The geometry of the space of branched rough paths

We construct an explicit transitive free action of a Banach space of Hölder functions on the space of branched rough paths, which yields in particular a bijection between these two spaces. This endows the space of branched rough paths with the structure of a principal homogeneous space over a Banach space and allows to characterize its automorphisms. The construction is based on the Baker–Campbell–Hausdorff formula, on a constructive version of the Lyons–Victoir extension theorem and on the Hairer–Kelly map, which allows to describe branched rough paths in terms of anisotropic geometric rough paths.


Introduction
The theory of rough paths has been introduced by Terry Lyons in the 1990s with the aim of giving an alternative construction of stochastic integration and stochastic differential equations. More recently, it has been expanded by Martin Hairer to cover stochastic partial differential equations, with the invention of regularity structures.
A rough path and a model of a regularity structure are mathematical objects which must satisfy some algebraic and analytical constraints. For instance, a rough path can be described as a Hölder function defined on an interval and taking values in a non-linear finite-dimensional Lie group; models of regularity structures are a generalization of this idea. A crucial ingredient of regularity structures is the renormalization procedure: given a family of regularized models, which fail to converge in an appropriate topology as the regularization is removed, one wants to modify it in a such a way that the algebraic and analytical constraints are still satisfied and the modified version converges. This procedure has been obtained in [6,9] for a general class of models with a stationary character.
The same question about rough paths has been asked recently in [3][4][5], and indeed it could have been asked much earlier. Maybe this has not happened because the motivation was less compelling; although one can construct examples of rough paths depending on a positive parameter which do not converge as the parameter tends to 0, this phenomenon is the exception rather than the rule. However, the problem of characterizing the automorphisms of the space of rough paths is clearly of interest; one example is the transformation from Itô to Stratonovich integration (see, for example, [1,16,17]). However, our aim is to put this particular example in a much larger context.
We recall that there are several possible notions of rough paths; in particular we have geometric rough paths (GRPs) and branched rough paths, two notions defined, respectively, by Terry Lyons [29] and Massimiliano Gubinelli [25] (see Sections 3 and 4). These two notions are intimately related to each other, as shown by Hairer and Kelly [28] (see Section 4). We note that regularity structures [27] are a natural and far-reaching generalization of branched rough paths.
In this paper we concentrate on the automorphisms of the space of branched rough paths (see below for a discussion of the geometric case). Let F be the collection of all non-planar rooted forests with nodes decorated by t1, . . . , du (see Section 4). For instance, the following forest is an element of F . We call T Ă F the set of rooted trees, namely of non-empty forests with a single connected component. Grading elements τ P F by the number |τ | of their nodes we set T n -tτ P T : |τ | ď nu, n P N.
Let now H be the linear span of F . It is possible to endow H with a product and a coproduct Δ: H Ñ H b H which make it a Hopf algebra, also known as the Butcher-Connes-Kreimer Hopf algebra (see Section 4.2). We let G denote the set of all characters over H , that is, elements of G are functionals X P H˚that are also multiplicative in the sense that xX, τ σy " xX, τ yxX, σy for all forests (and in particular trees) τ, σ P F . Furthermore, the set G can be endowed with a product ‹, dual to the coproduct, defined pointwise by xX ‹ Y, τ y " xX b Y, Δτ y. We work on the compact interval [0,1] for simplicity, and all results can be proved without difficulty on r0, T s for any T ě 0.
If γ|τ | ą 1, then Gubinelli's Sewing Lemma [24] yields that the function ps, tq Þ Ñ xX st , τy is uniquely determined by (1.4), that is, by the values of X on trees with at most |τ |´1 nodes, and therefore, applying a recursion, on trees with at most N -tγ´1u nodes. More explicitly, the Sewing Lemma is an existence and uniqueness result for r0, 1s 2 Q ps, tq Þ Ñ xX st , τy with γ|τ | ą 1, once the right-hand side of (1.1) is known. However, for γ|τ | ď 1 we have no uniqueness, as we have already seen, and existence is not trivial.
As we have seen in (1.3), the value of xX, τ y can be modified by adding the increment of a function in C γ|τ | pr0, 1sq, as long as γ|τ | ď 1. It seems reasonable to think that it is therefore possible to construct an action on the set of branched γ-rough paths of the abelian group (under pointwise addition) C γ -tpg τ q τ PTN : g τ 0 " 0, g τ P C γ|τ | pr0, 1sq, @ τ P T , |τ | ď N u, namely the set of all collections of functions pg τ P C γ|τ | pr0, 1sq : τ P T , |τ | ď N q indexed by rooted trees with fewer than N -tγ´1u nodes, such that g τ 0 " 0 and g τ P C γ|τ | pr0, 1sq. This is indeed the content of the following.
Theorem 1.2. Let γ P s0, 1r such that γ´1 P N. There is a transitive free action of C γ on BRP γ , namely a map pg, Xq Þ Ñ gX such that (1) for each g, g 1 P C γ and X P BRP γ the identity g 1 pgXq " pg`g 1 qX holds; (2) if pg τ q τ PTN P C γ is such that there exists a unique τ P T N with g τ " 0, then xpgXq st , τy " xX st , τy`g τ t´g τ s and xgX, σy " xX, σy for all σ P T not containing τ ; (3) for every pair X, X 1 P BRP γ there exists a unique g P C γ such that gX " X 1 .
We say that a tree σ P T contains a tree τ P T if there exists a subtree τ 1 of σ, not necessarily containing the root of σ, such that τ and τ 1 are isomorphic as rooted trees, where the root of τ 1 is its node which is closest to the root of σ. Note that every pg τ q τ PTN P C γ is the sum of finitely many elements of C γ having satisfying the property required in point (2) of Theorem 1.2.
The same argument applies for any γ ď 1{2 and any tree τ such that 2 ď |τ | ď N " tγ´1u, and the fact that the above Young integrals are not well defined shows why existence of the map X Ñ gX is not trivial.
Since Theorem 1.2 yields an action of C γ on BRP γ which is regular, that is, free and transitive, then BRP γ is a principal C γ -homogeneous space or C γ -torsor. In particular, BRP γ is a copy of C γ , but there is no canonical choice of an origin in BRP γ .
Therefore, Theorem 1.2 also yields the following.
Corollary 1.3. Given a branched γ-rough path X, the map g Ñ gX yields a bijection between C γ and the set of branched γ-rough paths.
Therefore Corollary 1.3 yields a complete parametrization of the space of branched rough paths. This result is somewhat surprising, since rough paths form a non-linear space, in particular because of the Chen relation; however, Corollary 1.3 yields a natural bijection between the space of branched γ-rough paths and the linear space C γ . Corollary 1.3 also gives a complete answer to the question of existence and characterization of branched γ-rough paths over a γ-Hölder path x. Unsurprisingly, for our construction we start from a result of Lyons and Victoir's [30] of 2007, which was the first general theorem of existence of a geometric γ-rough path over a γ-Hölder path x (see our discussion of Theorem 1.4).
An important point to stress is that the action constructed in Theorem 1.2 is neither unique nor canonical. In the proof of Theorem 3.4, some parameters have to be fixed arbitrarily, and the final outcome depends on them (see Remark 3.6). In this respect, the situation is similar to what happens in regularity structures with the reconstruction operator on spaces D γ with a negative exponent γ ă 0 (see [27,Theorem 3.10]).

Outline of our approach
A key point in Theorem 1.2 is the construction of branched γ-rough paths. In the case of geometric rough paths (GRPs; see Definition 4.1), the signature [11,29] of a smooth path x : r0, 1s Ñ R d yields a canonical construction. Other cases where GRPs over non-smooth paths have been constructed are Brownian motion and fractional Brownian motion (see [13] for the case H ą 1 4 and [33] for the general case), among others. However, until Lyons and Victoir's paper [30] in 2007, this question remained largely open in the general case. The precise result is as follows.
Our first result is a version of this theorem which holds for rough paths in a more general algebraic context (see Theorem 3.4). We use the Lyons-Victoir approach and an explicit form of the Baker-Campbell-Hausdorff formula by Reutenauer [34] (see formula (2.11)). Although Lyons and Victoir used in one passage the axiom of choice, our method is completely constructive.
Using the same idea we extend this construction to the case where the collection px 1 , . . . , x d q is allowed to have different regularities in each component, which we call anisotropic (geometric) rough paths (aGRP) (see Definition 4.8).
Theorem 1.5. To each collection px i q i"1,...,d , with x i P C γi pr0, 1sq, we can associate an anisotropic rough pathX over px i q i"1,...,d . For every collection pg i q i"1,...,d , with g i P C γi pr0, 1sq, denoting by gX the anisotropic geometric rough path (aGRP) over px i`gi q i"1,...,d , we have This kind of extension to rough paths has already been explored in the papers [2,26] in the context of isomorphisms between geometric and branched rough paths. It turns out that the additional property obtained by our method enables us to explicitly describe the propagation of suitable modifications from lower to higher degrees.
We then go on to describe the interpretation of the above results in the context of branched rough paths. The main tool is the Hairer-Kelly map [28], that we introduce and describe in Lemma 5.1 and then use to encode branched rough paths via aGRPs, along the same lines as in [2,Theorem 4.3].
Theorem 1.6. Let X be a branched γ-rough path. There exists an aGRPX indexed by words on the alphabet T N , with exponents pγ τ " γ|τ |, τ P T N q, and such that xX, τ y " xX, ψpτ qy, where ψ is the Hairer-Kelly map.
The main difference of this result with [28, Theorem 1.9] is that we obtain an aGRP instead of a classical GRP. This means that we do not construct unneeded components, that is, components with regularity larger than 1, and we also obtain the right Hölder estimates in terms of the size of the indexing tree. This addresses two problems mentioned in Hairer and Kelly's work, namely [28, Remarks 4.14 and 5.9].
We then use Theorems 1.5 and 1.6 to construct our action on branched rough paths. Given pg, Xq P C γˆB RP γ , we construct the aGRPsX and gX and then define the branched rough path gX P BRP γ as xgX, τ y " xgX, ψpτ qy, where ψ is the Hairer-Kelly map.
Our approach also does not make use of Foissy-Chapoton's Hopf algebra isomorphism [10,20] between the Butcher-Connes-Kreimer Hopf algebra and the shuffle algebra over a complicated set I of trees as is done in [2]. This allows us to construct an action of a larger group on the set of branched rough paths; indeed, using the above isomorphism one would obtain a transformation group parametrized by pg τ q τ PI where I is the aforementioned set of trees of Foissy-Chapoton's results and g τ P C γ|τ | ; on the other hand our approach yields a transformation group parametrized by pg τ q τ PTN . With the smaller set I X T N , transitivity of the action g Þ Ñ gX would be lost.
Finally we note that we use a special property of the Butcher-Connes-Kreimer Hopf algebra: the fact that it is freely generated as an algebra by the set of trees, so defining characters over it is significantly easier than in the geometric case. To define an element X P G it suffices to give the values xX, τ y for all trees τ P T ; by freeness there is a unique multiplicative extension to all of H . This is not at all the case for GRPs: the shuffle algebra T pAq over an alphabet A is not free over the linear span of words so if one is willing to define a character X over T pAq there are additional algebraic constraints that the values of X on words must satisfy.
Outline. We start by reviewing all the theoretical concepts needed to make the exposition in this section formal. In Section 3 we state and prove the main result of this chapter. We extend the notion of rough path and we give an explicit construction of such a generalized rough path above any given path x P C γ . Next, in Section 4.3 we extend this result to the class of aGRPs. Finally, in Section 4 we connect our construction with Gubinelli's branched rough paths, and we extend Hairer and Kelly's work in Section 5.1. We also explore possible connections with renormalization in Section 6 by studying how our construction behaves under modification of the underlying paths. Then, we connect this approach with a recent work by Bruned, Chevyrev, Friz and Preiß [4] in Section 6.1, who borrowed ideas from the theory of regularity structures [6,27] and proposed a renormalization procedure for geometric and branched rough paths [4] based on pre-Lie morphisms.
The main difference between our result and the Bruned-Chevyrev-Friz-Preiß (BCFP) procedure is that they consider translation only by time-independent factors, whereas -under reasonable hypotheses -we are also able to handle general translations depending on the time parameter. We also mention that some further algebraic aspects of renormalization in rough paths have been recently developed in [5].

Preliminaries
A Hopf algebra H is a vector space endowed with an associative product m : H b H Ñ H : and a coassociative coproduct Δ: satisfying moreover certain compatibility assumptions; H is also supposed to have a unit 1 P H , a counit ε P H˚and an antipode S : H Ñ H such that for all x P H . As usual we will use the more compact notation mpx b yq " xy. The reader is referred to the papers [8,31] for further details.
Definition 2.1. We say that the Hopf algebra H is graded if it can be decomposed as a direct sum In a graded Hopf algebra, each element x P H can be decomposed as a sum x " where only a finite number of the summands are non-zero. We call each x n the homogeneous part of degree n of x, and elements of H pnq are said to be homogeneous of degree n. In this case we write |x n | " n.
Definition 2.2. The graded Hopf algebra H is connected if the degree 0 part is onedimensional. It is locally finite if dim H pnq ă 8 for all n ě 0.
From now on we consider a graded connected locally finite Hopf algebra H . Then, for any homogeneous element x P H pnq the coproduct can be written as H ppq b H pqq and Δ 1 : H Ñ H b H is known as the reduced coproduct. Furthermore, the coassociativity of Δ and of Δ 1 , that is, the identity pΔ 1 b idqΔ 1 " pid b Δ 1 qΔ 1 , allows to unambiguously define their iterates Δ n , Δ 1 n : H Ñ H bpn`1q by setting for n ě 2 Δ n " pid b Δ n´1 qΔ, Δ 1 n " pid b Δ 1 n´1 qΔ 1 . Then we have, for a homogeneous element x P H pkq of degree k, Remark 2.3. These properties of the iterated coproduct imply that the bialgebra pH , Δq is conilpotent, that is, for each homogeneous x P H pkq there is an integer n ď k such that Δ 1 n x " 0. We obtain also the inclusion , that is, the n-fold reduced coproduct of a homogeneous element of degree n`1 is a sum of pn`1q-fold tensor products of homogeneous elements of degree 1.
We recall that in general the dual space H˚carries an algebra structure given by the convolution product ‹, dual to the coproduct Δ, defined by xf ‹ g, xy -xf b g, Δxy.
For a collection of maps f 1 , . . . , f k P H˚we have the formula We call g the set of all infinitesimal characters on H .
We observe that necessarily xX, 1y " 1 and xα, 1y " 0 for all X P G and α P g. It is well known that the pG, ‹, εq is a group with product ‹, unit ε and inverse X´1 " X˝S where S is the antipode defined above. Moreover pg, r¨,¨sq is a Lie algebra with bracket rα, βsα ‹ β´β ‹ α (see, for example, [31]).

Nilpotent Lie algebras
By Lemma 2.5 we can consider the dual algebra pHN , ‹, εq. This algebra is also graded and connected, since we have the natural grading Since H N is not a subalgebra of H , the notions of character and infinitesimal character on HN are not well defined. We can however introduce their truncated versions.
Definition 2.6. We say that X P HN zt0u is a truncated character on H N if xX, xyy " xX, xyxX, yy holds for all x P H pnq , y P H pmq with n`m ď N . We call G N the space of truncated characters on H N .
Likewise, we say that α P HN is a truncated infinitesimal character if xα, xyy " xα, xyxε, yy`xε, xyxα, yy holds for all x P H pnq , y P H pmq with n`m ď N . We call g N the space of truncated infinitesimal characters on H N .
There are a canonical inclusions HN ãÑ HN`1 ãÑ H˚, which induce canonical inclusion g N ãÑ g N`1 ãÑ g. Moreover, such canonical inclusions are right-inverse for the corresponding restriction maps H˚Ñ HN`1 Ñ HN . Proof. Using the notation (2.5), we can extend α P HN to α P HN`1 (respectively, H˚) by setting α pN`1q " 0 (respectively, α pkq " 0 for all k ě N`1). Trivially this extension takes HN to HN`1. If α P g N and x, y P H N are such that |x|`|y| ď N`1 then pxα, x j yxε, yy`xε, xyxα, y j yq " xα, xyxε, yy`xε, xyxα, yy, so that the extension of α is in g N`1 . The same argument yields the inclusion g N ãÑ g. l There are also the truncated exponential exp N : HN Ñ HN and logarithm log N : HN Ñ HN , defined by the sums The proof of the next result can be found for instance in [19,Theorem 77].
Lemma 2.8. pG N , ‹, εq is a group and pg N , r¨,¨sq is a Lie algebra. Moreover, exp N : For every k ě 0 we define now, using the notation (2.5), Proof. Let x P H . With the notation (2.3) we have for α P W n and β P W m By the canonical inclusion of Lemma 2.7, we observe that in the notation (2.5). With this decomposition g N becomes by Lemma 2.9 a graded Lie algebra. We recall that the center of g N is the subspace of all w P g N such that rα, ws " 0 for all α P g N , Proof. Let α P g N and w P W N . Clearly, xrα, ws, xy is zero unless |x| " N . In this case xrα, ws, xy " xα b w´w b α, Δxy " xα, 1yxw, xy´xw, xyxα, 1y " 0 since xw, yy " xw, y N y, in the notation (2.3). The second assertion follows easily: it is enough to write X " exp N pwq and Y " exp N pαq with α P g N and w P W N and use the explicit representation (2.6) of exp N and the fact that α ‹ w " w ‹ α. l The next (famous) result describes the group law on G N in terms of an operation on g N via the exponential/logarithmic map.
We define the map BCH N : Another way to interpret this theorem is to say that there exists an element It is a classical result that the map BCH N is formed by a sum of iterated Lie brackets of α and β, where the first terms are BCH N pα, βq " α`β`1 2 rα, βs`1 12 rα, rα, βss´1 12 rβ, rα, βss`¨¨¨, (2.9) and the following ones are explicit but difficult to compute. Nevertheless, fully explicit formulas have been known since 1947 by Dynkin [15]. For our purposes, however, Dynkin's formula is too complicated (for example, the regularity argument in step 2 of the proof of Theorem 3.4 would not be as evident) so we rely on a different expression first shown by Reutenauer [34]. In order to describe it, let ϕ k : pH˚q bk Ñ H˚be the linear map where S k denotes the symmetric group of order k, and a σ -p´1q dpσq k`k´1 dpσq˘´1 is a constant depending only on the descent number dpσq of the permutation σ P S k , namely the number of i P t1, . . . , k´1u such that σpiq ą σpi`1q.
Proof. Let us suppose first that T pV q is the (completed) tensor algebra over a twodimensional vector space V , with V linearly generated by te 1 , e 2 u. Then the result is contained in Reutenauer's paper [34] where the free step-N nilpotent Lie algebra L N plays the rôle of g N . We want now to show how this implies the same result in our more general setting.
In order to prove the first formula, we first note that Φ is not a graded morphism, since the generators e 1 and e 2 are homogeneous of degree 1 in T pV q N , but α and β are in general not homogeneous in HN . However, from the bilinearity of the Lie bracket and Lemma 2.9 we obtain From all these considerations we obtain the following result on the map: Note that BCH pn`1q takes indeed values in W n`1 rather than in g n`1 by (both assertions of) Lemma 2.12.
Lemma 2.13. Let x P H pn`1q and α, β P g n`1 . Then Then the result follows directly from the definition of ϕ k in (2.10) together with (2.4) and the fact that since xα j , 1y " 0 we can write α 1 ‹¨¨¨‹ α n`1 " pα 1 b¨¨¨b α n`1 qΔ 1 n (2.14) instead (note the reduced coproduct in place of the full coproduct). l

A distance on the group of truncated characters
Now we introduce a distance on G N which is well adapted to the notion of rough paths, to be introduced in Definition 3.1. We fix a basis B of H N and define a norm }¨} on this space by requiring that B is orthonormal. There is a unique function c : Then we define Then where for X P G N Ă HN we use the notation (2.5). We define Proposition 2.14. The map ρ N defines a left-invariant distance on the group G N such that the metric space pG N , ρ N q is complete.
Proof. We only need to prove that the function |¨| defined in (2.15) is subadditive, the other properties being clear. Note that for X, Y P G N , with the notation (2.5) we have The next result is the analog of [30,Proposition 7].
Proof. Using the notation (2.5), we have There are exactly`k´1 i´1˘ď Since X´1 " exp N p´w 1´¨¨¨´wN q, the bound for X´1 follows in the same way and we have therefore proved the desired upper bound for |X|. For the lower bound, we use the truncated logarithm Then we can estimatẽ and the proof is complete. l We now note that the function |¨| and the distance ρ N make G N a homogeneous group; see [22] for an extensive treatment of this subject, and [30] for the case of tensor algebras and the relation with GRPs.
To put it briefly, for all r ą 0 we can define the following linear operator, Ω r : H˚Ñ H˚: This family satisfies Ω r˝Ωs " Ω rs , r, s ą 0. Moreover, Ω r : g N Ñ g N is a Lie-algebra automorphism of g N for all r ą 0. Then they induce group automorphisms Λ rexp N˝Ωrl og N : G N Ñ G N , r ą 0. In the terminology of [22], pΩ r q rą0 is a family of dilations on the finite-dimensional Lie algebra g N and G N is a homogeneous group. Note that the function |¨|: G N Ñ R`is continuous, satisfies |Λ r X| " r|X| for all r ą 0 and X P G N , and |X| " 0 for X P G N if and only if X " 1. These three properties make |¨| a homogeneous norm on G N (see [22]). The homogeneity property plays an important role in the proof of Theorem 3.4.

Construction of rough paths
As in the previous section, we fix a locally finite graded connected Hopf algebra H . We also fix a number γ P s0, 1r and let N -tγ´1u be the biggest integer such that Nγ ď 1. Without loss of generality we can fix a basis B of H N consisting only of homogeneous elements and in particular we let te 1 , . . . , e d u " Definition 3.1. A pH , γq-rough path is a function X : r0, 1s 2 Ñ G N , with N " tγ´1u, which satisfies Chen's rule X su ‹ X ut " X st , s, u, t P r0, 1s, (3.1) and such that for all v P B s " xX st , e i y, s, t P r0, 1s, we say that X is a γ-rough path over px 1 , . . . , x d q.
Remark 3.2. By specializing this definition to different choices of H we recover both GRPs [29] where H is the shuffle Hopf algebra over an alphabet, branched rough paths [25] where H is the Butcher-Connes-Kreimer Hopf algebra on decorated non-planar rooted trees, and also planarly branched rough paths [14].
We remark that there is a bijection between (1) functions X : r0, 1s 2 Ñ G N such that X su ‹ X ut " X st , for all s, u, t P r0, 1s; (2) functions X : r0, 1s Ñ G N such that X 0 " 1, given by X Þ Ñ X, X t -X 0t , X Þ Ñ X, X st -X´1 s ‹ X t , s,t P r0, 1s. Proof. First note that the distance in (2.16) is defined with respect to a fixed (but arbitrary) basis so we use the basis B fixed at the beginning of this section. Also, due to the above remark we only have to verify that X is γ-Hölder with respect to ρ N if and only if X satisfies (3.2) using the same basis. In one direction, if X is γ-Hölder then, by definition |X st | " ρ N pX s , X t q À |t´s| γ and so, for a basis element v P B we have Conversely, if (3.2) holds then |X st | À |t´s| γ and so by definition also ρ N pX s , X t q À |t´s| γ , that is, X is γ-Hölder with respect to ρ N . l We now come to the problem of existence. Our construction of a rough path in the sense of Definition 3.1 over an arbitrary collection of γ-Hölder paths px 1 , . . . , x d q relies in the following extension theorem. We note that the proof is a reinterpretation of the approach of Lyons-Victoir [30,Theorem 1] in the context of a more general graded Hopf algebra H . Theorem 3.4 (Rough path extension). Let 1 ď n ď N´1 and γ P s0, 1r such that γ´1 P N. Suppose we have a γ-Hölder path X n : r0, 1s Ñ pG n , ρ n q. There is a γ-Hölder path X n`1 : r0, 1s Ñ pG n`1 , ρ n`1 q extending X n , that is, such that X n`1 | Hn " X n .
A key tool is the following technical lemma whose proof can be found in [30,Lemma 2]. Suppose y : D Ñ E is a path satisfying the bound ρpy t m k , y t m k`1 q À 2´γ m for some γ P p0, 1q. Then, there exists a γ-Hölder path x : r0, 1s Ñ E such that x| D " y.
Proof of Theorem 3.4. The construction of X n`1 is made in two steps.
Step 1. For m ě 0 and k P t0, . . . , 2 m u we define t m k -k2´m P r0, 1s. Then we define the following sets of dyadics in r0, 1s : Set X st " pX n s q´1 ‹ X n t P G n and L st " log n pX st q P g n where log n was defined in (2.6). Then, the Baker-Campbell-Hausdorff formula (2.8) and Chen's rule (3.1) imply that L st " BCH n pL su , L ut q. (3.4) We look for Y : r0, 1s 2 Ñ G n`1 such that Y satisfies Chen's rule (3.1) and Y | HN " X. We use throughout the proof that g n Ă g n`1 (see Lemma 2.7).
In a first step, we define Y : DˆD Ñ G n`1 . In the second step we show that Y has suitable uniform continuity properties and can thus be extended to r0, 1s 2 using Lemma 3.5.
The construction of Y : DˆD Ñ G n`1 goes through a construction of Y m : D mˆDm Ñ G n`1 by recursion on m ě 0. We claim that for all m ě 0 we can find Y m such that (3) Y m restricted to H n is equal to X : D mˆDm Ñ G n , in the sense that Y m ab | Hn " X ab , @ a, b P D m ; (4) for all k " 0, . . . , 2 m´1 , setting For m " 0, we set Y 0 01 " exp n`1 pL 01 q, Y 0 00 " Y 0 11ε and Z 0 01 -0 P W n`1 . For x P H n , we have xexp n`1 pL 01 q, xy " xexp n pL 01 q, xy, so that Y 0 restricted to H n is equal to X : D 0ˆD0 Ñ G n .
Let now m ě 1, and suppose that Y m´1 : D m´1ˆDm´1 Ñ G n`1 has been constructed with the above properties. We start by defining Y m tt " ε for all t P D pmq . Let us consider three consecutive points in D pmq of the form for some k " 0, . . . , 2 m´1´1 . Note that s " t m´1 k and t " t m´1 k`1 , so that Z m st -Z m´1 st P W n`1 is already defined by the recurrence hypothesis. We define Z m su and Z m ut as follows: st´B CH pn`1q pL su , L ut q˘, (3.5) where BCH pn`1q " BCH n`1´B CH n : g n`1ˆgn`1 Ñ W n`1 (see (2.12)). Since by recurrence Z m´1 st P W n`1 , we obtain that Z m su , Z m ut P W n`1 and Z m su`Z m ut " Z m´1 st´B CH pn`1q pL su , L ut q " L st`Z m st´B CH n`1 pL su , L ut q, (3.6) where in the last equality we have applied (3.4). Then we set Y m suexp n`1 pL su`Z m su q, Y m utexp n`1 pL ut`Z m ut q. Since exp n`1 pW n`1 q is in the center of G n`1 by Proposition 2.10, we obtain that Y m su " exp n`1 pL su q ‹ exp n`1 pZ m su q, Y m ut " exp n`1 pL ut q ‹ exp n`1 pZ m ut q.

By (2.8) and (3.6) the product is equal to
so that the identity Y m ab ‹ Y m bc " Y m ac is valid for any a, b, c P D pmq . We need now to check that this definition is compatible with the values already constructed on D m´1ˆDm´1 . By the recursion assumption, it is enough to show that for all k, P t0, If k " or |k´ | " 1, then this is true by construction. Otherwise, if for example k`1 ă then by the recursion property and the Chen relation satisfied by Y m (respectively, Y m´1 ) on D pmq (respectively, D pm´1q ).
We also have to check the extension property: for x P H n we have j`1 , xy. By recurrence, we have proved that Y m : D mˆDm Ñ G n`1 is well defined for all m ě 0, with the above properties. Therefore, we can unambiguously define Y : DˆD Ñ G n`1 , st , s,t P D m , and Y indeed satisfies the Chen relation on D, namely Y ab ‹ Y bc " Y ac for all a, b, c P D, and the restriction property xY ab , xy " xX ab , xy, @ a, b P D, x P H n .
Step 2. In order to have a pH n`1 , γq-Hölder path, Definition 3.1 requires us to construct a γ-Hölder path with values in G n`1 , and for this we will use Lemma 3.5. Set Then, if υ is a basis element in H pn`1q we have by (2.13), for s " t m k , u " t m k`1 and t " t m k`2 |xBCH pn`1q pL su , L ut q, υy| ď ÿ pυq ÿ i`j"n`1 Now, since υ pjq P H p1q for all j " 1, . . . , n`1 we actually have that for some coefficients υ k pjq P R such that υ pjq " ř d k"1 υ k pjq e k , and we have a similar estimate for L ut instead of L su . Therefore we obtain that BCH pn`1q pL su , L ut q n`1 ď C 2´m pn`1qγ , Therefore, from (3.5) we get hence a m ď 2 pn`1qγ´1 a m´1`C 2 , m ě 1.
Since a 0 " 0 we can show by recurrence on m ě 0 Since we are in the regime where pn`1qγ ă 1 (here we use that γ´1 R N) we obtain that Let now fix m ě 0, i P t0, . . . , 2 m´1 u, and set st m j , tt m j`1 . Then we want to prove that |Y st | À 2´m γ (see (2.15) for the definition of |¨|q. By subadditivity of |¨| with respect to the convolution product ‹ we have Moreover, using Lemma 2.15 again (first the upper bound, then the lower bound) and the fact that X n : r0, 1s Ñ G n is γ-Hölder by assumption, exp n`1 pL st qˇˇď C n`1 sup k"1,...,n`1~p Therefore, the path X n`1 : thus by Lemma 3.5 we obtain a γ-Hölder path X n`1 : r0, 1s Ñ G n`1 extending X n . l Remark 3.6. Our construction depends on a finite number of choices, namely we set Z 01 " 0 to start the recursion in (3.6), and this for each level; moreover in (3.6) we make the choice Z t m  Corollary 3.8. Given γ P s0, 1r with γ´1 R N and a collection of γ-Hölder paths x i : r0, 1s Ñ R, i " 1, . . . , d, there exists a γ-Hölder path X : r0, 1s Ñ G N such that xX, e i y " Proof. We start with the following observation: for n " 1, the group G 1 Ă Hp 1q is abelian, and isomorphic to the additive group Hp 1q . Indeed, let X, Y P G 1 and x P H p1q . Then, as Δx " x b 1`1 b x by the grading, we have that xX ‹ Y, xy " xX, xy`xY, xy, that is, X ‹ Y " X`Y . Moreover, in H 1 the product xy " 0. Therefore, we may set xX 1 t , e i yx i t´x i 0 where te 1 , . . . , e d u is a basis of H p1q and this path is γ-Hölder with respect to ρ 1 . By Theorem 3.4 there is a γ-Hölder path X 2 : r0, 1s Ñ pG 2 , ρ 2 q extending X 1 so in particular xX 2 t , e i y " x i t´x i 0 also. Continuing in this way we obtain successive γ-Hölder extensions X 3 , . . . , X N and we set X -X N . l The following result has already been proved in the case where the underlying Hopf algebra H is combinatorial by Curry, Ebrahimi-Fard, Manchon and Munthe-Kaas in [14,Theorem 4.3]. We remark that their proof works without modifications in our context so we have Theorem 3.9. Let X : r0, 1s Ñ G N be a γ-Hölder path with X 0 " 1 and suppose that γ´1 R N. There exists a pathX : r0, 1s Ñ G such that |xX´1 s ‹X t , vy| À |t´s| γ|v| for all homogeneous v P H and extending X, in the sense thatX| HN " X.
Remark 3.10. In view of Theorem 3.9 we can replace the truncated group in Definition 3.1 by the full group of characters G. What this means is that γ-rough paths are uniquely defined once we fix the first N levels and since H is locally finite, this amounts to a finite number of choices. This is of course a generalization of the extension theorem of [29] (see also [25,Theorem 7.3] for the branched case).

Applications
We now apply Theorem 3.4 to various kinds of Hopf algebras in order to link this result with the contexts already existing in the literature.

Geometric rough paths
In this setting we fix a finite alphabet A -t1, . . . , du. As a vector space H -T pAq is the linear span of the free monoid MpAq generated by A. The product on H is the shuffle product The coproductΔ: H Ñ H b H is obtained by deconcatenation of words, It turns out that pH , ⧢,Δq is a commutative unital Hopf algebra, and pH ,Δq is the cofree coalgebra over the linear span of A. The antipode is the linear map S : H Ñ H given by Finally, we recall that H is graded by the length pa 1¨¨¨an q " n and it is also connected. The homogeneous components H pnq are spanned by the sets ta 1¨¨¨an : a i P Au. Definition 3.1 specializes in this case to GRPs as defined in [28] (see just below for the precise definition) and Theorem 3.4 coincides with [30,Theorem 6].

Branched rough paths
Let T be the collection of all non-planar non-empty rooted trees with nodes decorated by t1, . . . , du. Elements of T are written as 2-tuples τ " pT, cq where T is a non-planar tree with node set N T and edge set E T , and c : N T Ñ t1, . . . , du is a function. Edges in E T are oriented away from the root, but this is not reflected in our graphical representation. Examples of elements of T include the following For τ P T write |τ | " #N T for its number of nodes. Also, given an edge e " px, yq P E T we set speq " x and tpeq " y. There is a natural partial-order relation on N T where x ď y if and only if there is a path in T from the root to y containing x.
We denote by F the collection of decorated rooted forests and we let H -H BCK denote the vector space spanned by F . There is a natural commutative and associative product on F , denoted by¨and given by the disjoint union of forests, where the empty forest 1 acts as the unit. Then, H is the free commutative algebra over T , with grading |τ 1¨¨¨τk | " |τ 1 |`¨¨¨`|τ k |. Given i P t1, . . . , du and a forest τ " τ 1¨¨¨τk we denote by rτ 1¨¨¨τk s i the tree obtained by grafting each of the trees τ 1 , . . . , τ k to a new root decorated by i, for example, The decorated Butcher-Connes-Kreimer coproduct [12,25] is the unique algebra morphism Δ: This coproduct admits a representation in terms of cuts. An admissible cut C of a tree T is a non-empty subset of E T such that any path from any vertex of the tree to the root contains at most one edge from C; we denote by ApT q the set of all admissible cuts of the tree T . Any admissible cut C containing k edges maps a tree T to a forest CpT q " T 1¨¨¨Tk`1 obtained by removing each of the edges in C. Observe that only one of the remaining trees T 1 , . . . , T k`1 contains the root of T , which we denote by R C pT q; the forest formed by the other k factors is denoted by P C pT q. This naturally induces a map on decorated trees by considering cuts of the underlying tree, and restriction of the decoration map to each of the rooted subtrees T 1 , . . . , T k`1 . Then, This, together with the counit map ε : F Ñ R such that εpτ q " 1 if and only if τ " 1 endows F with a connected graded commutative non-cocommutative bialgebra structure, hence a Hopf algebra structure [31]. As before we denote by H˚the linear dual of H which is an algebra via the convolution product xX ‹ Y, τ y " xX b Y, Δτ y and we denote by G the set of characters on H , that is, linear functionals X P H˚such that xX, σ¨τ y " xX, σyxX, τ y. For each n P N the finitedimensional vector space H n spanned by the set F n of forests with at most n nodes is a subcoalgebra of H , hence its dual is an algebra under the convolution product, and we let G n be the set of characters on H n .
We have already defined branched rough paths in Definition 1.1. Proposition 3.3 yields the following characterization: Directly applying Theorem 3.4 to the Butcher-Connes-Kreimer Hopf algebra H we obtain Corollary 4.3. Given γ P s0, 1r with γ´1 R N and a family of γ-Hölder paths px i : i " 1, . . . , dq, there exists a branched rough path X above px i : i " 1, . . . , dq, that is, Remark 4.4. Given the level of generality in which Theorem 3.4 is developed, our results also apply to the case when H is a combinatorial Hopf algebra as defined in [14]. In particular, we also have a construction theorem for planarly branched rough paths [14] which are characters over Munthe-Kaas and Wright's Hopf algebra of Lie group integrators [32].

Anisotropic geometric rough paths
We now apply our results to another class of rough paths which we call aGRPs. Gyurkó introduced a similar concept in [26], which he called Π-rough paths; unlike us, he uses a 'primal' presentation, that is, paths taking values in the tensor algebra T pR d q, and p-variation norms rather than Hölder norms. GRPs over a inhomogeneous (or anisotropic) set of paths can be traced back to Lyons' original paper [29].
As in the geometric case (see Section 4.1), fix a finite alphabet A " t1, . . . , du and denote by MpAq the free monoid generated by A. We denote again by H -T pAq the shuffle Hopf algebra over the alphabet A.
Let pγ a : a P Aq be a sequence of real numbers such that 0 ă γ a ă 1 for all a, and let γ " min aPA γ a . For a word v " a 1¨¨¨ak P MpAq of length k define ωpvqγ a1`¨¨¨`γa k and observe that ω is additive in the sense that ωpuvq " ωpuq`ωpvq for each pair of words u, v P MpAq. The set L -tv P MpAq : ωpvq ď 1u is finite; ifN -tγ´1u then L Ă HN . In analogy with Lemma 2.5, the additivity of ω implies Consequently, we will consider the dual algebra pHå , ‹, εq. In this case, we define g a to be the space of truncated infinitesimal characters on H a , namely the linear functionals α P Hå such that xα, x ⧢ yy " xα, xyxε, yy`xε, xyxα, yy for all x, y P H a such that x ⧢ y P H a , and let G a -tX " expN pαq| Ha : α P g a u. As before, there is a canonical injection Hå ãÑ H˚so we suppose that xX, vy " 0 for all X P H˚and v P L.
For each λ ą 0 there is a unique coalgebra automorphism Ω λ : H Ñ H such that Ω λ a " λ γa{γ a for all a P A. We also define }¨}: G a Ñ R, As at the end of Section 2, pΩ λ q λą0 is a one-parameter family of Lie-algebra automorphisms of g a and }Ω λ X} " λ}X} for all λ ą 0 and X P G a , namely }¨} is a homogeneous norm on G a . However, unlike~¨~this norm is not subadditive and it therefore does not define a distance on G a .

4.3.1.
Signatures. In order to construct an appropriate metric on G a we consider signatures of smooth paths. We observe that A Ă L. Let x " px a : a P Aq be a collection of (piecewise) smooth paths, and define a map Spxq : r0, 1s 2 Ñ H˚by In his seminal work [11], Chen showed that Spxq is a character of pT pAq, ⧢q; in particular, Consider the metric d a pX, Y q " ř aPA |xX´Y, ay|γ {γa on Hp 1q , where we recall that H p1q is the vector space spanned by A. The anisotropic length of a smooth curve θ : r0, 1s Ñ H1 is defined to be its length with respect to this metric and will be denoted by L a pθq. Observe that since d a pΩ λ X, Ω λ Y q " λd a pX, Y q we have that L a pΩ λ θq " λL a pθq.
We now define a homogeneous norm (see the end of Section 2) |¨| CC : G a Ñ R`, called the anisotropic Carnot-Carathéodory norm, by setting |X| CC -inftL a pxq : x a P C 8 , Spxq 01 " Xu.
Since curve length is invariant under reparametrization in any metric space we obtain, as in [23, Section 7.5.4]: Proposition 4.6. The infimum defining the anisotropic Carnot-Carathéodory norm is finite and attained at some minimizing pathx.
Proposition 4.7. The anisotropic Carnot-Carathéodory norm is homogeneous, that is, Proof. Letx be the curve such that |X| CC " L a pxq. For any λ ą 0 and word v P L we have xSpΩ λx q 01 , vy " λ ωpvq{γ xSpxq 01 , vy " xΩ λ Spxq 01 , vy " xΩ λ X, vy, thus |Ω λ X| CC ď L a pΩ λx q " λL a pxq " λ|X| CC . The reverse inequality is obtained by noting that X " pΩ λ´1˝Ωλ qX. l The anisotropic Carnot-Carathéodory norm can also be seen to satisfy |X| CC " |X´1| CC and |X ‹ Y | CC ď |X| CC`| Y | CC for all X, Y P G a (see, for example, the proof of [23, Proposition 7.40]); hence it induces a left-invariant metric ρ a pX, Y q -|X´1 ‹ Y | CC on G a . Moreover, arguing as in the proof of [23,Theorem 7.44] we see that there exist positive constants c, C such that Definition 4.8. An anisotropic geometric γ-rough path, with γ " pγ a , a P Aq, is a map X : r0, 1s 2 Ñ G a which satisfies (1) the Chen rule X su ‹ X ut " X st for all ps, u, tq P r0, 1s 3 ; (2) the bound |xX st , vy| À |t´s| ωpvq for all v P L. Proposition 4.9. Anisotropic geometric γ-rough paths are in one-to-one correspondence withγ-Hölder paths X : r0, 1s Ñ pG a , ρ a q with X 0 " 1.
Proof. Let X be an anisotropic geometric γ-rough path and v a word. By definition we have that |xX st , vy| À |t´s| ωpvq , hence }X st } À |t´s|γ. The equivalence between }¨} and |¨| CC of (4.3) implies that ρ a pX s , X t q " |X st | CC À |t´s|γ, hence t Þ Ñ X t isγ-Hölder with respect to ρ a . The other direction follows in a similar manner. l Theorem 3.4 also applies to this situation, and we obtain the following: Proof. We start by constructing the homogeneous GRP X given by theγ-Hölder path X : r0, 1s Ñ GN of Corollary 3.8. Then we restrict X to H a Ă HN and we show that on this space it satisfies the stronger bound |xX st , vy| À |t´s| ωpvq for all v P L.
Recalling the proof of Theorem 3.4, we consider v P H n X H a , and we proceed by recurrence on n. For n " 0 there is nothing to prove. Suppose we have proved the result for n and let v P H n`1 X H a . In this case xX n`1 st , vy " xexp n`1 pL st`Zst q, vy " We want to prove now thaťˇˇx Then, for s " t m k , u " t m k`1 and t " t m k`2 and v " v 1¨¨¨vn`1 Now, since v j P H p1q for all j " 1, . . . , n`1 we actually have that by the assumption x a P C γa |xL su , ay| " |x a u´x a s | À 2´m γa and we have a similar estimate for L ut instead of L su . Therefore we obtain thaťˇx BCH pn`1q pL su , L ut q, vyˇˇÀ 2´m ωpvq . Therefore, from (3.5) we get b m ď 2 mpωpvq´1q b m´1`C , m ě 1, hence since b 0 " 0 we can show by recurrence on m ě 0 Since we are in the regime where ωpvq ă 1 (here we use that 1 P ř aPA γ a N) we obtain that Analogously, since L st P g, arguing as in (2.14) we have 2´mγ.
Then we can use Lemma 3.5 and obtain that the path X n`1 constructed in the proof of Theorem 3.4 is in factγ-Hölder path with values in G a . l

The Hairer-Kelly construction
In this section we develop further results specifically for branched rough paths as introduced in Section 4.2 using our general results from Section 3. We analyze in detail the Hairer-Kelly map introduced in [28], which plays a very important role in our construction, and we use it to prove Theorem 1.2 and Corollary 1.3.

The Hairer-Kelly map
Recall that T denotes the set of all decorated rooted trees, F denotes the collection of all decorated rooted forests and H BCK is the Butcher-Connes-Kreimer Hopf algebra. As in Section 4.2, Δ denotes the Connes-Kreimer coproduct on H BCK . For each n P N, n ě 1, we denote by T n the set of (non-empty) trees with at most n vertices.
Recall also from Section 4.1 that given an alphabet A we denote by T pAq the shuffle Hopf algebra generated by A, and that Δ denotes the deconcatenation coproduct on it. We fix N P N and we consider the shuffle Hopf algebras T pT q and T pT N q, namely we choose as letters of our alphabet the (non-empty) decorated rooted trees (respectively, rooted trees with at most N vertices). Note that we can identify every non-empty tree τ P T with the word in T pT q composed by the single letter τ . We also remark that, in order to avoid confusion with the forest product on H BCK we denote the concatenation of letters in T pT q by a tensor symbol.
We note that T pT q and T pT N q admit two different natural gradings, both of which make them locally finite graded Hopf algebras. One grading, as in Section 4.1, is given by the number of letters (trees) of each word, namely the degree of v " τ 1 b¨¨¨b τ k is k. The other grading is given by the sum of the number of nodes of each letter (tree), namely the degree of v " τ 1 b¨¨¨b τ k is |τ 1 |`¨¨¨`|τ k |, where we recall that forests and trees are graded in H BCK by the number of nodes, with the notation |τ | " #N τ . We remark the latter grading is always greater or equal to the former. As an example, take v " i b j k ; then, as a word v has length 2 but the total number of nodes is 3.
We recall the following result from [28,Lemma 4.9].
Lemma 5.1. We grade T pT q according to the number of nodes. Then there exists a graded morphism of Hopf algebras ψ : H BCK Ñ T pT q satisfying ψpτ q " τ`ψ n´1 pτ q for all τ P T n , where ψ n´1 denotes the projection of ψ onto T pT n´1 q.
We call ψ the Hairer-Kelly map. Since ψ is graded, for any forest τ P F the image ψpτ q is a sum of words of the form τ 1 b¨¨¨b τ k where all terms satisfy |τ 1 |`¨¨¨`|τ k | " |τ |. Observe that since ψ is a Hopf algebra morphism, in particular a coalgebra morphism, then pψ b ψqΔ 1 τ "Δ 1 ψpτ q "Δ 1 ψ n´1 pτ q, τ P T n , since trees are primitive elements in T pT q, being single-letter words. From the proof of [28,Lemma 4.9] we are able to see that in fact ψ n´1 is given by the recursion ψ n´1 " m b pψ b idqΔ 1 on the linear span of T n (see also [3, Definition 1, Section 6]).

A special class of anisotropic geometric rough paths
We have already discussed aGRPs in Section 4.3. For the Hairer-Kelly construction we need a very particular subclass of aGRPs, where the base paths px a q aPA are such that each x a is γ a -Hölder and there exists γ P s0, 1r and pk a q aPA Ă N such that γ a " k a γ; therefore the Hölder exponents are all integer multiples of a fixed exponent γ. We may of course apply the extension result of Corollary 4.10, but it turns out that in this setting we can avoid using the Carnot-Carathéodory distance and rather use a more explicit metric, which is a simple generalization of the homogeneous case (2.16).
We have already seen that the space H -T pT N q can be graded in two ways. We can even define a bigrading on this space: for 1 ď n ď N and n ď j ď nN , we define the space H pn,jq as the linear span of the words τ 1 b¨¨¨b τ n P T pT N q such that |τ 1 |`¨¨¨`|τ n | " j.
In other words, H N,N is the linear span of all words τ 1 b¨¨¨b τ n with n ď N and |τ 1 |`¨¨¨`|τ n | ď N . Therefore, analogously to (2.3) and (2.5), we have decompositions and we can see that Proof. We only have to check the triangular inequality, which is equivalent to the subadditivity property |X ‹ Y | ď |X|`|Y | for all X, Y P G N,N . Arguing as in the proof of Proposition 2.14 pX ‹ Y q pn,jq ď n ÿ m"0 Let γ P s0, 1r and N -tγ´1u. In accordance with Definition 4.8, an anisotropic geometric γ-rough path in this setting is a map X : r0, 1s 2 Ñ G N,N which satisfies (1) the Chen rule X su ‹ X ut " X st for all ps, u, tq P r0, 1s 3 ; (2) |xX st , vy| À |t´s| jγ for all v P H pn,jq with 1 ď n ď N and j ď N .
Then, arguing as in Proposition 3.3, it is easy to show that X : r0, 1s Ñ G N,N is γ-Hölder with respect to the metric ρ N,N if and only if X : r0, 1s 2 Ñ G N,N , defined as X st -X´1 s ‹ X t , is an anisotropic geometric γ-rough path with γ v " jγ for v " τ 1 b¨¨¨b τ n with n ď N and |τ 1 |`¨¨¨`|τ n | " j ď N .
The next result is the analog of Corollary 4.10 in this setting. The proof is the same, with one exception: we can use the explicit norm (5.1) rather than the Carnot-Carathéodory norm |¨| CC and we do not need the equivalence of norms result (4.3).
theorem yields a γ-Hölder pathX 2 t Ñ G N pT 2 q such that π G N pT2qÑG N pT2q{K2 pX 2 q "X 2 . Finally they construct recursively in this wayX k andX k for all k ď N .
At this point we see the difference with our approach. We do not defineX 2 t norX k but rather we constructX step by step, namely on all G k pT n q with 1 ď k, n ď N , first by recursion on k for fixed n and then by recursion on n; at each step we enforce the Hölder continuity on G k pT n q and the compatibility with the previous levels. This is done using the Lyons-Victoir technique, but in a very explicit and constructive way, in particular without ever using the axiom of choice, since we have the explicit map exp k`1˝l og k : G k pT n q Ñ G k`1 pT n q which plays the role of the injection i G{K,G : G{K Ñ G in [30, Proposition 6].

An action on branched rough paths
In this section we prove Theorem 1.2.
Given γ P s0, 1r, let N " tγ´1u and denote by C γ the set of collections of functions pg τ q τ PTN such that g τ P C γ|τ | and g τ 0 " 0 for all τ P T N . It is easy to see that C γ is a group under pointwise addition in t, that is, As a consequence of Theorem 5.4, pg,Xq Þ Ñ gX is an action of C γ on the space of aGRPs.
We use the Hairer-Kelly map ψ of Lemma 5.1 to induce an action of C γ on branched rough paths. Given a branched rough path X and g P C γ we let gX be the branched rough path defined by xgX st , τy " xgX st , ψpτ qy, whereX is the aGRP given by Theorem 5.6. As a simple consequence of Theorem 5.4 we obtain Proposition 6.1. Let X P BRP γ .
(2) If pg τ q τ PTN P C γ is such that there exists a unique τ P T N with g τ " 0, then xpgXq st , τy " xX st , τy`g τ t´g τ s and xgX, σy " xX, σy for all σ P T not containing τ as a subtree.
where we regard τ as a linear functional on H , such that xτ, σy " 1 if σ " τ and zero else. The aforementioned modification procedure then acts as a translation of the series (6.2). Specifically, for each collection v " pv 0 , . . . , v d q : T Ñ R d`1 an operator M v : H˚Ñ H˚is defined, such that for a γ-branched rough path, pM v Xq st -M v pX st q is a γ{N -branched rough path.
In the particular case where v j " 0 except for v 0 , the action of this operator can be described in terms of an extraction/contraction map : Ψ : H Ñ H b H . This map acts on a tree τ by extracting subforests and placing them in the left factor; the right factor is obtained by contracting the extracted forest and decorating the resulting node with 0. As an example, consider Extending v " v 0 : T Ñ R to all of H˚as an algebra morphism it is shown that xpM v Xq st , τy " xX st , pv b idqΨpτ qy. |xX st , τy| |t´s| p1´γq|τ |0`γ|τ | ă 8, (6.4) where |τ | 0 counts the times the decoration 0 appears in τ . Essentially, this condition imposes that the components corresponding to the zero decoration be Lipschitz on the diagonal s " t.
We now show how this setting can be recovered from the results of Section 6. Let X be a γ-branched rough path on R d`1 satisfying (6.4). Since M v X is again a γ-branched rough path, by Theorem 6.2 there exists a collection of functions g P C γ such that gX " M v X. Moreover, this collection is the unique one satisfying g τ t´g τ s " xX st , pv b idqΨpτ qy´xX st , τy´xgX st , ψ |τ |´1 pτ qy (6.5) for all τ P T pR d`1 q where we have used (6.3) in order to express M v X in terms of Ψ. Theorem 28 in [4] ensures that the first term on the right-hand side is in C γ|τ | 2 hence g is actually in C γ|τ | as required.
The approach of [4] is based on pre-Lie morphisms and crucially on a cointeraction property, which has been explored by [7] (see in particular [4,Lemma 18]). The cointeraction property can be used for time-independent modifications, indeed note that the functional v in [4] is always constant.
Let us see why this is the case. The approach of [4] is based on a cointeraction property studied by [6,7,21] between the Butcher-Connes-Kreimer coproduct and another extractioncontraction coproduct δ : H Ñ H b H . The formula is the following: Let us consider now a character v P H˚. If we multiply both sides by pv b id b idq and set Mv " pv b idqδ : H Ñ H as in [4,Proposition 17], then we obtain Δ Mv " pMv b Mv qΔ, namely Mv is a coalgebra morphism on H . Then one can define a modified rough path as vX -M v X " X˝Mv . The crucial Chen property is still satisfied since : In [4] this map is named δ but we choose to call it Ψ in order to avoid confusion with the operator defined here. pvXq st " pv b X st qδ " pv b X su b X ut qpid b Δqδ " pv b X su b X ut qM 1,3 pδ b δqΔ " ppv b X su q b pv b X ut qqpδ b δqΔ " ppvXq su b pvXq ut qΔ.
However, this does not work if v : r0, 1s 2 Ñ H˚is a time-dependent character. Indeed in this case we set pvXq st -pv st b idqδ and we obtain pvXq st " pv st b X st qδ " pv st b X su b X ut qpid b Δqδ " pv st b X su b X ut qM 1,3 pδ b δqΔ " ppv st b X su q b pv st b X ut qqpδ b δqΔ but we cannot conclude that this is equal to ppvXq su b pvXq ut qΔ. Our construction, as explained after formula (1.8), is not purely algebraic but is based on a (non-canonical) choice of generalized Young integrals with respect to the rough path X. Moreover our transformation group, infinite-dimensional, is much larger than that finite-dimensional group studied in [4].

Perspectives
In this paper we have shown that the space of branched γ-rough paths is a principal homogeneous space with respect to the linear group C γ . This is related to the analytical properties of the operator δ defined in (1.2), which is invertible under the conditions of Gubinelli's Sewing Lemma, but not in general, and in particular not in the context of the Chen relation on trees with low degree.
It would be now interesting to see how this action can be translated on the level of controlled paths [24]. The space of paths controlled by a rough path X P BRP γ should be interpreted as the tangent space to BRP γ at X, and the action on rough paths should induce an action on controlled paths. In particular it should be possible to write an action on solutions to rough differential equations.
The proof of Theorem 6.2, and in particular (6.1), gives a recursive way of computing the unique g P C γ translating a given branched γ-rough path into another. An interesting feature of the BCFP scheme is that is given in terms of a coaction so explicit calculations are somewhat easier in this more restricted case as one can compute g τ for each tree τ P T N directly by extracting and contracting subforests of τ without doing any recursions (see (6.5).) However, we do not have a computational rule for an important case: suppose that X is branched rough path lift of a stochastic process with almost surely C γ´t rajectories; it would be nice to have a way of finding g P C γ such that gX is centered with respect to the underlying distribution of the process, provided this is possible. Even this last problem, namely giving precise conditions under which this centering is possible is interesting in itself. This should be related to the notion of Wick polynomials and deformations of products as considered in [18].
More generally, in the physics literature there are various renormalization procedures which allow to obtain convergent iterated integrals from divergent ones by subtracting suitable 'counterterms'. In the context of rough paths, implementing one of the most accepted such procedures due to Bogoliubov-Parasiuk-Hepp-Zimmermman (BPHZ) has been carried out by Unterberger in [35,36] by means of the Fourier normal ordering algorithm and using a technique relating the trees in the Butcher-Connes-Kreimer Hopf algebra to certain Feynman diagrams. In our context, this could provide a canonical choice for g P C γ implementing the BPHZ renormalization procedure in a way analogous to what is done in [6] for regularity structures.