Green Functions for Second Order Parabolic Integro-Differential Problems

440 Pages·1993·2.161 MB·English

by  Maria Giovanna Garroni, Jose Luis Menaldi

To Emilio
and
To Mar´ıa Cristina and Veronica Estela

Preface
The main purpose of this book is the construction and investigation of properties of the
Green function for a large class of parabolic integro–differential operators that arise from
diffusion processes with jumps. Important developments in the study of stochastic pro-
cesseswithjumpshavebeenachievedinrecentyearsthroughtheanalyticalapproach. In
thisapproachtheGreenfunctionplaysakeyrole.
We will construct Green functions for general boundary value problems in a smooth
(boundedorunbounded)cylindricaldomain(cf. (I.1.16))forintegro–differentialoperators
under the following two assumptions. First, a uniformly parabolic second–order differ-
ential operator not in divergence form, with bounded Ho¨lder continuous coefficients (i.e.
belongingtoCα,α2)isassumedaslocalpartoftheoperator. Second,anintegro–differential
operator of order at most γ < 2−α, 0 < α < 1, with H¨older continuous coefficients [cf.
(II.1.48)]isassumedasnon–local(integral)part.
Moreoverinthecaseoftheobliqueboundaryconditionwesupposethatthecoefficients
of the first order differential boundary operator are smooth (i.e. belonging to C1+α,1+2α).
Notice that the last assumption is the one which guarantees the existence of the classical
solution (i.e. in C2+α,2+2α) both for differential and integro–differential oblique boundary
problems.
In some cases the previous assumptions can be weakened. The most interesting case,
alsofromaprobabilisticpointofview,istheoneinwhichthecoefficientsappearinginthe
boundaryconditionshaveafairlylowregularity(Ho¨ldercontinuityonly). Thisimplies,for
instance,thattheclassicalsolutiondoesnotexist(thehighestderivativesofthesolutionof
the corresponding boundary valueproblem havean explosion at the boundary). However
alsointhiscaseitispossibletoconstructtheGreenfunctioneitherforintegro–differential
operatorsoforderγ≤1,withoutchangingtheotherassumptions,orforintegro–differential
oforderγ<2−α,butbyaddingsuitableconditionsonthejumpsandonthemeasureof
theintegralpart. Inbothcases,byusingtheconstructedGreenfunction,itispossibleto
generateaMarkov–Fellerjumpdiffusionprocesswithanobliquereflectionattheboundary,
whichunderourassumptionshasnotbeenconstructedinanyotherway.
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We would like to point out that the previous assumptions are focused on the actual
construction of the Green function. However, many intermediate results have an interest
inthemselves,andaregivenundermuchmoregeneralhypotheses,bothontheorderγ of
the integraloperator and on the regularityof the coefficients (for example, the maximum
principles,andtheexistenceanduniquenessresults). ThedefinitionoftheGreenfunction
and of the fundamental solution will also be given under more general assumptions. The
latterwillallowustoprovetheuniquenessandmanystructuralproperties,independently
oftheexistenceandtheactualconstruction.
InChapterIwerecallvariousresultsonaparabolicsecondorderdifferentialequation
withDirichlet,Neumannofobliqueboundaryconditions.
In Chapter II we extend most of these results to the case of a second order, integro–
differential operator. We make use of known estimates for parabolic second order dif-
ferential equations and sharp estimates (of compactness type) on the integro–differential
operators.
At this point, it would be interesting to investigate the asymptotic behaviour of the
solutionoftheparabolicsecondorderintegro–differentialequation. Howeverthisproblem
requiresaninvariantmeasurefortheunderlyingsemigroup. Inordertofindthismeasure
somesharpestimatesontheGreenfunctionarenecessary. But,evenifweconsiderasimple
Cauchy problem, as we do in Chapter III, for the Laplacian (associated with the Wiener
process)andthesimplest“integro–differential”operatoroforderzeroassociatedwiththe
Poissonprocess,werealizethattheclassicpointwiseestimatesofheat–kerneltypecannot
hold for the explicit expression of the transition function (i.e. fundamental solution) of a
Wiener–Poisson (Markov) process. This forces us to review all the arguments and all the
tools used to construct the Green function, where the presence of the exponential factor
playsafundamentalrole.
Therefore we start by presenting a definition of the (classic) Green function which is
well adapted to a general parabolic second order integro–differential operator (Chapter
IV). This definition allows us to obtain immediately most of the properties of the Green
function not involving sharp estimates. Moreover, we make use of the strong maximum
principletoprovethestrictpositivityoftheGreenfunction.
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In Chapter V we give a relatively detailed description of the construction with sharp
estimates of the fundamental solution for parabolic second order differential operators.
Next,wedescribethepassagetotheGreenfunctioninthatclassicsituation(ChapterVI).
At this point, in Chapter VII, we introduce the necessary tools to discuss the Green
functionforanintegro–differentialoperator,namelytheGreenfunctionspaces. Bymeans
of the latter we give a characterization of the non–explicit (or non–local) part of the
Greenfunction. Next,wemakeuseoftheseGreenfunctionspacestodevelopthemethod
of successive approximations. Thus, we construct the Green function, in Chapter VIII,
without using sharp estimates on the Green function. In Chapter IX, we discuss the
structureoftheGreenfunctioninvolvingestimatesofthefirstandsecondorderderivatives.
Each chapter begins with an introduction giving a detailed description of the topics
treatedinit. Moreover,whenpossible,thestochasticinterpretationsandtheprobabilistic
counterpart of these topics are given. The chapters that are either introductory to the
Green function or in which the classical results are dealt with are not meant to be ex-
haustive. Forexample,weobtainthemaximumprinciplesonlyfortheclassicalsolutionin
bounded and unbounded domains, because this is the only case we will make use of. On
the other hand, we deal in greater detail with a few classical results about fundamental
solutions, Green and Poisson functions. In particular, we obtain explicit expressions of
these two functions on the half–space for the differential operator with constant coeffi-
cients because these are essential for the subsequent constructions and estimates. In the
treatment of the classical case, special emphasis is laid on the crucial points and on the
toolsneededfortheconstructionoftheGreenfunction.
Theresultscontainedinthisbookprovidethetoolswithwhichtheinvariantmeasure
can be constructed. Then asymptotic problems for equations and inequations can be
studied as already done for integral operators of order ≤1 in Garroni and Menaldi [1,2].
Moreover,thesharpestimatesoftheGreenfunctionwithrespecttothecoefficientsallow
onetostudynonlinearproblems,asdoneinGarroniandVivaldi[1]andinGarronietal.
[1,2], for integral operators of order≤1. Finally, we would like to point out that most of
theresultsofthisbookarebasedonGarroniandMenaldi[1].
It is a pleasure to acknoweldge the help we have received from many colleagues and
March29,1996 iii
in particular from V.A. Solonnikov and E.B. Fabes. We are especially indebted to G.
BachelisandP.L.Chowwhoreadpartofthemanuscriptandofferedinvaluablecomments.
Letusmentionthatthelastpartofthisworkwaspartiallysupportedbytheinternational
cooperationCNR(Italy)andNSF(USA)undergrantINT–90228114andDMS–9101360.
MariaGiovannaGarroniandJos´eLuisMenaldi
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I Background 1
I.1 IntroductoryMaterial. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
I.1.1 DefinitionsofBasicFunctionSpaces . . . . . . . . . . . . . . . . . 2
I.1.2 SomeBasicResults . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
I.2 ExistenceandUniquenessResults . . . . . . . . . . . . . . . . . . . . . . . 18
I.3 ClassicMaximumPrinciples . . . . . . . . . . . . . . . . . . . . . . . . . . 27
I.4 Markov–FellerProcesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
I.4.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
I.4.2 DefinitionsandMainProperties . . . . . . . . . . . . . . . . . . . . 37
I.4.3 SomeStandardExamples . . . . . . . . . . . . . . . . . . . . . . . 42
I.4.4 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
I.4.5 Itˆo’sFormula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
II Integro–Differential Parabolic Equations 51
II.1 DefinitionoftheIntegro–DifferentialOperator . . . . . . . . . . . . . . . . 52
II.1.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
II.1.2 IntheWholeSpace . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
II.1.3 InaBoundedDomain . . . . . . . . . . . . . . . . . . . . . . . . . 64
II.1.4 InaUnboundedDomain . . . . . . . . . . . . . . . . . . . . . . . . 68
II.2 MaximumPrinciples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
II.3 ExistenceandUniquenessResults . . . . . . . . . . . . . . . . . . . . . . . 90
II.4 StochasticInterpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
IIIA Simple Cauchy Problem 103
III.1 WienerandPoissonProcesses . . . . . . . . . . . . . . . . . . . . . . . . . 103
III.2 EssentialProperties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
III.3 Dirichlet&NeumannProblemsinHalf-space. . . . . . . . . . . . . . . . . 113
III.3.1 SomeBasicProperties . . . . . . . . . . . . . . . . . . . . . . . . . 115
III.3.2 ConstructionofaSimpleGreenFunction . . . . . . . . . . . . . . . 120
III.4 StochasticRepresentation . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
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IVGreen and Poisson Functions 129
IV.1 DefinitionofaFundamentalSolution . . . . . . . . . . . . . . . . . . . . . 130
IV.1.1 HeatEquation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
IV.1.2 DefinitionandMainProperties . . . . . . . . . . . . . . . . . . . . 136
IV.2 DefinitionofaGreenFunction . . . . . . . . . . . . . . . . . . . . . . . . . 145
IV.2.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
IV.2.2 DefinitionandMainProperties . . . . . . . . . . . . . . . . . . . . 148
IV.3 DefinitionofaPoissonFunction . . . . . . . . . . . . . . . . . . . . . . . . 156
V Fundamental Solutions for Differential Equations 161
V.1 EstimatesfortheHeatPotentials . . . . . . . . . . . . . . . . . . . . . . . 163
V.2 OperatorswithConstantCoefficients . . . . . . . . . . . . . . . . . . . . . 171
V.2.1 ConstantCoefficients . . . . . . . . . . . . . . . . . . . . . . . . . . 171
V.2.2 CoefficientsDependingonParameters . . . . . . . . . . . . . . . . 174
V.3 VariableCoefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
V.3.1 GeneralTechniques . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
V.3.2 TheParametrixMethod . . . . . . . . . . . . . . . . . . . . . . . . 181
V.4 TheCauchyProblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
V.5 LayerPotentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
V.5.1 Single–LayerPotential: NeumannProblem . . . . . . . . . . . . . . 194
V.5.2 Double–LayerPotential: DirichletProblem . . . . . . . . . . . . . . 204
V.5.3 Single–LayerPotential: ObliqueDerivative . . . . . . . . . . . . . . 205
VIClassic Green and Poisson Functions 213
VI.1 ProblemsinHalf–Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
VI.1.1 PoissonFunctionfortheHeatEquation. . . . . . . . . . . . . . . . 215
VI.1.2 GreenFunctionfortheHeatEquation . . . . . . . . . . . . . . . . 220
VI.1.3 TheCaseofConstantCoefficients . . . . . . . . . . . . . . . . . . . 227
VI.1.4 EstimatesofSuperficialPotentials. . . . . . . . . . . . . . . . . . . 237
VI.1.5 TheCaseofVariableCoefficients . . . . . . . . . . . . . . . . . . . 239
VI.2 FirstandSecondBoundaryValueProblems . . . . . . . . . . . . . . . . . 246
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