Stochastic Partial Differential Equations - An Introduction

78 Pages·2021·1.11 MB·English

by  Étienne Pardoux

SPRINGER BRIEFS IN MATHEMATICS
Étienne Pardoux
Stochastic
Partial
Differential
Equations
An Introduction
123
SpringerBriefs in Mathematics
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É
tienne Pardoux
Stochastic Partial Differential
Equations
An Introduction
123
ÉtiennePardoux
Institut deMathématiques deMarseille
AixMarseille Université,CNRS
Marseille, France
ISSN 2191-8198 ISSN 2191-8201 (electronic)
SpringerBriefs inMathematics
ISBN978-3-030-89002-5 ISBN978-3-030-89003-2 (eBook)
https://doi.org/10.1007/978-3-030-89003-2
MathematicsSubjectClassification: 60H15,35R60,60G35,60H07,60J68
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Foreword
There is by now a growing interest in Stochastic Partial Differential Equations
(abbreviatedfromnowonasSPDEs).Onecanfindtworeasonsforthis.
First, more and more complex mathematical models are used in the applied
sciences,inordertodescribereality.Thehugeprogressincomputerpowerandour
abilitytosimulatehigh-dimensionaldynamicalsystemshasmadeitpossible toassess
highlycomplexmodels,whichtakeintoaccountbothrandomnessandthefactthat
mostsystemsaredistributedoverspace.ThisleadsnaturallytoPDEswithrandom
coefficients,andSPDEs.
Second, in recent decades we have seen the emergence of new sophisticated
mathematical techniques, which allow us to tackle new problems and classes of
equations. These include the theory of Rough Paths, first introduced by T. Lyons,
seethecoursebyHairerandFriz[11],thetheoryofregularitystructuresinvented
byM.Hairer[10],andthemethodofparacontrolleddistributions,duetoGubinelli,
ImkellerandPerkowski[8].
Theaimofthesenotesistopresentaconciseintroductiontothe“classicaltheory”
ofSPDEs,asitwasdevelopedduringthelast25yearsofthelastcentury.Webelieve
thatagoodunderstandingofthistheoryisusefulinordertostudyandunderstand
thenewapproaches.
Marseille,September2021 ÉtiennePardoux
v
Contents
1 IntroductionandMotivation .................................... 1
1.1 Introduction ............................................... 1
1.2 Motivation ................................................ 2
1.2.1 Turbulence.......................................... 2
1.2.2 Populationdynamics,populationgenetics................ 3
1.2.3 Neurophysiology .................................... 3
1.2.4 Evolutionofthecurveofinterestrate ................... 4
1.2.5 Nonlinearfiltering ................................... 4
1.2.6 Movementbymeancurvatureinarandomenvironment .... 5
1.2.7 Hydrodynamiclimitofparticlesystems ................. 5
1.2.8 Fluctuationsofaninterfaceonawall.................... 7
2 SPDEsasInfinite-DimensionalSDEs ............................. 9
2.1 Introduction ............................................... 9
2.2 ItôCalculusinHilbertspace ................................. 9
2.3 SPDEswithAdditiveNoise .................................. 12
2.3.1 ThesemigroupapproachtolinearparabolicPDEs......... 13
2.3.2 Thevariationalapproachtolinearandnonlinearparabolic
PDEs .............................................. 15
2.4 TheVariationalApproachtoSPDEs........................... 20
2.4.1 Monotone-coerciveSDPEs ............................ 20
2.4.2 Examples........................................... 29
2.4.3 CoerciveSPDEswithcompactness ..................... 31
2.5 SemilinearSPDEs.......................................... 36
3 SPDEsDrivenBySpace-TimeWhiteNoise........................ 41
3.1 Introduction ............................................... 41
3.2 RestrictiontoaOne-DimensionalSpaceVariable................ 41
3.3 AGeneralExistence-UniquenessResult........................ 43
3.4 AMoreGeneralExistenceandUniquenessResult ............... 51
3.5 PositivityoftheSolution .................................... 51
3.6 ApplicationsofMalliavinCalculustoSPDEs ................... 55
3.7 SPDEsandtheSuperBrownianMotion........................ 60
vii
viii Contents
3.7.1 Thecase𝛾 =1/2 .................................... 60
3.7.2 Othervaluesof𝛾 <1................................. 66
3.8 AReflectedSPDE.......................................... 66
References......................................................... 71
Index ............................................................. 73
Chapter 1
Introduction and Motivation
1.1 Introduction
In these lectures we shall study stochastic parabolic PDEs, most of which will be
nonlinear.Thegeneraltypeofequationswhichwehaveinmindareoftheform
𝜕𝑢
(𝑡,𝑥) = 𝐹(𝑡,𝑥,𝑢(𝑡,𝑥),D𝑢(𝑡,𝑥),D2𝑢(𝑡,𝑥))+𝐺(𝑡,𝑥,𝑢(𝑡,𝑥),D𝑢(𝑡,𝑥))𝑊˚ (𝑡,𝑥),
𝜕𝑡
orinthesemilinearcase
𝜕𝑢
(𝑡,𝑥) =Δ𝑢+ 𝑓(𝑡,𝑥,𝑢(𝑡,𝑥))+𝑔(𝑡,𝑥,𝑢(𝑡,𝑥))𝑊˚ (𝑡,𝑥).
𝜕𝑡
Weshallmakeprecisewhatwemeanby𝑊˚ (𝑡,𝑥).Wedistinguishtwocases
1. 𝑊˚ is white noise in time and colored noise in space. A particular case is that
wherethenoiseisoftheformÍ𝑘𝑁=1𝑒𝑘(𝑥)𝑊˚𝑘(𝑡).
2. 𝑊˚ iswhitenoisebothintimeandinspace.
Inbothcases,wecandefine𝑊˚ inthedistributionalsense,asacenteredgeneralized
Gaussianprocess,indexedbytestfunctionsℎ:R ×R𝑑 ↦→R:
+
𝑊˚ = {𝑊˚ (ℎ); ℎ ∈𝐶∞(R ×R𝑑)},
+
whosecovarianceisgivenby
E(cid:16)𝑊˚ (ℎ)𝑊˚ (𝑘)(cid:17) =∫ ∞d𝑡∫ d𝑥∫ d𝑦 ℎ(𝑡,𝑥)𝑘(𝑡,𝑦)𝜑(𝑥−𝑦) incase1
0 R𝑑 R𝑑
∫ ∞ ∫
= d𝑡 d𝑥 ℎ(𝑡,𝑥)𝑘(𝑡,𝑥) incase2.
0 R𝑑
Here𝜑isa“reasonable”kernel,whichmightblowuptoinfinityat0.Notethatthe
firstformulaconvergestothesecondone,ifwelet𝜑convergetotheDiracmassat
0.Ontheotherhand,thesolutionofaPDEoftheform
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 1
É. Pardoux, Stochastic Partial Differential Equations,
SpringerBriefs in Mathematics, https://doi.org/10.1007/978-3-030-89003-2_1
2 1 IntroductionandMotivation
𝜕𝑢
(𝑡,𝑥) =Δ𝑢(𝑡,𝑥)+ 𝑓(𝑡,𝑥,𝑢(𝑡,𝑥))
𝜕𝑡
canbeconsideredeitheras
1. a function of 𝑡 with values in an infinite-dimensional space of functions of 𝑥
(typicallyaSobolevspace);orelseas
2. areal-valuedfunctionof(𝑡,𝑥).
Likewise, in the case of an SPDE of one of the above types, we can consider the
solutioneitheras
1. a stochastic process indexed by 𝑡, and taking values in an infinite-dimensional
functionspace,i.e.thesolutionofaninfinite-dimensionalSDE;orelseas
2. aone-dimensionalrandomfield,i.e.thesolutionofamultiparameterSDE.
ThefirstpointofviewwillbepresentedinChapter2.Itappliesmainlytoequations
driven by Gaussian noises which are colored in space. The second one will be
presentedinChapter3forthestudyofspace-timewhitenoisedrivenSPDEs.
ThereareseveralseriousdifficultiesinthestudyofSPDEs,whichareduetothe
lackofregularitywithrespecttothetimevariable(resp.withrespecttoboththetime
and the space variable), and the interaction between the regularity in time and the
regularityinspace.Asaresult,aswewillsee,thetheoryofnonlinearSPDEsdriven
by space-time white noise, and with second order PDE operators, is limited to the
caseofaone-dimensionalspacevariable.Also,thereisnocompletelysatisfactory
theoryoffullynonlinearSPDEs,seetheworkofLionsandSouganidisonviscosity
solutionsofSPDEs[15].
New powerful methods have been introduced recently to deal with singular
SPDEs,namelythetheoryofregularitystructuresduetoM.Hairer[10],andtheno-
tionofparacontrolleddistributionsintroducedbyGubinelli,ImkellerandPerkowski
[8].Weshallnotdiscussthoseapproachesinthepresentnotes.
1.2 Motivation
Wenowintroduceseveralmodelsfromvariousfields,whichareexpressedasSPDEs.
1.2.1 Turbulence
SeveralmathematiciansandphysicistshaveadvocatedthattheNavier–Stokesequa-
tionwithadditivewhitenoiseforcingisasuitablemodelforturbulence.Thisequation
indimension𝑑 =2or3reads
𝜕𝜕𝑢𝑡 (𝑡,𝑥) =𝜈Δ𝑢(𝑡,𝑥)+∑︁𝑑 𝑢𝑖(𝑡,𝑥)𝜕𝜕𝑥𝑢 (𝑡,𝑥)+ 𝜕𝜕𝑊𝑡 (𝑡,𝑥)
𝑖
𝑖=1
 𝑢(0,𝑥) =𝑢0(𝑥),

1.2 Motivation 3
where𝑢(𝑡,𝑥) = (𝑢1(𝑡,𝑥),...,𝑢𝑑(𝑡,𝑥))isthevelocityofthefluidattime𝑡andpoint
𝑥.Thenoisetermisoftenchosenoftheform
ℓ
𝑊(𝑡,𝑥) =∑︁𝑊𝑘(𝑡)𝑒𝑘(𝑥),
𝑘=1
where {𝑊1(𝑡),...,𝑊ℓ(𝑡), 𝑡 ≥ 0} are mutually independent standard Brownian
motions.
1.2.2 Populationdynamics,populationgenetics
ThefollowingmodelwasproposedbyD.Dawsonin1972fortheevolutionofthe
densityofapopulation
𝜕𝑢 𝜕2𝑢 √
(𝑡,𝑥) =𝜈 (𝑡,𝑥)+𝛼 𝑢(𝑡,𝑥)𝑊˚ (𝑡,𝑥),
𝜕𝑡 𝜕𝑥2
where𝑊˚ isaspace-timewhitenoise.Inthiscase,onecanderiveclosedequations
forthefirsttwomoments
𝑚(𝑡,𝑥) =E[𝑢(𝑡,𝑥)], 𝑉(𝑡,𝑥,𝑦) =E[𝑢(𝑡,𝑥)𝑢(𝑡,𝑦)].
OnecanapproachthisSPDEbyamodelindiscretespaceasfollows.𝑢(𝑡,𝑖),𝑖 ∈ Z
denotesthenumberofindividualsinthecolony𝑖attime𝑡.Then
• 𝛼2𝑢(𝑡,𝑖)isboththebirthandthedeathrate;
2
• 𝜈𝑢(𝑡,𝑖)isthemigrationrate,bothfrom𝑖to𝑖−1andto𝑖+1.
W.Fleminghasproposedananalogousmodelinpopulationgenetics,wheretheterm
√
𝛼 𝑢isreplacedby𝛼√︁𝑢(1−𝑢).
1.2.3 Neurophysiology
The following model has been proposed by J. Walsh [30], in order to describe the
propagationofanelectricpotentialinaneuron(whichisidentifiedwiththeinterval
[0,𝐿]).
𝜕𝑉 𝜕2𝑉
(𝑡,𝑥) = (𝑡,𝑥)−𝑉(𝑡,𝑥)+𝑔(𝑉(𝑡,𝑥))𝑊˚ (𝑡,𝑥).
𝜕𝑡 𝜕𝑥2
Hereagain𝑊˚ (𝑡,𝑥)denotesaspace-timewhitenoise.