Invariant Probabilities of Transition Functions PDF
Preview Invariant Probabilities of Transition Functions
Probability and Its Applications Radu Zaharopol Invariant Probabilities of Transition Functions Probability and Its Applications PublishedinassociationwiththeAppliedProbabilityTrust Editors:S. Asmussen,J. Gani,P. Jagers,T.G. Kurtz Probability and Its Applications TheProbabilityandItsApplicationsseriespublishesresearchmonographs,withtheexpos- itory quality to make them useful and accessible to advanced students, in probability and stochasticprocesses,withaparticularfocuson: – FoundationsofprobabilityincludingstochasticanalysisandMarkovandotherstochastic processes – Applicationsofprobabilityinanalysis – Pointprocesses,randomsets,andotherspatialmodels – Branchingprocessesandothermodelsofpopulationgrowth – Geneticsandotherstochasticmodelsinbiology – Informationtheoryandsignalprocessing – Communicationnetworks – Stochasticmodelsinoperationsresearch Forfurthervolumes: http://www.springer.com/series/1560 Radu Zaharopol Invariant Probabilities of Transition Functions 123 RaduZaharopol AnnArbor Michigan USA ISSN1431-7028 ProbabilityandItsApplications ISBN978-3-319-05722-4 ISBN978-3-319-05723-1(eBook) DOI10.1007/978-3-319-05723-1 SpringerChamHeidelbergNewYorkDordrechtLondon LibraryofCongressControlNumber:2014942551 MathematicsSubjectClassification(2010):47A35,37A30,37A17,60J25,37A50,37A10,28D10 (cid:2)c SpringerInternationalPublishingSwitzerland2014 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’slocation,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer. PermissionsforusemaybeobtainedthroughRightsLinkattheCopyrightClearanceCenter.Violations areliabletoprosecutionundertherespectiveCopyrightLaw. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. While the advice and information in this book are believed to be true and accurate at the date of publication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityfor anyerrorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,with respecttothematerialcontainedherein. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) DedicatedtoMarinaand Daniel, andtothememoryofmymother andmy father Acknowledgements It is a great pleasure to acknowledge the various kinds of support that I received fromthefollowingpeople:EduardEmel’yanovshowedinterestinbothmymono- graph [143] and this book by making a series of useful comments including the improvementin the Lasota-Yorke lemma that appears in Theorem 1.2.3; Thomas Hempflinghasshowninterestinthemonographandhasofferedmeexpertguidance throughoutthepublicationprocess;MariusIosifescuhasalwaysbeenavailablefor ème extremelyusefuladvice;healsoinvitedmetothe9 ColloqueFranco-Roumain deMath.Appl.,28Août-2Septembre2008,Bras¸ov,Romania,togiveatalkonsome ofthetopicsdiscussedhere;MarinaReizakisandherteamatSpringerhaveexpertly prepared the book for publication; Daniel W. Stroock brought to my attention informationconcerningergodicmeasuresthatappearin hisbooks[119] and[29]; Tomasz Szarek sent me several of his works that I have used here; Daniël Worm showedgreatinterestintheresultsontheKBBY decompositionthatIobtainedin this bookand elsewhere (as I pointout several times, he has extendedthe KBBY decomposition to Polish spaces and has obtained various other very interesting resultsrelated to the decomposition);andlastly Marina Zaharopolhas offeredme veryusefuladvice. vii Contents 1 PreliminariesonTransitionProbabilities................................. 1 1.1 BasicDefinitionsandResults.......................................... 2 1.2 InvariantProbabilities.................................................. 20 1.3 TheErgodicDecompositionofKryloff,Bogoliouboff, BeboutoffandYosida .................................................. 25 1.4 FellerTransitionProbabilities ......................................... 36 1.4.1 SupportsofElementaryandErgodicInvariant Measures, Minimality,Unique Ergodicity, andGenericPoints............................................. 36 1.4.2 Equicontinuity................................................. 45 2 PreliminariesonTransitionFunctionsandTheirInvariant Probabilities .................................................................. 57 2.1 TransitionFunctions.................................................... 57 2.2 Examples................................................................ 68 2.2.1 TransitionFunctionsDefinedbyOne-Parameter SemigroupsorGroupsofMeasurableFunctions: GeneralConsiderations........................................ 69 2.2.2 Transition Functions Defined by Specific One-Parameter Semigroups or Groups ofMeasurableFunctions...................................... 72 2.2.3 TransitionFunctionsDefinedbyOne-Parameter ConvolutionSemigroupsofProbabilityMeasures........... 83 2.3 InvariantProbabilityMeasures ........................................ 89 3 PreliminariesonVectorIntegralsandAlmostEverywhere Convergence.................................................................. 97 3.1 TheBochnerandtheDunford-SchwartzIntegrals.................... 97 3.1.1 TheBochnerIntegral.......................................... 98 3.1.2 CompleteMeasureSpaces .................................... 102 3.1.3 TheDunford-SchwartzIntegral............................... 105 ix
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