The Parabolic Anderson Model: Random Walk in Random Potential

199 Pages·2016·2.48 MB·English

by  Wolfgang König (auth.)

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
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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
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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