Ornstein-Uhlenbeck-Cauchy Process

arXiv:chao-dyn/9910028v120Oct1999Ornstein–Uhlenbeck–CauchyProcessPiotrGarbaczewskiInstituteofPhysics,PedagogicalUniversityPL-65069ZielonaG´ora,PolandandRobertOlkiewiczInstituteofTheoreticalPhysics,UniversityofWroclawPL50-204Wroclaw,PolandFebruary5,2008AbstractWecombineearlierinvestigationsoflinearsystemswithL´evyfluc-tuations[Physica113A,203,(1982)]withrecentdiscussionsofL´evyflightsinexternalforcefields[Phys.Rev.E59,2736,(1999)].WegiveacompleteconstructionoftheOrnstein-Uhlenbeck-CauchyprocessasafullycomputablemodelofananomaloustransportandaparadigmexampleofDoob’sstablenoise-supportedOrnstein-Uhlenbeckpro-cess.Despitethenonexistenceofallmoments,wedeterminelocalcharacteristics(forwarddrift)oftheprocess,generatorsofforwardandbackwarddynamics,relevant(pseudodifferential)evolutionequa-tions.Finallyweprovethatthisrandomdynamicsisnotonlymixing(henceergodic)butalsoexact.TheinducednonstationaryspatialprocessisprovedtobeMarkovianandquiteapartfromitsinherentdiscontinuitydefinesanassociatedvelocityprocessinaprobabilisticsense.11MotivationThecasualunderstandingofthecentrallimittheorem(inreferencetotheBoltzmann-Gibbsthermostatics),combinedwiththeneedtohaveclearlyspecifiedthemeanfeatures(momentsandlocalconservationlaws)ofran-domlyimplementedtransport,atoroffthermalequlibrium,resultedinanobviouspredominanceofGaussianlawsofprobabilityanddiffusionprocessesintypicalstatistical(eventuallyprobabilistic,cf.theomnipresenceoftheBrownianmotionconceptualbackground)analysisofphysicalphenomena.Presently,weobserveacontinuallygrowingrecognitionoftheprofoundrˆole(ubiquity,[1])playedbynon-GaussianL´evydistributions(probabilitylaws)inbothaconsistentprobabilisticinterpretationofvariousexperimen-taldataandinastochasticmodellingofphysicalphenomena,followedbynumericalandrealisticexperimentationattemptstoverify(orratherfalsify)probabilistichypotheses.Generically,L´evy’sprobabilitylawsappearinthecontextofanomalousdiffusions(mostlysubdiffusionsthataremodelledintermsofcontinuousran-domwalks,[2]).Ontheotherhand,underthenameofL´evyflights,[2,3]weencounterstochasticjump-typeprocesseswhichareexplicitlyassociatedwiththosedistributions.Thatallowsinturntomodelquiteavarietyoftrans-portprocesses,cf.[3]whichareeitherregardedas(non)typicalphenomenaofnonequilibriumstatisticalphysicsorasmanifestationsofacomplexnon-lineardynamicswithsignaturesofchaos,yieldinganenhanceddiffusioninparticular.WefocusourattentiononL´evyflightswhichareconsideredaspossiblemodelsofprimordialnoise,[4,5].(Wienernoiseorprocessisnormallyin-terpretedtorepresentastatistical”stateofrest”oftherandommedium).Generically,thevarianceandhighercumulantsofthoseprocessesareinfi-nite(nonexistent).Thereisalsophysicallymoresingularsubclassofsuchprocessesforwhicheventhefirstmoment(meanvalue)isnonexistent.ThusweneedtorelaxthelimitationsofthestandardGaussianparadigm:wefacehereafundamentalproblemofestablishingothermeans(thanvariancesandmeanvalues)tocharacterisestatisticalpropertiesofL´evyprocesses(frac-2tionalmomentsofRef.[6]areinsufficienttoolsinthisrespect).Specifically,ifahabitualstatisticalanalysisisperformedonanyexper-imentallyavailablesetoffrequencydata,thereisnoobviousmethodtoextractareliableinformationabouttendencies(localmeanvalues)oftherandomdynamics.Nonexistenceofmeanvaluesandhighermomentsmayalsobeinterpretedasthenonexistenceofobservable(e.g.mean,likedriftsorlocalcurrents)regularitiesofthedynamics.Moreover,thejump-typepro-cessesusuallyadmitarbitrarilysmalljumps(withnolowerbound)andfinite,butarbitrarilylargejumpsizes(withnoupperbound).Anycomputersimu-lationutilizesboththelower(coarse-graining)andupperboundonthejumpsize,[8,9,10],andanyexperimentaldatacollectioninvolvessuchlimitationsaswell.Mathematically,thatputsusintheframeworkofstandardjumpprocessesforwhichthecentrallimittheoremisknowntoholdtrueinitsGaussianversion(evenifweaccountfortheabnormallyslowconvergencetoaGaussian,inviewoflongtailsoftheprobabilitydistribution,[8]).There-fore,thereisnoclear-cutproceduresallowingtoattributeanunambigousstatisticalinterpretationintermsofL´evyprocessestogivenphenomenolog-icaldata.InadrasticcontrasttoatraditionalGaussianmodelling.Merescalingarguments,reflectingtheself-similiarpatternsofsamplepaths,areinsufficientaswell.Althoughnorealisticformulationofafluctuation-dissipationtheoremispossibleinthatcase(nonexistenceofvariances),wecangiveameaningtoatheoryofL´evyflightsinexternalforcefields,[3],underasimplifyingassumptionthatforcefieldsdefinelinearprocesseswithL´evyfluctuations.ThecorrespondingvelocityprocesseswereintroducedinRef.[6](seealso[11]),butweshallgiveacompleteconstructionoftherelatedjump-typestochasticprocess,togetherwithadetailedcharacterizationofthedynamicsofinducedspatialdisplacements.Ourstrategyisthussubstantiallydifferentfromthattypicallyfollowedinthecurrentliterature,[3].Forexample,theconfigurationspaceLangevinequation,dx(t)dt=F(x)mγ+η(t)(1)wheremisthemassoftransportedparticle,γstandsforthefrictioncon-3stantandηrepresentsanyconceivablegeneralizationofthewhite-noisethatemploysL´evystablestatistics,[3],corroboratesatacitassumptio