Gaussian-type upper bound for the evolution kernels on nilpotent meta-abelian groups

25 Pages·2015·0.54 MB·English

by  Richard Penney

Positivity(2016)20:257–281 Positivity
DOI10.1007/s11117-015-0353-5
Gaussian-type upper bound for the evolution kernels on
nilpotent meta-abelian groups
RichardPenney1 · RomanUrban2
Received:6May2013/Accepted:6July2015/Publishedonline:5August2015
©TheAuthor(s)2015.ThisarticleispublishedwithopenaccessatSpringerlink.com
Abstract WegiveaGaussian-typeupperboundforthetransitionkernelsofthetime-
inhomogeneousdiffusionprocessesonanilpotentmeta-abelianLiegroupNgenerated
by the family of time dependent second order left-invariant differential operators.
These evolution kernels are related to the heat kernel for the left-invariant second
orderdifferentialoperatorsonhigherrank NAgroups.
Keywords Left invariant differential operators · Time-dependent parabolic
opearators·Brownianmotion·Evolutionkernel·Diffusionprocess·Meta-abelian
nilpotentLiegroups·Borell–TISinequality
Mathematics Subject Classification 43A85 · 31B05 · 22E25 · 22E30 · 60J25 ·
60J60
1 Introduction
Time-dependent parabolic equations and, in particular, the problem of finding the
upper and lower bounds for their fundamental solutions has attracted considerable
attentioninrecentyears(seee.g.[5,12–15,31]andthemonographsbyStroockand
Varadhan [27], and van Casteren [4]). The aim of this paper is to get a Gaussian-
B
RomanUrban
[email protected];[email protected]
RichardPenney
[email protected]
1 DepartmentofMathematics,PurdueUniversity,150N.UniversitySt,WestLafayette,IN47907,
USA
2 InstituteofMathematics,WroclawUniversity,PlacGrunwaldzki2/4,50-384Wrocław,Poland
258 R.Penney,R.Urban
type upper bound for the transition kernel of a particular kind of diffusion process
(evolution)onanilpotentmeta-abeliangroup N.Thetypeoftheevolutionequation
consideredherecomesfromthestudyoftheheatequationonaclassofsolvableLie
groups,thesocalledhigherrank NAgroupswhichare,bydefinition,thesemi-direct
productsofanilpotentandabelian(withdimensiongreaterthan1)groups(moreon
thatinSect.1.4).
1.1 Oursetting
Inwhatfollowsweassumethatthegroup N ismeta-abelian
N = M (cid:2)V,
where M andV areabelianLiegroupswiththecorrespondingLiealgebrasmandv.
Weconsiderafamilyofautomorphisms{(cid:2)(a)}a∈Rkofn,thatleavesmandvinvariant,
where a (cid:3)→ (cid:2)(a) is a homomorphism of Rk into Aut(n). Let m and v be spanned,
respectively, by {Y ,...,Y } and {X ,...,X }. We use these bases to identify m
1 d1 1 d2
andvwithRd1 andRd2 respectively.Wealsousetheexponentialmappingtoidentify
M andV withmandvandthuswithRd1 andRd2 respectively.Forx ∈ N wewrite
x =m(x)v(x)=mv =(m,v)wherem(x)=m ∈ M andv(x)=v ∈ V denotethe
componentsofx in M (cid:2)V.
Now we consider the action of an Lie abelian group A = Rk on N. We have a
semi-directproduct S = N (cid:2) A= N (cid:2)Rk withthemultiplicationinSgivenby
(x,a)(y,b)=(xya,a+b),
where,forx =expX, X ∈n,theactionofa ∈ A=expA=Rk on N isdefinedas
xa =exp((cid:2)(a)X).
Thegroup S isasolvableLiegroup.Therankof S is,bydefinition,equaltodimA.
Similarly,forg ∈ Swewriteg = x(g)a(g)= xa =(x,a),wherex(g)= x ∈ Nand
a(g)=a ∈ AdenotethecomponentsofginN (cid:2)A.Inwhatfollowsweidentifythe
group A,itsLiealgebraa,anda∗,thespaceoflinearformsona,withtheEuclidean
space Rk endowed with the usual scalar product (cid:6)·,·(cid:7) and the corresponding norm
(cid:8)a(cid:8)=(cid:6)a,a(cid:7)1/2.By(cid:8)·(cid:8)∞wedenotethemaximumnorm(cid:8)a(cid:8)∞ =max1≤j≤k|aj|.
Letσ beacontinuousfunctionfrom[0,+∞)to A=Rk,anddenote
(cid:2)σ(t)=(cid:2)(σ(t)).
Weassumealsothat
((AA12)) itnhethrees{tYriic}t1i≤oni≤Sd1σboafsi(cid:2)sσontomM,adcoXnissidloerweedratsriaanlginuelaarrfooprearalltoXro∈nvmanisdgivenin
the{Yi}1≤i≤d1 basisbyad1×d1lowertriangularmatrix:
Sσ(t)=(cid:2)σ(t)|M =[siσj]1≤i,j≤d1.
Gaussian-typeupperboundfortheevolution… 259
Specifically,fori ≥ j,
sσ(u)=hM(σ(u))eξj(σ(u)),
ij ij
wherehM ∈ R[a ,...,a ]arepolynomialsina ∈ A = Rk withhM = 1,for
ij 1 k jj
1≤ j ≤d ,andξ ,...,ξ ∈ A∗ =(Rk)∗.
1 1 d1
(A3) Thematrix
Tσ(t)=(cid:2)σ(t)|V =[tiσj]1≤i,j≤d2
isad ×d lowertriangularand,fori ≥ j,
2 2
tσ(u)=hV(σ(u))eϑj(σ(u)),
ij ij
wherehV ∈ R[a ,...,a ]arepolynomialsina ∈ A = Rk withhV = 1,for
ij 1 k jj
1≤ j ≤d ,andϑ ,...,ϑ ∈ A∗ =(Rk)∗.
2 1 d2
1.2 Evolutionkernel
Let,for Z ∈n,
Z(t)=(cid:2)σ(t)Z.
Let,
(cid:2)d2 (cid:2)d1
Lσ(t)= X (t)2+ Y (t)2.
N i j
i=1 j=1
NowweconsidertheevolutionprocessgeneratedbyLσ(t).ByC(N)wedenote
N
thesetofconinuousfunctionson N.Let
(cid:3) (cid:4)
C∞(N)= f ∈C(N): lim f(x)exists .
x→∞
Letd = dimn.For X ∈ n,welet X˜ denotethecorrespondingright-invariantvector
field.Foramulti-index I = (i ,...,i ),i ∈ Z+ andabasis X ,...,X oftheLie
1 d j 1 d
algebranwewrite XI = Xi1,...,Xid.Forκ,(cid:7)=0,1,2,...,∞wedefine
1 m
C(κ,(cid:7))(N)={f : X˜IXJ f ∈C∞(N) forevery|I|<κ +1and|J|<(cid:7)+1}
and
(cid:8)f(cid:8)0(κ,(cid:7)) = sup (cid:8)X˜IXJ f(cid:8)∞,
|I|=κ,|J|=(cid:7)
(cid:8)f(cid:8)(κ,(cid:7)) = sup (cid:8)X˜IXJ f(cid:8)∞.
|I|≤κ,|J|≤(cid:7)
260 R.Penney,R.Urban
In particular C(0,2)(N) with the norm (cid:8)f(cid:8)(0,2) is a Banach space. It is known (see
σ
[4,19,28])thatthereexiststhe(unique)familyofboundedoperatorsUs,tonC∞which
satisfies
(i) Usσ,s =Id,foralls ≥0,
(ii) limh→0Usσ,s+h f = f inC∞(N),
(iii) Usσ,rUrσ,t =Usσ,t,0≤s ≤r ≤t,
(iv) ∂sUsσ,t f =−LσN(s)Usσ,t f forevery f ∈C(0,2)(N),
(v) ∂tUsσ,t f =Usσ,tLσN(t)f forevery f ∈C(0,2)(N),
(vi) Usσ,t: C(0,2)(N)→C(0,2)(N)foralls ≤t.
The family Usσ,t is called the evolution generated by LσN(t). By Ptσ,s we denote the
correspondingkernel
(cid:5)
Uσ f(x)= Pσ (x;y)f(y)dy.
s,t t,s
N
SinceLσN(t)commuteswithlefttranslation,thesameistrueforUsσ,t.Hence,
Pσ (x;y)= Pσ (e;x−1y).
t,s t,s
Withasmallabuseofnotationwewrite
Pσ (x)= Pσ (e;x).
t,s t,s
σ
Hence,theoperatorUs,t isaconvolutionoperatorwithaprobabilitymeasure(witha
smoothdensity) Ptσ,s,
Uσ f = f ∗ Pσ .
s,t t,s
Wecall Ptσ,s(x)or Ptσ,s(x;y)theevolutionkernel.Sometimes Ptσ,s(x;y)iscalledthe
transitionkernelsinceinprobabilisticterms Ptσ,s(x;y)isthetransitionkernelforthe
time-dependent Markov process (or evolution), ω(t), on N defined by the operator
Lσ(t). Probability that starting from x at time s the proces ω(t) is in a given set
N
B ⊂ N is
(cid:5)
Ps,x(ω(t)∈ B)= Ptσ,s(x;y)dy.
B
By(iii),fors ≤r ≤t,
Pσ ∗ Pσ = Pσ .
t,r r,s t,s
Gaussian-typeupperboundfortheevolution… 261
1.3 Mainresult
Our aim is to estimate the evolution kernel Ptσ,s. In order to do this, first we disin-
tegratetheprocessω(t)intothecorrespondingprocesseson M and V respectively.
Specifically,let
(cid:2)d1 (cid:2)d2
Lσ (t)= Y (t)2 and Lσ(t)= X (t)2 (1.1)
M j V j
j=1 j=1
thoughtofasoperatorson M andV respectively.
Forv ∈ V,let
(cid:2)d1
Lσ (t)v = (Ad(v)Y (t))2. (1.2)
M j
j=1
ThentheoperatorLσ(t)istheskew-productoftheabovedefinedoperators,i.e.,
N
LσN(t)f(m,v)=LσV(t)f(m,·)|v +LσM(t)vf(·,v)|m, t ∈R+.
Thetime-dependentfamilyofoperatorsLσ(t)givesrisetoanevolutiononV =Rd2
V
that is described by a kernel PtV,s,σ which may be explicitly computed, since V is
abelian.Forη∈C∞([0,+∞),V)let
(cid:2)d1
Lσ (t)η = (Ad(η(t))Y (t))2.
M j
j=1
Thisfamilyofoperatorsgivesrisetoanevolutionon M =Rd1 thatisdescribedbya
M,σ,η
kernel Pt,s whichmayalsobeexplicitlycomputed(seeSect.4).
σ
Oneofourmaintoolsisthefollowingskew-productformulafor Pt,s (whichcan
beprovedalongthelinesof[23,Theorem1.2],wherediagonalactionof AonN was
considered).
Theorem1.1 Form ∈ M andv ∈ V,
(cid:5)
Pσ (m,v;m(cid:13),v(cid:13))f(m(cid:13),v(cid:13))dm(cid:13)dv(cid:13)
t,s
N (cid:5) (cid:5)
(cid:6) (cid:7)
= PtM,s,σ,η(m;m(cid:13))f m(cid:13),η(t) dm(cid:13)dWsV,,vσ(η)
M
whereWsV,,vσ istheprobabilitymeasureonthespaceC([s,+∞),V)generatedbythe
diffusionprocessη(t)startingfromv ∈ V attimes,withthegeneratorLσ(t).
V
A difficulty in applying the above formula is that the process η(t) does not have
independentcoordinates.ThisdifficultyisovercomewiththehelpofProposition3.1
262 R.Penney,R.Urban
whichgivestheestimateforthejointprobabilityofsupu∈[s,t](cid:8)η(u)(cid:8)∞andtheposition
oftheprocessηattimet,i.e.,η(t).Thismakesallthecomputationquiteinvolved.
In order to state our main theorem we need to introduce some notation. Let, for
1≤ j ≤d ,
2
V (τ,t)= max(Aσ(τ,s)− Aσ(τ,s)Aσ(τ,t)−1Aσ(τ,s)) , (1.3)
j s∈[τ,t] V V V V jj
where
(cid:5)
s
Aσ(τ,s)=2 Tσ(u)Tσ(u)∗du. (1.4)
V
τ
Set
(cid:5) (cid:5)
(cid:2) t (cid:8)d1 t
S(τ,t)= |siσj(u)|2du, S(cid:12)(τ,t)= e2ξj(σ(u))du,
τ τ
i≥j j=1
(cid:5) (cid:5)
(cid:2) t (cid:8)d2 t
T(τ,t)= |tiσj(u)|2du, T(cid:12)(τ,t)= e2ϑj(σ(u))du, (1.5)
τ τ
i≥j j=1
(cid:2)d2
V(τ,t)= V (τ,t).
j
j=1
Themainresultisthefollowingestimate.
Theorem1.2 Forevery T > 0 therearepositiveconstantsc ,c ,c andanatural
1 2 3
numberk suchthatforallT ≥t ≥τ ≥0andall(m,v)∈ N,
o
Ptσ,τ(m,v)
(cid:9) (cid:10)
≤c (cid:13)˜(τ,t,v)−(cid:8)v(cid:8)∞+2exp −c2(cid:8)v(cid:8)2 − c3(cid:8)m(cid:8)2
1S(cid:12)(τ,t)1/2T(cid:12)(τ,t)1/2 T(τ,t) ((cid:13)˜(τ,t,v)+1)2koS(τ,t)
(cid:11) (cid:12)
+c (cid:8)m(cid:8)2k1o exp −c2(cid:8)v(cid:8)2 − c3(cid:8)m(cid:8)2
1S(cid:12)(τ,t)1/2T(cid:12)(τ,t)1/2 T(τ,t) ((cid:13)˜(τ,t,v)+1+(cid:8)m(cid:8)2k1o)2koS(τ,t)
(cid:9) (cid:10)
+c1S(cid:12)(τ,t)−1/2T(cid:12)(τ,t)−1/2V(τ,t)1/2exp −Tc2((cid:8)τv,(cid:8)t2) − 2(cid:8)Vm((cid:8)τ1,/kto) , (1.6)
where
(cid:13)(τ,t,v)=sm∈[aτ,xt](cid:8)AσV(τ,s)AσV(τ,t)−1v(cid:8)∞, (1.7)
and
(cid:2)n
(cid:13)˜(τ,t,v)=(cid:13)(τ,t,v)+C V (τ,t)1/2. (1.8)
j
j=1
Gaussian-typeupperboundfortheevolution… 263
Remark InSect.7wegiveexplicitestimatesforthequantities(cid:13)˜(τ,t,v)andV(τ,t).
Remark GaussianestimatesinRnforthefundamentalsolutionofthetime-dependent
parabolic equations are usually obtained under the assumption that the operator is
(uniformly)elliptic(seee.g.classicalpapersbyAronson[2]andFabesandStroock
[11]). We do not require this condition and our estimate explicitly depends on the
coefficientsoftheoperator.
Remark IftheactionofAonN isdiagonal,i.e.,thepolynomialsinentriesofmatrices
Sσ(t)andTσ(t)[seetheassumptions(A2)and(A3)]satisfyhM =hV =0fori (cid:14)= j
ij ij
thenallthequantitiesappearinginTheorem1.2canbeeasilycomputed.Weget
(cid:5) (cid:5)
t (cid:2)d1 t
V (τ,t)=2 e2ϑj(σ(u))du, S(τ,t)= e2ξj(σ(u)du
j
τ τ
j=1
and
(cid:5)
(cid:2)d2 t
V(τ,t)=2 e2ϑj(σ(u))du T(τ,t)=V(τ,t)/2.
τ
j=1
Finally,
(cid:5) (cid:9)(cid:5) (cid:10)
s t −1
(cid:13)(τ,t,v)= max e2ϑj(σ(u))du e2ϑj(σ(u))du (cid:8)v(cid:8)∞ =(cid:8)v(cid:8)∞.
s∈[τ,t] τ τ
InthissettingTheorem1.2simplifiesandweobtain[23,Theorem4.1].
1.4 Applications
SincetheestimategivenbyTheorem1.2,atfirstglance,seemstobequitetechnical
andcomplicateditisworthtoexplainwhythisformulaisimportantandwhereitcan
beused.Firstofalltheestimatefor Ptσ,s,givenbyTheorem1.2,canbeappliedinthe
analysisofleft-invariant,second-orderdifferentialoperatorsonthehigherrank NA
groups,i.e.,thesemi-directproductN(cid:2)Rkasdescribedabove(atthismomentwedo
notassumethatN = M(cid:2)V).Consider,forα =(α ,...,α )∈Rk,theleft-invariant
1 k
differentialoperatoroftheform
(cid:2)d2 (cid:2)d1
Lα = Xj(a)2+ Yj(a)2+(cid:15)α, (1.9)
j=1 j=1
where
(cid:2)k
(cid:15)α = (∂a2j −2αj∂aj).
j=1
264 R.Penney,R.Urban
Inthissettingpropertiesofboundedharmonicfunctionson S iscertainlyofinterest.
Undersomeassumptiononthedriftvectorα thereexistsaPoissonkernel ν forLα
[6,7].Thatis,thereisaC∞ functionν on N suchthateveryboundedLα-harmonic
function F on S maybewrittenasaPoissonintegralagainstaboundedfunction f
onS/A= N,
(cid:5) (cid:5)
F(g)= f(gx)ν(x)dx = f(x)νˇa(x−1xo)dx, g =(xo,a),
S/A N
where
νˇa(x)=ν(a−1x−1a)χ(a)−1,
whereχ isthemodularfunctionforleftinvariantHaarmeasureonS,i.e.,
χ(g)=det(Ad(g)).
ConverselythePoissonintegralofany f ∈ L∞(N)isaboundedLα-harmonicfunc-
tion.
ItisknownthatthePoissonkernelν isequaltolimt→∞πN(μt),whereπN(g)=
x(g) is a projection from S onto N. To get some information on μ we use a well
t
knownformulawhichexpressT asaskew-productofthediffusionon N and A.For
t
f ∈C (N ×Rk)andt ≥0,
c
Tt f(x,a)=EaU0σ,t f(x,σt)=Ea(f ∗N Ptσ,0)(x,σt), (1.10)
where the expectation E is taken with respect to the distribution of the process σ
t
(Brownianmotionwithdrift)inRk generated by(cid:15)α.TheoperatorU0σ,t actsonthe
firstvariableofthefunction f (asaconvolutionoperator).Theideaofsuchadecom-
positiongoesbackto[16,17,29].Inthecontextof NA groupswithdimA = 1this
decompositionwasusedin[7–10],andlaterwasgeneralizedbytheauthorsandapplied
fordimA > 1,seee.g.[20,22].NotethatTheorem1.1isageneralizationof (1.10)
toevolutionoperators.
EstimatesforthePoissonkernelfortheoperator(1.9)wereobtainedbytheauthors
in a series of papers [20–24]. However, in all these papers the action of A on N is
diagonal.ThusTheorem1.2opensthedoortoconsidernon-diagonalactions.Thisis
goingtobethesubjectofourfutureresearch.
1.5 Structureofthepaper
Theoutlineoftherestofthepaperisasfollows.InSect.2westatetheformulaforthe
evolutionkernelinRn andrecalltheBorel(cid:6)l–TISinequalitywhichisinSect.3use(cid:7)din
theproofofanappropriateestimateforP sups∈[τ,t](cid:8)η(s)(cid:8)∞ ≥u andη(t)∈ B for
u ∈Rand B ⊂Rn.InSects.4and5westudyevolutionson M andV,respectively.
Finally in Sect. 6 we give the proof of Theorem 1.2 and in Sect. 7 we give some
estimatesforquantietiesgivenin(1.7)and(1.8).
Gaussian-typeupperboundfortheevolution… 265
2 Preliminaries
2.1 Gaussianvariablesandfields
We follow the presentation in [1]. For Rn-valued random variables X and Y their
covariancematrixisdefinedasCov(X,Y)=E(X −EX)(Y −EY)t.AnRn-valued
random variable X is said to be multivariate Gaussian if fo(cid:13)r every non-zero α =
(α ,...,α )∈Rn,therealvaluedrandomvariable(cid:6)α,X(cid:7)= n α X isGaussian.
1 n i=1 i i
Inthiscasethedensityof X isgivenbythemultivariatenormaldensity
(2π)−n/2(detC)−1/2e−21C−1(x−m)·(x−m),
where m = EX and C = Cov(X,X) is a positive semi-definite n ×n covariance
matrix.Inthiscasewewrite X ∼N (m,C)orsimply X ∼N(m,C).
n
Lemma2.1 Let X ∼N (m,C).Assumethatd <nandmakethepartition
n
X =(X1,X2)=((X1,...,Xd),(Xd+1,...,Xn)),
m =(m1,m2)=((m1,...,md),(md+1,...,mn))
and
(cid:14) (cid:15)
C C
C = 11 12 ,
C C
21 22
whereC isad×d-matrix.Theneach Xi ∼N(mi,C )andtheconditionaldistri-
11 ii
butionof Xi given Xj isalsoGaussian,withmeanvector
mi|j =mi +CijC−jj1(Xj −mj)
andcovariancematrix
Ci|j =Cii −CijC−jj1Cji.
Proof Seee.g.[1,p.8]. (cid:16)(cid:17)
A random field is a stochastic process, taking values in some space, usually in a
Euclidean space, and defined over a parametric space T. A real valued Gaussian
processisarandomfield f onaparameterset T forwhichthe(finitedimensional)
distributionsof (f ,..., f )are multivariate Gaussianfor each 1 ≤ n < +∞ and
t1 tn
each(t ,...,t )∈Tn.
1 n
2.2 Gaussianinequalities
Thefollowingpowerfulinequalitywasdiscoveredindependently,andwasprovedin
verydifferentways,byBorell[3]andTsirelsonetal.[30].Following[1]wecallthe
followinginequalityBorell–TISinequality.
266 R.Penney,R.Urban
Theorem2.2 (Borell–TISinequality) Let f beacenteredGaussianprocess,almost
t
surelyboundedonT.Write|f|T =supt∈T ft.ThenE|f|T <+∞and,forallu >0,
P(|f|T −E|f|T >u)≤e−u2/2σT2,
where
σ2 =supEf2.
T t
t∈T
Proof Fortheproofseetheoriginalpapers[3,30]or[1]. (cid:16)(cid:17)
Immediately,wegetthefollowing
Corollary2.3 Let f beacenteredGaussianprocess,almostsurelyboundedon T.
t
Thenforallu >E|f| ,
T
P(|f|T >u)≤e−(u−E|f|T)2/2σT2.
2.3 EvolutionequationinRn
Let
(cid:2)n (cid:2)n
1
L(t)= a (t)∂ ∂ + δ (t)∂ , (2.1)
ij i j j j
2
i,j=1 j=1
where∂ =∂ anda(t)=[a (t)]isasymmetric,positivedefinitematrixandthea
i xi ij ij
andδj belongtoC([0,∞),R).Fors > t,let Pt,s betheevolutionkernelgenerated
by L(t).Let,for1≤i, j ≤n,
(cid:16)(cid:5) (cid:17)
t
As,t =[Aij(s,t)]= aij(u)du ,
(cid:16)(cid:5)s (cid:17)
t
Ds,t =[Dj(s,t)]= δj(u)du . (2.2)
s
Proposition2.4 TheevolutionkernelPt,scorrespondingtotheoperatorL(t)defined
in(2.1)isgivenby
Pt,s(x)=(2π)−n2(detAs,t)−21e−12(A−s,t1(x−Ds,t))·(x−Ds,t).
Proof Seee.g.[23,Proposition2.9] (cid:16)(cid:17)
3 Mainprobabilisticestimate
Consider the operator L(t), defined in (2.1), without the drift vector δ(t) =
(δ (t),...,δ (t)),i.e,
1 n