Rough Paths Infdim

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Introduction Contents SPACES 26,5C7 01,6L0 60L50. 60L20, 60H10, 53C17, 22E65, 1 not-Carathéodory geometry. udmartingales applica- lued an As . tn.For tting. rtraare criteria e rsaetogether space or ie yrough by riven -valued rough lproperties. al il [ ticle α ∈ g path ugh Lyo98 akai 20 21 21 16 15 15 15 13 11 11 10 1 9 6 6 5 5 4 3 3 ] GEOMETRIC ROUGH PATHS ON INFINITE DIMENSIONAL SPACES 2 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 (al- though [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 infinite- dimensional 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 ∈ 2 t in signatures S (x)st =1+ xt xs + s (xr xs) dxr of curves xt of bounded − − (2)⊗ variation, while an α-rough path xst =1+R xst + xst is called weakly geometric if (2) 1 the symmetric part of xst equals 2 xst xst; a property that holds for all geometric rough paths in particular by an integration⊗ by parts argument. Our main result is the following.

C α C α Theorem 1.1. For α (1/3, 1/2), let g ([0,T ], E) and wg([0,T ], E) denote respectively geometric rough∈ paths and weakly geometric rough paths in a Hilbert space E, deﬁned on the interval [0,T ] and relative to the Schatten p-norm, 1 p on E E. Then for any β (1/3, α), we have inclusions ≤ ≤ ∞ ⊗ ∈

C α([0,T ], E) C α ([0,T ], E) C β([0,T ], E). g ⊂ wg ⊂ g GEOMETRIC ROUGH PATHS ON INFINITE DIMENSIONAL SPACES 3

We emphasize that this result includes the Hilbert-Schmidt norm, projective tensor norm and injective tensor norm as respectively p equal to 2, 1 and . The structure of the paper is as follows. In Section 2 we review the∞ infinite- dimensional framework for rough paths with values in Banach spaces. We continue 1 1 with a presentation of Banach space-valued α-rough paths for α ( 3 , 2 ) in Sec- tion 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 F for their algebraic tensor product. We use the convention that ⊗a E⊗a0 = R. For any k 0 we endow the k-fold algebraic tensor product E⊗ak with a family of norms ≥satisfying the following conditions, cf. [BGLY15]. k·kk 1. For every a E⊗ak,b E⊗aℓ, we have ∈ ∈ a b a b . k ⊗ kk+ℓ ≤k kk ·k kℓ 2. For any permutation σ of the integers 1, 2 ...k and for any x ,...,x E, 1 k ∈ x x x = x x . k 1 ⊗ 2 ⊗···⊗ kkk k σ(1) ⊗···⊗ σ(k)kk Inductively, for k,ℓ N we define the spaces E⊗k E⊗ℓ as the completion of E⊗k E⊗ℓ with respect∈ to the norm . From the⊗ inclusions ⊗a k·kk+ℓ E⊗a(k+ℓ) E⊗k E⊗ℓ E⊗(k+ℓ) ⊆ ⊗a ⊆ it follows that E⊗k E⊗ℓ = E⊗(k+ℓ) as Banach spaces. ⊗ ∼ 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 z := inf n x y : z = n x y . k kπ { i=1k ikEk ikF i=1 i ⊗ i} It is well known that the projectiveP tensor norm satisfiesP 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 ⊗ z = sup n ϕ(x )ψ(y ) : ϕ E∗, ψ F ∗, ϕ = ψ =1,z = n x y . k kǫ {| i=1 i i | ∈ ∈ k k k k i=1 i ⊗ i} satisfies 1. andP 2. Its completion is the injective tensor product [Rya02P, 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 GEOMETRIC ROUGH PATHS ON INFINITE DIMENSIONAL SPACES 4 identification allows one to identify the projective tensor as the space of nuclear operators (E, E) and the injective tensor product as the space of compact operators (E, E),N see [Rya02, Corollary 4.8 and Corollary 4.13] for details. K 2.2. Algebra of truncated tensor series. For N N , we define ∈ 0 ∪ {∞} N := E⊗k AN kY=0 as the the space of (truncated) formal tensor series of E. Elements in N will be (k) A ⊗k denoted as sequences (x )k≤N . A sequence concentrated in the k-th factor E is called homogeneous of degree k. The set N is an algebra with respect to degree wise addition and the multiplication A

(k) (k) (n) (m) (x )k≤N (y )k≤N := x y . · ⊗ ! n+Xm=k k≤N The algebras N turn out to be Banach algebras for N finite. For N = they are still continuousA 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 ∞), we adopt the notion of Bastiani calculus to define smooth maps. This meansA 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 is a Banach algebra for N < , while is a AN ∞ A∞ Fréchet algebra. Moreover, N is a continuous inverse algebra whose group of units × is a C0-regular infinite-dimensionalA Lie group for any N N . AN ∈ ∪ {∞} We recall the notion of regularity of a Lie group G. Let 1 denote the group’s r identity element and L(G) its Lie algebra. Then G is called C -regular, r N0 , if for each Cr-curve u: [0, 1] L(G) the initial value problem ∈ ∪ {∞} → γ˙ (t)= γ(t) u(t) γ(0) = 1 · has a (necessarily unique) Cr+1-solution Evol(u) := γ : [0, 1] G and the map → evol: Cr([0, 1], L(G)) G, u Evol(u)(1) → 7→ is smooth. A C∞-regular Lie group G is called regular (in the sense of Milnor). Every Banach Lie group is C0-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 (2.1) E⊗k E⊗ℓ E⊗(k+ℓ). · ⊆ By choice of tensor norms in section 2.1, is a Banach algebra for N < , so in AN ∞ particular a continuous inverse algebra. Now ∞ is a Fréchet space with respect to the product topology. The choice of tensorA norms shows that multiplication is separately continuous and by [Wae71, VII, Proposition 1] the multiplication is also jointly continuous. Since ∞ is a countable product of Banach spaces whose multiplication satisfies (2.1), weA conclude that is a densely graded locally convex A∞ algebra in the sense of [BDS16]. Due to [BDS16, Lemma B.8 (b)] ∞ is a continuous inverse algebra, i.e. inversion is continuous and the unit group ×Ais an open subset × A of ∞. Following [Glö02, GN12], the unit group N is a regular Banach (for N

× Remark 2.3. The unit group N of N is even a real analytic Lie group in the sense that the group operationsA extendA analytically to the complexiﬁcation. N R 2.3. Exponential map. Deﬁne the canonical projection π0 : N = 0 and the closed ideal := ker πN = E⊗k. Related to thisA ideal,→ we introduceA IAN 0 0

[Gl¨o02, Lemma 5.2], it suﬃces to prove that AN ( N ){|z|

Observe that for N < , the group GN (E) is a nilpotent group of step N generated by E. ∞

Proof. The group GN (E) is a closed subgroup of the locally exponential Lie group × . AN Due to Remark 2.5, the Lie group exponential of this group is expAN . Define (N) × N L := x A L( ) exp (Rx) G (E) . { ∈ I N ⊆ AN | AN ⊆ } Due to construction of the closed Lie subalgebra N (E), we have N (E) L(N). N P P ⊆ Conversely as N (E) and G (E) 1+ , we deduce from Lemma 2.4 P ⊆ IAN ⊆ IAN that also L(N) N (E) holds, hence the two sets coincide. It follows that GN (E) ⊆P × is a locally exponential Lie subgroup of N by [Nee06, Theorem IV.3.3]. N N A Since (E) AN , G (E) 1+ AN and the exponential expAN is a dif- feomorphismP between⊆ I those sets (Lemma⊆ I2.4), the Lie group exponential induces a GN diffeomorphism between Lie algebra and Lie group as exp = exp (E) due to AN |PN (E) [Nee06, Theorem IV.3.3]. The Banach Lie groups GN (E), N < are C0-regular, cf. also Remark 2.7 below, and we see that the canonical projection∞ mappings πM : GM (E) GN (E), N → N,M N0 , M N are smooth group homomorphisms. Hence we obtain a ∈ ∪ {∞} ≥ N N projective system of Lie groups (G (E), πN−1)N∈N whose Lie algebras also form a N projective system ( (E), L(πN )) N of Lie algebras. As sets P N−1 N∈ ∞(E) = lim N (E), G∞(E) = lim GN (E), P P ∞ ←− ←− and the limit maps πN 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 C0-regular.

Remark 2.7. The regularity of the Lie groups GN (E) can be strengthened by weak- ening the requirements on the curves in the Lie algebra. This results in a notion of Lp-regularity [Glö15] for infinite-dimensional Lie groups. One can show that Banach Lie groups such as GN (E) are L1-regular. Furthermore, as in the proof of Proposition 2.6, one sees that the limit G∞(E) is L1-regular. Note that L1- regularity 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 2 special case of N = 2. In this case (E) is the closure in 2 of sums of elements X, Y Z = Y Z Z Y with X,Y,ZP E and Lie bracketsA ∧ ⊗ − ⊗ ∈ [X + X, Y + Y]= X Y, X,Y E, X, Y 2(E) E⊗2. ∧ ∈ ∈P ∩ 3. 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.

1 1 Definition 3.1. Fix α ( 3 , 2 ) and a tensor product completion E E by a choice of a tensornorm satisfying∈ the assumptions from Section 2.1. An⊗ (E, )-valued k·k⊗ ⊗ α-rough path consists of a pair (x, x(2)) x: [0,T ] E, x(2) : [0,T ]2 E⊗2 = E E → → ⊗ where x is an α-Hölder continuous path and x(2) is “twice Hölder continuous”, i.e. (3.1) x x . t s α, x(2) . t s 2α. k t − sk | − | k st k2 | − | GEOMETRIC ROUGH PATHS ON INFINITE DIMENSIONAL SPACES 7

In addition, we require (3.2) x(2) x(2) x(2) = (x x ) (x x ) st − su − ut u − s ⊗ t − u usually called Chen’s relation. The set of rough paths equipped with the metric induced by (3.1) is denoted C α([0,T ], E).

(2) To be more precise about this distance, we write xst =1+ xt xs + xst in 2, the two step-truncated tensor algebra over E. Chen relation (−3.2) can thenA be rewritten as x = x x . Introduce a metric d on 1+ = x =1+ x + x(2) : st su ut IN { x E, x(2) E E , by ∈ ∈ ⊗ } x = max x , x 1/2 , | | {k k k k⊗ } d(x, y)= x−1 y = (1 + x + x(2))−1 (1 + y + y(2)) . | · | | · | 2 We then deﬁne the distance between two α-rough paths (s,t) xst, yst on [0,T ] as 7→ d(x , y ) (3.3) d (x, y) = sup st st . α t s α 0≤s

1 1 1 Let α ( 3 , 2 ) be given and let β ( 3 , α) be arbitrary. Assume that the following conditions∈ are satisfied. ∈ (I) For some C > 0 and any z G2(E), we have d(1, z) Cρ(1, z). ∈ ≤ (II) The metric space (Mcc,ρ) is a complete, geodesic space. (III) The set Cα([0,T ], G2(E)) C([0,T ],M ), ∩ cc α 2 is dense in C ([0,T ], G (E)) relative to the metric dβ. Then for any x Cα([0,T ], G(2)(E)) there exists a sequence of bounded variation paths xn : [0,T ] ∈ E such that → xn = S2(xn) x in C β([0,T ], E). → In particular, we have the inclusions C α([0,T ], E) Cα([0,T ], G2(E)) C β([0,T ], E). g ⊂ ⊂ g To explain condition (II) in more details, recall that if (M,ρ) is a metric space, then a curve γ : [0,T ] M is said to have constant speed if Length(γ [s,t])= c t s for any 0 s t →T and some c 0. A constant speed curve is| a geodesic| − if| ≤ ≤ ≤ ≥ Length(γ [s,t]) = ρ(γ(s),γ(t)) = t s ρ(γ(0),γ(T )). The metric space (M,ρ) 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.

Proof of Theorem 3.3. We ﬁrst consider the case when x Cα([0,T ], G2(E)) ∈ } ∩ C([0,T ],Mcc). As (Mcc,ρ) is a geodesic space, [FV10, Lemma 5.21] implies that n 2 n there exists a sequence of truncated signatures x = S (x ): [0,T ] Mcc of bounded variation paths xn such that → n sup ρ(xt, xt ) 0, for n , t∈[0,T ] → → ∞ and we have the uniform bound sup d(1, xn ) C t s α. From (I), we conclude n st ≤ | − | that xn converges to x in C([0,T ], G(2)(E)). To show the stronger convergence in Cβ([0,T ], G2(E)) we perform a classical interpolation argument. Since d is left invariant we see that d(xn , x ) d((xn)−1xn, (x )−1xn)+ d((x )−1xn, (x )−1x ) st st ≤ s t s t s t s t n n 2 sup d(xt , xt) 2C sup ρ(xt , xt), ≤ t∈[0,T ] ≤ t∈[0,T ] so that there exists a sequence of real numbers ε 0 with n → d(xn , x ) ε . st st ≤ n n n α From the construction of x we have d(1, xst), d(1, xst) C t s . Using the interpolation min a,b aθb1−θ for every a,b 0 and θ ≤[0, 1]| − we have| { }≤ ≥ ∈ d(xn , x ) ε C t s α εθ C1−θ t s α(1−θ) st st ≤ n ∧ | − | ≤ n | − | and by choosing θ such that α(1 θ)= β we get convergence − d(xn , x ) d (xn, x) = sup st st εθ C1−θ 0, n . β t s β ≤ n → → ∞ s,t∈[0,T ] | − | GEOMETRIC ROUGH PATHS ON INFINITE DIMENSIONAL SPACES 9

α 2 Finally, from the density of C ([0,T ], G (E)) C([0,T ],Mcc) by (III) it follows m α 2 ∩ that if x C ([0,T ], G (E)) C([0,T ],Mcc) is a sequence converging to an ∈ α 2 ∩ n,m arbitrary x C ([0,T ], G (E)) with respect to dβ, and x is a sequence of truncated signatures∈ of bounded variation curves converging to xm, then xm,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 Ba- nach space of sufficiently smooth functions, as systematically explored in [Kun97]. K Rd Rd Let (fk)k=0 be a collection of time-dependent vector fields fk : [0,T ] p Rd Rd × → of class Cb ( , ) in the x-variable for some p to be determined later, and let (ωt)t∈[0,T ] be a K-dimensional Brownian motion on some filtered probability space (Ω, , P). The study of the Stratonovich equation (for notational convenience we F 0 write ωt = t)

K (3.5) dy = f (t,y ) dωk t k t ◦ t Xk=0 is by now classical. The book [Kun97] stresses the importance of considering the p Rd Rd Cb ( , )-valued semi-martingale

K t k (3.6) mt(ξ) := fk(r, ξ)dωr 0 Xk=0 Z 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 dyt = m◦dt(yt). p Rd Rd Consider now the tensor product on Cb ( , ), (f g)(ξ, ζ) := f(ξ)g(ζ)T , ⊗ p Rd Rd ⊗2 p Rd Rd Rd×d which allows us to identify Cb ( , ) with a subspace of Cb ( , ). Let us deﬁne the iterated integral ×

t (3.7) m(2)(ξ, ζ) := (m m ) dm (ξ, ζ) st r − s ⊗◦ r Zs K t r T l k := fl(v, ξ)fk(r, ζ) dωv dωr , s s ◦ k,lX=0 Z Z p Rd Rd Rd×d as a Cb ( , )-valued random field. Checking the symmetry condition then boils down× to checking (3.4) for this tensor product. We have, for µ,ν 1,...,d ∈{ } (2),µ,ν (2),ν,µ mst (ξ, ζ)+ mst (ζ, ξ) t t = (mµ(ξ) mµ(ξ)) dmν (ζ)+ (mν (ζ) mν (ζ)) dmµ(ξ) r − s ◦ r r − s ◦ r Zs Zs (3.8) =(mµ(ξ) mµ(ξ))(mν (ζ) mν (ζ)) t − s t − s 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 GEOMETRIC ROUGH PATHS ON INFINITE DIMENSIONAL SPACES 10

p Rd Rd defined from infinite-dimensional, i.e. Cb ( , ), valued rough paths. In fact, the p p Rd p Rd Rd set of C -vector fields X ( ) is canonically identified with Cb ( , ) via Cp(Rd, Rd) Xp(Rd) b → f f = f µ ∂ . 7→ · ∇ µ ∂ξµ Moreover, define by linearity on the algebraic tensorP p Rd Rd ⊗a2 p Rd Cb ( , ) X ( ) → ν f g (f (g )) = f µ ∂g ∂ ⊗ 7→ · ∇ · ∇ µ,ν ∂ξµ ∂ξν and denote by ⊗ the extension to Cp(Rd Rd, Rd×dP). Moreover, for a matrix a we ∇2 b × let a 2 := aµ,ν ∂ ∂ . Then, given a rough path x C α([0,T ], Cp(Rd, Rd)), ∇ µ,ν ∂ξµ ∂ξν ∈ b if we let P (3.9) X (ξ) := x (ξ) , X (ξ) := ⊗x(2)(ξ, ξ)+ x(2)(ξ, ξ) 2, st st · ∇ st ∇2 st st ∇ then X := (X, X) is a weakly geometric rough driver in the sense of [BR19]. Con- cretely, 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 dyt = m◦dt(yt) 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-space7→ setting. Theorem 3.4. Let x C α ([0,T ],Hk(Rd, Rd)) for k > d + p +1 for some p 3 ∈ wg 2 ≥ and suppose y solves dyt = Xdt(yt) where Xt = (Xt, Xt) is the rough driver built from x. Then there exists a sequence of functions xn : [0,T ] Rd Rd of bounded variation of t such that the solution yn of × → n n n y˙t = xt (yt ) converges to y in Cβ([0,T ], C(Rd, Rd)) for any β ( 1 , α). ∈ 3 Proof. Since x is weakly geometric we have (3.10) x(2),µ,ν (ξ, ζ)+ x(2),ν,µ(ζ, ξ) = (xµ(ξ) xµ(ξ))(xν (ζ) xν (ζ)) st st t − s t − s (2) 2 1 T 2 for all µ,ν 1,...,d which gives xst = 2 (xt xs)(xt xs) . It follows that ∈ { } ∇ − − ∇ 1 1 X (ξ) X (X )(ξ)= ⊗ x(2) x x (ξ, ξ) X(Rd) st − 2 st st ∇2 st − 2 st ⊗ 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 Hk(Rd, Rd) Cp(Rd, Rd) is continuous. ⊂ b 3.3. 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 dyt = Xdt(yt) has a corresponding Eulerian dynamics described by the push-forward ut = (yt)∗φ, i.e. (3.11) du = X u , u = φ. t dt∇ t 0 GEOMETRIC ROUGH PATHS ON INFINITE DIMENSIONAL SPACES 11

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 K k of time and space, viz Xt(x) = k=0 fk(x)ωt . 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. TheP 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).

4. Finite dimensional Carnot-Caratheodory´ 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. h · i ∈ In the notation of Section 2.4, define a Lie algebra g(E)= 2(E). By Example 2.8, we can identify g(E) with E 2E equipped with a Lie bracketP structure ⊕ ∧ (4.1) [X + X, Y + Y]= X Y, X,Y E, X, Y 2E. ∧ ∈ ∈ ∧ We identify 2E with the space of skew-symmetric endomorphisms so(E) by writing ∧ (4.2) X Y = X∗ Y Y ∗ X. ∧ ⊗ − ⊗ Consider the corresponding simply connected Lie group G2(E). For the rest of this section, we will use the fact that exp : g(E) G2(E) is a diffeomorphism to identify these as spaces. Using group exponential→ coordinates G2(E) is then the space E so(E) with multiplication ⊕ 1 (4.3) (x + x(2)) (y + y(2))= x + y + x(2) + y(2) + x y, · 2 ∧ 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 2 − − the presentation of G (E) as a subset of 2 and its representation in exponential coordinates. A An absolutely continuous curve Γ(t) in G2(E) with an L1-derivative is called horizontal if for almost every t, Γ(t)−1 Γ˙ (t) E. · ∈ In other words, if we write Γ(t)= γ(t)+ γ(2)(t) with γ(t) E and γ(2)(t) 2E, then for some L1-function u(t) E, we have ∈ ∈ ∧ ∈ 1 γ˙ (t)= u(t), γ˙ (2)(t)= γ(t) u(t). 2 ∧ Since E is a generating subspace of g(E), it follows from the Chow-Rashevski¨ıThe- orem [Cho39, Ras38] that any pair of points in G2(E) can be connected by a horizontal curve. For any pair of points in x, y G2(E), define the Carnot-Carathéodory metric (CC-metric) by ∈

1 2 −1 Γ : [0, 1] G (E) horizontal, ρ(x, y)= Γ(t) Γ˙ (t) E dt : → . k · k Γ(0) = x, Γ(1) = y Z0 Note that if Γ(t) is horizontal, then so is x Γ(t). It follows that the distance ρ is left invariant. · GEOMETRIC ROUGH PATHS ON INFINITE DIMENSIONAL SPACES 12

From e.g. [ABB20, Section 7.3], length minimizers of ρ are all on the form, t 1 t (4.4) γ(t)= x + esΛu ds, γ(2)(t)= x(2) + γ(s) esΛu ds, 0 0 0 2 ∧ 0 Z0 Z0 for some constant element Λ so(H) and u E. ∈ 0 ∈ Example 4.1 (Heisenberg group). When E is two-dimensional, the group G2(E) is known as the Heisenberg group. For any choice of orthogonal frame X, Y , define 1 Z = 2 (X iY ). This means that we can represent any element y = aX+bY +cX Y as − ∧ y = (a + ib)Z + (a ib)Z¯ + cX Y. − ∧ We will use a similar notation in the rest of the paper. (2) If Λ= λX Y , u0 = u0Z +¯u0Z¯, γ(t)= z(t)Z +¯z(t)Z¯ and γ (t)= σ(t)X Y with z(t),u ∧C and σ(t), λ R, then (4.4) becomes ∧ 0 ∈ ∈ t eiλt 1 2 sin(λt/2) z(t)= z + eiλsu ds = z + − u = z + eiλ/2tu , 0 0 0 iλ 0 0 λ 0 Z0 1 t σ(t)= σ + Im(¯z(s)eiλsu ) ds 0 2 0 Z0 2 sin(λt/2) 1 u 2 sin(λt) = σ + Im eiλ/2tz¯ u | 0| t . 0 λ 0 0 − 2 λ − λ sin(λt) where we interpret λ as t if λ = 0. If the initial point is the identity 0, we have 2 sin(λt/2) u 2 sin(λt) z(t)= eitλ/2u , σ(t)= | 0| t . λ 0 − 2λ − λ If the above geodesic is defined on the intervall [0, 1], then it has length u0 . In particular, we observe the following. | | (a) A minimizing geodesic defined on [0, 1] from 0 to zZ +¯zZ¯ is given by the choice λ = 0. It follows that ρ(0,zZ +¯zZ¯)= z . | | (b) A minimizing geodesic defined on [0, 1] from 0 to σX Y is given by the choice λ = 2π depending on the sign of σ. Hence, we have∧ that ± ρ(0,σX Y )2 σ = ∧ . | | 4π (c) Note that since

1 1 z(1) = z = u(t)dt u(t) dt = u , | | | | ≤ | | | 0| Z0 Z0

we have z ρ(0,zZ +¯zZ¯ + σX Y ). It then also follows that | |≤ ∧ 2√π σ 1/2 = ρ(0,σX Y ) | | ∧ ρ(0, zZ +¯zZ¯)+ ρ( zZ z¯Z,σX¯ Y ) ≤ − − − ∧ = z + ρ(0,zZ +¯zZ¯ + σX Y ) 2ρ(0,zZ +¯zZ¯ + σX Y ). | | ∧ ≤ ∧ Using this fact along with the upper bound from the triangle inequality and left invariance, we have

(4.5) max z , √π σ 1/2 ρ(0,zZ +¯zZ¯ + σX Y ) z +2√π σ 1/2. {| | | | }≤ ∧ ≤ | | | | GEOMETRIC ROUGH PATHS ON INFINITE DIMENSIONAL SPACES 13

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, Appen- dix 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 iσ1,..., iσk for some k 0. We order∈ them in such a way that {± ± } ≥ σ σ > 0. 1 ≥···≥ k These are also the singular values of X as X = √ X2 has exactly these non-zero | | − X ∞ eigenvalues, with each σj appearing twice. Define a sequence σ( ) = (σj )j=1 of non-negative numbers such that σ = 0 for j > k. For 0

2. However, we will show that it is a quasi-norm.k·k 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 k≤ K( x + y ) for some K 1, [DF93, Section I.9]. From the deﬁnition of cc, wek notek thatk k ≥ k·k 1 1 (4.6) X 1 X X 1/2 . 2k kSch ≤k kcc ≤ 4k kSch The latter follows from the fact that for any k > 0, √a + kb √a + √b if b 0 2 ≤ ≥ and a (k−1) b. Hence ≥ 4 σ + + kσ σ + + (k 1)σ + √σ , 1 ··· k ≤ 1 ··· − k−1 k p k(k−p1) since σ1 + + (k 1)σk−1 2 σk. Deﬁne a··· homogeneous− norm≥

x + x(2) = max x , √π x(2) 1/2 . ||| ||| k kE k kcc n o We then have the following result. Theorem 4.2. Let E be an arbitrary ﬁnite dimensional Hilbert space. If ρ is the Carnot-Carath´eodory distance on G2(E), then

x + x(2) ρ(0, x + x(2)) 3 x + x(2) . ||| ||| ≤ ≤ ||| ||| Proof. The minimal geodesic from 0 to x E is just a straight line in E, and hence ∈ x = ρ(0, x). k kE We will show that we also have (4.7) ρ(0, x(2))=2√π x(2) 1/2, x(2) so(E), k kcc ∈ The result then follows from similar steps as in Example 4.1. GEOMETRIC ROUGH PATHS ON INFINITE DIMENSIONAL SPACES 14

We will use the geodesic equations in (4.4). Consider a general solution Γ(t)= γ(t)+ γ(2)(t) on G2(E) with Γ(0) = 0 and Γ(1) = x(2). Consider arbitrary initial values Λ = 0 and u = 0 for the geodesic equation as in (4.4). Choose an 6 0 6 orthonormal basis X1,...,Xk, Y1,...,Yk,T1,...,Tn−k such that we can write k Λ= λ X Y , λ > 0. j j ∧ j j j=1 X Introduce again complex notation Z = 1 (X iY ) and write j 2 j − j k k n−k u = w Z + w¯ Z¯ + c T , w C,c R. 0 j j j j j j j ∈ j ∈ j=1 j=1 j=1 X X X We will then have k k n−k u(t)= eiλj tw Z + e−iλj tw¯ Z¯ + c T , w C,c R. j j j j j j j ∈ j ∈ j=1 j=1 j=1 X X X We make the following simplifications. If wj = 0, then the value of λj 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 wj is non-zero. Next, if we have iλj t iλkt iλj t λj = λl for some 1 j, l k then e wj Zj + e wkZk = e (wj Zj + wkZk) =: iλj t wjl ≤ ≤ e 2 (Xjl iYjl) for some orthonormal pair of vectors Xjl, Yjl. Hence we again obtain the same− u(t) if we replace λ X Y +λ X Y with λ X Y . By repeating j j ∧ j l l∧ l j jl∧ jl such replacements, we may assume that all values of λ1,...,λk are different. If Γ(t)= γ(t)+ γ(2)(t) is the corresponding geodesic, then

k k n−k 2 sin(λj t/2) 2 sin(λj t/2) γ(t)= eiλj t/2w Z + e−iλj t/2w¯ Z¯ + tc T . λ j j λ j j j j j=1 j j=1 j j=1 X X X From the condition γ(1) = 0, it follows that c1,...,cn−k all vanish for every 1 j n k. Furthermore, since we assume that w = 0, it follows that λ = 2πn≤ ≤ − j 6 j j for some positive integers nj. (2) Computing x and using that the integers n1,...,nk are all diﬀerent, we obtain

k 2 1 k 2 1 wj 1 wj x(2) = Im | | (1 e2iπnj t)dt X Y = | | X Y . 4π in − j ∧ j −4π n j ∧ j j=1 j 0 j=1 j X Z X It follows that the endpoint x(2) has 2k non-zero eigenvalues iσ ,..., iσ with {± 1 ± k} 1 2 σj = wj . 4njπ | | In other words, any local length minimizer Γ(t) from 0 to the point x(2) has length

k k Length(Γ)2 = w 2 = 4πn σ . | j | j j j=1 j=1 X X In order to obtain the minimal value, we use nj = l if σj is the l-th largest eigenvalue. The result follows.

Using the identity (4.7) we also obtain the following result.

Corollary 4.3. cc is a quasi-norm on so(E), even a 1/2-norm [DF93, Section I.9], in that it satisﬁesk·k X + Y 1/2 X 1/2 + Y 1/2, X + Y 2( X + Y ). k kcc ≤k kcc k kcc k kcc ≤ k kcc k kcc GEOMETRIC ROUGH PATHS ON INFINITE DIMENSIONAL SPACES 15

5. 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 E which is assumed to satisfy properties k·k⊗ ⊗a 1. and 2. from Section 2.1. Moreover, we assume that ⊗ lies (pointwise) between the injective and projective tensor norms (cf. e.g. [Rya02k·k ]). 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 . ⊗ k·k⊗ In describing 2(E)= E 2 E, through (4.2) we identify the algebraic wedge P ⊕ product 2 E with the space of all finite rank skew-symmetric operators denoted by a V so (E). We identify 2(E) with E so (E) where so (E) are the skew-symmetric a V P ⊕ ⊗ ⊗ operators on E that are in the closure of soa(E) with respect to ⊗ and with brackets as in (4.1). If we give g(E) a norm k·k

x + x(2) = max x , x(2) . k kg(E) k kE k k⊗ n o 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 )∞ such that X = √ X2 has eigenvalues in non-increasing order σ = σ j j=1 | | − 1 1 ≥ σ2 = σ2 . For p (0, ], let sop(E) denote the space of compact skew- ≥ · · · ∈ ∞ 1/p symmetric operators X with finite Schatten p-norm X Schp = 2 σ(X) ℓp . As p = and p = 1 correspond to respectively the injectivek k norm andk the projectk ive norm,∞ we have so (E) so (E) so (E). 1 ⊆ ⊗ ⊆ ∞ Introduce the space socc(E) as the subspace of so∞(E) of elements X such that X := ∞ jσ is finite. Since all compact operators are limits of finite rank k kcc j=1 j operators ([MV97, Corollary 16.4]), all the previously mentioned inequalities from P Section 4.2 still hold. In particular, is a quasi-norm and we have inclusions k·kcc so (E) so (E) so (E). 1/2 ⊆ cc ⊆ 1 The group G2(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 G2(E) by d(x, y) = x−1y . Let t u(t) be any function in L1([0, 1], E), k kg(E) 7→ and let Γu be the solution of Γ (t)−1 Γ˙ (t)= u(t), Γ (0) = 0. u · u u Recall that 0 is the identity, since we are using exponential coordinates. This curve always exists from the L1-regularity property of the Banach Lie group G2(E) (see [Glö15] and also Remark 2.7). For any x, y G2(E), we define ρ(x, y) [0, ] by ∈ ∈ ∞ ρ(x, y)= ρ(0, x−1 y), · 1 ρ(0, x) = inf u 1 : u L ([0, 1], E), Γ (1) = x . k kL ∈ u 5.2. Properties of projections. Let F be a closed subspace of E. We write PrF : E F for the corresponding orthonormal projection. We then write the map pr →: G2(E) G2(F ) for the corresponding map F → pr (x + x(2))=Pr +(Pr x(2) Pr ), x E, x(2) so (E). F F F F ∈ ∈ ⊗ We then emphasize the following properties.

2 2 Lemma 5.1. (a) prF is a group homomorphism from G (E) and G (F ). GEOMETRIC ROUGH PATHS ON INFINITE DIMENSIONAL SPACES 16

2 (b) Let ρF denote the Carnot-Carathéodory distance defined on G (F ). For any x, y G2(E), we have ∈ ρ (pr x, pr y)= ρ(pr x, pr y) ρ(x, y). F F F F F ≤ In particular, if there is a geodesic from x to y in F with respect to ρF , then this is also the geodesic in G2(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 x) ρ(0, x) to prove (b) . We observe that F ≤ if Γ(t) is a horizontal curve from 0 to x, then prF Γ(t) is a horizontal curve of less or equal length with endpoint prF x. Lemma 5.2. If = is the Schatten p-norm, the following properties hold. k·k⊗ k·kp (a) For any closed subspace F of E and x, y G2(E), we have ∈ d(pr x, pr y) d(x, y). F F ≤ (2) (b) For any x = x + x , there is a sequence of finite dimensional subspaces F1 F such that ⊆ 2 ⊆ · · · x Fn for any n, lim d(x, prF x)=0. ∈ n→0 n

Proof. (a) Again it is sufficient to prove that d(0, prF x) d(0, x) for any x = x + (2) 2 ≤ (2) x G (E). We see that PrF x E x E and furthermore, PrF x PrF Schp (2)∈ k k ≤k k k k ≤ x p since k kSch (2) (2) σj+1(Pr x Pr)= max min PrF x PrF y E F F rank(E˜)=2j+1 y∈E˜ k k kykE =1 (2) (2) max min x y E = σj+1(x ). ≤ rank(E˜)=2j+1 y∈E˜ k k kykE =1 (b) Write x = x + x(2) and give the singular value decomposition ∞ (5.1) x(2) = σ X Y , σ σ , j j ∧ j 1 ≥ 2 ≥ · · · j=1 X with X1, Y1,X2, Y2,... all orthogonal unit vector fields. We define F˜ = span X , Y : j =1,...,n , F = span x, F˜ , n { j j } n { n} Then by left invariance 1/p (a) ∞ (2) (2) 1 1/p (2) p x x ⊥ ⊥ d( , prFn )= d(0, prF x ) d(0, prF˜ x )= 2 σ(x ) n ≤ n 2   j=n+1 X which converge to zero by definition.   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. Theorem 5.3. Let E be a Hilbert space and define G2(E) relative to a tensor norm satisfying 1. and 2. from Section 2.1 and bounded from below by the injective k·k⊗ tensor product ∞ . If we define G2(E) := exp(E so (E)), then k·kSch ⊕ cc G2 (E)= x G2(E): ρ(0, x) < , cc { ∈ ∞} 2 Furthermore, the metric space (Gcc(E),ρ) is a complete, geodesic space and if we define x + x(2) = max x , √π x(2) 1/2 , ||| ||| k kE k kcc n o GEOMETRIC ROUGH PATHS ON INFINITE DIMENSIONAL SPACES 17 then (5.2) x ρ(0, x) 3 x . ||| ||| ≤ ≤ ||| ||| We will do the proof of this theorem in two parts. In the first part, we will show 2 that Gcc(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, 2 which we will use in both parts of the proof. Recall the definition of prF : G (E) 2 2 → G (F ) G (E) for some closed subspace F from Section 5.2. Write prF,⊥ = prF ⊥ and write⊆ a projection operator (2) (2) (2) pr ⊥ (x)=Pr x Pr ⊥ +Pr ⊥ x Pr = x pr x pr x, x = x + x . F ∧F F F F F − F − F,⊥ 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 Ga(E) = exp(E (2) 2 (2) ⊕ soa(E) and consider an arbitrary element x = x + x Ga(E) with x = n σ X Y being the singular value decomposition as∈ in (5.1). Define the j=1 j j ∧ j finite dimensional subspace F = span x, X , Y ,...,X , Y . We then observe P 1 1 n n that since x G2(F ), ρ(0, x) < and there{ is a minimizing} geodesic from 0 to x. ∈ 2 ∞ Also, any element in Ga(E) satisfies the inequality (5.2). (2) (2) Step 2: Vertical elements. Consider an element x = x socc(E) with σ(x )= ∞ ∈ (σ ). Let x(2) = σ X Y be the singular value decomposition and define j j=1 j j ∧ j Z = 1 (X Y ). Consider the curve j 2 j − j P ∞ 1/2 −2πjt 2πjt u(t)=2√π (jσj ) (e Zj + e Z¯j ). j=1 X (2) We see that u(t) E = u L1 = 2 π x cc. Furthermore, if Fn is the span of X , Y , ... , Xk , Yk, thenk byk the proofk of Theoremk 4.2, it follows that pr Γ is a 1 1 n n p Fn u minimizing geodesic from 0 to xn := n σ X Y . Since pr u converges to u j=1 j j ∧ j Fn in L1([0, 1], E) and xn converges to x in the norm , it follows that Γ is a P g(E) u minimizing geodesic from 0 to x, and in particular,k·k ρ(0, x) = Length(Γ )=2√π x(2) 1/2. u k kcc

2 (2) Step 3: The CC-distance is exactly finite on Gcc(E). For any element x = x+x 2 ∈ Gcc(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 ρ(0, x) Length(Γ)= x +2√π x(2) 1/2 3 x < . ≤ k k k kcc ≤ ||| ||| ∞ Conversely if x G2(E) and x = , then using (5.2) and any sequence xn in G2(E) converging∈ to x in ||| ||| , we∞ see that ρ(0, x) = . G2 (E) is complete a k·kg(E) ∞ cc 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 (Gcc(E),ρ) is a geodesic space, we will need the following lemma. (2) 2 Lemma 5.4. Let x = x + x Gcc(E) be a fixed arbitrary element with singular ∞∈ value decomposition x(2) = σX Y as in (5.1). Define subspaces F F j=1 j ∧ j 0 ⊆ 1 ⊆ F2 by ⊆ · · · P (5.3) F = span x , F = span(F X , Y ). 0 { } n+1 n ∪{ n+1 n+1} For any n 0, define prn = prF , prn,⊥ = prF ⊥ and prn,∧ = prF ∧F ⊥ ≥ n n n n GEOMETRIC ROUGH PATHS ON INFINITE DIMENSIONAL SPACES 18

(a) The set

For any n 0, ≥    ρ(0, prn y) 2ρ(0, x)ρ(0, prn x)   ≤   2  (5.4) K(x)= y Gcc(E): p  ,  ∈ ρ(0, pr y) 2ρ(0, x)ρ(0, pr x)   n,⊥ ≤ n,⊥  q    pr y 4 2ρ(0, x)3ρ(0, pr x)   n,∧ ⊗ n,⊥   k k ≤   q  is relatively compact in G2(E).  (b) Any minimizing geodesic from 0 to x is contained in K(x).

Proof. To simplify notation in the proof, we write ρ(x) := ρ(0, x). (a) Define F = span x, X , Y ,X , Y ,... . From the definition of K(x) it follows ∞ { 1 1 2 2 } that prF∞ y = 0 for any y K(x). Considering the limit of prn, we also have that ∈ y ρ(y) √2ρ(x) k kg(E) ≤ ≤ 2 2 so K(x) is bounded in both Gcc(E) and G (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 k·k ∈ pr y ρ(pr y) √2ρ(x), k n kg(E) ≤ n ≤ pr y ρ(pr y) 2ρ(x)ρ(pr x), k n,⊥ kg(E) ≤ n,⊥ ≤ n,⊥ 1 q pr y = pr y 4 2ρ(x)3ρ(pr x). k n,∧ kg(E) 2k n,∧ 3k⊗ ≤ n,⊥ q Hence, for a given ε> 0, we can choose n sufficiently large such that ε max 2ρ(x)ρ(pr x), 2 2ρ(x)3ρ(pr x) . n,⊥ n,⊥ ≤ 3 nq q o Choose a finite set of points z ,..., z such that N B(z , ε ) covers the rel- 1 N ∪j=1 j 3 atively compact set Fn B(0,ρ(x)). By our choice of n, we then have that N ∩ j=1B(zj ,ε) covers all of K(x). ∪ 2 (b) We observe first that since prn,⊥ x is in the center of G (E), then by left invariance

ρ(x)= ρ((pr x) (pr x)) ρ(pr x)+ ρ(pr x) n · n,⊥ ≤ n n,⊥ Let Γ = Γ = γ + γ(2) : [0, 1] G2(E) be any minimizing geodesic with left u → logarithmic derivative u and write un = prn u and u⊥,n = prn,⊥ u. Since u is a minimizing geodesic, then by reparametrization, we may assume that

(5.5) u(t) = u (t) 2 + u (t) 2 = ρ(x), and note k kE k n kE k n,⊥ kE q 1 (5.6) ρ(pr x) Length(pr Γ)= u (t) dt ρ(x). n,⊥ ≤ n,⊥ k n,⊥ kE ≤ Z0 GEOMETRIC ROUGH PATHS ON INFINITE DIMENSIONAL SPACES 19

This leads to the following sequence of inequalities ρ(x)ρ(pr x) ρ(x) ρ(x) ρ(pr x) n ≥ − n,⊥ (5.5)+(5.6) 1 2 2 ρ(x) un(t) E + un,⊥(t) E un,⊥(t) E dt ≥ 0 k k k k −k k Z q 1 2 (5.5) u (t) = ρ(x) k n kE dt 2 2 0 un(t) + un,⊥(t) + un,⊥(t) E ! Z k kE k kE k k 2 (5.6) 1 1 p Jensen 1 1 1 u (t) 2 dt u (t) dt = Length(pr Γ)2. ≥ 2 k n kE ≥ 2 k n kE 2 n Z0 Z0 It follows that any point y on the curve Γ will have ρ(pr y) 2ρ(x)ρ(pr x). n ≤ n By a similar calculation, we have that ρ(pr y) 2ρ(x)ρ(pr x). n,⊥ ≤ pn,⊥ (2) We also see that prn,∧ Γ(t)=prn,∧ γ (t) with q 1 t pr γ(2)(t)= ((pr γ(s)) u (s) + (pr γ(s)) u (s))ds. n,∧ 2 n ∧ n,⊥ n,⊥ ∧ n Z0 t We note that 0 un(s) dt Length(prn Γ), while prn γ(t) ρ(prn γ(t)) Length(pr Γ). Sincek wek have≤ similar relation applyingk pr , Wek≤ finally use that≤ n R n pr γ(2)(t) Length(pr Γ(t)) Length(pr Γ(t)) 2ρ(x)3ρ(pr x). k ∧ k⊗ ≤ n n,⊥ ≤ n,⊥ q Hence the geodesic satisfies the pointwise bounds from the definition of K(x) and the result follows.

Lemma 5.5. For any r> 0 and x G2(E), we have that the set 0 ∈ B¯ (x , r)= x : ρ(x , x) r ρ 0 { 0 ≤ } is closed in G2(E).

n n n,(2) Proof. By left invariance, we only consider x0 = 0. Assume that y = y +y is 2 (2) a sequence contained in B¯ρ(0, r) converging in G (E) to some element y = y + y . We then have that y yn 0, y(2) yn,(2) 0. k − kE → k − k⊗ → In particular, we will have y(2) yn,(2) ∞ 0 implying that if σ(y(2)) = (σ ) k − kSch → j and σ(yn,(2)) = (σn), then σn σ . It follows that j j → j ∞ m n n y y E + √π jσj = lim y E + √π lim lim jσj 2r. ||| ||| ≤ k k n→∞ k k m→∞ n→∞  ≤ j=1 j=1 X X   Since y < , we can conclude the following. ||| ||| ∞ ∞ If y(2) = σ X Y is deﬁned with all vectors orthonormal, we deﬁne j=1 j j ∧ j F = span y,X , Y ,...,X , Y . Then m { P 1 1 m m} lim ρ(prF y, y) 3 lim y prF y =0. m→∞ m ≤ m→∞ ||| − m ||| Using that ρ(0, pr y) ρ(0, y) ρ(0, pr y)+ ρ(pr y, y), it follows that Fm ≤ ≤ Fm Fm lim ρ(0, prF y)= ρ(0, y). m→∞ m Furthermore, since n n n pr (y y ) ∞ y y ∞ y y , k Fm − kSch ≤k − kSch ≤k − k⊗ GEOMETRIC ROUGH PATHS ON INFINITE DIMENSIONAL SPACES 20

y yn ∞ we have that limn→∞ prFm ( ) Sch = 0. Since all left invariant homogeneous norms are equivalentk on− a ﬁnitek dimensional space, we have convergence lim ρ(pr yn, pr y) 0 for any ﬁxed m. Finally n→∞ Fm Fm → n ρ(0, y) = lim ρ(0, prF y) = lim lim ρ(0, prF y ) r, m→∞ m m→∞ n→∞ m ≤ so y B¯ (0, r). ∈ ρ 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+ ∞ σ X Y G2 (E) j=1 j j ∧ j ∈ cc xn x be arbitrary and define Fn as in (5.3). Ifwewrite = prFn , then by the definition n n 2 P in (5.4), we have that K(x ) K(x). Since x Ga(E), there exists a length minimizing geodesic Γn from 0⊆ to xn, which we know∈ is in K(x) by Lemma 5.4. Let sn denote the midpoint of each geodesic Γn. This satisfies 1 1 ρ(0, sn)= ρ(xn, sn)= ρ(0, xn) ρ(0, x) := r. 2 ≤ 2 m Write δm = ρ(x , x), and define balls B¯ = y G2(E): ρ(0, y) r , 0 { ∈ ≤ } B¯ = y G2(E): ρ(x, y) r + δ . m { ∈ ≤ m} n n By the definition of x , we have s B¯0 B¯m for any n m with δm 0. Since every sm is contained in K∈(x),∩ by compactness,≥ there is a subsequence→ 2 snk converging to a point s in G (E). This element hence has to be contained in B¯ B¯ for any m 1 by Lemma 5.5. It follows that 0 ∩ m ≥ 1 ρ(0, s)= ρ(x, s)= ρ(0, x), 2 2 i.e., s is a midpoint of x. Since (Gcc(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 p ,1 p on the tensor product, then for β (1/3, α) k·kSch ≤ ≤ ∞ ∈ C α([0,T ], E) C α ([0,T ], E) C β([0,T ], E). g ⊂ wg ⊂ g We can prove this by showing that the conditions (I), (II) and (III) in Theorem 3.3 are satisfied. By Theorem 5.3 it follows that (I) and (II) are satisfied for Hilbert spaces. Hence, we only need to prove that condition (III) holds. Recall the results of Lemma 5.2. Let x = x + x(2) Cα([0,T ], G2(E)) be an arbitrary weakly geometric α-rough path. For any fixed∈ t, define a sequence of increasing finite subspaces F ∞ , such that { t,n}n=1 (2) 1 xt Ft,n d(xt, pr xt)= d(0, pr ⊥ x ) . ∈ Ft,n Ft,n ≤ n

Consider a partition Π = t0 = 0 < t1 < t2 < < tk = T of the interval [0,T ]. Write { ··· } F = span F : t Π . Π,n { t,n ∈ } xΠ,n x Deﬁne t = prFΠ,n t. Since ρ and d are equivalent on the ﬁnite dimensional Π,n 2 FΠ,n, it follows that xt is a continuous function in Gcc(E) with respect to ρ. GEOMETRIC ROUGH PATHS ON INFINITE DIMENSIONAL SPACES 21

Since xt is uniformly continuous, we can for each r > 0 ﬁnd a number or such that or = Osc(xt; r) = sup d(xs, xt), 0≤s

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