Advances in Superprocesses and Nonlinear PDEs PDF
Preview Advances in Superprocesses and Nonlinear PDEs
Springer Proceedings in Mathematics & Statistics Volume 38 Forfurthervolumes: http://www.springer.com/series/10533 Springer Proceedings in Mathematics & Statistics Thisbookseriesfeaturesvolumescomposedofselectcontributionsfromworkshops and conferences in all areas of current research in mathematics and statistics, includingORandoptimization.Inadditiontoanoverallevaluationoftheinterest, scientific quality, and timeliness of each proposal at the hands of the publisher, individual contributions are all refereed to the high quality standards of leading journals in the field. Thus, this series provides the research community with well-edited, authoritative reports on developments in the most exciting areas of mathematicalandstatisticalresearchtoday. Janos Englander • Brian Rider Editors Advances in Superprocesses and Nonlinear PDEs 123 Editors JanosEnglander BrianRider DepartmentofMathematics DepartmentofMathematics UniversityofColorado UniversityofColorado Boulder,Colorado,USA Boulder,Colorado,USA ISSN2194-1009 ISSN2194-1017(electronic) ISBN978-1-4614-6239-2 ISBN978-1-4614-6240-8(eBook) DOI10.1007/978-1-4614-6240-8 SpringerNewYorkHeidelbergDordrechtLondon LibraryofCongressControlNumber:2013931995 MathematicsSubjectClassification(2010):60B15,60H25,60J80,60F05,60J85,60J68,60F25,60K35, 92B05,60J25,35J60 ©SpringerScience+BusinessMedia,LLC2013 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. 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Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Preface This bookgrew outof the conferenceAdvancesin Superprocessesand Nonlinear PDEs held between June 24 and June 26, 2010 at the University of Colorado Boulder.Themainspeakersatthemeetingswere Zhen-QingChen(U.Washington) DonaldDawson(Carleton,Ottawa) EugeneB.Dynkin(Cornell) SteveN.Evans(UCBerkeley) PatrickJ.Fitzsimmons(UCSanDiego) KlausFleischmann(WeierstrassInstitute,Berlin) SimonC.Harris(Bath,UK) AndreasE.Kyprianou(Bath,UK) RinaldoSchinazi(UniversityofColorado,ColoradoSprings) DanStroock(MIT) One of the motivationsof the conference was recent advancesin the theory of superprocesses. The last 10 years have witnessed intensive research on superpro- cesses,withimportantprogressmadeonsuperprocessesoverflows,backbonecon- structions, superprocesses in random media, interacting and branching-coalescing superprocesses,superprocesseswith immigration,scalinglimit theoremsandself- intersectionlocaltimes. The meeting was also dedicated to the 60th birthday of our colleague, Sergei Kuznetsov. Professor Kuznetsov is one of the top experts on measure-valued branching processes (or superprocesses) and their connection to nonlinear partial differen- tial operators. His research interests range from stochastic processes and partial differentialequationsto mathematicalstatistics, timeseriesanalysisandstatistical software.Hehaspublishedover90papersininternationaljournals. Here we mention just two of his remarkable results. In 1980, Kuznetsov provedthat every Markov processin a Borel state space (i.e., a measurable space v vi Preface isomorphictoaBorelsubsetinaPolishspace)hasatransitionfunction.Hismost well-known contribution to probability theory though is the so-called Kuznetsov- measure. Dualityisaveryimportantnotioninprobability:thestationaryMarkovprocesses withtransitionfunctionsp, pˆaredualswithrespecttoagivenσ-finitemeasuremif m(dx)p(x,dy)=m(dy)pˆ (y,dx). t t A closely related fact is that each Markov process can be considered in two time directions (this is the way Kolmogorov’s forward and backward equations are deduced).Infact,Dynkinsuggestedthatthefunctionspandpˆmaybeinterpretedas forwardandbackwardtransitionfunctionsofasingleMarkovprocesswithrandom birth time αand death time β. This approach was applied also to nonstationary transition functions p(s,x;t,dy), pˆ(s,x;t,dy) and measures m depending on the time interval (s,t). The process {Xt}t∈(α,β) can be given by its two-dimensional distributionsas m (dx,dy)=P(α<s,X ∈dx,X ∈ dy,t<β). st s t The problem, however, is that the family {m } should satisfy a normalization st condition that guarantees that P is a probability measure; in the stationary case, this condition holds if m is a probability measure, invariant for both processes. Since the definition of duality requires only σ-finiteness of m, this assumption is too restrictive. This problem was solved by Kuznetsov in 1973, who managed to get rid of the condition: the measure P (called the Kuznetsov measure) and the corresponding m are both just σ-finite. In the theory of Markov processes, considering a process with random birth and death times with the help of the Kuznetsov-measurehasproventobeaveryusefulalternativetoworkingwithdual processes. SergeiobtainedhisPh.D.in1976intheformerSovietUnionundertheguidance ofEugeneDynkin,whocontributedthefirstchapterinthisvolume,andeversince thattimeSergeihasbeenthemainresearchcollaboratorofhisformeradvisor.This extremelyfruitfulcollaborationresultedin17papersso far,inpremierjournalsin probabilityandfunctionalanalysis.SergeijoinedtheDepartmentofMathematicsat theUniversityofColoradoatBoulderin1998. Finally,we are gratefulto the NationalScienceFoundationfortheirsupportof the meeting and to Springerforinviting this proceedingsvolumeinto their series. Weofferourapologiesfortheunusuallylongeditingtime. Boulder,Colorado,USA JanosEnglander Boulder,Colorado,USA BrianRider Contents Markov Processes and Their Applications to Partial DifferentialEquations:Kuznetsov’sContributions.......................... 1 E.B.Dynkin StochasticEquationsonProjectiveSystemsofGroups...................... 11 StevenN.EvansandTatyanaGordeeva ModelingCompetitionBetweenTwoInfluenzaStrains ..................... 35 RinaldoB.Schinazi Asymptotic Results for Near CriticalBienayme´–Galton– WatsonandCatalyst-ReactantBranchingProcesses ........................ 41 AmarjitBudhirajaandDominikReinhold SomePathLarge-DeviationResults foraBranchingDiffusion....................................................... 61 RobertHardyandSimonC.Harris Longtime Behavior for Mutually Catalytic Branching withNegativeCorrelations...................................................... 93 LeifDo¨ringandLeonidMytnik Super-BrownianMotion: Lp-ConvergenceofMartingales ThroughthePathwiseSpineDecomposition.................................. 113 A.E.KyprianouandA.Murillo-Salas Index............................................................................... 123 vii ProfessorSergeiKuznetsov
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