Abstract
We introduce the class of “smooth rough paths” and study their main properties. Working in a smooth setting allows us to discard sewing arguments and focus on algebraic and geometric aspects. Specifically, a Maurer–Cartan perspective is the key to a purely algebraic form of Lyons’ extension theorem, the renormalization of rough paths following up on [Bruned et al.: A rough path perspective on renormalization, J. Funct. Anal. 277(11), 2019], as well as a related notion of “sum of rough paths”. We first develop our ideas in a geometric rough path setting, as this best resonates with recent works on signature varieties, as well as with the renormalization of geometric rough paths. We then explore extensions to the quasigeometric and the more general Hopf algebraic setting.
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1 Introduction
Recent years led to a remarkable convergence of different streams of mathematics. At the center of it is the notion of path \(X: [0, T]\to \mathbb {R}^{d}\) and the “stack” of its iterated integrals
commonly called the signature of X over [0,T], which takes values in \(T((\mathbb {R}^{d}))\), the space of tensor series over \(\mathbb {R}^{d}\). It is easy to see that the signature of a path segment actually takes its values in a very special curved subspace of the tensor (series) algebra, \(G(\mathbb {R}^{d}) \subset T((\mathbb {R}^{d}))\), with a natural group structure. This construction, which originates in Chen’s fundamental work [14], is central to the theory of rough paths and stochastic analysis [29, 41, 43]. Specifically, the path t↦X_{t} := Sig(X_{[0,t]}) is the canonical rough path lift of X, for any sufficiently smooth path X to make the signature welldefined. These ideas have proven useful in a remarkable variety of fields, stretching from machine learning [15] to algebraic geometry and renormalization theory. More specifically, we mention:
Signature varieties
In [1], Améndola, Sturmfels and one of us study the geometry of signatures tensors. The signatures of a given class of smooth paths parametrize an algebraic variety inside the space of tensors, derived from the free nilpotent Lie group, with surprising analogies with the Veronese variety from algebraic geometry. These signature varieties provide both new tools to investigate paths and new challenging questions about their behaviour. In [1] piecewise linear paths and polynomial paths are investigated. In a later work by Galuppi [30], and in the terminology of this paper, the role of classical smooth paths has been replaced by certain smooth rough paths, see Definition 2.1. Related recent works on signature varieties include [18, 49].
Rough paths and renormalization
Rough paths were famously used to solve the singular KPZ stochastic partial differential equation [33] and subsequently led to the theory of regularity structures for general singular SPDEs [34], with precise correspondences to rough paths highlighted e.g. in [26, Sec. 13.2.2]. The central topic of renormalization of singular SPDEs [8] was revisited from a rough path perspective in [6], which notably introduced preLie algebras, and further inspired progress in the field [7], see also Otto et al. [40].
We cannot possibly expose in full the subtle intertwining of probabilistic, analytic, geometric and algebraic techniques of the above works, but still, sketch the general idea in a simple setting. If X models a realization of noise, such as Brownian sample paths, then the very notion of Stieltjes integration against dX_{u} is illdefined (with probability one, such paths are not of locally bounded variation). The stochastic analysis provides probabilistic solutions: Stratonovich integration amounts to work with mollified X, followed by taken limits in probability, whereas Itô integration respects the martingale structure of such processes. Rough path theory (later: regularity structures) understands that these different calculi can be hardcoded imposing the first Niterated integrals of X. The resulting object, an enhancement of X, is called a rough path (resp. model in the context of regularity structures).
There are situations—notably the KPZ equation—when the Stratonovich solution diverges, whereas the correct and desired object is given by the Itô solution. From a rough paths perspective, this amounts to adjusting (“renormalize”) the higher levels of the aforementioned enhancement. Doing so in an algebraically consistent way is a highly nontrivial task, and was achieved in a rough path, resp. regularity structure, set in the aforementioned works. In particular, [8] develops the algebraic theory of renormalization exclusively for smooth models, to which our study of smooth rough paths is aligned.
In this paper, we look at the indefinite signature path t↦X_{t} := Sig(X_{[0,t]}), also known as canonical rough path lift, of a \(\mathbb {R}^{d}\)valued smooth path X, as the solution of the linear differential equation
with, as is wellknown (see e.g. [41, Section 2.1.1] or [29, Chapter 7]) \(\mathcal {G} := G(\mathbb {R}^{d} ) = \exp _{\otimes }(\mathfrak {g}) \subset T((\mathbb {R}^{d}))\), where \(\mathfrak {g} := {\mathscr{L}}((\mathbb {R}^{d})) = \mathbb {R}^{d} \oplus [\mathbb {R}^{d},\mathbb {R}^{d}] \oplus {\cdots } \) denotes the space of Lie series. The same construction in the quotient (or levelN truncated) algebra
yields a finitedimensional Lie group \(\mathcal {G}^{N} := G^{N}(\mathbb {R}^{d} )= \exp _{\otimes _{N}}(\mathfrak {g}^{N}) \subset T^{N} (\mathbb {R}^{d})\) with Lie algebra \(\mathfrak {g}^{N} := {\mathscr{L}}^{N} (\mathbb {R}^{d})\), the Lie polynomials of degree less equal N.
This yields an increasing family of Lie algebras (resp. groups) with inclusion maps i_{N} (resp. j_{N}), and Lie exponentials \(\exp _{\otimes N}\) and \(\exp _{\otimes }\), N in \(\mathbb {N}\), see Section 2.
Cartan’s classical development of a smooth \(\mathfrak {g}^{N}\)valued path \(\mathfrak {y}\) is precisely given by solving the differential equation
Basic results on linear differential equations in finite dimensions guarantee a unique and global solution. This gives a welldefined “projective” way to solve for \(\dot {\mathbf {X}}_{t} = \mathbf {X}_{t} \otimes \mathfrak {y} (t)\), \(\mathbf {X}_{0} = \mathbf {1} \in \mathcal {G}\), for any \(\mathfrak {g}\)valued path \(\mathfrak {y}\). This is a precise generalization of (1) and leads us to the class of smooth rough paths^{Footnote 1}. In infinitedimensional Lie group theory, solvability of such equations has led to the notion of regular Lie group [47]. However, this is of no concern to us and we refer to [3, 4] for the subtleties of infinite groups like \(\mathcal {G}\).
Conversely, every smooth \(\mathcal {G}^{N}\)valued path X_{t} is the Cartan development of
Here ω_{x} is the \(\mathfrak {g}^{N}\)valued (left invariant) Maurer–Cartan form, given at \(\mathbf {x} \in G^{N}(\mathbb {R}^{d})\), by
which can be viewed as left logarithmic derivative of the identity map of \(G^{N}(\mathbb {R}^{d})\), see also [42]. The reason we encounter the left invariant Maurer–Cartan form, rather than its right invariant counterpart ((dx) ⊗x^{− 1}) can be traced back to the order of interated integrals in the definition of the signatures, i.e. \(u_{1} \le {\dots } \le u_{n}\) rather than \(u_{n} \le {\dots } \le u_{1}\). We recall also that the Maurer–Cartan form has appeared in previous works of rough paths on manifolds [11], as well as signature based shape analysis [12]. We shall see in Sections 2.4 and 3.4 that renormalization of rough differential equations, in the spirit of [6], can much benefit from this geometric viewpoint. (For earlier use in the context of renormalization see also [16].) In the context of renormalization of rough differential equations however, its use appears to be new.
The geometry of \(\mathcal {G}\) encodes validity of a chain rule, equivalently expressed in terms of shuffle identities, that in turn exhibits \(\mathcal {G}\) as a character group of the shuffle Hopf algebra. This suggests correctly that the Maurer–Cartan perspective is not restrictive to \(G(\mathbb {R}^{d})\)valued (“geometric”) rough paths, but valid for “general” rough paths, in the sense of [54], with values in the character groups of a general graded Hopf algebra. However, too much generality does not allow for some of the concrete applications we have in mind, notably an understanding of differential equations driven by rough paths and their renormalization theory. We thus commence in Chapter 2 with smooth instances of geometric rough paths (in short: grp), which can be thought of as a multidimensional path enhanced with iterated integrals with classical integration by parts (shuffle) relations, together with suitable analytic conditions. A typical (nonsmooth) example is given by Brownian motion with iterated integrals in the Stratonovich sense, see e.g.[26, Sec. 2.2] for precise definitions, see also [26, 29, 35, 41, 43]. (We will review what we need in the main text below, cf. Definitions 2.1 and 2.3.) Another aspect concerns the subRiemannian structure of \(G^{N}(\mathbb {R}^{d} )\), the state space of (levelN) geometric rough paths, see e.g. in [29, Remark 7.43] or [25].
The indefinite signature of a \(\mathbb {R}^{d}\)valued path always stays tangent to the leftinvariant vector fields generated by the d coordinate vector fields. In subRiemannian geometry, such paths are called horizontal. The study of smooth geometric rough paths is effectively the study of (possibly nonhorizontal) smooth paths on \(\mathcal {G}\). It is classical in geometric rough path theory to equip this space with the Carnot–Caratheodory metric. A generic smooth geometric rough path then has infinite length and is thus a genuine and interesting example of rough paths. We then continue to extend smooth rough paths in a more general setting:
Quasigeometric theory
A quasigeometric rough path should then be thought of as a multidimensional path enhanced with iterated integrals with classical integration by parts (shuffle) relation replaced by a generalized integration by parts rule known as quasishuffle. A typical example is given by Brownian motion with iterated integrals in the Itô sense. This structure also arises naturally when dealing with discrete sums (cf. [22] for signature sums) or piecewise constant paths, as well as Lévy processes [19], general semimartingales and finally rough path analysis [2, 7]. We also note unpublished presentations on quasirough paths by D. Kelly, who first introduced the concept, following his work [35, 38]. The very influential article [35] focused on the interplay between geometric and branched rough paths, introduced in [32], not central to this work. We note that quasishuffle structures have emerged independently in renormalization theory, see e.g. [17, 39, 45, 46] and the references therein.
Hopf algebra constructions
We finally revisit the previous constructions from a general Hopf algebra construction. In view of [6] we do not single out the case of branched rough paths. Our gradual approach to treat first geometric (Chapter 2), then quasigeometric (Chapter 3) and finally the Hopf algebra case (Chapter 4) is a choice we made for essentially two reasons: (i) The material of Chapter 2 remains accessible to readers with a minimum on prerequisites and is also the setting that is most used in the context of signatures, including the recent developments in algebraic geometry. (ii) Not every result obtained in the (quasi)geometric setting has a precise counterpart in the Hopf algebraic generality. For instance, as already observed in [6] in the context of branched rough paths, one loses certain uniqueness properties of renormalization operators when passing to more general structures.
Let us list the main contributions of this work with some detailed pointers to the main text.

With Definitions 2.1, also 3.8, 4.1 we introduce the class of smooth rough paths in their respective setting and show in Theorems 2.8, also 3.10, 4.2 that levelN smooth rough paths have a lift whose uniqueness hinges on some algebraic/geometric minimality. (This is in contrast to the classical Lyons lift of levelN rough paths, where uniqueness depends on analytical conditions, see e.g. [29, Chapter 9].) We note that a related minimality condition appeared in the context of rough paths with jumps [27].

An interesting insight is then that smooth rough paths, the resulting space of which is by nature nonlinear, can be given a canonical linear structure. (This should be contrasted with ad hoc linearizations based on Lyons–Victoir extension, see e.g. [26, Ex. 2.4] and especially [54].)

We finally revisit differential equations driven by our classes of rough paths, followed by their renormalization as initiated in [6]. Specifically, in Theorem 2.26 we highlight the role of smooth rough paths in the argument. Our subsequent extensions in Sections 3 and 4 complements (and differ from) existing results, notably [6], with a sole focus on geometric and branched structures, and related works [7, 9] that involve/pass through branched constructions.
2 Smooth Geometric Rough Paths
2.1 Definitions and Fundamental Properties
Let \((T((\mathbb {R}^{d})), +, \otimes )\) denote the algebra of tensor series over \(\mathbb {R}^{d}\), equipped with the standard basis e_{1},…,e_{d}. Elements of \(T((\mathbb {R}^{d}))\) are of the form
with summation over all words w = ℓ_{1}…ℓ_{n} with letters ℓ_{j} ∈{1,…,d}, scalars x^{w} and \(e_{w} := e_{l_{1}} \otimes {\dots } \otimes e_{l_{n}}\). The summation includes also 1, the empty word. There is a natural pairing of \(T((\mathbb {R}^{d}))\) with \(T(\mathbb {R}^{d})\), the space of tensor polynomials linearly spanned by the e_{w} so that
The same pairing applies to the truncated space \(T^{N}(\mathbb {R}^{d})\), consisting of tensor polynomials of degree at most \(N\in \mathbb {N}\), spanned by pure tensors e_{w} whose word w has length w≤ N. We denote the canonical projection onto \(T^{N}(\mathbb {R}^{d})\) as \(\textsf {proj}_{N}\colon T((\mathbb {R}^{d}))\to T^{N}(\mathbb {R}^{d})\). Equivalently, we can introduce \(T^{N}(\mathbb {R}^{d})\) as a quotient algebra. Indeed, introducing the ideal \(T^{>N}((\mathbb {R}^{d})) := {\bigoplus }_{n> N}^{\infty }(\mathbb {R}^{d})^{\otimes n}\) one immediately has the following isomorphism
We write ⊗_{N} for the induced truncated tensor product.
Using the identification between words and tensors, we denote by the shuffle product on words , defined by for any word w and the recursive definition
for any couple of words w, v and letters i,j ∈{1,…,d}. The product induces a commutative algebra^{Footnote 2} on \(T(\mathbb {R}^{d})\).
There are several Lie algebras we want to look at which stem from the algebra \((T((\mathbb {R}^{d})),\otimes )\); we use the notation \({\mathscr{L}}(\mathbb {R}^{d})\) for the Lie algebra generated by the letters \(\mathbb {R}^{d}\) (the space of Lie polynomials), \({\mathscr{L}}^{N}(\mathbb {R}^{d}) \) for the projection of \({\mathscr{L}}(\mathbb {R}^{d})\) into \(T^{N} (\mathbb {R}^{d})\), and \({\mathscr{L}}((\mathbb {R}^{d})) \subset T((\mathbb {R}^{d}))\) the Lie series. We define the tensor and truncated tensor exponentials \(\exp _{\otimes }\colon T^{>0}((\mathbb {R}^{d})) \to T((\mathbb {R}^{d}))\), \(\exp _{\otimes _{N}}\colon T^{N}(\mathbb {R}^{d})\cap T^{>0}((\mathbb {R}^{d}))\to T^{N}(\mathbb {R}^{d})\) given respectively by
Then it is well known ([43, 53]) that \(G(\mathbb {R}^{d})= \exp _{\otimes } {\mathscr{L}}((\mathbb {R}^{d}))\) is a group with operation ⊗ which satisfies
Similar results hold for \(G^{N}(\mathbb {R}^{d})\), now in terms of words with joint length ≤ N and as the image of \({\mathscr{L}}^{N}(\mathbb {R}^{d})\) under \(\exp _{\otimes _{N}}\) respectively. We note that \((G^{N}(\mathbb {R}^{d}), \otimes _{N})\) is a bona fide finitedimensional Lie group with Lie algebra \({\mathscr{L}}^{N}(\mathbb {R}^{d})\).
The following definitions are standard (as e.g. found in [35]), but with analytic Hölder / variation type conditions replaced by a smoothness assumptions.
Definition 2.1
We call levelN smooth geometric rough path (in short: Nsgrp) over \(\mathbb {R}^{d}\) any nonzero path \(\mathbf {X}: [0,T] \to T^{N}(\mathbb {R}^{d})\) such that

(a.i)
The shuffle relation holds for all times t ∈ [0,T]
(5)for all words with joint length v + w≤ N.

(a.ii)
For every word of length w≤ N, the map t↦〈X_{t},w〉 is smooth. We write
$$ \dot{\mathbf{X}}_{t} = {\sum}_{w\le N} \langle\dot{\mathbf{X}}_{t},w\rangle e_{w} $$for the derivative of X.
By smooth geometric rough path (in short: sgrp) we mean a path with values in \(T((\mathbb {R}^{d}))\) with all defining “≤ N” restrictions on the word’s length omitted.
Remark 2.2
By Proposition 2.14 below Nsgrp are smooth \(G^{N}(\mathbb {R}^{d})\)valued paths, thus (with respect to the appropriate Carnot–Caratheodory metric [29]) genuine 1/NHölder regular rough paths, which justifies our terminology. Similarly, sgrps are nothing but smooth \(G(\mathbb {R}^{d})\)valued paths, provided \(G(\mathbb {R}^{d})\) is equipped with a suitable “weak” differential structure to make it a topological Lie group, see [4].
The shuffle relation applied to empty words and the demand that X is nonzero imply together with continuity in t that 〈X_{t},1〉 = 1 for all t ∈ [0,T], hence X is (similar to formal power series) invertible with respect to ⊗_{N}. Every Nsgrp gives then rise to increments, \((s,t) \mapsto \mathbf {X}_{s}^{1} \otimes _{N} \mathbf {X}_{t}\). This motivates the following definition.
Definition 2.3
We call levelN smooth geometric rough model (in short: Nsgrm) over \(\mathbb {R}^{d}\) any nonzero map \(\mathbf {X}: [0,T]^{2} \to T^{N}(\mathbb {R}^{d})\) such that

(b.i)
the shuffle relation (5) holds with X_{t} replaced by X_{s,t}, any s,t.

(b.ii)
Chen’s relation holds, by which we mean
$$ \mathbf{X}_{s,u} \otimes_{N} \mathbf{X}_{u,t}=\mathbf{X}_{s,t} $$(6)for any s,u,t ∈ [0,T].

(b.iii)
For every word of length w≤ N, the map ↦〈X_{s,t},w〉 is smooth, for one (equivalently: all) base point(s) s ∈ [0,T].
By smooth geometric rough model (in short sgrm) we mean a map with values in \(T((\mathbb {R}^{d}))\), with all “≤ N” quantifiers omitted and (6) with ⊗_{N} replaced by ⊗.
Remark 2.4
The terminology “model” is consistent with that of Hairer’s regularity structures. More specifically, given a Nsgrm X = X_{s,t} we have the map
which together with Chen’s relation yields precisely a model in the sense of regularity structures. See e.g. [26, Sec. 13.2.2], [50, Theorem 5.15] and [6, Proposition 48]. Our use of the adjective “smooth” is also consistent with the notion of smooth model, used by Hairer and coworkers and central to the algebraic renormalization theory of [8].
It follows from (5), resp. (b.i), and the nonzero condition that Nsmooth geometric rough paths and models actually take values in \(G^{N}(\mathbb {R}^{d})\); similarly for (tensor series) smooth rough paths and models and \(G(\mathbb {R}^{d})\). Clearly, every levelN smooth geometric rough path \(\mathbf {X}:[0,T]\to T^{N}(\mathbb {R}^{d})\) induces a levelN smooth geometric rough model \(\mathbf {X}: {[0,T]^{2}} \to T^{N}(\mathbb {R}^{d})\) in the above sense, by considering the increments
Conversely, every levelN smooth geometric rough model defines a levelN smooth geometric rough path X_{t} := X_{0,t} so that Nsgrp and Nsgrm are equivalent modulo a starting point in \(G^{N} (\mathbb {R}^{d})\).
Definition 2.5
A sgrp X is called extension of some Nsgrp Y if
if this holds for a \(N^{\prime }\)sgrp X, with \(N < N^{\prime } < \infty \), we call it \(N^{\prime }\)extension of Y. We adopt also the same denomination if Y is a Nsgrm and X is a sgrm (\(N^{\prime }\)sgrm) and one has the same relation above for any s,t ∈ [0,T].
When d = 1, the situation is trivial, and sgrps are in onetoone correspondence with smooth scalar paths: An arbitrary scalar path Y with initial point Y_{0} = 0 has a (unique) extension given by
Conversely, every sgrp with X_{0} = 1 must be of this form, as a consequence of the shuffle relations. When d > 1 such extensions are never unique. For instance, given the two basis vectors \(e_{1},e_{2}\in \mathbb {R}^{2}\) and denoting by 0 the common zero of \(T(\mathbb {R}^{2})\), we see that \(t \mapsto (1,\mathbf {0},\mathbf {0})\in T^{2}(\mathbb {R}^{2})\) and \(t \mapsto (1,\mathbf {0}, t[e_{1},e_{2}]) \in T^{2}(\mathbb {R}^{2})\) are both 2sgrps over \(\mathbb {R}\), and hence level2 extensions of the trivial 1sgrp t↦(1,0).
In classical rough path analysis [29, 41], a graded pvariation (or Hölder) condition is enforced, which guarantees a unique extension. We give here a novel algebraic condition that enforces uniqueness, somewhat similar in spirit to the minimal jump extension of cadlag rough paths in [27]. This condition is motivated by the following result.
Proposition 2.6
Given a sgrm X over \(\mathbb {R}^{d}\), the diagonal derivative defined for all times s by
lies in \({\mathscr{L}}((\mathbb {R}^{d})) \subset T((\mathbb {R}^{d}))\). The analogue statement holds for Nsgrm’s, with \({\mathscr{L}}((\mathbb {R}^{d}))\) replaced by its truncation \({\mathscr{L}}^{N}(\mathbb {R}^{d})\) (the Lie polynomials).
Proof
A geometric proof is not difficult. We define the \(G^{N}(\mathbb {R}^{d})\)valued path X_{t} := X_{0,t} so that \(\dot {\mathbf {X}}_{s,s}:= \mathbf {X}_{s}^{1} \otimes \dot {\mathbf {X}}_{s}\) which is exactly the Maurer–Cartan form at X_{s}, evaluated at the tangent vector \(\dot {\mathbf {X}}_{s}\). This results in an element in the Lie algebra, which here is identified with \({\mathscr{L}}^{N}(\mathbb {R}^{d})\). Since N is arbitrary, this also implies the first claim. □
Remark 2.7
Let us illustrate the previous proposition in the case N = 2. The defining shuffle relation of sgrm’s applied to single letter words i and j yields
Dividing this identity by t − s > 0, followed by sending t ↓ s, one has immediately
so that the second tensor level of \(\dot {\mathbf {X}}_{s,s}\) is antisymmetric, hence \(\dot {\mathbf {X}}_{s,s} \in \mathbb {R}^{d} \oplus [\mathbb {R}^{d},\mathbb {R}^{d}]\).
Theorem 2.8 (Fundamental theorem of sgrm)
Given an Nsgrm Y for some \(N \in \mathbb {N}\), there exists exactly one sgrm extension X of Y which is minimal in the following sense:
This unique choice then in fact satisfies \(\dot {\mathbf {X}}_{s,s}=\dot {\mathbf {Y}}_{s,s}\). We call MinExt(Y) := X the minimal extension of Y and also \(\mathrm {MinExt^{\textit {N}^{\prime }}(Y)} := \text {\textsf {proj}}_{N^{\prime }} \mathbf {X}\), for \(N^{\prime } > N\), the \(N^{\prime }\)minimal extension of Y. For a fixed interval [s,t] ⊂ [0,T], X_{s,t} only depends on {Y_{[u,v]} : s ≤ u ≤ v ≤ t} and we introduce the signature of Y on [s,t] defined by
Proof
(Existence) Thanks to the previous proposition
defines a smooth path with values in the Lie algebra \({\mathscr{L}}^{N} (\mathbb {R}^{d})\). Its Cartan development into \(G(\mathbb {R}^{d})\) amounts to solving \(\dot {\mathbf {X}}_{t} = \mathbf {X}_{t} \otimes \mathfrak {y} (t)\) with \(\mathbf {X}_{0} = \mathbf {1} \in G(\mathbb {R}^{d})\). It is enough to do this in finite dimensions, say in \(G^{N^{\prime }}(\mathbb {R}^{d})\) for arbitrary \(N^{\prime }>N\), in which case such differential equations have a unique and global solution. Indeed, existence of a unique local solution is clear from ODE theory, whereas nonexplosion is a consequence of linearity of this equation. See also a much more general reference [37]. Since the natural projections from \(G^{N^{\prime }+1}(\mathbb {R}^{d}) \to G^{N^{\prime }}(\mathbb {R}^{d})\) are Lie group morphisms, we obtain a consistent family of \(N^{\prime }\)extensions which defines t↦X_{t} in the projective limit. The minimal extension is then given by \(\mathbf {X}_{s,t} = \mathbf {X}_{s}^{1} \otimes \mathbf {X}_{t}\), where we note that
(Uniqueness) It is sufficient to consider \(N^{\prime }=N+1\). In this case two level\(N^{\prime }\) extensions X, \(\bar {\mathbf {X}}\), differ by an element in the center of \(G^{N^{\prime }}(\mathbb {R}^{d})\), so that \({\varPsi }_{s,t} := \langle \mathbf {X}_{s,t} \bar {\mathbf {X}}_{s,t}, w \rangle \) is additive for every word w≤ N + 1. We need to show that Ψ ≡ 0. By assumption, Ψ vanishes on words of length w≤ N, so we can assume \(w = N^{\prime }=N+1\). With Ψ_{t} := Ψ_{0,t} note that Ψ_{s,t} = Ψ_{t} −Ψ_{s}. Write X_{t} = X_{0,t}, and similarly for \(\bar {\mathbf {X}}\), and also \(\sim \) for equality in the limit t ↓ s. Then
and in view of the minimality assumption of order N of both X, \(\bar {\mathbf {X}}\), with the condition w = N + 1, we see that \(\dot {\varPsi }\) is zero, hence Ψ ≡Ψ_{0} = 0 which concludes the argument for uniqueness. □
Remark 2.9
(Computing the minimal extension) Note that the existence part of this proof is constructive and gives the minimal extension via a solution of a linear differential equation. To make this explicit in case of a Nsgrp Y, it suffices to solve
To compute the signature on [s,t], it suffices to start at the initial condition X_{s} = 1.
It follows automatically from the proof of Theorem 2.8 that, defining MinExt(N) as the class of sgrms which arise as minimal extension of some Nsgrms over \(\mathbb {R}^{d}\), we have the inclusions
When d > 1, all inclusions above are strict. For instance, take a nonfinite Lie series, i.e. \(\mathfrak {v} \in {\mathscr{L}}((\mathbb {R}^{d})) \backslash {\mathscr{L}}(\mathbb {R}^{d})\). Then \(\mathbf {X}_{s,t} = \exp _{\otimes }(\mathfrak {v}(ts))\) defines a sgrm, which is not a minimal extension of any Nsgrm. These strict inclusions motivate the following finer subset of sgrms.
Definition 2.10
A sgrm X over \(\mathbb {R}^{d}\) is called a good sgrm if X = MinExt(Y) for some Nsgrm Y, \(N \in \mathbb {N}\).
Recalling that a minimal extension has the same diagonal derivative as the underlying Nsgrm, one has the immediate characterisation of good sgrms.
Lemma 2.11
A sgrm X is good if and only if for all times s ∈ [0,T] one has
We link the notion of minimal extension with previous constructions in the literature.
Example 2.12
Every smooth path \(Y: [0,T] \to \mathbb {R}^{d}\) can be regarded (somewhat trivially) as 1sgrp Y = (1,Y ). The solution to
is then precisely given by the stack of iterated integrals
with nfold integration over the nsimplex over [0,t], i.e. \(0 \le u_{1} \le {\dots } \le u_{n} \le t\). The signature of Y on [s,t] coincides then with the usual definition of signature of Y, see e.g. [43], modulo a choice of unit initial data at time s, which entails integration over simplices over [s,t].
Example 2.13
Fix a step2 Lie element over \(\mathbb {R}^{d}\),
where \([\mathbb {R}^{d}, \mathbb {R}^{d}]\) stands for the set of antisymmetric 2tensors over \(\mathbb {R}^{d}\). Consider the example of a 2sgrm given by
One easily sees that \(\dot {\mathbf {Y}}_{t,t}= \mathfrak {v}\) for all t ∈ [0,T]. Since \(\mathfrak {v}\) is constant in time, the differential equation
has an explicit exponential solution at any time t given by
This is precisely the signature of Y on [s,t] and X is the minimal extension of Y. Modulo a starting point in the group such a construction is wellknown as a loglinear rough path, here the special case of second order logarithm. The case \(\mathfrak {v} = (\mathfrak {a},0) \) is covered in Example 2.12 via the constant velocity path \(\dot {Y} \equiv \mathfrak {a}\). By taking \(\mathfrak {v} = (0,\mathfrak {b})\) the minimal extension coincides with a pure area rough path, see [30] for an algebraic geometric perspective on these structures. Giving the explicit exponential solution (8) is possible here thanks to the constant velocity \(\mathfrak {v}\). In a general situation, with timedependent \(\mathfrak {v}=\mathfrak {v} (t)\) the solution has exponential form given by Magnus expansion and additional commutator terms will appear, see e.g. [37].
We finish this subsection by relating sgrp to (weakly) geometric rough paths in the sense of standard definitions as found e.g. in [29, 35].
Proposition 2.14
One has the following properties:

(i)
Every Nsgrp X is a 1/NHölder weakly geometric rough path. Consequently, X is also a γHölder geometric rough path, for any 1/γ ∈ (N,N + 1), and the set of Nsgrps is dense therein.

(ii)
The minimal extension of some Nsgrp X coincides with the Lyons lift of X, as constructed e.g. in [29, Chapter 9].
Proof
(i) We only need to discuss the analytic regularity, which is formulated for the increments \(\mathbf {X}_{s,t} = \mathbf {X}^{1}_{s}\otimes \mathbf {X}_{t}\), i.e. the associated rough model. Thanks to (b.iii), we have \(\langle \mathbf {X}_{s,t},w\rangle  \lesssim ts \lesssim ts^{w(1/N)}\) uniformly over s,t ∈ [0,T], using that w≤ N. The consequence follows from wellknown relations between geometric and weakly geometric rough paths [29, Chapter 9]. The final density statement is clear, since already the minimal levelN extensions of smooth paths (a.k.a. canonical lifts) are dense in this levelN rough paths space, cf. [29].
(ii) We remark that the minimal extension of X can be constructed by solving a linear differential equation driven by X, in the sense of Definition 2.18 below, whereas the Lyons lift solves the analogous rough differential equations. □
2.2 Canonical Sum and Minimal Coupling of Smooth Geometric Rough Models
As was pointed out, the state space of a sgrp is \(G(\mathbb {R}^{d})\), a nonlinear subset of \(T((\mathbb {R}^{d}))\). In particular, the pointwise sum X_{t} + Y_{t} of any given two sgrps X, Y does not generally make sense as a sgrp. From a classical rough path perspective, the obstruction to making sense of the addition of rough path increments lies in its possible dependence on (a priori) missing mixed iterated integrals. For smooth rough paths, however, it turns out that there is a canonical way to add smooth geometric rough models and, more generally a scalar multiplication.
Definition 2.15
For any fixed sgrms X, Y let \(t\mapsto \mathbf {Z}_{t} \in T ((\mathbb {R}^{d}))\) be the Cartan development of \(\dot {\mathbf {X}}_{s,s}+\dot {\mathbf {Y}}_{s,s}\), i.e. the unique solution to
We then write \(\mathbf {Z} := \mathbf {X} \boxplus \mathbf {Y}\) for the associated sgrm and call it the canonical sum of X and Y. For any \(\lambda \in \mathbb {R}\) we define also the sgrm \(\mathbf {Z}=\lambda \boxdot \mathbf {X}\) via the Cartan development of \(\lambda \dot {\mathbf {X}}_{s,s}\), we call it the canonical scalar multiplication.
We will see later that this addition of sgrm overlaps nontrivially with the renormalization of rough path. Operations \(\boxdot \) and \(\boxplus \) equip the space of sgrps with a vector space structure and by Lemma 2.11 good sgrps form a linear subspace.
For some general comments on these operations we refer to Remark 4.15 in the more general framework of roughs path associated to a Hopf algebra. We simply remark that in the smooth geometric setting the canonical sum coincides with a construction of Lyons for pvariation weakly geometric rough paths, see [41, Section 3.3.1 B].
Proposition 2.16
For any fixed couple of sgrms X, Y a map \(\mathbf {Z}\colon [0,T]^{2}\to G(\mathbb {R}^{d})\) coincides with \(\mathbf {X} \boxplus \mathbf {Y}\) if and only if Z satisfies Z_{s,t} = Z_{s,u} ⊗Z_{u,t} for s,u,t ∈ [0,T] and one has for any s ∈ [0,T]
for some \(R_{s,t}\in T((\mathbb {R}^{d}))\) such that for all \(x\in T(\mathbb {R}^{d})\) one has 〈R_{s,t},x〉 = o(t − s) as t → s. Moreover, we have the relations
for some r_{s,t}, \(r^{\prime }_{s,t} \in T((\mathbb {R}^{d}))\) such that for all \(x\in T(\mathbb {R}^{d})\) one has \(\langle r_{s,t},x\rangle ,\langle r^{\prime }_{s,t},x\rangle = o(ts)\) as t → s.
Proof
See the more general Proposition 4.16. □
Remark 2.17
Looking back at [41, Section 3.3.1 B] and Proposition 2.14, we actually conclude that it is possible to sum a Nsgrm X to any general γHölder weakly geometric rough path W for any γ ∈ (0,1) in a canonical way, though the construction is not obtained via diagonal derivatives but via the demand \((\mathbf {X}\boxplus \mathbf {W})_{s,t}=\mathbf {X}_{s,t}\otimes \mathbf {W}_{s,t}+o(ts)\) and a sewing lemma argument. This gives a strong motivation of looking at smooth rough paths as “universal perturbations” (accordingly to [41, Section 3.3.1 B]) of γHölder weakly geometric rough paths.
The canonical sum can now be used to define a minimal coupling in the case of a finite number of smooth geometric rough models \(\mathbf {X}^{i}:[0,T]\to G(\mathbb {R}^{d_{i}}),~i=1,\dots ,m\). Set d = d_{1} + ⋯ + d_{m} and consider the linear embedding
This extends uniquely to an injective algebra homomorphism
equipped with the tensor product ⊗, and hence (by restriction) also to a group homomorphism from \(G(\mathbb {R}^{d_{i}})\) into \(G(\mathbb {R}^{d})\). (See also [28, Section 7.5.6] for a more general discussion how to lift linear maps to group homomorphisms.) Then we can consider the sgrms \(\iota _{i}\mathbf {X}^{i}:[0,T]\to G(\mathbb {R}^{d})\) and we define the minimal coupling of the X^{i} as the canonical sum of the \(\iota _{i}\mathbf {X}^{i}\)
The name minimal coupling amounts to the fact that we are choosing a sgrm which involves the least possible information on mixed iterated integrals. This construction can also be seen as partial generalization of Lyons–Young “(p,q)” pairing discussed in [29].
2.3 Differential Equations Driven by SGRP
Let \((f_{1},\dots ,f_{d}) \in (\text {Vect}^{\infty }(\mathbb {R}^{e}))^{d}\) be a collection of smooth vector fields, with bounded derivatives of all orders, so that all stated operations and differential equations below make sense. For the empty word 1, set f_{1} = id, the identity vector field. For a word w = ℓ_{1}…ℓ_{n} with w = n > 0 letters ℓ_{j} ∈{1,…,d}, define the vector field
where \(\vartriangleright \) is the following operation of two vector fields: i.e. using Einstein summation over i,j = 1,…,e we set
where f = (f^{i}∂_{i}), and g = (g^{j}∂_{j}). Using tensor calculus notation, we can equivalently set \(f \vartriangleright g = (\nabla g) f\) where ∇g is the Jacobian matrix of g. The operation \(\vartriangleright \) is also a classical example of preLie product (see Section 4.2) for which one has the usual identity of commutators
for any couple of vector fields f,g. The resulting family of maps {y↦f_{w}(y)}_{w} is represented by a linear map \(f\colon T (\mathbb {R}^{d}) \to \text {Vect}^{\infty }(\mathbb {R}^{e})\) such that
where the final pairing amounts to viewing f as (formal) sum \(\sum f_{w} e_{w}\), with summation over all words. Importantly, this map, restricted to Lie polynomials induces a Lie algebra morphism
This morphism is uniquely determined by its values on the canonical basis of \(\mathbb {R}^{d}\), since \({\mathscr{L}}(\mathbb {R}^{d})\) is isomorphic to the free Lie algebra over \(\mathbb {R}^{d}\). From an intrinsic geometric point of view, \(f_{\mathfrak {u}}\) is a vector field over \(\mathbb {R}^{e}\) interpreted as a manifold. This holds only to the extent that one uses the flat connection on \(\mathbb {R}^{e}\).
In what follows, we equip \(T(\mathbb {R}^{d})\) with an inner product structure, by declaring orthonormal the basis vectors \(e_{w} \in T(\mathbb {R}^{d})\), induced by distinct words w. Viewing \(T(\mathbb {R}^{d})\) as subspace of \(T((\mathbb {R}^{d}))\), this is consistent with the readily used pairing 〈x,z〉 with \( \mathbf {x} \in T((\mathbb {R}^{d}))\), \(\mathbf {z} \in T(\mathbb {R}^{d})\). In particular {e_{w} : w≤ N} yields an orthonormal basis in the levelN truncated space \(T^{N}(\mathbb {R}^{d})\), with inner product 〈⋅,⋅〉 = 〈⋅,⋅〉_{N}. We now introduce a natural notion of differential equation associated to sgrms and a class of vector fields \((f_{1}, {\dots } ,f_{d})\).
Definition 2.18
Let X be Nsgrm or a good sgrm such that \(\dot {\mathbf {X}}_{s,s}\in {\mathscr{L}}^{N}(\mathbb {R}^{d})\). We say that a smooth path \(Y\colon [0,T]\to \mathbb {R}^{e}\) is the solution of a differential equation driven by X with vector fields \((f_{1},\dots ,f_{d})\in (\text {Vect}^{\infty }(\mathbb {R}^{e}))^{d}\) if it satisfies the differential equation
where f : x↦f_{x} is given in (12). We will refer to (15) with the shorthand notation
Remark 2.19
We remark that the equation (16) generalises the known notion of controlled ODE. Indeed, \(\langle \dot {\mathbf {X}}_{s,s},\mathbf {1}\rangle =0\) for any Nsgrm or a good sgrm and if X is the minimal Nextension of a smooth path \(X={\sum }_{i=1}^{d} X^{i} e_{i}\), obtained by classical iterated integration in Example 2.12, then Definition 2.18 collapses to the usual definition of controlled equation
thereby obtaining a consistent definition. However, the richer structure of smooth geometric rough paths in Example 2.13 extends this framework.
Given a generic initial condition \(Y_{0}\in \mathbb {R}^{e}\) and vector fields \((f_{1}, {\dots } , f_{d})\) on \(\mathbb {R}^{e}\), it follows from standard properties on classical differential equations that there exists a unique solution with initial condition Y_{0}. For a Nsgrp X, the notion of solution of differential equation driven by a sgrp is furthermore consistent with the notion of rough differential equation solution à la Davie [21] when the driver X is interpreted as a 1/NHölder weakly geometric rough path.
Proposition 2.20
A path \(Y\colon [0,T]\to \mathbb {R}^{e}\) is a solution of a differential equation driven by a Nsgrp X with vector fields \((f_{1}, {\dots } , f_{d})\) if and only if for all s,t ∈ [0,T]
where X_{s,t} is the geometric rough model associated to X_{t} and r is a remainder such that r_{s,t} = o(t − s) as t → s.
Proof
Supposing the property (17), we first observe, since f_{1} = 0 and 〈X_{s,s},w〉 = 0 for any nonempty word w,
We thus conclude that Y is differentiable and obtain (15). (With f and \(s \mapsto \dot {\mathbf {X}}_{s,s}\) smooth, it is then clear that Y is not only differentiable but in fact smooth.) Conversely, supposing that Y is a solution of (15), we apply Taylor’s formula to Y and t →〈X_{s,t},w〉 for any w≤ N obtaining for all s,t ∈ [0,T].
for some \(r^{\prime }_{s,t}\), \(r^{\prime \prime }_{s,t}\) satisfying both \(r^{\prime }_{s,t}, r^{\prime \prime }_{s,t}= o(ts)\) as t → s. □
Using the properties of the diagonal derivative, we can restate the identity (15) with respect to the Lie algebra \({{\mathcal L}}^{N}(\mathbb {R}^{d})\).
Lemma 2.21
Let X be Nsgrm or a good sgrm such that \(\dot {\mathbf {X}}_{s,s}\) lies in \({\mathscr{L}}^{N}(\mathbb {R}^{d})\) and consider an orthonormal basis \(\mathfrak {B}^{N}\) for \({\mathscr{L}}^{N}(\mathbb {R}^{d}) \subset T^{N}(\mathbb {R}^{d})\). The following equivalence holds:
Proof
Complete \(\mathfrak {B}=\mathfrak {B}^{N}\) to an orthonormal basis \(\bar { \mathfrak {B}} = \mathfrak {B} \cup \mathfrak {B}^{\perp }\) of \(T^{N}(\mathbb {R}^{d})\). In that basis
We now observe that \(\dot {\mathbf {X}}_{s,s} \in {\mathscr{L}}^{N}(\mathbb {R}^{d})\), thanks to Proposition 2.6. Consequently, there is no contribution from any \(\mathbf {x} \in \mathfrak {B}^{\perp } \subset {\mathscr{L}}^{N}(\mathbb {R}^{d})^{\perp }\). □
Example 2.22
In the case N = 2 a natural orthonormal basis for \({\mathscr{L}}^{2}(\mathbb {R}^{d})\) is given by
and the previous lemma asserts that dY = f(Y )dX is equivalent to the following higher order controlled equation
This also follows directly by applying Remark 2.7 to the equation (15) with N = 2.
Remark 2.23
Even if a differential equation driven by a Nsgrm can be viewed as a Davie solution, we note however that Lemma 2.21 is a particular property of differential equations driven by Nsmooth geometric rough paths which does not hold for RDEs driven by a generic 1/NHölder weakly geometric rough path.
Indeed, working with onedimensional 1/2Hölder weakly geometric rough path^{Footnote 3} and a generic vector field \(f\in \text {Vect}^{\infty }(\mathbb {R})\) one has the trivial identities
Therefore there is no r_{s,t} = o(t − s) such that
2.4 Algebraic Renormalization of SGRP
Consider a collection of Lie series \(v = (v_{1},\ldots , v_{d})\subset {\mathscr{L}} ((\mathbb {R}^{d}))\), and let T_{v} be the ⊗endomorphism on \(T((\mathbb {R}^{d}))\), obtained by extending the translation map e_{i}↦e_{i} + v_{i}. We call T_{v} the translation map. When the v_{i}’s are all Lie polynomials, we remark that T_{v} maps \(T^{N}(\mathbb {R}^{d})\) to \(T^{M}(\mathbb {R}^{d})\) with \(M=N\cdot N^{\prime }\) where \(N^{\prime }\) denotes the smallest integer such that \(v_{i}\in {\mathscr{L}}^{N^{\prime }}(\mathbb {R}^{d})\). By restriction to Lie series, \({\mathscr{L}} ((\mathbb {R}^{d}))\subset T((\mathbb {R}^{d}))\), we can and will view T_{v} also as Lie algebra endomorphism, still denoted by T_{v}. We first give a dynamic view on the higherorder translation of sgrp.
Theorem 2.24

(i)
Let \(v=(v_{1},\dots ,v_{d})\) be a collection of elements in \({\mathscr{L}}((\mathbb {R}^{d}))\). Given a sgrp X over \(\mathbb {R}^{d}\), the unique solution to
$$ \dot{\mathbf{Z}}_{t} = \mathbf{Z}_{t} \otimes (T_{v} \dot{\mathbf{X}}_{t,t}), \qquad \mathbf{Z}_{0} = T_{v} \mathbf{X}_{0} $$(18)takes values in \(G(\mathbb {R}^{d})\) and is again a sgrp, and we have the explicit form
$$ \mathbf{Z}_{t} = T_{v} (\mathbf{X}_{t}), \quad \mathbf{Z}_{s,t} = {T}_{v} (\mathbf{X}_{s,t}). $$(19)Starting with a sgrm X over \(\mathbb {R}^{d}\), the above applies to t↦X_{0,t} and we obtain a sgrm Z with explicit form given by
$$ \mathbf{Z}_{s,t} = {T}_{v} (\mathbf{X}_{s,t}). $$ 
(ii)
Let now \(v=(v_{1},\dots ,v_{d})\) be a collection of elements in \({\mathscr{L}}(\mathbb {R}^{d})\) and X be a good sgrm over \(\mathbb {R}^{d}\). Then Z as constructed in (i) is also a good sgrm. More specifically, let \(N^{\prime }\) be the smallest integer such that v_{i} lies in \({\mathscr{L}}^{N^{\prime }}(\mathbb {R}^{d})\) for all i and assume X is the minimal extension of some Nsgrm Y over \(\mathbb {R}^{d}\). Let \(M = N\cdot N^{\prime }\). The unique solution to
$$ \dot{\mathbf{W}}_{t} = \mathbf{W}_{t} \otimes_{M} (T_{v} \dot{\mathbf{Y}}_{t,t}), \qquad \mathbf{W}_{0} = 1, $$defines a Msgrp, given by \(\mathbf {W}_{s,t} = \mathbf {W}_{s}^{1}\otimes _{M} \mathbf {W}_{t}\), which we call \(\mathcal {T}_{v}[\mathbf {Y}]\). Moreover, we have the explicit form
$$ \mathcal{T}_{v}[\mathbf{Y}]_{s,t} = {T_{v}^{M}} (\mathbf{Y}_{s,t}^{M}) = \text{\textsf{proj}}_{M}{T}_{v}(\mathbf{X}_{s,t}) $$(20)with algebra endomorphism \({T_{v}^{M}} := \text {\textsf {proj}}_{M} T_{v}\mathfrak {i}^{M}\) of \((T^{M}(\mathbb {R}^{d}),\otimes _{M})\), using the (linear) embedding \(\mathfrak {i}^{M}: T^{M}(\mathbb {R}^{d})\to T((\mathbb {R}^{d}))\), and Y^{M} = MinExt^{M}(Y).
Remark 2.25
Note that the map \(\mathbf {Y} \mapsto \mathcal {T}_{v}[\mathbf {Y}]\) for an Nsgrm Y is neither a linear nor a pointwise map, in contrast to X↦T_{v}X for sgrps or sgrms, which underlies the convenience of working with “genuine” sgrm’s, with full state space \(T((\mathbb {R}^{d}))\) (rather than some levelN truncation thereof). With regard to [6, 8] we may view T_{v} and \(\mathcal {T}_{v}\) renormalization maps, acting on nonlinear spaces of sgrm’s and Nsgrm’s respectively. We shall discuss in Theorem 2.26 below how this action effects differential equations driven by sgrm’s.
Proof
(i) Fix a sgrp X and write
Since T_{v} commutes with derivation and is an algebra morphism, then Z_{t} = T_{v}(X_{t}) clearly satisfies the differential equation (18). The algebraic properties of T_{v} together with Proposition 2.6 imply that T_{v}(X_{t}) belongs to \(G(\mathbb {R}^{d})\) and \(T_{v}(\dot {\mathbf {X}}_{t,t})\) belongs to \( {\mathscr{L}}((\mathbb {R}^{d}))\) respectively. Therefore T_{v}(X_{t}) must coincide with the Cartan development of \(T_{v}(\dot {\mathbf {X}}_{t,t})\), thereby yielding (19). Similar results hold with a sgrm.
(ii) Fix a good sgrm and apply part (i) to the path t↦X_{0,t}. Since v contains only Lie polynomials the map T_{v} becomes an algebra morphism \(T_{v}\colon T(\mathbb {R}^{d})\to T(\mathbb {R}^{d}s)\). Using Lemma 2.11 and this last property one has \(T_{v}(\dot {\mathbf {X}}_{t,t})\in {\mathscr{L}}(\mathbb {R}^{d})\), which implies that T_{v}(X_{s,t}) is a good sgrm. In the specific case of a collections of elements belonging to \({\mathscr{L}}^{N^{\prime }}(\mathbb {R}^{d})\), we consider Y^{M} = MinExt^{M}(Y). The path \(t\mapsto \mathbf {Y}^{M}_{0,t}=\mathbf {Z}_{t} \) will satisfy the following equation
Using standard properties of tensors, \({T_{v}^{M}}\) is an algebra endomorphism of \((T^{M}(\mathbb {R}^{d}),\otimes _{M})\). Indeed for any \(x,y\in T^{M}(\mathbb {R}^{d})\)
where we used in the fourth equality the fact that \(T_{v} z\in T^{>M}((\mathbb {R}^{d}))\) for all \(z\in T^{>M}((\mathbb {R}^{d}))\) (T_{v} is not lowering the grade). Applying \({T^{M}_{v}}\) to both sides of (21), we obtain that \({T^{M}_{v}}(\mathbf {Z}_{t})\) coincides with the Cartan development in \(G^{M}(\mathbb {R}^{d})\) of \({T_{v}^{M}} \dot {\mathbf {Y}}^{M}\). By uniqueness of this we can express \(\mathcal {T}_{v}[\mathbf {Y}]_{s,t}\) as \({T^{M}_{v}}(\mathbf {Z}_{s})^{1}\otimes _{M}{T^{M}_{v}}(\mathbf {Z}_{t})= {T}_{v}^{M} (\mathbf {Y}_{s,t}^{M})\), obtaining (20). The identity \({T_{v}^{M}} (\mathbf {Y}_{s,t}^{M}) =\textsf {proj}_{M}{T}_{v}(\mathbf {X}_{s,t})\) follows trivially. □
Combining the properties of translation operators T_{v} with the differential equation (16), we can describe the effect of translation on smooth geometric rough paths in the same way as [6]. Given \(v=(v_{1},\dots ,v_{d})\) a collection of elements in \({\mathscr{L}}(\mathbb {R}^{d})\) and \((f_{1}, \ldots , f_{d})\in (\text {Vect}^{\infty }(\mathbb {R}^{e}))^{d}\) we consider the collection of vector fields \(({f^{v}_{1}},\ldots , {f^{v}_{d}})\in (\text {Vect}^{\infty }(\mathbb {R}^{e}))^{d}\) given for each i = 1,…,d by
where \(f_{v_{i}}\) was given in (14). Starting from this collection of translated vector fields on d directions, we can consider the linear map \(f^{v}\colon T (\mathbb {R}^{d}) \to \text {Vect}^{\infty }(\mathbb {R}^{e})\), extending the family \(({f^{v}_{1}},\ldots , {f^{v}_{d}})\) on any word like in (12). This operation allows to transfer the action of the translation at the level of differential equations.
Theorem 2.26
Let X be a good sgrm and W a Nsgrm. For any given collection v = (v_{1},…,v_{d}) of elements in \({\mathscr{L}}(\mathbb {R}^{d})\) a path \(Y\colon [0,T]\to \mathbb {R}^{e} \) solves one of the equations
if and only if it solves respectively
Remark 2.27
The differential equations in the above statement are understood, in the sense of Definition 2.18, as equations driven by a Nsgrm W, a \((N\cdot N^{\prime })\)sgrm \(\mathcal {T}_{v}[\mathbf {W}]\) (when the directions v = (v_{1},…,v_{d}) are all contained in \({\mathscr{L}}^{N^{\prime }}(\mathbb {R}^{d})\)) and a good sgrm T_{v}X, respectively.
Proof
Writing X as the minimal extension of a Msgrm for some \(M\in \mathbb {N}\), the result follows by showing only the equivalence for W. We present here two possible proofs.
(First argument) Fix s and y = Y_{s} and \(\mathfrak {w} := \dot {\mathbf {W}}_{s,s} \in {\mathscr{L}}^{N}(\mathbb {R}^{d})\) and \(M=N\cdot N^{\prime }\), where \(N^{\prime }\) is the minimal integer such that \(v_{i}\in {\mathscr{L}}^{N^{\prime }}(\mathbb {R}^{d})\) for all i = 1,…,d. By Theorem 2.24, the diagonal derivative of \(\mathcal {T}_{v}[\mathbf {W}]\) at time s is precisely \(T_{v} \mathfrak {w} \in {\mathscr{L}}^{M}(\mathbb {R}^{d})\) up to minimal extension. Using Lemma 2.21, the result follows once we check that
To prove this identity, we use an argument already contained in [6]. Let \(f : \mathfrak {x} \mapsto f_{\mathfrak {x}}\) be the induced Lie algebra morphism from \({\mathscr{L}}^{M}(\mathbb {R}^{d})\) into \(\text {Vect}^{\infty }(\mathbb {R}^{e})\). Hence, whenever f is smooth, the maps \(f_{T_{v}}\) and f^{v} are both Lie algebra morphisms from \({\mathscr{L}}^{M}(\mathbb {R}^{d})\) into \(\text {Vect}^{\infty }(\mathbb {R}^{e})\), which furthermore agree on the generators e_{i}. Thus \(f_{T_{v}\mathfrak {x}}(z) = f^{v}_{\mathfrak {x}}(z)\) for any \(z\in \mathbb {R}^{e}\) and any \(\mathfrak {x}\in {\mathscr{L}}^{M}(\mathbb {R}^{d})\) thanks to the universal property of \({\mathscr{L}}^{M}(\mathbb {R}^{d})\) so
(Second argument) The identity \(f_{T_{v}}(z)=f^{v}(z)\) does not hold simply over \({\mathscr{L}}(\mathbb {R}^{d})\) but over all \( T(\mathbb {R}^{d})\). For that, we first show that in fact for any \(u\in {\mathscr{L}}(\mathbb {R}^{d})\) and any \(x\in T^{>0}((\mathbb {R}^{d}))\) we have \(f_{u}\vartriangleright f_{x}=f_{u x}\). We do so completely analogously to [51, Lemma 2.5.11], proceeding by induction over the length of u. For u a letter, the statement holds by definition of w↦f_{w}. Assume the statement holds for all u of length n. Then, to check it for all homogeneous Lie elements of word length n + 1, since left bracketings span the free Lie algebra, it suffices to look at Lie elements of the form [u,i]. So,
where for the third equality, we used the identity (13). Then, since \(v_{i}\in {\mathscr{L}}(\mathbb {R}^{d})\), indeed we get, again via induction over the length of words,
We finally obtain the general identity by linearity. Thus for any \(\mathbf {x} \in T^{N}(\mathbb {R}^{d})\) we have
Then, by Proposition 2.20 we have the desired equivalence of differential equations through the following equality, for any smooth path \(Y:[0,T]\to \mathbb {R}^{e}\) and s,t ∈ [0,T], using the notation \(\mathbf {W}^{\infty }:=\text {MinExt}(\mathbf {W})\) one has
with
as \(\langle \mathbf {W}^{\infty }_{s,t},w\rangle =O(ts^{2})\) for all w with w > N, since then \((s,t)\mapsto \langle \mathbf {W}^{\infty }_{s,t},w\rangle \) is a smooth function on the compact domain [0,T]^{2} with \(\langle \mathbf {W}^{\infty }_{t,t},w\rangle =0\) and \(\partial _{s}_{s=t}\langle \mathbf {W}^{\infty }_{s,t},w\rangle =\langle \dot {\mathbf {W}}^{\infty }_{t,t},w\rangle =0\). □
Remark 2.28
The second argument in the above proof is valid for genuine (nonsmooth) γHölder weakly geometric rough paths, with N = ⌊1/γ⌋, \(Y:[0,T]\to \mathbb {R}^{e}\) an arbitrary continuous path and choosing MinExt(W) as the Lyons extension of W, see Proposition 2.14. There, it again correctly identifies the respective rough differential equations via Davies’ expansion.
This argument first presented in [51, Section 2.5.2] fills a gap left in the proof of [6, Theorem 38] where it is misleadingly suggested that a Lie algebraic argument can be used. Indeed, as noted in Remark 2.23 there is no Davietype definition of RDE solution where the sum is restricted to a Lie basis.
As final application of the section we combine Theorems 2.24 and 2.26 with the notion of minimal coupling and sum to obtain a “smooth geometric” renormalization along a time component, see [6, Theorem 38]. Consider a timespace path of the form
for some smooth path \(X^{1}\colon [0,T]\to \mathbb {R}^{d}\) (the first component of canonical basis of \(\mathbb {R}^{d+1}\) will be denoted by e_{0}). Assume also that there exists a good sgrm \(\mathbf {X}^{1}:[0,T]^{2} \to G (\mathbb {R}^{d})\) extending X^{1}. Then we can introduce the usual signature of time component \(\mathbf {X}^{0}:[0,T]^{2} \to G (\mathbb {R})\) defined by
and the choice
constitutes a canonical way to extend \(\bar {X}\). Choosing then a Lie polynomial \(v_{0}\in {\mathscr{L}}(\mathbb {R}^{d+1})\), we can also consider the translation map \(T_{v_{0}}\) obtained by the identity action on \(e_{1},\dots , e_{d}\), and e_{0}↦e_{0} + v_{0}. Then we can explicitly describe \(T_{v_{0}}\bar {\mathbf {X}}\) and differential equations driven by that.
Corollary 2.29
Let X^{1} be a good sgrm and \(v_{0}\in {\mathscr{L}}(\mathbb {R}^{d+1})\). Then the translation \(t\mapsto T_{v_{0}}\bar {\mathbf {X}}_{0,t}\) can be described as the sum \(\mathbf {V}_{0}\boxplus \bar {\mathbf {X}}\), where V_{0} is given by
Moreover, for any given family of vector fields (f_{0},f_{1},…,f_{d}) one has the equivalence
where \(\bar {f}\colon T(\mathbb {R}^{d+1})\to \text {Vect}^{\infty }(\mathbb {R}^{e})\) and \(f\colon T(\mathbb {R}^{d})\to \text {Vect}^{\infty }(\mathbb {R}^{e})\) are defined like in (14), starting respectively from \((f_{0}, {\dots } , f_{d})\) and \((f_{1}, {\dots } , f_{d})\).
Proof
The proof follows by applying the results in the section. By construction of \(\bar {\mathbf {X}}\), one has the following diagonal derivative \(\dot {\bar {\mathbf {X}}}_{t,t}=e_{0}+\dot {\mathbf {X}}^{1}_{t,t}\). Then by construction of \(T_{v_{0}}\) and Theorem 2.24, we have
Since the diagonal derivative identifies uniquely \(T_{v_{0}}(\bar {\mathbf {X}})\) we conclude. Passing to the differential equation, we remark from Definition 2.18 that a smooth path \(Y\colon [0,T]\to \mathbb {R}^{e}\) is a solution of \(dY=\bar {f}(Y)d\bar {\mathbf {X}}\) if and only if
Moreover, the map \(\bar {f}^{v_{0}}\) satisfies trivially \(\bar {f}^{v_{0}}_{0}=f_{0} + \bar f_{v_{0}}\), \(\bar {f}^{v_{0}}_{i}=f_{i}\) for any i = 1,…,d. We conclude then by Theorem 2.26. □
3 Smooth Quasigeometric Rough Paths
In the sequel, we extend the previous results and notions of smooth rough paths in the context of quasigeometric rough paths, the natural generalisation of rough paths when we consider an underlying quasishuffle algebra, see [36]. Quasigeometric rough paths were first introduced in talks by David Kelly and properly studied in [2].
3.1 Quasishuffle Algebras
We briefly review quasishuffle algebras in the version of [36]. Their origin can be traced back to [10] and we also refer the reader to [24, 46]. We shall use Hoffman’s isomorphism in the context of stochastic integration (‘Itô vs. Stratonovich’) which was discussed in detail in [23].
Let us start from a generic vector space \(\mathbb {R}^{A}\) where A is a locally finite (with respect to the weight function ω below) set called alphabet. To link this space with the previous section, we assume the inclusion {1,…,d}⊂ A, which induces a canonical inclusion \(\mathbb {R}^{d}\subset \mathbb {R}^{A}\). Given an alphabet A, with a slight abuse of notation we will denote by T(A) and T((A)) the spaces of tensor polynomials and tensor series over \(\mathbb {R}^{A}\) respectively. To simplify the notation, in what follows we will also identify elements of T((A)) as words, as explained in the previous section, i.e.
for any a_{1},…,a_{n} ∈ A. Moreover, we use the same symbol ⊗ to denote the tensor product operation of T((A)) and the “external” algebraic tensor product of two vector spaces.
The quasishuffle product relies on the existence of a commutative bracket, i.e. a map \(\{ ,\}\colon \overline {A}\times \overline {A}\to \overline {A}\), where \(\overline {A}=A\cup \{0\} \) such that
We adopt the notation {a} = a, for a ∈ A, and more generally, for any word w = a_{1}…a_{n} we set {a_{1}…a_{n}} := {a_{1},{…{a_{n− 1},a_{n}}}…} independently of the order of the letters in the word and the brackets. Together with the choice of a commutative bracket {,} we also fix a weight ω which is compatible with {,}, i.e. ω is a function \(\omega \colon A\to \mathbb {N}^{\ast }\) satisfying
when {ab}≠ 0. Note that there could be several weights compatible with a given bracket and the definition of bracket does not need a weight.
Given a word \(w=a_{1} {\dots } a_{n}\) we denote by w = n the length of w and introduce its weighted length ∥w∥ = ω(a_{1}) + ⋯ + ω(a_{n}). Since ω(a) ≥ 1 for all a ∈ A we have trivially w≤∥w∥. The weighted length induces a grading on T(A). To stress this property, we also use the notations T_{ω}(A) and T_{ω}((A)) when we grade these vector spaces according to the weighted length. e.g.
Moreover, for any \(N\in \mathbb {N}\) we define the truncated space \(T_{\omega }^{N}(A)\) by taking words of weighted length at most N, we denote the projection on \(T_{\omega }^{N}(A)\) by proj_{N,ω}. Using the weighted length, we can view \(T^{N}_{\omega }(A)\) as a quotient algebra and introduce the corresponding truncated tensor product ⊗_{N,ω}. The pairing 〈,〉 between \(T((\mathbb {R}^{d}))\) and \(T(\mathbb {R}^{d})\) generalises to the pairing \(\langle , \rangle \colon T_{\omega }((A))\times T_{\omega }(A)\to \mathbb {R}\) for any alphabet A. Its restriction T(A) × T(A) ⊂ T((A)) × T(A) yields a scalar product on T(A). We denote by 〈⋅,⋅〉_{N} the restriction to truncated spaces \(T^{N}_{\omega }(A)\times T^{N}_{\omega }(A)\). As before, the set {w : ∥w∥≤ N} is an orthonormal basis of \(T^{N}_{\omega }(A)\). Combining {,} and the shuffle product, we introduce the quasishuffle product.
Definition 3.1
The quasishuffle product is the unique bilinear map in T(A) satisfying the relation for any v ∈ T(A) and the recursive identity
for any words v, w in T(A) and letters a, b in A.
Example 3.2
If A = {1,…,d} and the commutative bracket is trivial, i.e. {ab} = 0 for any a, b in A, then the recursive relation (24) reduces to
so that coincides with the standard shuffle product in (3). Observe that setting ω(i) = 1 for all i ∈ A, one has ∥w∥ = w and T_{ω}({1,…,d}) is exactly \(T(\mathbb {R}^{d})\) but in general we can consider the shuffle algebra over a generic alphabet with a weight which is not identically 1, see e.g. in [35].
Example 3.3
For fixed integers M,d ≥ 1, we define the alphabet \(\mathbb {A}_{M}^{d}\) as
as well as the weight \(\omega (\alpha )= {\sum }_{i=1}^{d} \alpha _{i}\). We also define the bracket \(\{,\}_{M}\colon \overline {\mathbb {A}_{M}^{d}}\times \overline {\mathbb {A}_{M}^{d}}\to \overline {\mathbb {A}_{M}^{d}}\) as the unique bilinear map satisfying the property
for any \(\alpha , \beta \in \mathbb {A}_{M}^{d}\). The sum arising in this expression is the intrinsic sum between multiindices. The set {1,…,d} embeds in \(\mathbb {A}_{M}^{d}\) for any M ≥ 0 by simply sending every i ∈{1,…,d} to the ith element of the canonical basis. Using this inclusion the letter {112} is identified with the multiindex (2,1,0,…,0) and e have {i_{1}…i_{n}} = 0 for any n > M and i_{j} ∈{1,…,d}. It follows from standard properties on multiindices that {,} and ω satisfy properties (22) and (23). The associated quasishuffle algebra \(T_{\omega }(\mathbb {A}_{M}^{d})\) is used to define the notion of quasigeometric bracket extension in [2]. We can also drop the finite index M and consider the infinite alphabet \(\mathbb {A}^{d}= \mathbb {N}^{d}\setminus \{0\}\) with the same weight. This alphabet is then isomorphic to the free semigroup generated over d elements. The associated quasishuffle algebra was intensively used in [22].
For any choice of {,} the quasishuffle product is a welldefined commutative product on T(A), see [36]. It further defines a product on the graded algebra T_{ω}(A) for a given compatible weight function \(\omega \colon A\to \mathbb {N}^{\ast }\). The product is also compatible with the weighted length on T_{ω}(A). From a Hopf algebraic point of view, see Section 4 for the general framework, we can equip T_{ω}(A) with the deconcatenation coproduct Δ: T_{ω}(A) → T_{ω}(A) ⊗ T_{ω}(A) and the counit \(\mathbf {1}^{\ast }\colon T_{\omega }(A) \to \mathbb {R}\) defined respectively by
Both the triples and are graded Hopf algebras, see [36]. A fundamental result is the existence of an explicit isomorphism between these two Hopf algebras for any commutative bracket {,}, see [36, Theorem 3.3].
Theorem 3.4 (Hoffman Isomorphism)
For any commutative bracket {,} we define the Hoffman “exponential” and “logarithm” Φ_{H}, Ψ_{H}: T(A) → T(A) on any word as
where C(w) is the set of compositions of order w, i.e. the multiindices I = (i_{1},…,i_{m}) such that i_{1} + ⋯ + i_{m} = w and
Moreover, we denote by {w}_{I} the word
The map is an isomorphism of graded Hopf algebras such that \({\varPsi }_{H}= {\varPhi }_{H}^{1}\).
With Φ_{H} and Ψ_{H} as in (25), two explicit formulae were shown in [36] for the adjoint maps \( {\varPhi }_{H}^{\ast }, {\varPsi }_{H}^{\ast } \colon T_{\omega }(A)\to T_{\omega }(A)\) with respect to scalar product 〈,〉. Indeed, for any word w = a_{1}…a_{n} we have
and for each letter a ∈ A, the following identity holds:
where the underlying set of summation {a_{1}…a_{n}} = a involves all possible ways of writing the letter a in terms of brackets of the word a_{1}…a_{n}. By simply considering the graded dual of the shuffle and quasishuffle Hopf algebras (see Section 4) in Theorem 3.4, the map \({\varPhi }_{H}^{\ast }\) becomes an isomorphism where are the dual coproducts associated to the shuffle and quasishuffle operation. To extend \({\varPhi }^{\ast }_{H}\) and \({\varPsi }_{H}^{\ast }\) to the whole set T((A)), we observe that the concatenation product ⊗ easily extends to T_{ω}((A)). However, the coproducts and do not make sense on T_{ω}((A)) ⊗ T_{ω}((A)), see e.g. [50, Theorem 3.6] for a counterexample on \(T((\mathbb {R}^{1}))\). They need to be extended to the completed tensor space, see [53, Section 1.4]
leading to two topological graded coalgebras and .
Proposition 3.5
The maps \({\varPhi }^{\ast }_{H}\) and \({\varPsi }^{\ast }_{H}\) defined by (26) and (27) uniquely extend to two linear maps \({\varPhi }^{\ast }_{H},{\varPsi }^{\ast }_{H} \colon T_{\omega }((A))\to T_{\omega }((A))\), for which we shall use the same notation. The map \({\varPhi }^{\ast }_{H}\) gives rise to a graded automorphism of the algebra (T_{ω}((A)),⊗) as well as an isomorphism between the two topological graded coalgebras and . In both cases one has \({\varPsi }_{H}^{\ast }= ({\varPhi }_{H}^{\ast })^{1}\).
Proof
It is sufficient to build the extension of \({\varPhi }^{\ast }_{H}\). Combining properties (26) and (27), for any series \(S={\sum }_{w} \alpha _{w} w\in T_{\omega }((A))\) we define
where w = w_{1}…w_{m}. Thanks to property (23) and the assumption \(\omega (A)\subset \mathbb {N}^{\ast }\), the terms in the series \({\varPhi }^{\ast }_{H}(S)\) are locally finite, i.e. for any word u there is only a finite number of nonzero terms in the bracket \(\langle {\varPhi }^{\ast }_{H}(S),u\rangle \). It follows from property (26) that \({\varPhi }^{\ast }_{H}\) extended to infinite series is also an algebra morphism. The other properties easily follow from the fact that T_{ω}(A) and T_{ω}(A) ⊗ T_{ω}(A) are dense subsets of T_{ω}((A)) and \(T_{\omega }((A)){\overline {\otimes }}T_{\omega }((A))\). □
The Hoffman isomorphism allows to define Lie groups and Lie algebras from the corresponding Lie groups and Lie algebras built from the corresponding shuffle structures.
In what follows, we use the notations \({\mathscr{L}}^{N}(A)\), G^{N}(A), \({\mathscr{L}}(A)\), \({\mathscr{L}}((A))\) and \(G(A)= \exp _{\otimes } {\mathscr{L}}((A))\) to denote respectively the Nstep Lie polynomials, the free stepN Lie group, the Lie polynomials and Lie series generated by \(\mathbb {R}^{A}\) and \(G(A)= \exp _{\otimes } {\mathscr{L}}((A))\), where \(\exp _{\otimes }\) is given in (4). If we fix a generic weight ω and \(N \in \mathbb {N}\), we denote by \({\mathscr{L}}^{N}_{\omega }(A)\) and \(G^{N}_{\omega }(A)\) the following quotients
where \({\mathscr{L}}^{\omega }_{>N}\) and \(G^{\omega }_{>N}(A)\) are respectively the Lie ideal and normal subgroup
(These last two properties follow trivially from (23)). Moreover, passing to the quotient gives rise to the identification
and the property \(G^{N}_{\omega }(A) =\exp _{\otimes _{N,\omega }}({\mathscr{L}}^{N}_{\omega }(A))\) where \(\exp _{\otimes _{N,\omega }}\colon T^{N}_{\omega }(A)\to T^{N}_{\omega }(A)\) is the exponential associated with the truncated product ⊗_{N,ω}
thereby obtaining that \({\mathscr{L}}^{N}_{\omega }(A)\) is the Lie algebra of \(G^{N}_{\omega }(A)\). Similar objects can be defined in the quasishuffle context via the map \({\varPsi }^{\ast }_{H}\). The proof of the following corollary is a straightforward application of Proposition 3.5 combined with the usual properties of Lie series and the group G(A).
Corollary 3.6

(i)
The set \(\hat {G}(A) := {\varPsi }_{H}^{\ast } G(A)=\{{\varPsi }_{H}^{\ast } \beta , \beta \in G(A)\}\) with the operation ⊗ is a group which has the following description
We call \(\hat {G}(A)\) the group of quasishuffle characters.

(ii)
The sets \(\hat {{\mathscr{L}}}((A))= {\varPsi }_{H}^{\ast }{\mathscr{L}}((A))\) and \(\hat {{\mathscr{L}}}(A):= {\varPsi }_{H}^{\ast } {\mathscr{L}}(A)\) with the commutator of ⊗ are two Lie algebras, which satisfy \(\hat {{\mathscr{L}}}(A)\subset \hat {{\mathscr{L}}}((A))\) and \(\hat {G}(A)=\exp _{\otimes }{\mathscr{L}}((A))\). We call them the quasishuffle Lie series and quasishuffle Lie polynomials respectively and they coincide with the free Lie algebra and Lie series generated by the set \(\mathfrak {A}=\{{\varPsi }_{H}^{\ast }(a)\colon a\in A\}\).

(iii)
The set \(\hat {G}^{N}_{\omega }(A):= {\varPsi }_{H}^{\ast } G^{N}_{\omega }(A)\) with the operation ⊗_{N,ω} is a Lie group whose Lie algebra is given by \(\hat {{\mathscr{L}}}^{N}_{\omega }(A):= {\varPsi }_{H}^{\ast } {\mathscr{L}}^{N}_{\omega }(A)\) and one has \(\hat {G}^{N}_{\omega }(A)= \exp _{\otimes _{N,\omega }}(\hat {{\mathscr{L}}}^{N}_{\omega }(A))\) and the identification
Example 3.7
Let us spell out the primitive elements of \(\hat {{\mathscr{L}}}_{\omega }^{3}(\mathbb {A}_{2}^{d})\). Starting from the definition of \({\mathscr{L}}_{\omega }^{3}(\mathbb {A}_{2}^{d})\) and using the notations [,] for the Lie bracket, one has that \({\mathscr{L}}_{\omega }^{3}(\mathbb {A}_{2}^{d})\) is generated as vector space by the following elements
where i,j,k ∈{1,…,d}. It follows from the definition of \({\varPsi }^{\ast }_{H}\) that
We therefore obtain that \(\hat {{\mathscr{L}}}_{\omega }^{3}(\mathbb {A}_{2}^{d})\) is generated as vector space by
3.2 Smooth Quasigeometric Rough Paths
The notion and properties of smooth rough paths can be transposed to the quasishuffle context leading to equivalent concepts.
Definition 3.8
We call levelN smooth quasigeometric rough path over the alphabet A with weight ω (in short: Nsqgrp) any non zero path \(\mathbf {X}: [0,T] \to T^{N}_{\omega }(A)\) satisfying the following properties:

(a’ i)
For all times t ∈ [0,T] and all words v and w such that ∥w∥ + ∥v∥≤ N
(28) 
(a’ ii)
For every word of weighted length ∥w∥≤ N, the map t↦〈X_{t},w〉 is smooth and we write \(\dot {\mathbf {X}}_{t} \) for the derivative of X.
We call levelN smooth quasigeometric rough model (in short: Nsqgrm) any map \(\mathbf {X}: [0,T]^{2} \to T^{N}_{\omega }(A)\) such that the property (28) holds for all X_{s,t} and the properties (b.ii), (b.iii) in Definition 2.3 hold on a set of weighted words with the operation ⊗_{N,ω}. By smooth quasigeometric rough path (model) (in short: sqgrp and sqgrm) we mean a path (map) with values in the full space of tensor series T_{ω}((A)) where (28) hold for any w,v ∈ T_{ω}((A)) and relation (b.ii) hold with ⊗.
It follows immediately from Definition 3.8 that sqgrps and Nsqgrps are identified with smooth paths with values in the groups \(\hat {G}(A)\) or \(\hat {G}^{N}_{\omega }(A)\) and we can pass to the associated models by considering the increments with respect to ⊗_{N,ω}. Similarly we can easily adapt the notion of extension of a sqgrp and the diagonal derivative of a sqgrm, which takes value in \(\hat {{\mathscr{L}}}((A))\) or \(\hat {{\mathscr{L}}}^{N}_{\omega }(A)\). Since the shuffle product is a specific case of the quasishuffle product, Definition 3.8 extends also the notions of Nsgrps and Nsgrms in a presence of a generic alphabet A and weight ω, we will call these objects weighted Nsgrps and weighted Nsgrms.
Thanks to the properties of the Hoffman isomorphism, we can easily relate smooth (resp. truncated) quasigeometric rough paths and models with some corresponding weighted geometric objects by simply composing these objects with the appropriate version of the Hoffman isomorphism.
Theorem 3.9
Let N ≥ 0 and \(\mathbf {X}\colon [0,T]\to T^{N}_{\omega }(A)\). Then X is a Nsqgrp if and only if \(\hat {\mathbf {X}}={\varPhi }_{H}^{\ast }\mathbf {X}\) is a weighted Nsgrp. Conversely, \(\hat {\mathbf {X}}\) is a weighted Nsgrp if and only if \(\mathbf {X}={\varPsi }_{H}^{\ast }\hat {\mathbf {X}}\) is a Nsqgrp. The same result applies to Nsqgrms, sqgrps and sqgrms.
Proof
The result follows easily from combining the properties of the maps \({\varPhi }_{H}^{\ast }\) and \({\varPsi }_{H}^{\ast }\) in Theorem 3.4 with the conditions in Definition 2.1 and Definition 3.8. E.g. starting from a Nsqgrp X one has
for all words with joint weighted length ∥w∥ + ∥v∥≤ N. Moreover, restricting \({\varPhi }^{\ast }_{H}\mathbf {X}_{t}\) to all words with joint weighted length ∥w∥≤ N, it will be a finite linear combination of the functions 〈X_{t},v〉 with ∥v∥≤ N, which are all smooth functions. The same considerations apply to Nsqgrms, sqgrps and sqgrms trivially. The converse case with \({\varPsi }_{H}^{\ast }\) follows from the properties in Corollary 3.6. □
Thanks to this onetoone correspondence between quasigeometric rough paths and geometric rough paths, we can import all the constructions of the previous section by simply checking that they preserve the maps \({\varPhi }_{H}^{\ast }\) and \({\varPsi }_{H}^{\ast }\). In particular, the notion of minimal extension for sgrms can easily be transposed to sqgrms.
Proposition 3.10
Given an Nsqgrm Y for some \(N \in \mathbb {N}\), there exists exactly one sqgrm extension X of Y which is minimal in the sense that for all s ∈ [0,T] one has
We call q MinExt(Y) := X the quasigeometric minimal extension of Y and also \(q\text {MinExt}^{N^{\prime }}(Y) := \text {\textsf {proj}}_{N^{\prime },\omega } \mathbf {X}\), for \(N^{\prime } > N\), the \(N^{\prime }\)minimal extension of Y. For a fixed interval [s,t] ⊂ [0,T] it holds that X_{s,t} only depends on {Y_{[u,v]} : s ≤ u ≤ v ≤ t} and we introduce the quasisignature of Y on [s,t] by \(q\text {Sig}(\mathbf {Y} _{[s,t]}) := \mathbf {X}_{s,t} \in \hat {G}(A)\). Moreover, we have the identities
A sqgrm X is called a good sqgrm if X = q MinExt(Y) for some Nsqgrm Y.
Remark 3.11
The existence and uniqueness of a minimal extension for sqgrg hold on general grounds thanks to properties of Hopf algebras, see Theorem 4.2. However, Hoffman isomorphism gives us in (29) an explicit solution of the minimal extension in terms of the minimal extension related to a geometric rough path.
Proof
Thanks to Theorem 3.9, the map \(\hat {\mathbf {Y}}={\varPhi }^{\ast }_{H}\mathbf {Y}\) is a weighted Nsgrp. Since the proof of Theorem 2.8 applies to any alphabet A and any weight on letters ω, there exists a unique minimal extension \(\hat {\mathbf {X}}\) of \(\hat {\mathbf {Y}}\). Beyond the fact that the function \(\mathbf {X}:={\varPsi }_{H}^{\ast } \hat {\mathbf {X}}\) is a quasigeometric rough path, its diagonal derivative satisfies
The uniqueness of \(\hat {\mathbf {X}}\) follows from the uniqueness of the minimal extension in the geometric case and the isomorphism properties of \({\varPsi }^{\ast }_{H}\) and \({\varPhi }^{\ast }_{H}\). From this uniqueness we deduce also the identities (29) straightforwardly. □
In the same way as for geometric rough paths, the notion of smooth quasigeometric rough path is consistent with its equivalent γHölder version as introduced in [2]. The proof of the following corollary is left to the reader.
Corollary 3.12
Every Nsqgrm X is a 1/NHölder quasigeometric rough path and its minimal extension X coincides with the lift of X, as constructed in [2, Proposition 3.9].
Remark 3.13
Even if there is no distinction in the literature between γHölder quasigeometric rough paths and weakly γHölder quasigeometric rough paths, the techniques in [29] can easily be adapted to introduce these concepts. Once done, Nsqgrm’s \(\hat {\mathbf {X}}\) should also be 1/NHölder weakly quasigeometric rough paths and form a dense subset among γHölder quasigeometric rough paths for 1/γ ∈ (N,N + 1).
The existence of a minimal extension allows to define a vector space structure over sqgrms and weighted sgrps. Indeed given two smooth quasigeometric rough models \(\mathbf {X},\mathbf {Y}:[0,T]^{2}\to \hat {G}(A)\) and \(\lambda \in \mathbb {R}\), we define their canonical sum \(\mathbf {X}\boxplus \mathbf {Y}\) and their scalar multiplication \(\lambda \boxdot \mathbf {X}\) like in Definition 2.15.
Since the operation of taking the diagonal derivative of a sqgrm commutes with the linear maps \({\varPhi }_{H}^{\ast }\) and \({\varPsi }_{H}^{\ast }\), from Proposition 4.16 we deduce the following corollary.
Corollary 3.14
Let X, Y be smooth quasigeometric rough paths/models. Then the map (s,t) →X_{s,t} ⊗Y_{s,t} and \(\mathbf {X}\boxplus \mathbf {Y}\) satisfy the same relations in (10) and (9) respectively. Moreover \({\varPhi }_{H}^{\ast }\) is a linear isomorphism with inverse \({\varPsi }_{H}^{\ast }\) between the vector space of sqgrms and weighted sgrms endowed with \(\boxplus \) and \(\boxdot \).
Remark 3.15
The sum of smooth quasigeometric rough paths/models can be used to define a notion of minimal coupling like in (11). In this case, the elements of the sum are smooth geometric rough paths/models \(\mathbf {X}^{i}:[0,T]\to \hat {G}(A_{i})\), where A_{i}, i = 1,…,m form a partition of A where each A_{i} is closed under the commutative bracket {,}.
3.3 Differential Equations Driven by SQGRP
We introduce the notion of differential equation driven by a smooth quasigeometric rough path. Since we replaced our initial directions on \(\mathbb {R}^{d}\) with an alphabet A, we assume the existence of a wider collection of smooth vector fields \((f_{a}\colon a\in A)\subset \text {Vect}^{\infty }(\mathbb {R}^{e})\). Moreover, in the same way as {1,…,d}⊂ A we further fix a family (f_{1},…,f_{d}) ⊂ (f_{a}: a ∈ A). As before, we can apply the operation \(\vartriangleright \) to the elements (f_{a}: a ∈ A) obtaining again a linear map \(f\colon T_{\omega }(A)\to \text {Vect}^{\infty }(\mathbb {R}^{e})\) defined by (12). This map induces a morphism of Lie algebras \(f\colon {\mathscr{L}}(A)\to \text {Vect}^{\infty }(\mathbb {R}^{e})\), which is uniquely determined by his values on A. By means of the adjoint Hoffman exponential \({\varPhi }^{\ast }_{H}\), we can uniquely define a map similar to f, which preserves the quasishuffle Lie polynomials.
Proposition 3.16
Given a family of vector fields \((f_{a}\colon a\in A)\subset \text {Vect}^{\infty }(\mathbb {R}^{e})\), there exists a map \(\hat {f}\colon T_{\omega }(A)\to \text {Vect}^{\infty }(\mathbb {R}^{e})\) whose restriction on \(\hat {{\mathscr{L}}}(A)\) is the unique morphism of Lie algebras \(\hat {f}\colon \hat {{\mathscr{L}}}(A)\to \text {Vect}^{\infty }(\mathbb {R}^{e})\) which satisfies \(\hat {f}({\varPsi }^{\ast }_{H}(a))=f_{a}\) for any a ∈ A. This map is explicitly given by \(\hat {f}= f_{{\varPhi }^{\ast }_{H}}\).
Proof
By the definition of \(\hat {{\mathscr{L}}}(A)\), the map \(f_{{\varPhi }^{\ast }_{H}}\) satisfies clearly the properties of the statement. Moreover, thanks to the free structure of \(\hat {{\mathscr{L}}}(A)\) described in Corollary 3.6, the condition \(\hat {f}({\varPsi }^{\ast }_{H}(a))=f_{a}\) uniquely determines a morphism of Lie algebras \(\hat {f}\colon \hat {{\mathscr{L}}}(A)\to \text {Vect}^{\infty }(\mathbb {R}^{e})\). □
Using the explicit definitions of f and \({\varPhi }^{\ast }_{H}\), we can then describe \(\hat {f}\) as the unique linear map satisfying the conditions
for any words w = a_{1}…a_{n} and a ∈ A. We are now ready to introduce the notion of differential equation in the quasigeometric context.
Definition 3.17
Let X be Nsqgrm or a good sgrm such that \(\dot {\mathbf {X}}_{s,s}\in {\mathscr{L}}^{N}_{\omega }(A)\). We say that a smooth path \(Y\colon [0,T]\to \mathbb {R}^{e}\) is the solution of a differential equation driven by X and the vector fields (f_{a}: a ∈ A) if it satisfies
where \(\hat {f}:\mathbf {x}\mapsto \hat {f}_{\mathbf {x}}\) is given in (30). We will refer to (31) as
Remark 3.18
This definition leaves the open problem for future work how to canonically define a notion of differential equation, when only given d vector fields \(f_{1},\dots ,f_{d}\), without the need to always individually define a casespecific extension to (f_{a} : a ∈ A), and how to define this notion at best in such a way that it is consistent with geometric and branched d dimensional differential equations.
Combining the results in Proposition 2.20, Lemma 2.21 and Proposition 3.5, we can summarise three equivalent characterisations of (32) in the quasigeometric context.
Proposition 3.19
Given a path \(Y\colon [0,T]\to \mathbb {R}^{e}\) a Nsqgrp X and a family of vector fields (f_{a}: a ∈ A), one has the following equivalent conditions:

i)
Y solves \(dY=\hat {f}(Y)d\mathbf {X}\).

ii)
Y solves \(dY = f (Y) d \hat {\mathbf {X}}\), where \(\hat {\mathbf {X}}={\varPhi }_{H}^{\ast } \mathbf {X}\) is the Nlevel weighted geometric rough path obtained via Theorem 3.9.

iii)
For all s,t ∈ [0,T]
$$ Y_{t}Y_{s}={\sum}_{1\leq \w\\leq N}\hat{f}_{w}(Y_{s})\langle \mathbf{X}_{s,t},w\rangle+r_{s,t}, $$where X is the smooth quasigeometric rough model associated to X and r is a remainder such that r_{s,t} = o(t − s) as t → s.

iv)
For any \(\hat {\mathfrak {B}}^{N}\) orthonormal basis of \(\hat {{\mathscr{L}}}^{N}_{\omega }(A) \subset T^{N}_{\omega }(A)\) one has
$$ \dot{Y}_{t} = {\sum}_{\mathfrak{u} \in \hat{\mathfrak{B}}^{N}} \hat{f}_{\mathfrak{u}} (Y_{s})\langle \dot{\mathbf{X}}_{t,t},\mathfrak{u}\rangle. $$
The results i), ii) and iii) hold also if X is a good sqgrm.
Proof
The equivalence i) ⇔ iii) follows in the same way as in Proposition 2.20 and the equivalence i) ⇔ iv) is a consequence of the property \(\dot {\mathbf {X}}_{s,s}\in \hat {{\mathscr{L}}}_{\omega }^{N}(A)\). To prove the last equivalence i) ⇔ ii) it is sufficient to prove the identity
for any \(\mathfrak {x}\in \hat {{\mathscr{L}}}_{\omega }^{N}(A)\), \(y\in \mathbb {R}^{e}\) and any orthonormal basis \(\mathfrak {B}^{N}\) of \({\mathscr{L}}_{\omega }^{N}(A)\). However, by the definition of \(\hat {f}\) and using the property \({\varPhi }^{\ast }_{H}\mathfrak {x}\in {\mathscr{L}}_{\omega }^{N}(A)\) one has
The case of a smooth model follows straightforwardly. □
Example 3.20
Let us check how some properties of Proposition 3.19 appear when we fix \(A=\mathbb {A}_{2}^{d}\) from Example 3.3, a sqgrm X over A and a generic family of vector fields (f_{1},…,f_{d}) extended to A by setting f_{{ij}}≡ 0. The differential equation
is equivalent to
where w is a word with values in the alphabet {1,…,d}. Using the identity \(\hat {\mathbf {X}}:= {\varPhi }^{\ast }_{H} \mathbf {X}\), we can rewrite (33) as
It follows from the definition of Φ_{H} in (25) that it is possible to write down a general combinatorial formula for Φ_{H}(w), see [23, Proposition 4.10]
where {w} consists of the words we can construct from w by successively replacing any neighbouring pairs ij in w by {ij}. Switching the sum on w and u, (33) becomes
where the sum over {w}_{p} = u involves all the ways one can write a word u as {w}_{I} of a word w whose letters are in the alphabet {1,…,d} and I ∈ C(w). The element
then corresponds to the expression \(\hat {f}_{u}\) for any word u.
Using the shorthand notations \(\langle \dot {\mathbf {X}}_{s,s}, a \rangle =\langle \dot {\hat {\mathbf {X}}}_{s,s}, a\rangle =\dot {X}_{s}^{a}\) for any a ∈ A, the expressions (33) and (34) in case N = 2 become respectively
and
Interpreting X^{i} and X^{{ij}} as the components of a semimartingale and his quadratic variations, we obtain an algebraic version of the ItôStratonovich correction among semimartingales at level of SDEs, see [23] for further applications.
3.4 Algebraic Renormalization of SQGRP
We now want to adapt the notion of translation discussed for geometric rough paths to the context of quasigeometric rough paths. Adapting the same arguments in Section 2.4, for any family \((v_{a} \colon a\in A )\subset {\mathscr{L}}((A))\), a translation map T_{v}: T_{ω}((A)) → T_{ω}((A)) can easily be defined on any alphabet A. In the case of a family \((v_{a} \colon a\in A )\subset {\mathscr{L}}(A)\) it follows from the properties of any weight ω, that T_{v} sends \(T^{N}_{\omega }(A)\) to \(T^{M}_{\omega }(A)\) with \(M=N\cdot N^{\prime }\), where \(N^{\prime }\) is the smallest integer such that \(v_{i}\in {\mathscr{L}}^{N^{\prime }}_{\omega }(A)\). Passing to the quasishuffle context, the natural direction to perform a translation must be done along the Lie algebra \(\hat {{\mathscr{L}}}((A))\). To define a proper notion, Corollary 3.6 tells us the set \(\hat {{\mathscr{L}}}(A)\) consists of Lie series obtained from the set \(\mathfrak {A}= \{{\varPsi }^{\ast }_{H}(a)\colon a\in A\}\). Therefore, fixing a subset \(u = (u_{a} \colon a\in A)\subset \hat {{\mathscr{L}}}((A))\), there is a unique morphism of Lie algebras \(\hat {T}_{u}\colon \hat {{\mathscr{L}}}((A))\to \hat {{\mathscr{L}}}((A))\) such that
Using the translation maps T_{v} and the maps \({\varPsi }^{\ast }_{H}\), \({\varPhi }^{\ast }_{H}\) we see that \(\hat {T}_{u}\) uniquely extends to an endomorphism of T_{ω}((A)) with respect to the product ⊗.
Theorem 3.21
For any collection \(u = (u_{a} \colon a\in A )\subset \hat {{\mathscr{L}}}((A))\) there exists a unique ⊗endomorphism over T_{ω}((A)) which extends \(\hat {T}_{u}\colon \hat {{\mathscr{L}}}((A))\to \hat {{\mathscr{L}}} ((A))\). We call it the quasitranslation map and we denote it by the same symbol \(\hat {T}_{u}\). In particular, one has the explicit relation
where \(\hat {u}\) is given by the family \(\hat {u}= ({\varPhi }^{\ast }_{H}(u_{a}) \colon a\in A )\subset {\mathscr{L}}((A))\). Moreover, if \((u_{a} \colon a\in A)\subset \hat {{\mathscr{L}}} (A)\hat {T}_{u}\) sends \(T^{N}_{\omega }(A)\) to \(T^{M}_{\omega }(A)\) with \(M=N\cdot N^{\prime }\), where \(N^{\prime }\) is the smallest integer such that \(v_{i}\in \hat {{\mathscr{L}}}^{N^{\prime }}_{\omega }(A)\).
Proof
Thanks to Proposition 3.5 and the properties of the translation map, the map \({\varPsi }^{\ast }_{H} T_{\hat {u}}\) \({\varPhi }^{\ast }_{H}\) is a ⊗endomorphism satisfying
Moreover, from Corollary 3.6 we also deduce that \({\varPsi }^{\ast }_{H} T_{\hat {u}}{\varPhi }^{\ast }_{H}\) is a Lie endomorphism of \(\hat {{\mathscr{L}}}((A))\). Therefore we conclude from the freeness of \(\hat {{\mathscr{L}}}((A))\) that the definition (35) is a bona fide extension of \(\hat {T}_{u}\). Concerning the uniqueness of \(\hat {T}_{u}\), let Γ: T_{ω}((A)) → T_{ω}((A)) be an ⊗endomorphism extending \(\hat {T}_{u}\). Considering the map \({\varPhi }^{\ast }_{H} {\Gamma }{\varPsi }^{\ast }_{H}\) we can easily check on any a ∈ A the property
Using the fundamental property of T((A)), \({\varPhi }^{\ast }_{H} {\Gamma } {\varPsi }^{\ast }_{H}\) must coincide with \(T_{\hat {u}}\) and we obtain the uniqueness. The last properties of \(\hat {T}_{u}\) follows from the fact that \({\varPhi }^{\ast }_{H}\) and \({\varPsi }^{\ast }_{H}\) are graded maps. □
Remark 3.22
The procedure outlined in this section still leaves open the question as how to define a family of Lie endomorphisms \((\hat {T}_{u})_{u}\), \( \hat {T}_{u}: T_{\omega }((A))\to T_{\omega }((A))\) which only takes as input a finite family \(u=(u_{1},\dots ,u_{d})\) of primitive elements and the demand that T_{u}i = i + u_{i} for \(i=1,\dots ,d\). This further question, which is similar in nature and connected to what was described in Remark 3.18, is motivated by the fact that such translations coming from (only) d choices of Lie elements exist canonically in the geometric and branched case, see [6] where they were introduced. It would be desirable to have a translation map in the ddimensional quasigeometric setting (which is T_{ω}((A)) with {a ∈ Aω(a) = 1} = d) that works fully analogously, and at best is consistent with both the geometric and the branched ddimensional setting.
The quasitranslation maps now at hand can be applied to sqgrps/sqgrms and their truncated versions as in Theorem 2.24. Indeed for any \(u = (u_{a} \colon a\in A )\subset \hat {{\mathscr{L}}}((A))\) and sqgrp/sqgrm X we can actually define the compositions \(\hat {T}_{u} (\mathbf {X}_{t})\) and \(\hat {T}_{u} (\mathbf {X}_{s,t})\). In addition, for any \(u = (u_{a} \colon a\in A)\subset \hat {{\mathscr{L}}}(A)\) and Nsqgrm Y we define
where \(M=N\cdot N^{\prime }\) with \(N^{\prime }\) the smallest integer such that \(u_{a}\in \hat {{\mathscr{L}}}^{N^{\prime }}_{\omega }(A)\) for all a ∈ A and \(\hat {T}_{v}^{M} :=\textsf {proj}_{M,\omega } \hat {T}_{u}\mathfrak {i}^{M}\) with the embedding \(\mathfrak {i}^{M}: T^{M}_{\omega }(A)\to T_{\omega }((A))\). These two operations have equivalently a dynamical reinterpretation like Theorem 2.24.
Proposition 3.23
Given a sqgrp (sqgrm) X over A with weight ω and \(u = (u_{a} \colon a\in A )\subset \hat {{\mathscr{L}}} ((A))\), the composition \(\hat {T}_{u} (\mathbf {X}_{t})~(\hat {T}_{u} (\mathbf {X}_{s,t}))\) is again a sqgrp (sqgrm) which coincides with the solution of
The same result applies to good sqgrms when \(u\subset \hat {{\mathscr{L}}} (A)\). In addition, for any \(u = (u_{a} \colon a\in A)\subset \hat {{\mathscr{L}}} (A)\) and Nsqgrm Y the path \(t\to \hat {\mathcal {T}}_{u}[\mathbf {Y}]_{0,t}\) is a sqgrp coinciding with the solution of
where \(M\in \mathbb {N}\) is contained in the definition of \( \hat {\mathcal {T}}_{u}[\mathbf {Y}]\) in (36).
Proof
Choosing the family \(\hat {u}=({\varPhi }^{\ast }_{H}(u_{a}) \colon a\in A )\), we can apply Theorem 2.24 to the weighted sgrp (sgrm) \(\hat {\mathbf {X}}={\varPhi }^{\ast }_{H} \mathbf {X}\) and \(\hat {\mathbf {Y}}={\varPhi }^{\ast }_{H} \mathbf {Y}\). Then the paths associated to \({T}_{\hat {u}}(\hat {\mathbf {X}})\) and \(\mathcal {T}_{\hat {u}}[\hat {\mathbf {Y}}]\) solve respectively the equations
Using the identity (35) and the properties of \({\varPhi }^{\ast }_{H}\), \({\varPsi }^{\ast }_{H}\) of preserving the weighted length, we deduce that \(\hat {T}_{u}(\mathbf {X})={\varPsi }^{\ast }_{H}{T}_{\hat {u}}(\hat {\mathbf {X}})\) and \(\hat {\mathcal {T}}_{u}[\mathbf {Y}]={\varPsi }^{\ast }_{H}\mathcal {T}_{\hat {u}}[\hat {\mathbf {Y}}]\). Looking at the equation of the paths \({\varPsi }^{\ast }_{H} \hat {\mathbf {Z}}\) and \({\varPsi }^{\ast }_{H} \hat {\mathbf {W}}\), we obtain the equations (37) and (38). □
We conclude the section with a summary of the properties of translation operators \(\hat {T}_{u}\) together with the differential equation (16). For any collection of Lie polynomials with values in A given by \(v = (v_{a} \colon a\in A )\subset {\mathscr{L}}(A)\) and any family of vector fields (f_{a}: a ∈ A), we consider the collection of vector fields \(({f^{v}_{a}}\colon a\in A)\) defined on any a ∈ A as
where \(f\colon {\mathscr{L}}(A)\to \text {Vect}^{\infty }(\mathbb {R}^{e})\) is the natural definition on an alphabet A of the Lie algebra morphism defined in (12). Extending \(({f^{v}_{a}}\colon a\in A)\) to all T_{ω}(A), we obtain a map \(f^{v}\colon T_{\omega }(A)\to \text {Vect}^{\infty }(\mathbb {R}^{e})\) which satisfies the identity \(f^{v}= f_{T_{v}}\) over \({\mathscr{L}}(A)\), as it was already shown in the proof of Theorem 2.26, when A = {1,…,d}. By replacing the directions of translations v with \(u= (u_{a} \colon a\in A)\subset \hat {{\mathscr{L}}}(A)\) we can uniquely define a Lie algebra morphism \(\hat {f}^{u}\colon \hat {{\mathscr{L}}}(A)\to \text {Vect}^{\infty }(\mathbb {R}^{e})\) such that on any a ∈ A one has as
Extending \(\hat {f}^{u}\) to all T_{ω}(A) and applying the definition of \(\hat {f}\) with formula (35), we also obtain a map \(\hat {f}^{u}\colon T_{\omega }(A)\to \text {Vect}^{\infty }(\mathbb {R}^{e})\). Both maps \(\hat {f}^{u}\) and f^{v} allow to write the effect of translation at the level of differential equations.
Theorem 3.24
Let X be a good sqgrm and W a Nsqgrm. For any given \(u=\{u_{a}\colon a\in A\}\subset \hat {{\mathscr{L}}}(A)\) a path \(Y\colon [0,T]\to \mathbb {R}^{e}\) solves one of the equation
if and only if it solves respectively
Alternatively, by setting \(\hat {u}=\{{\varPhi }^{\ast }_{H}(u_{a})\colon a\in A\}\subset {\mathscr{L}}(A)\) and \(\hat {\mathbf {X}}= {\varPhi }^{\ast }_{H}\mathbf {X}\), \(\hat {\mathbf {W}}= {\varPhi }^{\ast }_{H}\mathbf {W}\) then Y solves also equivalently
Proof
Equivalences (39) and (40) follow by combining the proof of Theorem 2.26 and Proposition 3.19. The only thing to check is to describe the relations between the maps \(\hat {f}^{u}\) and \(f^{\hat {u}}\) and the translations \(T_{\hat {u}}\) and \(\hat {T}_{u}\). Indeed, it follows from the definition of \(\hat {T}_{u} \) and \(\hat {f}^{u}\) that one has
on any element \({\varPsi }^{\ast }_{H}(a)\) and consequently over all \(\hat {{\mathscr{L}}}(A)\). Following the first argument in Theorem 2.26 we get the first equivalence (39). Using the explicit definition of \(\hat {f}\) and identity (35), we deduce also the following equality on \(\hat {{\mathscr{L}}}(A)\)
Applying the same argument as in the proof of (ii) in Proposition 3.19, we conclude. □
We conclude the section by extending the time translation of Corollary 2.29 in the quasishuffle context. For these purposes, we suppose given a generic alphabet A^{1} with commutative bracket {,}_{1} and weight ω_{1}. Then we simply add an extra time component as new letter \(\hat {0}\) and we define the extended alphabet \(\bar {A}=\{\hat {0}\}\cup A^{1}\) together with weight \(\omega \colon \bar {A}\to \mathbb {N}^{\ast }\) defined by extending ω_{1} and putting \(\omega (\hat {0})=1\) and the extended commutative bracket {,} given for any a,b ∈ A^{1}
Under these conditions the signature of the time component \(\mathbf {X}^{0}\colon [0,T]^{2}\to G(\mathbb {R})\)
can be easily embedded in \(\hat {G}(\bar {A})\) and for any good sqgrm \(\mathbf {X}^{1}:[0,T]^{2} \to \hat {G} (A_{1})\) we can introduce again
(see also Remark 3.15). Since \({\varPsi }^{\ast }_{H}(\hat {0})= \hat {0}\), for any given a \(u_{0}\in \hat {{\mathscr{L}}}(\bar {A})\), we choose the translation \(\hat {T}_{u_{0}}\colon T(A)\to T(A)\) such that \(\hat {T}_{u_{0}}\hat {0}=\hat {0}+u_{0}\) and such that \(\hat {T}_{u_{0}}({\varPsi }^{\ast }_{H}(a))={\varPsi }^{\ast }_{H}(a)\). Then one has trivially \(\hat {T}_{u_{0}}=\) id over the subalgebra of \(T_{\omega }(\bar {A})\) isomorphic to \(T_{\omega _{1}}(A^{1})\) and we can describe the relations for equations driven by \(\bar {X}\). The proof follows easily from Corollary 2.29, Theorem 3.24 and Proposition 3.19.
Corollary 3.25
Let X^{1} be a good sqgrm and \(u_{0}\in \hat {{\mathscr{L}}}(A)\). Then the translation \(t\mapsto \hat {T}_{u_{0}}\bar {\mathbf {X}}_{0,t}\) can be described as the sum \(\mathbf {U}_{0}\boxplus \bar {\mathbf {X}}\), where U_{0} is given by
Moreover, for any given family \((f_{\hat {0}}, f_{a} \colon a\in A^{1})\) we define \(\bar {f}\colon T(A)\to \text {Vect}^{\infty }(\mathbb {R}^{e})\) and \(f\colon T(A_{1})\to \text {Vect}^{\infty }(\mathbb {R}^{e})\) like in (14) starting from \((f_{\hat {0}}, f_{a} \colon a\in A^{1})\) and (f_{a}: a ∈ A^{1}) respectively and using the notations \(\hat {u}_{0}= {\varPhi }^{\ast }_{H}u_{0}\), \(\hat {\mathbf {X}}^{1}= {\varPhi }^{\ast }_{H}\mathbf {X}^{1}\) equation
is equivalent to
Remark 3.26
Note that we recover in this smooth quasigeometric rough path setting a property like the final statement of [6, Theorem 30 (ii)]: The smooth rough path increment/model \(T_{v_{0}}\mathbf {Z}_{t}\) does not depend on the precise choice of the Hopf algebra homomorphism \(T_{u_{0}}\), as long as \(T_{u_{0}} \hat {0}=\hat {0}+u_{0}\) and \(T_{u_{0}}=\text {id}\) on \(T_{\omega _{1}}(A^{1})\).
4 Recast in an Abstract Hopf Algebra Framework
Many of the above constructions can be formulated in the abstract framework of Hopf algebras following the approach adopted in [54] and [20]. In this section, we introduce the notion smooth rough paths over a Hopf algebra and we will discuss their renormalization. An interesting operation arises at this level: the canonical sum of rough paths, which is applied to present an alternative construction to renormalization of branched rough paths, along the lines of [6].
4.1 Smooth Rough Paths on a Hopf Algebra
In what follows we fix \((\mathcal {H}, \cdot , {\Delta })\) a connected, \(\mathbb {N}\)graded and locally finite commutative Hopf algebra over \(\mathbb {R}\). That is to say there is a sequence of finitedimensional vector spaces \(\{{\mathscr{H}}_{n}\}_{n\geq 0}\) such that
where 1 is the unit of \({\mathscr{H}}\). For any given Hopf algebra in this class, we introduce two different notions of dual space: the full dual \({\mathscr{H}}^{\prime }\) and the graded dual \({\mathscr{H}}^{\ast }\) respectively defined by
where \(V^{\prime }=\text {Hom}(V, \mathbb {R})\) stands for the space of continuous \(\mathbb {R}\)valued linear forms on V. There is a canonical pairing 〈⋅,⋅〉 between \({\mathscr{H}}^{\prime }\) and \({\mathscr{H}}\) and the space \({\mathscr{H}}^{\ast }\) lies dense in \({\mathscr{H}}^{\prime }\) by equipping \({\mathscr{H}}^{\prime }\) with the weak topology, i.e. the weakest topology for which the evaluation maps v^{∗}↦〈v^{∗},u〉, \(u\in {\mathscr{H}}\) are continuous, see [3, Lemma 1.7], [4].
Using the grading of \({\mathscr{H}}\), we can also define for any \(N\in \mathbb {N}\) the truncated spaces
which yield two natural filtrations and \(\{({\mathscr{H}}^{\ast })^{N}\colon N\in \mathbb {N}\}\), \(\{{\mathscr{H}}^{N}\colon N\in \mathbb {N}\}\) for the vector spaces \({\mathscr{H}}^{\ast }\) and \({\mathscr{H}}\). We note that \(({\mathscr{H}}^{\ast })^{N}=({\mathscr{H}}^{N})^{\ast }\) and will henceforth write \({\mathscr{H}}^{\ast N}\).
The coproduct Δ of \({\mathscr{H}}\) induces by duality a product ⋆ in \({\mathscr{H}}^{\prime }\) and a fortiori on \({\mathscr{H}}^{\ast }\) defined on any \(\alpha , \beta \in {\mathscr{H}}^{\prime }\) and \(h\in {\mathscr{H}}\) via the identity
where 〈⋅,⋅〉_{2} is the canonical pairing between \({\mathscr{H}}^{\prime }\otimes {\mathscr{H}}^{\prime }\) and \({\mathscr{H}}\otimes {\mathscr{H}}\) induced by the canonical dual pairing 〈⋅,⋅〉. Moreover it is also possible to define a coproduct Δ^{∗} only on \({\mathscr{H}}^{\ast }\) from the identity
obtaining the dual Hopf algebra \(({\mathscr{H}}^{\ast }, \star , {\Delta }^{\ast })\). The coproduct Δ is compatible with the filtration and hence so is the product ⋆, which maps \({\mathscr{H}}^{\ast M}\times {\mathscr{H}}^{\ast N}\) to \({\mathscr{H}}^{*M+N}\). The projections \(\pi _{N}: {\mathscr{H}}\longrightarrow {\mathscr{H}}/\oplus _{n=N+1}^{\infty } {\mathscr{H}}_{n}\) onto the quotient by the ideal \(\oplus _{n=N+1}^{\infty } {\mathscr{H}}_{n}\) is an algebra morphism so that ⋆_{N} := π_{N} ⋆ restricted to \({\mathscr{H}}^{*N}\) defines a truncated product \(\star _{N}: {\mathscr{H}}^{*N}\times {\mathscr{H}}^{*N}\to {\mathscr{H}}^{*N}\). Much in the same way as rough paths over a Hopf algebra \({\mathscr{H}}\) were defined, see [20, 54], we now define smooth rough paths and models.
Definition 4.1
We call a levelN smooth \({\mathscr{H}}\) rough path (in short: Ns\({\mathscr{H}}\)rp) any non zero path \(\mathbf {X}: [0,T] \to {\mathscr{H}}^{*N}\) satisfying the following properties:

(a” i)
For all times t ∈ [0,T] and for all \( h\in {\mathscr{H}}^{K}\), \(k\in {\mathscr{H}}^{L}\), with K + L = N one has
$$ \langle\mathbf{X}_{t},v \cdot w\rangle = \langle \mathbf{X}_{t},v\rangle \langle\mathbf{X}_{t},w\rangle. $$(41) 
(a” ii)
For every word \( h\in {\mathscr{H}}^{N}\), the map t↦〈X_{t},h〉 is smooth.
We call levelN smooth \({\mathscr{H}}\) rough model (in short: Ns\({\mathscr{H}}\)rm) any map \(\mathbf {X}: [0,T]^{2} \to {\mathscr{H}}^{*N}\) which satisfies property (41) for all X_{s,t} as well as the following properties:

(b’ i)
The following abstract Chen relation holds:
$$ \mathbf{X}_{s,u} \star_{N} \mathbf{X}_{u,t}=\mathbf{X}_{s,t} $$(42)for any s,u,t ∈ [0,T].

(b’ ii)
For every \(h \in {\mathscr{H}}^{N}\), the map t↦〈X_{s,t},w〉 is smooth, for one (equivalently: all) base point(s) s ∈ [0,T].
By smooth \({\mathscr{H}}\) rough path (model) (in short: s\({\mathscr{H}}\)rp and s\({\mathscr{H}}\)rm), we mean a path (map) with values in \({\mathscr{H}}^{\prime }\) for which (41) holds for any \(w,v\in {\mathscr{H}}\) and relation (42) holds with ⋆.
Algebraic properties of smooth \({\mathscr{H}}\) rough paths are encoded in the specific Lie group (Lie algebra) structures of \({\mathscr{H}}^{\prime }\). Following [54, Definition 2.6], property (41) (in the case when \(v,w\in {\mathscr{H}}\)), amounts to X_{t} belonging to the group of Ntruncated characters \(G^{N}(\mathcal H)\) or the group of characters \(G(\mathcal H)\), for any t. The \((G({\mathscr{H}}), \star )\) is a topological Lie group when equipped with the topology of pointwise convergence, see [3] and by quotienting \((G^{N}(\mathcal H), \star _{N})\) is a finitedimensional Lie group. Their Lie algebras can be explicitly described by the set \({\mathfrak g}^{N}({\mathscr{H}})\) of levelN truncated infinitesimal characters, resp. the set \({\mathfrak g}({\mathscr{H}})\) of infinitesimal characters, i.e. of elements \(\alpha \in {\mathscr{H}}^{\star N}\), resp. \(\alpha \in {\mathscr{H}}^{\star }\) such that for all \( (h, k)\in {\mathscr{H}}^{K}\times {\mathscr{H}}^{L}\) with K + L = N, resp. \((h, k)\in {\mathscr{H}}^{2}\), the following property holds:
where 1^{∗} is the counit of \({\mathscr{H}}\). The Lie brackets \([\cdot , \cdot ]_{\star _{N}}\) on \({\mathfrak g}^{N}({\mathscr{H}})\) and [⋅,⋅]_{⋆} on \({\mathfrak g}({\mathscr{H}})\) are given by the commutators of ⋆_{N} and ⋆. The pair \(({\mathfrak g}({\mathscr{H}}),[\cdot , \cdot ]_{\star })\) defines a topological Lie algebra, see [4]. A special role is played by the set of primitive elements \(P({\mathscr{H}}^{\ast })= {\mathfrak g}({\mathscr{H}})\cap {\mathscr{H}}^{\ast }\) which is a Lie algebra with the induced Lie brackets. To relate these Lie algebras with the character groups we introduce the exponential and truncated exponential map \(\exp _{\star }:{\mathscr{H}}^{\prime }\to {\mathscr{H}}^{\prime }\), \(\exp _{\star _{N}}:{\mathscr{H}}^{*N}\to {\mathscr{H}}^{*N}\) defined by
When restricted to the Lie algebras, they induce bijections \(\exp _{\star _{N}}: {\mathfrak g}^{N}({\mathscr{H}})\to G^{N}({\mathscr{H}})\) and the \(\exp _{\star }: {\mathfrak g}({\mathscr{H}})\to G({\mathscr{H}})\), turning \(G({\mathscr{H}})\) into an analytic Lie group, see [3, Theorem 3.7 and Appendix B] and [4, Theorem 3.9] for the properties of \(\exp _{\star }\) and \(G({\mathscr{H}})\).
In practice, we can identify Ns\({\mathscr{H}}\)rp X_{t} and Ns\({\mathscr{H}}\)rm X_{s,t} with the same equivalence properties of Nsgrp and Nsgrm by using the group structure induced by ⋆, see (7). Similarly to Definition 2.5, we define the extension of a Ns\({\mathscr{H}}\)rp. The diagonal derivative
of any given X in s\({\mathscr{H}}\)rp lies in the Lie algebra \(g({\mathscr{H}})\). These properties allow to extend Theorem 2.8 on any smooth \({\mathscr{H}}\)rough path.
Theorem 4.2 (Fundamental Theorem of s\({\mathscr{H}}\)rp)
Let \(N \in \mathbb {N}\). Any Y in Ns\({\mathscr{H}}\)rp uniquely extends to some X in s\({\mathscr{H}}\)rp which is minimal in the sense that
for all s ∈ [0,T]. We call \({\mathscr{H}}\text {MinExt}(\mathbf {Y}) := \mathbf {X}\) the \({\mathscr{H}}\)minimal extension of Y and also \({\mathscr{H}}\text {MinExt}^{N^{\prime }}(Y) := \pi _{N^{\prime }} \mathbf {X}\), for \(N^{\prime } > N\), the \(N^{\prime }\)minimal extension of Y. Moreover, for any [s,t] ⊂ [0,T] the associated sgrm of X_{s,t} only depends on Y_{[s,t]}. We call this object the \({\mathscr{H}}\) signature of Y on [s,t], in symbols \(\text {Sig}_{{\mathscr{H}}}(\mathbf {Y}_{[s,t]})\). A s\({\mathscr{H}}\)rm X is called a good s\({\mathscr{H}}\)rm if it satisfies \(\mathbf {X}= {\mathscr{H}}\text {MinExt}(\mathbf {Y})\) for some Ns\({\mathscr{H}}\)rm Y.
Proof
The proof of the result goes as that of Theorem 2.8, modulo the replacement of ⊗ by ⋆. The only properties to check in this generalised context is the existence and uniqueness of a smooth solution for the initial value problem
for any given smooth curve \(\eta :[0, T]\to {\mathfrak g}({\mathscr{H}})\). Moreover, we need to prove that for any given couple \(X, \bar {X}\in G^{N+1}({\mathscr{H}})\) such that \(\langle X,h\rangle = \langle \bar {X}, h\rangle \) for any \(h\in {\mathscr{H}}_{N}\) then \(X\bar {X}\) belongs to the center of \(G^{N+1}({\mathscr{H}})\). The first property follows from the regularity in the sense of Milnor, see [47], of the Lie group \(G({\mathscr{H}})\), see [5, Theorem B] and references therein. The second property follows from [54, Proposition 2.10]. □
Example 4.3
Setting and we recover respectively Theorems 2.8 and 3.9. In this identification, the operation ⋆, depending on Δ is identified with the concatenation product ⊗ see [53]. As recalled in Theorem 3.4, the Hoffman’s exponential and logarithm (25) yield an isomorphism of graded Hopf algebras . Hence the basic properties of these maps follow from the general fact that for any given Hopf algebra isomorphism \({\Gamma }\colon {\mathscr{H}}\to \mathcal {\mathcal {K}}\) among two Hopf algebras \({\mathscr{H}}\) and \(\mathcal {K}\) with the properties listed at the beginning, the adjoint map \({\Gamma }^{\ast }\colon {\mathscr{H}}^{\ast }\to \mathcal {K}^{\ast }\) is also an isomorphism. We observe also that the scalar product 〈⋅,⋅〉 defined on \(T(\mathbb {R}^{d})\) and T_{ω}(A) can be used to identify the graded dual \({\mathscr{H}}^{\ast }\) with \({\mathscr{H}}\) by means of the Riesz lemma, so that it is not necessary to introduce the notion of graded dual in that context.
Example 4.4
Another relevant example in the context of renormalization of rough paths arising in [6] is the Butcher–Connes–Kreimer Hopf algebra \(\mathcal {H}_{BCK}(\mathbb {R}^{d})\) consisting of polynomials of rooted forests τ with nodes decorated by the finite set {1,…,d} together with the empty forest 1. A forest f is graded accordingly to f, the number of its nodes and the coproduct Δ is defined on each tree τ as
where the sum is taken over a specific set of admissible cuts over the tree. The result of each cut produces a polynomial of trees P_{c}(t) and R_{c}(t) a tree corresponding to the root. We call the Ns\({\mathscr{H}}\)rp or s\({\mathscr{H}}\)rp in this context levelN smooth branched rough paths and smooth branched rough paths (in short: Nsbrp and sbrp), see [32]. The operation ⋆ induced by the coproduct (43) coincide with the so called Grossman–Larson algebra of forests [31] and \(P({\mathscr{H}}^{\ast }_{BCK}(\mathbb {R}^{d}))\) coincides with the free vector space \(\langle \mathfrak {T}_{d}\rangle \) generated by \(\mathfrak {T}_{d}\), the set of dual trees decorated by the finite set {1,…,d} and \(\mathfrak {g}(\mathcal {H}_{BCK}(\mathbb {R}^{d}))\) is the vector space \(\langle \mathfrak {T}^{\prime }_{d}\rangle \) of tree series.
In adequacy with the previously results, levelN smooth \({\mathscr{H}}\)rough models are a special case of γHölder \({\mathscr{H}}\)rough paths introduced in [20].
Proposition 4.5
Every Ns\({\mathscr{H}}\)rm X is a 1/Nregular Ntruncated \({\mathscr{H}}\) rough path, see [20, Definition 4.3] and its minimal extension \(\tilde {\mathbf {X}}\) coincides with the lift of X, as constructed in [20, Theorem 4.4].
Remark 4.6
The same considerations of Remark 3.13 apply also to the class of γregular Ntruncated \({\mathscr{H}}\) rough paths.
4.2 Translation of Smooth Rough Paths on a Hopf Algebra
We now extend to the framework of smooth \({\mathscr{H}}\)rough paths the notion of translation discussed in Theorems 2.24 and 3.21. To specify in which direction we can perform a translation, we choose a generic finitedimensional subspace \( \mathcal {D}\subset P({\mathscr{H}}^{\ast })\). Fixing a basis {h_{1},…,h_{e}} of \(\mathcal {D}\), we obtain a fixed set of directions to apply a translation. By assigning an element of \(P({\mathscr{H}}^{\ast })\) to each direction, we introduce an abstract notion of translation.
Definition 4.7
Given a family of primitive elements \(v=\{v_{i}\colon i=1,\ldots ,e \}\subset \mathfrak {g}({\mathscr{H}})\) we call a translation map over \(\mathcal {D}\) any continuous Lie algebra homomorphism \(M_{v}\colon \mathfrak {g}({\mathscr{H}})\to \mathfrak {g}({\mathscr{H}})\) such that M_{v}h_{i} = h_{i} + v_{i} for any i = 1,…,e.
Remark 4.8
Using the topological properties of \(\mathfrak {g}({\mathscr{H}})\), if \( v=\{v_{i}\colon i=1,\ldots ,e\}\subset P({\mathscr{H}}^{\ast })\) to define a continuous Lie algebra homomorphism M_{v} it is sufficient to have a Lie homomorphism \(M_{v}\colon P({\mathscr{H}}^{\ast })\to \mathfrak {g}({\mathscr{H}})\) since the Lie algebra \(P({\mathscr{H}}^{\star })\) is dense in \({\mathfrak g}({\mathscr{H}})\), see [4, Remark 3.11]. Note that Definition 4.7 only gives a pointwise definition for a fixed v. As in the cases studied in this paper, in practice, we need a map v → M_{v} with certain consistency properties like \(M_{v} M_{u}=M_{v+M_{v} u}\), a condition we do not impose here. More generally we hope to investigate in future work, what properties one should impose on \(\mathcal {D}\) and v in a general Hopf algebra framework. A first suggestion of such a set of axioms was very recently given in [52, Definitions 6 and 7].
Remark 4.9
The previous translation maps T_{v} and \(\hat {T}_{\hat {v}}\) in the geometric and quasigeometric setting are specific examples of translation over two different subspaces of primitive elements, i.e. the vector space \(\mathbb {R}^{d}\) and the free vector space generated by \(\mathfrak {A}= \{{\varPsi }^{\ast }_{H}(a)\colon a\in A\}\). This is a very specific situation one uses the specific structure of \(P({\mathscr{H}}^{\ast })\) as a free Lie algebra, which ensures both the existence and the uniqueness of a translation map. However, it was shown in [6, Example 9] that one can construct two different translation maps over the same vector space \(\mathcal {D}\) when \({\mathscr{H}}=\mathcal {H}_{BCK}(\mathbb {R}^{d})\). The Lie structure of \(\mathfrak {g}({\mathscr{H}})\) is therefore not sufficient to determine a unique translation. Hence the idea of a definition which does not involve a uniqueness of the translation in its formulation.
Once given \( v=\{v_{i}\colon i=1,\ldots ,e \}\subset \mathfrak {g}({\mathscr{H}})\) and a translation map M_{v} over \(\mathcal {D}\), we can actually uniquely extend M_{v} to a continuous ⋆ morphism \(M_{v}\colon {\mathscr{H}}^{\prime }\to {\mathscr{H}}^{\prime }\) the full translation map which we denote in the same way. The extension is purely algebraic and follows by standard Milnor–Moore theorem [48]. Indeed any translation map defines uniquely a Lie algebra morphism \(M_{v}\colon P({\mathscr{H}}^{\ast })\to {\mathscr{H}}^{\prime }\) which is compatible with the product ⋆. Using the universal property of the universal enveloping algebra \(\mathcal {U}(P({\mathscr{H}}^{\ast }))\) and the Milnor–Moore theorem, the map M_{v} uniquely extends to a ⋆ morphism \(M_{v}\colon \mathcal {H}^{\ast }\to {\mathscr{H}}^{\prime }\) which by density can be defined over \({\mathscr{H}}^{\prime }\). This map allows us to perform translation of s\({\mathscr{H}}\)rms like in Theorem 2.24 and Proposition 3.23.
Theorem 4.10
Given a s\({\mathscr{H}}\)rp (s\({\mathscr{H}}\)rm), a family \( v=\{v_{i}\colon i=1,\ldots ,e \}\subset \mathfrak {g}({\mathscr{H}})\) and a translation map M_{v} over \(\mathcal {D}\), the composition M_{v}(X_{t}), (M_{v}(X_{s,t})) is again a s\({\mathscr{H}}\)rp (s\({\mathscr{H}}\)rm) which coincides with the solution of
The same result applies to good s\({\mathscr{H}}\)rms when \( v\subset P({\mathscr{H}}^{\ast })\). Supposing also that for any \(N\in \mathbb {N}\) there exists an integer L ≥ N depending on v such that \(M_{v}\colon {\mathscr{H}}^{*N}\to {\mathscr{H}}^{*L}\), then for any Ns\({\mathscr{H}}\)rp Y the unique solution to
defines a Ls\({\mathscr{H}}\)rm, given by \(\mathbf {W}_{s,t} = \mathbf {W}_{s}^{1}\otimes _{M} \mathbf {W}_{t}\), which we call \({\mathscr{M}}_{v}[\mathbf {Y}]\). Moreover, we have the explicit form
with algebra endomorphism \({M_{v}^{L}} := \pi _{M} M_{v}\mathfrak {i}^{M}\) of \(({\mathscr{H}}^{*M},\star _{M})\), using the (linear) embedding \(\mathfrak {i}^{M}: {\mathscr{H}}^{*M}\to {\mathscr{H}}^{\prime }\), and \(\mathbf {Y}^{M}={\mathscr{H}}\text {MinExt}^{M}(\mathbf {Y})\).
Proof
The theorem is then concluded by checking that M_{v}X and \({\mathscr{M}}_{v}[\mathbf {Y}]\) solves the equation (44) and (45) for any s\({\mathscr{H}}\)rp X and Ns\({\mathscr{H}}\)rp Y. This last check follows like in the proof of Theorem 2.24 by replacing ⊗ with ⋆. □
As direct application of Theorem 4.10, we present a characterisation of the renormalization of branched rough paths introduced in [6] when \({\mathscr{H}}=\mathcal {H}_{BCK}(\mathbb {R}^{d})\). In that case, the authors provided the existence a translation map \(M_{v}\colon \mathfrak {T}_{d}\to \mathfrak {T}_{d}\) over the subspace \(\mathcal {D}\) generated by the dual trees \(\{\bullet _{i}^{\ast }\colon i=1,\ldots ,d\}\) by using a specific property of the Lie algebra \((\langle \mathfrak {T}_{d}\rangle , [ ,]_{\star })\), which we briefly sketch.
Recall that a (left) preLie algebra is a vector space V equipped with a bilinear map \(\curvearrowright :V\otimes V\to V\) whose associator
is invariant under the exchange of the two variables y and z, see [44]. Given a preLie algebra, one can construct a Lie bracket via its antisymmetrisation
It turns out that \(\langle \mathfrak {T}_{d}\rangle \) admits an explicit preLie algebra structure \(\curvearrowright \) which as well as satisfying \([ ,]_{\curvearrowright }=[ ,]_{\star }\), also has the property that \((\langle \mathfrak {T}_{d}\rangle , \curvearrowright )\) is isomorphic to the free preLie algebra over d elements, see [13].
Thanks to this property for any given family of tree series \(v= (v_{1}, \ldots , v_{d} )\subset \mathfrak {T}^{\prime }_{d}\), it is then possible to fix a unique translation map \(M_{v}\colon \langle \mathfrak {T^{\prime }}_{d}\rangle \to \langle \mathfrak {T^{\prime }}_{d}\rangle \) which satisfies
for any i = 1,…,d. It follows from the standard properties of the operation \(\curvearrowright \) and the GrossmannLarson product on the grading on trees ⋅ that for any given \(v= (v_{1}, \ldots , v_{d} )\subset \mathfrak {T}_{d}\) M_{v} maps \(\mathcal {H}^{*N}_{BCK}(\mathbb {R}^{d})\) to \({\mathscr{H}}^{*L}_{BCK}(\mathbb {R}^{d})\) where \(L=N\cdot N^{\prime }\) with \(N^{\prime }\) the smallest integer such that \(v_{i}\leq N^{\prime }\) for any i = 1,…,d. From Theorem 4.10 we deduce the following corollary.
Corollary 4.11
Given a sbrp (sbrm) X and \(v = (v_{1},\ldots , v_{d})\subset \mathfrak {T}^{\prime }_{d}\), the composition M_{v}(X_{t}) (M_{v}(X_{s,t})) with M_{v} uniquely defined by (46) is again a sqgrp (sqgrm) which coincides with the solution of
The same result applies to good sqgrms when \(v\subset \mathfrak {T}_{d}\). Under the same restriction on v, for any Nsbrm Y the path \(t\to {\mathscr{M}}_{u}[\mathbf {Y}]_{0,t}\) is a Lsbrp coinciding with the solution of
where \(L\in \mathbb {N}\) is the smallest integer such that \(v_{i}\leq N^{\prime }\) for any i = 1,…,d.
Remark 4.12
Conditions (46) identify uniquely a full translation map M_{v} whose explicit calculation on forest is not direct. In [6] the authors obtained a dual description of the dual map \(M^{\ast }_{v}\colon {\mathscr{H}}_{BCK}(\mathbb {R}^{d})\to {\mathscr{H}}_{BCK}(\mathbb {R}^{d})\) using coalgebraic tools related to extraction and contraction of trees but an explicit expression of M_{v} is still unknown. Looking at (47), (48) we realize that in case of smooth branched rough paths it is sufficient to compute M_{v} only on trees, which should slightly simplify the computations. Moreover, when M_{v} coincides with the addition in (51), then we obtain an explicit equation which does not involve any coalgebraic tool.
Remark 4.13
Concerning the question of defining translation maps for more general Hopf algebras \({\mathscr{H}}\), the example of the free preLie algebra points towards the general approach of using some ’free’ algebraic structure on the Lie algebra \(P({\mathscr{H}}^{\ast })\), if it is available, to define the translation map as a universal homomorphism of the free object. Ideas to this regard were proposed in [51, Definition 5.2.1] and [52, Definition 8]. In the latter, this basic idea is also applied to the case of the free postLie algebra taking the role of \(P({\mathscr{H}}^{\ast })\).
4.3 Canonical Sum and Minimal Coupling of Smooth Rough Paths
We pass now to the notion of sum in the generalised context of s\({\mathscr{H}}\)rms, extending Definition 2.15 to a generic Hopf algebra as before.
Definition 4.14
For any fixed s\({\mathscr{H}}\)rms \(\mathbf {X}, \mathbf {Y}:[0,T]^{2}\to {\mathscr{H}}^{\prime }\) let \(t\mapsto \mathbf {Z}_{t} \in {\mathscr{H}}^{\prime }\) be the Cartan development of \(\dot {\mathbf {X}}_{s,s}+\dot {\mathbf {Y}}_{s,s}\), i.e. the unique solution to
We then write \(\mathbf {Z} := \mathbf {X} \boxplus \mathbf {Y}\) for the associated sgrm and call it the canonical sum of X and Y. For any \(\lambda \in \mathbb {R}\) we define also the sgrm \(\mathbf {Z}=\lambda \boxdot \mathbf {X}\) via the Cartan development of \(\lambda \dot {\mathbf {X}}_{s,s}\), we call it the canonical scalar multiplication.
Remark 4.15
This scalar multiplication yields a new scaling device for smooth rough paths which strongly differs from the wellknown dilation δ_{λ}, where the latter is defined for any \(x\in {\mathscr{H}}_{n}\) by \(\langle \delta _{\lambda }\mathbf {X}_{s,t},x\rangle =\langle \mathbf {X}_{s,t},\lambda ^{n} x\rangle =\lambda ^{n}\langle \mathbf {X}_{s,t},x\rangle \). In contrast to the pointwise scaling δ_{λ}, this new scaling is a dynamical way to scale a smooth rough path. However, we observe that \((0\boxdot \mathbf {X})_{s,t}=\delta _{0}\mathbf {X}_{s,t}=\mathbf {1}^{\ast }\) and that \(\overleftarrow {\mathbf {X}}\) (the backward^{Footnote 4} rough path), δ_{− 1}X and \((1)\boxdot \mathbf {X}\) are pairwise distinct. We expect this to have interesting applications to the theory of signatures of rough paths in the geometric setting.
From the vector space structure of the Lie algebra \({\mathfrak g}(\mathcal H)\), we easily deduce that the set of all smooth \({\mathscr{H}}\) rough models forms itself a vector space when equipped with the sum \(\boxplus \) and the scalar multiplication \(\boxdot \), with the set of all good smooth \({\mathscr{H}}\) rough models forming a subspace. In particular, for any real numbers λ_{1}, λ_{2}, λ we have
This is very much in contrast to the spaces of γHölder or bounded pvariation rough paths for γ ≤ 1/2 and p ≥ 2, where it is basic folklore by now that such a vector space structure does not exist in any meaningful sense.
It is then possible to characterise the sum \(\boxplus \) via the group operation ⋆ up to some small remainder. The following theorem is an extension in a Hopf algebra and smooth case of [41, Section 3.3.1 B], which was stated in the context of geometric pvariation rough paths.
Proposition 4.16
For any fixed couple of s\({\mathscr{H}}\)rms X, Y a map \(\mathbf {Z}\colon [0,T]^{2}\to G({\mathscr{H}})\) coincides with \(\mathbf {X} \boxplus \mathbf {Y}\) if and only if Z satisfies Z_{s,t} = Z_{s,u} ⋆ Z_{u,t} for s,u,t ∈ [0,T] and one has for any s ∈ [0,T]
for some \(R_{s,t}\in {\mathscr{H}}^{\prime }\) such that for all \(x\in {\mathscr{H}}\) one has 〈R_{s,t},x〉 = o(t − s) as t → s. Moreover, we have the relations
for some r_{s,t}, \(r^{\prime }_{s,t} \in {\mathscr{H}}^{\prime }\) such that for all \(x\in {\mathscr{H}}\) one has \(\langle r_{s,t},x\rangle ,\langle r^{\prime }_{s,t},x\rangle = o(ts)\) as t → s.
Proof
Let us start by formula (49). We fix \(\mathbf {Z}:[0,T]^{2}\to G({\mathscr{H}})\) any map with Z_{s,t} = Z_{s,u} ⋆ Z_{u,t} and for all \(x\in {\mathscr{H}}\) one has 〈Z_{s,t} −X_{s,t} ⋆ Y_{s,t},x〉 = o(t − s). Then, by considering the path t →Z_{0,t} and fixing s ∈ [0,T], we use the continuity of the map \({\mathscr{H}}^{\prime }\ni \mathbf {x}\mapsto \mathbf {Z}_{0,s}\star \mathbf {x}\), for the weak convergence with respect to the duality pairing of \({\mathscr{H}}^{\prime }\) with \({\mathscr{H}}\) to have the equalities
Thus the path s↦Z_{0,s} is differentiable with derivative \(s\mapsto \mathbf {Z}_{0,s}\star (\dot {\mathbf {X}}_{s,s}+\dot {\mathbf {Y}}_{s,s})\), implying \(\dot {\mathbf {Z}}_{s,s}=\dot {\mathbf {X}}_{s,s}+\dot {\mathbf {Y}}_{s,s}\), i.e. \(\mathbf {Z}=\mathbf {X}\boxplus \mathbf {Y}\). On the other hand, assuming that Z is given by \(\mathbf {X}\boxplus \mathbf {Y}\), for all \(x\in {\mathscr{H}}\) there are remainders \(\theta ^{x}_{s,t}= o(ts)\), \({\theta }^{\prime x}_{s,t}= o(ts)\) such that
Therefore the equivalence (49) will follow from equivalence (50). Recall that for X_{s,t}, Y_{s,t} in \(G({\mathscr{H}})\) and x in \( {\mathscr{H}}\), we have \(\langle \mathbf {X}_{s,t}\star \mathbf {Y}_{s,t},x\rangle = \langle \mathbf {X}_{s,t} \hat {\otimes } \mathbf {Y}_{s,t},{\Delta } x\rangle \), where \(\hat {\otimes }\) is the completed external tensor product, and furthermore 〈X_{s,t},1〉 = 〈Y_{s,t},1〉 = 1. Hence,
Furthermore, for every \(x\in {\mathscr{H}}^{\geq 1}={\sum }_{n=1}^{\infty }{\mathscr{H}}_{n}\), i.e. 〈1^{∗},x〉 = 0, we write \({\Delta } x=\mathbf {1}\otimes x+x\otimes \mathbf {1}+{\sum }_{i=1}^{n} y_{i}\otimes z_{i}\) for some n and \(y_{i},z_{i}\in {\mathscr{H}}^{\geq 1}\), thereby obtaining
The fact that this last term is of order o(t − s) follows from the smoothness of X and Y and 〈X_{t,t},y_{i}〉 = 〈Y_{t,t},z_{i}〉 = 0, which for any index i, actually yields the existence of a constant C_{i} > 0 such that 〈X_{s,t},y_{i}〉〈Y_{s,t},z_{i}〉≤ C_{i}(t − s)^{2} for all t,s ∈ [0,T]. Similarly, one shows that \(\langle \mathbf {Y}_{s,t}\star \mathbf {X}_{s,t}\mathbf {X}_{s,t}\mathbf {Y}_{s,t}+\mathbf {1}^{\ast },x\rangle = o(ts)\). □
Remark 4.17
As pointed out in Remark 2.17 with reference to [41, Section 3.3.1 B] for smooth geometric rough paths, we strongly conjecture that one can add the minimal extension of a Ns\({\mathscr{H}}\)rm X to any general γHölder \({\mathscr{H}}\) rough path W for any γ ∈ (0,1), via the requirement \(\langle (\mathbf {X}\boxplus \mathbf {W})_{s,t},x\rangle =\langle \mathbf {X}_{s,t}\star \mathbf {W}_{s,t},x\rangle +o(ts)\) using the sewing lemma.
The canonical sum can now be used to define a minimal coupling of smooth rough paths in the situation where we have connected graded commutative Hopf algebras \((\mathfrak {H}^{i})_{i=1,\dots ,m}\) of the form
together with embeddings \(\iota _{i}:(\mathfrak {H}^{i})^{\ast }\to ({\mathscr{H}})^{\ast }\) which are injective graded Hopf algebra homomorphisms, such that the sum of the nonunital parts (the kernels of the counit of \({\mathscr{H}}^{\ast }\)) of the images \(\hat {\mathfrak {H}}^{i}:=\iota _{i}(\mathfrak {H}^{i})^{\ast }\) and \(\hat {\mathfrak {H}}^{i}_{j}:=\iota _{i}(\mathfrak {H}^{i})^{\ast }_{j}\) inside \({\mathscr{H}}^{\ast }\) forms a direct product, i.e.
where \(\hat {\mathfrak {H}}^{i,\geq 1}:={\bigoplus }_{j=1}^{\infty } \hat {\mathfrak {H}}_{j}^{i}=\{x\in \hat {\mathfrak {H}}^{i}\colon \langle x,\mathbf {1}\rangle =0\}\), and such that the first level of \({\mathscr{H}}^{\ast }=\oplus _{n=0}^{\infty } {\mathscr{H}}^{\ast }_{n}\) is actually spanned by the first levels of the \(\hat {\mathfrak {H}}^{i}\), i.e.
Any s\(\mathfrak {H}^{i}\)rp \(\mathbf {X}^{i}:[0,T]\to G(\mathfrak {H}^{i})\) is trivially mapped via the embedding ι_{i} and extended by continuity inside \({\mathscr{H}}^{\prime }\) to a s\({\mathscr{H}}\)rp \(\iota _{i} \mathbf {X}^{i}:[0,T]\to G({\mathscr{H}})\). We define then the minimal coupling of the X^{i} as
As a special case, we consider two connected graded commutative Hopf algebras \(\mathfrak {H}^{0}\), \(\mathfrak {H}^{1}\) with embedded graded dual spaces \(\hat {\mathfrak {H}}^{0}=\iota _{0}(\mathfrak {H}^{0})^{\ast }\), \(\hat {\mathfrak {H}}^{1}=\iota _{1}(\mathfrak {H}^{1})^{\ast }\) such that
By taking \(\hat {\mathfrak {H}}^{0}\) the unital subalgebra generated by just a single element \(x_{0}\in ({\mathscr{H}})^{\ast }_{1}\) (which is automatically a graded sub Hopf algebra of \({\mathscr{H}}^{\ast }\)), the only smooth rough models \(\mathbf {X}^{0}:[0,T]\to G(\mathfrak {H}^{0})\) are of the form
for some smooth \(\psi \colon [0,T]\to \mathbb {R}\). Then one easily checks that
By taking ψ = 1, we find that for any s\(\mathfrak {H}^{1}\)rp \(\mathbf {X}^{1}:[0,T]\to G(\mathfrak {H}^{1})\) the minimal coupling \({\bar {\mathbf {X}}}:=(\mathbf {X}^{0},\mathbf {X}^{1})_{\min \limits }\) is uniquely determined by
In this situation, we can directly reinterpret \(\bar {\mathbf {X}}\) as a specific type of full translation.
Corollary 4.18
Let X^{1} be a good s\({\mathscr{H}}\)rm, \(v_{0}\in \mathfrak {g}({\mathscr{H}})\) and \(M_{v_{0}}\colon {\mathscr{H}}^{\ast }\to {\mathscr{H}}^{\ast }\) a full translation map such that \(M_{v_{0}}x_{0}=x_{0}+v_{0}\) and \(M_{v_{0}}\) restricted to \(\iota _{1}(\mathfrak {H}^{1})^{\ast }\) is the identity. Then the translation \(t\mapsto M_{v_{0}}\bar {\mathbf {X}}_{0,t}\) can be described as the sum \(\mathbf {V}_{0}\boxplus \bar {\mathbf {X}}\), where V_{0} is given by
Proof
We obviously have
which shows \(M_{v_{0}}{\bar {\mathbf {X}}}=\mathbf {V}_{0}\boxplus \bar {\mathbf {X}}\). □
Remark 4.19
In this general connected graded commutative Hopf algebra setting, we once again recover a property similar to the final statement of [6, Theorem 30 (ii)]: the s\({\mathscr{H}}\)rp \(M_{v_{0}}{\bar {\mathbf {X}}}_{t}\) does not depend on the precise choice of the Hopf algebra homomorphism \(M_{v_{0}}\), as long as \(M_{v_{0}} x_{0}=x_{0}+v_{0}\), \(M_{v_{0}}_{\hat {\mathfrak {H}}^{1}}=\text {id}\).
Notes
(Warning.) We insist again that smooth rough paths are much richer objects than canonical lifts of smooth paths, unfortunately also called smooth rough paths in the earlier stages of the theory. To wit, the pure area rough path, familiar in rough path theory and seen later on in the text, is a perfect example of a smooth rough path.
In fact, a Hopf algebra on \(T(\mathbb {R}^{d})\) together with the deconcatenation coproduct, see e.g. [53], though this will not play a role in this section.
Even though d = 1 weakly geometric rough paths are an outlier in the sense that they are fully determined by the underlying path, they can be trivially embedded into rough paths spaces of higher dimension and thus such a counterexample also applies there.
The backward path is given by \(\overleftarrow {\mathbf {X}}_{s,t}=\mathbf {X}_{(Ts),(Tt)}\).
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Acknowledgements
CB, PKF and SP were supported in part by DFG Research Unit FOR2402. PKF and RP were supported in part by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No. 683164). RP would like to thank Terry Lyons for a discussion that led to fundamental ideas for Section 2.2, Corollaries 2.29, 3.14, 3.25 and Section 4.3. The authors would also like to thank the anonymous referees for their detailed comments and suggestions which led to substantial improvements.
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Dedicated to Bernd Sturmfels on occasion of his 60th birthday.
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Bellingeri, C., Friz, P.K., Paycha, S. et al. Smooth Rough Paths, Their Geometry and Algebraic Renormalization. Vietnam J. Math. 50, 719–761 (2022). https://doi.org/10.1007/s10013022005707
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DOI: https://doi.org/10.1007/s10013022005707