introduction to regularity structures - arxiv.org · from a “philosophical” perspective, the...

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Introduction to Regularity Structures

March 26, 2014

Martin Hairer

The University of Warwick, Email:[email protected]

Abstract

These are short notes from a series of lectures given at the University of Rennes in June 2013, atthe University of Bonn in July 2013, at the XVIIth Brazilian School of Probability in Mambucabain August 2013, and at ETH Zurich in September 2013. They givea concise overview of thetheory of regularity structures as exposed in the article [Hai14]. In order to allow to focus on theconceptual aspects of the theory, many proofs are omitted and statements are simplified. We focuson applying the theory to the problem of giving a solution theory to the stochastic quantisationequations for the EuclideanΦ4

3quantum field theory.

Contents

1 Introduction 1

2 Definitions and the reconstruction operator 4

3 Examples of regularity structures 10

4 Products and composition by smooth functions 14

5 Schauder estimates and admissible models 17

6 Application of the theory to semilinear SPDEs 21

7 Renormalisation of the dynamicalΦ4

3model 24

1 Introduction

Very recently, a new theory of “regularity structures” was introduced [Hai14], unifying variousflavours of the theory of (controlled) rough paths (including Gubinelli’s theory of controlled roughpaths [Gub04], as well as his branched rough paths [Gub10]),as well as the usual Taylor expan-sions. While it has its roots in the theory of rough paths [Lyo98], the main advantage of this newtheory is that it is no longer tied to the one-dimensionalityof the time parameter, which makesit also suitable for the description of solutions to stochastic partial differential equations, ratherthan just stochastic ordinary differential equations. Theaim of this article is to give a concisesurvey of the theory while focusing on the construction of the dynamicalΦ4

3 model. While the

http://arxiv.org/abs/1401.3014v2

2 INTRODUCTION

exposition aims to be reasonably self-contained (in particular no prior knowledge of the theory ofrough paths is assumed), most of the proofs will only be sketched.

The main achievement of the theory of regularity structuresis that it allows to give a (path-wise!) meaning to ill-posed stochastic PDEs that arise naturally when trying to describe themacroscopic behaviour of models from statistical mechanics near criticality. One example ofsuch an equation is the KPZ equation arising as a natural model for one-dimensional interfacemotion [KPZ86, BG97, Hai13]:

∂th = ∂2xh+ (∂xh)2 + ξ − C .

Another example is the dynamicalΦ43 model arising for example in the stochastic quantisation of

Euclidean quantum field theory [PW81, JLM85, AR91, DPD03, Hai14], as well as a universalmodel for phase coexistence near the critical point [GLP99]:

∂tΦ = ∆Φ+ CΦ− Φ3 + ξ .

In both of these examples,ξ formally denotes space-time white noise,C is an arbitrary constant(which will actually turn out to be infinite in some sense!), and we consider a bounded squarespatial domain with periodic boundary conditions. In the case of the dynamicalΦ4

3 model, thespatial variable has dimension3, while it has dimension1 in the case of the KPZ equation. Whilea full exposition of the theory is well beyond the scope of this short introduction, we aim to givea concise overview to most of its concepts. In most cases, we will only state results in a ratherinformal way and give some ideas as to how the proofs work, focusing on conceptual rather thantechnical issues. The only exception is the “reconstruction theorem”, Theorem 2.10 below, whichis the linchpin of the whole theory. Since its proof (or rather a slightly simplified version of it) isrelatively concise, we provide a fully self-contained version. For precise statements and completeproofs of most of the results exposed here, we refer to the original article [Hai14].

Loosely speaking, the type of well-posedness results that can be proven with the help of thetheory of regularity structures can be formulated as follows.

Theorem 1.1 Letξε = δε∗ξ denote the regularisation of space-time white noise with a compactlysupported smooth mollifierδε that is scaled byε in the spatial direction(s) and byε2 in the timedirection. Denote byhε andΦε the solutions to

∂thε = ∂2xhε + (∂xhε)2 − Cε + ξε ,

∂tΦε = ∆Φε + CεΦε −Φ3ε + ξε .

Then, there exist choices of constantsCε andCε diverging asε → 0, as well as processesh andΦ such thathε → h andΦε → Φ in probability. Furthermore, while the constantsCε andCε dodepend crucially on the choice of mollifiersδε, the limitsh andΦ donot depend on them.

Remark 1.2 We made a severe abuse of notation here since the space-time white noise appearingin the equation forhε is onR × T1, while the one appearing in the equation forΦε is onR × T3.(Here we denote byTn then-dimensional torus.)

INTRODUCTION 3

Remark 1.3 We have not explicited the topology in which the convergencetakes place in theseexamples. In the case of the KPZ equation, one actually obtains convergence in some space ofspace-time Holder continuous functions. In the case of thedynamicalΦ4

3 model, convergencetakes place in some space of space-time distributions. One caveat that also has to be dealt within the latter case is that the limiting processΦ may in principle explode in finite time for someinstances of the driving noise.

From a “philosophical” perspective, the theory of regularity structures is inspired by the theoryof controlled rough paths [Lyo98, Gub04, LCL07], so let us rapidly survey the main ideas of thattheory. The setting of the theory of controlled rough paths is the following. Let’s say that we wantto solve a controlled differential equation of the type

dY = f (Y ) dX(t) , (1.1)

whereX ∈ Cα is a rather rough function (say a typical sample path for anm-dimensionalBrownian motion). It is a classical result by Young [You36] that the Riemann-Stieltjes integral(X,Y ) 7→

∫ ·0Y dX makes sense as a continuous map fromCα × Cα into Cα if and only ifα > 1

2.

As a consequence, “naıve” approaches to a pathwise solution to (1.1) are bound to fail ifX hasthe regularity of Brownian motion.

The main idea is to exploit the a priori “guess” that solutions to (1.1) should “look likeX atsmall scales”. More precisely, one would naturally expect the solutionY to satisfy

Yt = Ys + Y ′sXs,t +O(|t− s|2α) , (1.2)

where we wroteXs,t as a shorthand for the incrementXt − Xs. As a matter of fact, one wouldexpect to have such an expansion withY ′ = f (Y ). Denote byCα

X the space of pairs (Y, Y ′)satisfying (1.2) for a given “model path”X. It is then possible to simply “postulate” the valuesof the integrals

Xs,t =:

∫ t

sXs,r ⊗ dXr , (1.3)

satisfying “Chen’s relations”

Xs,t − Xs,u − Xu,t = Xs,u ⊗Xu,t , (1.4)

as well as the analytic bound|Xs,t| . |t − s|2α, and to exploit this additional data to give acoherent definition of expressions of the type

∫

Y dX, provided that the pathX is “enhanced”with its iterated integralsX andY is a “controlled path” of the type (1.2). See for example[Gub04] for more information or [Hai11] for a concise exposition of this theory.

Compare (1.2) to the fact that a functionf : R → R is of classCγ with γ ∈ (k, k + 1) if foreverys ∈ R there exist coefficientsf (1)

s , . . . , f (k)s such that

ft = fs +

k∑

ℓ=1

f (ℓ)s (t− s)ℓ +O(|t− s|γ) . (1.5)

4 DEFINITIONS AND THE RECONSTRUCTION OPERATOR

Of course,f (ℓ)s is nothing but theℓth derivative off at the points, divided byℓ!. In this sense,

one should really think of a controlled rough path (Y, Y ′) ∈ CαX as a2α-Holder continuous func-

tion, but with respect to a “model” determined by the functionX, rather than by the usual Taylorpolynomials. This formal analogy between controlled roughpaths and Taylor expansions sug-gests that it might be fruitful to systematically investigate what are the “right” objects that couldpossibly take the place of Taylor polynomials, while still retaining many of their nice properties.

Acknowledgements

Financial support from the Leverhulme trust through a leadership award is gratefully acknowledged.

2 Definitions and the reconstruction operator

The first step in such an endeavour is to set up an algebraic structure reflecting the propertiesof Taylor expansions. First of all, such a structure should contain a vector spaceT that willcontain the coefficients of our expansion. It is natural to assume thatT has a graded structure:T =

⊕

α∈A Tα, for some setA of possible “homogeneities”. For example, in the case of theusual Taylor expansion (1.5), it is natural to take forA the set of natural numbers and to haveTℓ contain the coefficients corresponding to the derivatives of order ℓ. In the case of controlledrough paths however, it is natural to takeA = {0, α}, to have againT0 contain the value of thefunctionY at any times, and to haveTα contain the Gubinelli derivativeY ′

s . This reflects the factthat the “monomial”t 7→ Xs,t only vanishes at orderα neart = s, while the usual monomialst 7→ (t− s)ℓ vanish at integer orderℓ.

This however isn’t the full algebraic structure describingTaylor-like expansions. Indeed, oneof the characteristics of Taylor expansions is that an expansion around some pointx0 can bere-expanded around any other pointx1 by writing

(x− x0)m =∑

k+ℓ=m

m!

k!ℓ!(x1 − x0)

k · (x− x1)ℓ . (2.1)

(In the case whenx ∈ Rd, k, ℓ andm denote multi-indices andk! = k1! . . . kd!.) Somewhatsimilarly, in the case of controlled rough paths, we have the(rather trivial) identity

Xs0,t = Xs0,s1 · 1 + 1 ·Xs1,t . (2.2)

What is a natural abstraction of this fact? In terms of the coefficients of a “Taylor expansion”,the operation of reexpanding around a different point is ultimately just a linear operation fromΓ: T → T , where the precise value of the mapΓ depends on the starting pointx0, the endpointx1, and possibly also on the details of the particular “model” that we are considering. In view ofthe above examples, it is natural to impose furthermore thatΓ has the property that ifτ ∈ Tα,thenΓτ − τ ∈ ⊕

β<α Tβ. In other words, when reexpanding a homogeneous monomial arounda different point, the leading order coefficient remains thesame, but lower order monomials mayappear.

These heuristic considerations can be summarised in the following definition of an abstractobject we call aregularity structure:

DEFINITIONS AND THE RECONSTRUCTION OPERATOR 5

Definition 2.1 LetA ⊂ R be bounded from below and without accumulation point, and let T =⊕

α∈A Tα be a vector space graded byA such that eachTα is a Banach space. Let furthermoreG be a group of continuous operators onT such that, for everyα ∈ A, everyΓ ∈ G, and everyτ ∈ Tα, one hasΓτ − τ ∈ ⊕

β<α Tβ. The tripleT = (A,T,G) is called aregularity structurewith model spaceT andstructure groupG.

Remark 2.2 Givenτ ∈ T , we will write ‖τ‖α for the norm of its component inTα.

Remark 2.3 In [Hai14] it is furthermore assumed that0 ∈ A, T0 ≈ R, andT0 is invariantunderG. This is a very natural assumption which ensures that our regularity structure is at leastsufficiently rich to represent constant functions.

Remark 2.4 In principle, the setA can be infinite. By analogy with the polynomials, it is thennatural to considerT as the set of all formal series of the form

∑

α∈A τα, where only finitely manyof theτα’s are non-zero. This also dovetails nicely with the particular form of elements inG. Inpractice however we will only ever work with finite subsets ofA so that the precise topology onT does not matter.

At this stage, a regularity structure is a completely abstract object. It only becomes usefulwhen endowed with amodel, which is a concrete way of associating to anyτ ∈ T andx0 ∈ Rd,the actual “Taylor polynomial based atx0” represented byτ . Furthermore, we want elementsτ ∈ Tα to represent functions (or possibly distributions!) that “vanish at orderα” around thegiven pointx0.

Since we would like to allowA to contain negative values and therefore allow elements inT torepresent actual distributions, we need a suitable notion of “vanishing at orderα”. We achieve thisby considering the size of our distributions, when tested against test functions that are localisedaround the given pointx0. Given a test functionϕ on Rd, we writeϕλ

x as a shorthand for

ϕλx(y) = λ−dϕ(λ−1(y − x)) .

Givenr > 0, we also denote byBr the set of all functionsϕ : Rd → R such thatϕ ∈ Cr with‖ϕ‖Cr ≤ 1 that are furthermore supported in the unit ball around the origin. With this notation,our definition of a model for a given regularity structureT is as follows.

Definition 2.5 Given a regularity structureT and an integerd ≥ 1, amodelfor T onRd consistsof maps

Π: Rd → L(T,S ′(Rd)) Γ: Rd × Rd → G

x 7→ Πx (x, y) 7→ Γxy

such thatΓxyΓyz = Γxz andΠxΓxy = Πy. Furthermore, givenr > | inf A|, for any compact setK ⊂ Rd and constantγ > 0, there exists a constantC such that the bounds

|(Πxτ)(ϕλx)| ≤ Cλ|τ |‖τ‖α , ‖Γxyτ‖β ≤ C|x− y|α−β‖τ‖α , (2.3)

hold uniformly overϕ ∈ Br, (x, y) ∈ K, λ ∈ (0, 1], τ ∈ Tα with α ≤ γ, andβ < α.


Remark 2.6 In principle, test functions appearing in (2.3) should be smooth. It turns out that ifthese bounds hold for smooth elements ofBr, thenΠxτ can be extended canonically to allow anyCr test function with compact support.

Remark 2.7 The identityΠxΓxy = Πy reflects the fact thatΓxy is the linear map that takes anexpansion aroundy and turns it into an expansion aroundx. The first bound in (2.3) states whatwe mean precisely when we say thatτ ∈ Tα represents a term that vanishes at orderα. (Notethatα can be negative, so that this may actually not vanish at all!)The second bound in (2.3) isvery natural in view of both (2.1) and (2.2). It states that when expanding a monomial of orderαaround a new point at distanceh from the old one, the coefficient appearing in front of lower-ordermonomials of orderβ is of order at mosthα−β .

Remark 2.8 In many cases of interest, it is natural to scale the different directions ofRd in adifferent way. This is the case for example when using the theory of regularity structures to buildsolution theories for parabolic stochastic PDEs, in which case the time direction “counts as” twospace directions. To deal with such a situation, one can introduce a scalings of Rd, which is justa collection ofd mutually prime strictly positive integers and to defineϕλ

x in such a way that theith direction is scaled byλsi . In this case, the Euclidean distance between two points should bereplaced everywhere by the corresponding scaled distance|x|s =

∑

i |xi|1/si . See also [Hai14]for more details.

With these definitions at hand, it is then natural to define an equivalent in this context of thespace ofγ-Holder continuous functions in the following way.

Definition 2.9 Given a regularity structureT equipped with a model (Π,Γ) overRd, the spaceDγ = Dγ(T ,Γ) is given by the set of functionsf : Rd → ⊕

α<γ Tα such that, for every compactsetK and everyα < γ, the exists a constantC with

‖f (x) − Γxyf (y)‖α ≤ C|x− y|γ−α (2.4)

uniformly overx, y ∈ K.

The most fundamental result in the theory of regularity structures then states that givenf ∈Dγ with γ > 0, there exists auniqueSchwartz distributionRf onRd such that, for everyx ∈ Rd,Rf “looks like Πxf (x) nearx”. More precisely, one has

Theorem 2.10 Let T be a regularity structure as above and let(Π,Γ) a model forT on Rd.Then, there exists a unique linear mapR : Dγ → S ′(Rd) such that

|(Rf −Πxf (x))(ϕλx)| . λγ , (2.5)

uniformly overϕ ∈ Br andλ as before, and locally uniformly inx.

Proof. The proof of the theorem relies on the following fact. Given any r > 0 (but finite!), thereexists a functionϕ : Rd → R with the following properties:


(1) The functionϕ is of classCr and has compact support.

(2) For every polynomialP of degreer, there exists a polynomialP of degreer such that, foreveryx ∈ Rd, one has

∑

y∈Zd P (y)ϕ(x− y) = P (x).

(3) One has∫

ϕ(x)ϕ(x − y) dx = δy,0 for everyy ∈ Zd.

(4) There exist coefficients{ak}k∈Zd such that2−d/2ϕ(x/2) =∑

k∈Zd akϕ(x− k).

The existence of such a functionϕ is highly non-trivial. This is actually equivalent to the existenceof a wavelet basis consisting ofCr functions with compact support, a proof of which was firstobtained by Daubechies in her seminal article [Dau88]. Fromnow on, we take the existence ofsuch a functionϕ as a given for somer > | inf A|. We also setΛn = 2−nZd and, fory ∈ Λn,we setϕn

y (x) = 2nd/2ϕ(2n(x − y)). Here, the normalisation is chosen in such a way that the set{ϕn

y}y∈Λn is again orthonormal inL2. We then denote byVn ⊂ Cr the linear span of{ϕny}y∈Λn ,

so that, by the property (4) above, one hasV0 ⊂ V1 ⊂ V2 ⊂ . . .. We furthermore denote byVntheL2-orthogonal complement ofVn−1 in Vn, so thatVn = V0 ⊕ V1 ⊕ . . . ⊕ Vn. In order tokeep notations compact, it will also be convenient to define the coefficientsank with k ∈ Λn byank = a2nk.

With these notations at hand, we then define a sequence of linear operatorsRn : Dγ → Cr by

(Rnf)(y) =∑

x∈Λn

(Πxf (x))(ϕnx)ϕn

x(y) .

We claim that there then exists a Schwartz distributionRf such that, for every compactly sup-ported test functionψ of classCr, one has〈Rnf, ψ〉 → (Rf)(ψ), and thatRf furthermoresatisfies the properties stated in the theorem.

Let us first consider the size of the components ofRn+1f − Rnf in Vn. Givenx ∈ Λn, wemake use of properties (3-4), so that

〈Rn+1f −Rnf, ϕnx〉 =

∑

k∈Λn+1

ank〈Rn+1f, ϕn+1x+k〉 − (Πxf (x))(ϕn

x)

=∑

k∈Λn+1

ank(Πx+kf (x+ k))(ϕn+1x+k) − (Πxf (x))(ϕn

x)

=∑

k∈Λn+1

ank((Πx+kf (x+ k))(ϕn+1x+k) − (Πxf (x))(ϕn+1

x+k))

=∑

k∈Λn+1

ank(Πx+k(f (x+ k) − Γx+k,xf (x)))(ϕn+1x+k) ,

where we used the algebraic relations betweenΠx andΓxy to obtain the last identity. Since onlyfinitely many of the coefficientsak are non-zero, it follows from the definition ofDγ that for thenon-vanishing terms in this sum we have the bound

‖f (x+ k) − Γx+k,xf (x)‖α . 2−n(γ−α) ,

uniformly overn ≥ 0 andx in any compact set. Furthermore, for anyτ ∈ Tα, it follows from thedefinition of a model that one has the bound

|(Πxτ)(ϕnx)| . 2−αn−nd

2 ,


again uniformly overn ≥ 0 andx in any compact set. Here, the additional factor2−nd2 comes

from the fact that the functionsϕnx are normalised inL2 rather thanL1. Combining these two

bounds, we immediately obtain that

|〈Rn+1f −Rnf, ϕnx〉| . 2−γn−nd

2 , (2.6)

uniformly overn ≥ 0 andx in compact sets. Take now a test functionψ ∈ Cr with compactsupport and let us try to estimate〈Rn+1f − Rnf, ψ〉. SinceRn+1f − Rnf ∈ Vn+1, we candecompose it into a partδRnf ∈ Vn and a partδRnf ∈ Vn+1 and estimate both parts separately.Regarding the part inVn, we have

|〈δRnf, ψ〉| =∣

∣

∣

∑

x∈Λn+1

〈δRnf, ϕnx〉〈ϕn

x , ψ〉∣

∣

∣. 2−γn−nd

2

∑

x∈Λn+1

|〈ϕnx , ψ〉| , (2.7)

where we made use of the bound (2.6). At this stage we use the fact that, due to the boundednessof ψ, we have|〈ϕn

x , ψ〉| . 2−nd/2. Furthermore, thanks to the boundedness of the support ofψ,the number of non-vanishing terms appearing in this sum is bounded by2nd, so that we eventuallyobtain the bound

|〈δRnf, ψ〉| . 2−γn . (2.8)

Regarding the second term, we use the standard fact coming from wavelet analysis [Mey92] thata basis ofVn+1 can be obtained in the same way as the basis ofVn, but replacing the functionϕby functionsϕ from some finite setΦ. In other words,Vn+1 is the linear span of{ϕn

x}x∈Λn;ϕ∈Φ.Furthermore, as a consequence of property (2), the functions ϕ ∈ Φ all have the property that

∫

ϕ(x)P (x) dx = 0 , (2.9)

for any polynomialP of degree less or equal tor. In particular, this shows that one has the bound

|〈ϕnx , ψ〉| . 2−

nd2−nr .

As a consequence, one has

|〈δRnf, ψ〉| =∣

∣

∣

∑

x∈Λn

ϕ∈Φ

〈Rn+1f, ϕnx〉〈ϕn

x , ψ〉∣

∣

∣. 2−

nd2−nr

∣

∣

∣

∑

x∈Λn

ϕ∈Φ

〈Rn+1f, ϕnx〉∣

∣

∣.

At this stage, we note that, thanks to the definition ofRn+1 and the bounds on the model (Π,Γ),we have|〈Rn+1f, ϕn

x〉| . 2−nd2−α0n, whereα0 = inf A, so that|〈δRnf, ψ〉| . 2−nr−α0n.

Combining this with (2.8), we see that one has indeedRnf → Rf for some Schwartz distributionRf .

It remains to show that the bound (2.5) holds. For this, givena distributionη ∈ Cα for someα > −r, we first introduce the notation

Pnη =∑

x∈Λn

η(ϕnx)ϕn

x , Pnη =∑

ϕ∈Φ

∑

x∈Λn

η(ϕnx) ϕn

x .


We also choose an integer valuen ≥ 0 such that2−n ∼ λ and we write

Rf −Πxf (x) = Rnf −PnΠxf (x) +∑

m≥n

(Rm+1f −Rmf − PmΠxf (x))

= Rnf −PnΠxf (x) +∑

m≥n

(δRmf − PmΠxf (x)) +∑

m≥n

δRmf . (2.10)

We then test these terms againstψλx and we estimate the resulting terms separately. For the first

term, we have the identity

(Rnf − PnΠxf (x))(ψλx ) =

∑

y∈Λn

(Πyf (y) −Πxf (x))(ϕny ) 〈ϕn

y , ψλx〉 . (2.11)

We have the bound|〈ϕny , ψ

λx〉| . λ−d2−dn/2 ∼ 2dn/2. Since one furthermore has|y − x| . λ for

all non-vanishing terms in the sum, one also has similarly tobefore

|(Πyf (y) −Πxf (x))(ϕny )| .

∑

α<γ

λγ−α2−dn2−αn ∼ 2−

dn2−γn . (2.12)

Since only finitely many (independently ofn) terms contribute to the sum in (2.11), it is indeedbounded by a constant proportional to2−γn ∼ λγ as required.

We now turn to the second term in (2.10), where we consider some fixed valuem ≥ n. Werewrite this term very similarly to before as

(δRmf − PmΠxf (x))(ψλx ) =

∑

ϕ∈Φ

∑

y,z

(Πyf (y) −Πxf (x))(ϕm+1y ) 〈ϕm+1

y , ϕmz 〉〈ϕm

z , ψλx〉 ,

where the sum runs overy ∈ Λm+1 andz ∈ Λm. This time, we use the fact that by the property(2.9) of the waveletsϕ, one has the bound

|〈ϕmz , ψ

λx〉| . λ−d−r2−rm−md

2 , (2.13)

and theL2-scaling implies that|〈ϕm+1y , ϕm

z 〉| . 1. Furthermore, for eachz ∈ Λm, only finitelymany elementsy ∈ Λm+1 contribute to the sum, and these elements all satisfy|y − z| . 2−m.Bounding the first factor as in (2.12) and using the fact that there are of the order ofλd2md termscontributing for every fixedm, we thus see that the contribution of the second term in (2.10) isbounded by

∑

m≥n

λd2md∑

α<γ

λγ−α−d−r2−dm−αm−rm ∼∑

α<γ

λγ−α−r∑

m≥n

2−αm−rm ∼ λγ .

For the last term in (2.10), we combine (2.7) with the bound|〈ϕmy , ψ

λx〉| . λ−d2−dm/2 and

the fact that there are of the order ofλd2−md terms appearing in the sum (2.7) to conclude thatthemth summand is bounded by a constant proportional to2−γm. Summing overm yields againthe desired bound and concludes the proof.

10 EXAMPLES OF REGULARITY STRUCTURES

Remark 2.11 Note that the spaceDγ depends crucially on the choice of model (Π,Γ). As aconsequence, the reconstruction operatorR itself also depends on that choice. However, the map(Π,Γ, f ) 7→ Rf turns out to be locally Lipschitz continuous provided that the distance between(Π,Γ, f ) and (Π, Γ, f ) is given by the smallest constant such that

‖f (x) − f (x) − Γxyf (y) + Γxyf (y)‖α ≤ |x− y|γ−α ,

|(Πxτ − Πxτ)(ϕλx)| ≤ λα‖τ‖ ,

‖Γxyτ − Γxyτ‖β ≤ |x− y|α−β‖τ‖ .Here, in order to obtain bounds on(Rf−Rf)(ψ) for some smooth compactly supported test func-tion ψ, the above bounds should hold uniformly forx andy in a neighbourhood of the supportof ψ. The proof that this stronger continuity property also holds is actually crucial when show-ing that sequences of solutions to mollified equations all converge to the same limiting object.However, its proof is somewhat more involved which is why we chose not to give it here.

Remark 2.12 In the particular case whereΠxτ happens to be a continuous function for everyτ ∈ T (and everyx ∈ Rd), Rf is also a continuous function and one has the identity

(Rf)(x) = (Πxf (x))(x) . (2.14)

This can be seen from the fact that

(Rf)(y) = limn→∞

(Rnf)(y) = limn→∞

∑

x∈Λn

(Πxf (x))(ϕnx)ϕn

x(y) .

Indeed, our assumptions imply that the function (x, z) 7→ (Πxf (x))(z) is jointly continuous andsince the non-vanishing terms in the above sum satisfy|x−y| . 2−n, one has2dn/2(Πxf (x))(ϕn

x) ≈(Πyf (y))(y) for largen. Since furthermore

∑

x∈Λn ϕnx(y) = 2dn/2, the claim follows.

3 Examples of regularity structures

3.1 The polynomial structure

It should by now be clear how the structure given by the usual Taylor polynomials fits into thisframework. A natural way of setting it up is to take forT the space of all abstract polynomialsin d commuting variables, denoted byX1, . . . ,Xd, and to postulate thatTk consists of the linearspan of monomials of degreek. As an abstract group, the structure groupG is then given byRd

endowed with addition as its group operation, which acts onto T via ΓhXk = (X − h)k, where

h ∈ Rd and we use the notationXk as a shorthand forXk11 · · ·Xkd

d for any multiindexk.The canonical polynomial model is then given by

(ΠxXk)(y) = (y − x)k , Γxy = Γy−x .

We leave it as an exercise to the reader to verify that this does indeed satisfy the bounds andrelations of Definition 2.5.

In the particular case of the canonical polynomial model andfor γ 6∈ N, the spacesDγ thencoincide precisely with the usual Holder spacesCγ . In the case of integer values, this should beinterpreted as bounded functions forγ = 0, Lipschitz continuous functions forγ = 1, etc.

EXAMPLES OF REGULARITY STRUCTURES 11

3.2 Controlled rough paths

Let us see now how the theory of controlled rough paths can be reinterpreted in the light of thistheory. For givenα ∈ (1

3, 12) andn ≥ 1, we can define a regularity structureT by setting

A = {α − 1, 2α − 1, 0, α}. We furthermore take forT0 a copy ofR with unit vector1, for TαandTα−1 a copy ofRn with respective unit vectorsWj andΞj, and forT2α−1 a copy ofRn×n

with unit vectorsWjΞi. The structure groupG is taken to be isomorphic toRn and, forx ∈ Rn,it acts onT via

Γx1 = 1 , ΓxΞi = Ξi , ΓxWi = Wi − xi1 , Γx(WjΞi) = WjΞi − xjΞi .

Let nowX = (X,X) be anα-Holder continuous rough path with values inRn. In other words,the functionsX andX are as in the introduction, satisfying the relation (1.4) and the analyticbounds|Xt −Xs| . |t − s|α, |Xs,t| . |t − s|2α. It turns out that this defines a model forT inthe following way (recall thatXs,t is a shorthand forXt −Xs):

Lemma 3.1 Given anα-Holder continuous rough pathX, one can define a model forT onR bysettingΓsu = ΓXs,u and

(Πs1)(t) = 1 , (ΠsWj)(t) = Xjs,t

(ΠsΞj)(ψ) =∫

ψ(t) dXjt , (ΠsWjΞi)(ψ) =

∫

ψ(t) dXi,js,t .

Here, both integrals are perfectly well-defined Riemann integrals, with the differential in the sec-ond case taken with respect to the variablet. Given a controlled rough path(Y, Y ′) ∈ Cα

X as in(1.2), this then defines an elementY ∈ D2α by setting

Y (s) = Y (s) 1+ Y ′i (s)Wi ,

with summation overi implied.

Proof. We first check that the algebraic properties of Definition 2.5are satisfied. It is clear thatΓsuΓut = Γst and thatΠsΓsuτ = Πuτ for τ ∈ {1,Wj ,Ξj}. RegardingWjΞi, we differentiateChen’s relations (1.4) which yields the identity

dXi,js,t = dXi,j

u,t +Xis,u dX

jt .

The last missing algebraic relation then follows at once. The required analytic bounds followimmediately from the definition of the rough path spaceDα.

Regarding the functionY defined in the statement, we have

‖Y (s) − ΓsuY (u)‖0 = |Y (s) − Y (u) + Y ′i (u)Xi

s,u| ,

‖Y (s) − ΓsuY (u)‖α = |Y ′(s) − Y ′(u)| ,

so that the condition (2.4) withγ = 2α does indeed coincide with the definition of a controlledrough path given in the introduction.

12 EXAMPLES OF REGULARITY STRUCTURES

In this context, the reconstruction theorem allows us to define an integration operator withrespect toW . We can formulate this as follows where one should really think of Z as providinga consistent definition of what one means by

∫

Y dXj .

Lemma 3.2 In the same context as above, letα ∈ (13, 12), and considerY ∈ D2α built as above

from a controlled rough path. Then, the mapY Ξi given by

(Y Ξj)(s) = Y (s)Ξj + Y ′i (s)WiΞj

belongs toD3α−1. Furthermore, there exists a functionZ such that, for every smooth test functionψ, one has

(RY Ξj)(ψ) =∫

ψ(t) dZ(t) ,

and such thatZs,t = Y (s)Xjs,t + Y ′

i (s)Xi,js,t +O(|t− s|3α).

Proof. The fact thatY Ξi ∈ D3α−1 is an immediate consequence of the definitions. Sinceα > 13

by assumption, we can apply the reconstruction theorem to it, from which it follows that thereexists a unique distributionη such that, ifψ is a smooth compactly supported test function, onehas

η(ψλs ) =

∫

ψλs (t)Y (s) dXj

t +

∫

ψλs (t)Y ′

i (s) dXi,js,t +O(λ3α−1) .

By a simple approximation argument, it turns out that one cantake forψ the indicator function ofthe interval [0, 1], so that

η(1[s,t] ) = Y (s)Xjs,t + Y ′

i (s)Xi,js,t +O(|t− s|3α) .

Here, the reason why one obtains an exponent3α rather than3α−1 is that it is really|t−s|−11[s,t]

that scales like an approximateδ-distribution ast → s.

Remark 3.3 Using the formula (2.14), it is straightforward to verify that if X happens to be asmooth function andX is defined fromX via (1.3), but this time viewing it as a definition for theleft hand side, with the right hand side given by a usual Riemann integral, then the functionZconstructed in Lemma 3.2 coincides with the usual Riemann integral ofY againstXj .

3.3 A classical result from harmonic analysis

The considerations above suggest that a very natural space of distributions is obtained in thefollowing way. For someα > 0, we denote byC−α the space of all Schwartz distributionsη suchthatη belongs to the dual ofCr with r > α some integer and such that

|η(ϕλx)| . λ−α ,

uniformly over allϕ ∈ Br andλ ∈ (0, 1], and locally uniformly inx. Given any compact setK, the best possible constant such that the above bound holds uniformly over x ∈ K yields aseminorm. The collection of these seminorms endowsC−α with a Frechet space structure.

EXAMPLES OF REGULARITY STRUCTURES 13

Remark 3.4 It turns out that the spaceC−α is independent of the choice ofr in the definitiongiven above, which justifies the notation. Different valuesof r give raise to equivalent seminorms.

Remark 3.5 In terms of the scale of classical Besov spaces, the spaceC−α is a local version ofB−α∞,∞. It is in some sense the largest space of distributions that is invariant under the scaling

ϕ(·) 7→ λ−αϕ(λ−1·), see for example [BP08].

It is then a classical result in the “folklore” of harmonic analysis that the product extendsnaturally toC−α × Cβ into S ′(Rd) if and only if β > α. The reconstruction theorem yields astraightforward proof of the “if” part of this result:

Theorem 3.6 There is a continuous bilinear mapB : C−α×Cβ → S ′(Rd) such thatB(f, g) = fgfor any two continuous functionsf andg.

Proof. Assume from now on thatξ ∈ C−α for someα > 0 and thatf ∈ Cβ for someβ > α. Wethen build a regularity structureT in the following way. For the setA, we takeA = N∪ (N−α)and forT , we setT = V ⊕W , where each one of the spacesV andW is a copy of the polynomialmodel ind commuting variables constructed in Section 3.1. We also chooseΓ as in the canonicalmodel, acting simultaneously on each of the two instances.

As before, we denote byXk the canonical basis vectors inV . We also use the suggestivenotation “ΞXk” for the corresponding basis vector inW , but we postulate thatΞXk ∈ Tα+|k|

rather thanΞXk ∈ T|k|. Given any distributionξ ∈ C−α, we then define a model (Πξ,Γ), whereΓ is as in the canonical model, whileΠξ acts as

(ΠξxX

k)(y) = (y − x)k , (ΠξxΞX

k)(y) = (y − x)kξ(y) ,

with the obvious abuse of notation in the second expression.It is then straightforward to verifythatΠy = Πx ◦ Γxy and that the relevant analytical bounds are satisfied, so that this is indeed amodel.

Denote now byRξ the reconstruction map associated to the model (Πξ ,Γ) and, forf ∈ Cβ ,denote byF the element inDβ given by the local Taylor expansion off of orderβ at each point.Note that even though the spaceDβ does in principle depend on the choice of model, in oursituationF ∈ Dβ for any choice ofξ. It follows immediately from the definitions that the mapx 7→ ΞF (x) belongs toDβ−α so that, provided thatβ > α, one can apply the reconstructionoperator to it. This suggests that the multiplication operator we are looking for can be defined as

B(f, ξ) = Rξ(ΞF ) .

By Theorem 2.10, this is a jointly continuous map fromCβ × C−α into S ′(Rd), provided thatβ > α. If ξ happens to be a smooth function, then it follows immediatelyfrom Remark 2.12 thatB(f, ξ) = f (x)ξ(x), so thatB is indeed the requested continuous extension of the usual product.

Remark 3.7 As a consequence of (2.5), it is actually easy to show thatB : C−α × Cβ → C−α.

14 PRODUCTS AND COMPOSITION BY SMOOTH FUNCTIONS

4 Products and composition by smooth functions

One of the main purposes of the theory presented here is to give a robust way to multiply distri-butions (or functions with distributions) that goes beyondthe barrier illustrated by Theorem 3.6.Provided that our functions / distributions are represented as elements inDγ for some model andregularity structure, we can multiply their “Taylor expansions” pointwise, provided that we giveourselves a table of multiplication onT .

It is natural to consider products with the following properties. Here, given a regularity struc-ture, we say that a subspaceV ⊂ T is asectorif it is invariant under the action of the structuregroupG and if it can furthermore be written asV =

⊕

α∈A Vα with Vα ⊂ Tα.

Definition 4.1 Given a regularity structure (T,A,G) and two sectorsV, V ⊂ T , a product on(V, V ) is a bilinear map⋆ : V × V → T such that, for anyτ ∈ Vα and τ ∈ Vβ , one hasτ ⋆ τ ∈ Tα+β and such that, for any elementΓ ∈ G, one hasΓ(τ ⋆ τ ) = Γτ ⋆ Γτ .

Remark 4.2 The condition that homogeneities add up under multiplication is very natural bear-ing in mind the case of the polynomial regularity structure.The second condition is also verynatural since it merely states that if one reexpands the product of two “polynomials” around adifferent point, one should obtain the same result as if one reexpands each factor first and thenmultiplies them together.

Given such a product, we can ask ourselves when the pointwiseproduct of an elementDγ1

with an element inDγ2 again belongs to someDγ . In order to answer this question, we introducethe notationDγ

α to denote those elementsf ∈ Dγ such that furthermore

f (x) ∈ T+α ≡

⊕

β≥α

Tβ ,

for everyx. With this notation at hand, it is not too difficult to verify that one has the followingresult:

Theorem 4.3 Let f1 ∈ Dγ1α1(V ), f2 ∈ Dγ2

α2(V ), and let⋆ be a product on(V, V ). Then, thefunctionf given byf (x) = f1(x) ⋆ f2(x) belongs toDγ

α with

α = α1 + α2 , γ = (γ1 + α2) ∧ (γ2 + α1) . (4.1)

Proof. It is clear thatf (x) ∈ ⊕

β>α Tβ, so it remains to show that it belongs toDγ . Furthermore,since we are only interested in showing thatf1 ⋆ f2 ∈ Dγ , we discard all of the components inTβfor β ≥ γ.

By the properties of the product⋆, it remains to obtain a bound of the type

‖Γxyf1(y) ⋆ Γxyf2(y) − f1(x) ⋆ f2(x)‖β . |x− y|γ−β .

By adding and subtracting suitable terms, we obtain

‖Γxyf (y) − f (x)‖β ≤ ‖(Γxyf1(y) − f1(x)) ⋆ (Γxyf2(y) − f2(x))‖β (4.2)

PRODUCTS AND COMPOSITION BY SMOOTH FUNCTIONS 15

+ ‖(Γxyf1(y) − f1(x)) ⋆ f2(x)‖β + ‖f1(x) ⋆ (Γxyf2(y) − f2(x))‖β .

It follows from the properties of the product⋆ that the first term in (4.2) is bounded by a constanttimes

∑

β1+β2=β

‖Γxyf1(y) − f1(x)‖β1‖Γxyf2(y) − f2(x)‖β2

.∑

β1+β2=β

‖x− y‖γ1−β1‖x− y‖γ2−β2 . ‖x− y‖γ1+γ2−β .

Sinceγ1 + γ2 ≥ γ, this bound is as required. The second term is bounded by a constant times∑

β1+β2=β

‖Γxyf1(y) − f1(x)‖β1‖f2(x)‖β2

. ‖x− y‖γ1−β1 1β2≥α2. ‖x− y‖γ1+α2−β ,

where the second inequality uses the identityβ1 + β2 = β. Sinceγ1 + α2 ≥ γ, this bound isagain of the required type. The last term is bounded similarly by reversing the roles played byf1andf2.

Remark 4.4 It is clear that the formula (4.1) forγ is optimal in general as can be seen fromthe following two “reality checks”. First, consider the case of the polynomial model and takefi ∈ Cγi . In this case, the truncated Taylor seriesFi for fi belong toDγi

0 . It is clear that in thiscase, the product cannot be expected to have better regularity thanγ1 ∧ γ2 in general, which isindeed what (4.1) states. The second reality check comes from the example of Section 3.3. In thiscase, one hasF ∈ Dβ

0 , while the constant functionx 7→ Ξ belongs toD∞−α so that, according to

(4.1), one expects their product to belong toDβ−α−α , which is indeed the case.

It turns out that if we have a product on a regularity structure, then in many cases this alsonaturally yields a notion of composition with smooth functions. Of course, one could in generalnot expect to be able to compose a smooth function with a distribution of negative order. As amatter of fact, we will only define the composition of smooth functions with elements in someDγ for which it is guaranteed that the reconstruction operatoryields a continuous function. Onemight think at this case that this would yield a triviality, since we know of course how to com-pose arbitrary continuous function. The subtlety is that wewould like to design our compositionoperator in such a way that the result is again an element ofDγ .

For this purpose, we say that a given sectorV ⊂ T is function-like if α < 0 ⇒ Vα = 0and if V0 is one-dimensional. (Denote the unit vector ofV0 by 1.) We will furthermore alwaysassume that our models arenormal in the sense that(Πx1)(y) = 1. I this case, it turns out thatif f ∈ Dγ(V ), thenRf is a continuous function and one has the identity(Rf)(x) = 〈1, f (x)〉,where we denote by〈1, ·〉 the element in the dual ofV which picks out the prefactor of1.

Assume now that we are given a regularity structure with a function-like sectorV and aproduct⋆ : V × V → V . For any smooth functionG : R → R and anyf ∈ Dγ(V ) with γ > 0,we can thendefineG(f ) to be theV -valued function given by

(G ◦ f)(x) =∑

k≥0

G(k)(f (x))k!

f (x)⋆k ,

16 PRODUCTS AND COMPOSITION BY SMOOTH FUNCTIONS

where we have setf (x) = 〈1, f (x)〉 , f (x) = f (x) − f (x)1 .

Here,G(k) denotes thekth derivative ofG andτ⋆k denotes thek-fold productτ ⋆ · · · ⋆ τ . We alsoused the usual conventionsG(0) = G andτ⋆0 = 1.

Note that as long asG is C∞, this expression is well-defined. Indeed, by assumption, thereexists someα0 > 0 such thatf (x) ∈ T+

α0. By the properties of the product, this implies that one

hasf (x)⋆k ∈ T+kα0

. As a consequence, when considering the component ofG◦f in Tβ for β < γ,the only terms that give a contribution are those withk < γ/α0. Since we cannot possibly hopein general thatG ◦ f ∈ Dγ′

for someγ′ > γ, this is all we really need.It turns out that ifG is sufficiently regular, then the mapf 7→ G ◦ f enjoys similarly nice

continuity properties to what we are used to from classical Holder spaces. The following result isthe analogue in this context to the well-known fact that the composition of aCγ function with asufficiently smooth functionG is again of classCγ .

Proposition 4.5 In the same setting as above, provided thatG is of classCk with k > γ/α0, themapf 7→ G ◦ f is continuous fromDγ(V ) into itself. Ifk > γ/α0 +1, then it is locally Lipschitzcontinuous.

The proof of this result can be found in [Hai14]. It is somewhat lengthy, but ultimately ratherstraightforward.

4.1 A simple example

A very important remark is that even if bothRf1 andRf2 happens to be continuous functions,this doesnot in general imply thatR(f1 ⋆ f2)(x) = (Rf1)(x) (Rf2)(x)! For example, fixκ < 0and consider the regularity structure given byA = (−2κ,−κ, 0), with eachTα being a copy ofR given byT−nκ = 〈Ξn〉. We furthermore take forG the trivial group. This regularity structurecomes with an obvious product by settingΞm ⋆ Ξn = Ξm+n provided thatm+ n ≤ 2.

Then, we could for example take as a model forT = (T,A,G):

(ΠxΞ0)(y) = 1 , (ΠxΞ)(y) = 0 , (ΠxΞ

2)(y) = c , (4.3)

wherec is an arbitrary constant. Let furthermore

F1(x) = f1(x)Ξ0 + f ′1(x)Ξ , F2(x) = f2(x)Ξ0 + f ′2(x)Ξ .

Since our groupG is trivial, one hasFi ∈ Dγ provided that each of thefi belongs toDγ andeach of thef ′i belongs toDγ+κ. (And one hasγ + κ < 1.) One furthermore has the identity(RFi)(x) = fi(x).

However, the pointwise product is given by

(F1 ⋆ F2)(x) = f1(x)f2(x)Ξ0 + (f ′1(x)f2(x) + f ′2(x)f1(x))Ξ + f ′1(x)f ′2(x)Ξ2 ,

which by Theorem 4.3 belongs toDγ−κ. Provided thatγ > κ, one can then apply the reconstruc-tion operator to this product and one obtains

R(F1 ⋆ F2)(x) = f1(x)f2(x) + cf ′1(x)f ′2(x) ,

SCHAUDER ESTIMATES AND ADMISSIBLE MODELS 17

which is obviously different from the pointwise productRF1 · RF2.How should this be interpreted? Forn > 0, we could have defined a modelΠ(n) by

(ΠxΞ0)(y) = 1 , (ΠxΞ)(y) =

√2c sin(nx) , (ΠxΞ

2)(y) = 2c sin2(nx) .

Denoting byR(n) the corresponding reconstruction operator, we have the identity

(R(n)Fi)(x) = fi(x) +√2cf ′i(x) sin(nx) ,

as well asR(n)(F1 ⋆ F2) = R(n)F1 · R(n)F2. As a model, the modelΠ(n) actually converges tothe limiting modelΠ defined in (4.3). As a consequence of the continuity of the reconstructionoperator, this implies that

R(n)F1 · R(n)F2 = R(n)(F1 ⋆ F2) → R(F1 ⋆ F2) 6= RF1 · RF2 ,

which is of course also easy to see “by hand”. This shows that in some cases, the “non-standard”models as in (4.3) can be interpreted as limits of “standard”models for which the usual rules ofcalculus hold. Even this is however not always the case.

5 Schauder estimates and admissible models

One of the reasons why the theory of regularity structures isvery successful at providing de-tailed descriptions of the small-scale features of solutions to semilinear (S)PDEs is that it comeswith very sharp Schauder estimates. Recall that the classical Schauder estimates state that ifK : Rd → R is a kernel that is smooth everywhere, except for a singularity at the origin that is(approximately) homogeneous of degreeβ − d for someβ > 0, then the operatorf 7→ K ∗ fmapsCα into Cα+β for everyα ∈ R, except for those values for whichα + β ∈ N. (See forexample [Sim97].)

It turns out that similar Schauder estimates hold in the context of general regularity structuresin the sense that it is in general possible to build an operator K : Dγ → Dγ+β with the propertythatRKf = K ∗ Rf . Of course, such a statement can only be true if our regularity structurecontains not only the objects necessary to describeRf up to orderγ, but also those required todescribeK ∗ Rf up to orderγ + β. What are these objects? At this stage, it might be useful toreflect on the effect of the convolution of a singular function (or distribution) withK.

Let us assume for a moment thatf is also smooth everywhere, except at some pointx0. Itis then straightforward to convince ourselves thatK ∗ f is also smooth everywhere, except atx0. Indeed, for anyδ > 0, we can writeK = Kδ + Kc

δ , whereKδ is supported in a ball ofradiusδ around0 andKc

δ is a smooth function. Similarly, we can decomposef asf = fδ + f cδ ,wherefδ is supported in aδ-ball aroundx0 andf cδ is smooth. Since the convolution of a smoothfunction with an arbitrary distribution is smooth, it follows that the only non-smooth componentof K ∗ f is given byKδ ∗ fδ, which is supported in a ball of radius2δ aroundx0. Sinceδ wasarbitrary, the statement follows. By linearity, this strongly suggests that the local structure of thesingularities ofK ∗f can be described completely by only using knowledge on the local structureof the singularities off . It also suggests that the “singular part” of the operatorK should be local,with the non-local parts ofK only contributing to the “regular part”.

18 SCHAUDER ESTIMATES AND ADMISSIBLE MODELS

This discussion suggests that we certainly need the following ingredients to build an operatorK with the desired properties:

• The canonical polynomial structure should be part of our regularity structure in order to beable to describe the “regular parts”.

• We should be given an “abstract integration operator”I on T which describes how the“singular parts” ofRf transform under convolution byK.

• We should restrict ourselves to models which are “compatible” with the action ofI inthe sense that the behaviour ofΠxIτ should relate in a suitable way to the behaviour ofK ∗ Πxτ nearx.

One way to implement these ingredients is to assume first thatour model spaceT contains abstractpolynomials in the following sense.

Assumption 5.1 There exists a sectorT ⊂ T isomorphic to the space of abstract polynomialsin d commuting variables. In other words,Tα 6= 0 if and only ifα ∈ N, and one can find basisvectorsXk ofT|k| such that every elementΓ ∈ G acts onT byΓXk = (X−h)k for someh ∈ Rd.

Furthermore, we assume that there exists an abstract integration operatorI with the followingproperties.

Assumption 5.2 There exists a linear mapI : T → T such thatITα ⊂ Tα+β , such thatIT = 0,and such that, for everyΓ ∈ G andτ ∈ T , one has

ΓIτ − IΓτ ∈ T . (5.1)

Finally, we want to consider models that are compatible withthis structure for a given kernelK. For this, we first make precise what we mean exactly when we said thatK is approximatelyhomogeneous of degreeβ − d.

Assumption 5.3 One can writeK =∑

n≥0Kn where each of the kernelsKn : Rd → R issmooth and compactly supported in a ball of radius2−n around the origin. Furthermore, weassume that for every multiindexk, one has a constantC such that the bound

supx

|DkKn(x)| ≤ C2n(d−β+|k|) , (5.2)

holds uniformly inn. Finally, we assume that∫

Kn(x)P (x) dx = 0 for every polynomialP ofdegree at mostN , for some sufficiently large value ofN .

Remark 5.4 It turns out that in order to define the operatorK onDγ , we will needK to annihilatepolynomials of degreeN for someN ≥ γ + β.

Remark 5.5 The last assumption may appear to be extremely stringent at first sight. In practice,this turns out not to be a problem at all. Say for example that we want to define an operator thatrepresents convolution withG, the Green’s function of the Laplacian. Then,G can be decomposed

SCHAUDER ESTIMATES AND ADMISSIBLE MODELS 19

into a sum of terms satisfying the bound (5.2) withβ = 2, but it does of course not annihilategeneric polynomials and it is not supported in the ball of radius 1.

However, for any fixed value ofN > 0, it is straightforward to decomposeG asG = K +R,where the kernelK is compactly supported and satisfies all of the properties mentioned above,and the kernelR is smooth. Lifting the convolution withR to an operator fromDγ → Dγ+β

(actually toDγ for any γ > 0) is straightforward, so that we have reduced our problem to that ofconstructing an operator describing the convolution byK.

Given such a kernelK, we can now make precise what we meant earlier when we said thatthe models under consideration should be compatible with the kernelK.

Definition 5.6 Given a kernelK as in Assumption 5.3 and a regularity structureT satisfyingAssumptions 5.1 and 5.2, we say that a model (Π,Γ) is admissibleif the identities

(ΠxXk)(y) = (y − x)k , ΠxIτ = K ∗ Πxτ −ΠxJ (x)τ , (5.3)

holds for everyτ ∈ T with |τ | ≤ N . Here,J (x) : T → T is the linear map given on homoge-neous elements by

J (x)τ =∑

|k|<|τ |+β

Xk

k!

∫

D(k)K(x− y) (Πxτ)(dy) . (5.4)

Remark 5.7 Note first that ifτ ∈ T , then the definition given above is coherent as long as|τ | < N . Indeed, sinceIτ = 0, one necessarily hasΠxIτ = 0. On the other hand, the propertiesof K ensure that in this case one also hasK ∗ Πxτ = 0, as well asJ (x)τ = 0.

Remark 5.8 WhileK ∗ ξ is well-defined for any distributionξ, it is not so cleara priori whetherthe operatorJ (x) given in (5.4) is also well-defined. It turns out that the axioms of a model doensure that this is the case. The correct way of interpreting(5.4) is by

J (x)τ =∑

|k|<|τ |+β

∑

n≥0

Xk

k!(Πxτ)(D

(k)Kn(x− ·)) .

Note now that the scaling properties of theKn ensure that2(β−|k|)nD(k)Kn(x−·) is a test functionthat is localised aroundx at scale2−n. As a consequence, one has

|(Πxτ)(D(k)Kn(x− ·))| . 2(|k|−β−|τ |)n ,

so that this expression is indeed summable as long as|k| < |τ |+ β.

Remark 5.9 As a matter of fact, it turns out that the above definition of anadmissible modeldovetails very nicely with our axioms defining a general model. Indeed, starting fromanyregular-ity structureT , anymodel (Π,Γ) for T , and a kernelK satisfying Assumption 5.3, it is usuallypossible to build a larger regularity structureT containingT (in the “obvious” sense thatT ⊂ T

20 SCHAUDER ESTIMATES AND ADMISSIBLE MODELS

and the action ofG onT is compatible with that ofG) and endowed with an abstract integrationmapI, as well as an admissible model (Π, Γ) on T which reduces to (Π,Γ) when restricted toT .See [Hai14] for more details.

The only exception to this rule arises when the original structureT contains some homoge-neous elementτ which does not represent a polynomial and which is such that|τ |+β ∈ N. Sincethe bounds appearing both in the definition of a model and in Assumption 5.3 are only upperbounds, it is in practice easy to exclude such a situation by slightly tweaking the definition ofeither the exponentβ or of the original regularity structureT .

With all of these definitions in place, we can finally build theoperatorK : Dγ → Dγ+β

announced at the beginning of this section. Recalling the definition of J from (5.4), we set

(Kf)(x) = If (x) + J (x)f (x) + (N f)(x) , (5.5)

where the operatorN is given by

(N f)(x) =∑

|k|<γ+β

Xk

k!

∫

D(k)K(x− y) (Rf −Πxf (x))(dy) . (5.6)

Note first that thanks to the reconstruction theorem, it is possible to verify that the right hand sideof (5.6) does indeed make sense for everyf ∈ Dγ in virtually the same way as in Remark 5.8.One has:

Theorem 5.10 LetK be a kernel satisfying Assumption 5.3, letT = (A,T,G) be a regularitystructure satisfying Assumptions 5.1 and 5.2, and let(Π,Γ) be an admissible model forT . Then,for everyf ∈ Dγ with γ ∈ (0, N − β) andγ + β 6∈ N, the functionKf defined in (5.5) belongstoDγ+β and satisfiesRKf = K ∗ Rf .

Proof. The complete proof of this result can be found in [Hai14] and will not be given here. Letus simply show that one has indeedRKf = K ∗ Rf in the particular case when our modelconsists of continuous functions so that Remark 2.12 applies. In this case, one has

(RKf)(x) = (Πx(If (x) + J (x)f (x)))(x) + (Πx(N f)(x))(x) .

As a consequence of (5.3), the first term appearing in the right hand side of this expression isgiven by

(Πx(If (x) + J (x)f (x)))(x) = (K ∗ Πxf (x))(x) .

On the other hand, the only term contributing to the second term is the one withk = 0 (which isalways present sinceγ > 0 by assumption) which then yields

(Πx(N f)(x))(x) =∫

K(x− y) (Rf −Πxf (x))(dy) .

Adding both of these terms, we see that the expression(K ∗Πxf (x))(x) cancels, leaving us withthe desired result.

APPLICATION OF THE THEORY TO SEMILINEARSPDES 21

6 Application of the theory to semilinear SPDEs

Let us now briefly explain how this theory can be used to make sense of solutions to very singularsemilinear stochastic PDEs. We will keep the discussion in this section at a very informal levelwithout attempting to make mathematically precise statements. The interested reader may findmore details in [Hai14].

For definiteness, we focus on the case of the dynamicalΦ43 model, which is formally given by

∂tΦ = ∆Φ− Φ3 + ξ , (6.1)

whereξ denotes space-time white noise and the spatial variable takes values in the3-dimensio-nal torus. The problem with such an equation is that even the solution to the linear part of theequation, namely

∂tΨ = ∆Ψ+ ξ ,

only admits solutions in some spaces of Schwartz distributions. As a matter of fact, one hasΨ(t, ·) ∈ C−α if and only if α > 1/2. As a consequence, it turns out that the only way of givingmeaning to (6.1) is to “renormalise” the equation by adding an “infinite” linear term “∞Φ” whichcounteracts the strong dissipativity of the term−Φ3. To be slightly more precise, one can prove astatement of the following kind:

Theorem 6.1 Consider the sequence of equations

∂tΦε = ∆Φε + CεΦε − Φ3ε + ξε , (6.2)

whereξε = δε ∗ ξ with δε(t, x) = ε−5(ε−2t, ε−1x), for some smooth and compactly supportedfunction, andξ denotes space-time white noise. Then, there exists a choiceof constantsCε suchthat the sequenceΦε converges in probability to a limiting (distributional) processΦ. Further-more, the limiting processΦ does not depend on the choice of mollifier.

Remark 6.2 It turns out that in order to obtain a limit that is independent of the choice of mollifier, one should takeCε of the form

Cε =c1ε

+ c logε+ c3 ,

wherec is universal, butc1 andc3 depend on the choice of.

Remark 6.3 The limiting solutionΦ is only local in time, so that the precise statement has to beslightly tweaked to allow for finite-time blow-ups. Regarding the initial condition, one can takeΦ0 ∈ C−β for anyβ < 2/3. This is expected to be optimal, even for the deterministic equation.

The aim of this section is to sketch how the theory of regularity structures can be used toobtain this kind of convergence results. First of all, we note that while our solutionΦ will be aspace-time distribution (or rather an element ofDγ for some regularity structure with a modeloverR4), the “time” direction has a different scaling behaviour from the three “space” directions.

22 APPLICATION OF THE THEORY TO SEMILINEARSPDES

As a consequence, it turns out to be effective to slightly change our definition of “localised testfunctions” by setting

ϕλ(s,x)(t, y) = λ−5ϕ(λ−2(t− s), λ−1(y − x)) .

Accordingly, the “effective dimension” of our space-time is actually5, rather than4. The theorypresented above extendsmutatis mutandisto this setting. (Note in particular that when consider-ing the degree of a regular monomial, powers of the time variable should now be counted double.)Note also that with this way of measuring regularity, space-time white noise belongs toC−α foreveryα > 5

2. This is because of the bound

(E〈ξ, ϕλx〉2)1/2 = ‖ϕλ

x‖L2 ≈ λ−5

2 ,

combined with an argument similar to the proof of Kolmogorov’s continuity lemma.

6.1 Construction of the associated regularity structure

Our first step is to build a regularity structure that is sufficiently large to allow to reformulate (6.1)as a fixed point inDγ for someγ > 0. Denoting byG the heat kernel (i.e. the Green’s function ofthe operator∂t −∆), we can write the solution to (6.1) with initial conditionΦ0 as

Φ = G ∗ (ξ − Φ3) + GΦ0 ,

where∗ denotes space-time convolution and where we denote byGΦ0 the harmonic extensionof Φ0. In order to have a chance of fitting this into the framework described above, we firstdecompose the heat kernelG as

G = K + K ,

where the kernelK satisfies all of the assumptions of Section 5 (withβ = 2) and the remainderK is smooth. If we consider any regularity structure containing the usual Taylor polynomials andequipped with an admissible model, is straightforward to associate toK an operatorK : Dγ →D∞ via

(Kf)(z) =∑

k

Xk

k!(D(k)K ∗ Rf)(z) ,

wherez denotes a space-time point andk runs over all possible4-dimensional multiindices. Sim-ilarly, the harmonic extension ofΦ0 can be lifted to an element inD∞ which we denote again byGΦ0 by considering its Taylor expansion around every space-time point. At this stage, we notethat we actually cheated a little: whileGΦ0 is smooth in{(t, x) : t > 0, x ∈ T3} and vanisheswhent < 0, it is of course singular on the time-0 hyperplane{(0, x) : x ∈ T3}. This problemcan be cured by introducing weighted versions of the spacesDγ allowing for singularities on agiven hyperplane. A precise definition of these spaces and their behaviour under multiplicationand the action of the integral operatorK can be found in [Hai14]. For the purpose of the informaldiscussion given here, we will simply ignore this problem.

This suggests that the “abstract” formulation of (6.1) should be given by

Φ = K(Ξ− Φ3) + K(Ξ− Φ3) + GΦ0 . (6.3)

APPLICATION OF THE THEORY TO SEMILINEARSPDES 23

In view of (5.5), this equation is of the type

Φ = I(Ξ−Φ3) + (. . .) , (6.4)

where the terms (. . .) consist of functions that take values in the subspaceT of T spanned by reg-ular Taylor polynomials. In order to build a regularity structure in which (6.4) can be formulated,it is natural to start with the structure given by abstract polynomials (again with the parabolicscaling which causes the abstract “time” variable to have homogeneity2 rather than1), and toadd a symbolΞ to it which we postulate to have homogeneity−5

2

−, where we denote byα− an

exponent strictly smaller than, but arbitrarily close to, the valueα.We then simply add toT all of the formal expressions that an application of the right hand side

of (6.4) can generate for the description ofΦ, Φ2, andΦ3. The homogeneity of a given expressionis completely determined by the rules|Iτ | = |τ | + 2 and|τ τ | = |τ | + |τ |. More precisely, weconsider a collectionU of formal expressions which is the smallest collection containing 1, X,andI(Ξ), and such that

τ1, τ2, τ3 ∈ U ⇒ I(τ1τ2τ3) ∈ U ,

where it is understood thatI(Xk) = 0 for every multiindexk. We then set

W = {Ξ} ∪ {τ1τ2τ3 : τi ∈ U} ,

and we define our spaceT as the set of all linear combinations of elements inW. (Note that since1 ∈ U , one does in particular haveU ⊂ W.) Naturally,Tα consists of those linear combinationsthat only involve elements inW that are of homogeneityα. It is not too difficult to convinceoneself that, for everyα ∈ R, W contains only finitely many elements of homogeneity less thanα, so that eachTα is finite-dimensional.

In order to simplify expressions later, we will use the following shorthand graphical notationfor elements ofW. ForΞ, we simply draw a dot. The integration map is then represented by adownfacing line and the multiplication of symbols is obtained by joining them at the root. Forexample, we have

I(Ξ) = , I(Ξ)3 = , I(Ξ)I(I(Ξ)3) = .

Symbols containing factors ofX have no particular graphical representation, so we will forex-ample writeXiI(Ξ)2 = Xi . With this notation, the spaceT is given by

T = 〈Ξ, , , , , , ,Xi ,1, , , . . .〉 ,

where we ordered symbols in increasing order of homogeneityand used〈·〉 to denote the linearspan. Given any sufficiently regular functionξ (say a continuous space-time function), there isthen a canonical way of liftingξ to a modelιξ = (Π,Γ) for T by setting

(ΠxΞ)(y) = ξ(y) , (ΠxXk)(y) = (y − x)k ,

and then recursively by(Πxτ τ)(y) = (Πxτ)(y) · (Πxτ)(y) , (6.5)

24 RENORMALISATION OF THE DYNAMICAL Φ43 MODEL

as well as (5.3). (Note that here we usedx andy as notations for generic space-time points inorder not to overload the notations.)

It turns out furthermore that there is a canonical way of building a structure groupG for Tand to also liftξ to a family of operatorsΓxy, in such a way that all of the algebraic and analyticproperties of an admissible model are satisfied. With such a model ιξ at hand, it follows from(6.5) and the admissibility ofιξ that the associated reconstruction operator satisfies the properties

RKf = K ∗ Rf , R(fg) = Rf · Rg ,

as long as all the functions to whichR is applied belong toDγ for someγ > 0. As a consequence,applying the reconstruction operatorR to both sides of (6.3), we see that ifΦ solves (6.3) then,provided that the model (Π,Γ) = ιξ was built as above starting from anycontinuousrealisationξof the driving noise,RΦ solves the equation (6.1).

At this stage, the situation is as follows. For anycontinuousrealisationξ of the driving noise,we have factored the solution map (Φ0, ξ) → Φ associated to (6.1) into maps

(Φ0, ξ) → (Φ0, ιξ) → Φ → RΦ ,

where the middle arrow corresponds to the solution to (6.3) in some weightedDγ-space. Theadvantage of such a factorisation is that the last two arrowsyield continuousmaps, even intopologies sufficiently weak to be able to describe driving noise having the lack of regularityof space-time white noise. The only arrow that isn’t continuous in such a weak topology is thefirst one. At this stage, it should be believable that a similar construction can be performed fora very large class of semilinear stochastic PDEs. In particular, the KPZ equation can also beanalysed in this framework.

Given this construction, one is lead naturally to the following question: given a sequenceξεof “natural” regularisations of space-time white noise, for example as in (6.2), do the liftsιξεconverge in probably in a suitable space of admissible models? Unfortunately, unlike in the caseof the theory of rough paths where this is very often the case,the answer to this question in thecontext of SPDEs is often an emphaticno. Indeed, if it were the case for the dynamicalΦ4

3 model,then one could have chosen the constantCε to be independent ofε in (6.2), which is certainly notthe case.

7 Renormalisation of the dynamicalΦ43 model

One way of circumventing the fact thatιξε does not converge to a limiting model asε → 0 is toconsider instead a sequence ofrenormalisedmodels. The main idea is to exploit the fact that ourabstract definitions of a model do not impose the identity (6.5), even in situations whereξ itselfhappens to be a continuous function. One question that then imposes itself is: what are the naturalways of “deforming” the usual product which still lead to lifts to an admissible model? It turnsout that the regularity structure whose construction was sketched above comes equipped witha naturalfinite-dimensionalgroup of continuous transformationsR on its space of admissiblemodels (henceforth called the “renormalisation group”), which essentially amounts to the spaceof all natural deformations of the product. It then turns outthat even thoughιξε does not converge,

RENORMALISATION OF THE DYNAMICAL Φ43 MODEL 25

it is possible to find a sequenceMε of elements inR such that the sequenceMειξε converges toa limiting model (Π, Γ). Unfortunately, the elementsMε no not preserve the image ofι in thespace of admissible models. As a consequence, when solving the fixed point map (6.3) withrespect to the modelMειξε and inserting the solution into the reconstruction operator, it is notcleara priori that the resultong function (or distribution) can again be interpreted as the solutionto some modified PDE. It turns out that in our case, at least fora certain two-parameter subgroupof R, this is again the case and the modified equation is preciselygiven by (6.2), whereCε issome linear combination of the two constants appearing in the description ofMε.

There are now three questions that remain to be answered:

1. How does one construct the renormalisation groupR?

2. How does one derive the new equation obtained when renormalising a model?

3. What is the right choice ofMε ensuring that the renormalised models converge?

7.1 The renormalisation group

In order to constructR, it is essential to first have some additional knowledge of the structuregroupG for the type of regularity structures considered above. Recall that the purpose of thegroupG is to provide a class of linear mapsΓ: T → T arising as possible candidates for theaction of “reexpanding” a “Taylor series” around a different point. In our case, in view of (5.3), thecoefficients of these reexpansions will naturally be some polynomials inx and in the expressionsappearing in (5.4). This suggests that we should define a spaceT+ whose basis vectors consist offormal expressions of the type

XkN∏

i=1

Jℓiτi , (7.1)

whereN is an arbitrary but finite number, theτi are basis elements ofT , and theℓi are d-dimensional multiindices satisfying|ℓi| < |τi| + 2. (The last bound is a reflection of the re-striction of the summands in (5.4) withβ = 2.) The spaceT+ also has a natural graded structureT+ =

⊕

T+α by setting

|Jℓτ | = |τ |+ 2− |ℓ| , |Xk| = |k| ,

and by postulating that the degree of a product is the sum of the degrees. Unlike in the case ofThowever, elements ofT+ all have strictly positive homogeneity, except for the empty product1which we postulate to have degree0.

To any given admissible model (Π,Γ), it is then natural to associate linear mapsfx : T+ → R

by fx(Xk) = xk, fx(σσ) = fx(σ)fx(σ), and

fx(Jℓiτi) =∫

D(ℓi)K(x− y) (Πxτi)(dy) . (7.2)

It then turns out that it is possible to build a linear map∆: T → T ⊗ T+ such that if we defineFx : T → T by

Fxτ = (I ⊗ fx)∆τ , (7.3)


whereI denotes the identity operator onT , then these maps are invertible andΠxF−1x is indepen-

dent ofx. Furthermore, there exists a map∆+ : T+ → T+ ⊗ T+ such that

(∆⊗ I)∆ = (I ⊗∆+)∆ , ∆+(σσ) = ∆+σ ·∆+σ . (7.4)

With this map at hand, we can define a product◦ on the space of linear functionalsf : T+ → Rby

(f ◦ g)(σ) = (f ⊗ g)∆+σ .

If we furthermore denote byΓf the operatorT associated to any such linear functional as in(7.3), the first identity of (7.4) yields the identityΓfΓg = Γf◦g. The second identity of (7.4)furthermore ensures that iff andg are both multiplicative in the sense thatf (σσ) = f (σ)f (σ),thenf ◦ g is again multiplicative. It also turns out that every multiplicative linear functionalfadmits a unique inversef−1 such thatf−1 ◦ f = f ◦ f−1 = e, wheree : T+ → R maps everybasis vector of the form (7.1) to zero, except fore(1) = 1. The elemente is neutral in the sensethatΓe is the identity operator.

It is now natural to define the structure groupG associated toT as the set of all multiplicativelinear functionals onT+, acting onT via (7.3). Furthermore, for any admissible model, one hasthe identity

Γxy = F−1x Fy = Γγxy , γxy = f−1

x ◦ fy .How does all this help with the identification of a natural class of deformations for the usual

product? First, it turns out that for every continuous function ξ, if we denote again by (Π,Γ) themodelιξ, then the linear mapΠ : T → C given by

Π = ΠyF−1y ,

which is independent of the choice ofy by the above discussion, is given by

(ΠΞ)(x) = ξ(x) , (ΠXk)(x) = xk , (7.5)

and then recursively by

Πτ τ = Πτ ·Πτ , ΠIτ = K ∗Πτ .

Note that this is very similar to the definition ofιξ, with the notable exception that (5.3) is replacedby the more “natural” identityΠIτ = K ∗ Πτ . It turns out that the knowledge ofΠ and theknowledge of (Π,Γ) are equivalent since one hasΠx = ΠFx and the mapFx can be recoveredfrom Πx by (7.2). (This argument appears circular but it is possibleto put a suitable recursivestructure onT andT+ ensuring that this actually works.) Furthermore, the translation (Π,Γ) ↔Π actually works forany admissible model and does not at all rely on the fact that it was builtby lifting a continuous function. However, in the general case, the first identity in (7.5) does notof course not make any sense anymore and might fail even if thecoordinates ofΠ consist ofcontinuous functions.

At this stage we note that ifξ happens to be a stationary stochastic process andΠ is built fromξ by following the above procedure, thenΠτ is a stationary stochastic process for everyτ ∈ T . In


order to defineR, it is natural to consider only transformations of the spaceof admissible modelsthat preserve this property. Since we are not in general allowed to multiply components ofΠ,the only remaining operation is to form linear combinations. It is therefore natural to describeelements ofR by linear mapsM : T → T and to postulate their action on admissible models byΠ 7→ Π

M withΠ

Mτ = ΠMτ .

It is not cleara priori whether given such a mapM and an admissible model (Π,Γ) there is a co-herent way of building a new model (ΠM ,ΓM ) such thatΠM is the map associated to (ΠM ,ΓM )as above. It turns out that one has the following statement:

Proposition 7.1 In the above context, for every linear mapM : T → T commuting withI andmultiplication byXk, there exist unique linear maps∆M : T → T ⊗ T+ and ∆M : T+ →T+ ⊗ T+ such that if we set

ΠMx τ = (Πx ⊗ fx)∆

Mτ , γMxy (σ) = (γxy ⊗ fx)∆Mσ ,

thenΠMx satisfies again (5.3) and the identityΠM

x ΓMxy = ΠM

y .

At this stage it may look likeany linear mapM : T → T commuting withI and multiplica-tion byXk yields a transformation on the space of admissible models byProposition 7.1. Thishowever is not true since we have completely disregarded theanalyticalbounds that every modelhas to satisfy. It is clear from Definition 2.5 that these are satisfied if and only ifΠM

x τ is a linearcombination of theΠxτj with |τj| ≥ |τ |. This suggests the following definition.

Definition 7.2 The renormalisation groupR consists of the set of linear mapsM : T → T com-muting withI and with multiplication byXk, such that forτ ∈ Tα andσ ∈ T+

α , one has

∆Mτ − τ ⊗ 1 ∈⊕

β>α

Tα ⊗ T+ , ∆Mσ − σ ⊗ 1 ∈⊕

β>α

T+α ⊗ T+ .

Its action on the space of admissible models is given by Proposition 7.1.

7.2 The renormalised equations

In the case of the dynamicalΦ4 model considered in this article, it turns out that we only need atwo-parameter subgroup ofR to renormalise the equations. More precisely, we consider elementsM ∈ R of the formM = exp(−C1L1 − C2L2), where the two generatorsL1 andL2 aredetermined by the substitution rules

L1 : 7→ 1 , L2 : 7→ 1 .

This should be understood in the sense that ifτ is an arbitrary formal expression, thenL1τ is thesum of all formal expressions obtained fromτ by performing a substitution of the type 7→ 1,and similarly forL2. For example, one has

L1 = 3 , L1 = , L2 = 3 .

One then has the following result:


Proposition 7.3 The linear mapsM of the type just described belong toR. Furthermore, if(Π,Γ) is an admissible model such thatΠxτ is a continuous function for everyτ ∈ T , then onehas the identity

(ΠMx τ)(x) = (ΠxMτ)(x) . (7.6)

Remark 7.4 Note that it it is the same valuex that appears twice on each side of (7.6). It is infactnot the case that one hasΠM

x τ = ΠxMτ ! However, the identity (7.6) is all we need to derivethe renormalised equations.

It is now rather straightforward to show the following:

Proposition 7.5 LetM = exp(−C1L1 − C2L2) as above and let(ΠM ,ΓM ) = Mιξ for somesmooth functionξ. Let furthermoreΦ be the solution to (6.3) with respect to the model(ΠM ,ΓM ).Then, the functionu(t, x) = (RMΦ)(t, x) solves the equation

∂tu = ∆u− u3 + (3C1 − 9C2)u+ ξ .

Proof. By Theorem 4.3, it turns out that (6.3) can be solved inDγ as soon asγ is a little bit greaterthan1. Therefore, we only need to keep track of its solutionΦ up to terms of homogeneity1. Byrepeatedly applying the identity (6.4), we see that the solution Φ is necessarily of the form

Φ = + ϕ1− − 3ϕ + 〈∇ϕ,X〉 , (7.7)

for some real-valued functionϕ and someR3-valued function∇ϕ. (Note that∇ϕ is treated as anindependent function here, we certainly do not suggest thatthe functionϕ is differentiable! Ournotation is only by analogy with the classical Taylor expansion...) Similarly, the right hand sideof the equation is given up to order0 by

Ξ− Φ3 = Ξ− − 3ϕ + 3 − 3ϕ2 + 6ϕ + 9ϕ − 3〈∇ϕ, X〉 − ϕ3 1 . (7.8)

Combining this with the definition ofM , it is straightforward to see that, modulo terms of strictlypositive homogeneity, one has

M (Ξ− Φ3) = Ξ− (MΦ)3 + 3C1 + 3C1ϕ1− 9C2 − 9C2ϕ1

= Ξ− (MΦ)3 + (3C1 − 9C2)MΦ .

Combining this with (7.6), the claim now follows at once.

7.3 Convergence of the renormalised models

It remains to argue why one expects to be able to find constantsCε1 andCε

2 such that the sequenceof renormalised modelsM ειξε converges to a limiting model. Instead of considering the actualsequence of models, we only consider the sequence of stationary processesΠ

ετ := Π

εM ετ ,whereΠε is associated to (Πε,Γε) = ιξε as before. Since there are general arguments available todeal with all the expressionsτ of positive homogeneity, we restrict ourselves to those of negativehomogeneity which, leaving outΞ which is easy to treat, are given by

, , , , , , Xi .


For this section, some elementary notions from the theory ofWiener chaos expansions arerequired, but we will try to hide this as much as possible. At aformal level, one has the identity

Πε = K ∗ ξε = Kε ∗ ξ ,

where the kernelKε is given byKε = K ∗ δε. This shows that, at least formally, one has

(Πε )(z) = (K ∗ ξε)(z)2 =

∫ ∫

Kε(z − z1)Kε(z − z2) ξ(z1)ξ(z2) dz1 dz2 .

Similar but more complicated expressions can be found for any formal expressionτ . This natu-rally leads to the study of random variables of the type

Ik(f ) =∫

· · ·∫

f (z1, . . . , zk) ξ(z1) · · · ξ(zk) dz1 · · · dzk . (7.9)

Ideally, one would hope to have an Ito isometry of the typeEIk(f )Ik(g) = 〈f sym, gsym〉, where〈·, ·〉 denotes theL2-scalar product andf sym denotes the symmetrisation off . This is unfortu-natelynot the case. Instead, one should replace the products in (7.9) by Wick products, which areformally generated by all possiblecontractionsof the type

ξ(zi)ξ(zj) 7→ ξ(zi) ⋄ ξ(zj) + δ(zi − zj) .

If we then set

Ik(f ) =∫

· · ·∫

f (z1, . . . , zk) ξ(z1) ⋄ · · · ⋄ ξ(zk) dz1 · · · dzk ,

One has indeedEIk(f )Ik(g) = 〈f sym, gsym〉 .

See [Nua95] for a more thorough description of this construction, which also goes under thename ofWiener chaos. It turns out that one has equivalence of moments in the sensethat, foreveryk > 0 andp > 0 there exists a constantCk,p such that

E|Ik(f )|p ≤ Ck,p‖f sym‖p ≤ Ck,p‖f‖p ,

where the second bound comes from the fact that symmetrisation is a contraction inL2. Finally,one hasEIk(f )Iℓ(g) = 0 if k 6= ℓ. Random variables of the formIk(f ) for somek ≥ 0 and somesquare integrable functionf are said to belong to thekth homogeneous Wiener chaos.

Returning to our problem, we first argue that it should be possible to chooseM in such a waythatΠ

εconverges to a limit asε→ 0. The above considerations suggest that one should rewrite

Πε as

(Πε )(z) = (K ∗ ξε)(z)2 =

∫ ∫

Kε(z − z1)Kε(z − z2) ξ(z1) ⋄ ξ(z2) dz1 dz2 + Cε , (7.10)

where the constantCε is given by

Cε =

∫

K2ε (z1) dz1 =

∫

K2ε (z − z1) dz1 .


Note now thatKε is anε-approximation of the kernelK which has the same singular behaviouras the heat kernel. In terms of the parabolic distance, the singularity of the heat kernel scales likeK(z) ∼ |z|−3 for z → 0. (Recall that we consider the parabolic distance|(t, x)| =

√

|t|+ |x|, sothat this is consistent with the fact that the heat kernel is bounded byt−3/2.) This suggests thatone hasK2

ε (z) ∼ |z|−6 for |z| ≫ ε. Since parabolic space-time has scaling dimension5 (timecounts double!), this is a non-integrable singularity. As amatter of fact, there is a whole power ofz missing to make it borderline integrable, which suggests that one has

Cε ∼1

ε.

This already shows that one should not expectΠε to converge to a limit asε → 0. However,

it turns out that the first term in (7.10) converges to a distribution-valued stationary space-timeprocess, so that one would like to somehow get rid of this diverging constantCε. This is ex-actly where the renormalisation mapM (in particular the factor exp(−C1L1)) enters into play.Following the above definitions, we see that one has

(Πε)(z) = (ΠεM )(z) = (Πε )(z) − C1 .

This suggests that if we make the choiceC1 = Cε, thenΠε

does indeed converge to a non-triviallimit as ε→ 0. This limit is a distribution given by

(Πε )(ψ) =∫ ∫

ψ(z)K(z − z1)K(z − z2) dz ξ(z1) ⋄ ξ(z2) dz1 dz2 .

Using again the scaling properties of the kernelK, it is not too difficult to show that this yieldsindeed a random variable belonging to the second homogeneous Wiener chaos for every choice ofsmooth test functionψ. Once we know thatΠ

εconverges, it is immediate thatΠ

εX converges

as well, since this amounts to just multiplying a distribution by a smooth function.A similar argument to what we did for allows to take care ofτ = since one then has

(Πε )(z) =∫ ∫

Kε(z − z1)Kε(z − z2)Kε(z − z3) ξ(z1) ⋄ ξ(z2) ⋄ ξ(z3) dz1 dz2 dz3

+ 3

∫ ∫

Kε(z − z1)Kε(z − z2)Kε(z − z3)δ(z1 − z2) ξ(z3) dz1 dz2 dz3 .

Noting that the second term in this expression is nothing but

3Cε

∫

Kε(z − z1) ξ(z1) dz1 = 3Cε(Πε )(z) ,

we see that in this case, provided again thatC1 = Cε, Πε

is given by only the first term in theexpression above, which turns out to converge to a non-degenerate limiting random distributionin a similar way to what happened for.

Turning to our list of terms of negative homogeneity, it remains to consider , , and .It turns out that the latter two are the more difficult ones, sowe only discuss these. Let us first


argue why we expect to be able to choose the constantsC1 andC2 in such a way thatΠε

converges to a limit. In this case, the “bad” terms comes fromthe part of(Πε )(z) belonging tothe homogeneous chaos of order0. This is simply a constant, which turns out to be given by

Cε = 2

∫

K(z)Q2ε(z) dz , (7.11)

where the kernelQε is given by

Qε(z) =∫

Kε(z)Kε(z − z) dz .

SinceKε is anε-mollification of a kernel with a singularity of order−3 and the scaling dimensionof the underlying space is5, we see thatQε behaves like anε-mollification of a kernel with asingularity of order−3 − 3 + 5 = −1 at the origin. As a consequence, the singularity of theintegrand in (7.11) is of order−5, which gives rise to a logarithmic divergence asε → 0. Thissuggests that one should chooseC2 = Cε in order to cancel out this diverging term and obtain anon-trivial limit for Π

εasε→ 0. This is indeed the case.

We finally turn to the symbol . In this case, the “bad” terms appearing in the Wiener chaosdecomposition ofΠε are the terms in the first homogeneous Wiener chaos, which areof theform

3

∫

Qε(z − z1)Kε(z1 − z2)ξ(z2) dz1 dz2 = 3

∫

(Qε ∗Kε)(z − z2)ξ(z2) dz2 , (7.12)

whereQε is the kernel given byQε(z) = 2K(z)Q2

ε(z) .

As already mentioned above, the problem here is that asε → 0, Qε converges to a kernelQ =2KQ2, which has a non-integrable singularity at the origin. In particular, the action of integratinga test function againstQε does not converge to a limiting distribution asε→ 0.

This is akin to the problem of making sense of integration against a one-dimensional kernelwith a singularity of type1/|x| at the origin. For the sake of the argument, let us consider afunctionW : R → R which is compactly supported and smooth everywhere except at the origin,where it diverges likeW (x) ∼ 1/|x|. It is then natural to associate toW a “renormalised”distributionRW given by

(RW )(ϕ) =∫

W (x)(ϕ(x) − ϕ(0)) dx .

Note thatRW has the property that ifϕ(0) = 0, then it simply corresponds to integration againstW , which is the standard way of associating a distribution to afunction. In a way, the extra termcan be interpreted as subtracting a Dirac distribution withan “infinite mass” located at the origin,thus cancelling out the divergence of the non-integrable singularity. It is also straightforward toverify that ifWε is a sequence of smooth approximations toW (say one hasWε(x) = W (x) for


|x| > ε andWε ∼ 1/ε otherwise), thenRW ε → RW in a distributional sense, and (using theusual correspondence between functions and distributions) one has

RW ε =W ε − Cεδ0 , Cε =

∫

W ε(x) dx .

The cure to the problem we are facing for showing the convergence ofΠε is virtually identical.Indeed,by choosingC2 = Cε as in (7.11), the term in the first homogeneous Wiener chaos forΠ

εcorresponding to (7.12) is precisely given by

3

∫

Qε(z − z1)Kε(z1 − z2)ξ(z2) dz1 dz2 − 3C2

∫

Kε(z − z2)ξ(z2) dz2

= 3

∫

(RQε ∗Kε)(z − z2)ξ(z2) dz2 .

It turns out that the convergence ofRQε to a limiting distributionRQ takes place in a sufficientlystrong topology to allow to conclude thatΠ

εdoes indeed converge to a non-trivial limiting

random distribution.It should be clear from this whole discussion that while the precise values of the constantsC1

andC2 depend on the details of the mollifierδε, the limiting (random) model (Π, Γ) obtained inthis way is independent of it. Combining this with the continuity of the solution to the fixed pointmap (6.3) and of the reconstruction operatorR with respect to the underlying model, we see thatthe statement of Theorem 6.1 follows almost immediately.

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