Self-averaging in time reversal for the parabolic

Self-averagingintimereversalfortheparaboliwaveequationGuillaumeBalGeorgePapaniolaouyLeonidRyzhikzMay8,2002AbstratWeanalyzetheself-averagingpropertiesoftime-reversedsolutionsoftheparaxialwaveequa-tionwithrandomoeÆients,whihwetaketobeMarkovianinthediretionofpropagation.ThisallowsustoonstrutanapproximatemartingaleforthephasespaeWignertransformoftwowaveelds.UsingaprioriL2-boundsavailableinthetime-reversalsetting,weprovethattheWignertransforminthehighfrequenylimitonvergesinprobabilitytoitsdeterministilimit,whihisthesolutionofatransportequation.1IntrodutionIntime-reversalexperimentsasignalemittedbyaloalizedsoureisreordedatanarrayoftrans-duers.Itisthenre-emittedintothemediumreversedintime,thatis,thepartofthesignalthatisreordedrstissentbaklast.Beauseofthetime-reversibilityofthewaveequationthebak-propagatedsignalrefousesapproximatelyattheloationoftheoriginalsourebeausethearrayislimitedinsize.Astrikingexperimentalobservationisthatthepreseneofinhomogeneitiesinthemediumimprovestherefousingresolution.Theexplanationforthissuper-resolutionismulti-pathing:wavesinomplexmediathatareapturedbythereordingarrayhaveundergonemultiplesatteringmakingiteetivelylargerthanitsphysialsize.Anotherimportantfeatureofsuper-resolutionintimereversalisthattherefousedsignaldoesnotdependontherealizationoftherandommedium.Thatis,therefousedsignalisdeterministi.Super-resolutionandself-averagingofrefousedsignalsinompliatedmediahasbeenobservedbothinlaboratoryexperiments(seereviews[15,17℄andreferenestherein)andinunderwateraoustiwavepropagationoverlongdis-tanes(tensofkilometers)[12,20℄.Time-reversaltehniqueshavenumerousappliationsrangingfrommediinetoommuniationsand,morereently,imaginginrandommedia[7,10,18℄.Therstmathematialanalysisoftime-reversalinrandommediawasgivenbyClouetandFouque[11℄,whoanalyzedrefousingandself-averagingoftime-reversedpulsesinaone-dimensionallayeredrandommedium.Theirresultwasextendedtoathree-dimensionallayeredmediumin[14℄.Super-resolutioninspatialrefousinganditsstatistialstabilityformulti-dimensionalwavesinrandommediawasanalyzedin[9,22℄,inaremote-sensingregimewheretheparaxialorparaboliwaveequationanbeused.Therefousingoftheaveragesignalinafullthree-dimensionalmedium,intheregimesofrandomgeometrialoptisandradiativetransfer(transport),wasstudiedin[2,3℄.Wealsomentionthatanothersoureofmultipathingisthemixingofwavesbytheboundariesinanergodiavity.Thishasbeenstudiedexperimentallyin[16℄andmathematiallyin[6℄.DepartmentofAppliedPhysisandAppliedMathematis,ColumbiaUniversity,NewYorkNY,10027;gb2030olumbia.eduyDepartmentofMathematis,StanfordCA,94305;papaniomath.stanford.eduzDepartmentofMathematis,UniversityofChiago,ChiagoIL,60637;ryzhikmath.uhiago.edu1Thepurposeofthispaperistoanalyzetimereversalintheradiativetransferregimeusingtheparaboliwaveequation,whenthewavesinteratfullywiththerandominhomogeneities.Weprovemathematiallythattherefousedsignalisself-averaging,whihmeansthatitdoesnotdependontherealizationoftherandommedium.ThemathematialquantitiythatweanalyzeistheWignermeasureofapairofosillatorysolutionsoftherandomShrodingerequation.Inthepresentsetting,therandompotentialdependsinaMarkovianwayonthevariablez,themaindiretionofpropagationofthewaves.Thisallowedustousein[4℄amartingalemethodtoprovethattheaverageoftheWignerdistributiononvergestoasolutionoftheradiativetransferequation.InthispaperweuseadditionalregularityoftheWignermeasure,availableintime-reversalwhenthereissomeblurringatthereordingarray,toshowthatthewholeWignerdistribution,andnotonlyitsaverage,onvergesweakly,asaShwartzdistributionandinprobability,tothedeterministisolutionofthetransportequation.TheblurringatthereordingarrayprovidesaprioriboundsfortheWignertransforminL2.TheseboundsandtheMarkovianityoftherandompotentialinthediretionofpropagationmakethetime-reversalproblemmoretratablemathematiallyandallowustoproveinafairlysimpleandstraightforwardmannerself-averagingofthetime-reversedsignal.WereallthattheWignertransformisaonvenienttooltoanalyzehighfrequenywaveprop-agationindeterministi[19,21,28℄andrandommedia[25℄.IntroduedbyWignerin1932[29℄,ithasbeenusedextensivelyinthemathematialliteraturereently.ConvergeneoftheaverageWignerdistributiontothesolutionoftheradiativetransferequationwasrstprovedbyH.Spohnin[26℄fortime-independentpotentialsonsmalltimeintervals.ThisresultwasextendedtoglobalintimeonvergenebyL.ErdosandH.-T.Yau[13℄.TheseproofsinvolveinniteNeumann(diagram-mati)expansionsforthesolutionoftheShrodingerequationandarequiteinvolvedtehnially.Theorrespondingproblemwithtime-dependentpotentialsismuhsimplermathematially.Itwastreatedbyusin[4℄intheMarkovianase,andbyF.PoupaudandA.Vasseur[23℄intheaseofnite-rangetimeorrelations.InthispaperweusethefatthattheWignerfamilyarisingintime-reversalproblemsismoreregularthantheusualonebeauseblurringisaddedatthereordingarray.Thisprovidessomeadditionalregularityusuallyobtainedbyonsideringmixturesofstatesas,forinstane,in[21,23,26℄.Thepaperisorganizedasfollows:wedesribethesalingandobtainanexpressionforthebak-propagatedsignalintermsoftheWignertransforminSeti