomwalk,Spitzer’scondition,functionallimittheorem1IntroductionBranchingprocessesinrandomenvironmentisoneofthetopics,whichwehavestudiedintheproject“VerzweigendePopulationen:GenealogischeB¨aumeundr¨aumlichesLangzeitverhalten”(GrantKe376/6)withintheDFG-Schwerpunk
6、t“InteragierendestochastischeSystemevonhoherKomplexit¨at”.Itisrepresentativeinthatacentralaspectofthewholeprojectwereproba-bilisticconstructionsofgenealogicaltreesandtheinterplaybetweenbranch-ingprocessesandrandomwalks.Theseconceptsturnouttobesignificantforbranchingprocessesinrandomenvironment
7、inaratherspecificway.Thisisduetothefactthatfromamethodicalpointofviewthesubjectofbranchingprocessesinrandomenvironmentissubstantiallyinfluencedbythetheoryof2Birkneretal.randomwalks.Manyoftheresultsrelyonastimulatinginterplaybetweenbranchingproc
