Weak universality of dynamical $\Phi^4_3$: non-Gaussian noise

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by  Hao Shen

Weak universality of dynamical Φ4: non-Gaussian noise
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February 3, 2016
1 2
Hao Shen and Weijun Xu
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1
1 ColumbiaUniversity,US,Email: [email protected]
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2 2 UniversityofWarwick,UK,Email: [email protected]

b
e
Abstract
F
2 Weconsider a class of continuous phase coexistence models in three spatial dimen-
sions. The fluctuations are driven by symmetric stationary random fields with suf-

]
h ficient integrability and mixing conditions, but not necessarily Gaussian. We show
p that,intheweaklynonlinear regime,iftheexternalpotentialisasymmetricpolyno-
-
h mialand acertain average ofitexhibits pitchfork bifurcation, then these models all
t rescaletoΦ4 neartheircriticalpoint.
a 3
m
[ 1 Introduction

2
v ThedynamicalΦ4 modelis formallygivenby
d
4
2 ∂ Φ = ∆Φ−λΦ3 +ξ, (1.1)
7 t
5
0 where ξ is the space-time white noise in d spatial dimensions, namely the Gaussian
.
1 randomfield withcovarianceformallygivenby
0
6 Eξ(s,x)ξ(t,y) = δ(s−t)δ(d)(x−y).
1
:
v This equation was initially derived from the stochastic quantisation of the Euclidean
i
X quantum field theory [PW81]. It is also believed to be the scaling limit for Kac-Ising
r model near critical temperature (conjectured in [GLP99]). Note that formally, one
a
could rescale Φ to turn the coefficient in front of the cubic term to be 1, but we still
retain thatλ herein ordertosimplifythescalinglater.
In dimension d = 1, the equation is well-posed, and it was shown in [BPRS93]
that the dynamic Ising model on the real line does rescale to it. However, as soon as
d ≥ 2,theequationbecomesill-posedinthesensethatthe“solution”Φisdistribution
valued so that the non-linearity Φ3 does not make sense. In d = 2, the solutioncould
beconstructedeitherviaDirichletforms([AR91])orviaWickordering([DPD03],by
turningthetermΦ3 intotheWickproduct :Φ3: withrespecttotheGaussianstructure
induced bythelinearised equation). Recently,Mourrat and Weber ([MW14]) showed
that 2D Ising model withGlauber dynamicsand Kac interactiondoes converge to the
Wick ordered solutionconstructedin [DPD03].
INTRODUCTION 2
The three dimensional case is much more involved as it exhibits further logarith-
mic divergence beyond Wick ordering. It became amenable to analysis only very
recently,eitherviathetheoryofregularitystructures([Hai14]),orviacontrolledpara-
products([GIP15,CC13]),orWilson’srenormalisationgroupapproach([Kup15]). In
all these cases, the solution is obtained as the limit of smooth solutions Φ to the
ǫ
equations
∂ Φ = ∆Φ −λΦ3 +ξ +C Φ , (1.2)
t ǫ ǫ ǫ ǫ ǫ ǫ
where ξ is a regularisation of the noise ξ at scale ǫ, and C → +∞ could be chosen
ǫ ǫ
suchthatthelimitisindependentoftheregularisation. Ford ≥ 4,onedoesnotexpect
toget anynontriviallimit([Aiz82, Fro¨82]).
Thethreedimensionalequationisalsobelievedtobetheuniversalmodelforphase
coexistence near criticality. In the recent work [HX16], Hairer and the second author
studiedphasecoexistencemodelsin threespatialdimensionsofthetype
∂ u = ∆u−ǫV′(u)+ζ, (1.3)
t θ
where ζ is a space-time Gaussian random field with correlation length 1, and V is
θ
a polynomial potential whose coefficients depends smoothly on a parameter θ. The
main result of [HX16], built on analogous results for the KPZ equation in [HQ15], is
that ifV is symmetricand its certain average exhibitsa pitchforkbifurcation near the
origin,thenthereexistsachoiceof
θ = aǫlogǫ+O(ǫ)
such thattherescaled process
uǫ(t,x) := ǫ−21u(t/ǫ2,x/ǫ), (1.4)
converge to the solution of the Φ4 equation, interpreted as the limit in (1.2), and the
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coefficient λ of the cubic term in the limit is a linear combination of all coefficients
a (0)’s of the polynomial V . We denote this limit as the Φ4(λ) family (where the
j 0 3
term “family”comes from thefact that onecould perturb thefinitepart oftheinfinite
renormalisation constant, and end up with a one-parameter family of limits). The
main purpose of this article is to show that when ζ is a symmetric, stationary noise
withstrongmixingbutnotnecessarilyGaussian,thenthemacroscopicprocessu still
ǫ
converges totheΦ4 familywitha properchoiceofθ (depending onǫ).
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Toappreciatethedifficultieswithnon-Gaussiannoise,wefirstnotethatthemacro-
scopicprocess u satisfies theequation
ǫ
∂tuǫ = ∆uǫ −ǫ−32Vθ′(ǫ12uǫ)+ζǫ, (1.5)
where
ζǫ(t,x) = ǫ−52ζ(t/ǫ2,x/ǫ) (1.6)
is an approximation to the space-time white noise at scale ǫ. Since the right hand
side of (1.5) involves diverging products of u as ǫ → 0, we recall from [Hai14],
ǫ
[HQ15] and [HX16] that the strategy is to lift for each ǫ > 0 the equation (1.5) to
INTRODUCTION 3
an abstract space of regularity structures associated with a (random) model, solvethe
lifted equation as an abstract fixed point problem, and then project down to the “real”
solution with the reconstruction operator. An essential ingredient in this procedure is
to bound arbitrarily high moments of the renormalised random models to prove their
convergence.
If the random models are all built from Gaussian noise, then by equivalence of
moments, one only needs to check the bounds for the second moments, and those for
all higher moments will follow immediately. However, for non-Gaussian noise, one
hastoboundthep-thmomentsoftherandommodelsbyhandforallp,whichinvolves
complicatedexpressionsofgeneralised convolutionsofsingularkernels.
In [HQ15], the authors developed a graphic machinery for generalised convolu-
tions,andreducedtheestimatesofthesecomplicatedconvolutionstotheverifications
of certain conditions on labelled graphs. The p-th moment of a random object is the
sumofcertaingraphs,eachobtainedbyglueingtogetherthenodesofpidenticaltrees
in a certain way, and each of the identical trees represents that random object. When
p is large, it is in general very hard to keep track of all the conditions that need to
be checked. In [HS15] and [CS16], the verification procedure was further reduced to
checking the conditions for every sub-tree of a single tree (instead of a large graph
consistingofptrees)thatrepresentstherandomobject(seeAssumption4.4andTheo-
rem 4.6 below). This significantlysimplifiesthe verification; see for example[HS15]
inthecaseofKPZ equation.
However, in the current problem, Eq. (1.5) will involve arbitrarily large powers
of u in contrast to [HS15] where only the second power appears. Thus, even for a
ǫ
singletree,itwillbeverydifficulttochecktheconditionsinAssumption4.4forevery
sub-tree of it. In addition, trees built from the non-linearity with high powers of u
ǫ
willin general fail theintegrabilityconditions(1)– (3)inAssumption4.4.
To overcome the second difficulty, we employ the positive “homogeneities” rep-
resented by the multiplication of ǫ, and incorporate them into the “non-integrable”
treesinacertainwaytocreate“integrable”trees(see(1.12)andthebeginningofSec-
tion4.4). On theotherhand, tosystematicallyandefficiently checking theconditions
for these new “integrable” trees, we make a key observation in Lemma 4.11 below,
which shows that one only needs to check the conditions for very few sub-trees, and
the verification for all other sub-trees will automatically follow from that. We expect
these treatments and simplifications will apply to other situations, for example prov-
inguniversalityoftheKPZ equationforpolynomialmicroscopicgrowthmodelswith
non-Gaussiannoise.
Before we state our main theorem, we first give precise assumptions on the noise
ζ as wellas thepotentialV .
θ
Assumption 1.1. We assume the random field ζ defined on R×(ǫ−1T)3 satisfies the
followingproperties:
1. ζ issymmetric(inthesensethatζ l=aw −ζ),stationary,continuous,andE|ζ(z)|p <
+∞for allp > 1.
2. ζ isnormalisedso that̺(z) = Eζ(0)ζ(z)integratesto 1.
INTRODUCTION 4
3. For any two compact sets K ,K ⊂ R×(ǫ−1T)3 with distance at least 1 away
1 2
fromeachother,thetwosigmafieldsF andF areindependent,whereF
K1 K2 Ki
istheσ-algebrageneratedbythepointevaluations{ζ(z) : z ∈ K }.
i
Typicalexamplesofsuchrandomfieldsaresmeared-outversionsofPoissonpoint
processes with uniform space-time intensity. We refer to [HS15] for more details of
theseandrelatedexamples. Givensucharandomfieldζ,weletΨtobethestationary
solutiontotheequation
∂ Ψ = ∆Ψ+ζ. (1.7)
t
Since we are in dimensionthree, such a stationary solutionexistsas the square of the
heat kernel is integrable at large scales. Let µ denote the distribution of Ψ evaluated
at aspace-timepoint,and definetheaveraged potentialhV i tobe
θ
hV i(x) := V (x+y)µ(dy). (1.8)
θ θ
Z
R
Sinceζ isassumedtobesymmetric,µisalsosymmetric,sohV iistheconvolutionof
θ
V and µ. OurmainassumptiononV isthefollowing.
θ
Assumption 1.2. Let µ denote the distribution of the stationary solution to (1.7),
where ζ is a random field satisfying Assumption 1.1. Let V : (θ,x) 7→ V (x) be a
θ
symmetricpolynomialinx with coefficients dependingsmoothlyon θ. Weassumethe
averaged potential hV i obtained from V via (1.8) has a pitchfork bifurcation near
θ θ
theoriginin thesensethat
∂4hVi ∂2hVi ∂3hVi
(0,0) > 0, (0,0) = 0, (0,0) < 0. (1.9)
∂x4 ∂x2 ∂θ ∂x2
Remark1.3. NotethatitisalwaysthederivativeV′ ratherthanV thatappears inthe
θ θ
dynamics,sotheconstanttermofV willnothaveany rolein thiscontext. Suppose
m
hV i′(x) = a (θ)x2j+1, (1.10)
θ j
Xj=0
b
thenthepitchforkbifurcationassumption(1.9)reads
a > 0, a = 0, a′ < 0, (1.11)
1 0 0
where we haveused the nobtationa = ba (0) and a′ b= a′(0) for simplicity. From now
j j j j
on, wewillassumetheaveraged potentialhVi hastheexpression(1.10), and satisfies
b b b b
thepitchforkbifurcationassumption(1.11).
Ourmaintheorem can belooselystatedas follows.
Theorem 1.4. Letζ andV satisfyAssumptions1.1and1.2. Letu(dependingonboth
ǫandθ)bethesolutionto(1.3). Then, thereexistsa < 0suchthatforθ = aǫ|logǫ|+
O(ǫ), therescaled processu in(1.4)converges to theΦ4(a )familysolutions.
ǫ 3 1
b
INTRODUCTION 5
Remark1.5. Thesymmetryassumptiononboththepotentialandthenoiseisessential
to get Φ4 limit. In [HX16], it was shown that with Gaussian (symmetric) noise but
3
asymmetric potential, the system would in general rescale to either OU process or
Φ3, depending on the value of θ. We expect the same phenomena to happen with
3
asymmetricnoiseevenifthepotentialissymmetric.
Before we proceed, we first briefly explain why the pitchfork bifurcation assump-
tion naturally appears on hVi but not V. To see this, we note that if u solves the
microscopic equation (1.3), then the rescaled process u defined in (1.4) solves the
ǫ
equation
m
∂ u = ∆u − a (θ)ǫj−1u2j+1+ζ , (1.12)
t ǫ ǫ j ǫ ǫ
Xj=0
where a (θ)’s are thecoefficientsofV′, and
j θ
ζǫ(t,x) = ǫ−52ζ(t/ǫ2,x/ǫ) (1.13)
is the rescaled noise. Heuristically, as u2 ∼ ǫ−1 as ǫ → 0, so each of the terms
ǫ
ǫj−1u2j+1 (j ≥ 1) has a “diverging part” of the order ǫ−1u , and the combined effect
ǫ ǫ
ofthesedivergencesisthenaconstantmultipleofthatorder. Itturnsoutthatthevalue
ofthisconstantisnothingbut
V′(y)µ(dy) =: a ,
0 0
Z
R
b
so a = 0 in (1.11) is a necessary condition for the right hand side of (1.12) to con-
0
verge. In fact, as we will see later, with a = 0, the Wick renormalisation (for non-
0
b
Gaussiannoiseζ )isautomaticallytakenaccountof,andthusthecoefficientsofthese
ǫ
divergentterms withorderǫ−1 willprecisbely cancel out each other.
Also, as in the standard Φ4 case, there will be further logarithmic divergence be-
3
yondtheWickordering. But thisdivergencecan berenormalisedbyasuitablechoice
ofthesmallparameterθ thankstoa′ 6= 0. Finally,a > 0isneeded forthefinal limit
0 1
tobetheΦ4 family.
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b b
The article is organised as follows. In Section 2, we briefly recall the facts of
non-Gaussian random variables and their Wick powers. Section 3 is devoted to the
construction of the regularity structures, the abstract fixed point equation as well as
the renormalisation group. Most of the set-up in these sections could be found in
[Hai14], [HS15], and [HX16], sowekeep thepresentationto aminimumand refer to
thosearticles formoredetails.
In Section 4, we prove the bounds for the renormalised models constructed in
Section 3.3 with proper renormalisation constants. These bounds are necessary in
provingtheconvergenceofrenormalisedmodels. Finally,in Section5, wecollectthe
results in all previous sections to prove the convergence of our models to the desired
Φ4 limitat large scales.
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Acknowledgements
WewouldliketothankAjayChandrafordiscussionsongeneralmomentbounds. WX
acknowledges the Mathematical Sciences Research Institute in Berkeley, California
NON-GAUSSIAN WICK POWERS AND AVERAGED POTENTIAL 6
for its warm hospitality, and the National Science Foundation for supporting his stay
here.
2 Non-Gaussian Wick powers and averaged potential
Inthissection,wereviewsomebasicpropertiesofWickpolynomialsfornon-Gaussian
random variables. We keep the presentation to the minimum here and only give defi-
nitions and statements that will be used later. More details can be found in Section 3
of[HS15].
2.1 Jointcumulants
Westartwiththedefinitionofjointcumulants. Givenacollectionofrandomvariables
X = {X } forsomeindexsetAand a subsetB ⊂ A, we writeX ⊂ X and XB
α α∈A B
as short-handsfor
X = {X : α ∈ B} , XB = X . (2.1)
B α α
Y
α∈B
Given a finite set B, we furthermore write P(B) for the collection of all partitions of
B, i.e. all sets π ⊂ P(B) (the power set of B) such that π = B and such that any
twodistinctelementsofπ are disjoint. S
Definition 2.1. Given a collectionX of random variablesas above and any finiteset
B ⊂ A, we definethecumulantE (X )inductivelyover |B|by
c B
E(XB) = E (X ) . (2.2)
c B¯
π∈XP(B)BY¯∈π
Remark 2.2. It is straightforward to check that the above definition does determine
thecumulantsuniquely.
From now on, we will use the notation C for the nth joint cumulant function of
n
thefield ζ:
C (z ,...,z ) = E ({ζ(z ),··· ,ζ(z )}) . (2.3)
n 1 n c 1 n
WesimilarlywriteC(ǫ)forthen-thcumulantbutwithζ(z )’sreplacedbyζ (z )’s. Note
n j ǫ j
thatC = 0sinceζ isassumedtobesymmetric. Also,C isitscovariancefunction.
2n+1 2
An importantproperty isthat thereexistsC > 0 suchthat
C(ǫ)(z ,··· ,z ) = 0 (2.4)
p 1 p
whenever|z −z | > Cǫfor somei,j.
i j
2.2 Wickpolynomials
ThenotionofWickproductsforrandomvariables(notnecessarilyGaussian)willplay
an essentialrolelater. Wegiveadefinitionbelow.
NON-GAUSSIAN WICK POWERS AND AVERAGED POTENTIAL 7
Definition 2.3. Given a collection X = {X } of randomvariables as before, the
α α∈A
Wickproduct :X : forA ⊂ A isbysetting :X : = 1 andpostulatingthat
A 6#
XA = :X : E (X ) (2.5)
B c B¯
BX⊂A π∈PX(A\B)BY¯∈π
recursively.
Remark 2.4. Again, (2.5) is sufficient to define :X : by induction over the size
A
of A. By the definition we can easily see that as soon as A 6= #6 , one always has
E :X : = 0. Note also that taking expectations on both sides of (2.5) yields exactly
A
theidentity(2.2).
Note that by (2.2), we can alternatively write XA = :X : E(XA\B), and
B⊂A B
there is also a formula to express the Wick product in tePrms of the usual products;
however, (2.5) is actually the identity frequently being used in the paper: in fact we
will frequently rewrite a product of generally non-Gaussian noises as a sum of terms,
eachtermcontainingaWickproductasRHSof(2.5)-calledthenon-Gaussianhomo-
geneouschaosof order|B|.
It isalso well-knownthat thereisan alternativecharacterisation ofWick products
viagenerating functionsby
∂n exp( n t X )
:X ···X : = i=1 i i . (2.6)
1 n ∂t1···∂tn Eexp(P ni=1tiXi)(cid:12)t1=...=tn=0
(cid:12)
P (cid:12)
In thiscaseX = ··· = X all withdistributionµ,then (2.6)can bereduced to
1 n
∂n etX
:Xn: = (2.7)
∂tn(cid:18)E etY (cid:19)(cid:12)
µ (cid:12)t=0
(cid:12)
(cid:12)
where Y is distributed according to µ. Note that there is no n in the exponential
generating function above, as the derivativeis taken with respect to the same t rather
thant ,··· ,t separately. Theform of(2.7)also suggeststhat wecan actuallydefine
1 n
then-thWick powerwithrespect toameasureµ as
∂n exp(tx)
W (x) = . (2.8)
n,µ ∂tn Eµexp(tY)(cid:12)t=0
(cid:12)
(cid:12)
Now, W (·) is a polynomial in the variable x ∈ R, rather than the random variable
n,µ
X. One can immediatelycheck using the definition (2.8) that d W (x) = nW ,
dx n,µ n−1,µ
and this means that the Wick powers form an Appell sequence. As a comparison to
theWickpowers,werecall thattheusualmonomialscan begenerated by
∂n
xn = etx . (2.9)
∂tn (cid:12)t=0
(cid:12)
(cid:12)
We thenhavethefollowingimportantlemma.
REGULARITY STRUCTURES AND ABSTRACT EQUATION 8
Lemma2.5. LetµbeaprobabilitymeasureonRwithfinitemomentsofallorder,and
V bea polynomial. Let
hVi(x) = V(x+y)µ(dy) = E V(x+Y)
µ
Z
R
be the average of V against the measure µ. Then, hVi(x) = xn if and only if V(x) =
W (x).
n,µ
Proof. We first prove the “if” part. Let V(x) = W (x), then by definitions of W
n,µ n,µ
and h·i,wehave
∂n etxetY ∂n etxE etY ∂n
hVi(x) = E = µ = etx = xn,
µ∂tn(cid:16)EµetZ(cid:17)(cid:12)t=0 ∂tn(cid:16) EµetZ (cid:17)(cid:12)t=0 ∂tn (cid:12)t=0
(cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12)
whereforthesecondequalitywehaveexchangedthedifferentiationwiththeexpecta-
tion with respect to Y. For the “only if” part, suppose U is another polynomial with
hUi = xn, thenU −V isapolynomialsuch that
(U −V)(x+y)µ(dy) = 0
Z
R
for all x ∈ R. It then follows easily that all coefficients of U −V must be 0, and we
haveU = V = W .
n,µ
3 Regularity structures and abstract equation
In this section, we build the appropriate regularity structures in order to solve the
equation
∂tuǫ = ∆uǫ −ǫ−23Vθ′(ǫ12uǫ)+ζǫ (3.1)
in an abstract space. The set-up follows essentially the same way as in [HX16] and
[Hai15].
3.1 Regularity structures andadmissiblemodels
Recall that a regularity structure consists of a pair (T ,G), where T = T is a
α∈A α
vector space graded by a (bounded below, locally finite) set A of homogLeneities, and
G is agroupoflineartransformationson T suchthat foreveryΓ ∈ G and τ ∈ T ,we
α
haveΓτ −τ ∈ T .
<α
Similaras[HX16],thebasiselementsinT arebuiltfromthesymbols1,Ξ,{X }3
i i=0
and the abstract integrationoperators I and Ek (unlike [HX16], we only need integer
powers of E here). In order to reflect the parabolic scaling of the equation, for any
multi-index of non-negative integers k = (k ,··· ,k ), we define an abstract mono-
0 3
mialofdegree|k|as
3
Xk = Xk0···Xk3, |k| = 2k + k , (3.2)
0 3 0 i
Xi=1
REGULARITY STRUCTURES AND ABSTRACT EQUATION 9
and theparabolicdistancebetween z,z′ ∈ R4 by
3
|z −z′| = |z0 −z0′|12 + |zi −zi′|. (3.3)
Xi=1
Inviewofthestructureof(3.1),wedefinetwosetsU andV suchthatXk ∈ U,Ξ ∈ V,
and suchthat foreveryk = 1,··· ,m, wedecree
τ ,···τ ∈ U ⇒ Ek−1(τ ···τ ) ∈ V,
1 2k+1 1 2k+1
(3.4)
τ ∈ V ⇒ I(τ) ∈ U.
The idea is that U consists of the formal symbols in the expansion of the abstract
solution of the equation, and V consists of symbols that will appear in the expansion
oftherighthandsideof(3.1). Wethen letW = U ∪V to bethesetofbasiselements
inT . Asforthehomogeneities,weset
5
|Ξ| = − −κ, |Xℓ| = |ℓ|,
2
and extendittoall ofW decreeing
|ττ¯| = |τ|+|τ¯|, |I(τ)| = |τ|+2, |Ek(τ)| = k +|τ|.
Here, κ is a small but fixed positive number, and |ℓ| denotes the parabolic degree of
the multi-index defined in (3.2). According to (3.4), we can keep adding new basis
elements into U and V, but for the purpose of this article, it suffices to restrict our
regularitystructurestoelementswith homogeneitieslessthan 3,i.e., T = T .
2 α<3 α
2
ThebasisvectorsinW generatedthiswaywithnegativehomogeneitiesLotherthan
Ξ areoftheform (withshorthandΨ = I(Ξ)):
1
τ = Ek−1Ψ2k+1−n , |τ| = (n−3)−(2k +1−n)κ ; (3.5)
2
k −1 ℓ 1
τ = E⌊(k−1)/2⌋(ΨkI(E⌊ℓ/2⌋−1Ψℓ)) , |τ| = ⌊ ⌋+⌊ ⌋−(k +ℓ)( +κ)+1
2 2 2
provided κ is sufficiently small, where in the first case n ∈ {0,1,2,3}, and in the
second onethesituationwhen k oddand ℓ evenisexcluded.
As for the transition group G, we define the extended regularity structure T and
ex
a linear map ∆ : T → T ⊗ T in the same way as in [HQ15, Section 3.1]. For
ex ex +
any linear functional g on T , one can obtain a linear map Γ : T → T by setting
+ g ex ex
Γ τ = (id⊗g)∆τ. The transition group G is defined by the collection of all Γ ’s
g g
for linear functional g on T with the further property that g(ττ¯) = g(τ)g(τ¯). The
restrictionofG onT givestheregularitystructure(T ,G).
Remark3.1. ThegradedvectorspaceofextendedregularitystructureT isthelinear
ex
span ofsymbolsin
W ∪{τ ,··· ,τ : τ ∈ U}.
1 2m+1 j
ThemainadvantageofintroducingT isthatEk canbeviewedasalinearmapdefined
ex
on T . As a consequence, the definition of the renormalisation group in Section 3.3
ex
willbesignificantlysimplified.
REGULARITY STRUCTURES AND ABSTRACT EQUATION 10
Givenaregularitystructure(T ,G), amodel foritis apair(Π,F)offunctions
Π : R1+3 → L(T,D′) F : R1+3 → G
z 7→ Π z 7→ F
z z
satisfyingtheidentity
Π F−1 = Π F−1, ∀z,z¯∈ R1+3
z z z¯ z¯
as wellas thebounds
|(Π τ)(ϕλ)| . λ|τ|, |F−1 ◦F | . |z −z¯||τ|−|σ| (3.6)
z z z z¯ σ
uniformlyoveralltestfunctionsϕ ∈ B,allspace-timepointsz,z¯inacompactdomain
K,and all |τ| ∈ W. Here, thesetB denotesthespaceoftestfunctions
B = {ϕ : kϕk ≤ 1,supp(ϕ) ⊂ B(0,1)},
C2
whereB(0,1)istheunitballcentredatorigin,andϕλ (t,x) = λ−5ϕ((t−t′)/λ2,(x−
t′,x′
x′)/λ). We will also use the notation f for the multiplicative element in T∗ (the
z +
adjointofT )suchthat F = Γ .
+ z fz
The norm of a model M = (Π,f) is defined to be the smallest proportionality
constantsuchthatbothboundsin(3.6)hold,whichwewilldenoteby|||M|||. Thenorm
ingeneraldependsonthecompactdomainK,butweomititfromthenotationjustfor
simplicity. Since in most situations the transition group F is completely determined
by Π,we willalso sometimeswrite|||Π|||insteadof|||M|||.
In addition to these minimal requirements, we also impose the admissibility on
our models. Let K : R1+3 \{0} → R be a function that coincides the heat kernel on
{|z| ≤ 1}, smooth everywhere except the origin, and annihilates all polynomials on
R1+3 up to parabolic degree 2. The existence of such a kernel can be easily checked
(seeforexample[Hai14, Section5]).
WiththistruncatedheatkernelK,wethenletM tobethesetofadmissiblemodels
(Π,f)suchthatonehas
(Π Xk)(z¯) = (z¯−z)k, f (Xk) = (−z)k, ∀k (3.7)
z z
as wellas
f (J τ) = − DℓK(z −z¯)(Π τ)(dz¯), 0 ≤ |ℓ| < |τ|+2
z ℓ z
Z
(3.8)
(·−z)ℓ
(Π I(τ))(·) = (K ∗Π τ)(·)+ ·f (J τ),
z z z ℓ
ℓ!
X
0≤|ℓ|<|τ|+2
and we set f (J τ) = 0 if |ℓ| ≥ |τ| + 2. Note that the above notion of admissible
z ℓ
modelsdonotimposeanyrestrictionson theoperatorEk.
Givenasmoothfunctionζ and ǫ ≥ 0,wenowbuildthecanonicalmodelL (ζ) =
ǫ
(Πǫ,fǫ)as follows. DefinetheactionofL (ζ)onXk according to (3.7), and set set
ǫ
(Π Ξ)(z¯) = ζ(z¯).
z