Table Of ContentSpringer 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. 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)
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
