Second order elliptic partial differential equations driven by Lévy white noise 2021 David Berger.pdf

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ModernStochastics:TheoryandApplications8(2)(2021)179–207
/21-VMSTA181
Secondorderellipticpartialdifferentialequations
drivenbyLévywhitenoise
DavidBergera,∗,FaridMohamedb
aTUDresden,InstituteofMathematicalStochastics,ZellescherWeg12-14,
01069Dresden,Germany
bUlmUniversity,InstituteofMathematicalFinance,Helmholtzstraße18,
89081Ulm,Germany
david.******@tu-(),farid.******@uni-()
Received:11February2021,Revised:21May2021,Accepted:21May2021,
Publishedonline:22June2021
AbstractThispaperdealswithlinearstochasticpartialdifferentialequationswithvariable
coefflcientsdrivenbyLé,anexistencetheoremforintegraltransformsof
Lévywhitenoiseisderivedandtheexistenceofgeneralizedandmildsolutionsofsecondorder
,thegeneralizedelectricSchrödinger
operatorfordifferentpotentialfunctionsVisdiscussed.
KeywordsStochasticpartialdifferentialequations,Lévywhitenoise
2010MSC60H15,60H40,35J15,35J10
1Introduction
SincethebeginningofstudyingpartialdifferentialequationstheLaplacianoperator
d
:=∂2wasofgreatinterestindifferentmathematicaltheoriesandapplications.
j
j=1
Forexample,thesolutionofthePoissonequation
−u=f
∗Correspondingauthor.
©2021TheAuthor(s)..
:.
,
forsomefunctionfcanbeinterpretedasastationarysolutionoftheheatequation
-
ityassumptionsinthespace,thedivergenceoperator
d
div(A(x)∇u):=∂i(aij(x)∂ju)
i,j=1
wasintroduced,
kindofoperatoris,forexample,usedintheMaxwellequationsingeneralmedia(see
[16]).
ThefundamentalsolutionoftheLaplaceequationiswell-known,butthereis
noexplicitformforafundamentalsolutionofageneraldivergenceformoperator,
althoughthereexistupperandlowerbounds,see,forexample,[9].
Thegoalofthispaperistoobtaingeneralizedsolutionsoftheequation
p(x,D)s=L,˙
whereL˙isaso-calledgeneralizedLévywhitenoiseandpisapartialdifferential
operatoroftheform
−div(A(x)∇u)+b(x)·∇u+V(x)u,u∈C∞(Rd),(1)
forauniformlyellipticRd-valuedmatrixfunctionAandfunctionsb:Rd→Rd,
V:Rd→
electricSchrödingeroperatordrivenbyaLévywhitenoise,
solutionuofthestochasticpartialdifferentialequation
−div(A(x)∇u)+V(x)u=L,˙(2)
whereAisauniformlyellipticd×dmatrix,thepotentialV>0belongstothere-
verseHölderclassandL˙isaLé
Schrödingeroperatorhasexponentialdecay,wewillderiveweakerassumptionson
theLévywhitenoiseincomparisontothegeneralcase(1)toshowtheexistenceof

in[2],
thisshortcomingwederiveexistenceresultsforgeneralizedrandomprocessescon-
structedbyintegraltransformsoftheunderlyingLé,we
studydifferentdistributionalpropertiesofthesesolutionsandshowthatwecancon-
structperiodicallystationarygeneralizedrandomprocesses.
Wearesolvingthestochasticpartialdifferentialequationsindistributionalsense,
s,p(x,D)∗ϕ=
L,ϕ˙-
lutionsofpartialdifferentialequationssee,forexample,[8].Untilnowtheredoesnot
existagoodunderstandingofLévywhitenoisedrivenstochasticpartialdifferential
equationsundergeneralmomentconditions,butthereexistsliteratureforthecaseof
GaussianwhitenoiseandLé[17]
-
tialdifferentialequationswithconstantcoefflcients,seealso[3]and[2].Ourmethod:.
SecondorderellipticpartialdifferentialequationsdrivenbyLévywhitenoise181
isinspiredbythepaper[5]andtheresultsof[12].Wealsomentionthemonograph
[11],whichgivesagoodoverviewaboutSPDEsdriven
byLévynoise,whereanotherapproachmotivatedbythesemigrouptheoryisusedto
considerparabolicandhyperbolicSPDEsdrivenbyLévynoiseinBanachspaces.
InSection2weprovidethegeneralframeworkneededtodiscussstochasticpar-
tialdifferentialequationsdrivenbyLévywhitenoise,whosesolutionsaredeflned
évywhitenoiseasageneralizedran-
(see[6]).Theorem1
impliesthatalargeclassoflinearstochasticpartialdifferentialequationsdrivenby
aLévywhitenoisehasageneralizedsolution,whereweusedamoregeneralkernel
G:Rm×Rd→[2].Furthermore,
westudythemomentpropertiesofgeneralizedrandomprocessessdrivenbyLévy
whitenoiseL˙.Forawell-deflnedrandomprocesss(ϕ)=L,G(ϕ)˙,ϕ∈D(Rd)
weshowinTheorem3thatifL˙hasflniteβ>0moment,thenshasalsoflniteβ-
,weshowthatifshas
flniteβ-moment,thenalsoL˙hasflniteβ-
example,thepartialdifferentialoperatorsoftheform(1)andgiveexistenceresults
,wediscussperiodicallystationarysolutionss
ödingeropera-
tordrivenbyLévywhitenoiseandshowunderweakerconditions,asintheexample
above,-
lutionof(2),
Lévywhitenoisewiththefundamentalsolutionof(2).InProposition3wemention

usedlateronisstandardorself-
LebesguemeasureonRdandD(Rd)thespaceoftestfunctionsonRd,
ofinflnitelydifferentiablerealvaluedfunctionsonRdwithcompactsupport,andD
itsdualspace,.
2IntegraltransformsandgeneralizedstochasticprocessesdrivenbyLévy
whitenoise
Weprovidethegeneralframeworkneededtodiscussstochasticpartialdifferential
equationsdrivenbyaLévywhitenoiseandintroduceaLévywhitenoiseasgeneral-
(see[6]).In[2]
itwasshownthataconvolutionoperator,withcertainpropertiesregardinghisinte-
grability,deflnesageneralizedrandomprocess,assuminglowmomentconditionson
theLé[2],wewillusethecharacterizationoftheextended
domain(see[5],)andachievenewresultsforamoregeneralkernel
G:Rm×Rd→R,whichallowsusinSection3tomodeldifferentkindsofstation-
arityassumptionsandalsotoobtaingeneralizedsolutionsofLévy-drivenstochastic
partialdifferentialequations.
Let(,F,P)beaprobabilityspace.
Deflnition1(See[5],).Ageneralizedrandomprocessisalinear
andcontinuousfunctions:D(Rd)→L0().Thelinearitymeansthat,forevery:.
,
ϕ,ϕ∈D(Rd)andμ∈R,
12
s(ϕ1+μϕ2)=s(ϕ1)+μs(ϕ2)almostsurely.
Thecontinuitymeansthatifϕn→ϕinD(Rd),thens(ϕn)convergestos(ϕ)in
probability.
DuetothenuclearstructureonD(Rd)itfollowsfrom[17],
ageneralizedrandomprocesshasaversionwhichisameasurablefunctionfrom
(,F)to(D (Rd),C)withrespecttothecylindricalσ-fleldCgeneratedbythesets
{u∈D (Rd)|(u,ϕ,...,u,ϕ)∈B}
1N
withN∈N,ϕ1,...,ϕN∈D(Rd)andB∈B(RN).Fromnowonsuchaversionis
meantalways.
Theprobabilitylawofageneralizedrandomprocesssistheprobabilitymeasure
onD (Rd)givenby
Ps(B):=P(s∈B)=P({ω∈:s(ω)∈B})
forB∈C,whereCisthecylindricalσ-fleldonD (Rd).
Thecharacteristicfunctionalofageneralizedrandomprocesssisthefunctional
P:D(Rd)→Cdeflnedby

Ps(ϕ)=exp(iu,ϕ)dPs(u).
D (Rd)
Thecharacteristicfunctionalcharacterizesthelawofsinthesensethattworandom
processesareequalinlawifandonlyiftheyhavethesamecharacteristicfunctional.
NowwedeflnetheLévywhitenoise,whichiscloselyconnectedtoaLévyprocess.
Ingeneral,aLévyprocessisastochasticallycontinuousprocesswithindependent
évyprocess(Lt)t≥0ischaracterizedby
itscharacteristicfunction
EeizLt=exp(tψ(z))
foreveryz∈Randt≥évyexponent;itcanbecharacterizedby
a≥0,γ∈RandaLévymeasureν,

ν({0})=0andmin{1,x2}ν(dx)<∞.
R\{0}
Forallz∈Ritholdsthat

12ixz
ψ(z)=iγz−az+(e−1−ixz1|x|≤1)ν(dx).
2
R
Thefunctionψisuniquelycharacterizedbythetriplet(a,γ,ν)knownasthechar-
:.
SecondorderellipticpartialdifferentialequationsdrivenbyLévywhitenoise183
évywhitenoiseL˙onRdisageneralizedrandomprocesswith
characteristicfunctionaloftheform
⎛⎞

⎜d⎟
PL˙(ϕ)=exp⎝ψ(ϕ(x))λ(dx)⎠
Rd
foreveryϕ∈D(Rd),whereψ:R→CisaLévyexponent.
TheexistenceoftheLévywhitenoisewasshownin[6].Anotherpossiblewayto
constructLévywhitenoisewouldbeviaindependentlyscatteredrandommeasures,

andareindependentlyscatteredwhentwoindicatorfunctionswithdisjointsupports
deflneindependentrandomvariables([12]).In[5]
évywhite
noise,deflnedasgeneralizedrandomprocesses,toindependentlyscatteredrandom
[5],whichmadeitpossibletoextend
thedomainofdeflnitionofLévywhitenoisetosomeBorel-measurablefunctions
f:Rd→˙ifthereexistsasequence
ofelementaryfunctionsfnconvergingalmosteverywheretofsuchthatL,f˙n1A
convergesinprobabilityforn→∞foreveryBorelsetAandsetL,f˙asthelimit
˙˙m˙
inprobabilityofL,fnforn→∞,whereL,fnisdeflnedbyj=1ajL,1Aj
m
foraelementaryfunctionfn:=j=1aj1Aj,seealso[5],
maximaldomainoftheLévywhitenoiseL˙wewriteD(L)˙.BysettingL(A):=
L,˙1AforboundedBorelsetsA,theextensionofaLévywhitenoiseL˙canbe
identifledwithaLévybasisLinthesenseofRajputandRosinski[12],see[5],
évybasiscanbeidentifledwithaLévy
whitenoiseinacanonicalway,.L,ϕ˙:=dϕ(x)dL(x)forϕ∈D(Rd),we
R
makenodifferencebetweenaLévywhitenoiseandaLé,a
Borel-measurablefunctionf:Rd→RisinD(L)˙ifandonlyiffisintegrable
withrespecttotheLévybasisLinthesenseofRajputandRosinski[12],see[5],
.
Deflnition3(See[7],.).Forameasurablefunctionf∈L0(Rd)we
deflnethedistributionfunctionoffas
df(α)=λd({x∈Rd:|f(x)|>α}),α>0.
Withtheaidofthedistributionfunctionwecannowobtainasufflcientcondition
fortheexistenceofthegeneralizedrandomprocesssdeflnedbys(ϕ)=L,G(ϕ)˙,
whereG:Rm×Rd→,weusethere-
sultsfrom[12]and[5]regardingintegrabilityconditionsforLé
contrastto[2],wheretheexistenceofthestationarygeneralizedrandomprocess
s(ϕ)=L,G˙∗ϕwasobtained,thismoregeneralkernelG:Rm×Rd→R
,thiswill:.
,
becrucialinSection3forprovingtheexistenceofgeneralizedprocessesassolutions
tostochasticpartialdifferentialequationsasin(1).
˙beaLévywhitenoiseonRmwithcharacteristictriplet(a,γ,ν)
andG:Rm×Rd→∈Rmand
R>0

GR(x):=|G(x,y)|λd(dy)∈[0,∞]
BR(0)
and
1/x
h(x):=xd(α)λ1(dα)forx>0.
RGR
0
AssumethatG∈L1(Rm)∩L2(Rm)and
R

1|r|>1hR(|r|)ν(dr)<∞(3)
R
d
foreveryR>(ϕ)(x):=G(x,y)ϕ(y)λ(dy)wehavethat
Rd
s(ϕ):=L,G(ϕ)˙,ϕ∈D(Rd),
deflnesageneralizedrandomprocess.
[2],-
(ϕ)∈D(L)˙and
L,G(ϕ˙n)→L,G(ϕ)˙asn→∞inprobabilityforasequence(ϕn)n∈Ncon-
vergingtoϕinD(Rd).AsL,G(˙·)islinear,thisisequivalenttocheckingthat
L,G(ϕ˙n−ϕ)→0asn→∞inprobability(see[5],).Nowgiven
[12],,wehavetoshow




m
γG(ϕn)(