Pathways in Mathematics Wolfgang König The Parabolic Anderson Model Random Walk in Random Potential Pathways in Mathematics Serieseditors T.Hibi Toyonaka,Japan W.KoRnig Berlin,Germany J.Zimmer Bath,UnitedKingdom Each“PathwaysinMathematics”bookoffersaroadmaptoacurrentlywelldevel- oping mathematical research field and is a first-hand information and inspiration for further study, aimed both at students and researchers. It is written in an educationalstyle, i.e., in a way that is accessible for advancedundergraduateand graduate students. It also serves as an introduction to and survey of the field for researcherswhowanttobequicklyinformedaboutthestateoftheart.Thepointof departureis typically a bachelor/masterslevelbackground,from which the reader is expeditiouslyguidedto the frontiers.Thisis achievedby focusingon ideasand conceptsunderlyingthedevelopmentofthesubjectwhilekeepingtechnicalitiesto aminimum.Eachvolumecontainsanextensiveannotatedbibliographyaswellasa discussionofopenproblemsandfutureresearchdirectionsasrecommendationsfor startingnewprojects Moreinformationaboutthisseriesathttp://www.springer.com/series/15133 Wolfgang KoRnig The Parabolic Anderson Model Random Walk in Random Potential WolfgangKoRnig Weierstraß-Institut fuRrAngewandteAnalysisundStochastik Berlin,Germany InstituteforMathematics TUBerlin Berlin,Germany ISSN2367-3451 ISSN2367-346X (electronic) PathwaysinMathematics ISBN978-3-319-33595-7 ISBN978-3-319-33596-4 (eBook) DOI10.1007/978-3-319-33596-4 LibraryofCongressControlNumber:2016940195 MathematicsSubjectClassification(2010):60-02,60J55,60F10,60K35,60K37,82B44,60J27,60J65, 60J80,60K40,80A20,82D30 ©SpringerInternationalPublishingSwitzerland2016 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper ThisBirkhäuserimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAGSwitzerland Preface ThisisasurveyoftheparabolicAndersonmodel(PAM),theCauchyproblemfor theheatequationwithrandompotential.Thismodelandmanyvariantsandrelated models are studied for decades by many authors from various points of view and with various intentions. The PAM has rich and deep connections with questions on random motions in random potential, trapping of random paths, branching processes in random medium,spectra of random operators,Anderson localisation andmore.Wearemainlyinterestedinthelong-timebehaviourofthesolutionofthe PAM, which shows interesting behaviours like intermittency, mass concentration, ageing,Poissonprocessconvergenceofeigenvaluesandeigenfunctionlocalisation centres and more. Its mathematical investigations require combinations of tools from various parts of probability and analysis, like spectral theory of random operators,largedeviationsorextremevaluestatistics. The research on the PAM and its variants has a high intensity since 1990 and continues to have. I felt that a survey text should be very useful, at a point at which most of the understanding of the basic model has been rigorously derived,andavarietyofvariantsandadditionalfeatures,likerandomenvironments and time-dependent potentials, and a number of related questions, like critically rescaledpotentials,transitionfromconcentratedtohomogenisedbehaviour,spatial branchingprocessesinrandomenvironmentandAndersonlocalisation,receivean increasinginterest. Thefocusofthisbookischaracterisedbytheintersectionofanumberoffeatures, whosemostimportantonesarethefollowing: (cid:129) The solution to the PAM admits explicit formulas (Feynman-Kac formula and eigenvalueexpansion), (cid:129) Its large-time behaviour can be investigated with the help of large-deviations theory, (cid:129) Thearisingvariationalformulasadmitadeeperstudyand (cid:129) There are deep connections with the spectral theory for a prominent random Schrödingeroperator,theAndersonoperator. v vi Preface All these aspects are more or less closely connectedwith the main propertyof thePAM,theintermittency,aconcentrationpropertyofthemainpartofthesolution insmallislands.Intermittencyisoneoftheleadingideasinthisbookandisalmost ubiquitous. For thisreason,suchimportanttopicsasdirectedpolymersin randomenviron- ment, PAM with drift and PAM with certain types of time-dependent potentials do not receive the space that they otherwise should have; they are just outside of the scopeof this book.Actually,thistextendsat a pointwhere itis getting really interesting, as the stochastic heat equation and the KPZ equation come into play (however, the account on PAM with time-dependent potential given in Chap.8 is quitecomprehensiveinasense). My intention was to provide a concise, but fairly complete, survey of the heuristic understanding of the PAM on one hand and of the state of the art of its mathematical treatment on the other hand. The goal is to quickly guide the reader to a good understanding of the essentials. I tried to give illuminating and nontechnicalexplanations,andIsometimesdecidedtoprovidesimplifiedversions of the main theorems, many of which are embedded in the running text. Where somebackgroundisneeded,theunderlyingtheoryissummarisedinamostcompact way,justatalengththatisnecessarytounderstandthefundamentals’allimportant connections. Therearealotofprecisereferencesgiventothefirst-handliterature,andmany side remarks hint at deeper results and open problems that emanate from the material. I also foundit usefulto isolate the essentials of proofmethodsfrom the originalpapers,iftimehasshownthattheyareusefulandcanbeadaptedtoseveral situations;notonlyChap.4isdevotedtothisbutalsoanumberofremarksthatare scatteredoverthetext. Originally, the text was meant to address experienced researchers, but in the course of writing, I felt that it would be desirable to attract also newcomers and young researchers to this field; therefore I added also explanations of terms, concepts, jargons and methods that are known to the community of the PAM and neighbouring fields. I hope that I found a style that is understandable and encouragingforallmathematicallyinterestedpeoplefromadvancedundergraduates onwards. In an appendix,I enumeratedsome openresearch directionsthatlie within the scope of this book or at its outer boundary. Certainly their choice relies on my personal taste, but I think that they each give rise to exciting new research, and hopefullytheyattractnewpeopletothefield. LetmeexpressmysincerethankstomyformerPhDstudentTilmanWolff,who helped me at an early stage in collecting some material, and to my current PhD studentFranziskaFlegel,whoproducedinstrumentalillustrations. Berlin,Germany WolfgangKönig March2016 Preface vii SomeRemarks on Notation Fordescribingasymptoticassertions,Iwillusethesymbol‘(cid:2)’todenoteasymptotic equivalence, i.e. that the quotient of the two sides converges to one, ‘(cid:3)’ for asymptotic comparability, i.e. that the ratio stays bounded and bounded away from zero, and (cid:4) and (cid:5) for asymptotic negligibility, i.e. that the ratio vanishes, respectively, and diverges to 1. Furthermore, I use the Landau symbols o.a / n for quantities whose ratio with a vanishes asymptotically and O.a / for positive n n quantities whose ratio with a stays bounded as n ! 1. When I do not want n to specify the sense of the asymptotic approximation,then I use the symbol ‘(cid:6)’, but often I indicate in words what I would like to mean by that. For expressing t!1 convergence, I often use the arrow !, or (cid:7)!, if I want to indicate the limiting parameter.Convergenceofrandomvariablesindistributionorweakconvergenceof measuresiswrittenusingH). ForintegralsandRinnerproductsbothonRdandonZd,IusethebrackRetsh(cid:8);(cid:8)i,e.g. h(cid:2);fi D hf;(cid:2)i D f d(cid:2)forfunctionsf andmeasures(cid:2)orhf;gRi D f.x/g.x/dx fortwofunctionsfPandg,whichIsometimesalsoabbreviateas fgifthedomain isRdandhf;giD z2Zdf.z/g.z/ifitisZd.Forp2Œ1;1/andB(cid:9)PZd,wedenote by`p.B/thevectorspaceoffunctionsfWB ! R suchthatkfkp D jf.z/jp is p z2B finiteandkfk isthenormoff. p For the parameter of some frequently used functions or processes, I use both the index notation and the bracket notation, depending on how much space the parameter requires. Hence a scale function ˛.t/ may be also written ˛, and the t simplerandomwalkattimet isdenotedbothbyX andbyX.t/.Likewise,Iwrite t both and AfortheindicatorfunctiononaneventA. A (cid:0) (cid:0) Contents 1 Background,ModelandQuestions ........................................ 1 1.1 IntroductionandScopeofThisText.................................. 1 1.2 TheParabolicAndersonModel....................................... 4 1.3 MainQuestions........................................................ 8 1.4 Intermittency........................................................... 9 1.5 ExamplesofPotentials................................................ 11 1.5.1 DiscreteSpace................................................ 12 1.5.2 ContinuousSpace............................................ 14 2 ToolsandConcepts .......................................................... 19 2.1 ProbabilisticAspects .................................................. 19 2.1.1 BranchingProcesswithRandomBranchingRates......... 19 2.1.2 Feynman-KacFormula ...................................... 21 2.1.3 Finite-SpaceFeynman-KacFormulas....................... 25 2.1.4 LocalTimesandMoments .................................. 26 2.1.5 QuenchedandAnnealedTransformedPathMeasures..... 27 2.2 FunctionalAnalyticAspects .......................................... 30 2.2.1 EigenvalueExpansion ....................................... 30 2.2.2 RelationBetweenEigenvalueExpansionandthePAM.... 31 2.2.3 AndersonLocalisation....................................... 33 2.2.4 IntermittencyandAndersonLocalisation................... 34 2.2.5 IntegratedDensityofStates ................................. 34 2.2.6 LifshitzTails ................................................. 36 2.3 FirstHeuristicObservations........................................... 36 2.3.1 TheTotalMassasanExponentialMoment ................ 37 2.3.2 MomentAsymptoticsVersusAlmostSureAsymptotics... 37 2.3.3 MassConcentration.......................................... 40 2.3.4 Time-EvolutionoftheMassFlow........................... 40 ix