Algebraic renormalisation of regularity structures

We give a systematic description of a canonical renormalisation procedure of stochastic PDEs containing nonlinearities involving generalised functions. This theory is based on the construction of a new class of regularity structures which comes with an explicit and elegant description of a subgroup of their group of automorphisms. This subgroup is sufficiently large to be able to implement a version of the BPHZ renormalisation prescription in this context. This is in stark contrast to previous works where one considered regularity structures with a much smaller group of automorphisms, which lead to a much more indirect and convoluted construction of a renormalisation group acting on the corresponding space of admissible models by continuous transformations. Our construction is based on bialgebras of decorated coloured forests in cointeraction. More precisely, we have two Hopf algebras in cointeraction, coacting jointly on a vector space which represents the generalised functions of the theory. Two twisted antipodes play a fundamental role in the construction and provide a variant of the algebraic Birkhoff factorisation that arises naturally in perturbative quantum field theory.


Introduction
In a series of celebrated papers [10][11][12][13] Kuo-Tsai Chen discovered that, for any finite alphabet A, the family of iterated integrals of a smooth path x : R + → R A has a number of interesting algebraic properties. Writing T = T (R A ) for the tensor algebra on R A , which we identify with the space spanned by all finite words {(a 1 . . . a n )} n≥0 with letters in A, we define the family of functionals X s,t on T inductively by X s,t () def = 1, X s,t (a 1 . . . a n ) def = t s X s,u (a 1 . . . a n−1 )ẋ a n (u) du where 0 ≤ s ≤ t. Chen showed that this family yields for fixed s, t a character on T endowed with the shuffle product , namely which furthermore satisfies the flow relation where : T → T⊗ T is the deconcatenation coproduct (a 1 . . . a n ) = n k=0 (a 1 . . . a k ) ⊗ (a k+1 . . . a n ) .
In other words, we have a function (s, t) → X s,t ∈ T * which takes values in the characters on the algebra (T, ) and satisfies the Chen relation X s,r X r,t = X s,t , s ≤ r ≤ t, (1.2) where is the product dual to . Note that T, endowed with the shuffle product and the deconcatenation coproduct, is a Hopf algebra. These two remarkable properties do not depend explicitly on the differentiability of the path (x t ) t≥0 . They can therefore serve as an important tool if one wants to consider non-smooth paths and still build a consistent calculus. This intuition was at the heart of Terry Lyons' definition [46] of a geometric rough path as a function (s, t) → X s,t ∈ T * satisfying the two algebraic properties above and with a controlled modulus of continuity, for instance of Hölder type |X s,t (a 1 . . . a n )| ≤ C|t − s| nγ , (1.3) with some fixed γ > 0 (although the original definition involved rather a p-variation norm, which is natural in this context since it is invariant under reparametrisation of the path x, just like the definition of X). Lyons realised that this setting would allow to build a robust theory of integration and of associated differential equations. For instance, in the case of stochastic differential equations of Stratonovich type with W : R + → R d a d-dimensional Brownian motion and σ : R d → R d ⊗ R d smooth, one can build rough paths X and W over X , respectively W , such that the map W → X is continuous, while in general the map W → X is simply measurable. The Itô stochastic integration was included in Lyons' theory although it can not be described in terms of geometric rough paths. A few years later Gubinelli [29] introduced the concept of a branched rough path as a function (s, t) → X s,t ∈ H * taking values in the characters of an algebra (H, ·) of rooted forests, satisfying the analogue of the Chen relation (1.2) with respect to the Grossman-Larsson -product, dual of the Connes-Kreimer coproduct, and with a regularity condition |X s,t (τ )| ≤ C|t − s| |τ |γ (1.4) where |τ | counts the number of nodes in the forest τ and γ > 0 is fixed. Again, this framework allows for a robust theory of integration and differential equations driven by branched rough paths. Moreover H, endowed with the forest product and Connes-Kreimer coproduct, turns out to be a Hopf algebra. The theory of regularity structures [32], due to the second named author of this paper, arose from the desire to apply the above ideas to (stochastic) partial differential equations (SPDEs) involving non-linearities of (random) spacetime distributions. Prominent examples are the KPZ equation [23,27,31], the 4 stochastic quantization equation [1,7,21,32,43,45], the continuous parabolic Anderson model [26,36,37], and the stochastic Navier-Stokes equations [20,53].
One apparent obstacle to the application of the rough paths framework to such SPDEs is that one would like to allow for the analogue of the map s → X s,t τ to be a space-time distribution for some τ ∈ H. However, the algebraic relations discussed above involve products of such quantities, which are in general ill-defined. One of the main ideas of [32] was to replace the Hopf-algebra structure with a comodule structure: instead of a single space H, we have two spaces (T, T + ) and a coaction + : T → T⊗ T + such that T is a right comodule over the Hopf algebra T + . In this way, elements in the dual space T * of T are used to encode the distributional objects which are needed in the theory, while elements of T * + encode continuous functions. Note that T admits neither a product nor a coproduct in general.
However, the comodule structure allows to define the analogue of a rough path as a pair: consider a distribution-valued continuous function as well as a continuous function The analogue of the Chen relation (1.2) is then given by γ xy γ yz = γ xz , y γ yz = z , (1.5) where the first -product is the convolution product on T * + , while the secondproduct is given by the dual of the coaction + . This structure guarantees that all relevant expressions will be linear in the y , so we never need to multiply distributions. To compare this expression to (1.2), think of ( y τ )(·) ∈ D (R d ) for τ ∈ T as being the analogue of z → X z,y (τ ). Note that the algebraic conditions (1.5) are not enough to provide a useful object: analytic conditions analogous to (1.4) play an essential role in the analytical aspects of the theory. Once a model X = ( , γ ) has been constructed, it plays a role analogous to that of a rough path and allows to construct a robust solution theory for a class of rough (partial) differential equations.
In various specific situations, the theory yields a canonical lift of any smoothened realisation of the driving noise for the stochastic PDE under consideration to a model X ε . Another major difference with what one sees in the rough paths setting is the following phenomenon: if we remove the regularisation as ε → 0, neither the canonical model X ε nor the solution to the regularised equation converge in general to a limit. This is a structural problem which reflects again the fact that some products are intrinsically ill-defined. This is where renormalisation enters the game. It was already recognised in [32] that one should find a group R of transformations on the space of models and elements M ε in R in such a way that, when applying M ε to the canonical lift X ε , the resulting sequence of models converges to a limit. Then the theory essentially provides a black box, allowing to build maximal solutions for the stochastic PDE in question.
One aspect of the theory developed in [32] that is far from satisfactory is that while one has in principle a characterisation of R, this characterisation is very indirect. The methodology pursued so far has been to first make an educated guess for a sufficiently large family of renormalisation maps, then verify by hand that these do indeed belong to R and finally show, again by hand, that the renormalised models converge to a limit. Since these steps did not rely on any general theory, they had to be performed separately for each new class of stochastic PDEs.
The main aim of the present article is to define an algebraic framework allowing to build regularity structures which, on the one hand, extend the ones built in [32] and, on the other hand, admit sufficiently many automorphisms (in the sense of [32,Def. 2.28]) to cover the renormalisation procedures of all subcritical stochastic PDEs that have been studied to date.
Moreover our construction is not restricted to the Gaussian setting and applies to any choice of the driving noise with minimal integrability conditions. In particular this allows to recover all the renormalisation procedures used so far in applications of the theory [32,[38][39][40]42,51]. It reaches however far beyond this and shows that the BPHZ renormalisation procedure belongs to the renormalisation group of the regularity structure associated to any class of subcritical semilinear stochastic PDEs. In particular, this is the case for the generalised KPZ equation which is the most natural stochastic evolution on loop space and is (formally!) given in local coordinates by where the ξ i are independent space-time white noises, α βγ are the Christoffel symbols of the underlying manifold, and the σ i are a collection of vector fields with the property that i L 2 σ i = , where L σ is the Lie derivative in the direction of σ and is the Laplace-Beltrami operator. Another example is given by the stochastic sine-Gordon equation [41] close to the Kosterlitz-Thouless transition. In both of these examples, the relevant group describing the renormalisation procedures is of very large dimension (about 100 in the first example and arbitrarily large in the second one), so that the verification "by hand" that it does indeed belong to the "renormalisation group" as done for example in [32,39], would be impractical.
In order to describe the renormalisation procedure of SPDEs we introduce a new construction of an associated regularity structure, that will be called extended since it contains a new parameter which was not present in [32], the extended decoration. As above, this yields spaces (T ex , T ex + ), such that T ex + is a Hopf algebra and T ex a right comodule over T ex + . The renormalisation procedure of distributions coded by T ex is then described by another Hopf algebra T ex − and coactions − ex : T ex → T ex − ⊗ T ex and − ex : T ex + → T ex − ⊗ T ex + turning both T ex and T ex + into left comodules over T ex − . This construction is, crucially, compatible with the comodule structure of T ex over T ex + in the sense that − ex and + ex are in cointeraction in the terminology of [25], see formulae (3.48)-(5.26) and Remark 3.28 below. Once this structure is obtained, we can define renormalised models as follows: given a functional g : T ex − → R and a model X = ( , γ ), we construct a new model X g by setting The cointeraction property then guarantees that X g satisfies again the generalised Chen relation (1.5). Furthermore, the action of T ex − on T ex and T ex + is such that, crucially, the associated analytical conditions automatically hold as well.
All the coproducts and coactions mentioned above are a priori different operators, but we describe them in a unified framework as special cases of a contraction/extraction operation of subforests, as arising in the BPHZ renormalisation procedure/forest formula [3,24,35,52]. It is interesting to remark that the structure described in this article is an extension of that previously described in [8,14,15] in the context of the analysis of B-series for numerical ODE solvers, which is itself an extension of the Connes-Kreimer Hopf algebra of rooted trees [16,18] arising in the abovementioned forest formula in perturbative QFT. It is also closely related to incidence Hopf algebras associated to families of posets [49,50].
There are however a number of substantial differences with respect to the existing literature. First we propose a new approach based on coloured forests; for instance we shall consider operations like −→ ⊗ −→ ⊗ of colouring, extraction and contraction of subforests. Further, the abovementioned articles deal with two spaces in cointeraction, analogous to our Hopf algebras T ex − and T ex + , while our third space T ex is the crucial ingredient which allows for distributions in the analytical part of the theory. Indeed, one of the main novelties of regularity structures is that they allow to study random distributional objects in a pathwise sense rather than through Feynman path integrals/correlation functions and the space T ex encodes the fundamental bricks of this construction. Another important difference is that the structure described here does not consist of simple trees/forests, but they are decorated with multiindices on both their edges and their vertices. These decorations are not inert but transform in a non-trivial way under our coproducts, interacting with other operations like the contraction of sub-forests and the computation of suitable gradings.
In this article, Taylor sums play a very important role, just as in the BPHZ renormalisation procedure, and they appear in the coactions of both T ex − (the renormalisation) and T ex + (the recentering). In both operations, the group elements used to perform such operations are constructed with the help of a twisted antipode, providing a variant of the algebraic Birkhoff factorisation that was previously shown to arise naturally in the context of perturbative quantum field theory, see for example [16,18,19,22,30,44].
In general, the context for a twisted antipode/Birkhoff factorisation is that of a group G acting on some vector space A which comes with a valuation. Given an element of A, one then wants to renormalise it by acting on it with a suitable element of G in such a way that its valuation vanishes. In the context of dimensional regularisation, elements of A assign to each Feynman diagram a Laurent series in a regularisation parameter ε, and the valuation extracts the pole part of this series. In our case, the space A consists of stationary random linear maps : T ex → C ∞ and we have two actions on it, by the group of characters G ex ± of T ex ± , corresponding to two different valuations. The renormalisation group G ex − is associated to the valuation that extracts the value of E( τ )(0) for every homogeneous element τ ∈ T ex of negative degree. The structure group G ex + on the other hand is associated to the valuations that extract the values ( τ )(x) for all homogeneous elements τ ∈ T ex of positive degree.
We show in particular that the twisted antipode related to the action of G ex + is intimately related to the algebraic properties of Taylor remainders. Also in this respect, regularity structures provide a far-reaching generalisation of rough paths, expanding Massimiliano Gubinelli's investigation of the algebraic and analytic properties of increments of functions of a real variable achieved in the theory of controlled rough paths [28].

A general renormalisation scheme for SPDEs
Regularity Structures (RS) have been introduced [32] in order to solve singular SPDEs of the form where u = u(t, x) with t ≥ 0 and x ∈ R d , ξ is a random space-time Schwartz distribution (typically stationary and approximately scaling-invariant at small scales) driving the equation and the non-linear term F(u, ∇u, ξ) contains some products of distributions which are not well-defined by classical analytic methods. We write this equation in the customary mild formulation where G is the heat kernel and we suppose for simplicity that u(0, ·) = 0. If we regularise the noise ξ by means of a family of smooth mollifiers ( ε ) ε>0 , setting ξ ε := ε * ξ , then the regularised PDE u ε = G * (F(u ε , ∇u ε , ξ ε )) is well-posed under suitable assumptions on F. However, if we want to remove the regularisation by letting ε → 0, we do not know whether u ε converges. The problem is that ξ ε → ξ in a space of distributions with negative (say) Sobolev regularity, and in such spaces the solution map ξ ε → u ε is not continuous.
The theory of RS allows to solve this problem for a class of equations, called subcritical. The general approach is as in Rough Paths (RP): the discontinuous solution map is factorised as the composition of two maps: where (M , d) is a metric space that we call the space of models. The main point is that the map : M → D (R d ) can be chosen in such a way that its is continuous, even though M is sufficiently large to allow for elements exhibiting a local scaling behaviour compatible with that of ξ . Of course this means that ξ ε → X ε is discontinuous in general. In RP, the analogue of the model X ε is the lift of the driving noise as a rough path, the map is called the Itô-Lyons map, and its continuity (due to T. Lyons [46]) is the cornerstone of the theory. The construction of : M → D (R d ) in the general context of subcritical SPDEs is one of the main results of [32].
The construction of , although a very powerful tool, does not solve alone the aforementioned problem, since it turns out that the most natural choice of X ε , which we call the canonical model, does in general not converge as we remove the regularisation by letting ε → 0. It is necessary to modify, namely renormalise, the model X ε in order to obtain a familyX ε which does converge in M as ε → 0 to a limiting modelX. The continuity of then implies thatû ε := (X ε ) converges to some limitû := (X), which we call the renormalised solution to our equation, see Fig. 1. A very important fact is thatû ε is itself the solution of a renormalised equation, which differs from the original equation only by the presence of additional local counterterms, the form of which can be derived explicitly from the starting SPDE, see [2].
The transformation X ε →X ε is described by the so-called renormalisation group. The main aim of this paper is to provide a general construction of the space of models M together with a group of automorphisms G − S : M → M which allows to describe the renormalised modelX ε = S ε X ε for an appropriate choice of S ε ∈ G − .
Starting with the ϕ 4 We also see that in the space of models M we have several possible lifts of ξ ε ∈ S (R d ), e.g. the canonical model X ε and the renormalised modelX ε ; it is the latter that converges to a modelX, thus providing a lift of ξ . Note thatû ε = (X ε ) andû = (X) renormalised model and its convergence as the regularisation is removed are based on ad hoc arguments which have to be adapted to each equation. The present article, together with the companion "analytical" article [9] and the work [2], complete the general theory initiated in [32] by proving that virtually every 1 subcritical equation driven by a stationary noise satisfying some natural bounds on its cumulants can be successfully renormalised by means of the following scheme: • Algebraic step: Construction of the space of models (M , d) and renormalisation of the canonical model M X ε →X ε ∈ M , this article. • Analytic step: Continuity of the solution map : M → D (R d ), [32]. • Probabilistic step: Convergence in probability of the renormalised model X ε toX in (M , d), [9]. • Second algebraic step: Identification of (X ε ) with the classical solution map for an equation with local counterterms, [2].
We stress that this procedure works for very general noises, far beyond the Gaussian case. 1 There are some exceptions that can arise when one of the driving noises is less regular than white noise. For example, a canonical solution theory for SDEs driven by fractional Brownian motion can only be given for H > 1 4 , even though these equations are subcritical for every H > 0. See in particular the assumptions of [9, Thm 2.14].

Overview of results
We now describe in more detail the main results of this paper. Let us start from the notion of a subcritical rule. A rule, introduced in Definition 5.7 below, is a formalisation of the notion of a "class of systems of stochastic PDEs". More precisely, given any system of equations of the type (1.7), there is a natural way of assigning to it a rule (see Sect. 5.4 for an example), which keeps track of which monomials (of the solution, its derivatives, and the driving noise) appear on the right hand side for each component. The notion of a subcritical rule, see Definition 5.14, translates to this general context the notion of subcriticality of equations which was given more informally in [32,Assumption 8.3].
Suppose now that we have fixed a subcritical rule. The first aim is to construct an associated space of models M ex . The superscript 'ex' stands for extended and is used to distinguish this space from the restricted space of models M , see Definition 6.24, which is closer to the original construction of [32]. The space M ex extends M in the sense that there is a canonical continuous injection M → M ex , see Theorem 6.33. The reason for considering this larger space is that it admits a large group G ex − of automorphisms in the sense of [32, Def. 2.28] which can be described in an explicit way. Our renormalisation procedure then makes use of a suitable subgroup G − ⊂ G ex − which leaves M invariant. The reason why we do not describe its action on M directly is that although it acts by continuous transformations, it no longer acts by automorphisms, making it much more difficult to describe without going through M ex .
To define M ex , we construct a regularity structure (T ex , G ex + ) in the sense of [32,Def. 2.1]. This is done in Sect. 5, see in particular Definitions 5. 26-5.35 and Proposition 5.39. The corresponding structure group G ex + is constructed as the character group of a Hopf algebra T ex + , see (5.23), Proposition 5.34 and Definition 5.36. The vector space T ex is a right-comodule over T ex + , namely there are linear operators such that the identity holds both between operators on T ex and on T ex + . The fact that the two operators have the same name but act on different spaces should not generate confusion since the domain is usually clear from context. When it isn't, as in (1.8), then the identity is assumed by convention to hold for all possible meaningful interpretations.
Next, the renormalisation group G ex − is defined as the character group of the Hopf algebra T ex − , see ( The action of G ex − on the corresponding dual spaces is given by Crucially, these separate actions satisfy a compatibility condition which can be expressed as a cointeraction property, see (5.26) in Theorem 5.37, which implies the following relation between the two actions above: Proposition 3.33 and (5.27). This result is the algebraic linchpin of Theorem 6.16, where we construct the action of G ex − on the space M ex of models. The next step is the construction of the space of smooth models of the regularity structure (T ex , G ex + ). This is done in Definition 6.7, where we follow [32,Def. 2.17], with the additional constraint that we consider smooth objects. Indeed, we are interested in the canonical model associated to a (regularised) smooth noise, constructed in Proposition 6.12 and Remark 6.13, and in its renormalised versions, namely its orbit under the action of G ex − , see Theorem 6.16.
Finally, we restrict our attention to a class of models which are random, stationary and have suitable integrability properties, see Definition 6.17. In this case, we can define a particular deterministic element of G ex − that gives rise to what we call the BPHZ renormalisation, by analogy with the corresponding construction arising in perturbative QFT [3,24,35,52], see Theorem 6.18. We show that the BPHZ construction yields the unique element of G ex − such that the associated renormalised model yields a centered family of stochastic processes on the finite family of elements in T ex with negative degree. This is the algebraic step of the renormalisation procedure. This is the point where the companion analytical paper [9] starts, and then goes on to prove that the BPHZ renormalised model does converge in the metric d on M , thus achieving the probabilistic step mentioned above and thereby completing the renormalisation procedure.
The BPHZ functional is expressed explicitly in terms of an interesting map that we call negative twisted antipode by analogy to [17], see Proposition 6.6 and (6.25). There is also a positive twisted antipode, see Proposition 6.3, which plays a similarly important role in (6.12). The main point is that these twisted antipodes encode in the compact formulae (6.12) and (6.25) a number of nontrivial computations.
How are these spaces and operators defined? Since the analytic theory of [32] is based on generalised Taylor expansions of solutions, the vector space T ex is generated by a basis which codes the relevant generalised Taylor monomials, which are defined iteratively once a rule (i.e. a system of equations) is fixed. Definitions 5.8, 5.13 and 5.26 ensure that T ex is sufficiently rich to allow one to rewrite (1.7) as a fixed point problem in a space of functions with values in our regularity structure. Moreover T ex must also be invariant under the actions of G ex ± . This is the aim of the construction in Sects. 2, 3 and 4, that we want now to describe.
The spaces which are constructed in Sect. 5 depend on the choice of a number of parameters, like the dimension of the coordinate space, the leading differential operator in the equation (the Laplacian being just one of many possible choices), the non-linearity, the noise. In the previous sections we have built universal objects with nice algebraic properties which depend on none of these choices, but for the dimension of the space, namely an (arbitrary) integer number d fixed once for all.
The spaces T ex , T ex + and T ex − are obtained by considering repeatedly suitable subsets and suitable quotients of two initial spaces, called F 1 and F 2 and defined in and after Definition 4.1; more precisely, F 1 is the ancestor of T ex and T ex − , while F 2 is the ancestor of T ex + . In Sect. 4 we represent these spaces as linearly generated by a collection of decorated forests, on which we can define suitable algebraic operations like a product and a coproduct, which are later inherited by T ex , T ex + and T ex − (through other intermediary spaces which are called H • , H 1 andĤ 2 ). An important difference between T ex − and T ex + is that the former is linearly generated by a family of forests, while the latter is linearly generated by a family of trees; this difference extends to the algebra structure: T ex − is endowed with a forest product which corresponds to the disjoint union, while T ex + is endowed with a tree product whereby one considers a disjoint union and then identifies the roots.
The content of Sect. 4 is based on a specific definition of the spaces F 1 and F 2 . In Sects. 2 and 3 however we present a number of results on a family of spaces (F i ) i∈I with I ⊂ N, which are supposed to satisfy a few assumptions; Sect. 4 is therefore only a particular example of a more general theory, which is outlined in Sects. 2 and 3. In this general setting we consider spaces F i of decorated forests, and vector spaces F i of infinite series of such forests. Such series are not arbitrary but adapted to a grading, see Sect. 2.3; this is needed since our abstract coproducts of Definition 3.3 contain infinite series and might be ill-defined if were to work on arbitrary formal series.
The family of spaces (F i ) i∈I are introduced in Definition 3.12 on the basis of families of admissible forests A i , i ∈ I . If (A i ) i∈I satisfy Assumptions 1, 2, 3, 4, 5 and 6, then the coproducts i of Definition 3.3 are coassociative and moreover i and j for i < j are in cointeraction, see (3.27). As already mentioned, the cointeraction property is the algebraic formula behind the fundamental relation (1.9) between the actions of G ex + and G ex − on T ex + . "Appendix A" contains a summary of the relations between the most important spaces appearing in this article, while "Appendix B" contains a symbolic index.

Rooted forests and bigraded spaces
Given a finite set S and a map : S → N, we write and we define the corresponding binomial coefficients accordingly. Note that if 1 and 2 have disjoint supports, then ( 1 + 2 )! = 1 ! 2 !. Given a map π : S →S, we also define π :S → N by π (x) = y∈π −1 (x) (y).
For k, : S → N we define with the convention k = 0 unless 0 ≤ ≤ k, which will be used throughout the paper. With these definitions at hand, one has the following slight reformulation of the classical Chu-Vandermonde identity.

Rooted trees and forests
Recall that a rooted tree T is a finite tree (a finite connected simple graph without cycles) with a distinguished vertex, = T , called the root. Vertices of T , also called nodes, are denoted by N = N T and edges by E = E T ⊂ N 2 .
Since we want our trees to be rooted, they need to have at least one node, so that we do not allow for trees with N T = . We do however allow for the trivial tree consisting of an empty edge set and a vertex set with only one element. This tree will play a special role in the sequel and will be denoted by •. We will always assume that our trees are combinatorial meaning that there is no particular order imposed on edges leaving any given vertex. Given a rooted tree T , we also endow N T with the partial order ≤ where w ≤ v if and only if w is on the unique path connecting v to the root, and we orient edges in E T so that if (x, y) = (x → y) ∈ E T , then x ≤ y. In this way, we can always view a tree as a directed graph.
Two rooted trees T and T are isomorphic if there exists a bijection ι : E T → E T which is coherent in the sense that there exists a bijection ι N : N T → N T such that ι(x, y) = (ι N (x), ι N (y)) for any edge (x, y) ∈ e and such that the roots are mapped onto each other.
We say that a rooted tree is typed if it is furthermore endowed with a function t : E T → L, where L is some finite set of types. We think of L as being fixed once and for all and will sometimes omit to mention it in the sequel. In particular, we will never make explicit the dependence on the choice of L in our notations. Two typed trees (T, t) and (T , t ) are isomorphic if T and T are isomorphic and t is pushed onto t by the corresponding isomorphism ι in the sense that t • ι = t.
Similarly to a tree, a forest F is a finite simple graph (again with nodes N F and edges E F ⊂ N 2 F ) without cycles. A forest F is rooted if every connected component T of F is a rooted tree with root T . As above, we will consider forests that are typed in the sense that they are endowed with a map t : E F → L, and we consider the same notion of isomorphism between typed forests as for typed trees. Note that while a tree is non-empty by definition, a forest can be empty. We denote the empty forest by either 1 or .
Given a typed forest F, a subforest A ⊂ F consists of subsets E A ⊂ E F and tree whose root is defined to be the minimal node in the partial order inherited from F. We say that subforests A and B are disjoint, and write A ∩ B = , if one has N A ∩ N B = (which also implies that E A ∩ E B = ). Given two typed forests F, G, we write F G for the typed forest obtained by taking the disjoint union (as graphs) of the two forests F and G and adjoining to it the natural typing inherited from F and G. If furthermore A ⊂ F and B ⊂ G are subforests, then we write A B for the corresponding subforest of F G.
We fix once and for all an integer d ≥ 1, dimension of the parameter-space R d . We also denote by Z(L) the free abelian group generated by L.

Coloured and decorated forests
Given a typed forest F, we want now to consider families of disjoint subforests of F, denoted by (F i , i > 0). It is convenient for us to code this family with a single functionF : E F N F → N as given by the next definition.
We say thatF is a colouring of F. For i > 0, we define the subforest of F as well asÊ = i>0Ê i . We denote by C the set of coloured forests.
The condition onF guarantees that everyF i is indeed a subforest of F for i > 0 and that they are all disjoint. On the other hand,F −1 (0) is not supposed to have any particular structure and 0 is not counted as a colour.
Example 2.4 This is an example of a forest with two colours: red for 1 and blue for 2 (and black for 0) We then haveF 1 The set C is a commutative monoid under the forest product where colouringss defined on one of the forests are extended to the disjoint union by setting them to vanish on the other forest. The neutral element for this associative product is the empty coloured forest 1.
We add now decorations on the nodes and edges of a coloured forest. For this, we fix throughout this article an arbitrary "dimension" d ∈ N and we give the following definition.

Remark 2.6
The reason why o takes values in the space Z d ⊕Z(L) will become apparent in (3.33) below when we define the contraction of coloured subforests and its action on decorations.
We identify (F,F, n, o, e) and (F ,F , n , o , e ) whenever F is isomorphic to F , the corresponding isomorphism mapsF toF and pushes the three decoration functions onto their counterparts. We call elements of F decorated forests. We will also sometimes use the notation (F,F) n,o e instead of (F,F, n, o, e). , and on all remaining (black) nodes and edgesF is set equal to 0. Every edge has a type t ∈ L, but only black edges have a possibly non-zero decoration e ∈ N d . All nodes have a decoration n ∈ N d , but only coloured nodes have a possibly non-zero decoration o ∈ Z d ⊕ Z(L).
Example 2.7 is continued in Examples 3.2, 3.4 and 3.5.

Definition 2.8
For any coloured forest (F,F), we define an equivalence relation ∼ on the node set N F by saying that x ∼ y if x and y are connected inÊ; this is the smallest equivalence relation for which x ∼ y whenever (x, y) ∈Ê. Definition 2.8 will be extended to a decorated forest (F,F, n, o, e) in Definition 3.18 below. Remark 2. 9 We want to show the intuition behind decorated forests. We think of each τ = (F,F, n, o, e) as defining a function on (R d ) N F in the following way. We associate to each type t ∈ L a kernel ϕ t : R d → R and we define the domain where ∼ is the equivalence relation of Definition 2.8. Then we set H τ ∈ C ∞ (U F ), In this way, a decorated forest encodes a function: every node in N F / ∼ represents a variable in R d , every uncoloured edge of a certain type t a function ϕ t(e) of the difference of the two variables sitting at each one of its nodes; the decoration n(v) gives a power of x v and e(e) a derivative of the kernel ϕ t(e) . In this example the decoration o plays no role; we shall see below that it allows to encode some additional information relevant for the various algebraic manipulations we wish to subject these functions to, see Remarks 3.7, 3.19, 5.38 and 6.26 below for further discussions.
Remark 2.10 Every forest F = (N F , E F ) has a unique decomposition into non-empty connected components. This property naturally extends to decorated forests (F,F, n, o, e), by considering the connected components of the underlying forest F and restricting the colouringF and the decorations n, o, e.

Remark 2.11
Starting from Sect. 4 we are going to consider a specific situation where there are only two colours, namelyF → {0, 1, 2}; all examples throughout the paper are in this setting. However the results of Sects. 2 and 3 are stated and proved in the more general settingF → N without any additional difficulty.

Bigraded spaces and triangular maps
It will be convenient in the sequel to consider a particular category of bigraded spaces as follows.

Definition 2.12
For a collection of vector spaces {V n : n ∈ N 2 }, we define the vector space as the space of all formal sums n∈N 2 v n with v n ∈ V n and such that there exists k ∈ N such that v n = 0 as soon as n 2 > k. Given two bigraded spaces V and W , we write V⊗ W for the bigraded space One has a canonical inclusion V ⊗ W ⊂ V⊗ W given by However in general V⊗ W is strictly larger since its generic element has the form Note that all tensor products we consider are algebraic.

Definition 2.13
We introduce a partial order on N 2 by Given two such bigraded spaces V andV , a family {A mn } m,n∈N 2 of linear maps Proof Let v = n v n ∈ V and k ∈ N such that v n = 0 whenever n 2 > k. First we note that, for fixed m ∈ N 2 , the family (A mn v n ) n∈N 2 is zero unless n ∈ [0, m 1 ] × [0, k]; indeed if n 2 > k then v n = 0, while if n 1 > m 1 then A mn = 0. Therefore the sum n A mn v n is well defined and equal to somē v m ∈V m .
We now prove thatv m = 0 whenever m 2 > k, so that indeed mv m ∈ m∈N 2Vm . Let m 2 > k; for n 2 > k, v n is 0, while for n 2 ≤ k we have n 2 < m 2 and therefore A nm = 0 and this proves the claim.
A linear function A : V →V which can be obtained as in Lemma 2.14 is called triangular. The family (A mn ) m,n∈N 2 defines an infinite lower triangular matrix and composition of triangular maps is then simply given by formal matrix multiplication, which only ever involves finite sums thanks to the triangular structure of these matrices.

Remark 2.15
The notion of bigraded spaces as above is useful for at least two reasons: 1. The operators i built in (3.7) below turn out to be triangular in the sense of Definition 2.13 and are therefore well-defined thanks to Lemma 2.14, see Remark 2.15 below. This is not completely trivial since we are dealing with spaces of infinite formal series. 2. Some of our main tools below will be spaces of multiplicative functionals, see Sect. 3.6 below. Had we simply considered spaces of arbitrary infinite formal series, their dual would be too small to contain any non-trivial multiplicative functional at all. Considering instead spaces of finite series would cure this problem, but unfortunately the coproducts i do not make sense there. The notion of bigrading introduced here provides the best of both worlds by considering bi-indexed series that are infinite in the first index and finite in the second. This yields spaces that are sufficiently large to contain our coproducts and whose dual is still sufficiently large to contain enough multiplicative linear functionals for our purpose.

Remark 2.16
One important remark is that this construction behaves quite nicely under duality in the sense that if V and W are two bigraded spaces, then it is still the case that one has a canonical inclusion V * ⊗ W * ⊂ (V⊗ W ) * , see e.g. (3.46) below for the applications we have in mind. Indeed, the dual V * consists of formal sums n v * n with v * n ∈ V * n such that, for every k ∈ N there exists f (k) such that v * n = 0 for every n ∈ N 2 with n 1 ≥ f (n 2 ). The set F, see Definition 2.5, admits a number of different useful gradings and bigradings. One bigrading that is well adapted to the construction we give below is and |F \ (F ∪ F )| denotes the number of edges and vertices on whichF vanishes that aren't roots of F. For any subset A ⊆ F let now A denote the space built from A with this grading, namely where Vec S denotes the free vector space generated by a set S. Note that in general M is larger than Vec M. The following simple fact will be used several times in the sequel. Here and throughout this article, we use as usual the notation f A for the restriction of a map f to some subset A of its domain.

Lemma 2.17
Let V = n V n be a bigraded space and let P : V → V be a triangular map preserving the bigrading of V (in the sense that there exist linear maps P n : V n → V n such that P V n = P n for every n) and satisfying P • P = P. Then, the quotient spaceV = V / ker P is again bigraded and one has canonical identificationŝ V = n (V n / ker P n ) = n (P n V n ) .

Bialgebras, Hopf algebras and comodules of decorated forests
In this section we want to introduce a general class of operators on spaces of decorated forests and show that, under suitable assumptions, one can construct in this way bialgebras, Hopf algebras and comodules.
We recall that (H, M, • the coproduct and the counit are homomorphisms of algebras (or, equivalently, multiplication and unit are homomorphisms of coalgebras).
A Hopf algebra is a bialgebra (H, M, 1, , 1 ) endowed with a linear map A left comodule over a bialgebra (H, M, 1, , 1 ) is a pair (M, ψ) where M is a vector space and ψ : M → H ⊗ M is a linear map such that Right comodules are defined analogously. For more details on the theory of coalgebras, bialgebras, Hopf algebras and comodules we refer the reader to [6,47].

Incidence coalgebras of forests
Denote by P the set of all pairs (G; F) such that F is a typed forest and G is a subforest of F and by Vec(P) the free vector space generated by P. Suppose that for all (G; F) ∈ P we are given a (finite) collection A(G; F) of subforests A of F such that G ⊆ A ⊆ F. Then we define the linear map : Vec(P) → Vec(P) ⊗ Vec(P) by We also define the linear functional 1 : Vec(P) → R by 1 (G; F) := 1 (G=F) . If A(G; F) is equal to the set of all subforests A of F containing G, then it is a simple exercise to show that (Vec(P), , 1 ) is a coalgebra, namely (3.1) holds. In particular, since the inclusion G ⊆ F endows the set of typed forests with a partial order, (Vec(P), , 1 ) is an example of an incidence coalgebra, see [49,50]. However, if A(F; G) is a more general class of subforests, then coassociativity is not granted in general and holds only under certain assumptions. Suppose now that, given a typed forest F, we want to consider not one but several disjoint subforests G 1 , . . . , G n of F. A natural way to code (G 1 , . . . , G n ; F) is to use a coloured forest (F,F) wherê Then, in the notation of Definition 2.3, we haveF i = G i for i > 0 and In order to define a generalisation of the operator of formula (3.3) to this setting, we fix i > 0 and assume the following. Assumption 1 Let i > 0. For each coloured forest (F,F) as in Definition 2.3 we are given a collection A i (F,F) of subforests of F such that for every for all 0 < j < i and every connected component T ofF j , one has either We also assume that A i is compatible with the equivalence relation ∼ given by forest isomorphisms described above in the sense that if A ∈ A i (F,F) and It is important to note that colours are denoted by positive integer numbers and are therefore ordered, so that the forestsF j ,F i andF k can play different roles in Assumption 1 if j < i < k. This becomes crucial in our construction below, see Proposition 3.27 and Remark 3.29.

Lemma 3.1 Let (F,F) ∈ C be a coloured forest and A ∈ A i (F,F). Write •F A for the restriction ofF to N A E A •F ∪ i A for the function on E F N F given by
Then, under Assumption 1, (A,F A) and (F,F ∪ i A) are coloured forests.
Proof The claim is elementary for (A,F A); in particular, settingĜ def =F A, we haveĜ j =F j ∩ A for all j > 0. We prove it now for (F,F ∪ i A). We must prove that, settingĜ def =F ∪ i A, the setsĜ j def =Ĝ −1 ( j) define subforests of F for all j > 0. We have by the definitionŝ and these are subforests of F by the properties 1 and 2 of Assumption 1.
We denote by Vec(C) the free vector space generated by all coloured forests. This allows to define the following operator for fixed i > 0, i : Vec(C) → Vec(C) ⊗ Vec(C) Note that if i = 1 andF ≤ 1 then we can identify • the coloured forest (F,F) with the pair of subforests (F 1 ; F) ∈ P, Example 3.2 Let us continue Example 2.7, forgetting the decorations but keeping the same labels for the nodes and in particular for the leaves. We recall thatF is equal to 1 on the red subforest, to 2 on the blue subforest and to 0 elsewhere. Then Note that in this example, one hasF 2 ⊂ A, so that A / ∈ A 1 (F,F) since A violates the first condition of Assumption 1. A valid example of B ∈ A 1 (F,F) could be such that In the rest of this section we state several assumptions on the family A i (F,F) yielding nice properties for the operator i such as coassociativity, see e.g. Assumption 2. However, one of the main results of this article is the fact that such properties then automatically also hold at the level of decorated forests with a non-trivial action on the decorations which will be defined in the next subsection.

Operators on decorated forests
The set F, see Definition 2.5, is a commutative monoid under the forest product (3.5) where decorations defined on one of the forests are extended to the disjoint union by setting them to vanish on the other forest. This product is the natural extension of the product (2.1) on coloured forests and its identity element is the empty forest 1.
Note that for any F, G ∈ F, where |·| bi is the bigrading defined in (2.4) above. Whenever M is a submonoid of F, as a consequence of (3.6) the forest product · defined in (3.5) can be interpreted as a triangular linear map from M ⊗ M into M , thus turning ( M , ·) into an algebra in the category of bigraded spaces as in Definition 2.12; this is in particular the case for M = F. We recall that M is defined in (2.5). We generalise now the construction (3.4) to decorated forests.

Definition 3.3 The triangular linear maps
(b) The sum over n A runs over all maps n A : that we call the boundary of A in F. This notation is consistent with point a).
We will henceforth use these notational conventions for sums over node/edge decorations without always spelling them out in full.
Example 3. 4 We continue Examples 2.7 and 3.2, by showing how decorations are modified by i . We consider first i = 2, corresponding to a blue subforest where we refer to the labelling of edges and nodes fixed in the Example 2.7, and Note that the edge 1 was black in (F,F) and becomes blue in the value of e on 1 is set to 0 and e(1) is subtracted from o(a). In accordance with Assumption 1, A ∈ A 2 (F,F) contains one of the two connected components ofF 1 and is disjoint from the other one.
Example 3. 5 We continue Example 3.4 for the choice of B made in Example 3.2 and for i = 1, corresponding to a red subforest B ∈ A 1 (F,F). Then Here we have that ∂(B, F) = {7}, where we refer to the labelling of edges and nodes fixed in the Example 2.7. Therefore πε F B (d) = ε F B (7). Note that the edge 8 was black in (F,F) and becomes red in the value of e on 8 is set to 0 and e(8) is subtracted from o(d). In accordance with Assumption 1, B ∈ A 1 (F,F) is disjoint from the blue subforestF 2 and, accordingly, all decorations onF 2 are unchanged. Finally, note that the edge 1 is not in Remark 3.6 From now on, in expressions like (3.7) we are going to use the simplified notation namely the restrictions of o and e will not be made explicit. This should generate no confusion, since by Definition 2.
On the other hand, the notationF A refers to a slightly less standard operation, see Lemma 3.1 above, and will therefore be made explicitly throughout. Note also that n A is not defined as the restriction of n to N A .
Remark 3.7 It may not be obvious why Definition 3.3 is natural, so let us try to offer an intuitive explanation of where it comes from. First note that (3.7) reduces to (3.4) if we drop the decorations and the combinatorial coefficients.
If we go back to Remark 2.9, and we recall that a decorated forest encodes a function of a set of variables in R d indexed by the nodes of the underlying forest, then we can realise that the operator i in (3.7) is naturally motivated by Taylor expansions.
Let us consider first the particular case of τ = (F,F, 0, o, e). Then n A has to vanish because of the constraint 0 ≤ n A ≤ n and (3.7) becomes Consider a single term in this sum and fix an edge e = (v, w) ∈ ∂(A, F). Then, in the expression the decoration of e is changing from e(e) to e(e) + ε F A (e). Recalling (2.2), this should be interpreted as differentiating ε F A (e) times the kernel encoded by the edge e. At the same time, in the expression If we take into account the factor 1/ε F A (e)!, we recognise a (formal) Taylor sum If n is not zero, then we have a similar Taylor sum given by The role of the decoration o is still mysterious at this stage: we ask the reader to wait until the Remarks 3.19, 5.38 and 6.26 below for an explanation. The connection between our construction and Taylor expansions (more precisely, Taylor remainders) will be made clear in Lemma 6.10 and Remark 6.11 below.
Remark 3.8 Note that, in (3.7), for each fixed A the decoration n A runs over a finite set because of the constraint 0 ≤ n A ≤ n.
On the other hand, ε F A runs over an infinite set, but the sum is nevertheless well defined as an element of F ⊗ F , even though it does not belong to the algebraic tensor product F ⊗ F . Indeed, since |e it is the case that if |τ | bi = n, then the degree of each term appearing on the right hand side of (3.7) is of the type (n 1 + k 1 , Since furthermore the sum is finite for any given value of |ε F A |, this is indeed a triangular map on F , see Remark 2.15 above.
There are many other ways of bigrading F to make the i triangular, but the one chosen here has the advantage that it behaves nicely with respect to the various quotient operations of Sects. 3.5 and 4.1 below.

Remark 3.9
The coproduct i defined in (3.7) does not look like that of a combinatorial Hopf algebra since for ε F A the coefficients are not necessarily integers. This could in principle be rectified easily by a simple change of basis: if we set then we can write (3.7) equivalently as for τ = (F,F, n, o, e) • . Note that with this notation it is still the case that However, since this lengthens some expressions, does not seem to create any significant simplifications, and completely destroys compatibility with the notations of [32], we prefer to stick to (3.7).

Remark 3.10
As already remarked, the grading | · | bi defined in (3.6) is not preserved by the i . This should be considered a feature, not a bug! Indeed, the fact that the first component of our bigrading is not preserved is precisely what allows us to have an infinite sum in (3.7). A more natural integer-valued grading in that respect would have been given for example by which would be preserved by both the forest product · and i . However, since e can take arbitrarily large values, this grading is no longer positive. A grading very similar to this will play an important role later on, see Definition 5.3 below.

Coassociativity
Assumption 2 For each coloured forest (F,F) as in Definition 2.3, the collection A i (F,F) of subforests of F satisfies the following properties.

One has
Assumption 2 is precisely what is required so that the "undecorated" versions of the maps i , as defined in (3.4), are both multiplicative and coassociative. The next proposition shows that the definition (3.7) is such that this automatically carries over to the "decorated" counterparts.

Proposition 3.11
Under Assumptions 1 and 2, the maps i are coassociative and multiplicative on F , namely the identities Proof The multiplicativity property (3.16b) is an immediate consequence of property 1 in Assumption 2 and the fact that the factorial factorises for functions with disjoint supports, so we only need to verify (3.16a). Applying the definition (3.7) twice yields the identity , but in this as in other cases we prefer the lighter notation if there is no risk of confusion. Analogously, one has This is the reason, in particular, why the term π((ε F A ) F C ) appears in the last line of (3.18). In the proof of (3.18) we also make use of the fact that, since A ⊂ C, one has We now make the following changes of variables. First, we set with the naming conventions (3.19). Note that the support ofε given by (3.20) is invertible on its image, with inverse given by Furthermore, the only restriction on its image besides the constraints on the supports is the fact thatε F A,C ≤ε F C , which is required to guarantee that, with Since the factorial factorises for functions with disjoint supports, we can rewrite the combinatorial prefactor as In this way, the constraintε F A,C ≤ε F C is automatically enforced by our convention for binomial coefficients, so that (3.18) can be written as where ε F A and ε F C are determined by (3.21). We now make the further change of variables It is clear that, givenε F A,C , this is again a bijection onto its image and that the latter is given by those functions with the relevant supports such that furthermore With these new variables, (3.21) immediately yields Rewriting the combinatorial factor in this way, our convention on binomial coefficients once again enforces the condition (3.24), so that (3.23) can be written as with the summation only restricted by the conditions on the supports implicit in the notations. At this point, we note that the right hand side depends on ε F A,C only via the combinatorial factor and that, as a consequence of Chu-Vandermonde, one has Inserting (3.28) into (3.27), using the fact that (F C) ∪ i A = (F ∪ i A) C and comparing to (3.17) (with B replaced by C) completes the proof.

Bialgebra structure
Fix throughout this section i > 0.

Definition 3.12
For A i a family satisfying Assumptions 1 and 2, we set We also define the set In particular, one has |τ | bi = 0 for every τ ∈ U i . Finally we define 1 i : F → R by setting For instance, the following forest belongs to U 1 where 1 corresponds to red: We also define 1 i :

Assumption 3 For every coloured forest
Under Assumptions 1 and 3 it immediately follows from (3.7) that, setting

Lemma 3.13
Under Assumptions 1, 2 and 3, is a bialgebra in the category of bigraded spaces as in Definition 2.12.
Proof We consider only ( F i , ·, i , 1, 1 i ), since the other case follows in the same way. By the first part of Assumption 2, F i is closed under the forest product, so that ( F i , ·, 1) is indeed an algebra. Since we already argued that i : The required compatibility between the algebra and coalgebra structures is given by (3.16b), thus concluding the proof.

Contraction of coloured subforests and Hopf algebra structure
The bialgebra ( F i , ·, i , 1, 1 i ) does not admit an antipode. Indeed, for any In other words τ is grouplike. If a linear map A : F i → F i must satisfy (3.2), then , which is impossible since F is non-empty while 1 is the empty decorated forest. A way of turning F i into a Hopf algebra (again in the category of bigraded spaces as in Definition 2.12) is to take a suitable quotient in order to eliminate elements which do not admit an antipode, and this is what we are going to show now.
To formalise this, we introduce a contraction operator on coloured forests. Given a coloured forest (F,F), we recall thatÊ, defined in Definition 2.3, is the union of all edges inF j over all j > 0. Definition 3.14 For any coloured forest (F,F), we write KF F for the typed forest obtained in the following way. We use the equivalence relation ∼ on the node set N F defined in Definition 2.8, namely x ∼ y if x and y are connected inÊ. Then KF F is the quotient graph of (N F , E F \Ê) by ∼. By the definition of ∼, each equivalence class is connected so that KF F is again a typed forest.
Finally,F is constant on equivalence classes with respect to ∼, so that the coloured forest (KF F,F) is well defined and we denote it by If G := KF F, then there is a canonical projection π : N F → N G . This allows to define a canonical map K F from subforests of KF F to subforests of F as Note that in (KF F,F) all non-empty coloured subforests are reduced to single nodes.
We are going to restrict our attention to collections A i satisfying the following assumption.

Assumption 4 For all coloured forests
We recall that we have defined in (3.4) the operator acting on linear combinations of coloured forests (F,F) → i (F,F). Then we have Lemma 3.15 If A i satisfies Assumption 4, then which follow from the definitions. Moreover for the choice A ∈ A 2 (KF F,F) given by Then in accordance with Lemma 3.15 we have Then in accordance with Lemma 3.15 we have and both are equal to Contraction of couloured subforests leads us closer to a Hopf algebra, but there is still a missing element. Indeed, an element like (F,F) = (• •, 1), namely two red isolated roots with no edge, is grouplike since it satisfies 1 (F,F) = (F,F) ⊗ (F,F) and therefore it can not admit an antipode, see the discussion after (3.31) above.
We recall that C i has been introduced in Definition 3.12. We define first the factorisation of C i τ = μ · ν where the forest product · has been defined in (2.1) and • ν ∈ C i is the disjoint union of all non-empty connected componens of τ of the form (A, i) For instance Note that by the first part of Assumption 2, we know that if τ = μ · ν ∈ C i , then μ ∈ C i and ν ∈ C i . Then, we know by Assumption 4 that if μ ∈ C i , then K(μ) ∈ C i . Then, using this factorisation, we define K i : as the linear operator such that Proof The first assertion follows from the fact that K i is an algebra morphism, and from Lemma 3.15.
For the second assertion, we note that the vector space B i is isomorphic to where M denotes the forest product. The latter space is a Hopf algebra since it is a connected graded bialgebra with respect to the grading |(F,F)| i def = |F \F i |, namely the number of nodes and edges which are not coloured with i.
We now extend the above construction to decorated forests. Definition 3.18 Let K : F → F be the triangular map given by and [e] are defined as follows: • if x is an equivalence class of ∼ as in Definition 3.14, then [n](x) = y∈x n(y). Compare this forest with that in (3.30), which belongs to U 1 ; in (3.36) the decoration n can be non-zero, while it has to be identically zero in (3.30).
We define then an operator k i : We define first the factorisation of F i τ = μ · ν where the forest product · has been defined in (3.5) and • ν ∈ M i is the disjoint union of all non-empty connected componens of τ of the form (A, i, n, o, e) • μ ∈ F i is the unique element such that τ = μ · ν.
For instance, in (3.34) and (3.35), we have two forests in F 2 ; in both cases we have τ = μ · ν as above, where μ is the product of the first two trees (from left to right) and ν ∈ M 2 is the product of the two remaining trees.
By the first part of Assumption 2, we know that if τ = μ · ν ∈ F i , then μ ∈ F i and ν ∈ F i . We also know by Assumption 4 that if μ ∈ F i , then K(μ) ∈ F i . Therefore, using this factorisation, we define i : (3.37) In (3.34) and (3.35), the action of 2 corresponds to merging the third and fourth tree into a single decorated node (•, 2, n 3 + n 4 , 0, 0) with all other components remaining unchanged. We also defineˆ i : Finally For instance, if τ is the forest of (3.34) and σ = K(τ ) is that of (3.35), then Note that in K 2 (τ ) the roots of the connected components which do not belong to M 2 may have a non-zero o decoration, while the unique connected component in M 2 (reduced to a blue root with a possibly non-zero n decoration) always has a zero o decoration. InK 2 (τ ) all roots have zero o decoration. Since K commutes with i (as well as withˆ i ), is multiplicative, and is the identity on the image of k i in M i , it follows that for τ = μ · ν as above, we have Moreover K i andK i are idempotent and extend to triangular maps on F i since K, i andˆ i are all idempotent and preserve our bigrading. We then have the following result.
and similarly forÎ i .
Proof Although K i is not quite an algebra morphism of ( F i , ·), it has the The same can easily be verified for i andP i , so that it also holds for K i and K i , whence the claim follows.
If we define that the subspace spanned by (X k , k ∈ N d ) is isomorphic to the Hopf algebra of polynomials in d commuting variables, provided that we set  For τ = (F,F, n, o, e) and k : For such a τ with |τ | i = N and |τ | bi = M, we then note that one has Note that the first term in the right hand side above corresponds to the choice of A = F, while the second term contains the sum over all possible A = F. Here, the property |τ (1) | i < N holds because these terms come from terms with A = F in (3.7). Since for τ = 1 i we want to have this forces us to choose A i τ in such a way that (1) ) · τ m (2) . (3.45) In the case n = 0, this uniquely defines A i τ by the induction hypothesis since every one of the terms τ (1) appearing in this expression satisfies |τ (1) In the case where n = 0, A i τ is also easily seen to be uniquely defined by performing a second inductive step over |n| ∈ N. All terms appearing in the right hand side of (3.45) do indeed satisfy that their total |·| bi -degree is at least M by using the induction hypothesis. Furthermore, our definition immediately guarantees that M(A i ⊗ id) i = 1 i 1 i . It remains to verify that one also has M(id ⊗ A i ) i = 1 i 1 i . For this, it suffices to verify that A i is multiplicative, whence the claim follows by mimicking the proof of the fact that a semigroup with left identity and left inverse is a group.
Multiplicativity of A i also follows by induction over N = |τ | i . Indeed, it follows from (3.44) that it is the case for N = 0. It is also easy to see from (3.45) that if τ is of the form τ · X k for some τ and some k > 0, then one has Assuming that it is the case for all values less than some N , it therefore suffices to verify that A i is multiplicative for elements of the type τ = σ ·σ with |σ | i ∧ |σ | i > 0. If we extend A i multiplicatively to elements of this type then, as a consequence of the multiplicativity of i , one has as required. Since the map A i satisfying this property was uniquely defined by our recursion, this implies that A i is indeed multiplicative.

Characters group
Recall that an element g ∈ H * i is a character if g(τ ·τ ) = g(τ )g(τ ) for any τ,τ ∈ H i . Denoting by G i the set of all such characters, the Hopf algebra structure described above turns G i into a group by where the former operation is guaranteed to make sense by Remark 2.16. It is then easy to see that for every τ ∈ H i there exists a unique (possibly empty) collection {τ 1 , . . . , τ N } ⊂ P i such that τ = K i (τ 1 · . . . · τ N ). As a consequence, a multiplicative functional on H i is uniquely determined by the collection of values {g(τ ) : τ ∈ P i }. The following result gives a complete characterisation of the class of functions g : P i → R which can be extended in this way to a multiplicative functional on H i . Proof We first show that, under this condition, the unique multiplicative extension of g defines an element of H * i . By Remark 2.16, we thus need to show that there exists a functionm : N → N such that g(τ ) = 0 for every τ ∈ H i with |τ | bi = n and n 1 >m(n 2 ).
If σ = (F,F, n, o, e) ∈ P i satisfies n 2 = 0, thenF is nowhere equal to 0 on F by the definition (2.4); by property 2 in Definition 2.3,F is constant on F, since we also assume that F has a single connected component; in this case e ≡ 0 by property 3 in Definition 2.5; therefore, if n 2 = 0 then n 1 = 0 as well. Therefore we can setm(0) = 0.

Comodule bialgebras
Let us fix throughout this section 0 < i < j. We want now to study the possible interaction between the structures given by the operators i and j . For the definition of a comodule, see the beginning of Sect. 3.

Assumption 5
Let 0 < i < j. For every coloured forest (F,F) such that F ≤ j and {F,F j } ⊂ A j (F,F), one hasF i ∈ A i (F,F).

Lemma 3.26 Let 0 < i < j. Under Assumptions 1-4 for i and under Assump-
which endows F j with the structure of a left comodule over the bialgebra F i .
Proof Let (F,F, n, o, e) ∈ F j and A ∈ A i (F,F); by Definition 3.12, we haveF ≤ j and {F,F j } ⊂ A j (F,F), so that by Assumption 5 we havê (A,F A). Now, since A ∩F j = by property 1 in Assumption 1, we have Finally, the co-associativity (3.16a) of i on F shows the required compatibility between the coaction i : F j → F i ⊗ F j and the coproduct i : We now introduce an additional structure which will yield as a consequence the cointeraction property (3.48) between the maps i and j , see Remark 3.28.

Assumption 6
Let 0 < i < j. For every coloured forest (F,F), one has

47a)
if and only if We then have the following crucial result.

Proposition 3.27
Under Assumptions 1 and 6 for some 0 < i < j, the identity holds on F, where we used the notation The proof is very similar to that of Proposition 3.11, but using (3.47) instead of (3.15). Using (3.47) and our definitions, for τ = (F,F, n, o, e) ∈ F one has We claim that A 2 ∩ B = . Indeed, as noted in the proof of Lemma 3.1, since since ε F B has support in ∂(B, F) which is disjoint from E A 2 . This is because, for e = (e + , e − ) ∈ ∂(B, F) we have by definition e + ∈ N B ⊂ N F \ N A 2 and therefore e / ∈ E A 2 . Similarly, one has we only need to consider the decorations and the combinatorial factors. For this purpose, we definē as well as As before, the supports of these functions are consistent with our notations, with the particular case ofε F A 1 ,C whose support is contained in is invertible on its image, given by the functions with the correct supports and the additional constraintn Its inverse is given by Following a calculation virtually identical to (3.22) and (3.26), combined with the fact that n A + n C =n C +n A 2 , we see that Since A 2 ∩ C = and A 1 ⊂ C, we can simplify this expression further and obtain Following the same argument as (3.28), we conclude that so that (3.51) can be rewritten as We have also used the fact that On the other hand, since A 2 and B are disjoint, one has so that (3.50) can be rewritten as Remark 3.29 Note that the roles of i and j are asymmetric for 0 < i < j: F i is in general not a comodule bialgebra over F j . This is a consequence of the asymmetry between the roles played by i and j in Assumption 1. In particular, every A ∈ A i (F,F) has empty intersection withF j , while any B ∈ A j (F,F) can contain connected components ofF i .

Skew products and group actions
We assume throughout this subsection that 0 < i < j and that Assumptions 1-6 hold. Following [47], we define a space H i j = H i H j as follows. As a vector space, we set H i j = H i⊗ H j , and we endow it with the product and coproduct We also define 1 i j

Proposition 3.30
The 5-tuple (H i j , ·, i j , 1 i j , 1 i j ) is a Hopf algebra.
Proof We first note that, for every τ ∈ M j , one has i τ = 1⊗τ since one has A i (F, j) = { } by Assumptions 1 and 5. It follows that one has the identity see also (3.41). Combining this with Lemma 3.26, we conclude that one can indeed view i as a map i : H j → H i⊗ H j , so that (3.54) is well-defined. By Proposition 3.27, i j is coassociative, and it is multiplicative with respect to the product, see also [47,Thm 2.14]. Note also that on H j one has the identity where 1 i is the unit in H i . As a consequence, 1 i j is the counit for H i j , and one can verify that is the antipode turning H i j into a Hopf algebra.
Let us recall that G i denotes the character group of H i .

Lemma 3.31 Let us set for g
Then this defines a left action of G i onto G j by group automorphisms.
Proof The dualization of the cointeraction property (3.48) yields that g( , which means that this is indeed an action.
defines a sub-group of the group of characters of H i j .
Proof Note that (3.55) is the dualisation of i j in (3.54). The inverse is given by

Proposition 3.33
Let V be a vector space such that G i acts on V on the left and G j acts on V on the right, and we assume that Then G i j acts on the left on V by which is exactly what we wanted.
For instance, we can choose as V the dual space H * j of H j . For all h ∈ H * j , g ∈ G i and f ∈ G j we can set In this case (3.56) is the dualisation of the cointeraction property (3.48). The space H j is a left comodule over H i j with coaction given by β i j : Note that (3.57) is the dualisation of (3.58).

A specific setting suitable for renormalisation
We now specialise the framework described in the previous section to the situation of interest to us. We define two collections A 1 and A 2 as follows.

Definition 4.1 For any coloured forest (F,F) as in Definition 2.3 we define the collection
We also define A 2 (F,F) to consist of all subforests A of F with the following properties: 1. A containsF 2 2. for every non-empty connected component T of F, T ∩ A is connected and contains the root of T 3. for every connected component S ofF 1 , one has either S ⊂ A or S∩ A = .
The images in Examples 3.2 and 3.16 above are compatible with these definitions. We recall from Definition 3.12 that C i and F i are given for i = 1, 2 by Proof Let (F,F) ∈ C. IfF ≤ 1 thenF 2 = and therefore F ∈ A 1 (F,F); moreover A =F 1 clearly satisfiesF 1 ⊂ A and A ∩F 2 = , so thatF 1 ∈ A 1 (F,F) and therefore (F,F) n,o e ∈ C 1 . The converse is obvious.
Let us suppose now thatF ≤ 2 and for every connected component T of F,F 2 ∩ T is a subtree of T containing the root of T . Then A = F clearly satisfies the properties 1-3 of Definition 4.1. If now A =F 2 , then A satisfies the properties 1 and 2 since for every non-empty connected component T of F, F 2 ∩ T is a subtree of T containing the root of T , while property 3 is satisfied sinceF 1 ∩F 2 = . The converse is again obvious.
On the other hand,  Proof The first statement concerning A 1 is elementary. The only non-trivial property to be checked about A 2 is (3.15); note that A 2 has the stronger property that for any two subtrees F A), so that property (3.15) follows at once. Assumption 5 is easily seen to hold, since for every coloured forest (F,F) such thatF ≤ 2 and {F,F 2 } ⊂ A 2 (F,F), for A def =F 1 one hasF 1 ⊂ A and F 2 ∩ A = , so thatF 1 ∈ A 1 (F,F).
We check now that A 1 and A 2 satisfy Assumption 6. Let A ∈ A 1 (F,F) and B ∈ A 2 (F,F ∪ 1 A); then A ∩F 2 = and therefore B ∈ A 2 (F,F); moreover every connected component of A is contained in a connected component ofF 1 and therefore is either contained in B or disjoint from B, i.e. A ∈ A 1 (F,F ∪ 2 B) A 1 (B,F B). Conversely, let B ∈ A 2 (F,F) and A ∈ A 1 (F,F ∪ 2 B) A 1 (B,F B); thenF 1 = (F ∪ 2 B) 1 (F B) 1 andF 2 ⊂ (F ∪ 2 B) 2 so that A containsF 1 and is disjoint fromF 2 and therefore A ∈ A 1 (F,F); moreover (F ∪ 1 A) 2 ⊆F 2 so that B contains (F ∪ 1 A) 2 ; finally (F ∪ 1 A) 1 = A and by the assumption on A we have that every connected component of (F ∪ 1 A) 1 is either contained in B or disjoint from B. The proof is complete.
In view of Propositions 3.17, 3.23 and 3.27, we have the following result. We note that B 1 can be canonically identified with Vec(C 1 ), where C 1 = K 1 C 1 , see the definition of K i before Proposition 3.17, and C 1 is the set of (possibly empty) coloured forests (F,F) such thatF ≤ 1 andF 1 is a collection of isolated nodes, namely E 1 = . For instance Analogously, B 2 can be canonically identified with Vec(C 2 ), where C 2 = K 2 C 2 , and C 2 is the set of non-empty coloured forests (F,F) such thatF ≤ 2, F 1 is a collection of isolated nodes, namely E 1 = , andF 2 coincides with the set of roots of F. For instance The action of 1 on B i , i = 1, 2, can be described on Vec(C i ) as the action of (K 1 ⊗ K i ) 1 , namely: on a coloured forest (F,F) ∈ C i , one chooses a subforest B of F which containsF 1 and is disjoint fromF 2 , which is empty if i = 1 and equal to the set of roots of F if i = 2; then one has (B,F B) ∈ C 1 and K i (F, F ∪ 1 B) ∈ C i . Summing over all possible B of this form, we find This describes the coproduct of B 1 if i = 1 and the coaction on B 2 if i = 2. In both cases, we have a contraction/extraction operator of subforests: indeed, in (B,F B) we have the extracted subforest B, with colouring inherited from F, while in K i (F, F ∪ 1 B) we have extended the red colour to B and then contracted B to a family of red single nodes. For instance, using Example 3.16 since by (3.32) the red node labelled k on the left side of the tensor product is killed by K 1 .
The action of 2 on B 2 can be described on Vec(C 2 ) as the action of (K 2 ⊗ K 2 ) 2 , namely: on a coloured tree (F,F) ∈ C 2 , one chooses a subtree A of F which contains the root of F; then one has (A,F A) ∈ C 2 and K 2 (F, F ∪ 2 A) ∈ C 2 . Summing over all possible A of this form, we find If (F,F) = τ ∈ C 2 is a coloured forest, one decomposes τ in connected components, and then uses the above description and the multiplicativity of the coproduct. This describes the coproduct of B 2 as a contraction/extraction operator of rooted subtrees. For instance, using Example 3.16 The operators { 1 , 2 } on the spaces {H 1 , H 2 } act in the same way on the coloured subforests, and add the action on the decorations.

Joining roots
While the product given by "disjoint unions" considered so far is very natural when considering forests, it is much less natural when considering spaces of trees. There, the more natural thing to do is to join trees together by their roots. Given a typed forest F, we then define the typed tree J (F) by joining all the roots of F together. In other words, we set J (F) = F/ ∼, where ∼ is the equivalence relation on nodes in N F given by x ∼ y if and only if either x = y or both x and y belong to the set F of nodes of F. For example Example 4.6 Using as usual red for 1 and blue for 2, we have We can then extend J to D(J ) in a natural way as follows.
The following coloured forests belong toD 2 (J ) It is clear that the D i 's are closed under multiplication and that one has for every i ≥ 0. Furthermore, J is idempotent and preserves our bigrading.
The following fact is also easy to verify, where K,K i , i ,ˆ i andP i were defined in Sect. 3.5.
In particularK i J is idempotent onD i (J ).
Proof The spaces D i (J ) andD i (J ) are invariant under K, i andP i because these operations never change the colours of the roots. The invariance under J follows in a similar way. The fact that J commutes with Kis obvious. The reason why it commutes withP i is that o vanishes on colourless nodes by the definition of F. Regarding (4.2), sinceK i =P i i K, and all three operators are idempotent and commute with each other, we havê For this, consider an element τ ∈D i (J ) and write τ = μ · ν as in (3.37). By the definition of this decomposition and of K, there exist k ≥ 0 and labels n j ∈ N d , o j ∈ Z d ⊕ Z(L) with j ∈ {1, . . . , k} such that with n = k j=1 n j . On the other hand, by (4.1), one has with o defined from the o i similarly to n. Comparing this to (4.4), it follows that J Kτ differs from J i Kτ only by its o-decoration at the root of one of its connected components in the sense of Remark 2.10. Since these are set to 0 byˆ i , (4.3) follows.
Finally, we show that the operation of joining roots is well adapted to the definitions given in the previous subsection. In particular, we assume from now on that the A i for i = 1, 2 are given by Definition 4.1. Our definitions guarantee that We then have the following, where J is extended to the relevant spaces as a triangular map.

Proposition 4.10 One has the identities
Proof Extend J to coloured trees by J (F,F) = (J (F), [F]) with [F] as in Definition 4.7. The first identity then follows from the following facts. By the definition of A 2 , one has where J F A is the subforest of J F obtained by the image of the subforest A of F under the quotient map. The map J F is furthermore injective on A 2 (F,F), thus yielding a bijection between A 2 (J (F,F)) and A 2 (F,F). Finally, as a consequence of the fact that each connected component of A contains a root of F, there is a natural tree isomorphism between J F A and J A. Combining this with an application of the Chu-Vandermonde identity on the roots allows to conclude. The identity (4.5) fails to be true for A 1 in general. However, if (F,F, n, o, e) ∈ F 2 , then each of the roots of F is covered byF −1 (2), so that (4.5) with A 2 replaced by A 1 does hold in this case. Furthermore, one then has a natural forest isomorphism between J F A and A (as a consequence of the fact that A does not contain any of the roots of F), so that the second identity follows immediately.
We now use the "root joining" map J to definê H 2 def = F 2 / ker(JK 2 ) H 2 / ker(JP 2 ) . (4.6) Note here that JP 2 is well-defined on H 2 by (4.2), so that the last identity makes sense. The identity (4.2) also implies that ker(JK 2 ) = ker(K 2 J ), so the order in which the two operators appear here does not matter. We define alsoB 2 def = Vec(C 2 )/ ker(J K 2 ) B 2 / ker(J ) , (4.7) where J : C 2 → C 2 is defined by (J (F),F), which makes sense since all roots in F have the same (blue) colour. Finally, we define the tree product for i ≥ 0 Then we have the following complement to Corollary 4.5 Proof The Hopf algebra structure of H 2 turnsĤ 2 into a Hopf algebra as well by the first part of Proposition 4.10 and (4.1), combined with [48, Thm 1 (iv)], which states that if H is a Hopf algebra over a field and I a bi-ideal of H such that H/I is commutative, then H/I is a Hopf algebra. ForB 2 , the same proof holds.
The second assertion in Proposition 4.11 is in fact the same result, just written differently, as [8,Thm 8]. Indeed, our space B 2 is isomorphic to the Connes-Kreimer Hopf algebra H CK , and B 1 is isomorphic to an extension of the extraction/contraction Hopf algebra H. The difference between our B 1 and Hin [8] is that we allow extraction of arbitrary subforests, including with connected components reduced to single nodes; a subspace of B 1 which turns out to be exactly isomorphic to H is the linear space generated by coloured forests (F,F) ∈ C 1 such that N F ⊂F 1 .

Algebraic renormalisation
We set (4.9) Then, H • is an algebra when endowed with the tree product (4.8) in the special case i = 1. Note that this product is well-defined on H • since Kis multiplicative and J commutes with K. Furthermore, one has τ ·τ ∈ D 1 (J ) for any τ,τ ∈ F • . As a consequence of (4.1) and the fact that · is associative, we see that the tree product is associative, thus turning H • into a commutative algebra with unit (•, 0, 0, 0, 0).

Remark 4.12
The main reason why we do not define H • similarly toĤ 2 by setting H • = F 1 / ker(J K) is that 1 is not well-defined on that quotient space, while it is well-defined on H • as given by (4.9), see Proposition 4.14.
Remark 4.13 Using Lemma 2.17 as in Remark 3.22, we have canonical isomorphisms In particular, we can view H • andĤ 2 as spaces of decorated trees rather than forests. In both cases, the original forest product · can (and will) be interpreted as the tree product (4.8) with, respectively, i = 1 and i = 2.
We denote byĜ 2 the group of characters ofĤ 2 and by G 1 the group of characters of H 1 .
Combining all the results we obtained so far, we see that we have constructed the following structure.
and a right action ofĜ 2 on H * by Then we have Proof The first, the second and the third assertions follow from the coassociativity of 1 , respectively 2 , proved in Proposition 3.11, combined with Proposition 4.10 to show that these maps are well-defined on the relevant quotient spaces. The multiplicativity of 2 with respect to the tree product (4.8) follows from the first identity of Proposition 4.10, combined with the fact that H 2 is a quotient by ker J . In order to prove the last assertion, we show first that the above definitions yield indeed actions, since by the coassociativity of 1 and 2 proved in Proposition 3.11 and

Following (3.57), the natural definition is for
We prove now (4.11). By the definitions, we have and we conclude by Proposition 3.27.
Proposition 4.14 and its direct descendant, Theorem 5.36, are crucial in the renormalisation procedure below, see Theorem 6.16 and in particular (6.20).

and coaction
where σ (132) (a ⊗b⊗c) def = a ⊗c⊗b andÂ 2 is the antipode ofĤ 2 . Equivalently, the semi-direct product G 1 Ĝ 2 acts on the left on the dual space H * • by the formula In other words, with this action H * • is a left module on G 1 Ĝ 2 , see Proposition 3.33.

Remark 4.15
The action of 1 onĤ 2 differs from the action on {H • , H 1 } because of the following detail:Ĥ 2 is generated (as bigraded space) by a basis of rooted trees whose root is blue; since 1 acts by extraction/contraction of subforests which containF 1 and are disjoint fromF 2 , such subforests can never contain the root. Since on the other hand in H • and H 1 one has coloured forests with emptyF 2 , no such restriction applies to the action of 1 on these spaces.

Recursive formulae
We now show how the formalism developed so far in this article links to the one developed in [32,Sec. 8]. For that, we use the canonical identifications given in Remarks 3.22 and 4.13. We furthermore introduce the following notations.
1. For k ∈ N d , we write X k as a shorthand for (•, 0) k,0 0 ∈ H • . We also interpret this as an element ofĤ 2 , although its canonical representative there is (•, 2) k,0 0 ∈Ĥ 2 . As usual, we also write 1 instead of X 0 , and we write X i with i ∈ {1, . . . , d} as a shorthand for X k with k equal to the i-th canonical basis element of N d . 2. For every type t ∈ L and every k ∈ N d , we define the linear operator in the following way. Let τ = (F,F) n,o e ∈ H • , so that we can assume that F consists of a single tree with root . Then, I t k (τ ) = (G,Ĝ)n ,ō e ∈ H • is given by The decorations of I t k (τ ), as well asĜ, coincide with those of τ , except on the newly added edge/vertex whereĜ,n andō vanish, whileē( G , ) = k. This gives a triangular operator and I t k : H • → H • is therefore well defined. 3. Similarly, we define operatorsĴ t k : H • →Ĥ 2 (4.13) in exactly the same way as the operators I t k defined in (4.12), except that the root ofĴ t k (τ ) is coloured with the colour 2, for instance

Remark 4.16
With these notations, it follows from the definition of the sets H • , H 1 andĤ 2 that they can be constructed as follows.
• Every element of H • \ {1} can be obtained from elements of the type X k by successive applications of the maps I t k , R α , and the tree product (4.8). • Every element of H 1 is the forest product of a finite number of elements of H • . • Every element ofĤ 2 is of the form (4.14) for some finite collection of elements τ i ∈ H • \ {1}, t i ∈ L and k i ∈ N d .
Then, one obtains a simple recursive description of the coproduct 2 .
and it is completely determined by these properties. Likewise, 2 :Ĥ 2 → H 2⊗Ĥ 2 is multiplicative, satisfies the identities on the first line of (4.15) and and it is completely determined by these properties.
Proof The operator 2 is multiplicative on H • as a consequence of the first identity of Proposition 4.10 and its action on X k was already mentioned in (3.43). It remains to verify that the recursive identities hold as well.
We first consider 2 σ with σ = I t k (τ ) and τ = (T,T ) n,o e . We write σ = (F,F) n,o e+k1 e , where e is the "trunk" of type t created by I t k and is the root of F; moreover we extend n to N F and o to NF by setting n( ) = o( ) = 0. It follows from the definitions that Indeed, if e does not belong to an element A of A 2 (F,F) then, since A has to contain and be connected, one necessarily has A = { }. If on the other hand e ∈ A, then one also has ∈ A and the remainder of A is necessarily a connected subtree of T containing its root, namely an element of A 2 (T,T ).
Given A ∈ A 2 (T,T ), since the root-label of σ is 0, the set of all possible node-labels n A for σ appearing in (3.7) for 2 σ coincides with those appearing in the expression for 2 τ , so that we have the identity This is because n( ) = 0, so that the sum over n contains only the zero term. Since 2 : H • → H •⊗Ĥ 2 , we are implicitly applying the appropriate contraction K ⊗ JK 2 , see (4.6)-(4.9). We now consider 2 σ with σ = R α (τ ). In this case, we write τ = (T,T ) n,o e so that, denoting by the root of T , one has σ = (T,T ∨ 1 , n, o + α1 , e).
We claim that in this case one has This is non-trivial only in the caseT ( ) = 0. In this case however, it is necessarily the case thatT (e) = 0 for every edge e incident to the root. This in turn guarantees that the family A 2 (T,T ) remains unchanged by the operation of colouring the root. This implies that one has This appears slightly different from the desired identity, but the latter then follows by observing that, for everyτ ∈Ĥ 2 , one has R ατ =τ as elements of H 2 , thanks to the fact that we quotiented by the kernel ofK 2 which sets the value of o to 0 on the root.
We finally have the following results on the antipode ofĤ 2 : Proposition 4.18 LetÂ 2 :Ĥ 2 →Ĥ 2 be the antipode ofĤ 2 . Then • The algebra morphismÂ 2 :Ĥ 2 →Ĥ 2 is defined uniquely by the fact that A 2 X i = −X i and for allĴ t where M:Ĥ 2⊗Ĥ 2 →Ĥ 2 denotes the (tree) product. • OnĤ 2 , one has the identity Proof By (4.14) and by induction over the number of edges in τ , this uniquely determines a morphismÂ 2 ofĤ 2 , so it only remains to show that The formula is true for τ = X k , so that, since both sides are multiplicative, it is enough to consider elements of the formĴ t k (τ ) for some τ ∈ H • . Exploiting the identity (4.17), one then has In this section we have shown several useful recursive formulae that characterize 2 , see also Sect. 6.4 below. The paper [4] explores in greater detail this recursive approach to Regularity Structures, and includes a recursive formula for 1 , which is however more complex than that for 2 .

Rules and associated regularity structures
We recall the definition of a regularity structure from [32, Def. 2.1]

Definition 5.1 A regularity structure T = (A, T, G) consists of the following elements:
• An index set A ⊂ R such that A is bounded from below, and A is locally finite. • A model space T , which is a graded vector space T = α∈A T α , with each T α a Banach space.
• A structure group G of linear operators acting on T such that, for every ∈ G, every α ∈ A, and every a ∈ T α , one has a − a ∈ β<α T β .
The aim of this section is to relate the construction of the previous section to the theory of regularity structures as exposed in [32,34]. For this, we first assign real-valued degrees to each element of F. scaling is a map s : {1, . . . d} → [1, ∞) and a degree assignment is a map | · | s : L → R \ {0}. By additivity, we then assign a degree to each (k, v) ∈ Z d ⊕ Z(L) by setting

Definition 5.3
Given a scaling s as above, for τ = (F,F, n, o, e) ∈ F 2 , we define two different notions of degree |τ | − , |τ | + ∈ R by where we recall that o takes values in Z d ⊕ Z(L) and t : E F → L is the map assigning to an edge its type in F, see Sect. 2.1.
Note that both of these degrees are compatible with the contraction operator Kof Definition 3.18, as well as the operator J , in the sense that |τ | ± = |τ | ± if and only if |Kτ | ± = |Kτ | ± and similarly for J . In the case of | · | + , this is true thanks to the definition (3.33), while the coloured part of the tree is simply ignored by | · | − . We furthermore have Lemma 5. 4 The degree | · | − is compatible with the operators K i andK i of (3.39), while | · | + is compatible with K 2 andK 2 . Furthermore, both degrees are compatible with J and K, so that in particular H 1 is | · | − -graded and H 2 and H • are both | · | − and | · | + -graded.
Proof The first statement is obvious since | · | − ignores the coloured part of the tree, except for the labels n whose total sum is preserved by all these operations. For the second statement, we need to verify that | · | + is compatible withˆ 2 as defined just below (3.37). which is the case when acting on a tree with ∈F 2 since the o-decoration of nodes inF 2 does not contribute to the definition of | · | + . As a consequence, | · | − yields a grading for H 1 , | · | + yields a grading for H 2 , and both of them yield gradings for H • . With these definitions, we see that we obtain a structure resembling a regularity structure by taking H • to be our model space, with grading given by | · | + and structure group given by the character groupĜ 2 ofĤ 2 acting on H • via The second statement of Proposition 4.14 then guarantees that this action is multiplicative with respect to the tree product (4.8) on H • , so that we are in the context of [32,Sec. 4]. There are however two conditions that are not met: 1. The action ofĜ 2 on H • is not of the form "identity plus terms of strictly lower degree", as required for regularity structures. 2. The possible degrees appearing in H • have no lower bound and might have accumulation points.
We will fix the first problem by encoding in our context what we mean by considering a "subcritical problem". Such problems will allow us to prune our structure in a natural way so that we are left with a subspace of H • that has the required properties. The second problem will then be addressed by quotienting a suitable subspace ofĤ 2 by the terms of negative degree. The group of characters of the resulting Hopf algebra will then turn out to act on H • in the desired way.

Trees generated by rules
From now to Sect. 5.4 included, the colourings and the labels o will be ignored. It is therefore convenient to consider the space In order to lighten notations, we write elements of T as (T, n, e) = T n e with T a typed tree (for some set of types L) and n : N T → N d , e : E T → N d as above. Similarly to before, T is a monoid for the tree product (4.8). Again, this product is associative and commutative, with unit (•, 0, 0).

Definition 5.5
We say that an element T n e ∈ T is trivial if T consists of a single node •. It is planted if T has exactly one edge incident to its root and furthermore n( ) = 0.
In other words, a planted T n e ∈ T is necessarily of the form I t k (τ ) with τ ∈ T, see (4.12). For example, a planted tree: and a non-planted tree: .
With this definition, each τ ∈ T has by (4.14) a unique (up to permutations) factorisation with respect to the tree product (4.8) for some n ∈ N d , where each τ i is planted and • n denotes the trivial element (•, n, 0) ∈ T.
In order to define a suitable substructure of the structure described in Proposition 4.14, we introduce the notion of "rules". Essentially, a "rule" describes what behaviour we allow for a tree in the vicinity of any one of its nodes.
In order to formalise this, we first define the set of edge types E and the set of node types N by where [E] n denotes the set of unordered E-valued n-uples, namely [E] n = E n /S n , with the natural action of the symmetric group S n on E n . In other words, given any set A,P(A) consists of all finite multisets whose elements are elements of A. Remark 5.6 The fact that we consider multisets and not just n-uples is a reflection of the fact that we always consider the situation where the tree product (4.8) is commutative. This condition could in principle be dropped, thus leading us to consider forests consisting of planar trees instead, but this would lead to additional complications and does not seem to bring any advantage.
Given two sets A ⊂ B, we have a natural inclusionP(A) ⊂P(B). We will usually write elements of [E] n as n-uples with the understanding that this is just an arbitrary representative of an equivalence class. In particular, we write () for the unique element of [E] 0 . Given any T n e ∈ T, we then associate to each node x ∈ N T a node type N(x) ∈ N by N(x) = s(e 1 ), . . . , s(e n ) , s(e) def = (t(e), e(e)) ∈ E, e ∈ E T , (5.6) where (e 1 , . . . , e n ) denotes the collection of edges leaving x, i.e. edges of the form (x, y) for some node y. We will sometimes use set-theoretic notations. In particular, given N = (s 1 , . . . , s n ) ∈ N and M = (r 1 , . . . , r ) ∈ N, we write . . . , r , s 1 , . . . , s n ) , and we say that M ⊂ N if there existsN such that N = M N . When we write a sum of the type M⊂N , we take multiplicities into account. For example (a, b) is contained twice in (a, b, b), so that such a sum always contains 2 n terms if N is an n-tuple. Similarly, we write t ∈ N if (t) ⊂ N and we also count sums of the type t∈N with the corresponding multiplicities. For example we may have L = {t 1 , t 2 } and Then, according to the rule R, an edge of type t 1 or t 2 can be followed in a tree by, respectively, no edge, or a single edge of type t i with decoration e i with i ∈ {1, 2}, or by two edges, one of type t 1 with decoration e 1 and one of type t 2 with decoration e 2 . We do not expect however to find two edges both of type t 1 (or t 2 ) sharing a node which is not the root.

Definition 5.8
Let R be a rule and τ = T n e ∈ T. We say that • τ conforms to R at the vertex x if either x is the root and there exists t ∈ L such that N(x) ∈ R(t) or one has N(x) ∈ R(t(e)), where e is the unique edge linking x to its parent in T . • τ conforms to R if it conforms to R at every vertex x, except possibly its root. • τ strongly conforms to R if it conforms to R at every vertex x.
In particular, the trivial tree • strongly conforms to every normal rule since, as a consequence of Definition 5.7, there exists at least one t ∈ L with () ∈ R(t).
The first tree does not conform to the rule R since the bottom left edge of type t 2 is followed by three edges. The second tree conforms to R but not strongly, since the root is incident to three edges. The third tree strongly conforms to R. If we call i the root of the i-th tree, then we have N( 1 ) = {(t 2 , e 2 ), (t 2 , e 2 )}, Finally, note that R is normal.

Remark 5.10
If R is a normal rule, then by Definition 5.7 we have in particular that () ∈ R(t) for every t ∈ L. This guarantees that L contains no useless labels in the sense that, for every t ∈ L, there exists a tree conforming to R containing an edge of type t: it suffices to consider a rooted tree with a single edge e = (x, y) of type t; in this case, N(y) = {()} ∈ R(t). More importantly, this also guarantees that we can build any tree conforming to R from the root upwards (start with an edge of type t, add to it a node of some type in R(t), then restart the construction for each of the outgoing edges of that node) in finitely many steps.

Remark 5.11
A rule R can be represented by a directed bipartite multigraph G(R) = (V (R), E(R)) as follows. Take as the vertex set V (R) = E N. Then, connect N ∈ N to t ∈ E if t ∈ N . If t is contained in N multiple times, repeat the connection the corresponding number of times. Conversely, connect (t, k) ∈ E to N ∈ N if N ∈ R(t). The conditions then guarantee that () ∈ N can be reached from every vertex in the graph. Given a tree τ ∈ T, every edge of τ corresponds to an element of Eand every node corresponds to an element of N via the map x → N(x) defined above. A tree then conforms to R if, for every path joining the root to one of the leaves, the corresponding path in V always follows directed edges in G(R). It strongly conforms to R if the root corresponds to a vertex in V with at least one incoming edge. This definition is compatible with both notions of degree given in Definition 5.3, since we view T as a subset of F withF and o identically 0. This also allows us to give the following definition.

Definition 5.13 Given a rule R, we write
• T • (R) ⊂ T for the set of trees that strongly conform to R • T 1 (R) ⊂ F for the submonoid of F (for the forest product) generated by T • (R) • T 2 (R) ⊂ T for the set of trees that conform to R.
The second restriction on the definition of τ ∈ T − (R) is related to the definition (5.22) of the Hopf algebra T ex − and of its characters group G ex − , that we call the renormalisation group and which plays a fundamental role in the theory, see e.g. Theorem 6.16.

Subcriticality
Given a map reg : L → R we will henceforth interpret it as maps reg : E → R and reg : N → R as follows: for (t, k) ∈ E and N ∈ N with the convention that the sum over the empty word () ∈ N is 0.
Definition 5.14 A rule R is subcritical with respect to a fixed scaling s if there exists a map reg : L → R such that where we use the notation (5.9).
We will see in Sect. 5.4 below that classes of stochastic PDEs generate rules. In this context, the notion of subcriticality given here formalises the one given somewhat informally in [32]. In particular, we have the following result which is essentially a reformulation of [32,Lem. 8.10] in this context.
Proof Fix γ ∈ R and let T n e ∈ T • (R) with |T n e | s ≤ γ . Since there exists c > 0 such that |T n e | s ≥ |T 0 e | s + c|n| and there exist only finitely many trees in T • (R) of the type |T 0 e | for a given number of edges, it suffices to show that the number |E T | of edges of T is bounded by some constant depending only on γ .
Since the set L is finite, (5.10) implies that there exists a constant κ > 0 such that the bound holds for every t ∈ L with the notation (5.9). We claim that for every planted T n e ∈ T • (R) such that the edge type of its trunk e = ( , x) is (t, k) ∈ E, we have We denote the space of such planted trees by T (t,k) • (R). We verify (5.12) by induction on the number of edges |E T | of T . If |E T | = 1, namely the unique element of E T is the trunk e = ( , x), then N(x) = () ∈ R(t) in the notation of (5.6) and by (5.11) For a planted T n e ∈ T • (R) with |E T | > 1, then N(x) = (s(e 1 ), . . . , s(e n )) ∈ R(t) and by (5.11) and the induction hypothesis where s(e i ) = (t i , k i ). Therefore (5.12) is proved for planted trees. Given an arbitrary tree T n e of degree at most γ strongly conforming to the rule R, there exists t 0 ∈ L such that e ∈ N( T ) = R(t 0 ). We can therefore consider the planted treeT n e containing a trunk of type t 0 connected to the root of T , and with vanishing labels on the root and trunk respectively. It then follows that and the latter expression is finite since L is finite. The claim follows at once.
Remark 5. 16 The inequality (5.10) encodes the fact that we would like to be able to assign a regularity reg(t) to each component u t of our SPDE in such a way that the "naïve regularity" of the corresponding right hand side obtained by a power-counting argument is strictly better than reg(t) − |t|. Indeed, inf N ∈R(t) reg(N ) is precisely the regularity one would like to assign to F t (u, ∇u, ξ). Note that if the inequality in (5.10) is not strict, then the conclusion of Proposition 5.15 may fail to hold.

Remark 5.17
Assuming that there exists a map reg satisfying (5.11) for a given κ > 0, one can find a map reg κ that is optimal in the sense that it saturates the bound (5.12): We proceed as follows. Set reg 0 κ (t) = +∞ for every t ∈ L and then define recursively By recurrence we show that n → reg n κ (t) is decreasing and reg ≤ reg n κ ; then the limit reg κ (t) = lim n→∞ reg n κ (t) exists and has the required properties. If we extend reg n κ to E CN by (5.9), the iteration (5.13) can be interpreted as a min-plus network on the graph G(R) with arrows reversed, see Remark 5.11.

Completeness
Given an arbitrary rule (subcritical or not), there is no reason in general to expect that the actions of the analogues of the groups G 1 andĜ 2 constructed in Sect. 4 leave the linear span of T • (R) invariant. We now introduce a notion of completeness, which will guarantee later on that the actions of G 1 and G 2 do indeed leave the span of T • (R) (or rather an extension of it involving again labels o on nodes) invariant. This eventually allows us to build, for large classes of subcritical stochastic PDEs, regularity structures allowing to formulate them, endowed with a large enough group of automorphisms to perform the renormalisation procedures required to give them canonical meaning. N = ((t 1 , k 1 ), . . . , (t n , k n )) ∈ N and m ∈ N d , we define ∂ m N ⊂ N as the set of all n-tuples of the form ((t 1 , k 1 +  m 1 ), . . . , (t n , k n + m n )) where the m i ∈ N d are such that i m i = m.

Definition 5.19
Given a rule R, for any tree T n e ∈ T • (R) we associate to each edge e ∈ E T a setN(e) ⊂ Nin the following recursive way. If e = (x, y) and y is a leaf, namely the node-type N(y) of the vertex y is equal to the empty word () ∈ N, then we setN It is easy to see that, if we explore the tree from the leaves down, this specifies N(e) and M(y) uniquely for all edges and nodes of T .

Definition 5.20
A rule R is -complete with respect to a fixed scaling s if, whenever τ ∈ T − (R) and t ∈ L are such that there exists N ∈ R(t) with N( τ ) ⊂ N , one also has for every M ∈ M( τ ) and for every multiindex m with |m| s + |τ | s < 0.
At first sight, the notion of -completeness might seem rather tedious to verify and potentially quite restrictive. Our next result shows that this is fortunately not the case, at least when we are in the subcritical situation.

Proposition 5.21
Let R be a normal subcritical rule. Then, there exists a normal subcritical ruleR which is -complete and extends R in the sense that R(t) ⊂R(t) for every t ∈ L.
Proof Given a normal subcritical rule R, we define a new rule QR by setting where R − (t; τ ) is the union of all collections of node types of the typê , and some multiindex m with |m| s + |τ | s ≤ 0. Since QR (t) ⊃ R(t) and T − (R) is finite by Proposition 5.15, this is again a valid rule. Furthermore, by definition, a rule R is -complete if and only if QR = R. We claim that the desired ruleR can be obtained by settinḡ It is straightforward to verify thatR is -complete. (This follows from the fact that the sequence of rules Q n R is increasing and Q is closed under increasing limits.) It remains to show thatR is again normal and subcritical. To show normality, we note that if R is normal, then QR is again normal. This is because, by Definition 5.19, the setsN(e) used to build M( τ ) also have the property that if N ∈N(e) and M ⊂ N , then one also has M ∈N(e). As a consequence, Q n R is normal for every n, from which the normality ofR follows.
To show thatR is subcritical, we first recall that by Remark 5.17, for κ as in (5.11), we can find a maximal function reg κ : L → R such that Furthermore, the extension of reg κ to node types given by (5.9) is such that, for every node type N and every multiindex m, one has We used a small abuse of notation here since ∂ m N is really a collection of node types. Since reg κ takes the same value on each of them, this creates no ambiguity.) We claim that the same function reg κ also satisfies (5.10) for the larger rule QR. In view of (5.16) and of the definition (5.15) of QR, it is enough to prove that Arguing by induction as in the proof of (5.12), one can first show the following. Let σ ∈ T • (R) any every planted tree whose trunk e has edge type (t, k). Then one has the bound where σ i is the largest planted subtree of σ with trunk e i . Then Combining this with (5.16) we obtain, since |t| s − |k| s + i=1 |σ i | s = |σ | s , and (5.19) is proved.
We prove now (5.18). Let τ ∈ T − (R), N ∈ R(t) with N( τ ) ⊂ N , M = (M 1 , . . . , M ) ∈ M( τ ), and m ∈ N d with |m| s +|τ | s ≤ 0. Let τ = τ 1 . . . τ be the decomposition of τ into planted trees. Recalling (5.17) and Definitions 5.19 and 5.18, we have where s i is the edge type of the trunk of τ i . Combining this with (5.19) yields with the last inequality a consequence of the condition |m| s + |τ | s ≤ 0. This proves (5.18). We conclude that (5.16) also holds when considering N ∈ (QR)(t), thus yielding the desired claim. Iterating this, we conclude that reg κ satisfies (5.10) for each of the rules Q n R and therefore also forR as required.

Definition 5.22
We say that a subcritical rule R is complete (with respect to a fixed scaling s) if it is both normal and -complete. If R is only normal, we call the ruleR constructed in the proof of Proposition 5.21 the completion of R.

Three prototypical examples
Let us now show how, concretely, a given stochastic PDE (or system thereof) gives rise to a rule in a natural way. Let us start with a very simple example, the KPZ equation formally given by One then chooses the set L so that it has one element for each noise process and one for each convolution operator appearing in the equation. In this case, using the variation of constants formula, we rewrite the equation in integral form as where P denotes the heat kernel and * is space-time convolution. We therefore need two types in L in this case, which we call { , I} in order to be consistent with [32].
We assign degrees to these types just as in [32]. In our example, the underlying space-time dimension is d = 2 and the equation is parabolic, so we fix the parabolic scaling s = (2, 1) and then assign to a degree just below the exponent of self-similarity of white noise under the scaling s, namely | | s = − 3 2 − κ for some small κ > 0. We also assign to each type representing a convolution operator the degree corresponding to the amount by which it improves regularity in the sense of [32,Sec. 4]. In our case, this is given by |I| s = 2.
It then seems natural to assign to such an equation a ruleR bỹ where I 1 is a shorthand for the edge type (I, (0, 1)) and we simply write t as a shorthand for the edge type (t, 0). In other words, for every noise type t, we setR(t) = {()} and for every kernel type t we include one node type intõ R(t) for each of the monomials in our equation that are convolved with the corresponding kernel. The problem is that such a rule is not normal. Therefore we define rather which turns out to be normal and complete. It is simple to see that the function makes R subcritical for sufficiently small κ > 0. One can also consider systems of equations. Consider for example the system of coupled KPZ equations formally given by In this case, we have two noise types 1,2 as well as two kernel types, which we call I for the heat kernel with diffusion constant 1 and I ν for the heat kernel with diffusion constant ν. There is some ambiguity in this case whether the term u 1 appearing in the second equation should be considered part of the linearisation of the equation or part of the nonlinearity. In this case, it turns out to be more convenient to consider this term as part of the nonlinearity, and we will see that the corresponding rule is still subcritical thanks to the triangular structure of this system. Using the same notations as above, the normal and complete rule R naturally associated with this system of equations is given by In this case, we see that R is again subcritical for sufficiently small κ > 0 with Our last example is given by the following generalisation of the KPZ equation: which is motivated by (1.6) above, see [33]. In this case, the set L is again given by Again, it is straightforward to verify that R is subcritical and that one can use the same map reg κ as in the case of the standard KPZ equation. Even though in this case there are infinitely many node types appearing in R(I), this is not a problem because reg κ (I) > 0, so that repetitions of the symbol I in a node type only increase the corresponding degree.

Regularity structures determined by rules
Throughout this section, we assume that we are given • a finite type set L together with a scaling s and degrees | · | s as in Definition 5.2, • a normal rule R for L which is both subcritical and complete, in the sense of Definition 5.22, • the integer d ≥ 1 which has been fixed at the beginning of the paper.
We show that the above choices, when combined with the structure built in Sects. 3 and 4, yield a natural substructure with the same algebraic properties (the only exception being that the subspace of H • we consider is not an algebra in general), but which is sufficiently small to yield a regularity structure. Furthermore, this regularity structure contains a very large group of automorphisms, unlike the slightly smaller structure described in [32]. The reason for this is the additional flexibility granted by the presence of the decoration o, which allows to keep track of the degrees of the subtrees contracted by the action of G 1 .
Definition 5. 23 We define for every τ = (G, n , e ) ∈ T and every node Definition 5.13) and We define S : F → T ⊂ F by S(F,F, n, o, e) def = (F, n, e).

Definition 5.24
We denote by = (L, R, s, d) the set of all τ = (F,F, n, o, e) ∈ F such that τ = K 1 τ and, for all x ∈ N F , exactly one of the following two mutually exclusive statements holds.
as in (5.20). Setting F = F n and A = A n we have the required representation. Now the first assertion follows easily from the second one.
We now define spaces of coloured forests τ = (F,F, n, o, e) such that (F, 0, n, 0, e) is compatible with the rule R in a suitable sense, and such that τ ∈ .

Remark 5.27
The superscript "ex" stands for "extended", see Sect. 6.4 below for an explanation of the reason why we choose this terminology. The identification of these spaces as suitable subspaces ofĤ 2 , H 1 and H • is done via the canonical basis (4.10).
Note that bothT ex − andT ex + are algebras for the products inherited from H 1 andĤ 2 respectively. On the other hand, T ex is in general not an algebra anymore.
Lemma 5. 28 We have as well as 2 : H → H⊗T ex + for H ∈ {T ex ,T ex + }. Moreover,T ex + is a Hopf subalgebra ofĤ 2 and T ex is a right Hopf-comodule overT ex + with coaction 2 .
Proof By the normality of the rule R, if a tree conforms to R then any of its subtrees does too. On the other hand, contracting subforests can generate non-conforming trees in the case of 1 , while, since 2 extracts only subtrees at the root, completeness of the rule implies that this can not happen in the case of 2 , thus showing that the maps i do indeed behave as claimed.
The fact thatT ex + is in fact a Hopf algebra, namely that the antipodeÂ 2 ofĤ 2 leavesT ex + invariant, can be shown by induction using (4.17) and Remark 4.16.
Note thatT ex − is a sub-algebra but in general not a sub-coalgebra of H 1 (and a fortiori not a Hopf algebra). Recall also that, by Lemma 5.4, the grading | · | − of Definition 5.3 is well defined onT ex − and on T ex , and that | · | + is well defined on bothT ex + and T ex . Furthermore, these gradings are preserved by the corresponding products and coproducts.

Definition 5.29
Let J ∓ ⊂T ex ± be the ideals given by Then, we set with canonical projections p ex ± :T ex ± → T ex ± . Moreover, we define the operator J t k : With these definitions at hand, it turns out that the map (p ex − ⊗ id) 1 is much better behaved. Indeed, we have the following.

Lemma 5.30 The map
Proof This follows immediately from Lemma 5.28, combined with the fact that completeness of R has beed defined in Definition 5.20 in terms of extraction of τ ∈ T − (R), which in particular means that |τ | s = |τ | − < 0.
Analogously to Lemma 3.21 we have Lemma 5. 31 We have Proof We note that the degrees |·| ± have the following compatibility properties with the operators i . For 0 < i ≤ j ≤ 2, τ ∈ F j and i τ = τ (1) i ⊗ τ (2) i (with the summation variable suppressed), one has |τ (1) 2 | + + |τ (2) 2 | + = |τ | + . (5.25) The first identity of (5.24) then follows from the first identity of (5.25) and from the following remark: if B − τ = I t k (σ ), then for each term appearing in the sum over A ∈ A 1 in the expression (3.7) for 1 τ , one has two possibilities: • either A does not contain the edge incident to the root of τ , and then the second factor is a tree with only one edge incident to its root, • or A does contain the edge incident to the root, in which case the first factor contains one connected component of that type.
The second identity of (5.24) follows from the second identity of (5.25) combined with the fact that, for τ ∈ F 2 , 1 τ contains no term of the form σ ⊗ 1 2 , even when quotiented by ker(JK 2 ). The third identity of (5.24) finally follows from the third identity of (5.25), combined with the fact that if τ ∈ B + \ {1 2 } with |τ | + ≤ 0, then the term 1 2 ⊗ 1 2 does not appear in the expansion for 2 τ .
As a corollary, we have the following.

Corollary 5.32 The operator
Similarly, the operator + ex = (id ⊗ p ex + ) 2 is well-defined as a map Proof We already know thatT ex + is a Hopf sub-algebra ofĤ 2 with antipodê A 2 satisfying (4.17). Since J − is a bialgebra ideal by Lemma 5.31, the first claim follows from [48,Thm 1.(iv)].
The fact that + ex : T ex → T ex ⊗ T ex + is a co-action and turns T ex into a right comodule for T ex + follows from the coassociativity of 2 .

Proposition 5.35 There exists an algebra morphism
Proof One difference between T ex − and T ex + is thatT ex − is not in general a sub-coalgebra of H 1 and therefore it does not possess an antipode. However we can see that the antipode A 1 of H 1 satisfies for all τ = 1 where M is the product map. By the second formula of (5.25), it follows that if |τ | − > 0 then A 1 τ ∈ J + and therefore, since A 1 is an algebra morphism, A 1 (J + ) ⊆ J + . We obtain that A 1 defines a unique algebra morphism A ex − : T ex − → T ex − which is an antipode for T ex − .
Definition 5. 36 We call G ex ± the character group of T ex ± .
We have therefore obtained the following analogue of Proposition 4.14: Theorem 5.37 1. On T ex , we have the identity and a right action of G ex Then we have Proof By the second identity of (5.25), the action of − ex preserves the degree | · | + . In particular we have − ex p ex From this property, one has: and we conclude by applying the Proposition 3.27. Now the proof of (5.27) is the same as that of (4.11) above.
Remark 5. 38 We can finally see here the role played by the decoration o: were it not included, the cointeraction property (5.26) of Theorem 5.37 would fail, since it is based upon (5.28), which itself depends on the second identity of (5.25). Now recall that | · | + takes the decoration o into account, and this is what makes the second identity of (5.25) true. See also Remark 6.26 below.
As in the discussion following Proposition 4.14, we see that T ex is a left comodule over the Hopf algebraT ex

Proposition 5.39
The above construction yields a regularity structure T ex = (A ex , T ex , G ex + ) in the sense of Definition 5.1.
Proof By the definitions, every element τ ∈ B • has a representation of the type (5.21) for some σ = (T, 0, n, 0, e) ∈ T. Furthermore, it follows from the definitions of | · | + and | · | s that one has |τ | + = |σ | s . The fact that, for all γ ∈ R, the set {a ∈ A ex : a ≤ γ } is finite then follows from Proposition 5.15. The space T ex is graded by |·| + and G ex + acts on it by g def = (id⊗g) + ex . The property (5.1) then follows from the fact + ex preserves the total | · | + -degree by the third identity in (5.25) and all terms appearing in the second factor of + ex τ − τ ⊗ 1 have strictly positive | · | + -degree by Definition 5.29.

Remark 5.40
Since T ex − is finitely generated as an algebra (though infinitedimensional as a vector space), its character group G ex − is a finite-dimensional Lie group. In contrast, G ex + is not finite-dimensional but can be given the structure of an infinite-dimensional Lie group, see [5].

Renormalisation of models
We now show how the construction of the previous sections can be applied to the theory of regularity structures to show that the "contraction" operations one would like to perform in order to renormalise models are "legitimate" in the sense that they give rise to automorphisms of the regularity structures built in Sect. 5.5. Throughout this section, we are in the framework set at the beginning of Sect. 5.5. We furthermore impose the additional constraint that, writing L = L − L + with t ∈ L + if and only if |t| s > 0, one has (6.1) Remark 6.1 Labels in L + represent "kernels" while labels in L + represent "noises", which naturally leads to (6.1). (We could actually have defined L − by L − = {t : The condition that elements of L − are of negative degree and those in L + are of positive degree is also natural in this context. It could in principle be weakened, which corresponds to allowing kernels with a non-integrable singularity at the origin. This would force us to slightly modify Definition 6.8 below in order to interpret these kernels as distributions but would not otherwise lead to any additional complications.
Note now that we have a natural identification of T ex ± with the subspaces Denote by i ex ± : T ex ± →T ex ± the corresponding inclusions, so that we have direct sum decompositionsT ex For instance, with this identification, the mapĴ t k : T ex →T ex + defined in (4.13) associates to τ ∈ T ex an elementĴ t k (τ ) ∈T ex + which can be viewed as J t k (τ ) ∈ T ex + \ {0} if and only if its degree |J t k (τ )| + is positive, namely |τ | + + |t| s − |k| s > 0.
where M ex + : T ex + ⊗T ex + → T ex + denotes the (tree) product and + ex : Proof The claims follow easily from Propositions 4.18 and 5.34.

Twisted antipodes
We define now the operator P + :T ex + →T ex + given on τ ∈ B + by Note that this is quite different from the projection i ex + • p ex + . However, for elements of the formĴ t k (τ ) ∈T ex + for some τ ∈ T ex , we have . The difference is that i ex + • p ex + is multiplicative under the tree product, while P + is not.

Proposition 6.3
There exists a unique algebra morphismÃ ex + : T ex + →T ex + , which we call the "positive twisted antipode", such thatÃ ex + X i = −X i and furthermore for all J t k (τ ) ∈ T ex whereĴ t k : T ex →T ex + is defined in (4.13), similarly to aboveM ex + is the product inT ex + and + ex : T ex → T ex ⊗ T ex + is as in Corollary 5.32.
Proof Proceeding by induction over the number of edges appearing in τ , one easily verifies that such a map exists and is uniquely determined by the above properties.
Comparing this to the recursion for A ex + given in (6.3), we see that they are very similar, but the projection p ex + in (6.3) is inside the multiplication M ex + , while P + in (6.5) is outsideM ex + . We recall now that the antipode A ex + is characterised among algebramorphisms of T ex + by the identity where + ex : T ex + → T ex + ⊗ T ex + is as in Corollary 5.32. The following result shows thatÃ ex + satisfies a property close to (6.6), which is where the name "twisted antipode" comes from.

Proposition 6.4
The mapÃ ex + : T ex + →T ex + satisfies the equation where + ex :T ex + →T ex + ⊗ T ex + is as in Corollary 5.32.
Proof Since both sides of (6.7) are multiplicative and since the identity obviously holds when applied to elements of the type X k , we only need to verify that the left hand side vanishes when applied to elements of the form J t k (τ ) for some τ ∈ T ex with |τ | + + |t| s − |k| s > 0, and then use Remark 4.16. Similarly to the proof of (4.17), we havê A very useful property of the positive twisted antipodeÃ ex + is that its action is intertwined with that of − ex in the following way.

Lemma 6.5 The identity
− exÃ ex holds between linear maps from T ex + to T ex − ⊗T ex + .
Proof Since both sides of the identity are multiplicative, by using Remark 4.16 it is enough to prove the result on X i and on elements of the form J k (τ ) ∈ T ex + . The identity clearly holds on the linear span of X k since − ex acts trivially on them andÃ ex + preserves that subspace.
Using the recursion (6.5) forÃ ex + , the identity − ex P + = (id ⊗ P + ) − ex on T ex + , followed by the fact that − ex is multiplicative, we obtain − exÃ ex Using the fact that − exĴ t Here, the passage from the penultimate to the last line crucially relies on the fact that the action of G − ex onto T ex + preserves the |·| + -degree, i.e. on the second formula in (5.25).
We have now a similar construction of a negative twisted antipode. Proposition 6.6 There exists a unique algebra morphismÃ ex − : T ex − →T ex − , that we call the "negative twisted antipode", such that for τ ∈ T ex − ∩ ker 1 1 Similarly to (6.7), the morphismÃ ex Proof Proceeding by induction over the number of colourless edges appearing in τ , one easily verifies that such a morphism exists and is uniquely determined by (6.8). The property (6.9) is a trivial consequence of (6.8).

Models
We now recall (a simplified version of) the definition of a model for a regularity structure given in [32,Def. 2.17]. Given a scaling s as in Definition 5.2 and interpreting our constant d ∈ N as a space(-time) dimension, we define a metric d s on R d by Note that · s is not a norm since it is not 1-homogeneous, but it is still a distance function since s i ≥ 1. It is also homogeneous with respect to the (inhomogeneous) scaling in which the ith component is multiplied by λ s i .

Definition 6.7
A smooth model for a given regularity structure T = (A, T, G) on R d with scaling s consists of the following elements: such that x x = id, the identity operator, and such that xy yz = xz for every x, y, z in R d .
Furthermore, for every ∈ A and every compact set K ⊂ R d , we assume the existence of a constant C ,K such that the bounds (6.11) hold uniformly over all x, y ∈ K, all m ∈ A with m < and all τ ∈ T .
Here, recalling that the space T in Definitions 5.1 and 6.7 is a direct sum of Banach spaces (T α ) α∈A , the quantity σ m appearing in (6.11) denotes the norm of the component of σ ∈ T in the Banach space T m for m ∈ A. We also note that Definition 6.7 does not include the general framework of [32,Def. 2.17], where x takes values in D (R d ) rather than C ∞ (R d ); however this simplified setting is sufficient for our purposes, at least for now. The condition (6.11) on x is of course relevant only for > 0 since x τ (·) is assumed to be a smooth function at this stage.
Recall that we fixed a label set L = L − L + . We also fix a collection of kernels {K t } t∈L + , K t : R d \ {0} → R, satisfying the conditions of [32, Ass. 5.1] with β = |t| s . We use extensively the notations of Sect. 4.3.
The proof of the second bound in (6.11) for xy is virtually identical to the one given in [32,Prop. 8.27], combined with Lemma 6.10. Formally, the main difference comes from the change of basis (6.31) mentioned in Sect. 6.4, but this does not affect the relevant bounds since it does not mix basis vectors of different | · | + -degree. Remark 6.13 If a map : T ex → C ∞ is admissible and furthermore satisfies (6.16), then it is uniquely determined by the functions ξ l def = l for l ∈ L − . In this case, we call the canonical lift of the functions ξ l .

Renormalised Models
We now use the structure built in this article to provide a large class of renormalisation procedures, which in particular includes those used in [32,39,42]. For this, we first need a topology on the space of all models for a given regularity structure. Given two smooth models ( , ) and (¯ ,¯ ), for all ∈ A and K ⊂ R d a compact set, we define the pseudo-metrics Here, the set B ∈ C ∞ 0 (R d ) denotes the set of test functions with support in the centred ball of radius one and all derivatives up to oder 1 + | inf A| bounded by 1. Given ϕ ∈ B, ϕ λ x : R d → R denotes the translated and rescaled function for x ∈ R d and λ > 0 as in [32]. Finally, ·, · is the usual L 2 scalar product.
Definition 6.14 We denote by M ex ∞ the space of all smooth models of the form Z ex ( ) for some admissible linear map : T ex → C ∞ in the sense of Definition 6.9. We endow M ex ∞ with the system of pseudo-metrics (|||·; ·||| ;K ) ;K and we denote by M ex 0 the completion of this metric space.
When we apply gp ex − to the terms corresponding to case 2, the result is 0 since A contains one planted tree (with same root as that of I t k (τ )) and p ex − I t k = 0 by the definition (5.22) of J + . Therefore we have Therefore Since X k has positive degree, with a similar computation we obtain and this shows that g is admissible. Now we verify that, writing M ex g as before and Z ex ( g ) = ( g , g ), we have To show this, one first uses (6.4) to show that f g z = (g ⊗ f z ) − ex , where f and f g are defined from and g as in (6.12). Indeed, one has One then uses (5.26) on T ex to show that the required identity (6.19) for g z holds. Indeed, it follows that In other words, we have applied (5.27) for (g, f, h) = (g, f z , ). Regarding γ zz , we have analogously Every random stationary map : T ex → C ∞ in the sense of Definition 6.17 then naturally determines a (deterministic) character g − ( ) ofT ex − by setting for τ ∈ B • , where the symbol E on the right hand side denotes expectation over the underlying probability space. This is extended multiplicatively to all ofT ex − . Then we can define a renormalised mapˆ : T ex → C ∞ bŷ whereÃ ex − : T ex − →T ex − is the negative twisted antipode defined in (6.8) and satisfying (6.9).
Let us also denote by B − • the (finite!) set of basis vectors τ ∈ B • such that |τ | − < 0. The specific choice of g = g − ( )Ã ex − used to defineˆ is very natural and canonical in the following sense. Theorem 6.18 Let : T ex → C ∞ be stationary and admissible such that Then, among all random functions g : T ex → C ∞ of the form with M ex g as in (6.18),ˆ is the only one such that, for all h ∈ R d , we have We callˆ the BPHZ renormalisation of .
Proof We first show thatˆ does indeed have the desired property. We first consider h = 0 and we write 0 : T ex → R for the map (not to be confused with 0 ) Let us denote by B • the set of τ ∈ B − • which are not of the form I t k (σ ) with |t| s > 0. The main point now is that, thanks to the definitions of g − ( ) and − ex , we have the identity Combining this with (6.25), we obtain for all by the defining property (6.9) of the negative twisted antipode, since ι • τ belongs both to the image of i ex − and to the kernel of 1 1 .
Arguing as in the proof of Theorem 6.16 we see that It then follows that The definition of g − ( ) combined with the fact that is admissible and the definition ofˆ now implies that where D k K t should be interpreted in the sense of distributions. In particular, one has For σ = (F,F, n, o, e) andn : N F → N d withn ≤ n, we now write Lnσ = (F,F, n −n, o, e) and we note that for g h as in (6.22) one has the identity so that the stationarity of implies that so that Lnσ ∈ B − • and has strictly less colourless edges than τ = I t k (σ ). If σ has only one colourless edge, then σ belongs to B • ; therefore the proof follows by induction over the number of colourless edges of τ .
Let us now turn to the case h = 0. First, we claim that, settingˆ h =T h (ˆ ), one has This follows from the fact thatˆ is stationary since the actionT commutes with that of G ex − as a consequence of (5.26), combined with the fact that for every f ∈ G ex − , every τ ∈ T ex + and every g h of the form (6.22).
On the other hand, we haveˆ It follows immediately from the expression for the action ofT thatˆ τ is a deterministic linear combination of terms of the formˆ h σ with |σ | − ≤ |τ | − , so that the claim (6.26) follows from (6.28). It remains to show thatˆ is the only function of the type g with this property. For this, note that every such function is also of the formˆ g for some different g ∈ G ex − , so that we only need to show that for every element g different from the identity, there exists τ such that E ˆ g τ (0) = 0.
Using Definitions 5.26 and 5.29, Remark 4.16 and the identification (6.2), T ex − can be canonically identified with the free algebra generated by B • . Therefore the character g is completely characterised by its evaluation on B • and it is the identity if and only if this evaluation vanishes identically. Fix now such a g different from the identity and let τ ∈ B • be such that g(τ ) = 0, and such that g(σ ) = 0 for all σ ∈ B • with the property that either |σ | − < |τ | − or |σ | − = |τ | − , but σ has strictly less colourless edges than τ . Since B • is finite and g doesn't vanish identically, such a τ exists.
We can then also view τ as an element of T ex and we write Moreover Q commutes with the maps K i ,K i and J , and preserves the | · | −degree, so that it is in particular also well-defined on T ex ,T ex + ,T ex − and T ex − . It does however not preserve the | · | + -degree so that it is not well-defined on T ex + ! Indeed, the | · | + -degree depends on the o decoration, which is set to 0 by Q, see Definition 5.3. We also set T + = p ex +T + , where p ex ± :T ex ± → T ex ± is defined after (5.23).
The reason why we define T + in this slightly more convoluted way instead of setting it equal to {τ ∈ T ex + : Qτ = τ } is that although Q is well-defined on T ex + , it is not well-defined on T ex + since it does not preserve the | · | + -degree, as already mentioned above. Since Q is multiplicative, T + is a subalgebra of T ex + . We set def = + ex : T → T⊗ T + , + def = + ex : T + → T + ⊗ T + . (6.30) Looking at the recursive definition (6.3) of the antipode A ex + , it is clear that it also maps T + into itself, so that T + is a Hopf subalgebra of T ex + . Moreover turns T into a co-module over T + .
We can therefore define G + as the characters group of T + and introduce the action of G + on T: If we grade T by | · | + and we define T = (A, T, G + ) where A def = {|τ | + : τ ∈ B • , τ = Qτ } and T ex = B • as in Definition 5.26, then arguing as in the proof of Proposition 5.39, we see that the action of G + on Tsatisfies (5.1). Therefore T is a regularity structure as in Definition 5.1.
We set nowJ t k : T ex → T ex + andJ t k : T → T + , Suppose that {t, i} ⊆ L with |t| s > 0 and |i| s < 0. We set i := I i 0 (1). Then we have by (4.15) and (4.16) for all τ ∈ T as well as with the additional property that both maps are multiplicative with respect to the tree product. We see therefore that the operators : T → T ⊗ T + and + : T + → T + ⊗ T + are isomorphic to those defined in [32, Eq. (8.8)- (8.9)]. This shows that the regularity structure T , associated to a subcritical complete rule R, is isomorphic to the regularity structure associated to a subcritical equation constructed in [32,Sec. 8], modulo a simple change of coordinates. Note that this change of coordinates is "harmless" as far as the link to the analytical part of [32] is concerned since it does not mix basis vectors of different degrees.
As explained in Remark 5.27, the superscript 'ex' stands for extended: the reason is that the regularity structure T ex is an extension of T in the sense that T ⊂ T ex with the inclusion interpreted as in [32,Sec. 2.1]. By contrast, we call T the reduced regularity structure.
By the definition of Q, the extended structure T ex encodes more information since we keep track of the effect of the action of G − by storing the (negative) homogeneity of the contracted subtrees in the decoration o and by colouring the corresponding nodes; both these details are lost when we apply Q and therefore in the reduced structure T.
Note that if : T ex → C ∞ is such that Z ex ( ) = ( , ) is a model of T ex , then the restriction Z( ) of Z ex ( ) to T is automatically again a • We set G − def = {g ∈ G ex − : g(τ ) = g(Qτ ), ∀ τ ∈ T ex − }. This is the most natural subgroup of G ex − since it contains the characters g − ( )Ã ex − used for the definition ofˆ in (6.25), as soon as = Q. The fact that G − is a subgroup follows from the property (6.34).
• We set G a − def = {g ∈ G ex − : g(τ ) = 0, ∀τ ∈ T c − } where T c − is the bialgebra ideal of T ex − generated by {τ ∈ B − , Qτ = τ }. Then one can identify G a − with the group of characters of the Hopf algebra T ex − /T c − , − ex . It turns out that this is simply the polynomial Hopf algebra with generators {τ ∈ B − : |τ | − < 0, Qτ = τ }, so that G a − is abelian. We then have the following result.

Theorem 6.29
There is a continuous action R of G − onto M 0 with the property that, for every g ∈ G − and every reduced and admissible : T ex → C ∞ with Z ex ( ) ∈ M ex 0 , one has R g Z( ) = Z( M g ). Proof We already know by Theorem 6.16 that G − acts continuously onto M ex 0 . Furthermore, by the definition of G − , it preserves the subset M r 0 ⊂ M ex 0 of reduced models, i.e. the closure in M ex 0 of all models of the form Z ex ( ) for admissible and reduced. Since T ⊂ T ex , we already mentioned that we have a natural projection π ex : M ex 0 → M 0 given by restriction (so that Z( ) = π ex Z ex ( )), and it is straightforward to see that π ex is injective on M r 0 . It therefore suffices to show that there is a continuous map ι ex : M 0 → M ex 0 which is a right inverse to π ex , and this is the content of Theorem 6.33 below.
Remark 6. 30 We'll show in Sect. 6.4.3 below that the action of G − onto M 0 is given by elements of the "renormalisation group" defined in [32, Sec. 8.3].

An example
We consider the example of the stochastic quantization given in dimension 3 by: This equation has been solved first in [32] with regularity structures and then in [7]. One tree needed for its resolution reveals the importance of the extended decoration. Using the symbolic notation, it is given by τ = I( ) 2 I(I( ) 3 ). Then we use the following representation: where e i is the ith canonical basis element of N d and a belongs to {α, β, γ } with α = 2I + 2 , β = 2I + 2 + 1 and γ = 5I + 4 . Then we have − ex with summation over i and j implied. In (...), we omit terms of the form τ (1) ⊗ τ (2) where τ (1) may contain planted trees or where τ (2) has an edge of type I finishing on a leaf. The planted trees will disappear by applying an element of G ex − and the others are put to zero through the evaluation of the smooth model see [32,Ass. 5.4] where the kernels {K t } t∈L + are chosen such that they integrate polynomials to zero up to a certain fixed order. If g ∈ G ex − is the character associated to the BPHZ renormalisation for a Gaussian driving noise with a covariance that is symmetric under spatial reflections, we obtain and all other renormalisation constants vanish. Applying Q, we indeed recover the renormalisation map given in [32,Sec 9.2]. The main interest of the extended decorations is to shorten some Taylor expansions which allows us to get the co-interaction between the two renormalisations. In the computation below, we show the difference between a term having extended decoration and the same without:

Construction of extended models
In general if, for some sequence (n) : T ex → C ∞ , Z ex ( (n) ) ∈ M ex ∞ converges to a limiting model in M ex 0 , it does not follow that the characters g + ( (n) ) ofT ex + converge to a limiting character. However, we claim that the characters f (n) x of T ex + given by (6.12) do converge, which is not so surprising since our definition of convergence implies that the characters γ (n) xy of T ex + given by (6.13) do converge. More surprising is that the convergence of the characters f (n) x follows already from a seemingly much weaker type of convergence. Writing D for the space of distributions on R d , we have the following. Proposition 6.31 Let (n) : T ex → C ∞ be an admissible linear map with Z ex ( (n) ) = ( (n) , (n) ) ∈ M ex ∞ and assume that there exist linear maps x : T ex → D (R d ) such that, with the notation of (6.17), (n) − ,K → 0 for every ∈ R and every compact set K. Then, the characters f (n) x defined as in (6.12) converge to a limit f x . Furthermore, defining xy by (6.13), one has Z = ( , ) ∈ M ex 0 and Z ex ( (n) ) → Z in M ex 0 . Finally, one has : T ex → D (R d ) such that x = ( ⊗ f x ) + ex and such that (n) τ → τ in D (R d ) for every τ ∈ T ex .
Proof The convergence of the f (n) x follows immediately from the formula given in Lemma 6.10, combined with the convergence of the (n) x and [32,Lem. 5.19]. The fact that ( , ) satisfies the algebraic identities required for a model follows immediately from the fact that this is true for every n. The convergence of the Remark 6.32 This relies crucially on the fact that the maps under consideration are admissible and that the kernels K t satisfy the assumptions of [32,Sec. 5]. If one considers different notions of admissibility, as is the case for example in [40], then the conclusion of Proposition 6.31 may fail.
For a linear : T → C ∞ we define ex : T ex → C ∞ by simply setting ex = Q. Then we say that is admissible if ex is. We have the following crucial fact Theorem 6.33 If : T → C ∞ is admissible and Z( ex ) belongs to M ∞ , then Z ex ( ex ) belongs to M ex ∞ . Furthermore, the map Z( ex ) → Z ex ( ex ) extends to a continuous map from M 0 to M ex 0 .
Before proving this Theorem, we define a linear map L : T ex → T ⊗ T + such that L l k, = l k, ⊗ 1 , L X k = X k ⊗ 1 , and then recursively as well as (6.35) where M + is the tree product (4.8) on T + andJ is as in (6.31). Moreover L + : T ex + → T + is the algebra morphism such that L + X k = X k and for J t k (τ ) ∈ T ex + with |J t k (τ )| + > 0 The reason for these definitions is that these map will provide the required injection M 0 → M ex 0 by (6.38) below. Before we proceed to show this, we state the following preliminary identity. Proof We prove (6.37) by recursion. Both maps in (6.37) agree on elements of the form l k, or X k and both maps are multiplicative for the tree product. Consider now a tree of the form I t k (τ ) and assume that (6.37) holds when applied to τ . Then we have by (6.32) where f z = f ex z T + and similarly for γ zz . Defineˆ z : T ex → C ∞ and f z ∈ G ex + byˆ z def = ( z ⊗ f z )L ,f z def = f z L + , (6.38) where L , L + are defined in (6.35)-(6.36). We want to show that ex = (ˆ z ⊗ f z A ex + ) + ex for all z. By the definitions By (6.37)

Renormalisation group of the reduced structure
In this section, we show that the action of the renormalisation group G − on M 0 given by Theorem 6.29 is indeed given by elements of the "renormalisation group" R as defined in [32,Sec. 8.3]. This shows in particular that the BPHZ renormalisation procedure given in Theorem 6.18 does always fit into the framework developed there. We recall that, by [32,Lem. 8.43,Thm 8.44] and [40,Thm B.1], R is the set of linear operators M : T → T satisfying the following properties.
• One has I t k Mτ = M I t k τ and M X k τ = X k Mτ for all t ∈ L + , k ∈ N d , and τ ∈ T.

Remark 6.35
Despite what a cursory inspection may suggest, the condition (6.39) is not equivalent to the same expression withJ t k replaced by J t k . This is because (6.39) will typically fail to hold when |J t k (σ )| + ≤ 0.
We recall that the group G − def = {g ∈ G ex − : g(τ ) = g(Qτ ), ∀ τ ∈ T ex − } has beed defined after Remark 6.28. Theorem 6.36 Given g ∈ G − , define M ex g on T ex and T ex + as in (6.18) and let M g : T → T be given by M g = QM ex g . Then M g ∈ R, g → M g is a group homomorphism, and one has the identitieŝ M g = L + M ex g : T + → T + , M g = L M ex g : T → T⊗ T + , (6.41) where the maps L , L + are given in (6.35)-(6.36).
Proof In order to check (6.39), it suffices by (6.41) to use (6.36) and the fact that M ex g preserves the | · | + -degree. It remains to check (6.40). We have on T that where we have used the co-interaction property in the last line. It follows from (6.37) that these two terms are indeed equal. The triangularity of L and M ex g , combined with (6.41), implies the triangularity of M g .
The homomorphism property follows from (6.34) and the definition of G − since as required. Proof This follows immediately from (6.34), Theorem 6.36, the definition of G − , the fact that Q is an algebra morphism on T ex − , and the same argument as in the proof of Proposition 4.11.
By the Remarks 6.19 and 6.28, the renormalisation procedures of [31,32,39,42] can be described in this framework. a workshop held in Bergen in April 2017, where the results of this paper were presented and discussed in detail.
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