The parabolic Anderson model

83 Pages·2012·0.42 MB·English

WeierstrassInstitutefor
AppliedAnalysisandStochastics
The parabolic Andersonmodel
BasedonjointworkswithMarekBiskup(CˇeskéBudeˇjoviceandLosAngeles),JürgenGärtner(Berlin),RemcovanderHofstad
(Eindhoven),StanislavMolchanov(Charlotte),PeterMörters(Bath)andNadiaSidorova(London)
WolfgangKönig
WIASBerlinandTUBerlin
Mohrenstrasse39·10117Berlin·Germany·Tel.[phone]·www.wias-berlin.de·Langeoog,7November2012
RandomMotionsinRandomMedia
Importantmodelsforavarietyofsituationsandreal-worldapplications.Examples:
(cid:4) randomwalkinrandomenvironment
(cid:4) randomwalkinrandomscenery
(cid:4) randomwalkamongrandomconductances
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RandomMotionsinRandomMedia
Importantmodelsforavarietyofsituationsandreal-worldapplications.Examples:
(cid:4) randomwalkinrandomenvironment
(cid:4) randomwalkinrandomscenery
(cid:4) randomwalkamongrandomconductances
Butwewillbeconcernedwith
(cid:4) randommotionsinrandompotential,
whicharecloselyconnectedwith
(cid:4) spectraofrandomoperators.
Theoperatorsthatweconsiderhaveakineticpartandarandompotential.More
precisely,theyarerandomSchrödingeroperators.
Welookatthetimedependentproblemandstudylong-timeproperties.Thisis
closelyconnectedwithspectraltheory,inparticular,Andersonlocalisationproperties,
butonlyclosetothetopofthespectrumoftherandomoperator.
Warning:Weuseprobabilisticsignconvention.
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TheParabolicAndersonModel
WeconsidertheCauchyproblemfortheheatequationwithrandomcoefficientsand
localisedinitialdatum:
∂
u(t,z) = ∆du(t,z)+ξ(z)u(t,z), for(t,z)∈(0,∞)×Zd, (1)
∂t
u(0,z) = δ0(z), forz∈Zd. (2)
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TheParabolicAndersonModel
WeconsidertheCauchyproblemfortheheatequationwithrandomcoefficientsand
localisedinitialdatum:
∂
u(t,z) = ∆du(t,z)+ξ(z)u(t,z), for(t,z)∈(0,∞)×Zd, (1)
∂t
u(0,z) = δ0(z), forz∈Zd. (2)
(cid:4) ξ=(ξ(z): z∈Zd)i.i.d.randompotential,[−∞,∞)-valued.
(cid:4) ∆df(z)=∑y∼z f(y)−f(z) discreteLaplacian
(cid:2) (cid:3)
(cid:4) ∆d+ξAndersonHamiltonian(acelebratedrandomSchrödingeroperator)
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TheParabolicAndersonModel
WeconsidertheCauchyproblemfortheheatequationwithrandomcoefficientsand
localisedinitialdatum:
∂
u(t,z) = ∆du(t,z)+ξ(z)u(t,z), for(t,z)∈(0,∞)×Zd, (1)
∂t
u(0,z) = δ0(z), forz∈Zd. (2)
(cid:4) ξ=(ξ(z): z∈Zd)i.i.d.randompotential,[−∞,∞)-valued.
(cid:4) ∆df(z)=∑y∼z f(y)−f(z) discreteLaplacian
(cid:2) (cid:3)
(cid:4) ∆d+ξAndersonHamiltonian(acelebratedrandomSchrödingeroperator)
Thesolutionu(t,·)isarandomtime-dependentshift-invariantfield.
Itsa.s.existenceisguaranteedunderamildmomentconditiononthepotential.
Ithasallmomentsfiniteifallpositiveexponentialmomentsofξ(0)arefinite.
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MotivationsandComments
Interpretations/Motivations:
(cid:4) Randommasstransportthrougharandomfieldofsinksandsources.
(cid:4) Expectedparticlenumberinabranchingrandomwalkmodelinafieldof
randombranchingandkillingrates.
(cid:4) AndersonHamiltonian∆d+ξdescribesconductancepropertiesofalloysof
metals,oropticalpropertiesofglasseswithimpurities.Manyopenquestions
aboutdelocalisedversusextendedstates.
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MotivationsandComments
Interpretations/Motivations:
(cid:4) Randommasstransportthrougharandomfieldofsinksandsources.
(cid:4) Expectedparticlenumberinabranchingrandomwalkmodelinafieldof
randombranchingandkillingrates.
(cid:4) AndersonHamiltonian∆d+ξdescribesconductancepropertiesofalloysof
metals,oropticalpropertiesofglasseswithimpurities.Manyopenquestions
aboutdelocalisedversusextendedstates.
Comments:
(cid:4) Inthespecialcaseξ(z)∈{−∞,0},wecallsiteszwithξ(z)=−∞a(hard)trap.
Thenu(t,x)isequaltothesurvivalprobabilityuptotimetinx.
(cid:4) Thespatiallycontinuousversion(Brownianmotioninsteadofrandomwalk)is
alsohighlyinteresting.
Backgroundliteratureandsurveys:[MOLCHANOV1994],[CARMONA/MOLCHANOV
1994],[SZNITMAN1998],[GÄRTNER/K.2005].
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Maintools
Feynman-Kacformula
t
u(t,z)=E0hexpnZ0 ξ(X(s))dso1l{X(t)=z}i, z∈Zd,t>0,
where(X(s)) isthesimplerandomwalkonZd withgenerator∆d,startingfromz
s∈[0,∞)
underPz.
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Maintools
Feynman-Kacformula
t
u(t,z)=E0hexpnZ0 ξ(X(s))dso1l{X(t)=z}i, z∈Zd,t>0,
where(X(s)) isthesimplerandomwalkonZd withgenerator∆d,startingfromz
s∈[0,∞)
underPz.
Eigenvalueexpansion
t
u(t,z)≈E0hexpnZ0 ξ(X(s))dso1l{X(t)=z}1l{X[0,t]⊂B(2)(t)}i
=∑etλk(ξ,B(2)(t))ϕk(0)ϕk(z),
k
where(λk(ξ,B(2)(t)),ϕk)k isasequenceofeigenvaluesλ1>λ2≥λ3≥... and
L2-orthonormaleigenfunctionsϕ1,ϕ2,ϕ3,... of∆+ξinsomebox
B(2)(t)=tlog2t×[−1,1]d withzeroboundarycondition.
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