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SPRINGER BRIEFS IN MATHEMATICS Étienne Pardoux Stochastic Partial Differential Equations An Introduction 123 SpringerBriefs in Mathematics Series Editors Nicola Bellomo, Torino, Italy Michele Benzi, Pisa, Italy Palle Jorgensen, Iowa, USA Tatsien Li, Shanghai, China Roderick Melnik, Waterloo, Canada Otmar Scherzer, Linz, Austria Benjamin Steinberg, New York, USA Lothar Reichel, Kent, USA Yuri Tschinkel, New York, USA George Yin, Detroit, USA Ping Zhang, Kalamazoo, USA SpringerBriefsinMathematicsshowcasesexpositionsinallareasofmathematics andappliedmathematics.Manuscriptspresentingnewresultsorasinglenewresult inaclassicalfield,newfield,oranemergingtopic,applications,orbridgesbetween newresultsandalreadypublishedworks,areencouraged.Theseriesisintendedfor mathematicians and applied mathematicians. All works are peer-reviewed to meet the highest standards of scientific literature. More information about this series at http://www.springer.com/series/10030 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 ©TheAuthor(s),underexclusivelicensetoSpringerNatureSwitzerlandAG2021 Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether thewholeorpartofthematerialisconcerned,specificallytherightsofreprinting,reuseofillustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. 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ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland 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.