Geometric rough paths on infinite dimensional spaces
aa r X i v : . [ m a t h . P R ] J un GEOMETRIC ROUGH PATHS ON INFINITE DIMENSIONALSPACES
ERLEND GRONG , TORSTEIN NILSSEN AND ALEXANDER SCHMEDING Abstract.
Similar to ordinary differential equations, rough paths and roughdifferential equations can be formulated in a Banach space setting. For α ∈ (1 / , / α -rough paths by signatures of curves of bounded variation,given some tuning of the H¨older parameter. We show that these criteria aresatisfied for weakly geometric rough paths on Hilbert spaces. As an applica-tion, we obtain Wong-Zakai type result for function space valued martingalesusing the notion of (unbounded) rough drivers. Contents
1. Introduction 22. The infinite-dimensional framework for rough paths 32.1. Tensor products of Banach spaces 32.2. Algebra of truncated tensor series 42.3. Exponential map 52.4. Free nilpotent groups 62.5. Solutions to the regularity problem as iterated integrals 73. Applications to infinite dimensional rough paths 83.1. Rough paths and geometric rough paths in Banach space 83.2. Wong-Zakai for stochastic flows 113.3. Applications to unbounded rough drivers 134. Finite dimensional Carnot-Carath´eodory geometry 134.1. Sub-Riemannian manifolds and the Carnot-Carath´eodory distance 134.2. Invariant sub-Riemannian structure on Lie groups and geodesics 144.3. Carnot groups 155. Carnot-Caratheodory geometry and weakly geometric rough paths onHilbert spaces 165.1. Free step 2 sub-Riemannian groups 165.2. Dimension-free inequality on step 2 Carnot groups 185.3. Free Lie groups of step 2 from Hilbert spaces 205.4. Proof of Theorem 1.1 255.5. Generalizing the result to Banach spaces 27Appendix A. Infinite-dimensional calculus 28A.1. Differentiable and smooth maps 28A.2. Manifolds modeled on infinite dimensional spaces 28References 28
Mathematics Subject Classification.
Key words and phrases.
Rough paths on Banach spaces, geometric rough paths, Wong-Zakairesult for rough flows, infinite-dimensional Lie groups, Carnot-Carath´eodory geometry. 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 roughsignals. A rough path consists of a H¨older continuous path in a vector space togetherwith higher level information satisfying certain algebraic and analytical properties.The algebraic identities in turn allow one to conveniently formulate a rough pathas a path in nilpotent groups of truncated tensor series, cf. [FH14] for a detailedaccount. Similar to the well-known theory of ordinary differential equations, itmakes sense to formulate rough paths and rough differential equations with values ina Banach space, [LCL07, FV06]. It is expected that the general theory carries overto this infinite-dimensional setting, yet a number of results which are elementarycornerstones of rough path theory are still unknown in the Banach setting.In [BR19] the authors introduce the notion of a rough driver , which are vec-tor fields with an irregular time-dependence. Rough drivers provides a somewhatrelaxed description of necessary conditions for the well-posedness of a rough differ-ential equation and the authors use this for the construction of flows generated bythese equations. The push-forward of the flow, at least formally, satisfies a (rough)partial differential equation, and this equation is studied rigorously in [BG17] wherethe authors introduce the notion of unbounded rough drives . This theory was fur-ther developed in [DGHT19, HH18, HN20] in the linear setting (although [DGHT19]also tackles the kinetic formulation of conservation laws) as well as nonlinear per-turbations in [CHLN20, HLN19b, HLN19a, Hoc18, HNS20]. Still, the unboundedrough drivers studied in these papers assume a factorization of time and space inthe sense that the vector fields lies in the algebraic tensor of the time and spacedependence.Our main motivation for this paper is the observation in [CN19] that rough dri-vers can be understood as rough paths taking values in the space of sufficientlysmooth functions, see Section 3.2. Moreover, in [CN19] the authors needed un-bounded 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 smoothapproximations can be easily checked and is the so-called weakly geometric roughpaths . In [CN19] and ad-hoc method was introduced to tackle the lack of a similarresult in infinite dimensions.For other papers dealing with infinite-dimensional rough paths, let us also men-tion [Der10, Bai14, CDLL16].In the present paper we address the characterization of weakly geometric roughpaths in Banach spaces. Our aims are twofold. Firstly, we describe and developthe infinite-dimensional geometric framework for Banach space-valued rough pathsand weakly geometric rough paths. These rough paths take their values in infinite-dimensional groups of truncated tensor products. Some care needs to be taken inthis setting, as the tensor product of two Banach spaces will depend on choice ofof norm on the product. Secondly, we characterize the geometric rough paths thattake their values in an Hilbert space and their relationship to weakly geometricrough paths. Our main result is to prove the following well-known relationshipfor finite dimensional rough paths in an infinite dimensional setting. Recall thatfor α ∈ (1 / , / α -rough path is the is an element of the closurein signatures S ( x ) st = 1 + x t − x s + R ts ( x r − x s ) ⊗ dx r of curves x t of boundedvariation, while an α -rough path x st = 1 + x st + x (2) st is called weakly geometric ifthe symmetric part of x (2) st equals x st ⊗ x st ; a property that holds for all geometricrough paths in particular by an integration by parts argument. EOMETRIC ROUGH PATHS ON INFINITE DIMENSIONAL SPACES 3
Theorem 1.1.
For α ∈ (1 / , / , let C αg ([0 , T ] , E ) and C αwg ([0 , T ] , E ) denoterespectively geometric rough paths and weakly geometric rough paths in E , definedon the interval [0 , T ] and relative to a reasonable crossnorm on E ⊗ E . Then forany β ∈ (1 / , α ) , we have inclusions C αg ([0 , T ] , E ) ⊂ C αwp ([0 , T ] , E ) ⊂ C βg ([0 , T ] , E ) . 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 beginwith a review of basics from functional analysis used. In particular, an overviewof tensor products, choice of norm and the induced topology is given. We thenconstruct Lie groups sitting inside of infinite-dimensional tensor algebras. Thesegroups and their analytical structure provide the right framework to understandthe geometric properties of rough paths and their signatures. Certain parts ofthis material will be well known to experts in the field, however, we were notable to locate a complete presentation in the literature. In particular, the infinite-dimensional Lie structures are novel.We continue with a presentation of Banach space-valued α -rough paths for α ∈ ( , ) in Section 3. This leads to the three prerequisite assumptions in Theo-rem 3.3 which states when weakly geometric rough paths can be approximated bysignatures of bounded variation path after some tuning of the H¨older parameter.In Section 3.2, we apply Theorem 1.1 to prove Wong-Zakai type results for roughflows; a rough generalization of flows of time-dependent vector fields. This yields aconcrete application for rough paths on infinite dimensional space.The remainder of the paper is dedicated to proving Theorem 1.1 by showingthat the criteria of Theorem 3.3 are indeed satisfied in the Hilbert space setting.All of these criteria depends on considering Carnot-Carath´eodory geometry or sub-Riemannian geometry of our infinite dimensional groups. We will therefore reviewsome prerequisites from this topic in Section 4. This section gives an overview ofthe topic in finite dimensions while at the same time highlighting those parts ofthe theory that will no longer be available in the infinite dimensional setting. Wethen do the proof of Theorem 1.1 in several steps throughout Section 5, includinga result in Theorem 5.4 where we prove that the Carnot-Carath´eodory metric onthe free step 2 nilpotent group generated by a Hilbert space becomes a geodesicdistance when restricted to the subset where it is finite. We conclude the proofof Theorem 1.1 in Section 5.4. In Section 5.5 we give some comments about thegeneralization of such results to arbitrary Banach spaces.This paper uses tools from rough path theory, infinite dimensional Lie groupsand sub-Riemannian geometry, and has therefore been written to accommodate thereader that might be unfamiliar with one or more of these topics. To this end, wehave also added Appendix A which includes some basic definitions from infinitedimensional calculus.2. The infinite-dimensional framework for rough paths
Tensor products of Banach spaces. If E and F are two Banach spaces,we write E ⊗ a F for their algebraic tensor product. Using the convention that E ⊗ a = R , we will for any k ≥ k -fold algebraic tensor product E ⊗ a k with a family of norms k·k k satisfying the following conditions, cf. [BGLY15].1. For every a ∈ E ⊗ a k , b ∈ E ⊗ a ℓ , we have k a ⊗ b k k + ℓ ≤ k a k k · k b k ℓ .
2. For any permutation σ of the integers 1 , . . . k and for any x , . . . , x k ∈ E , k x ⊗ x ⊗ · · · ⊗ x k k k = k x σ (1) ⊗ · · · ⊗ x σ ( k ) k k . EOMETRIC ROUGH PATHS ON INFINITE DIMENSIONAL SPACES 4
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·k k + ℓ . From the inclusions E ⊗ a ( k + ℓ ) ⊆ E ⊗ k ⊗ a E ⊗ ℓ ⊆ E ⊗ ( k + ℓ ) it follows that E ⊗ k ⊗ E ⊗ ℓ ∼ = E ⊗ ( k + ℓ ) as Banach spaces.By [Rya02, Section 6], such a family of norms on each tensor power can beiteratively constructed using reasonable crossnorms . Definition 2.1. A reasonable crossnorm on E ⊗ a F is a norm k · k E ⊗ F , such that • k x ⊗ y k E ⊗ F ≤ k x k E k y k F for all x ∈ E, y ∈ F , • for all continuous linear ϕ ∈ E ∗ , ψ ∈ F ∗ the functional ϕ ⊗ ψ on E ⊗ a F is boundedand satisfies k ϕ ⊗ ψ k ≤ k ϕ kk ψ k relative to their respective induced norms.Example . The projective tensor product of Banach spaces is the completion ofthe algebraic tensor product with respect to the projective tensor norm k z k π := inf { P ni =1 k x i k E k y i k F : z = P ni =1 x i ⊗ y i } . It is well known that the projective tensor norm is a reasonable crossnorm on E ⊗ a F .Similarly, the injective tensor norm, defined by k z k ǫ = sup {| P ni =1 ϕ ( x i ) ψ ( y i ) | : ϕ ∈ E ∗ , ψ ∈ F ∗ , k ϕ k = k ψ k = 1 , z = P ni =1 x i ⊗ y i } . is a reasonable crossnorm, for which the completion is called the injective tensorproduct [Rya02, Section 3]. Every reasonable crossnorm k · k E ⊗ F satisfies k z k ǫ ≤k z k E ⊗ F ≤ k z k π , [Rya02, Theorem 6.1].If E is a Hilbert space, then we can identify E ⊗ a E with finite rank operatorsfrom 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, thisidentification allows one to identify the projective tensor as the space of nuclear op-erators 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.
Remark . Reasonable crossnorms are symmetric in their arguments, i.e. we haveequality k x ⊗ y k E ⊗ F = k x k E k y k F , for any x ∈ E , y ∈ F , [Rya02, Theorem 6.1].2.2. Algebra of truncated tensor series.
For N ∈ N ∪ {∞} , we define A N := N Y k =0 E ⊗ k as the the space of (truncated) formal tensor series of E . Elements in A N will bedenoted 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 degreewise addition and the multiplication( x ( k ) ) k ≤ N · ( y ( k ) ) k ≤ N := X n + m = k x ( n ) ⊗ y ( m ) ! k ≤ N . The algebras A N turn out to be Banach algebras for N finite. For N = ∞ theyare still continuous inverse algebras, i.e. topological algebras such that inversion iscontinuous and the unit group is an open subset. Continuous inverse algebras andtheir unit groups can be seen as an infinite-dimensional generalization of matrixalgebras and their unit Lie groups. We summarize the details in the followingresult. Lemma 2.4.
The algebra A N is a Banach algebra for N < ∞ , while A ∞ is aFr´echet algebra. Moreover, A N is a continuous inverse algebra whose group ofunits A × N is a C -regular infinite-dimensional Lie group for any N ∈ N ∪ {∞} . EOMETRIC ROUGH PATHS ON INFINITE DIMENSIONAL SPACES 5
Before stating the proof, let us recall the notion of regularity of an infinite-dimensional Lie group G . Let 1 denotes the group’s identity element and write L ( G ) for the Lie algebra of G . Then G is called C r -regular , r ∈ N ∪ {∞} , if foreach C r -curve u : [0 , → L ( G ) the initial value problem ( ˙ γ ( t ) = γ ( t ) · u ( t ) γ (0) = 1has a (necessarily unique) C r +1 -solution Evol( u ) := γ : [0 , → G and the mapevol: C r ([0 , , 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 -regular (cf. [Nee06]). Several important results ininfinite-dimensional Lie theory are only available for regular Lie groups, cf. [KM97]. Proof of Lemma 2.4.
By construction of the algebra structure we have for elementsof degree k and ℓ that E ⊗ k · E ⊗ ℓ ⊆ E ⊗ ( k + ℓ ) . (2.1)By choice of tensor norms in section 2.1, A N is a Banach algebra for N < ∞ , so inparticular a continuous inverse algebra. Now A ∞ is a Fr´echet space with respectto the product topology. The choice of tensor norms shows that multiplicationis separately continuous and by [Wae71, VII, Proposition 1] the multiplication isalso jointly continuous. Since A ∞ is a countable product of Banach spaces whosemultiplication satisfies (2.1), we conclude that A ∞ is a densely graded locally convexalgebra in the sense of [BDS16]. Due to [BDS16, Lemma B.8 (b)] A ∞ is a continuousinverse algebra, i.e. inversion is continuous and the unit group A × is an open subsetof A ∞ . Following [Gl¨o02, GN12], the unit group A × N is a regular Banach (for N < ∞ ) or Fr´echet Lie group ( N = ∞ ). (cid:3) Remark . The unit group A × of A N is even a real analytic Lie group in thesense that the group operations extend analytically to the complexification, cf.Appendix A. However, for our purpose analyticity will not be important.2.3. Exponential map.
Define the canonical projection π N : A N → K = A andthe closed ideal I A N := ker π N = Y Lemma 2.6 (Exponential and logarithm) . The exponential and logarithm series exp N : I A N → I A N , X X ≤ n ≤ N X ⊗ n n ! , log N : 1 + I A N → I A N , Y X ≤ n ≤ N ( − n +1 Y ⊗ n n , yield mutually inverse real analytic isomorphisms. In this paper we work only with algebras over the real numbers. However,in the following proof we need a holomorphic functional calculus which is mostconveniently formulated in complex algebras. This presents no problem as we canalways pass to the complexification of an algebra, work there and return to theoriginal algebra, see [Gl¨o02]. EOMETRIC ROUGH PATHS ON INFINITE DIMENSIONAL SPACES 6 Proof of Lemma 2.6. We follow [Gl¨o02] and define the spectrum of x ∈ A N as σ ( x ) := C \ { z ∈ C | z · − x ∈ A × N } , where 1 is the unit of A N . For Ω ⊆ C open we let ( A N ) Ω := { x ∈ A N | σ ( x ) ⊆ Ω } .In view of the holomorphic functional calculus developed in [Gl¨o02, Section 4] and[Gl¨o02, Lemma 5.2], it suffices to prove that I A ⊆ ( A N ) {| z | < log(2) } and 1 + I A N ⊆{| z − | < } . However, from [BDS16, Lemma B.8 (a)], the element z · − x isinvertible if and only if z = π N ( x ). Thus the statement follows from holomorphicfunctional calculus. (cid:3) Remark . Due to [Gl¨o02, Theorem 5.6] the Lie group exponential of A × N is givenby the exponential seriesexp A N : I A N = L ( A × N ) → A × N , x X n ∈ N x ⊗ n n ! . Thus the Lie group exponential induces a local diffeomorphism between Lie algebraand Lie group and one can even prove that the groups A × N are Baker-Campbell-Hausdorff (BCH) Lie groups, i.e. for X, Y in some neighborhood of 0, the BCH-series ∞ X k =1 ( − k +1 k X r i + s i > r + s + ··· + r k + s k ≤ N [ X r Y s · · · X r k Y s k ]( P kj =1 ( r j + s j )) · Q ki =1 r i ! s i !with [ X r Y s · · · X r k Y s k ] := [ X [ X, · · · [ X | {z } r , [ Y [ Y [ · · · [ Y | {z } s [ X, [ · · · ]]] · · · ]]]]]]converges and defines an analytic function of X, Y there, cf. [Nee06, IV.1].2.4. Free nilpotent groups. Using the exponential map, we are ready to definethe subgroups of A × N we are interested in. Observe that A N = L ( A × N ) is a Liealgebra with respect to the commutator bracket [ x, y ] := x ⊗ y − y ⊗ x . Wedefine inductively the space P na ( E ) of Lie polynomials over E of degree n ∈ N by P a ( E ) := E and P n +1 a ( E ) := P na ( E ) + span { [ x, y ] | x ∈ P na ( E ) , y ∈ E } ⊆ A n +1 , P ∞ a ( E ) := ( X n ∈ N P n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P n ∈ E ⊗ n is a Lie polynomial ) . Elements in the set P ∞ a ( E ) are called Lie series . The set of all Lie polynomialsor Lie series is a Lie subalgebra of ( I A N , [ · , · ]), [Reu93, Chapter 1.2]. Since A N is a topological Lie algebra, we see that also P N ( E ) := P Na ( E ) is a closed Liesubalgebra of ( A N , [ · , · ]). Due to [Reu93, Theorem 1.4], we have P N ( E ) ⊆ I A N . The set G N ( E ) := exp A N ( P N ( E )) = exp A N ( P N ( E )) , forms a closed subgroup of A × N due to [Reu93, Corollary 3.3]. Note that the equalityis due to Lemma 2.6.Closed subgroups of infinite-dimensional Lie groups are in general not again Liesubgroups [Nee06, Remark IV.3.17]. So indeed the next proposition is non-trivial. Proposition 2.8. The group G N ( E ) is a closed submanifold of A N and this struc-ture turns it into a Banach Lie group for N < ∞ and into a Fr´echet Lie group for EOMETRIC ROUGH PATHS ON INFINITE DIMENSIONAL SPACES 7 N = ∞ . Moreover, G N ( E ) is a C -regular Lie group and the exponential map exp : P N ( E ) → G N ( E ) is a diffeomorphism.Proof. The group G N ( E ) is a closed subgroup of the locally exponential Lie group A × N . Due to Remark 2.7, the Lie group exponential of this group is exp A N . Definethe set L ( N ) := { x ∈ I A N = L ( A × N ) | exp A N ( R x ) ⊆ G N ( E ) } . Due to construction of the closed Lie subalgebra P N ( E ), we have P N ( E ) ⊆ L ( N ) .Conversely as P N ( E ) ⊆ I A N and G N ( E ) ⊆ I A N , we deduce from Lemma 2.6that 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].We have to show that the Lie group exponential exp is a diffeomorphism. Wealready know that P N ( E ) ⊆ I A N and G N ( E ) ⊆ I A N and the exponentialexp A N is a diffeomorphism between those sets due to Lemma 2.6. This impliesthat the Lie group exponential is a diffeomorphism as exp = exp A N | G N ( E ) P N ( E ) due to[Nee06, Theorem IV.3.3].The Banach Lie groups G N ( E ), N < ∞ are C -regular, cf. also Remark 2.9below, and we see that the canonical projection mappings π MN : G M ( E ) → G N ( E ), N, M ∈ N ∪ {∞} , M ≥ N are smooth group homomorphisms. Hence we obtain aprojective system of Lie groups ( G N ( E ) , π NN − ) N ∈ N whose Lie algebras also form aprojective system ( P N ( E ) , L ( π NN − )) N ∈ N of Lie algebras. As sets P ∞ ( E ) = lim ←− P N ( E ) , G ∞ ( E ) = lim ←− G N ( E ) , and the limit maps π ∞ N are smooth group homomorphisms. Then we deduce from[Gl¨o15, Lemma 7.6] that G ∞ ( E ) admits a projective limit chart, hence [Gl¨o15,Proposition 7.14] shows that G ∞ ( E ) is C -regular. (cid:3) Remark . The regularity of the Lie groups G N ( E ) can be strengthened by weak-ening the requirements on the curves in the Lie algebra. This results in a notionof L p -regularity [Gl¨o15] for infinite-dimensional Lie groups. One can show thatBanach Lie groups such as G N ( E ) are L -regular. Furthermore, as in the proofof Proposition 2.8, one sees that the limit G ∞ ( E ) is L -regular. Note that L -regularity implies all other known types of measurable regularity for Lie groups.Observe that for N < ∞ , the group G N ( E ) is a nilpotent group of step N generated by E . Example . For the remainder of the paper, we will mostly focus on thespecial case of N = 2. In this case P ( E ) is the closure in A of sums of elements Y , Y ∧ Z = Y ⊗ Z − Z ⊗ Y with X, Y, Z ∈ E and Lie brackets[ X + X , Y + Y ] = X ∧ Y, X, Y ∈ E, X , Y ∈ P ( E ) ∩ E ⊗ . Solutions to the regularity problem as iterated integrals. We will givean alternate perspective on the regularity of the Lie group G N ( E ) which will beuseful in our application in the context of rough paths.From the regularity of the Lie group G N ( E ), we can solve the differential equa-tion(2.2) ˙ γ ( t ) = γ ( t ) · u ( t ) , γ (0) = 1 , in G N ( E ) for all curves u : [0 , → L ( G N ( E )) of a given regularity. Recall that G N ( E ) is a closed regular Lie subgroup of the Lie group A × N . In [GN12, Lemma EOMETRIC ROUGH PATHS ON INFINITE DIMENSIONAL SPACES 8 A × N is given bythe Volterra type series(2.3) γ ( t ) = 1 + N X n =1 Z t Z t n − · · · Z t u ( t ) ⊗ · · · ⊗ u ( t n )d t . . . d t n . Since u is a curve into L ( G N ( E )) ⊆ A N = L ( A × N ), we can identify the solution of(2.2) in G N ( E ) with a solution of the equation in A N . By uniqueness these solutionscoincide, hence the Volterra series (2.3) takes its values in G N ( E ). We conclude thatthe solution to the differential equation (2.2) is given by a series of iterated integrals.Assume now that the curve u takes its values in the subspace E ⊆ L ( G N ( E )),identified as the degree 1 elements in the (truncated) tensor algebra. We thenidentify the component of degree k as the iterated integral π k ( γ ( t )) = Z t Z t k − · · · Z t u ( t ) ⊗ · · · ⊗ u ( t k )d t . . . d t k . In other words, if we interpret u : [0 , → E as a rough path with values in the Ba-nach space E , the solution of equation (2.2) is the signature of u , cf. also Section 3.3. Applications to infinite dimensional rough paths Rough paths and geometric rough paths in Banach space. Let usfirst 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 acompleted tensor product. Definition 3.1. Fix α ∈ ( , ) and a tensor product completion E ⊗ E by a choiceof reasonable crossnorm k · k . An ( E, ⊗ ) -valued α -rough path consists of a pair ( x, x (2) ) x : [0 , T ] → E, x (2) : [0 , T ] → E ⊗ = E ⊗ E where x is an α -H¨older continuous path and x (2) is ”twice H¨older continuous”, i.e. (3.1) k x t − x s k . | t − s | α , k x (2) st k . | t − s | α . In addition, we require (3.2) x (2) st − x (2) su − x (2) ut = ( x u − x s ) ⊗ ( x t − x u ) usually called Chen’s relation. The set of rough paths equipped with the metricinduced by (3.1) is denoted C α ([0 , T ] , E ) . To be more precise about this distance, we write x st = 1 + x t − x s + x (2) st in A ,the two step-truncated tensor algebra over E . The Chen relations (3.2) can thenwe rewritten as x st = x su x ut . Introduce a metric d on 1 + I N = { x = 1 + x + x (2) : x ∈ E, x (2) ∈ E ⊗ E } , by | x | = max {k x k , k x k / } ,d ( x , y ) = | x − · y | = | (1 + x + x (2) ) − · (1 + y + y (2) ) | . We then define the distance between two α -rough paths ( s, t ) x st , y st defined on[0 , T ] as(3.3) d α ( x , y ) = sup ≤ s Recall the interpretation of the second component of a rough path. It is thoughtof as the iterated integral of x , x (2) st ”=” Z ts ( x r − x s ) ⊗ dx r and is usually defined by probabilistic arguments when there is no canonical analyticargument for the construction of x (2) . More precicely, it is by now well-known (see[You36]) that the iterated integral R x ⊗ dx can be defined using usual analysisarguments provided x is α -H¨older continuous with α ∈ ( , α < . On the other handthe Brownian motion has additional probabilistic structure, namely the martingalestructure, which allows the definition of an iterated integral.More specifically, let E = R K and let ( ω t ) t ∈ [0 ,T ] be a K -dimensional Brownianmotion on some probability space (Ω , F , P ). Using the aforementioned martingalestructures of the Brownian motion one can show that the Riemann sum Z T ω r ⊗ d θ ω r = lim | P |→ X P =( t i ) ω t i + θ ( t +1 − t i ) ⊗ ( ω t i +1 − ω t i )converges in L (Ω , R K × K ). Above P is a partition of [0 , T ], | P | denotes its meshand θ is some parameter in [0 , R ω r ⊗ d θ ω r is not independent of θ . The choice θ = 0 leavesthe martingale structure invariant and is referred to as the Itˆo integral, whereas θ = is compatible with regular calculus rules and is referred to as the Stratonovichintegral.To elaborate on the phrase “compatible with regular calculus”, notice that if x is smooth, then if we define x (2) st := R ts ( x r − x s ) ⊗ dx r using classical calculusand ( x, x (2) ) is referred to as the canonical lift of x to a rough path ( x, x (2) ). Theintegration by parts formula gives us(3.4) Z ts ( x r − x s ) ⊗ dx r + Z ts dx r ⊗ ( x r − x s ) = ( x t − x s ) ⊗ ( x t − x s ) . This additional structure is then captured by regarding x as a path with values in G ( E ) and we note that log ( x st ) = x t − x s + a (2) st where(3.5) a (2) st = 12 (cid:18)Z ts ( x r − x s ) ⊗ dx r − Z ts dx r ⊗ ( x r − x s ) (cid:19) = 12 Z ts ( x r − x s ) ∧ dx r , which is the so-called Levy area of x . Definition 3.2 (Weakly geometric and geometric rough paths) . We say that α -rough path x t is weakly geometric if it takes values in G ( E ) . These can againcan be given the structure of a metric space C αwg ([0 , T ] , E ) with the metric d α as in (3.3) and can be identified with C α ([0 , T ] , G ( E )) .The space of geometric rough paths is defined as the closure in the rough pathtopology of the set canonical lift of smooth paths and is denoted C αg ([0 , T ] , E ) . Since (3.4) is stable under limits, we get that the set of geometric rough pathscan be regarded as a subspace of C α ([0 , T ] , G ( E )). The reversed question, namelyif any x ∈ C α ([0 , T ] , G ( E )) can be approximated by a sequence of smooth pathsis answered positively modulo some tuning of the H¨older parameter α given thefollowing conditions.We recall the definition of the Carnot-Caratheodory metric , which we will oftenabbreviate as the CC-metric. We define this metric ρ on G ( E ) by ρ ( y , z ) = EOMETRIC ROUGH PATHS ON INFINITE DIMENSIONAL SPACES 10 ρ (1 , y − · z ) and ρ (1 , y ) = inf (Z T k ˙ x t k dt : x ∈ C ([0 ,T ] ,E ) ,x =0 , x t has bounded variation y = S ( x ) t :=1+ x T + R T x t ⊗ dx t ) . Theorem 3.3. Write M cc = { z ∈ G ( E ) : ρ (1 , z ) < ∞} , Write C ([0 , T ] , M cc ) for the space of continuous curves in M cc with respect to ρ .Let α ∈ ( , ) be given and let β ∈ ( , α ) be arbitrary. Assume that the followingconditions are satisfied. (I) For some C > and any z ∈ G ( E ) , we have d (1 , z ) ≤ Cρ (1 , z ) . (II) The metric space ( M cc , ρ ) is a complete, geodesic space. (III) The set C α ([0 , T ] , G ( E )) ∩ C ([0 , T ] , M cc ) , is dense in C α ([0 , T ] , G ( E )) relative to the metric d β .Then for any x ∈ C α ([0 , T ] , G (2) ( E )) there exists a sequence of bounded variationpaths x n : [0 , T ] → E such that x n = S ( x n ) → x in C β ([0 , T ] , E ) .In particular, we have the inclusions C αg ([0 , T ] , E ) ⊂ C α ([0 , T ] , G ( E )) ⊂ C βg ([0 , T ] , E ) . 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 ifLength( γ | [ s,t ] ) = ρ ( γ ( s ) , γ ( t )) = | t − s | ρ ( γ (0) , γ ( T )). The metric space ( M, ρ ) iscalled 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 arethen 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 infinitedimensional case, yet assumptions (I), (II) and (III) will be satisfied, giving us theresult in Theorem 1.1. We will prove this statement in Section 5, finishing the proofin Section 5.4. Proof of Theorem 3.3. We first consider the case when x ∈ C α ([0 , T ] , G ( E )) } ∩ C ([0 , T ] , M cc ). As ( M cc , ρ ) is a geodesic space, [FV10, Lemma 5.21] implies thatthere exists a sequence of truncated signatures x n = S ( x n ) : [0 , T ] → M cc ofbounded variation paths x n such thatsup t ∈ [0 ,T ] ρ ( x t , x nt ) → , for n → ∞ , and we have the uniform bound sup n d (1 , x nst ) ≤ C | t − s | α . From (I), we concludethat x n converges to x in C ([0 , T ] , G (2) ( E )). To show the stronger convergencein C β ([0 , T ] , G ( E )) we perform a classical interpolation argument. Since d is leftinvariant we see that d ( x nst , x st ) ≤ d (( x ns ) − x nt , ( x s ) − x nt ) + d (( x s ) − x nt , ( x s ) − x t ) ≤ t ∈ [0 ,T ] d ( x nt , x t ) ≤ C sup t ∈ [0 ,T ] ρ ( x nt , x t ) , so that there exists a sequence of real numbers ε n → d ( x nst , x st ) ≤ ε n . EOMETRIC ROUGH PATHS ON INFINITE DIMENSIONAL SPACES 11 From the construction of x n we have d (1 , x nst ) , d (1 , x st ) ≤ C | t − s | α . Using theinterpolation min { a, b } ≤ a θ b − θ for every a, b ≥ θ ∈ [0 , 1] we have d ( x nst , x st ) ≤ ε n ∧ C | t − s | α ≤ ε θn C − θ | t − s | α (1 − θ ) and by choosing θ such that α (1 − θ ) = β we get convergence d β ( x n , x ) = sup s,t ∈ [0 ,T ] d ( x nst , x st ) | t − s | β ≤ ε θn C − θ → , n → ∞ . Finally, from the density of C α ([0 , T ] , G ( E )) ∩ C ([0 , T ] , M cc ) by (III) it followsthat if x m ∈ C α ([0 , T ] , G ( E )) ∩ C ([0 , T ] , M cc ) is a sequence converging to anarbitrary x ∈ C α ([0 , T ] , G ( E )) with respect to d β , and x n,m is a sequence oftruncated signatures of bounded variation curves converging to x m , then x m,m converge to x . This completes the proof. (cid:3) Remark . Note that by (I), we will have C ([0 , T ] , M cc ) ⊆ { x ∈ C ([0 , T ] , G ) : x t ∈ M cc , ≤ t ≤ T } . We will see in Theorem 5.4 that in the case of Hilbert spaces, M cc will be a propersubset of G ( E ), as will C ([0 , T ] , M cc ) be of the set { x ∈ C ([0 , T ] , G ) : x t ∈ M cc , ≤ t ≤ T } by Remark 5.6.3.2. Wong-Zakai for stochastic flows. As an application of Theorem 3.3 andTherorem 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].Let ( f k ) Kk =0 be a collection of time-dependent vector fields f k : [0 , T ] × R d → R d of class C pb ( R d , R d ) 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 wewrite ω t = t )(3.6) dy t = K X k =0 f k ( t, y t ) ◦ dω kt is by now classical. The book [Kun97] stresses the importance of considering the C pb ( R d , R d )-valued semi-martingale(3.7) m t ( ξ ) := K X k =0 Z t f k ( r, ξ ) dω kr which allows for a one-to-one characterization of stochastic flows (see [Kun97] forprecise statement and result). Equation (3.6) is then understood as dy t = m ◦ dt ( y t ).Consider now the tensor product on C pb ( R d , R d ),( f ⊗ g )( ξ, ζ ) := f ( ξ ) g ( ζ ) T , which allows us to identify C pb ( R d , R d ) ⊗ with a subspace of C pb ( R d × R d , R d × d ).Let us define the iterated integral m (2) st ( ξ, ζ ) := Z ts ( m r − m s ) ⊗ ◦ dm r ( ξ, ζ )(3.8) := K X k,l =0 Z ts Z rs f l ( v, ξ ) f k ( r, ζ ) T dω lv ◦ dω kr , EOMETRIC ROUGH PATHS ON INFINITE DIMENSIONAL SPACES 12 as a C pb ( R d × R d , R d × d )-valued random field. Checking the symmetry condition thenboils down to checking (3.4) for this tensor product. We have, for µ, ν ∈ { , . . . , d } m (2) ,µ,νst ( ξ, ζ ) + m (2) ,ν,µst ( ζ, ξ )= Z ts ( m µr ( ξ ) − m µs ( ξ )) ◦ dm νr ( ζ ) + Z ts ( m νr ( ζ ) − m νs ( ζ )) ◦ dm µr ( ξ )=( m µt ( ξ ) − m µs ( ξ ))( m νt ( ζ ) − m νs ( ζ ))(3.9)by the well-known integration by parts formula for the Stratonovich integral. Wenote that the particular decomposition of (3.7) and (3.8) in terms of the vectorfields f and ω are not important for this property; only the choice of Stratonovichintegration in the definition of m (2) plays a role.The thread of [Kun97] was picked up in the rough path setting in [BR19] wherethe authors introduce so-called “rough drivers”, which are vector field analoguesof rough paths. It was noted in [CN19] that these vector fields can be canonicallydefined from infinite-dimensional, i.e. C pb ( R d , R d ), valued rough paths. In fact, theset of C p -vector fields X p ( R d ) is canonically identified with C pb ( R d , R d ) via C pb ( R d , R d ) → X p ( R d ) f f · ∇ = P µ f µ ∂∂ξ µ . Moreover, define by linearity on the algebraic tensor C pb ( R d , R d ) ⊗ a → X p ( R d ) f ⊗ g ( f · ∇ ( g · ∇ )) = P µ,ν f µ ∂g ν ∂ξ µ ∂∂ξ ν and denote by ∇ ⊗ the extension to C pb ( R d × R d , R d × d ). Moreover, for a matrix a welet a ∇ := P µ,ν a µ,ν ∂∂ξ µ ∂∂ξ ν . Then, given a rough path x ∈ C α ([0 , T ] , C pb ( R d , R d )),if we let(3.10) X st ( ξ ) := x st ( ξ ) · ∇ , X st ( ξ ) := ∇ ⊗ x (2) st ( ξ, ξ ) + x (2) st ( ξ, ξ ) ∇ , then X := ( X, X ) is a weakly geometric rough driver in the sense of [BR19]. Con-cretely, we will assume p ≥ dy t = m ◦ dt ( y t ) byusing linear interpolation of the Banach-space martingale m , showing that thecorresponding iterated integral converges to m (2) in the appropriate sense and usingcontinuity of the Itˆo-Lyons map, see [BR19] for details. The proposition below isproved in a similar way, except the martingale structure is replaced by Theorem1.1 and the continuity of the mapping x X . Notice that we use the Sobolevembedding to put ourselves in a Hilbert-space setting. Theorem 3.5. Let x ∈ C αwg ([0 , T ] , H k ( R d , R d )) for k > d + p + 1 for some p ≥ and suppose y solves dy t = X dt ( y t ) where X t = ( X t , X t ) is the rough driver builtfrom x . Then there exists a sequence of functions x n : [0 , T ] × R d → R d of boundedvariation of t such that the solution y n of ˙ y nt = x nt ( y nt ) converges to y in C β ([0 , T ] , C ( R d , R d )) for any β ∈ ( , α ) .Proof. Since x is weakly geometric we have(3.11) x (2) ,µ,νst ( ξ, ζ ) + x (2) ,ν,µst ( ζ, ξ ) = ( x µt ( ξ ) − x µs ( ξ ))( x νt ( ζ ) − x νs ( ζ )) EOMETRIC ROUGH PATHS ON INFINITE DIMENSIONAL SPACES 13 for all µ, ν ∈ { , . . . , d } which gives x (2) st ∇ = ( x t − x s )( x t − x s ) T ∇ . It followsthat X st ( ξ ) − X st ( X st )( ξ ) = ∇ ⊗ (cid:18) x (2) st − x st ⊗ x st (cid:19) ( ξ, ξ ) ∈ X ( R d )so it is a weakly geometric rough driver in the sense [BR19]. From Theorem 1.1we get can approximate the infinite dimensional rough path ( x, x (2) ) by a sequenceof smooth paths. The result now follows from [BR19, Theorem 2.6] since theembedding H k ( R d , R d ) ⊂ C pb ( R d , R d ) is continuous. (cid:3) Applications to unbounded rough drivers. We briefly mention, at a for-mal level, how infinite dimensional rough path can be used in the study of roughpath partial differential equations. To avoid technicalities we refrain from intro-ducing the full and rather large machinery needed for stating precise results.Formally, the Eulerian dynamics dy t = X dt ( y t ) has a corresponding Lagrangiandynamics described by the push-forward u t = ( y t ) ∗ φ , i.e.(3.12) du t = X dt ∇ u t , u = φ. The notion unbounded rough drivers was introduced in [BG17] to give rigorousmeaning to (3.12). In [HN20] the notion of geometric differential rough drivers wasused 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 differentialrough driver was restricted to the setting when X belongs to the algebraic tensorof time and space, viz X t ( x ) = P Kk =0 f k ( x ) ω kt . It was noted in [CN19] that alsounbounded rough drivers can be thought of as infinite dimensional rough paths, see[CN19, Section 5] for details. The present paper thus yields approximation resultsfor rough path partial differential equations for more general unbounded roughdrivers using [HN20, Theorem 2.1] by simply checking the corresponding symmetrycondition (3.11).4. Finite dimensional Carnot-Carath´eodory geometry Sub-Riemannian manifolds and the Carnot-Carath´eodory distance. We review some of the theory of finite dimensional sub-Riemannian manifolds. Formore details, see e.g. [Mon02] and [ABB20].An n -dimensional sub-Riemannian manifold is a triple ( M, E , g ) where M is an n -dimensional manifold, E is a subbundle of the tangent bundle and g = h · , · i g isa metric tensor defined only on E . An absolutely continuous curve γ : [0 , T ] → M is called horizontal if ˙ γ ( t ) ∈ E γ ( t ) for almost every t . For each such curve, we defineits length by Length( γ ) = Z T | ˙ γ | g ( t ) dt, | ˙ γ | g = h ˙ γ, ˙ γ i / g . We assume that E satisfies the bracket-generating condition , meaning that sectionsof E and their iterated brackets span the entire T M . This is a sufficient conditionfor any pair of points x, y ∈ M to be connected by a horizontal curve by the Chow-Rashevski˘ı theorem [Cho39, Ras38]. Hence, if we define the Carnot-Carath´eodorymetric ρ , or the CC-metric for short, such that ρ ( x, y ) is the infimum of the lengthof all horizontal curves connecting x and y , then this distance is finite. Furthermore,the bracket-generating condition implies that the topology of ρ coincides with themanifold topology.In order to describe length minimizers of ρ , we introduce the following notation.Associated to a sub-Riemannian structures ( E , g ), define a corresponding map ♯ : EOMETRIC ROUGH PATHS ON INFINITE DIMENSIONAL SPACES 14 T ∗ M → E ⊆ T M by p ( v ) = h ♯p, v i g , for any v ∈ E , p ∈ T ∗ M .Introduce a sub-Riemannian Hamiltonian function H : T ∗ M → R by H ( p ) = 12 h ♯p, ♯p i g , p ∈ T ∗ M. A projection γ ( t ) = π ( λ ( t )) of a solution λ ( t ) = e t ~H ( p ) of the Hamiltonian sys-tem is called a normal geodesics . Such curves are smooth and always local lengthminimizers, but there might be minimizers that do not appear in this way. For agiven initial point x ∈ M , we define Ω x as the Hilbert manifold of all horizontal L -curves defined on an interval [0 , T ] with initial value x . Let end : Ω x → M be the endpoint map γ γ ( T ). A curve γ ∈ Ω x is called an abnormal curve if it is a singular point of end. Abnormal curves only depend on the horizontalbundle E and not the metric g . An abnormal curve is not necessarily a minimizer,even locally, but any length minimizer is either a normal geodesic, an abnormalcurve or both. In the following special case, we can disregard the abnormal curves[ABB20, Corollary 12.15]. Theorem 4.1 (The Goh condition) . Assume that E has step , i.e. assume that E + [ E , E ] = T M . Then all length minimizers are normal geodesics.Remark . We could also talk about a generalizationof sub-Riemannian metrics called sub-Finsler manifolds, which replaces the metrictensor g by a Finsler metric ρ on a bracket-generating subbundle E . This gener-alization will then also contain Riemannian and Finsler manifolds as special cases.By [Ber89], any locally compact, locally contractible homogeneous length space isisometric to a quotient G/H of a Lie group with a closed subgroup equipped withthe Carnot-Carath´eodory metric of an invariant sub-Finsler metric. Remark . The Chow-Rashevski˘ı theorem relat-ing the bracket-generation condition to finiteness of the CC-distance does not havea counterpart in infinite dimensions. The closest result exists on Hilbert manifolds,where we are assured that we can reach a dense subset by horizontal curves froma given point if the horizontal distribution is bracket-generating, see [Led04]. Formore on sub-Riemannian manifolds of infinite dimensions, see e.g. [GMV15, AT17].4.2. Invariant sub-Riemannian structure on Lie groups and geodesics. Wewill now give a more explicit description of the candidates for length minimizers insub-Riemannian manifold. Consider an affine connection ∇ on T M with torsion T ( X, Y ) = ∇ X Y − ∇ Y X − [ X, Y ] , It is called compatible with ( E , g ) if for every X ∈ X ( T M ), Y, Y ∈ X ( E ), we havethat ∇ X Y | x ∈ E x , and X h Y, Y i g = h∇ X Y, Y i g + h Y, ∇ X Y i g . We note the following way to describe normal geodesics and abnormal curvesthrough compatible connections, see [GMG17] and [Gro20]. Proposition 4.4. Let ∇ be any connection compatible with the sub-Riemannianstructure. (a) A curve γ ( t ) is a normal geodesic if and only if there is some one-form λ ( t ) along γ ( t ) such that ♯λ ( t ) = ˙ γ ( t ) , ∇ ˙ γ λ ( t ) = − λ ( t )( T ( ˙ γ ( t ) , · )) . EOMETRIC ROUGH PATHS ON INFINITE DIMENSIONAL SPACES 15 (b) A curve γ ( t ) is an abnormal curve if and only if there is some one-form λ ( t ) along γ ( t ) such that ♯λ ( t ) = 0 , ∇ ˙ γ λ ( t ) = − λ ( t )( T ( ˙ γ ( t ) , · )) . We will focus on the case of invariant structures on Lie groups which will bemost important for us. Let G be an n -dimensional Lie group with Lie algebra g .Let e be a generating subspace of g equipped with an inner product. Define a sub-Riemannian structure ( E , g ) by left translation of e with its inner product. Usingthe left invariant connection in Proposition 4.4, we obtain the following corollary.Recall that ad ∗ is the adjoint representation of g on g ∗ , defined by the formula. h ad ∗ ( X ) µ, Y i = −h µ, [ X, Y ] i , X, Y ∈ g , µ ∈ g ∗ . Corollary 4.5. Let γ ( t ) be a horizontal curve with left logarithmic derivative u ( t ) = γ ( t ) − · ˙ γ ( t ) . (a) γ is a normal geodesic if and only if there is a curve t λ ( t ) in g ∗ satisfying (4.1) ♯λ ( t ) = u ( t ) , ˙ λ ( t ) = − ad ∗ ( u ( t )) λ ( t ) . (b) γ is an abnormal curve if and only if there is a curve t λ ( t ) in g ∗ satisfying ♯λ ( t ) = 0 , ˙ λ ( t ) = − ad ∗ ( u ( t )) λ ( t ) . Carnot groups. A stratified Lie algebra is a finite dimensional Lie algebrawith decomposition g = g ⊕ · · · ⊕ g N such that for k = 1 , . . . , N − g , g k ] = g k +1 , [ g , g N ] = 0 . Let G be the corresponding simply connected Lie group of g . Assume that g is equipped with a norm k · k . Define a sub-Finsler structure ( E , ρ ) of G by lefttranslation of g and its norm. The sub-Finsler manifold ( G, E , ρ ) is then called aCarnot group .Let g = g ⊕ · · · ⊕ g N be a stratified Lie algebra with a norm k · k on g . Forany s > 0, we define a Lie algebra automorphism dil s : g → g as the linear mapsatisfying dil s ( X ) = s k X, for any X ∈ g k , k = 1 , . . . , N .Let ( G, E , ρ ) be the corresponding Carnot group and define Dil s : G → G as thecorresponding Lie group automorphism of dil s . If ρ is the corresponding Carnot-Carath´eodory metric, then for any x , y ∈ G , we have ρ (Dil s x , Dil s y ) = sρ ( x , y ) . The maps Dil s are called dilations . Carnot groups are the only metric spaces thatare locally compact, geodesic, isometrically homogeneous and that admit dilations[LD15].We consider homogeneous norms on Carnot groups, adapting definitions from[FV10, Chapter 7.5.3]. Definition 4.6. Let G be a group with a family of dilations (Dil s ) s> . A homoge-neous norm is a continuous map ||| · ||| : G → R ≥ satisfying (i) For any x ∈ G , ||| x ||| = 0 if and only x is the identity G in G . (ii) It is homogeneous with respect to the dilations, i.e. ||| Dil s x ||| = s ||| x ||| .A homogeneous norm on G is said to be symmetric if ||| x ||| = ||| x − ||| . It is called sub-additive if ||| x · y ||| ≤ ||| x ||| + ||| y ||| . EOMETRIC ROUGH PATHS ON INFINITE DIMENSIONAL SPACES 16 Example . (i) Choose norms on each g k for 2 ≤ k ≤ N as well. Then we candefine a homogeneous norm | x | = max ≤ k ≤ N k pr g k log x k /k . which is symmetric, but not necessarily sub-additive.(ii) If 1 is the unit of G , then ||| x ||| = ρ (1 , x ) is a homogeneous norm which issymmetric and sub-additive. Example . Let ( E, k · k ) be a finite dimensional normedvector space and define the free nilpotent Lie algebra P N ( E ) of step N < ∞ asin Section 2.4. This is then the free algebra of elements in E , divided out bythe relations of anti-symmetry, Jacobi identity and words in E of length longerthan N . The Lie algebra P N ( E ) has a natural structure of a stratified Lie algebra P N ( E ) = f ⊕ · · · ⊕ f N where elements in f n are spanned by brackets of length n .On the corresponding simply connected Lie group G N ( E ), we define a sub-Finslerstructure given by left translation of ( E, k · k ).Any Carnot group can be considered as the quotient of such a group. Let ( G, E , ρ )be a Carnot group with Lie algebra g = g ⊕ · · · ⊕ g N . Let k · k be the norm on g .By definition, there is a canonical surjective Lie algebra homomorphism P N ( g ) → g , by dividing out with the extra relations of g . This map induces a submersion π : G N ( g ) → G such that every horizontal curve in ( G, E , ρ ) is the image of ahorizontal curve of the same length in G N ( g ). It follows that π is a length-shortening map. Remark . Invariant sub-Finsler structures on Lie groups are complete, and arehence geodesic spaces by the Hopf-Rinow theorem. This holds in particular forCarnot groups.5. Carnot-Caratheodory geometry and weakly geometric roughpaths on Hilbert spaces Free step 2 sub-Riemannian groups. Let E be a finite dimensional innerproduct space and use the notation X ∗ = h X, · i for any X ∈ E . Define a Liealgebra g ( E ) = P ( E ). By Example 2.10, we can identify g ( E ) with E ⊕ ∧ E equipped with a Lie bracket structure[ X + X , Y + Y ] = X ∧ Y, X, Y ∈ E, X , Y ∈ ∧ E. We identify ∧ E with the space of skew-symmetric endomorphisms so ( E ) by writing(5.1) X ∧ Y = X ∗ ⊗ Y − Y ∗ ⊗ X. Consider the corresponding simply connected Lie group G ( E ) with its inducedsub-Riemannian structure ( E , g ). For the rest of this section, we will use the factthat exp : g ( E ) → G ( E ) is a diffeomorphism to identify these as spaces. Thegroup G ( E ) is hence the space E ⊕ so ( E ) with multiplication(5.2) ( x + x (2) ) · ( y + y (2) ) = x + y + x (2) + y (2) + 12 x ∧ y,x, y ∈ E, x (2) , y (2) ∈ so ( E ). With this identification the identity is 0 and inversesare given by ( x + x (2) ) − = − x − x (2) . Recall the identity in (3.5) for relatingthe presentation of G ( E ) as a subset of A and its representation in exponential EOMETRIC ROUGH PATHS ON INFINITE DIMENSIONAL SPACES 17 coordinates. A curve Γ ( t ) = γ ( t ) + γ (2) ( t ) in G ( E ) is then horizontal with leftlogarithmic derivative u ( t ) in E when( γ ( t ) + γ (2) ( t )) − · ddt ( γ ( t ) + γ (2) ( t )) = ˙ γ ( t ) + ˙ γ (2) ( t ) − γ ( t ) ∧ ˙ γ ( t ) = u ( t ) . In other words ˙ γ ( t ) = u ( t ) , ˙ γ (2) ( t ) = 12 γ ( t ) ∧ u ( t ) . Since our sub-Riemmannian manifold ( G ( E ) , E , g ) is step 2, we know that alllength minimizers are normal geodesics. As a tool to present the normal geodesicequations, introduce first the inner product h X ⊕ X , Y ⊕ Y i = h X, Y i + h X , Y i = h X, Y i E − 12 tr E XY , which allows us to identify g ( E ) and its dual. Let Γ ( t ) be a normal geodesicwith left logarithmic derivative Γ ( t ) − · ˙ Γ ( t ) = u ( t ). Using our inner product andCorollary 4.5, we know that there is a curve λ ( t ) = u ( t ) + Λ( t ), Λ( t ) ∈ so ( E ) in g ( E ), such that for any Y ∈ E , Y ∈ so ( E ), h ˙ λ ( t ) , Y + Y i = h ˙ u ( t ) , Y i + h ˙Λ( t ) , Y (2) i = h λ ( t ) , [ u ( t ) , Y + Y ] i = h λ ( t ) , u ( t ) ∧ Y i = h Λ( t ) u ( t ) , Y i . It follows that ˙Λ( t ) = 0 , ˙ U ( t ) = Λ U ( t ) . As a result, for some constant element Λ ∈ so ( H ) and u ∈ E , any normal geodesicis on the form(5.3) γ ( t ) = x + Z t e s Λ u ds, γ (2) ( t ) = x (2)0 + 12 Z t γ ( s ) ∧ e s Λ u ds. Example . In the special case when E is two-dimensional, wecall G ( E ) the Heisenberg group. Introduce a complex structure on E by choosingan orthonormal basis and writing this as respectively X and iX where i = √− λX ∧ iX , u = u X , γ ( t ) = z ( t ) w and γ (2) ( t ) = σ ( t ) X ∧ iX with z ( t ) , u ∈ C and σ ( t ) , λ ∈ R . The following rewritten version of the geodesicequations in (5.3) then becomes z ( t ) = z + Z t e iλs u ds = z + e iλt − iλ u ,σ ( t ) = σ + 12 Z t Im(¯ z ( s ) e iλs u ) ds = σ + Im (cid:18) e iλt − iλ ¯ z u (cid:19) + | u | Im (cid:18) i λt − i ( e − iλt − λ (cid:19) . where we interpret e iλt − iλ and i λt − i ( e iλt − λ as respectively t and t if λ = 0. Ifthe initial point is the identity 0, we have z ( t ) = 1 − e iλt iλ u = 2 sin( λt/ λ e itλ/ u , σ ( t ) = | u | λ (cid:18) t − sin( λt ) λ (cid:19) . If the above geodesic is defined on the intervall [0 , | u | . Inparticular, we observe the following.(a) A minimizing geodesic defined on [0 , 1] from 0 to zX is given by the choice λ = 0. It follows that ρ (0 , zX ) = | z | . EOMETRIC ROUGH PATHS ON INFINITE DIMENSIONAL SPACES 18 (b) A minimizing geodesic defined on [0 , 1] from 0 to σX ∧ iX is given by the choice λ = ± π depending on the sign of σ . Hence, we have that | σ | = ρ (0 , σX ∧ iX ) π . Using the triangle inequality it follows that(5.4) max {| z | , √ π | σ | / } ≤ ρ (0 , zX + σX ∧ iX ) ≤ | z | + 2 √ π | σ | / . Dimension-free inequality on step 2 Carnot groups. We want to gener-alize the inequality (5.4) to free nilpotent groups of step 2 of arbitrary dimensions.The inequality can be concluded from formulas of the CC-distance to the verticalspace in [RS17, Appendix A], but we include some more details here for the sake ofcompletion and for applications to infinite dimensional vector spaces in Section 5.3.Consider the case of a finite dimensional Hilbert space E of arbitrary finitedimension 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 , . . . , ± iσ k } forsome k ≥ 0. We order them in such a way that σ ≥ · · · ≥ σ k > . These are also the singular values of X as | X | = √− X has exactly these non-zeroeigenvalues, with each σ j appearing twice. Define a sequence σ ( X ) = ( σ j ) ∞ j =1 ofnon-negative numbers such that σ j = 0 for j > k . For 0 < p ≤ ∞ , we define k X k Sch p = k σ ( X ) k ℓ p . For p ≥ 1, these are norms called the Schatten p -norms, [MV97, 16], or to bemore precise, the Schatten p -norm is the map X /p k X k Sch p , since we omit theduplicated singular values in our definition of σ ( X ). We will also introduce thefollowing special map k X k cc = k σ ( X ) k ℓ ( R ; N ) = ∞ X j =1 jσ j . It is simple to see that k · k cc is not a norm when dim E > 2, however, we willshow that it is a quasi-norm. Recall that a quasi-norm is a map satisfying thenorm axioms except the triangle inequality which is assumed in the form k x + y k ≤ K ( k x k + k y k ) for some K ≥ 1, [DF93, Section I.9]. Furthermore, from the definition,(5.5) k X k Sch ≤ k X k cc ≤ k X k Sch / . The latter follows from the fact that for any k > √ a + kb ≤ √ a + √ b if b ≥ a ≥ ( k − b , hence p σ + · · · + kσ k ≤ p σ + · · · + ( k − σ k − + √ σ k , since σ + · · · + ( k − σ k − ≥ k ( k − σ k .Define a homogeneous norm ||| x + x (2) ||| = max n k x k E , √ π k x (2) k / cc o . We then have the following result. Theorem 5.2. Let E be an arbitrary finite dimensional Hilbert space and consider G ( E ) with its sub-Riemannian structure. If ρ is the Carnot-Carath´eodory distanceof g , then ||| x + x (2) ||| ≤ ρ (0 , x + x (2) ) ≤ ||| x + x (2) ||| . EOMETRIC ROUGH PATHS ON INFINITE DIMENSIONAL SPACES 19 Proof. We will use the geodesic equations in (5.3). We note first that the minimalgeodesic from 0 to x ∈ E is just a straight line in E , and hence k x k E = ρ (0 , x ) . These correspond to the geodesics in (5.3). We will show that we also have(5.6) ρ (0 , x (2) ) = 2 √ π k x (2) k / cc , x (2) ∈ so ( E ) , which will give us the result from the triangle inequality.Consider a general solution of the geodesic equations Γ ( t ) = γ ( t ) + γ (2) ( t ) on G ( E ) with Γ (0) = 0 and Γ (1) = x (2) . Consider arbitrary initial values Λ = 0and u = 0 for the geodesic equation as in (5.3). Choose an orthonormal basis X , . . . , X k , Y , . . . , Y k , T , . . . , T n − k such that we can writeΛ = k X j =1 λ j X j ∧ Y j , λ j > . Introduce again complex notation iX j = Y j , i = √− 1, and write u = k X j =1 w j X j + n − k X j =1 c j T j , w j ∈ C , c j ∈ R . We will then have u ( t ) = k X j =1 e iλ j t w j X j + n − k X j =1 c j T j , w j ∈ C , c j ∈ R . We make the following simplifications. If w j = 0, then the value of λ j has no effecton u ( t ). We may hence set it to zero and reduce the value of k . Without any loss ofgenerality, 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 iλ j t w j X j + e iλ k t w k X k = e iλ j t ( w j X j + w k X k ), and sowe again obtain the same u ( t ) if we replace λ j X j ∧ Y j + λ l X l ∧ Y l with λ j | w j | + | w k | ( w j X j + w k X k ) ∧ i ( w j X j + w k X k ) . By repeating such replacements, we may assume that all values of λ , . . . , λ k aredifferent.If Γ ( t ) = Γ u ( t ) = γ ( t ) + γ (2) ( t ) is the corresponding geodesic, then γ ( t ) = k X j =1 λ j t ) λ j e iλ j t/ w j X j + n − k X j =1 tc j T j . From the condition γ (1) = 0, it follows that c , . . . , c n − k all vanish for every 1 ≤ j ≤ n − k . Furthermore, since we assume that w j = 0, it follows that λ j = 2 πn j for some positive integers n j .Computing x (2) , and using that all of the integers n , . . . , n k are different, weobtain x (2) = 12 Z γ ( t ) ∧ u ( t ) dt = 14 π k X j =1 n j ( − iw j X j ) ∧ ( w j X j ) . It follows that the endpoint x (2) has 2 k non-zero eigenvalues {± iσ , . . . , ± iσ k } with σ s = 14 n s π | w j | . EOMETRIC ROUGH PATHS ON INFINITE DIMENSIONAL SPACES 20 In other words, any normal geodesic Γ ( t ) from 0 to the point x (2) has lengthLength( Γ ) = k X j =1 | w j | = k X j =1 πn j σ j . In order to obtain the minimal value, we use n j = l if σ j is the l -th largest eigenvalue.The result follows. (cid:3) Using the identity (5.6) we also obtain the following result. Corollary 5.3. k · k cc is a quasi-norm on so ( E ) , even a / -norm [DF93, SectionI.9] , in that it satisfies k X + Y k / cc ≤ k X k / cc + k Y k / cc , k X + Y k cc ≤ k X k cc + k Y k cc ) . Free Lie groups of step 2 from Hilbert spaces. Let E be a real Hilbertspace, not necessarily finite dimensional. We choose and fix a reasonable crossnorm k · k ⊗ on the algebraic tensor product E ⊗ a E . As mentioned in Example 2.2, wecan identity E ⊗ a E with finite rank operators, and we can consider E ⊗ E as theclosure of finite rank operator with respect to k · k ⊗ .In describing P ( E ) = E ⊕ V E , we first note that we can identify the algebraicwedge product V a E with the space of all finite rank skew-symmetric operators,which we denote by so a ( E ). We then identify P ( E ) with E ⊕ so ⊕ ( E ) where so ⊗ ( E )denotes the skew-symmetric operators on E that are in the closure of so a ( E ) withrespect to k · k ⊗ .We note the following consequence of the fact that reasonable crossnorms arebounded between the injective and projective norm. For any compact skew symmet-ric map X : E → E , define a sequence σ ( X ) = ( σ j ) ∞ j =1 such that | X | = √− X haseigenvalues in non-increasing order order σ = σ ≥ σ = σ ≥ · · · .For p ∈ (0 , ∞ ],let so p ( E ) denote the space of compact skew symmetric operators X with finiteSchatten p -norm k X k Sch p = k σ ( X ) k ℓ p . As p = ∞ and p = 1 correspond to respec-tively the injective norm and (one half of) the projective norm, we have so ( E ) ⊆ so ⊗ ( E ) ⊆ so ∞ ( E ) , with so ∞ ( E ) consisting of all compact skew-symmetric operators, reflecting the factthat k X k Sch ≤ k X k ⊗ ≤ k X k Sch for any X ∈ so ∞ ( E ).Also introduce the space so cc ( E ) as the subspace of so ∞ ( E ) of elements X suchthat k X k cc := P ∞ j =1 jσ j is finite. Since all compact operators are limits of finiterank operators ([MV97, Corollary 16.4]), all the previously mentioned inequalitiesfrom Section 5.2 still hold. In particular, k · k cc is a quasi-norm and we haveinclusions so / ( E ) ⊆ so cc ( E ) ⊆ so ( E ) . Consider Lie algebras g a ( E ) = E ⊕ so a ( E ) ⊆ g cc ( E ) = E ⊕ so cc ( E ) ⊆ g ( E ) = E ⊕ so ⊗ ( E ) , with Lie brackets,[ X + X , Y + Y ] = X ∧ Y, X ∧ Y = X ∗ ⊗ Y − Y ∗ ⊗ X.X, Y ∈ E, X , Y ∈ so ⊗ ( E ). If we give g ( E ) a norm k x + x (2) k g ( E ) = max {k x k E , k x (2) k ⊗ } . then it has the structure of a Banach Lie algebra. This Lie algebra have an associ-ated Banach Lie group: The corresponding group G ( E ) is the set g ( E ) with groupoperation as in (5.2) as a consequence, making the exponential map exp equal to theidentity on the set g ( E ). We write G a ( E ) = exp( g a ( E )) and G cc ( E ) = exp( g cc ( E )). EOMETRIC ROUGH PATHS ON INFINITE DIMENSIONAL SPACES 21 Let t u ( t ) be any function in L ([0 , , E ), and Γ u be the solution of Γ − u ( t ) · ˙ Γ u ( t ) = u ( t ) , γ u (0) = 0 . This curve always exists from the L -regularity property of the Banach Lie group G ( E )(see [Gl¨o15] and also Remark 2.9). For any x , y ∈ G ( E ), we define ρ ( x , y ) ∈ [0 , ∞ ]by ρ ( x , y ) = ρ (0 , x − · y ) ,ρ (0 , x ) = inf (cid:8) k u k L : u ∈ L ([0 , , E ) , Γ u (1) = x (cid:9) . Relative to this notation, we can express the main technical step in provingTheorem 1.1 as the following result. Theorem 5.4. For any Hilbert space E and every reasonable crossnorm k · k ⊗ , wehave G cc ( E ) = { x ∈ G ( E ) : ρ (0 , x ) < ∞} , Furthermore, The metric space ( G cc ( E ) , ρ ) is a complete, geodesic space and if wedefine ||| x + x (2) ||| = max n k x k E , √ π k x (2) k / cc o , then (5.7) ||| x ||| ≤ ρ (0 , x ) ≤ ||| x ||| . We will do the proof of this theorem in two parts. In the first part, we will showthat G cc ( E ) is indeed exactly the set with finite ρ -distance and that the inequality(5.7) holds. In the second part, we show that it is a geodesic space. Proof of Theorem 5.4, Part I. We will begin by introducing the following notation,that we will use in both parts of the proof. If F is any closed subspace of E , wewrite pr F : E → F for the corresponding orthogonal projection and write pr F, ⊥ =id E − pr F for the projection to the complement. By slight abuse of notation, wewill extend this projection to a linear map pr F : G ( E ) → G ( F ), denoted by thesame symbol, and determined by,pr F X ∧ Y = (pr F X ) ∧ (pr F Y ) . We use similar convention for pr F, ⊥ : G ( E ) → G ( F ⊥ ). Finally, we write aprojection operator pr F ∧ F ⊥ : G ( E ) → G ( E ) by pr F ∧ F ⊥ : X X ∧ (pr F X ) ∧ (pr F, ⊥ Y ) + (pr F, ⊥ X ) ∧ (pr F Y ) . We have already shown the result for finite dimensional spaces, so we assume that E is infinite dimensional. Step 1: Properties of projections to closed subspaces. Let F ⊆ E be any closedsubspace and let x ∈ G ( F ) ⊆ G ( E ) be arbitrary. Assume that there is a curveLet Γ u ( t ) from 0 to x in G ( E ). We observe then thatpr F Γ ( t ) = Γ pr F u ( t ) , is a curve in G ( F ) from 0 to x of less or equal length. Hence, we obtain ρ (0 , x ) = inf {k u k L : u ∈ L ([0 , , F ) ⊆ L ([0 , , E ) , Γ u (1) = x } and in particular, if there is a minimizing geodesic from 0 in G ( F ) it will also beminimizing in the larger space. In particular, if F is a finite dimensional subspace,there is a minimizing geodesic from 0 to any x ∈ G ( F ) ⊆ G ( E ). EOMETRIC ROUGH PATHS ON INFINITE DIMENSIONAL SPACES 22 Step 2: The CC-distance is finite on G a ( E ) . Consider an arbitrary element x ∈ G a ( E ) with x = x ⊕ x (2) , x (2) = n X j =1 σ j X j ∧ Y j . such that X , Y , . . . , X n , Y n are mutually orthogonal unit vectors. Define the finitedimensional subspace F = span { x, X , Y , . . . , X n , Y n } . We then observe that since x ∈ G ( F ), ρ (0 , x ) < ∞ and there is a minimizing geodesic from 0 to x . Also, anyelement in G a ( E ) satisfies the inequality (5.7). Step 3: Vertical elements. Consider an element x = x (2) ∈ so cc ( E ) with σ ( x (2) ) =( σ j ). Find mutually orthogonal unit vectors X , Y , X , Y , . . . such that x (2) = ∞ X j =1 σ j X j ∧ Y j . Consider the curve u ( t ) = 2 √ π ∞ X j =1 ( jσ j ) / (cos(2 πjt ) X j + sin(2 πjt ) Y j ) . We see that k u ( t ) k E = k u k L = 2 p π k y (2) k cc . Furthermore, if pr ≤ n denotes theorthogonal projections to F ≤ n = span { X , Y , . . . , X n , Y n } , then by the proof ofTheorem 5.2, it follows that pr ≤ n Γ u is a minimizing geodesic from 0 to y n = y n, (2) with y n, (2) = n X j =1 σ j X j ∧ Y j . Since pr ≤ n u converges to u in L ([0 , , H ) and y n converges to x in the norm k · k g ( E ) , it follows that Γ u is a minimizing geodesic from 0 to x , and in particular, ρ (0 , x ) = Length( Γ u ) = 2 √ π k x (2) k / cc . Step 4: The CC-distance is exactly finite on G cc ( E ) . For any element x = x + x (2) ∈ G cc ( E ), we can construct a horizontal curve Γ from 0 to x by a concatenation of thestraight line from 0 to x with a minimizing geodesic from 0 to x (2) left translatedby x . The result is that ρ (0 , x ) ≤ Length( Γ ) = k x k + 2 √ π k x (2) k / cc ≤ ||| x ||| < ∞ . Conversely if x ∈ G ( E ) and ||| x ||| = ∞ , then using (5.7) and any sequence y n in G a ( E ) converging to x in k · k g ( E ) , we see that ρ (0 , x ) = ∞ . G cc ( E ) is completewith the distance ρ as it is complete with respect to ||| · ||| by definition. (cid:3) We will need the following lemma to prove that ( G, ρ ) is a geodesic space. Lemma 5.5. Let x ∈ G cc ( E ) be a fixed arbitrary element. (a) The set (5.8) K ( x ) = y ∈ G cc ( E ) : For any closed subspace F ⊆ Eρ (0 , pr F y ) ≤ p ρ (0 , x ) ρ (0 , pr F x ) k pr F ∧ F ⊥ y k ⊗ ≤ p ρ (0 , x ) ρ (0 , pr F x ) , is relatively compact in G ( E ) . (b) Any minimizing geodesic from to x is contained in K ( x ) .Proof. To simplify notation in the proof, we write ρ ( x ) := ρ (0 , x ). EOMETRIC ROUGH PATHS ON INFINITE DIMENSIONAL SPACES 23 (a) By definition, for any y ∈ K ( x ), we have k y k g ( E ) ≤ ρ ( y ) ≤ √ ρ ( x )so K ( x ) is bounded in both G cc ( E ) and G ( E ). Write x = x + x (2) with σ ( x (2) ) = ( σ j ). Define mutually orthogonal unit vectors T, X , Y , X , Y , . . . , such that x ∈ ˇ E := span { T, X , Y , . . . } and such that x (2) = P ∞ j =1 σ n X n ∧ Y n .Observe that since pr E, ⊥ x = 0, we must have K ( x ) ⊆ G cc ( ˇ E ) and hence it issufficient to consider subspaces of this space.Identify ˇ E by the space R × ℓ ( C ) by y r × ( z j ) , r = h y, T i , z j = h y, X j i + i h y, Y j i . 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 ev-ery ε > 0, there is a finite set of balls of radius ε > F n = span { T, X , Y . . . , X n , Y n } with orthogonal complement F >n =span { X n +1 , Y n +1 , X n +2 , Y n +2 , . . . , } in F . Write pr n and pr n, ⊥ for their cor-responding orthogonal projections. We can decompose any element y ∈ K ( x )by y = y + y + y , with y = pr n y , y = pr n, ⊥ y and y = pr F n ∧ F >n y . We observe that ρ ( y ) ≤ ρ ( x ) .ρ ( y ) ≤ q ρ ( x ) ρ (pr n, ⊥ x ) . k y k ≤ q ρ ( x ) ρ (pr n, ⊥ x ) . Hence, for a given ε > 0. we can choose n sufficiently large such that y and y are contained in the ε -ball B (0 , ε ) centered at 0 ∈ g ( E ). We can then finda finite set of ε balls to cover the set F n ∩ B (0 , ρ ( x )). Hence, we have coveredall of K ( x ) for the result.(b) Let F be a fixed closed subspace and write pr F = pr, pr F, ⊥ = pr ⊥ andpr F ∧ F ⊥ = pr ∧ . Let Γ = Γ u = γ + γ (2) : [0 , → M be any minimizing ge-odesic with left logarithmic derivative u and write u F = pr u and u ⊥ = pr ⊥ u .Note that since u is a minimizing geodesic, then by reparametrization, we mayassume that k u ( t ) k = p k u F ( t ) k + k u ⊥ ( t ) k = ρ ( x ) , and note(5.9) ρ (pr ⊥ x ) ≤ Length(pr ⊥ Γ ) = Z k u ⊥ ( t ) k dt ≤ ρ ( x ) . (5.10)This leads to the following sequence of inequalities ρ ( x ) ρ (pr x ) ≥ ρ ( x ) ( ρ ( x ) − ρ (pr ⊥ x )) (5.9)+(5.10) ≥ ρ ( x ) Z (cid:16)p k u F ( t ) k + k u ⊥ ( t ) k − k u ⊥ ( t ) k (cid:17) dt (5.9) = ρ ( x ) Z k u F ( t ) k p k u F ( t ) k + k u ⊥ ( t ) k + k u ⊥ ( t ) k ! dt (5.10) ≥ Z k u F ( t ) k dt Jensen ≥ (cid:18)Z k u F ( t ) k dt (cid:19) = 12 Length(pr Γ ) . EOMETRIC ROUGH PATHS ON INFINITE DIMENSIONAL SPACES 24 It follows that any point y on the curve Γ will have ρ (pr y ) ≤ p ρ ( x ) ρ (pr x ). Wealso see that pr ∧ Γ t = pr ∧ γ (2) ( t ) withpr ∧ γ (2) ( t ) = 12 Z t ((pr γ ( s )) ∧ u ⊥ ( s ) + (pr ⊥ γ ( s )) ∧ u F ( s )) ds. We finally use that k pr ∧ γ (2) ( t ) k ≤ k pr ∧ γ (2) ( t ) k Sch ≤ Z ( k pr γ ∧ ¯ u ( t ) k Sch + k ¯pr γ ∧ u F ( t ) k Sch ) dt ≤ √ p ρ ( x ) ρ (pr x ) Z k ¯ u ( t ) k E dt + ρ ( x ) 12 Z k u F ( t ) k E dt ≤ p ρ ( x ) ρ (pr x ) . Hence the geodesic satisfies pointwise the bounds from the definition of K ( x ) andthe result follows. (cid:3) Proof of Theorem 5.4, Part II. We are now ready to complete the proof. For thispart, the following notation will be practical. If y (2) ∈ so ⊗ ( E ) is an arbitraryelement with σ ( y (2) ) = ( σ j ), and we can write in a collection of mutually orthogonalunit vectors as X , Y , X , Y . . . , y (2) = ∞ X j =1 σ j X j ∧ Y j , then we write y (2) ( m ) := m X j =1 σ j X j ∧ Y j , If y = y ⊕ y (2) ∈ G ( E ), we write y ( m ) = y ( m ) ⊕ y (2) ( m ) where y ( m ) is theprojection of y to the orthogonal complement of { X m +1 , Y m +1 , X m +2 , Y m +2 , . . . , } .Observe the identity k y (2) ( m ) k cc ≤ m ( m + 1)2 k y (2) ( m ) k Sch ∞ ≤ m ( m + 1)2 k y (2) ( m ) k ⊗ . Step 5: Closed balls in different metrics. For fixed r ≥ x ∈ G cc ( E ), let¯ B r = { x ∈ G cc ( E ) : ρ ( x , x ) ≤ r } , be the closed ball centered at x with respect to ρ . We will prove that this isa closed set in G ( E ). By left invariance, we only consider x = 0. Assume that y n = y n + y n, (2) is a sequence contained in ¯ B r converging in G ( E ) to some element y = y + y (2) . We then have that k y − y n k E → , k y (2) − y n, (2) k ⊗ → . In particular, we will have k y (2) − y n, (2) k Sch ∞ → σ ( y (2) ) = ( σ j )and σ ( y n, (2) ) = ( σ nj ), then σ nj → σ j . It follows that ||| y ||| ≤ k y k E + ∞ X j =1 jσ j = lim n →∞ k y n k E + lim m →∞ lim n →∞ m X j =1 jσ nj ≤ r. As a consequence, we have lim m →∞ ρ ( y ( m ) , y ) ≤ m →∞ ||| y − y ( m ) ||| = 0. Usingthat ρ (0 , y ( m )) ≤ ρ (0 , y ) ≤ ρ (0 , y ( m )) + ρ ( y ( m ) , y ), it follows thatlim m →∞ ρ (0 , y ( m )) = ρ (0 , y ) . EOMETRIC ROUGH PATHS ON INFINITE DIMENSIONAL SPACES 25 Continuing using that projections are contractions, it follows that k y (2) ( m ) − y n, (2) ( m ) k ⊗ → k y (2) ( m ) − y n, (2) ( m ) k cc ≤ m ( m + 1)2 k y (2) ( m ) − y n, (2) ( m ) k ⊗ , it follows that ρ ( y n ( m ) , y ( m )) → m . As a consequence ρ (0 , y ) = lim m →∞ ρ (0 , y ( m )) = lim m →∞ lim n →∞ ρ (0 , y n ( m )) ≤ r. Hence, y ∈ ¯ B r . Step 6: Every point has a midpoint. Let x = x + x (2) be an arbitrary element in G cc ( E ) and define x n = x ( n ). Since x n ∈ G a ( E ), there exists a length minimizinggeodesic Γ n from 0 to x n . Using the notation of (5.8) and Lemma 5.5, we knowthat Γ n is contained in the set K ( x n ). Furthermore, since for any closed subspace F of E , we have ρ (0 , pr F x n ) ≤ ρ (0 , pr F x ), if follows that for any 1 ≤ n < ∞ , Γ n is contained in K ( x ).Let s n denote the midpoint of each geodesic Γ n . This satisfies ρ (0 , s n ) = ρ ( x n , s n ) = 12 ρ (0 , x n ) ≤ ρ (0 , x ) := r. Write δ m = ρ ( x m , x ), and define balls¯ B = { y ∈ G ( E ) : ρ (0 , y ) ≤ r } , ¯ B m = { y ∈ G ( E ) : ρ ( x , y ) ≤ r + δ m } . By the definition of x n , we have s n ∈ ¯ B ∩ ¯ B m for any n ≥ m with δ m → s m is contained in K ( x ), by compactness, there is a subsequence s n k converging to a point s in G ( E ). This element hence has to be contained in¯ B ∩ ¯ B m for any m ≥ 1. It follows that ρ (0 , s ) = ρ ( x , s ) = 12 ρ (0 , x ) , i.e., s is a midpoint of x . Since ( G cc ( E ) , ρ ) is a complete length space, it follows fromleft-invariance of the metric together with [BBI01, Theorem 2.4.16] that existenceof such midpoint for any element is equivalent to the space being a geodesic space.This completes the proof. (cid:3) Proof of Theorem 1.1. We will now want to show our main result, which isthat if E is a Hilbert space and we define α -weak geometric rough path relative tothe reasonable crossnorm k · k ⊗ on the tensor product, then for β ∈ (1 / , α ) C αg ([0 , T ] , E ) ⊂ C αwp ([0 , T ] , E ) ⊂ C βg ([0 , T ] , E ) . We can prove this by showing that (I), (II) and (III) in Theorem 3.3. By Theo-rem 5.4 it follows that (I) and (II) are satisfied for Hilbert spaces. Hence, we onlyneed to prove that assumption (III) holds. To this end, we will exploit that thereasonable crossnorm satisfies k · k Sch ∞ ≤ k · k ⊗ ≤ k · k Sch .Let x = x + x (2) ∈ C α ([0 , T ] , G ( E )) be an arbitrary weakly geometric α -roughpaths. We write x (2) t = ∞ X j =1 σ j,t X j,t ∧ Y j,t . with each σ j,t depending continuously on t . Define z nt = ∞ X j = n +1 σ j,t X j,t ∧ Y j,t . EOMETRIC ROUGH PATHS ON INFINITE DIMENSIONAL SPACES 26 We observe that by continuity on [0 , T ], we have that d (0 , z nt ) = k z nt k ⊗ ≤ k z nt k Sch = P ∞ j = n σ j,t converges uniformly to zero. It follows that we find a sequence ε n → t ∈ [0 ,T ] d (0 , z nt ) < ε n . Define x nt = ( z nt ) − x . Since both d ( z ns , z nt ) and d ( x ns , x nt ) are bounded fromabove by d ( x s , x t ) for any s, t , we have that x n , z n ∈ C α (0 , G ( E )). Furthermore,by definition, for every s, t and n ≥ 1, there exists a subspace F nst of rank at most4 n + 2 such that x ns , x nt ∈ G ( F nst ). By using (5.11), we have1(4 n + 2)(4 n + 3) ρ ( x ns , x nt ) ≤ d ( x ns , x nt ) ≤ ρ ( x s , x t ) . for any s, t ∈ [0 , T ]. It follows that x nt is a continuous function in G cc with respectto ρ for each n ≥ z nt takes values in the center of G ( E ), then d ( x st , x nst ) = d ( x st , x st · z ns · ( z nt ) − ) = d ( z ns , z nt ) < ε n . Using now the α -H¨older property of x n and x and an interpolation argument as inthe proof of Theorem 3.3, we obtain that d β ( x , x n ) → β ∈ ( , α ). Thiscompletes the proof. Remark ρ ) . We note that even though in the aboveproof, each x n takes values in G cc ( E ), the separate statement about continuity isalso a necessary part of the proof. Indeed, for any infinite dimensional Hilbert space E , there are elements in C α ([0 , T ] , G ( E )) that take values in G cc ( E ) but are notcontinuous with respect to the CC-metric.In the following, we choose the Schatten 1-norm, i.e. we compute in the projectivetensor product. Let X , Y , X , Y , . . . be an arbitrary countably infinite collectionof mutually orthogonal unit vector fields. Consider the sequence y m = m X j =1 m X j ∧ Y j , n ≥ . Then for m < n , d (0 , y m ) = k y m k Sch = 1 m and d ( y n , y m ) = ( n − m ) m n ( n + 2 m ) , In particular, d ( y n , y m ) ≤ n − m ) m n . On the other hand ρ (0 , y m ) = k y n k cc = ( m +1)2 m ≥ , so every y m is outside theCC-ball of radius . Consider the path x t = t = 0, y m +1 + t − m +11 m − m +1 ( y m − y m +1 ) if m +1 ≤ t ≤ m , y if t = 1.This path is obviously not continuous at t = 0, however, it will in fact be -H¨olderwith respect to the distance d . We show this through considering three cases.(i) If 0 < m +1 ≤ t ≤ m , then d (0 , x t ) ≤ m ≤ m + 1) ≤ t . EOMETRIC ROUGH PATHS ON INFINITE DIMENSIONAL SPACES 27 (ii) Observe that if m +1 ≤ s < t ≤ m , then x st = m ( m + 1)( t − s )( y m − y m +1 ) . Furthermore, d ( x s , x t ) = m ( m + 1)( t − s ) d ( y m +1 , y m ) ≤ t − sm + 1 ≤ t − s ) . (iii) We finally assume that 0 < n +1 ≤ s ≤ n ≤ m +1 ≤ t ≤ m . If m + 1 < n , thenit follows that t − s > n and also d ( x s , x t ) ≤ d ( y n +1 , y m ) ≤ n + 1 − m ) m ( n + 1) ≤ m n − m − m + 1) n ≤ t − s ) . If m + 1 = n , we observe that x t = y m +1 + (( m + 1) t − m ( y m − y m +1 ) , x s = y m +1 + (( m + 1) s − m + 2)( y m +1 − y m +2 ) . so x st = (cid:18) t − m + 1 (cid:19) m ( m + 1)( y m − y m +1 )+ (cid:18) m + 1 − s (cid:19) ( m + 1)( m + 2)( y m +1 − y m +2 ) . If follows that d ( x s , x t ) ≤ (cid:18) t − m + 1 (cid:19) m ( m + 1) d ( y m , y m +1 ) + 2 (cid:18) m + 1 − s (cid:19) ( m + 1)( m + 2) d ( y m +1 , y m +2 ) ≤ t − s ) ( m + 1) (cid:0) md ( y m , y m +1 ) + ( m + 2) d ( y m , y m +1 ) (cid:1) ≤ t − s ) ( m + 1) (cid:18) m m ( m + 1) + ( m + 2) 6( m + 1) ( m + 2) (cid:19) = 12 ( t − s ) (cid:18) m + 1 m + 1 (cid:19) ≤ t − s ) . Through these calculations, we obtain a concrete counterexample.5.5. Generalizing the result to Banach spaces. In our approach to char-acterize geometric rough paths for H¨older index α ∈ (1 / , / 2) with values inHilbert spaces we have exploited that we could solve the problem for every finite-dimensional subspace of the Hilbert space. This led to dimension independentestimates which could then be lifted to the Hilbert space. Note that the liftingprocedure works only because the orthogonal projection onto finite-dimensionalsubspaces is contractive, i.e. it shortens distances between points. Equivalently,this property can be characterized as the projections being of operator norm 1. Asthis condition makes also sense for projections onto subspaces of Banach spaces,one may ask oneself if we could not just copy the approach for Hilbert spaces byemploying contractive projections instead of the orthogonal ones.Let us first mention that contractive projections might in general be rare inBanach spaces. Namely, we have from [Ran01, Theorem 3.1] the following charac-terization: Theorem 5.7 (Contractive projections characterise Hilbert spaces) . For a Banachspace E with dim E ≥ , the following statements are equivalent: (i) E is isometrically isomorphic to a Hilbert space, EOMETRIC ROUGH PATHS ON INFINITE DIMENSIONAL SPACES 28 (ii) every -dimensional subspace of E is the range of a projection of norm , (iii) every subspace of E is the range of a projection of norm . A generalization to Banach spaces hence need an an approach that does not leanso heavily on projections or choose subspaces carefully in order to have contractiveprojection, see [Ran01] for examples. Appendix A. Infinite-dimensional calculus We include some basic definitions and standard notation of differential calculusin locally convex spaces. For more details, we refer to [Gl¨o03, Kel74].A.1. Differentiable and smooth maps. Let E and F be locally convex K -vectorspaces with K ∈ { R , C } . For some open set U ⊆ E , let f : U → F be a continuousmap. For any ( x, y ) ∈ U × E , we define the directional derivative at x in thedirection of x by df ( x, y ) := D y f ( x ) := lim t → t − ( f ( x + ty ) − f ( x )) , whenever the limit is defined. We say that f is C r K , for 1 ≤ r ≤ ∞ , if the iterateddirectional derivatives d ( k ) f ( x, y , . . . , y k ) := ( D y k D y k − · · · D y f )( x )exist for all finite k ≤ r , x ∈ U and y , . . . , y k ∈ E and define continuous maps d ( k ) f : U × E k → F . We say that f smooth if it is C ∞ K . We write C r K simply as C r if K is clear from the context. Definition A.1. Let E , F be real locally convex spaces and f : U → F definedon an open subset U . We call f real analytic (or C ω R ) if f extends to a C ∞ C -map ˜ f : ˜ U → F C on an open neighborhood ˜ U of U in the complexification E C . For r ∈ N ∪ {∞ , ω } the composition of C r K -maps (if possible) is again a C r K -map(cf. [Gl¨o03, Propositions 2.7 and 2.9]).A.2. Manifolds modeled on infinite dimensional spaces. Fix a Hausdorfftopological space M and a locally convex space E over K ∈ { R , C } . An E -manifoldchart ( U κ , κ ) on M is an open set U κ ⊆ M together with a homeomorphism κ : U κ → V κ ⊆ E onto an open subset of E . Two such charts are called C r -compatible for r ∈ N ∪ {∞ , ω } if the change of charts map ν − ◦ κ : κ ( U κ ∩ U ν ) → ν ( U κ ∩ U ν )is a C r -diffeomorphism. A C r K -atlas of M is a family of pairwise C r -compatiblemanifold charts, whose domains cover M . Two such C r -atlases are equivalent iftheir union is again a C r -atlas.A locally convex C r -manifold M modeled on E is a Hausdorff space M with anequivalence class of C r -atlases of E -manifold charts. 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Box 7803, 5020 Bergen,Norway Pure and Applied Mathematical Analysis Research group (PAMAR), University ofAgder, P.O. Box 422, 4604 Kristiansand E-mail address : [email protected] E-mail address : [email protected] E-mail address ::