Stochastic Partial Differential Equations

517 Pages·2017·5.717 MB·English

by  Sergey V. Lototsky, Boris L. Rozovsky (auth.)

Universitext
Sergey V. Lototsky
Boris L. Rozovsky
Stochastic
Partial
Differential
Equations
Universitext
Universitext
Serieseditors
SheldonAxler
SanFranciscoStateUniversity
CarlesCasacuberta
UniversitatdeBarcelona
AngusMacIntyre
QueenMary,UniversityofLondon
KennethRibet
UniversityofCalifornia,Berkeley
ClaudeSabbah
Écolepolytechnique,CNRS,UniversitéParis-Saclay,Palaiseau
EndreSüli
UniversityofOxford
WojborA.Woyczyn´ski
CaseWesternReserveUniversity
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Sergey V. Lototsky (cid:129) Boris L. Rozovsky
Stochastic Partial
Differential Equations
SergeyV.Lototsky BorisL.Rozovsky
DepartmentofMathematics DivisionofAppliedMathematics
UniversityofSouthernCalifornia BrownUniversity
LosAngeles Providence
California,USA RhodeIsland,USA
ISSN0172-5939 ISSN2191-6675 (electronic)
Universitext
ISBN978-3-319-58645-8 ISBN978-3-319-58647-2 (eBook)
DOI10.1007/978-3-319-58647-2
LibraryofCongressControlNumber:2017942781
MathematicsSubjectClassification(2010):60H15,35R60
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To ourfamilies
Preface
There are textbooks and there are research monographs. Some believe that a
researchmonographissupposedtoboreratherthanbatter.Thefollowingquotation
is attributed to Voltaire: “Le secret d’etre ennuyeuxs’est de tout dire”—the art of
beingaboreconsistsintellingeverything.AslightvariationisattributedtoaFrench
proverb,which,in Englishtranslation,saysthatthe artof beinga boreconsistsin
tellingeverydetail.Thereadermightwanttostopforamomentandthinkaboutthe
differencebetweeneverythingandevery detail, aswellas the two mainmeanings
oftheverbtobore.
OurobjectiveinthistextbookistodiscussasmuchoftheSPDE-relatedmaterial
as possible withoutgoing too much into the details: learning is not very effective
whenitisboring,andso,unlikearesearchmonograph,webelievethatatextbook
should batter rather than bore. While the objective of a research monograph is
to document the results, the objective of a graduate-level textbook is to prepare
the reader for independent research. This preparation means providing the tools
necessary not only to read and understand the current paper and monographs on
thesubjectbutalsotocreatenewmathematics.Inotherwords,ourhopeisthatthis
book can help the reader not only to understand the proofs of existing theorems
aboutSPDEsbutalsotoformulateandprovenewtheorems.
Most of the book is about linear equations; a separate volume dealing with
nonlinearequationisplannedforthefuture.
Our intention was to make the presentation as self-contained as possible; a
knowledgablereader coming across a definition of the standard Brownian motion
orthestatementofParseval’sidentityshouldnottakeitpersonally.Wealsotriedto
makethebookreadablenotjustfromthebeginningbutalso froma randomplace
(for possible use as a reference). As a result, there are inevitable repetitions and
redundancies, as well as page references in addition to the number of the cited
formula or theorem. A reader with a good memory reading the book from the
beginningshouldnottakethispersonallyeither.Someredundancyisalsobuiltinto
thesubjectindex.
vii
viii Preface
A subtle message is sent to the reader at the end of many proofs: a completed
proofissomewhatmorecompletethanaconcludedone,andareadermightwantto
spendsomeextratimethinkingabouttheconcludedproofs.
Problemsand exercisesfacilitate transitionof a subjectfromthe research level
to the level of a graduate or even undergraduatecourse. As the subject of SPDEs
is currently making this transition, the book attempts to present enough exercise
materialtofillenoughpotentialexamsandhomeworkassignments.Thedistinction
betweena problemandanexerciseisnotalwaysclear.Onepossibledistinctionis
providedbythefollowingexample:tosolvethequadraticequationax2CbxCcD0
isaproblem;tosolvex2CxC1 D 0isanexercise.Inthisbook,theapproachto
problemsandexercisesissomewhatdifferent.Exercisesappearthroughoutthetext
andareusuallydirectlyconnectedtothe materialdiscussed attheparticularplace
in the text. The question is usually to verifysomething,so that the readeralready
knows the answer and, if pressed for time, can move on. Accordingly, there are
no solutions for the exercises (but there are often hints on how to proceed). We
alsorealizethatexercisescanpresentanextrachallengeforthosebraveenoughto
actuallyteachaclassusingthisbook.
Theletterlabelofeachexerciseissupposedtoconveythreepiecesofinformation
atonce:thelevelofdifficulty,thedegreeofrelevance,andtheambitionofthereader.
Inotherwords,exerciseslabeledwitha“C”aretheeasiest,mostrelevanttothecore
material,andareintendedforallreaders;infact,manyofthe“C” exercisescould
havebeenlabeledasexamplesor(easy)lemmas,andtheideaisthatadifferentlabel
mightforcethereadertothinkabitmoreaboutthecorrespondingresult.Exercises
labeled with an “A” are the most difficult, least relevant, and are intended for the
mostambitiousreaders.Exerciseslabeledwitha“B”fallsomewhereinbetweenin
allthreecategories.Ofcourse,thegradationrepresentsthesubjectiveviewsofthe
authors.
Problemsarecollectedattheendofeachchapterandusuallypointtothetopics
that were not discussed in the main text. While most problems have at least an
outlineofthesolutionattheendofthebook,
(cid:129) Some of the solutions are not at the level that would earn the reader a perfect
scoreonahomeworkassignment,
(cid:129) Even if the solution looks complete, the reader should not treat it as a final
judgementbutratherasacallforfurtherthinkingandinvestigation.
Andevenifnotattemptingtosolveanyproblems,thereaderisencouragedtolook
through both the problems and solutions, as some of the potentially interesting
informationishiddenthere.Whenaproblem(or,forthatmatter,anexercise)asks
to showsomething,the solutionassumesa derivationofthe result.An alternative,
whichisnotnecessarilyaneasy(orefficient)wayout,1istoverifythattheresultis
truebytakingitforgrantedandsomehowconfirmingthatitmakessense.
1Forexample,wouldthereaderratherpluginthefunctiony.t/Detcos.t/intoy00(cid:2)2y0C2yD0
toverifythatitsatisfiestheequationorsolvetheequationusingthegeneraltheoryandrecognize
thefunctionasaparticularsolution?
Preface ix
Asfarasteachingaclass,bothauthorsdidsoonseveraloccasions,moreorless
followingtheorderofthematerialinthebook.Itisindeedpossibletocovermost
of the materialin about40h of lectures, as long as nottoo muchtime is spent on
thegeneraldiscussionofstochasticanalysisininfinitedimensions.Amoredetailed
description,bothofthestructureofthisbookandofthecorrespondingclass,isin
Sect.1.2.7.
To summarize, the objective of this book is not to present all the result about
stochastic partial differentialequations, but instead to provide the reader with the
necessary tools to understand those results (by reading other sources) and maybe
eventodiscoverafewnewones.
The authors gratefully acknowledge the support of several grants during the
periodthebookwaswritten:ARO(DAAD19-02-1-0374)andNSF(DMS-0237724
(CAREER) and DMS-0803378) [SL]; AFOSR (5-21024 (inter), FA9550-09-1-
0613) ARO (DAAD19-02-1-0374, W911NF-07-1-0044, W911NF-13-1-0012,
W911-16-1-0103), NSF (DMS 0604863, DMS 1148284), ONR (N0014-03-1-
0027, N0014-07-1-0044, OSD/AFOSR 9550-05-1-0613, and SD Grant 5-21024
(inter.)) [BR]. SL gratefully acknowledges hospitality of the Division of Applied
Mathematics at Brown University on several occasions that were crucial to the
successoftheproject.
LosAngeles,CA,USA SergeyV.Lototsky
Providence,RI,USA BorisL.Rozovsky
March2017
Contents
1 Introduction .................................................................. 1
1.1 GettingStarted .......................................................... 1
1.1.1 ConventionsandNotations..................................... 1
1.1.2 DealingwithNoise ............................................. 2
1.1.3 AFewUsefulEqualities........................................ 6
1.1.4 AFewUsefulInequalities...................................... 7
1.2 SomeSourcesofSPDEs................................................ 12
1.2.1 Biology.......................................................... 12
1.2.2 ClassicalProbabilityTheory................................... 12
1.2.3 EconomicsandFinance ........................................ 15
1.2.4 Engineering ..................................................... 16
1.2.5 Physics .......................................................... 18
1.2.6 Literature........................................................ 21
1.2.7 TheStructureofThisBook .................................... 22
2 BasicIdeas.................................................................... 25
2.1 SomeUsefulFacts ...................................................... 25
2.1.1 ContinuityofRandomFunctions .............................. 25
2.1.2 ConnectionBetweentheItôandStratonovichIntegrals...... 29
2.1.3 RandomChangeofVariablesinRandomFunctions.......... 32
2.1.4 Problems ........................................................ 35
2.2 ClassificationofSPDEs ................................................ 36
2.2.1 SPDEsasStochasticEquations................................ 36
2.2.2 SPDEsasPartialDifferentialEquations....................... 38
2.2.3 VariousNotionsofaSolution.................................. 41
2.2.4 Problems ........................................................ 50
2.3 Closed-FormSolutions ................................................. 52
2.3.1 HeatEquation................................................... 52
2.3.2 WaveEquation.................................................. 57
2.3.3 PoissonEquation ............................................... 60
xi