Algebraic renormalisation of regularity structures
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Abstract
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.
Mathematics Subject Classification
16T05 82C28 60H151 Introduction
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 nonlinearities of (random) space–time distributions. Prominent examples are the KPZ equation [23, 27, 31], the \(\Phi ^4\) 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 \(s\mapsto \mathbb {X}_{s,t}\tau \) to be a space–time distribution for some Open image in new window . However, the algebraic relations discussed above involve products of such quantities, which are in general illdefined. One of the main ideas of [32] was to replace the Hopfalgebra structure with a comodule structure: instead of a single space Open image in new window , we have two spaces Open image in new window and a coaction Open image in new window such that Open image in new window is a right comodule over the Hopf algebra Open image in new window . In this way, elements in the dual space Open image in new window of Open image in new window are used to encode the distributional objects which are needed in the theory, while elements of Open image in new window encode continuous functions. Note that Open image in new window admits neither a product nor a coproduct in general.
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 \(\mathbb {X}^\varepsilon \). Another major difference with what one sees in the rough paths setting is the following phenomenon: if we remove the regularisation as \(\varepsilon \rightarrow 0\), neither the canonical model \(\mathbb {X}^\varepsilon \) 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 illdefined.
This is where renormalisation enters the game. It was already recognised in [32] that one should find a group \(\mathfrak {R}\) of transformations on the space of models and elements \(M_\varepsilon \) in \(\mathfrak {R}\) in such a way that, when applying \(M_\varepsilon \) to the canonical lift \(\mathbb {X}^\varepsilon \), 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 \(\mathfrak {R}\), 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 \(\mathfrak {R}\) 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.
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 Bseries for numerical ODE solvers, which is itself an extension of the ConnesKreimer 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].
of colouring, extraction and contraction of subforests. Further, the abovementioned articles deal with two spaces in cointeraction, analogous to our Hopf algebras Open image in new window and Open image in new window , while our third space Open image in new window 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 Open image in new window 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 nontrivial way under our coproducts, interacting with other operations like the contraction of subforests 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 Open image in new window (the renormalisation) and Open image in new window (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 \(\varepsilon \), and the valuation extracts the pole part of this series. In our case, the space A consists of stationary random linear maps Open image in new window and we have two actions on it, by the group of characters Open image in new window of Open image in new window , corresponding to two different valuations. The renormalisation group Open image in new window is associated to the valuation that extracts the value of \(\mathbf{E}(\varvec{\Pi }\tau )(0)\) for every homogeneous element Open image in new window of negative degree. The structure group Open image in new window on the other hand is associated to the valuations that extract the values \((\varvec{\Pi }\tau )(x)\) for all homogeneous elements Open image in new window of positive degree.
We show in particular that the twisted antipode related to the action of Open image in new window is intimately related to the algebraic properties of Taylor remainders. Also in this respect, regularity structures provide a farreaching 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].
1.1 A general renormalisation scheme for SPDEs
The transformation \({\mathbb {X}}^\varepsilon \mapsto \hat{\mathbb {X}}^\varepsilon \) is described by the socalled renormalisation group. The main aim of this paper is to provide a general construction of the space of models \(\mathscr {M}\) together with a group of automorphisms Open image in new window which allows to describe the renormalised model \(\hat{\mathbb {X}}^\varepsilon =S_\varepsilon {\mathbb {X}}^\varepsilon \) for an appropriate choice of Open image in new window .

Algebraic step: Construction of the space of models \((\mathscr {M},\mathrm{d})\) and renormalisation of the canonical model \(\mathscr {M}\ni {\mathbb {X}}^\varepsilon \mapsto \hat{\mathbb {X}}^\varepsilon \in \mathscr {M}\), this article.

Analytic step: Continuity of the solution map Open image in new window , [32].

Probabilistic step: Convergence in probability of the renormalised model \(\hat{\mathbb {X}}^\varepsilon \) to \(\hat{\mathbb {X}}\) in \((\mathscr {M},\mathrm{d})\), [9].

Second algebraic step: Identification of \(\Phi (\hat{\mathbb {X}}^\varepsilon )\) with the classical solution map for an equation with local counterterms, [2].
1.2 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 \(\mathscr {M}^\mathrm {ex}\). The superscript ‘\(\mathrm {ex}\)’ stands for extended and is used to distinguish this space from the restricted space of models \(\mathscr {M}\), see Definition 6.24, which is closer to the original construction of [32]. The space \(\mathscr {M}^\mathrm {ex}\) extends \(\mathscr {M}\) in the sense that there is a canonical continuous injection \(\mathscr {M}\hookrightarrow \mathscr {M}^\mathrm {ex}\), see Theorem 6.33. The reason for considering this larger space is that it admits a large group Open image in new window 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 Open image in new window which leaves \(\mathscr {M}\) invariant. The reason why we do not describe its action on \(\mathscr {M}\) 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 \(\mathscr {M}^\mathrm {ex}\).
The next step is the construction of the space of smooth models of the regularity structure Open image in new window . 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 Open image in new window , 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 Open image in new window 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 Open image in new window such that the associated renormalised model yields a centered family of stochastic processes on the finite family of elements in Open image in new window 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 \(\mathrm{d}\) on \(\mathscr {M}\), 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 Open image in new window 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 Open image in new window 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 Open image in new window must also be invariant under the actions of Open image in new window . 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 nonlinearity, 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 Open image in new window , Open image in new window and Open image in new window are obtained by considering repeatedly suitable subsets and suitable quotients of two initial spaces, called \(\mathfrak {F}_1\) and \(\mathfrak {F}_2\) and defined in and after Definition 4.1; more precisely, \(\mathfrak {F}_1\) is the ancestor of Open image in new window and Open image in new window , while \(\mathfrak {F}_2\) is the ancestor of Open image in new window . 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 Open image in new window , Open image in new window and Open image in new window (through other intermediary spaces which are called Open image in new window , Open image in new window and Open image in new window ). An important difference between Open image in new window and Open image in new window 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: Open image in new window is endowed with a forest product which corresponds to the disjoint union, while Open image in new window 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 \(\mathfrak {F}_1\) and \(\mathfrak {F}_2\). In Sects. 2 and 3 however we present a number of results on a family of spaces \((\mathfrak {F}_i)_{i\in I}\) with \(I\subset \mathbf{N}\), 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 \(\mathfrak {F}_i\) of decorated forests, and vector spaces \({\langle \mathfrak {F}_i\rangle }\) 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 illdefined if were to work on arbitrary formal series.
The family of spaces \((\mathfrak {F}_i)_{i\in I}\) are introduced in Definition 3.12 on the basis of families of admissible forests \(\mathfrak {A}_i\), \(i\in I\). If \((\mathfrak {A}_i)_{i\in I}\) satisfy Assumptions 1, 2, 3, 4, 5 and 6, then the coproducts \(\Delta _i\) of Definition 3.3 are coassociative and moreover \(\Delta _i\) and \(\Delta _j\) 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 Open image in new window and Open image in new window on Open image in new window . “Appendix A” contains a summary of the relations between the most important spaces appearing in this article, while “Appendix B” contains a symbolic index.
2 Rooted forests and bigraded spaces
Lemma 2.1
Remark 2.2
These notations are also consistent with the case where the maps k and \(\ell \) are multiindex valued under the natural identification of a map \(S \rightarrow \mathbf{N}^d\) with a map \(S \times \{1,\ldots ,\infty \} \rightarrow \mathbf{N}\) given by \(\ell (x)_i \leftrightarrow \ell (x,i)\).
2.1 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, \(\varrho =\varrho _T\), called the root. Vertices of T, also called nodes, are denoted by \(N=N_T\) and edges by \(E=E_T\subset N^2\). 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 \(N_T = \varnothing \). 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 \(\bullet \). 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 \(\le \) where \(w \le v\) 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 \((x,y) = (x \rightarrow y) \in E_T\), then \(x \le y\). 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 \(\iota :E_T \rightarrow E_{T'}\) which is coherent in the sense that there exists a bijection \(\iota _N :N_T \rightarrow N_{T'}\) such that \(\iota (x,y) = (\iota _N(x),\iota _N(y))\) for any edge \((x,y) \in e\) 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 \(\mathfrak {t}:E_T \rightarrow \mathfrak {L}\), where \(\mathfrak {L}\) is some finite set of types. We think of \(\mathfrak {L}\) 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 \(\mathfrak {L}\) in our notations. Two typed trees \((T,\mathfrak {t})\) and \((T',\mathfrak {t}')\) are isomorphic if T and \(T'\) are isomorphic and \(\mathfrak {t}\) is pushed onto \(\mathfrak {t}'\) by the corresponding isomorphism \(\iota \) in the sense that \(\mathfrak {t}'\circ \iota = \mathfrak {t}\).
Similarly to a tree, a forestF is a finite simple graph (again with nodes \(N_F\) and edges \(E_F \subset N_F^2\)) without cycles. A forest F is rooted if every connected component T of F is a rooted tree with root \(\varrho _T\). As above, we will consider forests that are typed in the sense that they are endowed with a map \(\mathfrak {t}:E_F \rightarrow \mathfrak {L}\), and we consider the same notion of isomorphism between typed forests as for typed trees. Note that while a tree is nonempty by definition, a forest can be empty. We denote the empty forest by either \(\mathbf {1}\) or \(\varnothing \).
Given a typed forest F, a subforest \(A \subset F\) consists of subsets \(E_A \subset E_F\) and \(N_A \subset N_F\) such that if \((x,y) \in E_A\) then \(\{x,y\}\subset N_A\). Types in A are inherited from F. A connected component of A is a 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 \(A \cap B = \varnothing \), if one has \(N_A \cap N_B = \varnothing \) (which also implies that \(E_A\cap E_B = \varnothing \)). Given two typed forests F, G, we write \(F\sqcup G\) 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 \(A \subset F\) and \(B \subset G\) are subforests, then we write \(A \sqcup B\) for the corresponding subforest of \(F \sqcup G\).
We fix once and for all an integer \(d\ge 1\), dimension of the parameterspace \(\mathbf{R}^d\). We also denote by \(\mathbf{Z}(\mathfrak {L})\) the free abelian group generated by \(\mathfrak {L}\).
2.2 Coloured and decorated forests
Given a typed forest F, we want now to consider families of disjoint subforests of F, denoted by \((\hat{F}_i, i>0)\). It is convenient for us to code this family with a single function \(\hat{F}:E_F \sqcup N_F \rightarrow \mathbf{N}\) as given by the next definition.
Definition 2.3
 1.
\(F = (E_F,N_F,\mathfrak {t})\) is a typed rooted forest
 2.
\(\hat{F} :E_F \sqcup N_F \rightarrow \mathbf{N}\) is such that if \(\hat{F}(e) \ne 0\) for \(e=(x,y) \in E_F\) then \(\hat{F}(x) = \hat{F}(y) = \hat{F}(e)\).
The condition on \(\hat{F}\) guarantees that every \(\hat{F}_i\) is indeed a subforest of F for \(i>0\) and that they are all disjoint. On the other hand, \(\hat{F}^{1}(0)\) is not supposed to have any particular structure and 0 is not counted as a colour.
Example 2.4
We then have \(\hat{F}_1= \hat{F}^{1}(1) = A_1\sqcup A_3 \) and \(\hat{F}_2=\hat{F}^{1}(2) = A_2\sqcup A_4\).
We add now decorations on the nodes and edges of a coloured forest. For this, we fix throughout this article an arbitrary “dimension” \(d \in \mathbf{N}\) and we give the following definition.
Definition 2.5
 1.
\((F,\hat{F})\in \mathfrak {C}\) is a coloured forest in the sense of Definition 2.3.
 2.
One has \(\mathfrak {n}:N_F \rightarrow \mathbf{N}^d\)
 3.
One has \(\mathfrak {o}:N_F \rightarrow \mathbf{Z}^d\oplus \mathbf{Z}(\mathfrak {L})\) with \(\text {supp}\mathfrak {o}\subset \text {supp}\hat{F}\).
 4.
One has \(\mathfrak {e}:E_F \rightarrow \mathbf{N}^d\) with \(\text {supp}\mathfrak {e}\subset \{e\in E_F :\, \hat{F}(e)=0\}=E_F\setminus \hat{E}\).
Remark 2.6
The reason why \(\mathfrak {o}\) takes values in the space \(\mathbf{Z}^d\oplus \mathbf{Z}(\mathfrak {L})\) will become apparent in (3.33) below when we define the contraction of coloured subforests and its action on decorations.
We identify \((F,\hat{F},\mathfrak {n},\mathfrak {o},\mathfrak {e})\) and \((F',\hat{F}',\mathfrak {n}',\mathfrak {o}',\mathfrak {e}')\) whenever F is isomorphic to \(F'\), the corresponding isomorphism maps \(\hat{F}\) to \(\hat{F}'\) and pushes the three decoration functions onto their counterparts. We call elements of \(\mathfrak {F}\)decorated forests. We will also sometimes use the notation \((F,\hat{F})^{\mathfrak {n},\mathfrak {o}}_\mathfrak {e}\) instead of \((F,\hat{F},\mathfrak {n},\mathfrak {o},\mathfrak {e})\).
Example 2.7
In this figure, the edges in \(E_F\) are labelled with the numbers from 1 to 13 and the nodes in \(N_F\) with the letters \(\{a,b,c,f,e,f,g,h,i,j,k,l,m,p\}\). We set \(\hat{F}^{1}(1)=\{b,d,e,j,k\}\sqcup \{3,4,9\}\) (red subforest), \(\hat{F}^{1}(2)=\{a,c,f,g,l,m\}\sqcup \{2,5,6,11,12\}\) (blue subforest), and on all remaining (black) nodes and edges \(\hat{F}\) is set equal to 0. Every edge has a type \(\mathfrak {t}\in \mathfrak {L}\), but only black edges have a possibly nonzero decoration \(\mathfrak {e}\in \mathbf{N}^d\). All nodes have a decoration \(\mathfrak {n}\in \mathbf{N}^d\), but only coloured nodes have a possibly nonzero decoration \(\mathfrak {o}\in \mathbf{Z}^d\oplus \mathbf{Z}(\mathfrak {L})\).
Example 2.7 is continued in Examples 3.2, 3.4 and 3.5.
Definition 2.8
For any coloured forest \((F,\hat{F})\), we define an equivalence relation \(\sim \) on the node set \(N_F\) by saying that \(x \sim y\) if x and y are connected in \(\hat{E}\); this is the smallest equivalence relation for which \(x \sim y\) whenever \((x,y) \in \hat{E}\).
Definition 2.8 will be extended to a decorated forest \((F,\hat{F},\mathfrak {n},\mathfrak {o},\mathfrak {e})\) in Definition 3.18 below.
Remark 2.9
In this example the decoration \(\mathfrak {o}\) 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 nonempty connected components. This property naturally extends to decorated forests \((F, \hat{F},\mathfrak {n},\mathfrak {o},\mathfrak {e})\), by considering the connected components of the underlying forest F and restricting the colouring \(\hat{F}\) and the decorations \(\mathfrak {n},\mathfrak {o},\mathfrak {e}\).
Remark 2.11
Starting from Sect. 4 we are going to consider a specific situation where there are only two colours, namely \(\hat{F}\rightarrow \{0,1,2\}\); 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 setting \(\hat{F}\rightarrow \mathbf{N}\) without any additional difficulty.
2.3 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
Definition 2.13
Lemma 2.14
Proof
Let \(v=\sum _nv_n\in V\) and \(k\in \mathbf{N}\) such that \(v_n=0\) whenever \(n_2>k\).
First we note that, for fixed \(m\in \mathbf{N}^2\), the family \((A_{mn}v_n)_{n\in \mathbf{N}^2}\) is zero unless \(n\in [0,m_1]\times [0,k]\); indeed if \(n_2>k\) then \(v_n=0\), while if \(n_1>m_1\) then \(A_{mn}=0\). Therefore the sum \(\sum _n A_{mn}v_n\) is well defined and equal to some \(\bar{v}_m\in \bar{V}_m\).
We now prove that \(\bar{v}_m=0\) whenever \(m_2>k\), so that indeed Open image in new window . Let \(m_2> k\); for \(n_2>k\), \(v_n\) is 0, while for \(n_2\le k\) we have \(n_2< m_2\) and therefore \(A_{nm}=0\) and this proves the claim. \(\square \)
A linear function \(A:V\rightarrow \bar{V}\) which can be obtained as in Lemma 2.14 is called triangular. The family \((A_{mn})_{m,n\in \mathbf{N}^2}\) 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
 1.
The operators \(\Delta _i\) built in (3.7) below turn out to be triangular in the sense of Definition 2.13 and are therefore welldefined 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 nontrivial multiplicative functional at all. Considering instead spaces of finite series would cure this problem, but unfortunately the coproducts \(\Delta _i\) do not make sense there. The notion of bigrading introduced here provides the best of both worlds by considering biindexed 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 \(V^* \otimes W^* \subset (V {\hat{\otimes }}W)^*\), see e.g. (3.46) below for the applications we have in mind. Indeed, the dual \(V^*\) consists of formal sums \(\sum _n v^*_n\) with \(v_n^* \in V_n^*\) such that, for every \(k \in \mathbf{N}\) there exists f(k) such that \(v^*_n = 0\) for every \(n \in \mathbf{N}^2\) with \(n_1 \ge f(n_2)\).
The following simple fact will be used several times in the sequel. Here and throughout this article, we use as usual the notation \(f {\upharpoonright }A\) for the restriction of a map f to some subset A of its domain.
Lemma 2.17
3 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.

H is a vector space over \(\mathbf{R}\)

there are a linear map Open image in new window (product) and an element \(\mathbf {1}\in H\) (identity) such that Open image in new window is a unital associative algebra, where \(\eta :\mathbf{R}\rightarrow H\) is the map \(r\mapsto r\mathbf {1}\) (unit)
 there are linear maps \(\Delta :H\rightarrow H\otimes H\) (coproduct) and \(\mathbf {1}^\star :H\rightarrow \mathbf{R}\) (counit), such that \((H,\Delta ,\mathbf {1}^\star )\) is a counital coassociative coalgebra, namely$$\begin{aligned} (\Delta \otimes \mathrm {id})\Delta =(\mathrm {id}\otimes \Delta )\Delta , \qquad (\mathbf {1}^\star \otimes \mathrm {id})\Delta = (\mathrm {id}\otimes \mathbf {1}^\star )\Delta =\mathrm {id}\end{aligned}$$(3.1)

the coproduct and the counit are homomorphisms of algebras (or, equivalently, multiplication and unit are homomorphisms of coalgebras).
For more details on the theory of coalgebras, bialgebras, Hopf algebras and comodules we refer the reader to [6, 47].
3.1 Incidence coalgebras of forests
In order to define a generalisation of the operator \(\Delta \) of formula (3.3) to this setting, we fix \(i>0\) and assume the following.
Assumption 1
 1.
\(\hat{F}_i \subset A\) and \(\hat{F}_{j} \cap A=\varnothing \) for every \(j>i\),
 2.
for all \(0<j<i\) and every connected component T of \(\hat{F}_{j}\), one has either \(T \subset A\) or \(T \cap A=\varnothing \).
It is important to note that colours are denoted by positive integer numbers and are therefore ordered, so that the forests \(\hat{F}_j\), \(\hat{F}_i\) and \(\hat{F}_k\) 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

\(\hat{F}{\upharpoonright }A\) for the restriction of \(\hat{F}\) to \(N_A\sqcup E_A\)
 \(\hat{F} \cup _i A\) for the function on \(E_F\sqcup N_F\) given by$$\begin{aligned} (\hat{F} \cup _i A)(x) = \left\{ \begin{array}{ll} i &{} \text {if }x \in E_A \sqcup N_A, \\ \hat{F}(x) &{} \text {otherwise.} \end{array}\right. \end{aligned}$$
Proof
Example 3.2
In the rest of this section we state several assumptions on the family \(\mathfrak {A}_i(F,\hat{F})\) yielding nice properties for the operator \(\Delta _i\) 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 nontrivial action on the decorations which will be defined in the next subsection.
3.2 Operators on decorated forests
We generalise now the construction (3.4) to decorated forests.
Definition 3.3
 (a)
For \(A\subseteq B \subseteq F\) and \(f:E_F\rightarrow \mathbf{N}^d\), we use the notation Open image in new window .
 (b)
The sum over \(\mathfrak {n}_A\) runs over all maps \(\mathfrak {n}_A:N_F \rightarrow \mathbf{N}^d\) with \(\text {supp}\mathfrak {n}_A\subset N_A\).
 (c)The sum over \(\varepsilon _A^F\) runs over all \(\varepsilon _A^F:E_F \rightarrow \mathbf{N}^d\) supported on the set of edgesthat we call the boundary of A in F. This notation is consistent with point a).$$\begin{aligned} \partial (A,F) {\mathop {=}\limits ^{ \text{ def }}}\left\{ (e_+,e_) \in E_F\setminus E_{A} \,:\, e_+ \in N_A \right\} , \end{aligned}$$(3.8)
 (d)For all \(\varepsilon : E_F\rightarrow \mathbf{Z}^d\) we denote$$\begin{aligned} \pi \varepsilon :N_F\rightarrow \mathbf{Z}^d, \qquad \pi \varepsilon (x){\mathop {=}\limits ^{ \text{ def }}}\sum _{e=(x,y)\in E_F} \varepsilon (e). \end{aligned}$$
We will henceforth use these notational conventions for sums over node/edge decorations without always spelling them out in full.
Example 3.4
Example 3.5
Here we have that \(\partial (B,F)=\{7\}\), where we refer to the labelling of edges and nodes fixed in the Example 2.7. Therefore \(\pi \varepsilon _B^F(d)=\varepsilon _B^F(7)\). Note that the edge 8 was black in \((F,\hat{F})\) and becomes red in \((F,\hat{F} \cup _1 B)\); accordingly, in \((F,\hat{F} \cup _1 B,\mathfrak {n} \mathfrak {n}_B,\ \mathfrak {o}+\mathfrak {n}_B+\pi (\varepsilon _B^F\mathfrak {e}_\varnothing ^B),\mathfrak {e}_B^F + \varepsilon _B^F)\) the value of \(\mathfrak {e}\) on 8 is set to 0 and \(\mathfrak {e}(8)\) is subtracted from \(\mathfrak {o}(d)\). In accordance with Assumption 1, \(B\in \mathfrak {A}_1(F,\hat{F})\) is disjoint from the blue subforest \(\hat{F}_2\) and, accordingly, all decorations on \(\hat{F}_2\) are unchanged. Finally, note that the edge 1 is not in \(\partial (B,F)\) since it is equal to (a, b) with \(b\in B\) and \(a\notin B\).
Remark 3.6
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 \(\mathbf{R}^d\) indexed by the nodes of the underlying forest, then we can realise that the operator \(\Delta _i\) in (3.7) is naturally motivated by Taylor expansions.
Remark 3.8
Note that, in (3.7), for each fixed A the decoration \(\mathfrak {n}_A\) runs over a finite set because of the constraint \(0\le \mathfrak {n}_A\le \mathfrak {n}\).
There are many other ways of bigrading \(\mathfrak {F}\) to make the \(\Delta _i\) 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
Remark 3.10
3.3 Coassociativity
Assumption 2
 1.One has$$ \begin{aligned} \mathfrak {A}_i(F\sqcup G,\hat{F} + \hat{G}) = \{C \sqcup D \,:\, C \in \mathfrak {A}_i(F,\hat{F})\; \& \; D \in \mathfrak {A}_i(G,\hat{G})\}\;. \end{aligned}$$(3.14)
 2.One hasif and only if$$ \begin{aligned} A \in \mathfrak {A}_i(F,\hat{F})\quad \& \quad B \in \mathfrak {A}_i(F,\hat{F} \cup _i A)\;, \end{aligned}$$(3.15a)$$ \begin{aligned} B \in \mathfrak {A}_i(F,\hat{F})\quad \& \quad A \in \mathfrak {A}_i(B,\hat{F}{\upharpoonright }B). \end{aligned}$$(3.15b)
Assumption 2 is precisely what is required so that the “undecorated” versions of the maps \(\Delta _i\), 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
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).
3.4 Bialgebra structure
Fix throughout this section \(i>0\).
Definition 3.12
We also define \(\mathbf {1}_i^\star :\mathfrak {C}\rightarrow \mathbf{R}\) as Open image in new window .
Assumption 3
 1.
\(\{A,\hat{F}_i\}\subset \mathfrak {A}_i(A,\hat{F}{\upharpoonright }A)\)
 2.
if \(\hat{F} \le i\) then \(\{F, A\}\subset \mathfrak {A}_i(F,\hat{F}\cup _i A)\).
Lemma 3.13

\(({\mathrm {Vec}}(\mathfrak {C}_i), \cdot ,\Delta _i, \mathbf {1}, \mathbf {1}_i^\star )\) is a bialgebra

\(({\langle \mathfrak {F}_i\rangle }, \cdot ,\Delta _i, \mathbf {1}, \mathbf {1}_i^\star )\) is a bialgebra in the category of bigraded spaces as in Definition 2.12.
Proof
We consider only \(({\langle \mathfrak {F}_i\rangle }, \cdot ,\Delta _i, \mathbf {1}, \mathbf {1}_i^\star )\), since the other case follows in the same way. By the first part of Assumption 2, \(\mathfrak {F}_i\) is closed under the forest product, so that \(({\langle \mathfrak {F}_i\rangle }, \cdot , \mathbf {1})\) is indeed an algebra.
The required compatibility between the algebra and coalgebra structures is given by (3.16b), thus concluding the proof. \(\square \)
3.5 Contraction of coloured subforests and Hopf algebra structure
To formalise this, we introduce a contraction operator on coloured forests. Given a coloured forest \((F,\hat{F})\), we recall that \(\hat{E}\), defined in Definition 2.3, is the union of all edges in \(\hat{F}_j\) over all \(j>0\).
Definition 3.14
Note that in Open image in new window all nonempty coloured subforests are reduced to single nodes.
We are going to restrict our attention to collections \(\mathfrak {A}_i\) satisfying the following assumption.
Assumption 4
For all coloured forests \((F,\hat{F})\), the map Open image in new window is a bijection between Open image in new window and \(\mathfrak {A}_i(F,\hat{F})\).
We recall that we have defined in (3.4) the operator acting on linear combinations of coloured forests \((F,\hat{F})\mapsto \Delta _i(F,\hat{F})\). Then we have
Lemma 3.15
Proof
Example 3.16
Contraction of couloured subforests leads us closer to a Hopf algebra, but there is still a missing element. Indeed, an element like \((F,\hat{F})=(\bullet \sqcup \bullet ,1)\), namely two red isolated roots with no edge, is grouplike since it satisfies \(\Delta _1(F,\hat{F})=(F,\hat{F})\otimes (F,\hat{F})\) and therefore it can not admit an antipode, see the discussion after (3.31) above.

\(\nu \in \mathfrak {C}_i\) is the disjoint union of all nonempty connected componens of \(\tau \) of the form (A, i)

\(\mu \in \mathfrak {C}_i\) is the unique element such that \(\tau =\mu \cdot \nu \).
Then
Proposition 3.17
Proof
The first assertion follows from the fact that Open image in new window is an algebra morphism, and from Lemma 3.15.
For the second assertion, we note that the vector space \(\mathfrak {B}_i\) is isomorphic to \({\mathrm {Vec}}(C_i)\), where Open image in new window . Moreover Open image in new window as a bialgebra is isomorphic to Open image in new window , where Open image in new window denotes the forest product. The latter space is a Hopf algebra since it is a connected graded bialgebra with respect to the grading \((F,\hat{F})_i{\mathop {=}\limits ^{ \text{ def }}}F\setminus \hat{F}_i\), namely the number of nodes and edges which are not coloured with i. \(\square \)
We now extend the above construction to decorated forests.
Definition 3.18

if x is an equivalence class of \(\sim \) as in Definition 3.14, then \([\mathfrak {n}](x) = \sum _{y \in x} \mathfrak {n}(y)\).

\([\mathfrak {e}]\) is defined by simple restriction of \(\mathfrak {e}\) on \(E_F\setminus \hat{E}\).
 \([\mathfrak {o}](x)\) is defined by$$\begin{aligned}{}[\mathfrak {o}](x) {\mathop {=}\limits ^{ \text{ def }}}\sum _{y \in x} \mathfrak {o}(y) + \sum _{e\in E_F\cap x^2} \mathfrak {t}(e) . \end{aligned}$$(3.33)
The definition (3.33) explains why \(\mathfrak {o}\) is defined as a function taking values in \(\mathbf{Z}^d\oplus \mathbf{Z}(\mathfrak {L})\), see Remark 2.6 above.
Remark 3.19
The contraction of a subforest entails a loss of information. We use the decoration \(\mathfrak {o}\) in order to retain part of the lost information, namely the types of the edges which are contracted. This plays an important role in the degree \(\cdot _+\) introduced in Definition 5.3 below and is the key to one of the main results of this paper, see Remark 5.38.
Example 3.20
Note that the types \(\mathfrak {t}\) of edges which are erased by the contraction are stored inside the decoration \([\mathfrak {o}]\) of the corresponding node.
Compare this forest with that in (3.30), which belongs to \(\mathfrak {U}_1\); in (3.36) the decoration \(\mathfrak {n}\) can be nonzero, while it has to be identically zero in (3.30).

\(\nu \in \mathfrak {M}_i\) is the disjoint union of all nonempty connected componens of \(\tau \) of the form \((A,i, \mathfrak {n}, \mathfrak {o}, \mathfrak {e})\)

\(\mu \in \mathfrak {F}_i\) is the unique element such that \(\tau =\mu \cdot \nu \).
namely the red node which is not in the red connected component of the root is left unchanged.
Note that in Open image in new window the roots of the connected components which do not belong to \(\mathfrak {M}_2\) may have a nonzero \(\mathfrak {o}\) decoration, while the unique connected component in \(\mathfrak {M}_2\) (reduced to a blue root with a possibly nonzero \(\mathfrak {n}\) decoration) always has a zero \(\mathfrak {o}\) decoration. In Open image in new window all roots have zero \(\mathfrak {o}\) decoration.
Lemma 3.21
Proof
Remark 3.22
 1.
the coloured subforests \(\hat{F}_k\), \(0<k\le i\), contain no edges, namely \(\hat{E}=\varnothing \),
 2.
there is one and only one connected component of F which has the form \((\bullet ,i,\mathfrak {n},\mathfrak {o},0)\) and moreover \(\mathfrak {o}(\bullet )=0\).
Proposition 3.23
Under Assumptions 1–4, the space Open image in new window is a Hopf algebra.
Proof
By Lemma 3.13, \(\mathbf {1}_i^\star \) is a counit in Open image in new window . We only need now to show that this space admits an antipode Open image in new window , that we are going to construct recursively.
If \(\tau _i = 0\) then, by definition, one has \(\tau \in \mathfrak {M}_i\) so that \(\tau = X^k\) for some k and (3.44) defines Open image in new window . Let now \(N > 0\) and assume that Open image in new window has been defined for all Open image in new window with \(\tau _i < N\). Assume also that it is such that if \(\tau _{\mathrm {bi}}= m\), then Open image in new window only if \(n \ge m\), which is indeed the case for (3.44) since all the terms appearing there have degree (0, 0). (This latter condition is required if we want Open image in new window to be a triangular map.)
In the case where \(\mathfrak {n}\ne 0\), Open image in new window is also easily seen to be uniquely defined by performing a second inductive step over \(\mathfrak {n} \in \mathbf{N}\). All terms appearing in the right hand side of (3.45) do indeed satisfy that their total \(\cdot _{\mathrm {bi}}\)degree is at least M by using the induction hypothesis. Furthermore, our definition immediately guarantees that Open image in new window . It remains to verify that one also has Open image in new window . For this, it suffices to verify that Open image in new window 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.
3.6 Characters group
Definition 3.24

F has exactly one connected component

either \(\hat{F}\) is not identically equal to i or Open image in new window for some \(n \in \{1,\ldots ,d\}\), where \((\delta _n(\bullet ))_j=\delta _{nj}\).
It is then easy to see that for every \(\tau \in H_i\) there exists a unique (possibly empty) collection \(\{\tau _1,\ldots ,\tau _N\} \subset \mathfrak {P}_i\) such that Open image in new window . As a consequence, a multiplicative functional on Open image in new window is uniquely determined by the collection of values \(\{g(\tau )\,:\, \tau \in \mathfrak {P}_i\}\). The following result gives a complete characterisation of the class of functions \(g :\mathfrak {P}_i \rightarrow \mathbf{R}\) which can be extended in this way to a multiplicative functional on Open image in new window .
Proposition 3.25
A function \(g :\mathfrak {P}_i \rightarrow \mathbf{R}\) determines an element of Open image in new window as above if and only if there exists \(m :\mathbf{N}\rightarrow \mathbf{N}\) such that \(g(\tau ) = 0\) for every \(\tau \in \mathfrak {P}_i\) with \(\tau _{\mathrm {bi}}= n\) such that \(n_1 > m(n_2)\).
Proof
We first show that, under this condition, the unique multiplicative extension of g defines an element of Open image in new window . By Remark 2.16, we thus need to show that there exists a function \(\tilde{m}:\mathbf{N}\rightarrow \mathbf{N}\) such that \(g(\tau ) = 0\) for every \(\tau \in H_i\) with \(\tau _{\mathrm {bi}}= n\) and \(n_1 > \tilde{m}(n_2)\).
If \(\sigma =(F,\hat{F},\mathfrak {n},\mathfrak {o},\mathfrak {e}) \in \mathfrak {P}_i\) satisfies \(n_2 =0\), then \(\hat{F}\) is nowhere equal to 0 on F by the definition (2.4); by property 2 in Definition 2.3, \(\hat{F}\) is constant on F, since we also assume that F has a single connected component; in this case \(\mathfrak {e}\equiv 0\) by property 3 in Definition 2.5; therefore, if \(n_2=0\) then \(n_1=0\) as well. Therefore we can set \(\tilde{m}(0)=0\).
3.7 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 \(\Delta _i\) and \(\Delta _j\). For the definition of a comodule, see the beginning of Sect. 3.
Assumption 5
Let \(0<i < j\). For every coloured forest \((F,\hat{F})\) such that \(\hat{F} \le j\) and \(\{F,\hat{F}_j\} \subset \mathfrak {A}_j(F,\hat{F})\), one has \(\hat{F}_i \in \mathfrak {A}_i(F,\hat{F})\).
Lemma 3.26
Proof
Let \((F,\hat{F}, \mathfrak {n}, \mathfrak {o}, \mathfrak {e}) \in \mathfrak {F}_j\) and \(A\in \mathfrak {A}_i(F,\hat{F})\); by Definition 3.12, we have \(\hat{F}\le j\) and \(\{F,\hat{F}_j\}\subset \mathfrak {A}_j(F,\hat{F})\), so that by Assumption 5 we have \(\hat{F}_i\in \mathfrak {A}_i(F,\hat{F})\). Then, by property 1 in Assumption 3, we have \(\hat{F}_i\cap A=\hat{F}_i\in \mathfrak {A}_i(A,\hat{F}{\upharpoonright }A)\). Now, since \(A\cap \hat{F}_j=\varnothing \) by property 1 in Assumption 1, we have \((\hat{F}\cup _i A)_j=\hat{F}_j\setminus A=\hat{F}_j\in \mathfrak {A}_j(F,\hat{F}\cup _i A)\) by the Definition 3.12 of \(\mathfrak {F}_j\); all this shows that \(\Delta _i :{\langle \mathfrak {F}_{j}\rangle } \rightarrow {\langle \mathfrak {F}_i\rangle } {\hat{\otimes }}{\langle \mathfrak {F}_{j}\rangle }\).
For \(A\in \mathfrak {A}_i(F,\hat{F})\), we have \((A,\hat{F}{\upharpoonright }A,\mathfrak {n}',\mathfrak {o}',\mathfrak {e}')\in \mathfrak {U}_i\) if and only if \(\hat{F}\equiv i\) on A, i.e. \(A\subseteq \hat{F}_i\); since \(\hat{F}_i\subseteq A\) by Assumption 1, then the only possibility is \(A=\hat{F}_i\). By Assumption 5 we have \(\hat{F}_i \in \mathfrak {A}_i(F,\hat{F})\) and therefore \((\mathbf {1}_i^\star \otimes \mathrm {id})\Delta _i = \mathrm {id}\).
Finally, the coassociativity (3.16a) of \(\Delta _i\) on \(\mathfrak {F}\) shows the required compatibility between the coaction \(\Delta _i :{\langle \mathfrak {F}_{j}\rangle } \rightarrow {\langle \mathfrak {F}_i\rangle } {\hat{\otimes }}{\langle \mathfrak {F}_{j}\rangle }\) and the coproduct \(\Delta _i :{\langle \mathfrak {F}_{i}\rangle } \rightarrow {\langle \mathfrak {F}_i\rangle } {\hat{\otimes }}{\langle \mathfrak {F}_{i}\rangle }\). \(\square \)
We now introduce an additional structure which will yield as a consequence the cointeraction property (3.48) between the maps \(\Delta _i\) and \(\Delta _j\), see Remark 3.28.
Assumption 6
We then have the following crucial result.
Proposition 3.27
Proof
Remark 3.28
Let \(0<i < j\). If Assumptions 1–6 hold, then the space \({\langle \mathfrak {F}_{j}\rangle }\) is a comodule bialgebra over the bialgebra \({\langle \mathfrak {F}_i\rangle }\) with coaction \(\Delta _i\), in the sense of [47, Def 2.1(e)]. In the terminology of [25, Def. 1], \({\langle \mathfrak {F}_{j}\rangle }\) and \({\langle \mathfrak {F}_{i}\rangle }\) are in cointeraction.
Remark 3.29
Note that the roles of i and j are asymmetric for \(0<i < j\): \({\langle \mathfrak {F}_{i}\rangle }\) is in general not a comodule bialgebra over \({\langle \mathfrak {F}_j\rangle }\). This is a consequence of the asymmetry between the roles played by i and j in Assumption 1. In particular, every \(A\in \mathfrak {A}_i(F,\hat{F})\) has empty intersection with \(\hat{F}_j\), while any \(B\in \mathfrak {A}_j(F,\hat{F})\) can contain connected components of \(\hat{F}_i\).
3.8 Skew products and group actions
Proposition 3.30
The 5tuple Open image in new window is a Hopf algebra.
Proof
Let us recall that Open image in new window denotes the character group of Open image in new window .
Lemma 3.31
Proof
The dualization of the cointeraction property (3.48) yields that \(g(f_1f_2) = (g f_1)(g f_2)\), which means that this is indeed an action. \(\square \)
Proposition 3.32
Proof
Proposition 3.33
Proof
4 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 \(\mathfrak {A}_1\) and \(\mathfrak {A}_2\) as follows.
Definition 4.1
 1.
A contains \(\hat{F}_2\)
 2.
for every nonempty connected component T of F, \(T\cap A\) is connected and contains the root of T
 3.
for every connected component S of \(\hat{F}_1\), one has either \(S \subset A\) or \(S \cap A = \varnothing \).
Lemma 4.2

\(\tau \in \mathfrak {C}_1\) if and only if \(\hat{F} \le 1\)

\(\tau \in \mathfrak {C}_2\) if and only if \(\hat{F}\le 2\) and, for every nonempty connected component T of F, \(\hat{F}_2\cap T\) is a subtree of T containing the root of T.
Proof
Let \((F,\hat{F})\in \mathfrak {C}\). If \(\hat{F}\le 1\) then \(\hat{F}_2=\varnothing \) and therefore \(F\in \mathfrak {A}_1(F,\hat{F})\); moreover \(A=\hat{F}_1\) clearly satisfies \(\hat{F}_1\subset A\) and \(A\cap \hat{F}_2=\varnothing \), so that \(\hat{F}_1\in \mathfrak {A}_1(F,\hat{F})\) and therefore \((F,\hat{F})_\mathfrak {e}^{\mathfrak {n},\mathfrak {o}}\in \mathfrak {C}_1\). The converse is obvious.
Let us suppose now that \(\hat{F}\le 2\) and for every connected component T of F, \(\hat{F}_2\cap T\) is a subtree of T containing the root of T. Then \(A=F\) clearly satisfies the properties 13 of Definition 4.1. If now \(A=\hat{F}_2\), then A satisfies the properties 1 and 2 since for every nonempty connected component T of F, \(\hat{F}_2\cap T\) is a subtree of T containing the root of T, while property 3 is satisfied since \(\hat{F}_1\cap \hat{F}_2=\varnothing \). The converse is again obvious. \(\square \)
Example 4.3
because \(\hat{F}_2\) does not contain the root in the first case, and in the second \(\hat{F}_2\) has two disjoint connected components inside a connected component of F. The decorated forests (3.11), (3.30), (3.36) and (3.38) are in \(\mathfrak {C}_1\), while the decorated forests in (3.9), (3.10), (3.12), (3.34) and (3.35) are in \(\mathfrak {C}_2\).
Lemma 4.4
Proof
The first statement concerning \(\mathfrak {A}_1\) is elementary. The only nontrivial property to be checked about \(\mathfrak {A}_2\) is (3.15); note that \(\mathfrak {A}_2\) has the stronger property that for any two subtrees \(B \subset A \subset F\), one has \(A \in \mathfrak {A}_2(F,\hat{F})\) if and only if \(A \in \mathfrak {A}_2(F,\hat{F} \cup _2 B)\) and \(B \in \mathfrak {A}_2(F,\hat{F})\) if and only if \(B \in \mathfrak {A}_2(A,\hat{F} {\upharpoonright }A)\), so that property (3.15) follows at once.
Assumption 5 is easily seen to hold, since for every coloured forest \((F,\hat{F})\) such that \(\hat{F} \le 2\) and \(\{F,\hat{F}_2\} \subset \mathfrak {A}_2(F,\hat{F})\), for \(A{\mathop {=}\limits ^{ \text{ def }}}\hat{F}_1\) one has \(\hat{F}_1\subset A\) and \(\hat{F}_2\cap A=\varnothing \), so that \(\hat{F}_1 \in \mathfrak {A}_1(F,\hat{F})\).
We check now that \(\mathfrak {A}_1\) and \(\mathfrak {A}_2\) satisfy Assumption 6. Let \(A \in \mathfrak {A}_1(F,\hat{F})\) and \(B \in \mathfrak {A}_2(F, \hat{F} \cup _1 A)\); then \(A\cap \hat{F}_2=\varnothing \) and therefore \(B \in \mathfrak {A}_2(F,\hat{F})\); moreover every connected component of A is contained in a connected component of \(\hat{F}_1\) and therefore is either contained in B or disjoint from B, i.e. \(A \in \mathfrak {A}_1(F, \hat{F} \cup _2 B) \sqcup \mathfrak {A}_1(B, \hat{F} {\upharpoonright }B)\). Conversely, let \(B \in \mathfrak {A}_2(F,\hat{F})\) and \(A \in \mathfrak {A}_1(F, \hat{F} \cup _2 B) \sqcup \mathfrak {A}_1(B, \hat{F} {\upharpoonright }B)\); then \(\hat{F}_1=(\hat{F} \cup _2 B)_1\sqcup (\hat{F} {\upharpoonright }B)_1\) and \(\hat{F}_2\subset (\hat{F} \cup _2 B)_2\) so that A contains \(\hat{F}_1\) and is disjoint from \(\hat{F}_2\) and therefore \(A\in \mathfrak {A}_1(F,\hat{F})\); moreover \((\hat{F} \cup _1 A)_2\subseteq \hat{F}_2\) so that B contains \((\hat{F} \cup _1 A)_2\); finally \((\hat{F} \cup _1 A)_1=A\) and by the assumption on A we have that every connected component of \((\hat{F} \cup _1 A)_1\) is either contained in B or disjoint from B. The proof is complete. \(\square \)
In view of Propositions 3.17, 3.23 and 3.27, we have the following result.
Corollary 4.5
 1.
The space Open image in new window is a Hopf algebra and a comodule bialgebra over the Hopf algebra Open image in new window with coaction \(\Delta _1\) and counit \(\mathbf {1}^\star _1\).
 2.
The space Open image in new window is a Hopf algebra and a comodule bialgebra over the Hopf algebra Open image in new window with coaction \(\Delta _1\) and counit \(\mathbf {1}^\star _1\).
since by (3.32) the red node labelled k on the left side of the tensor product is killed by Open image in new window .
The operators \(\{\Delta _1,\Delta _2\}\) on the spaces Open image in new window act in the same way on the coloured subforests, and add the action on the decorations.
4.1 Joining roots
then we do not know how to define a colouring of \(\mathscr {J}(F)\) which is compatible with \(\hat{F}\). This justifies the definition of the subset Open image in new window as the set of all forests \((F,\hat{F},\mathfrak {n},\mathfrak {o},\mathfrak {e})\) such that \(\hat{F}(\varrho ) \in \{0,i\}\) for every root \(\varrho \) of F. We also write Open image in new window and Open image in new window for the set of forests such that every root has colour i.
Example 4.6
We can then extend \(\mathscr {J}\) to Open image in new window in a natural way as follows.
Definition 4.7
Example 4.8
Lemma 4.9
Proof
The spaces Open image in new window and Open image in new window are invariant under Open image in new window , \(\Phi _i\) and \(\hat{P}_i\) because these operations never change the colours of the roots. The invariance under \(\mathscr {J}\) follows in a similar way.
Proposition 4.10
Proof
The identity (4.5) fails to be true for \(\mathfrak {A}_1\) in general. However, if \((F, \hat{F}, \mathfrak {n}, \mathfrak {o}, \mathfrak {e}) \in \mathfrak {F}_2\), then each of the roots of F is covered by \(\hat{F}^{1}(2)\), so that (4.5) with \(\mathfrak {A}_2\) replaced by \(\mathfrak {A}_1\) does hold in this case. Furthermore, one then has a natural forest isomorphism between \(\mathscr {J}_F A\) 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. \(\square \)
Proposition 4.11
 1.
Open image in new window is a Hopf algebra and a comodule bialgebra over the Hopf algebra Open image in new window with coaction \(\Delta _1\) and counit \(\mathbf {1}^\star _1\).
 2.
Open image in new window is a Hopf algebra and a comodule bialgebra over the Hopf algebra Open image in new window with coaction \(\Delta _1\) and counit \(\mathbf {1}^\star _1\).
Proof
The Hopf algebra structure of Open image in new window turns Open image in new window 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 biideal of H such that H / I is commutative, then H / I is a Hopf algebra. For \(\hat{\mathfrak {B}}_2\), the same proof holds. \(\square \)
The second assertion in Proposition 4.11 is in fact the same result, just written differently, as [8, Thm 8]. Indeed, our space \(\mathfrak {B}_2\) is isomorphic to the ConnesKreimer Hopf algebra \({\mathcal H}_\mathrm{CK}\), and \(\mathfrak {B}_1\) is isomorphic to an extension of the extraction/contraction Hopf algebra \({\mathcal {H}}\). The difference between our \(\mathfrak {B}_1\) and \({\mathcal H}\) in [8] is that we allow extraction of arbitrary subforests, including with connected components reduced to single nodes; a subspace of \(\mathfrak {B}_1\) which turns out to be exactly isomorphic to \(\mathcal {H}\) is the linear space generated by coloured forests \((F,\hat{F})\in C_1\) such that \(N_F\subset \hat{F}_1\).
4.2 Algebraic renormalisation
Remark 4.12
The main reason why we do not define Open image in new window similarly to Open image in new window by setting Open image in new window is that \(\Delta _1\) is not welldefined on that quotient space, while it is welldefined on Open image in new window as given by (4.9), see Proposition 4.14.
Remark 4.13
We denote by Open image in new window the group of characters of Open image in new window and by Open image in new window the group of characters of Open image in new window .
Combining all the results we obtained so far, we see that we have constructed the following structure.
Proposition 4.14
 1.
Open image in new window is a left comodule over Open image in new window with coaction \(\Delta _1\) and counit \(\mathbf {1}^\star _1\).
 2.
Open image in new window is a left comodule over Open image in new window with coaction \(\Delta _1\) and counit \(\mathbf {1}^\star _1\).
 3.
Open image in new window is a right comodule algebra over Open image in new window with coaction \(\Delta _2\) and counit \(\mathbf {1}^\star _2\).
 4.Let Open image in new window . We define a left action of Open image in new window on Open image in new window by and a right action of Open image in new window on Open image in new window by Then we have(4.11)
Proof
The first, the second and the third assertions follow from the coassociativity of \(\Delta _1\), respectively \(\Delta _2\), proved in Proposition 3.11, combined with Proposition 4.10 to show that these maps are welldefined on the relevant quotient spaces. The multiplicativity of \(\Delta _2\) with respect to the tree product (4.8) follows from the first identity of Proposition 4.10, combined with the fact that Open image in new window is a quotient by \(\ker \mathscr {J}\).
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).
Remark 4.15
The action of \(\Delta _1\) on Open image in new window differs from the action on Open image in new window because of the following detail: Open image in new window is generated (as bigraded space) by a basis of rooted trees whose root is blue; since \(\Delta _1\) acts by extraction/contraction of subforests which contain \(\hat{F}_1\) and are disjoint from \(\hat{F}_2\), such subforests can never contain the root. Since on the other hand in Open image in new window and Open image in new window one has coloured forests with empty \(\hat{F}_2\), no such restriction applies to the action of \(\Delta _1\) on these spaces.
4.3 Recursive formulae
 1.
For \( k \in \mathbf{N}^d\), we write \(X^k\) as a shorthand for \((\bullet ,0)_{0}^{k,0} \in H_\circ \). We also interpret this as an element of Open image in new window , although its canonical representative there is \((\bullet ,2)_{0}^{k,0} \in \hat{H}_2\). As usual, we also write \(\mathbf {1}\) instead of \(X^0\), and we write \(X_i\) with \(i \in \{1,\ldots ,d\}\) as a shorthand for \(X^k\) with k equal to the ith canonical basis element of \(\mathbf{N}^d\).
 2.For every type \( \mathfrak {t}\in \mathfrak {L}\) and every \( k \in \mathbf{N}^d \), we define the linear operator(4.12)the root of G is \(\varrho _G\), the type of the edge \((\varrho _G,\varrho )\) is \(\mathfrak {t}\). For instance The decorations of Open image in new window , as well as \(\hat{G}\), coincide with those of \(\tau \), except on the newly added edge/vertex where \(\hat{G}\), \(\bar{\mathfrak {n}}\) and \(\bar{\mathfrak {o}}\) vanish, while \(\bar{\mathfrak {e}}(\varrho _G,\varrho ) = k\). This gives a triangular operator and Open image in new window is therefore well defined.$$\begin{aligned} N_G = N_F \sqcup \{\varrho _G\}\;,\qquad E_G = E_F \sqcup \{(\varrho _G,\varrho )\}\;, \end{aligned}$$
 3.Similarly, we define operators(4.13)
 4.
For \(\alpha \in \mathbf{Z}^d\oplus \mathbf{Z}(\mathfrak {L})\), we define linear triangular maps Open image in new window in such a way that if \(\tau = (T,\hat{T})_\mathfrak {e}^{\mathfrak {n},\mathfrak {o}}\in H_\circ \) with root \(\varrho \in N_T\), then Open image in new window coincides with \(\tau \), except for \(\mathfrak {o}(\varrho )\) to which we add \(\alpha \) and \(\hat{T}(\varrho )\) which is set to 1. In particular, one has Open image in new window .
Remark 4.16

Every element of \(H_\circ \setminus \{\mathbf {1}\}\) can be obtained from elements of the type \(X^k\) by successive applications of the maps Open image in new window , Open image in new window , and the tree product (4.8).

Every element of \(H_1\) is the forest product of a finite number of elements of \(H_\circ \).
 Every element of \(\hat{H}_2\) is of the form(4.14)
Then, one obtains a simple recursive description of the coproduct \(\Delta _2\).
Proposition 4.17
Proof
The operator \(\Delta _2\) is multiplicative on Open image in new window 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 finally have the following results on the antipode of Open image in new window :
Proposition 4.18
 The algebra morphism Open image in new window is defined uniquely by the fact that Open image in new window and for all Open image in new window with Open image in new window(4.17)
 On Open image in new window , one has the identity(4.18)
Proof
A similar proof by induction yields (4.18): see the proof of Lemma 6.5 for an analogous argument. Note that (4.18) is also a direct consequence of Proposition 3.27 and more precisely of the fact that the bialgebras Open image in new window and Open image in new window are in cointeraction, as follows from Remark 3.28: see [25, Prop. 2] for a proof. Having this property, the antipode Open image in new window is a morphism of the Open image in new window comodule Open image in new window . \(\square \)
In this section we have shown several useful recursive formulae that characterize \(\Delta _2\), 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 \(\Delta _1\), which is however more complex than that for \(\Delta _2\).
5 Rules and associated regularity structures
We recall the definition of a regularity structure from [32, Def. 2.1]
Definition 5.1

An index set \(A \subset \mathbf{R}\) such that A is bounded from below, and A is locally finite.

A model spaceT, which is a graded vector space \(T = \bigoplus _{\alpha \in A} T_\alpha \), with each \(T_\alpha \) a Banach space.
 A structure groupG of linear operators acting on T such that, for every \(\Gamma \in G\), every \(\alpha \in A\), and every \(a \in T_\alpha \), one has$$\begin{aligned} \Gamma a  a \in \bigoplus _{\beta < \alpha } T_\beta \;. \end{aligned}$$(5.1)
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 realvalued degrees to each element of \(\mathfrak {F}\).
Definition 5.2
Definition 5.3
Note that both of these degrees are compatible with the contraction operator Open image in new window of Definition 3.18, as well as the operator \(\mathscr {J}\), in the sense that \(\tau _\pm = \bar{\tau }_\pm \) if and only if Open image in new window and similarly for \(\mathscr {J}\). In the case of \(\cdot _+\), this is true thanks to the definition (3.33), while the coloured part of the tree is simply ignored by \(\cdot _\). We furthermore have
Lemma 5.4
The degree \(\cdot _\) is compatible with the operators Open image in new window and Open image in new window of (3.39), while \(\cdot _+\) is compatible with Open image in new window and Open image in new window . Furthermore, both degrees are compatible with \(\mathscr {J}\) and Open image in new window , so that in particular Open image in new window is \(\cdot _\)graded and Open image in new window and Open image in new window are both \(\cdot _\) and \(\cdot _+\)graded.
Proof
The first statement is obvious since \(\cdot _\) ignores the coloured part of the tree, except for the labels \(\mathfrak {n}\) whose total sum is preserved by all these operations. For the second statement, we need to verify that \(\cdot _+\) is compatible with \(\hat{\Phi }_2\) as defined just below (3.37). which is the case when acting on a tree with \(\varrho \in \hat{F}_2\) since the \(\mathfrak {o}\)decoration of nodes in \(\hat{F}_2\) does not contribute to the definition of \(\cdot _+\). \(\square \)
 1.
The action of Open image in new window on Open image in new window is not of the form “identity plus terms of strictly lower degree”, as required for regularity structures.
 2.
The possible degrees appearing in Open image in new window have no lower bound and might have accumulation points.
5.1 Trees generated by rules
Definition 5.5
We say that an element \(T_\mathfrak {e}^{\mathfrak {n}}\in \mathfrak {T}\) is trivial if T consists of a single node \(\bullet \). It is planted if T has exactly one edge incident to its root \(\varrho \) and furthermore \(\mathfrak {n}(\varrho ) = 0\).
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.
Remark 5.6
The fact that we consider multisets and not just nuples 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 \(A \subset B\), we have a natural inclusion Open image in new window . We will usually write elements of Open image in new window as nuples with the understanding that this is just an arbitrary representative of an equivalence class. In particular, we write () for the unique element of Open image in new window .
Definition 5.7
Denoting by Open image in new window the powerset of Open image in new window , a rule is a map Open image in new window . A rule is said to be normal if, whenever \(M \subset N \in R(\mathfrak {t})\), one also has \(M \in R(\mathfrak {t})\).
Then, according to the rule R, an edge of type \(\mathfrak {t}_1\) or \(\mathfrak {t}_2\) can be followed in a tree by, respectively, no edge, or a single edge of type \(\mathfrak {t}_i\) with decoration \(\mathfrak {e}_i\) with \(i\in \{1,2\}\), or by two edges, one of type \(\mathfrak {t}_1\) with decoration \(\mathfrak {e}_1\) and one of type \(\mathfrak {t}_2\) with decoration \(\mathfrak {e}_2\). We do not expect however to find two edges both of type \(\mathfrak {t}_1\) (or \(\mathfrak {t}_2\)) sharing a node which is not the root.
Definition 5.8

\(\tau \)conforms toR at the vertex x if either x is the root and there exists \(\mathfrak {t}\in \mathfrak {L}\) such that Open image in new window or one has Open image in new window , where e is the unique edge linking x to its parent in T.

\(\tau \)conforms to R if it conforms to R at every vertex x, except possibly its root.

\(\tau \)strongly conforms to R if it conforms to R at every vertex x.
In particular, the trivial tree \(\bullet \) strongly conforms to every normal rule since, as a consequence of Definition 5.7, there exists at least one \(\mathfrak {t}\in \mathfrak {L}\) with \(() \in R(\mathfrak {t})\).
Example 5.9
The first tree does not conform to the rule R since the bottom left edge of type \( \mathfrak {t}_2\) 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 \(\varrho _i\) the root of the ith tree, then we have Open image in new window , Open image in new window , Open image in new window , see (5.6). 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 \(()\in R(\mathfrak {t})\) for every \(\mathfrak {t}\in \mathfrak {L}\). This guarantees that \(\mathfrak {L}\) contains no useless labels in the sense that, for every \(\mathfrak {t}\in \mathfrak {L}\), there exists a tree conforming to R containing an edge of type \(\mathfrak {t}\): it suffices to consider a rooted tree with a single edge \(e=(x,y)\) of type \(\mathfrak {t}\); in this case, Open image in new window . More importantly, this also guarantees that we can build any tree conforming to R from the root upwards (start with an edge of type \(\mathfrak {t}\), add to it a node of some type in \(R(\mathfrak {t})\), 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 Open image in new window as follows. Take as the vertex set Open image in new window . Then, connect Open image in new window to Open image in new window if \(t \in N\). If t is contained in N multiple times, repeat the connection the corresponding number of times. Conversely, connect Open image in new window to Open image in new window if \(N \in R(\mathfrak {t})\). The conditions then guarantee that Open image in new window can be reached from every vertex in the graph. Given a tree \(\tau \in \mathfrak {T}\), every edge of \(\tau \) corresponds to an element of Open image in new window and every node corresponds to an element of Open image in new window via the map Open image in new window 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 Open image in new window . It strongly conforms to R if the root corresponds to a vertex in V with at least one incoming edge.
Definition 5.12
This definition is compatible with both notions of degree given in Definition 5.3, since we view \(\mathfrak {T}\) as a subset of \(\mathfrak {F}\) with \(\hat{F}\) and \(\mathfrak {o}\) identically 0. This also allows us to give the following definition.
Definition 5.13

\(\mathfrak {T}_\circ (R) \subset \mathfrak {T}\) for the set of trees that strongly conform to R

\(\mathfrak {T}_1(R) \subset \mathfrak {F}\) for the submonoid of \(\mathfrak {F}\) (for the forest product) generated by \(\mathfrak {T}_\circ (R)\)

\(\mathfrak {T}_2(R) \subset \mathfrak {T}\) for the set of trees that conform to R.

\(\tau _\mathfrak {s}< 0\), \(\mathfrak {n}(\varrho _\tau ) = 0\),

if \(\tau \) is planted, namely Open image in new window with \(\bar{\tau }\in \mathfrak {T}\), see (4.12), then \(\mathfrak {t}_\mathfrak {s}< 0\).
The second restriction on the definition of \(\tau \in \mathfrak {T}_(R)\) is related to the definition (5.22) of the Hopf algebra Open image in new window and of its characters group Open image in new window , that we call the renormalisation group and which plays a fundamental role in the theory, see e.g. Theorem 6.16.
5.2 Subcriticality
Definition 5.14
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.
Proposition 5.15
If R is a subcritical rule, then, for every \(\gamma \in \mathbf{R}\), the set \(\{\tau \in \mathfrak {T}_\circ (R)\,:\, \tau _\mathfrak {s}\le \gamma \}\) is finite.
Proof
Remark 5.16
The inequality (5.10) encodes the fact that we would like to be able to assign a regularity \({\mathrm {reg}}(\mathfrak {t})\) to each component \( u_{\mathfrak {t}} \) of our SPDE in such a way that the “naïve regularity” of the corresponding right hand side obtained by a powercounting argument is strictly better than \({\mathrm {reg}}(\mathfrak {t})  \mathfrak {t}\). Indeed, \(\inf _{N \in R(\mathfrak {t})} {\mathrm {reg}}(N) \) is precisely the regularity one would like to assign to \( F_{\mathfrak {t}}(u,\nabla u,\xi ) \). 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
5.3 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 Open image in new window and Open image in new window constructed in Sect. 4 leave the linear span of \(\mathfrak {T}_\circ (R)\) invariant. We now introduce a notion of completeness, which will guarantee later on that the actions of Open image in new window and Open image in new window do indeed leave the span of \(\mathfrak {T}_\circ (R)\) (or rather an extension of it involving again labels \(\mathfrak {o}\) 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.
Definition 5.18
Given Open image in new window and \(m \in \mathbf{N}^d\), we define Open image in new window as the set of all ntuples of the form \(((\mathfrak {t}_1,k_1+m_1),\ldots ,(\mathfrak {t}_n,k_n+m_n))\) where the \(m_i\in \mathbf{N}^d\) are such that \(\sum _i m_i = m\).
Definition 5.19
It is easy to see that, if we explore the tree from the leaves down, this specifies Open image in new window and Open image in new window uniquely for all edges and nodes of T.
Definition 5.20
At first sight, the notion of \(\ominus \)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 rule \(\bar{R}\) which is \(\ominus \)complete and extends R in the sense that \(R(\mathfrak {t}) \subset \bar{R}(\mathfrak {t})\) for every \(\mathfrak {t}\in \mathfrak {L}\).
Proof
It remains to show that \(\bar{R}\) is again normal and subcritical. To show normality, we note that if R is normal, then Open image in new window is again normal. This is because, by Definition 5.19, the sets Open image in new window used to build Open image in new window also have the property that if Open image in new window and \(M \subset N\), then one also has Open image in new window . As a consequence, Open image in new window is normal for every n, from which the normality of \(\bar{R}\) follows.
We conclude that (5.16) also holds when considering Open image in new window , thus yielding the desired claim. Iterating this, we conclude that \({\mathrm {reg}}_\kappa \) satisfies (5.10) for each of the rules Open image in new window and therefore also for \(\bar{R}\) as required. \(\square \)
Definition 5.22
We say that a subcritical rule R is complete (with respect to a fixed scaling \(\mathfrak {s}\)) if it is both normal and \(\ominus \)complete. If R is only normal, we call the rule \(\bar{R}\) constructed in the proof of Proposition 5.21 the completion of R.
5.4 Three prototypical examples
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 \(\mathfrak {s}= (2,1)\) and then assign to \(\Xi \) a degree just below the exponent of selfsimilarity of white noise under the scaling \(\mathfrak {s}\), namely \(\Xi _\mathfrak {s}= {3\over 2}  \kappa \) for some small \(\kappa > 0\). 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 Open image in new window .
5.5 Regularity structures determined by rules

a finite type set \(\mathfrak {L}\) together with a scaling \(\mathfrak {s}\) and degrees \(\cdot _\mathfrak {s}\) as in Definition 5.2,

a normal rule R for \(\mathfrak {L}\) which is both subcritical and complete, in the sense of Definition 5.22,

the integer \(d\ge 1\) which has been fixed at the beginning of the paper.
Definition 5.23

\(\sigma =(F,\mathfrak {n},\mathfrak {e})\in \mathfrak {T}\)

\(A\subset F\) is a subtree such that \(\sigma \) conforms to the rule R at every node \(y \in A\)

functions \(\mathfrak {n}_A :N_A \rightarrow \mathbf{N}^d\) with \(\mathfrak {n}_A\le \mathfrak {n}{\upharpoonright }N_A\) and \(\varepsilon _A^F:\partial (A,F)\rightarrow \mathbf{N}^d\)
We define Open image in new window by Open image in new window .
Definition 5.24

One has \(\hat{F}(x) \in \{0,2\}\) and \(\mathfrak {o}(x)=0\).

One has \(\hat{F}(x) = 1\) and Open image in new window .
Lemma 5.25
Conversely, every element \(\tau \) of \(\Lambda \) is of the form (5.21) for an element \(\sigma \) with \(\hat{F}(x) \in \{0,2\}\) and \(\mathfrak {o}\equiv 0\).
Proof
Now the first assertion follows easily from the second one. \(\square \)
We now define spaces of coloured forests \(\tau = (F,\hat{F}, \mathfrak {n},\mathfrak {o},\mathfrak {e})\) such that \((F,0,\mathfrak {n},0,\mathfrak {e})\) is compatible with the rule R in a suitable sense, and such that \(\tau \in \Lambda \).
Definition 5.26
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 of Open image in new window , Open image in new window and Open image in new window is done via the canonical basis (4.10).
Note that both Open image in new window and Open image in new window are algebras for the products inherited from Open image in new window and Open image in new window respectively. On the other hand, Open image in new window is in general not an algebra anymore.
Lemma 5.28
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 nonconforming trees in the case of \(\Delta _1\), while, since \(\Delta _2\) extracts only subtrees at the root, completeness of the rule implies that this can not happen in the case of \(\Delta _2\), thus showing that the maps \(\Delta _i\) do indeed behave as claimed.
The fact that Open image in new window is in fact a Hopf algebra, namely that the antipode Open image in new window of Open image in new window leaves Open image in new window invariant, can be shown by induction using (4.17) and Remark 4.16. \(\square \)
Note that Open image in new window is a subalgebra but in general not a subcoalgebra of Open image in new window (and a fortiori not a Hopf algebra). Recall also that, by Lemma 5.4, the grading \(\cdot _\) of Definition 5.3 is well defined on Open image in new window and on Open image in new window , and that \(\cdot _+\) is well defined on both Open image in new window and Open image in new window . Furthermore, these gradings are preserved by the corresponding products and coproducts.
Definition 5.29
With these definitions at hand, it turns out that the map \((\mathfrak {p}_^\mathrm {ex}\otimes \mathrm {id})\Delta _1\) is much better behaved. Indeed, we have the following.
Lemma 5.30
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 \(\tau \in \mathfrak {T}_(R)\), which in particular means that \(\tau _\mathfrak {s}=\tau _<0\). \(\square \)
Analogously to Lemma 3.21 we have
Lemma 5.31
Proof

either A does not contain the edge incident to the root of \(\tau \), 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.
As a corollary, we have the following.
Corollary 5.32
Remark 5.33
The operators \(\Delta _\mathrm {ex}^{\pm }\) of Corollary 5.32 are now given by finite sums so that for all of these choices of Open image in new window , the operators \(\Delta ^{\!}_\mathrm {ex}\) and \(\Delta ^{\!+}_\mathrm {ex}\) actually map Open image in new window into Open image in new window and Open image in new window respectively.
Proposition 5.34
There exists an algebra morphism Open image in new window so that Open image in new window , where Open image in new window is the tree product (4.8), is a Hopf algebra. Moreover the map Open image in new window , turns Open image in new window into a right comodule for Open image in new window with counit \(\mathbf {1}^\star _2\).
Proof
We already know that Open image in new window is a Hopf subalgebra of Open image in new window with antipode Open image in new window satisfying (4.17). Since Open image in new window is a bialgebra ideal by Lemma 5.31, the first claim follows from [48, Thm 1.(iv)].
The fact that Open image in new window is a coaction and turns Open image in new window into a right comodule for Open image in new window follows from the coassociativity of \(\Delta _2\). \(\square \)
Proposition 5.35
There exists an algebra morphism Open image in new window so that Open image in new window is a Hopf algebra. Moreover the map Open image in new window turns Open image in new window into a left comodule for Open image in new window with counit \(\mathbf {1}^\star _1\).
Proof
Definition 5.36
We call Open image in new window the character group of Open image in new window .
We have therefore obtained the following analogue of Proposition 4.14:
Theorem 5.37
 1.On Open image in new window , we have the identity(5.26)
 2.Let Open image in new window . We define a left action of Open image in new window on Open image in new window by and a right action of Open image in new window on Open image in new window by Then we have(5.27)
Proof
Formula (5.26) yields the cointeraction property see Remark 3.28.
Remark 5.38
We can finally see here the role played by the decoration \(\mathfrak {o}\): 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 \(\cdot _+\) takes the decoration \(\mathfrak {o}\) into account, and this is what makes the second identity of (5.25) true. See also Remark 6.26 below.
We define \(A^\mathrm {ex}{\mathop {=}\limits ^{ \text{ def }}}\{\tau _+:\tau \in B_\circ \}\), where Open image in new window as in Definition 5.26.
Proposition 5.39
The above construction yields a regularity structure Open image in new window in the sense of Definition 5.1.
Proof
By the definitions, every element \(\tau \in B_\circ \) has a representation of the type (5.21) for some \(\sigma = (T,0,\mathfrak {n},0,\mathfrak {e}) \in \mathfrak {T}\). Furthermore, it follows from the definitions of \(\cdot _+\) and \(\cdot _\mathfrak {s}\) that one has \(\tau _+ = \sigma _\mathfrak {s}\). The fact that, for all \(\gamma \in \mathbf{R}\), the set \(\{a\in A^\mathrm {ex}: a\le \gamma \}\) is finite then follows from Proposition 5.15.
The space Open image in new window is graded by \(\cdot _+\) and Open image in new window acts on it by \(\Gamma _g{\mathop {=}\limits ^{ \text{ def }}}(\mathrm {id}\otimes g)\Delta ^{\!+}_\mathrm {ex}\). The property (5.1) then follows from the fact \(\Delta ^{\!+}_\mathrm {ex}\) preserves the total \(\cdot _+\)degree by the third identity in (5.25) and all terms appearing in the second factor of \(\Delta ^{\!+}_\mathrm {ex}\tau  \tau \otimes \mathbf {1}\) have strictly positive \(\cdot _+\)degree by Definition 5.29. \(\square \)
Remark 5.40
Since Open image in new window is finitely generated as an algebra (though infinitedimensional as a vector space), its character group Open image in new window is a finitedimensional Lie group. In contrast, Open image in new window is not finitedimensional but can be given the structure of an infinitedimensional Lie group, see [5].
6 Renormalisation of models
Remark 6.1
Labels in \(\mathfrak {L}_+\) represent “kernels” while labels in \(\mathfrak {L}_+\) represent “noises”, which naturally leads to (6.1). (We could actually have defined \(\mathfrak {L}_\) by \(\mathfrak {L}_ = \{\mathfrak {t}\,:\, R(\mathfrak {t}) = \{()\}\}\).) The condition that elements of \(\mathfrak {L}_\) are of negative degree and those in \(\mathfrak {L}_+\) are of positive degree is also natural in this context. It could in principle be weakened, which corresponds to allowing kernels with a nonintegrable 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.
Proposition 6.2
 Open image in new window is defined uniquely by the fact that Open image in new window and for all Open image in new window(6.3)
 On Open image in new window , one has the identity(6.4)
6.1 Twisted antipodes
Proposition 6.3
Proof
Proceeding by induction over the number of edges appearing in \(\tau \), one easily verifies that such a map exists and is uniquely determined by the above properties. \(\square \)
Comparing this to the recursion for Open image in new window given in (6.3), we see that they are very similar, but the projection \(\mathfrak {p}_+^\mathrm {ex}\) in (6.3) is inside the multiplication Open image in new window , while \(P_+\) in (6.5) is outside Open image in new window .
Proposition 6.4
Proof
A very useful property of the positive twisted antipode Open image in new window is that its action is intertwined with that of \(\Delta ^{\!}_\mathrm {ex}\) in the following way.
Lemma 6.5
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 Open image in new window . The identity clearly holds on the linear span of \(X^k\) since \(\Delta ^{\!}_\mathrm {ex}\) acts trivially on them and Open image in new window preserves that subspace.
We have now a similar construction of a negative twisted antipode.
Proposition 6.6
6.2 Models
Definition 6.7

A map \(\Gamma :\mathbf{R}^d\times \mathbf{R}^d \rightarrow G\) such that \(\Gamma _{xx} = \mathrm {id}\), the identity operator, and such that \(\Gamma _{xy}\, \Gamma _{yz} = \Gamma _{xz}\) for every x, y, z in \(\mathbf{R}^d\).

A collection of continuous linear maps Open image in new window such that \(\Pi _y = \Pi _x \circ \Gamma _{xy}\) for every \(x,y \in \mathbf{R}^d\).
Here, recalling that the space T in Definitions 5.1 and 6.7 is a direct sum of Banach spaces \((T_\alpha )_{\alpha \in A}\), the quantity \(\Vert \sigma \Vert _m\) appearing in (6.11) denotes the norm of the component of \(\sigma \in T\) in the Banach space \(T_m\) for \(m\in A\). We also note that Definition 6.7 does not include the general framework of [32, Def. 2.17], where \(\Pi _x\) takes values in Open image in new window rather than Open image in new window ; however this simplified setting is sufficient for our purposes, at least for now. The condition (6.11) on \(\Pi _x\) is of course relevant only for \(\ell >0\) since \(\Pi _x \tau (\cdot )\) is assumed to be a smooth function at this stage.
Recall that we fixed a label set \(\mathfrak {L}= \mathfrak {L}_ \sqcup \mathfrak {L}_+\). We also fix a collection of kernels \(\{K_{\mathfrak {t}}\}_{\mathfrak {t}\in \mathfrak {L}_+}\), \(K_{\mathfrak {t}}:\mathbf{R}^d\setminus \{0\}\rightarrow \mathbf{R}\), satisfying the conditions of [32, Ass. 5.1] with \(\beta = \mathfrak {t}_\mathfrak {s}\). We use extensively the notations of Sect. 4.3.
Definition 6.8
 a character Open image in new window by extending multiplicatively for \(\mathfrak {t}\in \mathfrak {L}_+\) and setting Open image in new window for \(\mathfrak {l}\in \mathfrak {L}_\).
 a linear map Open image in new window and a character Open image in new window by(6.12)
We do not want to consider arbitrary maps \(\varvec{\Pi }\) as above, but we want them to behave in a “nice” way with respect to the natural operations we have on Open image in new window . We therefore introduce the following notion of admissibility. For this, we note that, as a consequence of (6.1), the only basis vectors of the type Open image in new window with \(\mathfrak {t}\in \mathfrak {L}_\) belonging to Open image in new window are those with \(\tau = X^\ell \) for some \(\ell \in \mathbf{N}^d\), so we give them a special name by setting Open image in new window and \(\Xi ^\mathfrak {l}= \Xi _{0,0}^\mathfrak {l}\).
Definition 6.9
It is then simple to check that, with these definitions, \(\Pi _z \Gamma _{\!z\bar{z}} = \Pi _{\bar{z}}\) and \((\Pi ,\Gamma )\) satisfies the algebraic requirements of Definition 6.7. However, \((\Pi ,\Gamma )\) does not necessarily satisfy the analytical bounds (6.11), although one has the following.
Lemma 6.10
Proof
It follows immediately from (4.16) and the admissibility of \(\varvec{\Pi }\) that Open image in new window is a polynomial of degree Open image in new window . On the other hand, it follows from (6.7) that Open image in new window and its derivatives up to the required order (because taking derivatives commutes with the action of the structure group) vanish at z, so there is no choice of what that polynomial is, thus yielding the second identity. The first identity then follows by comparing the second formula to (6.12). \(\square \)
Remark 6.11
Lemma 6.10 shows that the positive twisted antipode Open image in new window is intimately related to Taylor remainders, see Remark 3.7 and (6.12).
Lemma 6.10 shows that \((\Pi ,\Gamma )\) satisfies the analytical property (6.11) on planted trees of the form Open image in new window . However this is not necessarily the case for products of such trees, since neither \(\varvec{\Pi }\) nor \(\Pi _z\) are assumed to be multiplicative under the tree product (4.8). If, however, we also assume that \(\varvec{\Pi }\) is multiplicative, then the map Open image in new window always produces a bona fide model.
Proposition 6.12
Proof
The proof of the second bound in (6.11) for \(\Gamma _{xy}\) 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 \(\cdot _+\)degree. \(\square \)
Remark 6.13
If a map Open image in new window is admissible and furthermore satisfies (6.16), then it is uniquely determined by the functions \(\xi _\mathfrak {l}{\mathop {=}\limits ^{ \text{ def }}}\varvec{\Pi }\Xi ^\mathfrak {l}\) for \(\mathfrak {l}\in \mathfrak {L}_\). In this case, we call \(\varvec{\Pi }\) the canonical lift of the functions \(\xi _\mathfrak {l}\).
6.3 Renormalised Models
Definition 6.14
We denote by \(\mathscr {M}^\mathrm {ex}_{\infty }\) the space of all smooth models of the form Open image in new window for some admissible linear map Open image in new window in the sense of Definition 6.9. We endow \(\mathscr {M}^\mathrm {ex}_{\infty }\) with the system of pseudometrics \((\!\!\cdot ; \cdot \!\!_{\ell ;\mathfrak {K}})_{\ell ;\mathfrak {K}}\) and we denote by \(\mathscr {M}^\mathrm {ex}_0\) the completion of this metric space.
We refer to [32, Def. 2.17] for the definition of the space \(\mathscr {M}^\mathrm {ex}\) of models of a fixed regularity structure. With that definition, \(\mathscr {M}^\mathrm {ex}_0\) is nothing but the closure of \(\mathscr {M}^\mathrm {ex}_{\infty }\) in \(\mathscr {M}^\mathrm {ex}\).
In many singular SPDEs, one is naturally led to a sequence of models Open image in new window which do not converge as \(\varepsilon \rightarrow 0\). One would then like to be able to “tweak” this model in such a way that it remains an admissible model but has a chance of converging as \(\varepsilon \rightarrow 0\). A natural way of “tweaking” \(\varvec{\Pi }^{(\varepsilon )}\) is to compose it with some linear map Open image in new window . This naturally leads to the following question: what are the linear maps \(M^{\mathrm {ex}}\) which are such that if Open image in new window is an admissible model, then Open image in new window is also a model? We then give the following definition.
Definition 6.15

for every admissible Open image in new window such that Open image in new window , \(\varvec{\Pi }M\) is admissible and Open image in new window

the map Open image in new window extends to a continuous map from \(\mathscr {M}^\mathrm {ex}_0\) to \(\mathscr {M}^\mathrm {ex}_0\).
Theorem 6.16
Proof
 1.
A is a subforest of \(\tau \)
 2.
A contains the edge of type \(\mathfrak {t}\) added by the operator Open image in new window or the root of Open image in new window as an isolated node (which has however positive degree and is therefore killed by the projection \(\mathfrak {p}^\mathrm {ex}_\) in \(\Delta ^{\!}_\mathrm {ex}\)).
The exact same argument also shows that if we extend the action of Open image in new window to all of \(\mathscr {M}^\mathrm {ex}\) by (6.20) and (6.21), then this yields a continuous action, which in particular leaves \(\mathscr {M}^\mathrm {ex}_0\) invariant as required by Definition 6.15. \(\square \)
Definition 6.17
We say that a random linear map Open image in new window is stationary if, for every (deterministic) element \(h \in \mathbf{R}^d\), the random linear maps \(T_h(\varvec{\Pi })\) and \(\tilde{T}_h(\varvec{\Pi })\) are equal in law. We also assume that \(\varvec{\Pi }\) and its derivatives, computed at 0 have moments of all orders.
Let us also denote by \(B_{\circ }^\) the (finite!) set of basis vectors \(\tau \in B_\circ \) such that \(\tau _ < 0\). The specific choice of Open image in new window used to define Open image in new window is very natural and canonical in the following sense.
Theorem 6.18
Proof
It remains to show that \(\hat{\varvec{\Pi }}\) is the only function of the type \(\varvec{\Pi }^g\) with this property. For this, note that every such function is also of the form Open image in new window for some different Open image in new window , so that we only need to show that for every element g different from the identity, there exists \(\tau \) such that Open image in new window .
Using Definitions 5.26 and 5.29, Remark 4.16 and the identification (6.2), Open image in new window can be canonically identified with the free algebra generated by \(B_\circ ^\sharp \). Therefore the character g is completely characterised by its evaluation on \(B_\circ ^\sharp \) and it is the identity if and only if this evaluation vanishes identically. Fix now such a g different from the identity and let \(\tau \in B_\circ ^\sharp \) be such that \(g(\tau ) \ne 0\), and such that \(g(\sigma ) = 0\) for all \(\sigma \in B_\circ ^\sharp \) with the property that either \(\sigma _ < \tau _\) or \(\sigma _ = \tau _\), but \(\sigma \) has strictly less colourless edges than \(\tau \). Since \(B_\circ ^\sharp \) is finite and g doesn’t vanish identically, such a \(\tau \) exists.
Remark 6.19
The rigidity apparent in (6.26) suggests that for a large class of random admissible maps Open image in new window built from some stationary processes \(\xi _\mathfrak {t}^{(\varepsilon )}\) by (6.14) and (6.16), the corresponding collection of models built from Open image in new window defined as in (6.25) should converge to a limiting model, provided that the \(\xi _\mathfrak {t}^{(\varepsilon )}\) converge in a suitable sense as \(\varepsilon \rightarrow 0\). This is indeed the case, as shown in the companion “analytical” article [9]. It is also possible to verify that the renormalisation procedures that were essentially “guessed” in [31, 32, 39, 42] are precisely of BPHZ type, see Sects. 6.4.1 and 6.4.3 below.
Remark 6.20
One immediate consequence of Theorem 6.18 is that, for any Open image in new window and any admissible \(\varvec{\Pi }\), if we set \(\varvec{\Pi }^g = (g \otimes \varvec{\Pi })\Delta ^{\!}_\mathrm {ex}\) as in Theorem 6.16, then the BPHZ renormalisation of \(\varvec{\Pi }^g\) is Open image in new window . In particular, the BPHZ renormalisation of the canonical lift of a collection of stationary processes \(\{\xi _\mathfrak {l}\}_{\mathfrak {l}\in \mathfrak {L}_}\) as in Remark 6.13 is identical to that of the centred collection \(\{\tilde{\xi }_\mathfrak {l}\}_{\mathfrak {l}\in \mathfrak {L}_}\) where \(\tilde{\xi }_\mathfrak {l}= \xi _\mathfrak {l} \mathbf{E}\xi _\mathfrak {l}(0)\).
Remark 6.21
Although the map Open image in new window selects a “canonical” representative in the class of functions of the form \(\varvec{\Pi }^g\), this does not necessarily mean that every stochastic PDE in the class described by the underlying rule R can be renormalised in a canonical way. The reason is that the kernels \(K_\mathfrak {t}\) are typically some truncated version of the heat kernel and not simply the heat kernel itself. Different choices of the kernels \(K_\mathfrak {t}\) may then lead to different choices of the renormalisation constants for the corresponding SPDEs.
6.4 The reduced regularity structure
In this section we study the relation between the regularity structure \(\mathscr {T}^\mathrm {ex}\) introduced in this paper and the one originally constructed in [32, Sec. 8].
Definition 6.22
An admissible map is reduced if and only if Open image in new window for every Open image in new window . Moreover Open image in new window commutes with the maps Open image in new window , Open image in new window and \(\mathscr {J}\), and preserves the \(\cdot _\)degree, so that it is in particular also welldefined on Open image in new window , Open image in new window , Open image in new window and Open image in new window . It does however not preserve the \(\cdot _+\)degree so that it is not welldefined on Open image in new window ! Indeed, the \(\cdot _+\)degree depends on the \(\mathfrak {o}\) decoration, which is set to 0 by Open image in new window , see Definition 5.3.
Definition 6.23
We see therefore that the operators Open image in new window and Open image in new window are isomorphic to those defined in [32, Eq. (8.8)–(8.9)]. This shows that the regularity structure \(\mathscr {T}\), 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 ‘\(\mathrm {ex}\)’ stands for extended: the reason is that the regularity structure \(\mathscr {T}^\mathrm {ex}\) is an extension of \(\mathscr {T}\) in the sense that \(\mathscr {T}\subset \mathscr {T}^\mathrm {ex}\) with the inclusion interpreted as in [32, Sec. 2.1]. By contrast, we call \(\mathscr {T}\) the reduced regularity structure.
By the definition of Open image in new window , the extended structure Open image in new window encodes more information since we keep track of the effect of the action of Open image in new window by storing the (negative) homogeneity of the contracted subtrees in the decoration \(\mathfrak {o}\) and by colouring the corresponding nodes; both these details are lost when we apply Open image in new window and therefore in the reduced structure Open image in new window .
Note that if Open image in new window is such that Open image in new window is a model of \(\mathscr {T}^\mathrm {ex}\), then the restriction Open image in new window of Open image in new window to \(\mathscr {T}\) is automatically again a model. This is always the case, irrespective of whether \(\varvec{\Pi }\) is reduced or not, since the action of Open image in new window leaves Open image in new window invariant. This allows to give the following definition.
Definition 6.24
We denote by \(\mathscr {M}_{\infty }\) the space of all smooth models for \(\mathscr {T}\), in the sense of Definition 6.7, obtained by restriction to Open image in new window of Open image in new window for some reduced admissible linear map Open image in new window . We endow \(\mathscr {M}_{\infty }\) with the system of pseudometrics (6.17) and we denote by \(\mathscr {M}_0\) the completion of this metric space.
Remark 6.25
The restriction that \(\varvec{\Pi }\) be reduced may not seem very natural in view of the discussion preceding the definition. It follows however from Theorem 6.33 below that lifting this restriction makes no difference whatsoever since it implies in particular that every smooth admissible model on \(\mathscr {T}\) is of the form Open image in new window for some reduced \(\varvec{\Pi }\).
Remark 6.26
By restriction of Open image in new window to \(\mathscr {T}\) for Open image in new window , we get a renormalised model Open image in new window which covers all the examples treated so far in singular SPDEs. It is however not clear a priori whether we really have an action of a suitable subgroup of Open image in new window onto \(\mathscr {M}_{\infty }\) or \(\mathscr {M}_0\). This is because the coaction of \(\Delta ^{\!}_\mathrm {ex}\) on Open image in new window and Open image in new window fails to leave the reduced sector invariant. If on the other hand we tweak this coaction by setting Open image in new window , then unfortunately \(\Delta ^{\!+}\) and \(\Delta ^{\!}\) do not have the cointeraction property (3.48), which was crucial for our construction, see Remark 5.38. See Corollary 6.37 below for more on \(\Delta ^{\!}\).
Remark 6.27
Remark 6.28

We set Open image in new window . This is the most natural subgroup of Open image in new window since it contains the characters Open image in new window used for the definition of \( \hat{\varvec{\Pi }} \) in (6.25), as soon as Open image in new window . The fact that Open image in new window is a subgroup follows from the property (6.34).

We set Open image in new window where Open image in new window is the bialgebra ideal of Open image in new window generated by Open image in new window . Then one can identify Open image in new window with the group of characters of the Hopf algebra Open image in new window . It turns out that this is simply the polynomial Hopf algebra with generators Open image in new window , so that Open image in new window is abelian.
Theorem 6.29
There is a continuous action R of Open image in new window onto \(\mathscr {M}_0\) with the property that, for every Open image in new window and every reduced and admissible Open image in new window with Open image in new window , one has Open image in new window .
Proof
We already know by Theorem 6.16 that Open image in new window acts continuously onto \(\mathscr {M}_0^\mathrm {ex}\). Furthermore, by the definition of Open image in new window , it preserves the subset \(\mathscr {M}_0^r\subset \mathscr {M}_0^\mathrm {ex}\) of reduced models, i.e. the closure in \(\mathscr {M}_0^\mathrm {ex}\) of all models of the form Open image in new window for \(\varvec{\Pi }\) admissible and reduced. Since \(\mathscr {T}\subset \mathscr {T}^\mathrm {ex}\), we already mentioned that we have a natural projection \(\pi ^\mathrm {ex}:\mathscr {M}_0^\mathrm {ex}\rightarrow \mathscr {M}_0\) given by restriction (so that Open image in new window ), and it is straightforward to see that \(\pi ^\mathrm {ex}\) is injective on \(\mathscr {M}_0^r\). It therefore suffices to show that there is a continuous map \(\iota ^\mathrm {ex}:\mathscr {M}_0 \rightarrow \mathscr {M}_0^\mathrm {ex}\) which is a right inverse to \(\pi ^\mathrm {ex}\), and this is the content of Theorem 6.33 below. \(\square \)
Remark 6.30
We’ll show in Sect. 6.4.3 below that the action of Open image in new window onto \(\mathscr {M}_0\) is given by elements of the “renormalisation group” defined in [32, Sec. 8.3].
6.4.1 An example
6.4.2 Construction of extended models
In general if, for some sequence Open image in new window , Open image in new window converges to a limiting model in \(\mathscr {M}_0^\mathrm {ex}\), it does not follow that the characters \(g_+(\varvec{\Pi }^{(n)})\) of Open image in new window converge to a limiting character. However, we claim that the characters \(f_x^{(n)}\) of Open image in new window given by (6.12) do converge, which is not so surprising since our definition of convergence implies that the characters \(\gamma _{xy}^{(n)}\) of Open image in new window given by (6.13) do converge. More surprising is that the convergence of the characters \(f_x^{(n)}\) follows already from a seemingly much weaker type of convergence. Writing Open image in new window for the space of distributions on \(\mathbf{R}^d\), we have the following.
Proposition 6.31
Finally, one has Open image in new window such that \(\Pi _x = (\varvec{\Pi }\otimes f_x)\Delta ^{\!+}_\mathrm {ex}\) and such that \(\varvec{\Pi }^{(n)}\tau \rightarrow \varvec{\Pi }\tau \) in Open image in new window for every Open image in new window .
Proof
The convergence of the \(f_x^{(n)}\) follows immediately from the formula given in Lemma 6.10, combined with the convergence of the \(\Pi _x^{(n)}\) and [32, Lem. 5.19]. The fact that \((\Pi ,\Gamma )\) satisfies the algebraic identities required for a model follows immediately from the fact that this is true for every n. The convergence of the \(\Gamma _{xy}^{(n)}\) and the analytical bound on the limit then follow from [32, Sec. 5.1]. \(\square \)
Remark 6.32
This relies crucially on the fact that the maps \(\varvec{\Pi }\) under consideration are admissible and that the kernels \(K_\mathfrak {t}\) 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 Open image in new window we define Open image in new window by simply setting Open image in new window . Then we say that \(\varvec{\Pi }\) is admissible if \(\varvec{\Pi }^\mathrm {ex}\) is. We have the following crucial fact
Theorem 6.33
If Open image in new window is admissible and Open image in new window belongs to \(\mathscr {M}_\infty \), then Open image in new window belongs to \(\mathscr {M}_\infty ^\mathrm {ex}\). Furthermore, the map Open image in new window extends to a continuous map from \(\mathscr {M}_0\) to \(\mathscr {M}_0^\mathrm {ex}\).
Lemma 6.34
Proof
Proof of Theorem 6.33
6.4.3 Renormalisation group of the reduced structure
In this section, we show that the action of the renormalisation group Open image in new window on \(\mathscr {M}_0\) given by Theorem 6.29 is indeed given by elements of the “renormalisation group” \({\mathfrak R}\) 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.

One has Open image in new window and \(MX^k \tau =X^k M \tau \) for all \(\mathfrak {t}\in \mathfrak {L}_+\), \(k\in \mathbf{N}^d\), and Open image in new window .
 Consider the (unique) linear operators Open image in new window and Open image in new window such that \(\hat{M}\) is an algebra morphism, \(\hat{M} X^k=X^k\) for all k, and such that, for every Open image in new window and every Open image in new window and \(k \in \mathbf{N}^d\) with Open image in new window ,(6.39)(6.40)
Remark 6.35
Despite what a cursory inspection may suggest, the condition (6.39) is not equivalent to the same expression with Open image in new window replaced by Open image in new window . This is because (6.39) will typically fail to hold when Open image in new window .
We recall that the group Open image in new window has beed defined after Remark 6.28.
Theorem 6.36
Proof
Corollary 6.37
Proof
This follows immediately from (6.34), Theorem 6.36, the definition of Open image in new window , the fact that Open image in new window is an algebra morphism on Open image in new window , and the same argument as in the proof of Proposition 4.11. \(\square \)
By the Remarks 6.19 and 6.28, the renormalisation procedures of [31, 32, 39, 42] can be described in this framework.
Footnotes
 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 \(H > {1\over 4}\), even though these equations are subcritical for every \(H>0\). See in particular the assumptions of [9, Thm 2.14].
Notes
Acknowledgements
We are very grateful to Christian Brouder, Ajay Chandra, Alessandra Frabetti, Dominique Manchon and Kurusch EbrahimiFard for many interesting discussions and pointers to the literature. MH gratefully acknowledges support by the Leverhulme Trust and by an ERC consolidator grant, project 615897 (Critical). LZ gratefully acknowledges support by the Institut Universitaire de France and the project of the Agence Nationale de la Recherche ANR15CE40002001 grant LSD. The authors thank the organisers and the participants of a workshop held in Bergen in April 2017, where the results of this paper were presented and discussed in detail.