Stochastic ordinary and stochastic partial differential equations: transition from microscopic to macroscopic equations

458 Pages·2008·2.265 MB·English

by  Peter Kotelenez (auth.)

Stochastic Mechanics Stochastic Modelling
Random Media and Applied Probability
Signal Processing and Image Synthesis (Formerly:
Mathematical Economics and Finance Applications of Mathematics)
Stochastic Optimization
58
Stochastic Control
Stochastic Models in Life Sciences
Edited by B. Rozovskii
G. Grimmett
Advisory Board D. Dawson
D. Geman
I. Karatzas
F. Kelly
Y. Le Jan
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E. Pardoux
StochasticModellingandAppliedProbability
formerly:ApplicationsofMathematics
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(continuedafterReferences)
Peter Kotelenez
Stochastic Ordinary and Stochastic
Partial Differential Equations
Transition from Microscopic
to Macroscopic Equations
Author
PeterKotelenez
DepartmentofMathematics
CaseWesternReserveUniversity
10900EuclidAve.
Cleveland, OH44106–7058
USA
[email protected]
ManagingEditors
B.Rozovskii G.Grimmett
DivisionofAppliedMathematics CentreforMathematicalSciences
182GeorgeSt. WilberforceRoad
Providence,RI01902 CambridgeCB30WB
USA UK
[email protected] [email protected]
ISBN978-0-387-74316-5 e-ISBN978-0-387-74317-2
DOI:10.1007/978-0-387-74317-2
LibraryofCongressControlNumber:2007940371
Mathematics Subject Classification (2000): 60H15, 60H10, 60F99, 82C22, 82C31, 60K35, 35K55,
35K10,60K37,60G60,60J60
(cid:1)c 2008SpringerScience+BusinessMedia,LLC
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Contents
Introduction ....................................................... 1
PartI FromMicroscopicDynamicstoMesoscopicKinematics
1 Heuristics:MicroscopicModelandSpace–TimeScales ............. 9
2 Deterministic Dynamics in a Lattice Model and a Mesoscopic
(Stochastic)Limit .............................................. 15
3 ProofoftheMesoscopicLimitTheorem .......................... 31
PartII MesoscopicA:StochasticOrdinaryDifferentialEquations
4 StochasticOrdinaryDifferentialEquations:Existence,Uniqueness,
andFlowsProperties ........................................... 59
4.1 Preliminaries .............................................. 59
4.2 TheGoverningStochasticOrdinaryDifferentialEquations ........ 64
4.3 EquivalenceinDistributionandFlowPropertiesforSODEs....... 73
4.4 Examples ................................................. 78
5 QualitativeBehaviorofCorrelatedBrownianMotions ............. 85
5.1 UncorrelatedandCorrelatedBrownianMotions ................. 85
5.2 ShiftandRotationalInvarianceofw(dq,dt) .................... 92
5.3 SeparationandMagnitudeoftheSeparationofTwoCorrelated
BrownianMotionswithShift-Invariant
andFrame-IndifferentIntegralKernels......................... 94
5.4 AsymptoticsofTwoCorrelatedBrownianMotions
withShift-InvariantandFrame-IndifferentIntegralKernels .......105
vii
viii Contents
5.5 DecompositionofaDiffusionintotheFluxandaSymmetric
Diffusion .................................................110
5.6 LocalBehaviorofTwoCorrelatedBrownianMotions
withShift-InvariantandFrame-IndifferentIntegralKernels .......116
5.7 ExamplesandAdditionalRemarks ............................121
5.8 AsymptoticsofTwoCorrelatedBrownianMotions
withShift-InvariantIntegralKernels...........................128
6 ProofoftheFlowProperty ......................................133
6.1 ProofofStatement3ofTheorem4.5 ..........................133
6.2 SmoothnessoftheFlow .....................................138
7 CommentsonSODEs:AComparisonwithOtherApproaches ......151
7.1 PreliminariesandaComparisonwithKunita’sModel ............151
7.2 ExamplesofCorrelationFunctions............................156
PartIII MesoscopicB:StochasticPartialDifferentialEquations
8 StochasticPartialDifferentialEquations:
FiniteMassandExtensions .....................................163
8.1 Preliminaries ..............................................163
8.2 APrioriEstimates..........................................171
8.3 NoncoerciveSPDEs ........................................174
8.4 CoerciveandNoncoerciveSPDEs.............................189
8.5 GeneralSPDEs ............................................197
8.6 Semilinear Stochastic Partial Differential Equations
inStratonovichForm .......................................198
8.7 Examples .................................................200
9 StochasticPartialDifferentialEquations:InfiniteMass ............203
9.1 NoncoerciveQuasilinearSPDEsforInfiniteMassEvolution ......203
9.2 Noncoercive Semilinear SPDEs for Infinite Mass Evolution
inStratonovichForm .......................................219
10 Stochastic Partial Differential Equations: Homogeneous
andIsotropicSolutions .........................................221
11 ProofofSmoothness,Integrability,
andItoˆ’sFormula ..............................................229
11.1 BasicEstimatesandStateSpaces .............................229
11.2 ProofofSmoothnessof(8.25)and(8.73).......................246
11.3 ProofoftheItoˆ formula(8.42)................................269
12 ProofofUniqueness ............................................273
Contents ix
13 CommentsonOtherApproachestoSPDEs .......................291
13.1 Classification ..............................................291
13.1.1 LinearSPDEs .......................................294
13.1.2 BilinearSPDEs......................................297
13.1.3 SemilinearSPDEs ...................................299
13.1.4 QuasilinearSPDEs...................................301
13.1.5 NonlinearSPDEs ....................................301
13.1.6 StochasticWaveEquations ............................302
13.2 Models ...................................................302
13.2.1 NonlinearFiltering...................................302
13.2.2 SPDEsforMassDistributions .........................303
13.2.3 FluctuationLimitsforParticles.........................304
13.2.4 SPDEsinGenetics ...................................305
13.2.5 SPDEsinNeuroscience...............................305
13.2.6 SPDEsinEuclideanFieldTheory ......................306
13.2.7 SPDEsinFluidMechanics ............................306
13.2.8 SPDEsinSurfacePhysics/Chemistry ...................308
13.2.9 SPDEsforStrings....................................308
13.3 BooksonSPDEs ...........................................308
PartIV Macroscopic:DeterministicPartialDifferentialEquations
14 PartialDifferentialEquationsasaMacroscopicLimit..............313
14.1 LimitingEquationsandHypotheses ...........................313
14.2 TheMacroscopicLimitford 2 .............................316
≥
14.3 Examples .................................................327
14.4 ARemarkond 1 ........................................330
=
14.5 ConvergenceofStochasticTransportEquations
toMacroscopicParabolicEquations...........................331
PartV GeneralAppendix
15 Appendix .....................................................335
15.1 Analysis ..................................................335
15.1.1 Metric Spaces: Extension by Continuity, Contraction
Mappings,andUniformBoundedness...................335
15.1.2 SomeClassicalInequalities............................336
15.1.3 TheSchwarzSpace ..................................340
15.1.4 MetricsonSpacesofMeasures.........................348
15.1.5 RiemannStieltjesIntegrals ............................357
15.1.6 TheSkorokhodSpace D( 0, ) B) ....................359
[ ∞ ;
15.2 Stochastics ................................................362
15.2.1 RelativeCompactnessandWeakConvergence............362
x Contents
15.2.2 RegularandCylindricalHilbertSpace-ValuedBrownian
Motions ............................................366
15.2.3 Martingales,QuadraticVariation,andInequalities.........371
15.2.4 RandomCovarianceandSpace–timeCorrelations
forCorrelatedBrownianMotions.......................380
15.2.5 StochasticItoˆ Integrals................................387
15.2.6 StochasticStratonovichIntegrals .......................403
15.2.7 Markov-DiffusionProcesses...........................411
15.2.8 Measure-ValuedFlows:ProofofProposition4.3 ..........418
15.3 TheFractionalStepMethod..................................422
15.4 Mechanics:Frame-Indifference...............................424
SubjectIndex ......................................................431
Symbols...........................................................439
References.........................................................445
Introduction
Thepresentvolumeanalyzesmathematicalmodelsoftime-dependentphysicalphe-
nomena on three levels: microscopic, mesoscopic, and macroscopic. We provide a
rigorous derivation of each level from the preceding level and the resulting meso-
scopic equations are analyzed in detail. Following Haken (1983, Sect. 1.11.6) we
deal, “at the microscopic level, with individual atoms or molecules, described by
their positions, velocities, and mutual interactions. At the mesoscopic level, we
describe the liquid by means of ensembles of many atoms or molecules. The ex-
tension of such an ensemble is assumed large compared to interatomic distances
but small compared to the evolving macroscopic pattern... . At the macroscopic
levelwewishtostudythecorrespondingspatialpatterns.”Typically,atthemacro-
scopic level, the systems under consideration are treated as spatially continuous
systemssuchasfluidsoracontinuousdistributionofsomechemicalreactants,etc.
Incontrast,onthemicroscopiclevel,Newtonianmechanicsgovernstheequationsof
motionoftheindividualatomsormolecules.1 Theseequationsarecastintheform
ofsystemsofdeterministiccouplednonlinearoscillators.Themesoscopiclevel2 is
probabilistic innature and many models may be faithfullydescribed by stochastic
ordinaryandstochasticpartialdifferentialequations(SODEsandSPDEs),3 where
thelatteraredefinedonacontinuum.Themacroscopiclevelisdescribedbytime-
dependentpartialdifferentialequations(PDE’s)anditsgeneralizationandsimplifi-
cations.
Inourmathematicalframeworkwetalkofparticlesinsteadofatomsandmole-
cules. The transition from the microscopic description to a mesoscopic (i.e., sto-
chastic)descriptionrequiresthefollowing:
Replacementofspatiallyextendedparticlesbypointparticles
•
Formationofsmallclusters(ensembles)ofparticles(iftheirinitialpositionsand
•
velocitiesaresimilar)
1Werestrictourselvesinthisvolumeto“classicalphysics”(cf.,e.g.,Heisenberg(1958)).
2Fortherelationbetweennanotechnologyandmesoscales,werefertoRoukes(2001).
3Inthisvolume,mesoscopicequationswillbeidentifiedwithSODEsandSPDEs.
1