Geometric rough paths on infinite dimensional spaces

Similar to ordinary differential equations, rough paths and rough differential equations can be formulated in a Banach space setting. For $\alpha\in (1/3,1/2)$, we give criteria for when we can approximate Banach space-valued weakly geometric $\alpha$-rough paths by signatures of curves of bounded variation, given some tuning of the H\"older parameter. We show that these criteria are satisfied for weakly geometric rough paths on Hilbert spaces. As an application, we obtain Wong-Zakai type result for function space valued martingales using the notion of (unbounded) rough drivers.


Introduction
The theory of rough paths was invented by T. Lyons in his seminal article [Lyo98] and provides a fresh look at integration and differential equations driven by rough signals. A rough path consists of a Hölder continuous path in a vector space together with higher level information satisfying certain algebraic and analytical properties. The algebraic identities in turn allow one to conveniently formulate a rough path as a path in nilpotent groups of truncated tensor series, cf. [FV06] for a detailed account. Similar to the well-known theory of ordinary differential equations, it makes sense to formulate rough paths and rough differential equations with values in a Banach space, [LCL07]. It is expected that the general theory carries over to this infinite-dimensional setting, yet a number of results which are elementary cornerstones of rough path theory are still unknown in the Banach setting.
In [BR19] the authors introduce the notion of a rough driver, which are vector fields with an irregular time-dependence. Rough drivers provides a somewhat generalized description of necessary conditions for the well-posedness of a rough differential equation and the authors use this for the construction of flows generated by these equations. The push-forward of the flow, at least formally, satisfies a (rough) partial differential equation, and this equation is studied rigorously in [BG17] where the authors introduce the notion of unbounded rough drives. This theory was further developed in [DGHT19,HH18,HN20] in the linear setting (although [DGHT19] also tackles the kinetic formulation of conservation laws) as well as nonlinear perturbations in [CHLN20,HLN19b,HLN19a,Hoc18,HNS20]. Still, the unbounded rough drivers studied in these papers assume a factorization of time and space in the sense that the vector fields lies in the algebraic tensor of the time and space dependence.
Our main motivation for this paper is the observation in [CN19] that rough drivers can be understood as rough paths taking values in the space of sufficiently smooth functions, see Section 3.2. Moreover, in [CN19] the authors needed unbounded rough drivers for which the factorization of time and space was not valid, and in particular approximating the unbounded rough driver by smooth drivers. In finite dimensions, sufficient conditions that guarantee the existence of smooth approximations can be easily checked and is the so-called weakly geometric rough paths. In [CN19] and ad-hoc method was introduced to tackle the lack of a similar result in infinite dimensions. For other papers dealing with infinite-dimensional rough paths, let us also mention [Der10,Bai14,CDLL16].
In the present paper we address the characterization of weakly geometric rough paths in Banach spaces. Our aims are twofold. Firstly, we describe and develop the infinite-dimensional geometric framework for Banach space-valued rough paths and weakly geometric rough paths. These rough paths take their values in infinitedimensional groups of truncated tensor products. Some care needs to be taken in this setting, as the tensor product of two Banach spaces will depend on choice of of norm on the product. Secondly, we characterize the geometric rough paths that take their values in an Hilbert space and their relationship to weakly geometric rough paths. Our main result is to prove the following well-known relationship for finite dimensional rough paths in an infinite dimensional setting. Recall that for α ∈ (1/3, 1/2), a geometric α-rough path is the is an element of the closure in signatures S 2 (x) st = 1 + x t − x s + t s (x r − x s ) ⊗ dx r of curves x t of bounded variation, while an α-rough path st is called weakly geometric if the symmetric part of x (2) st equals 1 2 x st ⊗ x st ; a property that holds for all geometric rough paths in particular by an integration by parts argument. Our main result is the following.
The structure of the paper is as follows. In Section 2 we review the infinitedimensional framework for rough paths with values in Banach spaces. We continue with a presentation of Banach space-valued α-rough paths for α ∈ ( 1 3 , 1 2 ) in Section 3. This leads to the three prerequisite assumptions in Theorem 3.3 which states when weakly geometric rough paths can be approximated by signatures of bounded variation path after some tuning of the Hölder parameter. In Section 3.2, we apply Theorem 1.1 to prove Wong-Zakai type results for rough flows; a rough generalization of flows of time-dependent vector fields. This yields a concrete application for rough paths on infinite dimensional space.
The remainder of the paper is dedicated to proving Theorem 1.1 by showing that the criteria of Theorem 3.3 are indeed satisfied in the Hilbert space setting. All of these criteria depends on considering Carnot-Carathéodory geometry or sub-Riemannian geometry of our infinite dimensional groups. Section 4 establishes the necessary prerequisite results from finite dimensional Hilbert spaces. We then do the proof of Theorem 1.1 in several steps throughout Section 5, including a result in Theorem 5.3 where we prove that the Carnot-Carathéodory metric on the free step 2 nilpotent group generated by a Hilbert space becomes a geodesic distance when restricted to the subset of finite distance from the identity. We conclude the proof of Theorem 1.1 in Section 5.4.
2. The infinite-dimensional framework for rough paths 2.1. Tensor products of Banach spaces. If E and F are two Banach spaces, we write E ⊗ a F for their algebraic tensor product. We use the convention that E ⊗a0 = R. For any k ≥ 0 we endow the k-fold algebraic tensor product E ⊗ak with a family of norms · k satisfying the following conditions, cf. [BGLY15].
2. For any permutation σ of the integers 1, 2 . . . k and for any x 1 , . . . , x k ∈ E, Inductively, for k, ℓ ∈ N we define the spaces E ⊗k ⊗ E ⊗ℓ as the completion of E ⊗k ⊗ a E ⊗ℓ with respect to the norm · k+ℓ . From the inclusions Example 2.1. The projective tensor product of Banach spaces is the completion of the algebraic tensor product with respect to the projective tensor norm It is well known that the projective tensor norm satisfies properties 1. and 2. since it is a reasonable crossnorm on E ⊗ a F (cf. [Rya02, Section 6]). Similarly, the injective tensor norm, defined by satisfies 1. and 2. Its completion is the injective tensor product [Rya02,Section 3].
If E is a Hilbert space, then we can identify E ⊗ a E with finite rank operators from E to itself. In this case, the projective and injective norm of z : E → E correspond respectively to the trace norm and the operator norm. Moreover, this identification allows one to identify the projective tensor as the space of nuclear operators N (E, E) and the injective tensor product as the space of compact operators K(E, E), see [Rya02, Corollary 4.8 and Corollary 4.13] for details.
2.2. Algebra of truncated tensor series. For N ∈ N 0 ∪ {∞}, we define as the the space of (truncated) formal tensor series of E. Elements in A N will be denoted as sequences (x (k) ) k≤N . A sequence concentrated in the k-th factor E ⊗k is called homogeneous of degree k. The set A N is an algebra with respect to degree wise addition and the multiplication The algebras A N turn out to be Banach algebras for N finite. For N = ∞ they are still continuous inverse algebras (CIAs), i.e. topological algebras such that inversion is continuous and the unit group is an open subset. CIAs and their unit groups can be seen as an infinite-dimensional generalization of matrix algebras and their unit Lie groups. In the case of locally convex spaces more general then Banach spaces (such as A ∞ ), we adopt the notion of Bastiani calculus to define smooth maps. This means that we require the existence and continuity of directional derivatives, see [Glö03,Kel74] for more information. The relevant results on tensor algebras and their unit groups are summarized in the following result.
Lemma 2.2. The algebra A N is a Banach algebra for N < ∞, while A ∞ is a Fréchet algebra. Moreover, A N is a continuous inverse algebra whose group of units A × N is a C 0 -regular infinite-dimensional Lie group for any N ∈ N ∪ {∞}. We recall the notion of regularity of a Lie group G. Let 1 denote the group's identity element and L(G) its Lie algebra. Then G is called C r -regular, r ∈ N 0 ∪ {∞}, if for each C r -curve u : [0, 1] → L(G) the initial value probleṁ has a (necessarily unique) C r+1 -solution Evol(u) := γ : [0, 1] → G and the map evol: C r ([0, 1], L(G)) → G, u → Evol(u)(1) is smooth. A C ∞ -regular Lie group G is called regular (in the sense of Milnor ). Every Banach Lie group is C 0 -regular (cf. [Nee06]). Several important results in infinite-dimensional Lie theory are only available for regular Lie groups, cf. [KM97].
Proof of Lemma 2.2. By construction of the algebra structure we have for elements of degree k and ℓ that By choice of tensor norms in section 2.1, A N is a Banach algebra for N < ∞, so in particular a continuous inverse algebra. Now A ∞ is a Fréchet space with respect to the product topology. The choice of tensor norms shows that multiplication is separately continuous and by [Wae71, VII, Proposition 1] the multiplication is also jointly continuous. Since A ∞ is a countable product of Banach spaces whose multiplication satisfies (2.1), we conclude that A ∞ is a densely graded locally convex algebra in the sense of [BDS16]. Due to [BDS16, Lemma B.8 (b)] A ∞ is a continuous inverse algebra, i.e. inversion is continuous and the unit group A × is an open subset of A ∞ . Following [Glö02,GN12], the unit group A × N is a regular Banach (for N < ∞) or Fréchet Lie group (N = ∞).
Remark 2.3. The unit group A × N of A N is even a real analytic Lie group in the sense that the group operations extend analytically to the complexification.
2.3. Exponential map. Define the canonical projection π N 0 : A N → R = A 0 and the closed ideal I AN := ker π N 0 = 0<k≤N E ⊗k . Related to this ideal, we introduce the following maps.
Lemma 2.4 (Exponential and logarithm). The exponential and logarithm series exp N : yield mutually inverse real analytic isomorphisms.
Proof. We follow [Glö02] and define the spectrum of x ∈ A N as Remark 2.5. Due to [Glö02, Theorem 5.6] the Lie group exponential of A × N is given by the exponential series x ⊗n n! .
2.4. Free nilpotent groups. Using the exponential map, we are ready to define the subgroups of A × N we are interested in. Observe that A N = L(A × N ) is a Lie algebra with respect to the commutator bracket [x, y ] := x ⊗ y − y ⊗ x. We define inductively the space P n a (E) of Lie polynomials over E of degree n ∈ N by P 1 a (E) := E and P n+1 a (E) := P n a (E) + span{[x, y ] | x ∈ P n a (E), y ∈ E} ⊆ A n+1 , P ∞ a (E) := n∈N0 P n P n ∈ E ⊗n is a Lie polynomial . Proposition 2.6. The group G N (E) is a closed submanifold of A N and this structure turns it into a Banach Lie group for N < ∞ and into a Fréchet Lie group for N = ∞. Moreover, G N (E) is a C 0 -regular Lie group and the exponential map exp : Observe that for N < ∞, the group G N (E) is a nilpotent group of step N generated by E.
Proof. The group G N (E) is a closed subgroup of the locally exponential Lie group A × N . Due to Remark 2.5, the Lie group exponential of this group is exp AN . Define Conversely as P N (E) ⊆ I AN and G N (E) ⊆ 1 + I AN , we deduce from Lemma 2.4 that also L (N ) ⊆ P N (E) holds, hence the two sets coincide. It follows that G N (E) is a locally exponential Lie subgroup of A × N by [Nee06, Theorem IV.3.3]. Since P N (E) ⊆ I AN , G N (E) ⊆ 1 + I AN and the exponential exp AN is a diffeomorphism between those sets (Lemma 2.4), the Lie group exponential induces a diffeomorphism between Lie algebra and Lie group as exp = exp AN | The Banach Lie groups G N (E), N < ∞ are C 0 -regular, cf. also Remark 2.7 below, and we see that the canonical projection mappings π M N : Hence we obtain a projective system of Lie groups (G N (E), π N N −1 ) N ∈N whose Lie algebras also form a projective system (P N (E), L(π N N −1 )) N ∈N of Lie algebras. As sets and the limit maps π ∞ N are smooth group homomorphisms. Then we deduce from [Glö15, Lemma 7.6] that G ∞ (E) admits a projective limit chart, hence [Glö15, Proposition 7.14] shows that G ∞ (E) is C 0 -regular.
Remark 2.7. The regularity of the Lie groups G N (E) can be strengthened by weakening the requirements on the curves in the Lie algebra. This results in a notion of L p -regularity [Glö15] for infinite-dimensional Lie groups. One can show that Banach Lie groups such as G N (E) are L 1 -regular. Furthermore, as in the proof of Proposition 2.6, one sees that the limit G ∞ (E) is L 1 -regular. Note that L 1regularity implies all other known types of measurable regularity for Lie groups.
Example 2.8 (Step 2). For the remainder of the paper, we will mostly focus on the special case of N = 2. In this case P 2 (E) is the closure in A 2 of sums of elements

Applications to infinite dimensional rough paths
3.1. Rough paths and geometric rough paths in Banach space. Let us first recall the notion of a Banach-space valued rough path, see e.g. [CDLL16]. The definition of a rough path involves higher level components with values in a completed tensor product.
Definition 3.1. Fix α ∈ ( 1 3 , 1 2 ) and a tensor product completion E ⊗ E by a choice of a tensornorm · ⊗ satisfying the assumptions from Section 2.1. An (E, ⊗)-valued α-rough path consists of a pair (x, x (2) ) x where x is an α-Hölder continuous path and x (2) is "twice Hölder continuous", i.e. (3.1) In addition, we require To be more precise about this distance, we write st in A 2 , the two step-truncated tensor algebra over E. Chen relation (3.2) can then be rewritten as We then define the distance between two α-rough paths ( Rephrasing these properties, we can define and regard t → x t as a α-Hölder continuous path with values in A 2 . The relations (3.2) tells us that If x t is a smooth path in E, then we can lift it to a rough path Definition 3.2 (Weakly geometric and geometric rough paths). We say that αrough path x t is weakly geometric if it takes values in G 2 (E). These can again can be given the structure of a metric space C α wg ([0, T ], E) with the metric d α as in (3.3) and can be identified with C α ([0, T ], G 2 (E)).
The space of geometric rough paths is defined as the closure in the rough path topology of the set canonical lift of smooth paths and is denoted C α g ([0, T ], E). Since (3.4) is stable under limits, we get that the set of geometric rough paths can be regarded as a subspace of C α ([0, T ], G 2 (E)). The reversed question, namely if any x ∈ C α ([0, T ], G 2 (E)) can be approximated by a sequence of smooth paths is answered positively modulo some tuning of the Hölder parameter α given the following conditions.
We recall the definition of the Carnot-Caratheodory metric, which we will often abbreviate as the CC-metric. We define this metric ρ on G 2 (E) by ρ(y, z) = ρ(1, y −1 · z) and and C([0, T ], M cc ) for the space of continuous curves in M cc with respect to ρ.
(I) For some C > 0 and any Then for any In particular, we have the inclusions is called geodesic if any pair of points can be connected by a geodesic.
If E is finite dimension, the assumptions (I), (II) and (III) hold as ρ and d are then equivalent and we have access to the Hopf-Rinow theorem, see e.g. [FV06]. If E is a general Hilbert space, the Hopf-Rinow theorem is no longer available [Eke78]. We will also show that the metrics ρ and d will not be equivalent in the infinite dimensional case, yet assumptions (I), (II) and (III) will be satisfied, giving us the result in Theorem 1.1. We will prove this statement in Section 5, finishing the proof in Section 5.4. ρ(x t , x n t ) → 0, for n → ∞, and we have the uniform bound sup n d(1, x n st ) ≤ C|t − s| α . From (I), we conclude that x n converges to x in C([0, T ], G (2) (E)). To show the stronger convergence in C β ([0, T ], G 2 (E)) we perform a classical interpolation argument. Since d is left invariant we see that so that there exists a sequence of real numbers ε n → 0 with ) with respect to d β , and x n,m is a sequence of truncated signatures of bounded variation curves converging to x m , then x m,m converge to x. This completes the proof.
3.2. Wong-Zakai for stochastic flows. As an application of Theorem 3.3 and Therorem 1.1 we prove a Wong-Zakai type result for martingales with values in a Banach space of sufficiently smooth functions, as systematically explored in [Kun97].
in the x-variable for some p to be determined later, and let (ω t ) t∈[0,T ] be a K-dimensional Brownian motion on some filtered probability space (Ω, F , P). The study of the Stratonovich equation (for notational convenience we write ω 0 t = t) is by now classical. The book [Kun97] stresses the importance of considering the which allows for a one-to-one characterization of stochastic flows (see [Kun97] for precise statement and result). Equation (3.5) is then understood as dy t = m •dt (y t ).
Consider now the tensor product on C p b (R d , R d ), Checking the symmetry condition then boils down to checking (3.4) for this tensor product. We have, for µ, ν ∈ {1, . . . , d} by the well-known integration by parts formula for the Stratonovich integral. We note that the particular decomposition of (3.6) and (3.7) in terms of the vector fields f and ω are not important for this property; only the choice of Stratonovich integration in the definition of m (2) plays a role.
The thread of [Kun97] was picked up in the rough path setting in [BR19] where the authors introduce so-called "rough drivers", which are vector field analogues of rough paths. It was noted in [CN19] that these vector fields can be canonically defined from infinite-dimensional, i.e. C p b (R d , R d ), valued rough paths. In fact, the set of C p -vector fields Moreover, define by linearity on the algebraic tensor and denote by ∇ ⊗ 2 the extension to C p b (R d ×R d , R d×d ). Moreover, for a matrix a we let a∇ 2 := µ,ν a µ,ν ∂ ∂ξ µ ∂ ∂ξ ν . Then, given a rough path st (ξ, ξ)∇ 2 , then X := (X, X) is a weakly geometric rough driver in the sense of [BR19]. Concretely, we will assume p ≥ 3 to be an integer for simplicity. This could in principle be relaxed at the expense of introducing vector fields which are Hölder continuous in space, but we stick to the simpler case which is also in line with the regularity assumptions in [FH14].
In [BR19] the authors prove Wong-Zakai approximations of dy t = m •dt (y t ) by using linear interpolation of the Banach-space martingale m, showing that the corresponding iterated integral converges to m (2) in the appropriate sense and using continuity of the Itô-Lyons map, see [BR19] for details. The proposition below is proved in a similar way, except the martingale structure is replaced by Theorem 1.1 and the continuity of the mapping x → X. Notice that we use the Sobolev embedding to put ourselves in a Hilbert-space setting.
for k > d 2 + p + 1 for some p ≥ 3 and suppose y solves dy t = X dt (y t ) where X t = (X t , X t ) is the rough driver built from x. Then there exists a sequence of functions x n : [0, T ] × R d → R d of bounded variation of t such that the solution y n oḟ y n t = x n t (y n t ) converges to y in C β ([0, T ], C(R d , R d )) for any β ∈ ( 1 3 , α). Proof. Since x is weakly geometric we have (3.10) x (2),µ,ν st so it is a weakly geometric rough driver in the sense [BR19]. From Theorem 1.1 we get can approximate the infinite dimensional rough path (x, x (2) ) by a sequence of smooth paths. The result now follows from [BR19, Theorem 2.6] since the embedding Applications to unbounded rough drivers. We briefly mention, at a formal level, how infinite dimensional rough path can be used in the study of rough path partial differential equations. To avoid technicalities we refrain from introducing the full and rather large machinery needed for stating precise results.
Formally, the Lagrangian dynamics dy t = X dt (y t ) has a corresponding Eulerian dynamics described by the push-forward u t = (y t ) * φ, i.e.
The notion unbounded rough drivers was introduced in [BG17] to give rigorous meaning to (3.11). In [HN20] the notion of geometric differential rough drivers was used to characterize a relaxed sufficient condition for the so-called renormalizability of unbounded rough drivers. Still, the examples where one could verify the condition of a geometric differential rough driver was restricted to the setting when X belongs to the algebraic tensor of time and space, viz X t (x) = K k=0 f k (x)ω k t . It was noted in [CN19] that also unbounded rough drivers can be thought of as infinite dimensional rough paths, see [CN19, Section 5] for details. The present paper thus yields approximation results for rough path partial differential equations for more general unbounded rough drivers using [HN20, Theorem 2.1] by simply checking the corresponding symmetry condition (3.10).

Finite dimensional Carnot-Carathéodory geometry
4.1. Free nilpotent groups of step 2 in finite dimensions. Let E be a finite dimensional inner product space and use the notation X * = X, · for any X ∈ E. In the notation of Section 2.4, define a Lie algebra g(E) = P 2 (E). By Example 2.8, we can identify g(E) with E ⊕ ∧ 2 E equipped with a Lie bracket structure We identify ∧ 2 E with the space of skew-symmetric endomorphisms so(E) by writing Consider the corresponding simply connected Lie group G 2 (E). For the rest of this section, we will use the fact that exp : g(E) → G 2 (E) is a diffeomorphism to identify these as spaces. Using group exponential coordinates G 2 (E) is then the space E ⊕ so(E) with multiplication (4.3) (x + x (2) ) · (y + y (2) ) = x + y + x (2) + y (2) + 1 2 x ∧ y, x, y ∈ E, x (2) , y (2) ∈ so(E). With this identification the identity is 0 and inverses are given by (x + x (2) ) −1 = −x − x (2) . Recall the identity in (??) for relating the presentation of G 2 (E) as a subset of A 2 and its representation in exponential coordinates. An absolutely continuous curve Γ(t) in G 2 (E) with an L 1 -derivative is called horizontal if for almost every t, In other words, if we write Γ(t) = γ(t) + γ (2) (t) with γ(t) ∈ E and γ (2) (t) ∈ ∧ 2 E, then for some L 1 -function u(t) ∈ E, we havė Since E is a generating subspace of g(E), it follows from the Chow-Rashevskiï Theorem [Cho39,Ras38] that any pair of points in G 2 (E) can be connected by a horizontal curve. For any pair of points in x, y ∈ G 2 (E), define the Carnot-Carathéodory metric (CC-metric) by Note that if Γ(t) is horizontal, then so is x · Γ(t). It follows that the distance ρ is left invariant. Example 4.1 (Heisenberg group). When E is two-dimensional, the group G 2 (E) is known as the Heisenberg group. For any choice of orthogonal frame X, Y , define Z = 1 2 (X−iY ). This means that we can represent any element y = aX+bY +cX∧Y as We will use a similar notation in the rest of the paper.

4.2.
Dimension-free inequality. We want to generalize the inequality (4.5) to free nilpotent groups of step 2 of arbitrary dimensions. The inequality can be concluded from formulas of the CC-distance to the vertical space in [RS17, Appendix A], but we include some more details here for the sake of completion and for applications to infinite dimensional vector spaces in Section 5.1. Consider the case of a Hilbert space E of arbitrary finite dimension n ≥ 2. We want to introduce a class of norms and quasi-norms on so(E). Any element X ∈ so(E) will have non-zero eigenvalues {±iσ 1 , . . . , ±iσ k } for some k ≥ 0. We order them in such a way that These are also the singular values of X as |X| = √ −X 2 has exactly these non-zero eigenvalues, with each σ j appearing twice. Define a sequence σ(X) = (σ j ) ∞ j=1 of non-negative numbers such that σ j = 0 for j > k. For 0 < p ≤ ∞, we define For p ≥ 1, these are norms called the Schatten p-norms [MV97,16]. We will also introduce the following map It is simple to see that · cc is not a norm when dim E > 2. However, we will show that it is a quasi-norm. Recall that a quasi-norm is a map satisfying the norm axioms except the triangle inequality which is assumed in the form x + y ≤ K( x + y ) for some K ≥ 1, [DF93,Section I.9]. From the definition of · cc , we note that (4.6) 1 2 X Sch 1 ≤ X cc ≤ 1 4 X Sch 1/2 .
The latter follows from the fact that for any k > 0, We then have the following result.
Theorem 4.2. Let E be an arbitrary finite dimensional Hilbert space. If ρ is the Carnot-Carathéodory distance on G 2 (E), then Proof. The minimal geodesic from 0 to x ∈ E is just a straight line in E, and hence x E = ρ(0, x).
We will use the geodesic equations in (4.4). Consider a general solution Γ(t) = γ(t) + γ (2) (t) on G 2 (E) with Γ(0) = 0 and Γ(1) = x (2) . Consider arbitrary initial values Λ = 0 and u 0 = 0 for the geodesic equation as in (4.4). Choose an orthonormal basis X 1 , . . . , X k , Y 1 , . . . , Y k , T 1 , . . . , T n−k such that we can write Introduce again complex notation Z j = 1 2 (X j − iY j ) and write We will then have We make the following simplifications. If w j = 0, then the value of λ j has no effect on u(t). We may hence set it to zero and reduce the value of k. Without any loss of generality, we can hence assume that every w j is non-zero. Next, if we have λ j = λ l for some 1 ≤ j, l ≤ k then e iλj t w j Z j + e iλ k t w k Z k = e iλj t (w j Z j + w k Z k ) =: e iλj t w jl 2 (X jl − iY jl ) for some orthonormal pair of vectors X jl , Y jl . Hence we again obtain the same u(t) if we replace λ j X j ∧Y j +λ l X l ∧Y l with λ j X jl ∧Y jl . By repeating such replacements, we may assume that all values of λ 1 , . . . , λ k are different.
From the condition γ(1) = 0, it follows that c 1 , . . . , c n−k all vanish for every 1 ≤ j ≤ n − k. Furthermore, since we assume that w j = 0, it follows that λ j = 2πn j for some positive integers n j . Computing x (2) and using that the integers n 1 , . . . , n k are all different, we obtain It follows that the endpoint x (2) has 2k non-zero eigenvalues {±iσ 1 , . . . , ±iσ k } with In other words, any local length minimizer Γ(t) from 0 to the point x (2) has length In order to obtain the minimal value, we use n j = l if σ j is the l-th largest eigenvalue. The result follows.
Using the identity (4.7) we also obtain the following result.

Geometric rough paths on Hilbert spaces
5.1. Free nilpotent groups on step 2 from Hilbert spaces. Let E be a real Hilbert space, not necessarily finite dimensional. We choose and fix a tensor norm · ⊗ on the algebraic tensor product E ⊗ a E which is assumed to satisfy properties 1. and 2. from Section 2.1. Moreover, we assume that · ⊗ lies (pointwise) between the injective and projective tensor norms (cf. e.g. [Rya02]). As mentioned in Example 2.1, we can identity E ⊗ a E with finite rank operators, and we can consider E ⊗ E as the closure of finite rank operators with respect to · ⊗ .
In describing P 2 (E) = E ⊕ 2 E, through (4.2) we identify the algebraic wedge product 2 a E with the space of all finite rank skew-symmetric operators denoted by so a (E). We identify P 2 (E) with E ⊕ so ⊗ (E) where so ⊗ (E) are the skew-symmetric operators on E that are in the closure of so a (E) with respect to · ⊗ and with brackets as in (4.1). If we give g(E) a norm then it has the structure of a Banach Lie algebra. For any compact skew-symmetric map X : E → E, define a sequence σ(X) = (σ j ) ∞ j=1 such that |X| = √ −X 2 has eigenvalues in non-increasing order σ 1 = σ 1 ≥ σ 2 = σ 2 ≥ · · · . For p ∈ (0, ∞], let so p (E) denote the space of compact skewsymmetric operators X with finite Schatten p-norm X Sch p = 2 1/p σ(X) ℓ p . As p = ∞ and p = 1 correspond to respectively the injective norm and the projective norm, we have Introduce the space so cc (E) as the subspace of so ∞ (E) of elements X such that X cc := ∞ j=1 jσ j is finite. Since all compact operators are limits of finite rank operators ([MV97, Corollary 16.4]), all the previously mentioned inequalities from Section 4.2 still hold. In particular, · cc is a quasi-norm and we have inclusions The group G 2 (E) corresponding to g(E) can be considered in exponential coordinates as the set g(E) with group operation as in (4.3). We define the distance d on G 2 (E) by d(x, y) = x −1 y g(E) . Let t → u(t) be any function in L 1 ([0, 1], E), and let Γ u be the solution of Recall that 0 is the identity, since we are using exponential coordinates. This curve always exists from the L 1 -regularity property of the Banach Lie group G 2 (E) (see [Glö15] and also Remark 2.7). For any x, y ∈ G 2 (E), we define ρ(x, y) ∈ [0, ∞] by

Properties of projections.
Let F be a closed subspace of E. We write Pr F : E → F for the corresponding orthonormal projection. We then write the map pr F : G 2 (E) → G 2 (F ) for the corresponding map We then emphasize the following properties.
Lemma 5.1. (a) pr F is a group homomorphism from G 2 (E) and G 2 (F ).
(b) Let ρ F denote the Carnot-Carathéodory distance defined on G 2 (F ). For any x, y ∈ G 2 (E), we have In particular, if there is a geodesic from x to y in F with respect to ρ F , then this is also the geodesic in G 2 (E) with respect ρ.
Proof. (a) follows from the definition of the definition of the group operation. Using (a), we only need to prove that ρ(0, pr F x) ≤ ρ(0, x) to prove (b) . We observe that if Γ(t) is a horizontal curve from 0 to x, then pr F Γ(t) is a horizontal curve of less or equal length with endpoint pr F x.
Lemma 5.2. If · ⊗ = · p is the Schatten p-norm, the following properties hold.
(a) For any closed subspace F of E and x, y ∈ G 2 (E), we have , there is a sequence of finite dimensional subspaces F 1 ⊆ F 2 ⊆ · · · such that x ∈ F n for any n, lim x (2) y E = σ j+1 (x (2) ).
5.3. Geodesic completeness. One of the main steps in completing Theorem 1.1 will be to establish that ρ makes a subset into a geodesic space.
We will do the proof of this theorem in two parts. In the first part, we will show that G 2 cc (E) is indeed exactly the set with finite ρ-distance and that the inequality (5.2) holds. In the second part, we show that it is a geodesic space.
Proof of Theorem 5.3, Part I. We will begin by introducing the following notation, which we will use in both parts of the proof. Recall the definition of pr F : G 2 (E) → G 2 (F ) ⊆ G 2 (E) for some closed subspace F from Section 5.2. Write pr F,⊥ = pr F ⊥ and write a projection operator We have already shown the result for finite dimensional spaces, so we assume that E is infinite dimensional.
Step 1: The CC-distance is finite on algebraic elements. Let G a (E) = exp(E ⊕ so a (E) and consider an arbitrary element x = x + x (2) ∈ G 2 a (E) with x (2) = n j=1 σ j X j ∧ Y j being the singular value decomposition as in (5.1). Define the finite dimensional subspace F = span{x, X 1 , Y 1 , . . . , X n , Y n }. We then observe that since x ∈ G 2 (F ), ρ(0, x) < ∞ and there is a minimizing geodesic from 0 to x. Also, any element in G 2 a (E) satisfies the inequality (5.2).
Step 2: Vertical elements. Consider an element x = x (2) ∈ so cc (E) with σ(x (2) ) = (σ j ). Let x (2) = ∞ j=1 σ j X j ∧ Y j be the singular value decomposition and define Z j = 1 2 (X j − Y j ). Consider the curve We see that u(t) E = u L 1 = 2 π x (2) cc . Furthermore, if F n is the span of X 1 , Y 1 , . . . , X n , Y n , then by the proof of Theorem 4.2, it follows that pr Fn Γ u is a minimizing geodesic from 0 to x n := n j=1 σ j X j ∧ Y j . Since pr Fn u converges to u in L 1 ([0, 1], E) and x n converges to x in the norm · g(E) , it follows that Γ u is a minimizing geodesic from 0 to x, and in particular, Step 3: The CC-distance is exactly finite on G 2 cc (E). For any element x = x+x (2) ∈ G 2 cc (E), we can construct a horizontal curve Γ from 0 to x by a concatenation of the straight line from 0 to x with a minimizing geodesic from 0 to x (2) left translated by x. The result is that Conversely if x ∈ G 2 (E) and |||x||| = ∞, then using (5.2) and any sequence x n in G 2 a (E) converging to x in · g(E) , we see that ρ(0, x) = ∞. G 2 cc (E) is complete with the distance ρ as it is complete with respect to ||| · ||| by definition.
In order for us to complete Part II of the proof of Theorem 5.3 and show that (G cc (E), ρ) is a geodesic space, we will need the following lemma.
(a) Define F ∞ = span{x, X 1 , Y 1 , X 2 , Y 2 , . . . }. From the definition of K(x) it follows that pr F∞ y = 0 for any y ∈ K(x). Considering the limit of pr n , we also have that so K(x) is bounded in both G 2 cc (E) and G 2 (E). Recall (e.g. from [Eng77, Theorem 4.3.29]) that for a complete metric space, a set is relatively compact if and only if it is totally bounded, i.e. for every ε > 0 there is a finite set of balls of radius ε > 0 covering the set. Let B(z, r) be the ball of radius r centered at z ∈ g(E) with respect to the · g(E) -norm. We observe that for any y ∈ K(x), we have pr n y g(E) ≤ ρ(pr n y) ≤ √ 2ρ(x), pr n,⊥ y g(E) ≤ ρ(pr n,⊥ y) ≤ 2ρ(x)ρ(pr n,⊥ x), pr n,∧ y g(E) = 1 2 pr n,∧ y 3 ⊗ ≤ 4 2ρ(x) 3 ρ(pr n,⊥ x).
Hence the geodesic satisfies the pointwise bounds from the definition of K(x) and the result follows.
Proof of Theorem 5.3, Part II. We are now ready to complete the proof.
Step 5: Every point has a midpoint. Let x = x+x (2) = x+ ∞ j=1 σ j X j ∧Y j ∈ G 2 cc (E) be arbitrary and define F n as in (5.3). If we write x n = pr Fn x, then by the definition in (5.4), we have that K(x n ) ⊆ K(x). Since x n ∈ G 2 a (E), there exists a length minimizing geodesic Γ n from 0 to x n , which we know is in K(x) by Lemma 5.4. Let s n denote the midpoint of each geodesic Γ n . This satisfies ρ(0, s n ) = ρ(x n , s n ) = 1 2 ρ(0, x n ) ≤ 1 2 ρ(0, x) := r.
By the definition of x n , we have s n ∈B 0 ∩B m for any n ≥ m with δ m → 0. Since every s m is contained in K(x), by compactness, there is a subsequence s n k converging to a point s in G 2 (E). This element hence has to be contained in B 0 ∩B m for any m ≥ 1 by Lemma 5.5. It follows that ρ(0, s) = ρ(x, s) = 1 2 ρ(0, x), i.e., s is a midpoint of x. Since (G 2 cc (E), ρ) is a complete length space, it follows from left-invariance of the metric together with [BBI01, Theorem 2.4.16] that existence of such midpoint for any element is equivalent to the space being a geodesic space. This completes the proof. 5.4. Proof of Theorem 1.1. We now come to the proof of our main result. Namely, if E is a Hilbert space and we define α-weak geometric rough path relative to the tensor norm · Sch p , 1 ≤ p ≤ ∞ on the tensor product, then for β ∈ (1/3, α) Recall the results of Lemma 5.2. Let x = x + x (2) ∈ C α ([0, T ], G 2 (E)) be an arbitrary weakly geometric α-rough path. For any fixed t, define a sequence of increasing finite subspaces {F t,n } ∞ n=1 , such that x t ∈ F t,n d(x t , pr Ft,n x t ) = d(0, pr F ⊥ t,n x (2) ) ≤ 1 n .
Define x Π,n t = pr FΠ,n x t . Since ρ and d are equivalent on the finite dimensional F Π,n , it follows that x Π,n t is a continuous function in G 2 cc (E) with respect to ρ.
Since x t is uniformly continuous, we can for each r > 0 find a number o r such that o r = Osc(x t ; r) = sup with o r approaching 0 as r → 0. We now see that for every t ∈ [t i , t i+1 ], n Defining x n t = x n,Πn t where Π n is a partition with |Π n | = 1 n , we have that d(x n t , x t ) converges uniformly to 0. Using now the α-Hölder property of x n and x and an interpolation argument as in the proof of Theorem 3.3, we obtain that d β (x, x n ) → 0 for any β ∈ ( 1 3 , α). This completes the proof. Remark 5.6 (Other cross-norms). As one can see from the proof in Section 5.4, what is needed for our result is the properties of Lemma 5.2 and Theorem 5.3. Hence, for any norm on the tensor product which satisfy these two results, the result in Theorem 1.1 holds.

5.5.
Generalizing the result to Banach spaces. One of the central tools in our proof for geometric rough paths when E is a Hilbert space, is that we can use orthogonal projections Pr F : E → F , which all shorten lengths and hence have norm 1. Such contractive projections are in general rare in Banach spaces as we have the following characterisation from [Ran01, Theorem 3.1].
Theorem 5.7. For a Banach space E with dim E ≥ 3, the following statements are equivalent: (i) E is isometrically isomorphic to a Hilbert space, (ii) every 2-dimensional subspace of E is the range of a projection of norm 1, (iii) every subspace of E is the range of a projection of norm 1.