Large scale behaviour of 3D continuous phase coexistence models

0.5 MB·

by  Martin Hairer

Large scale behaviour of 3D continuous
phase coexistence models
January21, 2016
6
1
0 Martin Hairer1 and Weijun Xu2
2

n 1 UniversityofWarwick,UK,Email: [email protected]
a 2 UniversityofWarwick,UK,Email: [email protected]
J

9
1 Abstract

] We study a class of three dimensional continuous phase coexistence models, and
h
show that, under different symmetry assumptions on the potential, the large-scale
p
- behaviour ofsuchmodelsnearabifurcation pointisdescribed bythedynamical Φp
h 3
t modelsforp 2,3,4 .
a ∈ { }
m
Contents
[

1
1 Introduction 1
v
8
2 Constructionoftheregularitystructure 9
3
1 3 Abstractfixedpointproblem 15
5
0 4 Convergence oftherenormalised models 25
.
1
5 Identificationofthelimits 39
0
6
1
1 Introduction
:
v
i
X Theaimofthisarticleistostudythelargescalebehaviorofphasecoexistencemodels
r ofthetype
a
∂ u = ∆u ǫV′(u)+δξ , (1.1)
t − θ
in three spatial dimensions, where V denotes a potenbtial depending on some param-
θ
eter θ and ǫ, δ are two small parameters. Throughout this article, ξ is assumed to be
a continuousspace-timeGaussian random field modellingthelocal fluctuations,with
b
covariancehavingcompact supportand integratingto 1. Thepotential(θ,u) V (u)
θ
7→
is a sufficiently regular function (depending on the regime, we will actually assume
thatitispolynomialinu). Regardingthetwoparametersǫandδ,wewillconsidertwo
extremal regimes: either ǫ = o(1),δ 1, which we call the weakly nonlinearregime,
≈
or δ = o(1),ǫ 1, which we call the weak noiseregime. However,our results would
≈
INTRODUCTION 2
easily carry overto intermediateregimes as well. Also, thespatial domainof the pro-
cess uisalargethreedimensionaltoruswhosesizedependsonǫ (seeRemark 1.2for
moredetails).
For the sake of the present discussion, consider the weakly nonlinear regime, i.e.
set δ = 1 in (1.1). It is then natural to consider scalings of the type u (t,x) =
λ
λ−1/2u(tλ−2,xλ−1) whichleaveinvariantthestochasticheat equation,sothat u sat-
ǫα
isfies
∂ u = ∆u ǫ1−5α/2V′(ǫα/2u )+ξ , (1.2)
t ǫα ǫα − θ ǫα ǫα
whereξ denotesasuitablerescalingofξ whichapproximatesspace-timewhitenoise
ǫ¯
at scales largerthan ¯ǫ.
b
Remark 1.1. Since the process u in (1.1) itselfdepends on ǫ, oneshould really write
u for the rescaled process in (1.2) to avoid ambiguity. However, we still write the
ǫ,ǫα
ambiguousoneu here inorderto keep thenotationssimple.
ǫα
The form (1.2) suggests that if we start (1.1) with an initial condition located at
a local minimum of V, then at scales of order ǫ−1/2 (i.e. setting α = 1 in (1.2))
2
solutionsshouldbewellapproximatedbysolutionstoanOrnstein-Uhlenbeckprocess
ofthetype
∂ v = ∆v cv +ξ , (1.3)
t
−
for some c > 0 and ξ a space-time white noise. As we will see in Theorem 5.2
below, this is in general false, unless V is harmonic to start with. Instead, one should
computefromV aneffectivepotential V inthefollowingway. Considerthespace-
θ θ
h i
timestationarysolutionΨ tothelinearisedequation
∂ Ψ = ∆Ψ+ξ . (1.4)
t
Sinceweareindimension3,suchasolutionexistbsandisGaussianwithfinitevariance
C . Wethen set
0
V (x) = V (x+y)µ(dy),
θ θ
h i Z
R
where µ = (0,C ). In other words, V is the effective potential obtained by aver-
0 θ
N h i
agingV againstthestationarymeasureofΨ. WeshowinTheorem5.2thatifwestart
with an initialconditionlocated at a local minimumof V , then itis indeed thecase
θ
h i
thatthebehaviourat scales oforderǫ−1/2 is describedby (1.3).
These considerations suggest that more interesting nonlinear scaling limits can
arise in regimes where θ V undergoes a bifurcation, and this is the main object
θ
7→ h i
of study of this article. In particular, if V is symmetric and undergoes a pitchfork
θ
h i
bifurcation at some θ = θ , then one would expect the large-scale behaviour to be
0
describednearθ bythedynamicalΦ4modelbuiltin[Hai14b]andfurtherinvestigated
0 3
in [CC13, Kup15]. Similarly, near a saddle-node bifurcation, one would expect the
large-scale behaviour to be described by the dynamical Φ3 model built in [EJS13]
3
usingthetechniquesdevelopedin[DPD02, DPD03].
Recall that,at leastformally,thedynamicalΦp modelis givenby
3
∂ Φ = ∆Φ Φp−1 +ξ, (1.5)
t
−
INTRODUCTION 3
where ξ is the space-time white noise, and the spatial variable belongs to the three-
dimensional torus T3. In this article, we will only ever consider p 2,3,4 , with
∈ { }
p = 2 corresponding to the Ornstein-Uhlenbeck process (1.3). For p 3,4 , the
∈ { }
interpretation of (1.5) is not clear a priori since solutions are distribution-valued so
that theterm Φp−1 lacks a canonical interpretation. However, theycan beconstructed
as limitsofsolutionsto
∂ Φ = ∆Φ Φp−1 +C Φp−3 +ξ , (1.6)
t ǫ ǫ − ǫ ǫ ǫ ǫ
for a regularisation ξ of space-time white noise and a suitable diverging sequence
ǫ
of constants C . In the case p = 3, this turns the term Φ2 into the Wick product
ǫ
:Φ2: with respect to the Gaussian structure induced by the stationary solution to the
corresponding linearised equation (see [EJS13] for more details). In the case p =
4, the situation is more delicate and additional logarithmic divergencies arise due to
higherorder effects,see [GJ73, Fel74, Hai14b].
At this stage, it is important to note that the notation (1.5), even when interpreted
as limit of processes of the type (1.6), is really an abuse of notation: since one could
always change the value of C in (1.6) by a finite quantity, one actually ends up with
ǫ
a one-parameter family of solutions indexed by that finite quantity, and we call the
resulting family of solutions Φp family. Let us point out that, without the diverging
3
counter-termC ,thesequenceΦ wouldconvergeto0inasufficientlyweaktopology
ǫ ǫ
dependingonthedimensiond (see[HRW12]formoredetails).
Formally,the equilibriummeasure of thedynamics (1.5) for p = 4 is the measure
on Schwartz distributionsassociated to Bosonic Euclidean quantum field theory. The
construction of this measure was one of the main achievements of constructive field
theory; see the articles [EO71, Fel74, FO76, GJ73, Gli68] and references therein. In
two spatial dimensions, the equation (1.5) was treated in [AR91, DPD03, MW15].
For d 4, one does not expect to be able to obtain any non-trivial scaling limit, see
≥
[Fro¨82, Aiz82, BBS14].
Another reason why the dynamical Φ4 is interesting is that it is expected to de-
3
scribe the 3D Ising model with Glauber dynamics and Kac interactions near critical
temperature (as conjectured in [GLP99]). In fact, the one dimensionalversion of this
result was shown in [BPRS93] at the critical temperature. The two dimensional case
ismoredifficult,astheequationitselfrequiresrenormalisation. Itwasshownrecently
in [MW14] that the 2D Kac-Ising model does rescale to Φ4 near critical temperature,
2
and the renormalisation constant has a nice interpretation as the shift of critical tem-
peraturefromitsmeanfieldvalue. Seealsothearticle[GS73]whichhoweverrequired
atwo-stepprocedureto obtainΦ4 froman Isingmodel.
2
We now turn back to the rescaled process (1.2). As suggested by the form of
renormalisation in (1.6), it is reasonable to expect that the behaviour of u at scale
ǫ
α = 1 and θ at (or near) a pitchfork bifurcation should be well approximated by
the dynamical Φ4 model. However, it turns out that this is not true in full generality.
3
The main result of this article is that, although u converges to Φ4 for all symmetric
ǫ 3
polynomialpotentials,forgenericasymmetricpotentials,afterproperre-centeringand
rescaling,thelargescalebehaviourofthesystemwillalwaysbedescribedeitherbyΦ3
3
or by the O.U. process of the type (1.3). One way to understand this is that pitchfork
INTRODUCTION 4
bifurcations are structurally unstable: small generic perturbations tend to turn them
into a saddle-node bifurcation taking place very close to a local minimum. One can
then argue (this is quite clear in Wilson’s renormalisation group picture which has
recently been applied to theconstructionof thedynamicalΦ4 modelin [Kup15]) that
3
the effective potential experienced by the process at large scale is not V but some
θ
h i
smallperturbationthereof, thusreconcilingourresultswithintuition.
1.1 Weaklynonlinear regime
We startwiththeweaklynonlinearregimegivenby
∂ u = ∆u ǫV′(u)+ξ, (1.7)
t − θ
where we assume that V is a polynomial whose coebfficients depend smoothly on θ.
θ
Defining V as above,wethuswrite
θ
h i
m
V′ (u) = a (θ)uj ,
h θi j
Xj=0
b
for some smooth functions a . For notational simplicity, we let a ,a′ and a′′ denote
j j j j
the value and first two derivativesof a (θ) at 0. We will always assume that V has
j θ
b b b bh i
a critical point at the origin (which could easily be enforced by just translating u), so
b
thata = 0.
0
Remark 1.2. From now on, we willalways assumethat (1.7) is considered on a peri-
b
odicdomainoftherelevantsize. Inparticular,wedefineu directlyasthesolutionto
ǫα
(1.2)onadomainofsize (1)(theprecisesizeisirrelevant,butitshouldbebounded
O
and no longerdepend on ǫ). Ideally, onewould liketo extend theconvergence results
of this article to all of R3, which would be much more canonical, but this requires
somecontrolat infinitywhichis lackingat present.
Remark 1.3. In principle,thenoiseξ appearing in (1.7) also depends on ǫ, sinceit is
defined on a torus of size ǫ−α for some α > 0 depending on the regime we consider.
b
However, since we assume that its correlation function is fixed (independent of ǫ)
and has compact support, the noises on domains of different sizes agree in law when
considered on an identical patch, as long as a suitable fattening of that patch remains
simplyconnected.
Inthesimplestcasewhena = 0,itisnotverydifficulttoshowthatatscaleα = 1,
1 6 2
u converges in probability to the O.U. process. Interesting phenomena occur when
ǫα
b
(0,0)is abifurcationpointfor V , whichgivesthenecessary bifurcationcondition
θ
h i
a = a = 0 . (1.8)
0 1
The saddle-node bifurcation furtherbrequibres that a′ = 0 and a = 0, and in this case
0 6 2 6
one should choose α = 2 so that as long as θ = (ǫ23), the macroscopic process
u converges to Φ3 fami3ly. In fact, the terms ibn VO′(ǫα/2ub) in (1.2) are Hermite
ǫα 3 θ ǫα
polynomialsinu whosecoefficientsarepreciselya (θ)’swithcorrespondingpowers
ǫα j
ofǫ. Thus,theWickrenormalisationisalreadytakenaccountof,andthisisthereason
b
whythebifurcationassumptionnaturallyappears for V but notV .
θ θ
h i
INTRODUCTION 5
Themostinterestingcaseariseswhen(0,0)isapitchforkbifurcationpointof V
θ
h i
sothat inadditionto(1.8), onehas
a′ = 0, a′ < 0, a = 0, a > 0 . (1.9)
0 1 2 3
As mentionedabove, frbom (1.6), ibt is naturabl to expectbthat at scale α = 1, and witha
suitablechoiceofθ,theprocessesu shouldconvergetothesolutionoftheΦ4 model.
ǫα 3
As already alludedto earlier, thisturnoutto betrueifand onlyifthequantity
A = P(z) E(V′(Ψ(0))V′′(Ψ(z)))dz (1.10)
0 0
Z
vanishes, where P is the heat kernel, z denotes the space time variable (t,x), and the
expectation is taken with respect to the stationary measure of Ψ as defined in (1.4).
Notethatthisexpressionis finiteonlyif
a a = a = 0,
0 1 2
whichiscertainlythecasewhenVθbhabsapibtchforkbifurcationattheorigin. Also,the
quantityA couldbewritteninterms ofthecoefficients of V as
h i
m−1
A = (j +1)! a a C , (1.11)
j j+1 j
·
Xj=3
b b
where theC (to be defined in Section 4 below)are explicitconstants dependingonly
j
on the covariance of ξ. It is clear from this expression that A is automatically 0 if V
issymmetric.
b
If A = 0 then, in order to obtain a nontrivial limit, it is necessary to slightly shift
6
thepotentialfrom theorigin,so weset
uǫα(t,x) = ǫ−α2(u(t/ǫ2α,x/ǫα) hǫ) , (1.12)
−
forsomesmallh . Theprocessu abovethen satisfiestheequation
ǫ ǫα
∂ u = ∆u ǫ1−5α/2V′(ǫα/2u +h )+ξ . (1.13)
t ǫα ǫα − θ ǫα ǫ ǫα
From now on, in both weakly nonlinear and weak noise regimes, we will use u to
ǫα
denotethere-centredprocess,andtheprocessin(1.2)isaspecialcaseof(1.13)when
h = 0.
If one then takes θ ǫβ for some β < 2, then there are three different choices of
∼ 3
h ’s such that the shifted process u converges to O.U. for α = 1+β. As expected,
ǫ ǫα 2
twoofthepossiblelimitingO.U.processesarestable,andthethirdoneisunstable. If
θ ǫβ forsomeβ > 2 ontheotherhand,thenthereisauniquechoiceofh suchthat
∼ 3 ǫ
at scaleα = 5, theprocess u converges toa stableO.U.process.
6 ǫα
At the critical case θ = cǫ23, there is a constant c∗ such that for c > c∗ and c < c∗,
at scale α = 5, u either converges to three O.U.’s or just one O.U., respectively. At
6 ǫα
c = c∗, there are two possible choices of h . One of them again yields a stable O.U.
ǫ
processatscale 5 inthelimit,buttheotheroneyieldsΦ3 atscaleα = 8. Notethatthis
6 3 9
scale is much larger than the scale 2 at which one obtains Φ3 in the case of a simple
3 3
saddle-nodebifurcation. Wesummarisethemin thefollowingtheorem.
INTRODUCTION 6
Theorem 1.4. Let V have a pitchfork bifurcation at the origin, and let u be the
θ ǫα
h i
solutionto(1.2)on [0,T] T3.
×
If the quantity A given by (1.10) is 0, then there exists µ < 0 such that at the
distancetocriticality
θ = µǫ logǫ +λǫ+ (ǫ2),
| | O
scaleα = 1and h = 0, theprocessu converges to theΦ4 familyindexed byλ.
ǫ 3
If A = 0, then thelargescalebehaviourof uα dependsonthevalue
6 ǫ
θ = ρǫβ,ρ > 0.
In fact, thereexistsρ∗ > 0 suchthatif β < 2, or ifβ = 2 and ρ > ρ∗, then thereexist
3 3
three choices of h ’s such that at scale α = 1+β, two of the resulting processes u
ǫ 2 ǫα
convergeto a stableOU process,andtheotherconverges to anunstableone.
Ifβ > 2,orifβ = 2 andρ < ρ∗,thenthereexistsachoiceofh suchthatatscale
3 3 ǫ
α = 5, theprocessu converges toa stableOU process.
6 ǫα
At the critical value β = 2 and ρ = ρ∗, there exist two choices of h ’s such that
3 ǫ
one of the resulting process converges to a stable OU at scale α = 5, and the other
6
converges to Φ3 at α = 8.
3 9
Remark 1.5. The coefficient of the Wick term :u2: in the critical Φ3 case is propor-
3
1
tionaltoA3. IfA = 0,thentheprocessbecomesafreefield,andonecan thenfurther
enlargethescaleto1, and adjustθ and htoget Φ4.
3
Also, the coefficient of theterm Φp−1 in the limitingequation depends on various
coefficients of V , but we can rescale them while leaving invariant the white noise
0
h i
such thattheyallbecome1.
Remark 1.6. In the asymmetric case (A = 0), one can actually expand θ to the
6
second order such that in the branch containing the saddle point, the scale increases
continuously from 0 up to 8 with respect to θ (see Remark 5.5). Similar results also
9
holdinthesymmetriccase, butthisisnot importanthere, so weomitthedetails.
All theseresultswillbeformulatedprecisely inSection 5 below.
1.2 Weaknoiseregime
Thereisanotherregimeofmicroscopicmodelsinwhichthenonlineardynamicsdom-
inatesthenoise. Thelocalmean field fluctuationis givenbytheequation
∂tu = ∆u−Vθ′(u)+ǫ12ξ, (1.14)
b
where now V could bea potentialwithsufficient regularity,not necessarily apolyno-
θ
mial. More precisely, we assume V : θ V ( ) is a smooth function in the space of
θ
7→ ·
8 functions. Thus,wecould TaylorexpandV′ around x = 0 as
C θ
6
V′(x) = a (θ)xj +F (x), (1.15)
θ j θ
Xj=0
INTRODUCTION 7
where a ’s are smooth functions in θ, and F (x) . x 7 uniformly over θ < 1 and
j θ
| | | | | |
x < 1.
| |
1
Since the noise now has strength of order ǫ2, the large scale behaviour of (1.14)
willbedeterminedbythebehaviourofV itselfneartheoriginratherthanitsaverage.
θ
Again, in order to observe an interesting limit, we assume that V has a pitchfork
bifurcationat (0,0),namely onehas
a = a′ = a = a = 0, a′ < 0, a > 0, (1.16)
0 0 1 2 1 3
where the a (θ) are the coefficients of the Taylor series of V′(φ) around φ = 0. For
j θ
λ > 0, wesetsimilarlytobefore
u (t,x) = λ−1(u(tλ−2,xλ−1) h ) ,
λ ǫ
−
where h is a small parameter as before. By setting λ = ǫα, we see that this time u
ǫ ǫα
solvesthePDE
∂tuǫα = ∆uǫα −ǫ−(21+52α)Vθ′(ǫ12+α2uǫα +hǫ)+ξǫα . (1.17)
While this appears to be identical to (1.2) modulo the substitution α α + 1, it
7→
genuinelydiffersfromitinthatthedrivingnoisestillhascorrelationlengthǫα andnot
ǫα+1. In order for u to converge to Φ4, it then seems natural to choose α = 1, thus
ǫα 3
guaranteeing that the coefficient of the cubic term in the Taylor expansion of V′ is of
θ
order1. Butthiscreates thedivergencesinbothlinearandconstanttermsontheright
hand sideoftheequation. Since a = a = 0, and wehavetwoparameters θ and hto
0 1
tune, it looks like that we could kill the divergences by choosing the proper values of
θ and hand getΦ4 in thelimit.
3
Unfortunately,thisturnsouttobeimpossible. Whentuningθtoitscorrectvalueto
kill the linear divergence, the terms involving the leading order of h will be precisely
be canceled out so that h could only have a second order effect, which is far from
enough to kill the divergence in the constant term. Thus, one could not make both
linear and constant terms convergent unless the coefficients of V itself are balanced.
Itturnsoutthatsimilarasbefore,whetheru convergestoΦ4 dependsonthequantity
ǫ 3
3a′′a2 a′a
B = a + 0 3 2 3 .
4 2a′2 − a′
1 1
Themainstatementisthefollowing.
Theorem1.7. AssumeV : θ V ( )isasmoothfunctioninthespaceof 8 functions,
θ
7→ · C
and exhibits a pitchfork bifurcation at the origin (θ,x) = (0,0). Let u solves the
ǫα
PDE (1.17).
If B = 0, then thereexist choicesof θ andhoftheform
θ = aǫ+bǫ2logǫ+ (ǫ2), h = (ǫ)
O O
suchthatu converges to Φ4(a )withan additionalconstantintheequation.
ǫα 3 3
INTRODUCTION 8
If B = 0, then thereexist ρ∗ > 0 forj = 1,2,3suchthatif
6 j
θ = θ∗ = ρ∗ǫ+ρ∗ǫ43 +ρ∗ǫ35 + (ǫ196),
1 2 3 O
then there exists two choices of h such that one of the resulting processes u con-
ǫ ǫα
verges to Φ3 at scaleα = 7,and theother converges toa stableOU atα = 2.
3 9 3
If θ > θ∗ (or θ < θ∗), then there exists three (or one) choices of h ’s such that
ǫ
the resulting u ’s converge to OU processes. In the former case, two of the OU are
ǫα
stableandtheothernon-stable,andinthelattercasetheOU isstable.
Remark 1.8. Similar as the weakly nonlinear case, the coefficient of the Wick term
forΦ33 isproportionaltoB13. Also,onecouldrescalethesolutionleavinginvariantthe
whitenoisesuchthat allthelimitsareoftheform (1.5).
Theprecisestatementwillbegivenin Theorems5.8and 5.10.
1.3 Someremarks and structure ofthearticle
Before describing the structure of this article, we discuss two possible natural gener-
alisationsofourresults.
1. We expect that analogous results still hold when the noise ξ is not assumed to
beGaussian, but stillsatisfies good enough integrabilityand mixingconditions.
b
The techniques developed in [HS15] should apply here as well. Note however
that if the noise is asymmetric, then we do not expect to see Φ4 at large scales
3
generically,evenifV is symmetric.
θ
2. The assumption that V is a polynomial can probably be relaxed. It is however
θ
notclearatallatthisstagehowthemethodsinthisarticlecouldbecarriedover
tohandlethiscase.
It turns outthat,as in [HQ15], theweak noiseregimecan betreated as aperturba-
tion of the weakly nonlinear regime, so we will mainly focus on the latter case. The
main strategy to provetheaboveresults is therecently developedtheory ofregularity
structures([Hai14b]),combinedwiththeresultsof([HQ15]),whereresultsanalogous
to ours are obtained for the KPZ equation. The idea is to lift and solve(1.2) in an ab-
stract regularity structure space that is purposed built for this equation, and then pull
thesolutionback totheusualdistributionspaces after suitablerenormalisation.
The articleis organised as follows. In Section 2, we construct theregularitystruc-
tureaswellastherenormalisationmapsthatallowustotreattheequationsoftheform
(1.2). Section 3 is devoted to construction of the solution to the abstract equation. In
Section 4, we prove the convergence of the renormalised models. Finally, in Section
5,wecollectallthepreviousresultstoidentifythelimitoftherenormalisedsolutions.
Acknowledgements
MHgratefullyacknowledges financialsupportfromthePhilipLeverhulmetrustandtheEuro-
peanResearchCouncil.
CONSTRUCTION OF THE REGULARITY STRUCTURE 9
2 Construction of the regularity structure
Inthissection,webuildaregularitystructurethatissufficientlyrichtosolvethefixed
pointproblemfortheequation
∂tuǫ = ∆uǫ −ǫ−23Vθ′(ǫ12uǫ)+ξǫ (2.1)
in the abstract space of modelled distributions. Here, ξ is a mollified version of the
ǫ
space-time white noise ξ at scale ǫ, and V′ is a polynomial of degree m. Note that
θ
(2.1) corresponds to the weakly nonlinear regime with scale α = 1, and we do not
restrictV tobesymmetrichere. Sincethisisthelargestscalewewilllookat,allother
situations(includingtheweaknoiseregime)willfollowasaperturbationoftheabove
equation.
Theconstructionoftheregularitystructuremainlyfollowsthemethodologiesand
set up in [Hai14b] and [HQ15], with some slight modifications to accommodate the
particularformoftheequation(2.1). Moregentleintroductionstoregularitystructures
can befoundin [Hai15a], [Hai15b], [Hai14a]and [CW15].
2.1 The (extended) regularity structure
Recall that a regularity structure is a pair ( , ), where = is a vector
T G T α∈ATα
space that is graded by some (bounded below, locally finite) sLet A R of homo-
⊂
geneities, and is a group of linear transformations of such that, for every Γ ,
G T ∈ G
Γ idisstrictlyuppertriangularwithrespect tothegraded structure.
−
For the purpose of this article, we build basis vectors similarly to [HQ15] as a
T
collection of formal expressions built from the symbols 1, Ξ, X 3 and operators
{ i}i=0
and β for half integers β > 0. As usual, we assume that all symbols and sub-
I E
expressionscommuteandthat1 isneutralfortheproduct,soweidentifyforexample
(ΞX )Ξ and Ξ1 (X Ξ). Givenamulti-indexk = (k , ,k ), we also writeXk as
1 1 0 3
aIshorthand for XIk0 Xk3 (with the convention X0 =·1·)·, and k = 2k + 3 k
0 ··· 3 i | | 0 i=1 i
foritsparabolicdegree. P
Withthesenotations,wedefinetwosets and ofsuchexpressionsasthesmall-
U V
est sets such that Xk , Ξ , and such that for every k 1,...,m 3 ,
∈ U ∈ V ∈ { − }
k
{τ1,··· ,τk} ⊂ U ⇒ {τ1τ2τ3 , E2(τ1···τk+3)} ⊂ V , (2.2)
τ (τ) .
∈ V ⇒ I ∈ U
Wethenset = andweassociatetoeach elementof ahomogeneityinthe
W U ∪V W
followingway. Weset
5
Ξ = κ, Xk = k ,
| | −2 − | | | |
whereκisasmallpositivenumbertobefixedlater,andweextendthistoeveryformal
expressionin by
W
ττ¯ = τ + τ¯ , (τ) = τ +2, β(τ) = β + τ . (2.3)
| | | | | | |I | | | |E | | |
We then write for the free vector space generated by τ : τ = α . In this
α
T { ∈ W | | }
article,wewillonlyeverusebasisvectorswithhomogeneitylessthan2,wetherefore
CONSTRUCTION OF THE REGULARITY STRUCTURE 10
takefor thespaceofallfinitelinearcombinationsofelementsof ofhomogeneity
T W
lessthan 2,i.e. = .
T α<2Tα
The main reasonLfor introducing β as in (2.2) rather than treating ǫ as a fixed
E
real number is the following crucial fact. It reflects that (2.1) is subcritical under the
scalingreflected by ourregularitystructure.
Lemma 2.1. If κ < 1 , thenfor everyγ > 0,theset τ : τ < γ isfinite.
8m { ∈ W | | }
As in [HQ15], it will be convenient to consider β as a linear map such that β :
E E
τ β(τ). The problem is that the product τ τ appearing in (2.2) does in
1 ℓ+3
7→ E ···
general not belong to . Just as in [HQ15], one way to circumvent this problem is to
T
introducetheextendedregularitystructure , givenby thelinearspan of
ex
T
= τ τ : τ .
ex 1 m j
W W ∪{ ··· ∈ U}
In thisway,wecan view β as alinearmapdefined on (asubspaceof) .
ex
E T
We now start to describe the structure group for . For this, we introduce the
ex
G T
followingthreesetsofformal symbols:
= 1,X , = J (τ) : τ Xk , τ +2 > ℓ ,
1 2 ℓ
F { } F { ∈ W \{ } | | }
k k (2.4)
= E 2(τ τ ) : τ , + τ > ℓ τ .
F3 { ℓ 1··· k+3 j ∈ U 2 | j| | | ≥ | j|}
Xj Xj
Wethenlet bethecommutativealgebrageneratedbytheelementsin
+ 1 2 3
T F ∪F ∪F
and we define a linear map ∆ : in the same way as in [HQ15, Section
+
T → T ⊗T
3.1].
For any linear functional g : R, one obtains a linear map Γ : by
+ g
T → T → T
Γ τ = (id g)∆τ. Denoting by the set of multiplicative linear functionals g on
g +
⊗ G
, wethen set
+
T
= g ∗ : g(ττ¯) = g(τ)g(τ¯), τ,τ¯ ,
G+ { ∈ T+ ∀ ∈ T+}
and wedefine by
G
= Γ : g . (2.5)
g +
G { ∈ G }
It is straightforward to verify that has the desired properties, including the fact
G
that its elements respect the product structure of in the sense that Γ(ττ¯) = Γτ Γτ¯.
T ·
Furthermore, preservesnotonly ,butalso ,sothatitalsoservesasthestructure
ex
G T T
groupfor .
T
2.2 Admissiblemodels
Wenowstarttointroduceaclassofadmissiblemodelsforourregularitystructure. As
in [Hai14b], we fix a truncation K of the heat kernel which coincides with it near the
origin and annihilates polynomials of degree up to 3. The existence of such a kernel
K iseasy to show,and can befound,forexample,in[Hai14b, Sec. 5.1].
We let ′ denote the space of Schwartz distributions on R1+3 and ( , ′) the
D L T D
space oflinearmaps from to ′. Furthermore, forany test function ϕ : R1+3 R,
T D →