Polynomial mixing for the complex Ginzburg--Landau

arXiv:math-ph/0609041v114Sep2006PolynomialmixingforthecomplexGinzburg–LandauequationperturbedbyarandomforceatrandomtimesVahagnNersesyanLaboratoiredeMath´ematiques,Universit´edeParis-SudXIBˆatiment425,91405OrsayCedex,FranceE-mail:[e-mail]blemofergodicityforthecomplexGinzburg–Landau(CGL)equationperturbedbyanunboundedrandomkick-force.Randomnessisintroducedboththroughthekicksandthroughthetimesbetweenthekicks.WeshowthattheMarkovprocessassociatedwiththeequationinquestionpossessesauniquestationarydistributionandsatisfiesapropertyofpolynomialmixing.1IntroductionWeconsidertheCGLequationperturbedbyarandomkick-forceonadomainD⋐Rn,n≤4with∂D∈C2:˙u−νΔu+iβ|u|2u=η(t,x),x∈D,(1.1)u|∂D=0,(1.2)u(0,x)=u0(x),(1.3)whereu=u(t,x)andν,β0.Weassumethatη(t,x)isarandomprocessoftheformη(t,x)=∞Xk=1ηk(x)δ(t−τk),(1.4)whereδ(t)istheDiracmeasure,ηkareindependentidenticallydistributed(i.i.d.)randomvariableswithrangeinthespaceH:=H10(D),andthewaitingtimestk=τk−τk−1,k≥2andt1=τ1arei.i.d.randomvariablesexponentiallydistributedwithparameterλ.Moreover,weassumethatthesequencesηk,tkareindependent.Supposethat{gk}∞k=1isanorthonormalbasisinH.ThemainresultofthepresentpaperisTheorem4.2,whichstatesthat,ifthelowofηkisnon-degenerateonthespacespannedby{gk}Nk=1forsufficientlylargeN,thenthereis1auniquestationarymeasureforthecontinuoustimeMarkovprocessassociatedwith(1.1),(1.2),(1.4).Moreover,anysolutionoftheproblempolynomiallyconvergestothestationarymeasureinthedualLipschitznorm.ManyauthorshavestudiedsimilarproblemsforvariousPDE’swithdifferentrandomperturbations(e.g.,see[14,1,15,16,19,13,20,12,23]fordiscreteforcingand[5,7,2,17,6,8,22,24,3]forwhitenoise).Severalideasofthisarticlearetakenfrom[16,13,23].TheproblemofergodicityforrandomlyforcedGinzburg–Landauequationwasstudiedinthefollowingarticles.In[8],HairerconsideredarealGinzburg–Landauequationonmultidimensionaltorus.Odasso[22]studiedaclassofCGLequationswithstrongnonlineardissipation.Inbothoftheseworksthepropertyofexponentialmixingisestablished.In[24],ShirikyanusedasufficientconditionforergodicityofMarkovprocessestoshowuniquenessandmixingforaclassofCGLequationswithlineardispersion.Finally,in[3],DebusscheandOdassoprovedthepolynomialmixingpropertyforadamped1DSchr¨odingerequation.Themainnoveltyofthepresentpaperistheconditionoverthewaitingtimes.Notethattherestrictionofthesolutionattimesτklooksliketherandomdy-namicalsystemsconsideredbyKuksin,Shirikyan[12],[14],[23]andMasmoudi,Young[19]:uτk=Stk(uτk−1)+ηk,(1.5)buttherearesomeessentialdifferences.Asthewaitingtimescanbearbitrarilysmall,duringanytimeintervalthesystemcanreceiveanynumberofkicks.Thischangesthedynamicsoftheassociatedprocess,forexample:•Thedistancebetweentwotrajectorieshavingcloseinitialdatacanbearbitrarylargeatanytimet0.•Thephasespaceoftheproblemisnotboundedeveninthecaseofboundedkicks.LetusgiveinafewwordstheideasoftheproofofTheorem4.2.AnimportanttoolfortheproofoftheresultistheFoia¸s–Proditypeestimate.ThiskindofestimatesareoftenusedtoproveergodicpropertiesofPDE’s.Supposethattherearetwosequencesofkicksζkandζ′k,havingequalhighFouriermodesfork≥l,suchthatthesolutionsofcorrespondingproblemshaveequallowFouriermodesatkickingtimesτk,k≥l(seeLemma2.1fortheexactformulation).LetNtbethenumberofkicksbeforetimet,i.e.Nt=max{k:τk≤t}.Then,byFoia¸s–ProdiLemma,wehavethefollowingestimateforthedistencebetweensolutionsattimet,ift≥τl:kut−u′tk1≤e−C(Nt−l)NtYi=l+1ti
−12eϕkuτl−u′τlk1,(1.6)wherek·k1standsforthenorminH,utandu′taresolutionscorrespondingtothesequencesζkandζ′krespectively,ϕisapolynomialfunctionof{kuτik1}Nti=l2and{ku′τik1}Nti=landC0isalargeconstant.Followingtheideasfrom[23],weconstructtwosequencesζkandζ′kofi.i.d.randomvariablesinHdistributedasη1suchthattheconditionsofFoia¸s–ProdiLemmaaresatisfiedforarandomintegerℓ≥1.Moreover,usingthelowoflargenumbersandsomemartingaleinequalities,weshowthatℓcanbechooseninasuchwaythatthefollowingpropertiesalsohold:(i)QNti=ℓ+1ti
−12eϕ≤e(Nt−ℓ),ifNt≥ℓ+1,(ii)kuτℓk1+ku′τℓk1≤1,(iii)Eℓp≤Cp.AsweshowinSection4,properties(i)-(iii)and(1.6)implythepolynomialmixingproperty.Therandomvariablesζk,ζ′kandℓareconstructedinProposition4.3.InSection4,weshowproperlyhowTheorem4.2isderivedfromProposition4.3.TheproofofProposition4.3iscarriedoutinSections5and6.Notethatanexponentialestimatefortherandomvariableℓin(iii)impliesimmediatelytheexponentialmixingpropertyforthesystem.Finally,using(i)-(iii),onecanshowthattheembeddedMarkovchainuτkalsosatisfiesapropertyofpolynomialmixing.ThestationarymeasureoftheoriginalprocessandthatofembeddedchainareconnectedwiththeKhasminskiirelation(seeSection4).Acknowledgments.TheauthorthanksA.Shirikyanforattractinghisat-tentiontothisproblemandforhelpfuldiscussionsandencouragements.ApartofthispaperwaswrittenwhentheauthorwasvisitingtheDeGiorgiCenter(Pisa);hethankstheCenterforhospitality.NotationLetD⊂Rnbeaboundeddomainwithsmoothboundaryandlet{gj}j∈NbeanorthonormalbasisinH.LetHNbethevectorspanof{g1,...,gN}andH⊥NbeitsorthogonalcomplementinH.WedenotebyPNandQNtheor-thogonalprojectionsontoHNandH⊥NinH.Denot