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ROBUSTNESS OF THE PATHWISE STRUCTURE OF FLUCTUATIONS IN STOCHASTIC HOMOGENIZATION英文资料

ROBUSTNESS OF THE PATHWISE STRUCTURE OF FLUCTUATIONS IN STOCHASTIC HOMOGENIZATION英文资料

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时间：2019-08-08

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1、ROBUSTNESSOFTHEPATHWISESTRUCTUREOFFLUCTUATIONSINSTOCHASTICHOMOGENIZATIONMITIADUERINCKX,ANTOINEGLORIA,ANDFELIXOTTOAbstract.Weconsideralinearellipticsystemindivergenceformwithrandomcoeffi-cientsandstudytherandomfluctuationsoflarge-scaleaveragesofthefieldandthefluxofthesolution.Inapreviouscontri

2、bution,fordiscreteellipticequationswithindependentandidenticallydistributedconductances,wedevelopedatheoryoffluctuationsbasedonthenotionofhomogenizationcommutator,definedasthefluxminusthehomogenizedcoefficientstimesthefieldofthesolution:weprovedthatthetwo-scaleexpansionofthisspecialquantityisa

3、ccurateatleadingorderwhenaveragedonlargescales(asopposedtothetwo-scaleexpansionofthefieldandfluxtakenseparately)andthatthelarge-scalefluctuationsofboththefieldandthefluxcanberecoveredfromthoseofthecommutator.Thisimpliesthatthelarge-scalefluctuationsofthecommutatorofthecorrectordriveallotherlar

4、ge-scalefluctuationstoleadingorder,whichwerefertoasthepathwisestructureoffluctuationsinstochastichomogenization.Inthepresentcontributionweextendthisresultintwodirections:wetreatcontinuumelliptic(possiblynon-symmetric)systemsandwithstronglycorrelatedcoefficientfields(Gaussian-likewithacovarian

5、cefunctionthatdisplaysanarbitrarilyslowalgebraicdecayatinfinity).Ourmainresultshowsthatthetwo-scaleexpansionofthehomogenizationcommutatorisstillaccuratetoleadingorderwhenaveragedonlargescales,whichillustratestherobustnessofthepathwisestructureoffluctuations.Contents1.Introduction12.Mainres

6、ultsandstructureoftheproof52.1.Notationandstatementofthemainresults52.2.Structureoftheproof83.Proofs123.1.ProofofLemma2.4:Representationformulas123.2.Dyadicpartitionsandreformulationofweightednorms14arXiv:1807.11781v2[math.AP]2Aug20183.3.ProofofProposition2.5:Mainestimates153.4.ProofofPr

7、oposition2.6:Mainestimates(cont’d)193.5.Proofofthemainresults24Acknowledgements28References281.IntroductionThisarticleconstitutesthesecondpartofaseriesthatdevelopsatheoryoffluctuationsinstochastichomogenizationoflinearelliptic(non-necessarilysymmetric)systemsindi-vergencef

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