Low-$x$ physics at LHCb

Thomas Boettcher

University of Cincinnati
on behalf of the LHCb collaboration

> The LHCb detector’s forward geometry provides unprecedented kinematic
> coverage at low Bjorken-$x$. LHCb’s excellent momentum resolution, vertex
> reconstruction, and particle identification enable precision measurements at
> low transverse momentum and high rapidity in proton-lead collisions, probing
> $x$ as small as $10^{-6}$. In this contribution, we present recent studies
> of low-$x$ physics using the LHCb detector. These studies include charged
> hadron, neutral pion, and $D^{0}$ production in proton-lead collisions, as
> well as charmonium production in ultraperipheral lead-lead collisions.
> Future prospects and implications for the understanding of low-$x$ nuclear
> PDFs and parton saturation are also discussed.

> PRESENTED AT
>
>
>
>
> DIS2023: XXX International Workshop on Deep-Inelastic Scattering and Related
> Subjects,
> Michigan State University, USA, 27-31 March 2023
>

##  1 Introduction

High-energy collisions at the LHC provide sensitivity to partons carrying
small fractions $x$ of the colliding nucleons’ momenta. In this low-$x$
regime, parton densities, and particularly the gluon density, in the nucleon
are large. As a result, interactions at a low momentum scale $Q$ can have a
distance scale larger than the typical spatial separation between gluons. This
is illustrated in Figure 1a, which shows the gluon density in a nucleon
divided by $Q^{2}$. At large $x$ and large $Q^{2}$, partons are well-separated
relative to the length scale of the interaction. In this case, the evolution
of parton densities is dominated by QCD radiation and can be described using
the linear DGLAP equation. At small $x$ and small $Q^{2}$, parton separations
become small relative to the length scale of the interaction. This is
particularly true in heavy nuclei, as the gluon density scales with $A^{1/3}$.
Consequently, the study of nuclei at low $x$ is the study of matter at high
gluon densities. In this low-$x$ regime, which is often described using the
color-glass condensate (CGC) effective field theory [1], parton recombination
competes with radidiation, resulting in saturation of the parton density.

The LHCb detector is a fully instrumented detector in the forward region at
the LHC [2]. Its forward acceptance of $2<\eta<5$ provides sensitivity to very
high- and low-$x$ partons. The LHCb detector features excellent vertex
resolution, charged-particle momentum resolution, and particle identification
capabilities, making it well-suited for studying the low-$p_{\rm T}$ signals
that are particularly sensitive to low-$x$ partons at low $Q^{2}$. The LHCb
detector collects $p{\rm Pb}$ data in two configurations: the forward
configuration, in which the proton travels toward the spectrometer at the time
of the collision; and the backward configuration in which the lead travels
toward the spectrometer. The forward configuration covers positive
pseudorapidity and is sensitive to low-$x$ partons in the nucleus, while the
backward configuration covers negative pseudorapidity and is sensitive to
moderate-to-high $x$. The LHCb detector’s kinematic coverage in the $x-Q^{2}$
plane is shown in Figure 1b.

(a)  (b)

Figure 1: (a) Illustration of the gluon area density in a nucleon divided by
$Q^{2}$ as a function of $x$ and $Q^{2}$. For regions with values larger than
one, saturation effects may be important. (b) The LHCb detector’s coverage in
the $x-Q^{2}$ plane compared to other LHC experiments and past fixed-target
experiments.

Parton densities in hadrons are often described using parton distribution
functions (PDFs), and the parton densities of bound nucleons are described by
nuclear PDFs (nPDFs) [3, 5, 4, 6, 7]. PDFs and nPDFs encode the $x$
distributions of partons at a particular $Q^{2}$, and their evolution in
$Q^{2}$ is assumed to be governed by the linear DGLAP equation. Because they
encode the nonperturbative structure of protons and nuclei, PDFs and nPDFs
must be determined from data. Until recently, the gluon nPDF was almost
completely unconstrained for $x\lesssim 10^{-4}$. However, recent LHCb
measurements of open charm production in $p{\rm Pb}$ collisions at
$\sqrt{s_{\rm NN}}=5.02\,{\rm TeV}$ [9] have resulted in large reductions in
uncertainties in the gluon nPDF at low $x$ in state-of-the-art nPDF fits [5,
4]. The effect of these data is shown in Figure 2, which shows the ratio
$R^{g}_{\rm Pb}$ of the gluon distribution in a bound nucleon to the gluon
distribution in the free proton in fits with and without LHCb data. The LHCb
data tightly constrain the gluon nPDF at $x$ as low as about $10^{-5}$.
Consequently, additional measurements will serve to overconstrain nPDFs,
potentially revealing the effects of nonlinear QCD at low $x$.

Figure 2: Comparison of the EPPS16 [8] and EPPS21 [4] gluon nuclear
modification factors. The EPPS21 fit uses LHCb $D^{0}$ production data, while
the EPPS16 fit does not.

##  2 Light hadron production

The minimum $Q^{2}$ of open charm production is limited by the relatively
large $D^{0}$ mass. In order to study lower $x$ and $Q^{2}$, the LHCb
collaboration has measured inclusive charged-particle production in $p{\rm
Pb}$ collisions at $\sqrt{s_{\rm NN}}=5.02\,{\rm TeV}$ [10]. The measured
nuclear modification factor is shown in Figure 3. At forward rapditiy, the
results agree with calculations using nPDFs, although the nPDF uncertainties
are large. The forward results show some tension with CGC calculations, but
this tension disappears when comparing to more recent next-to-leading order
(NLO) calculations [11]. The backward data, however, show a large enhancement
of charged particle production and are not successfully described by any
available models.

Figure 3: The charged-particle nuclear modification factor measured by LHCb in
$p{\rm Pb}$ collisions at $\sqrt{s_{\rm NN}}=5.02\,{\rm TeV}$ [10].

Understanding this backward charged-particle enhancement is necessary in order
to understand the effects of low-$x$ phenomena on the forward charged-particle
data. The backward enhancement could be caused by a final-state effect, such
as radial flow. In this case, the production of different particle species
could be enhanced by different amounts [12]. As a result, studies of
identified particle production are necessary to untangle initial- and final-
state effects. The LHCb collaboration measured $\pi^{0}$ production in $p{\rm
Pb}$ collisions at $\sqrt{s_{\rm NN}}=8.16\,{\rm TeV}$ [13]. The resulting
nuclear modification factor is shown in Figure 4. The forward data agree with
both nPDF predictions and the LHCb charged-particle data. The backward data,
however, show a larger enhancement than predicted by nPDF calculations, but a
smaller enhancement than that observed in the LHCb backward charged-particle
data. This pattern is qualitatively consistent with expectations from radial
flow, and further studies of identified particles will help clarify the source
of the enhancement.

Figure 4: The $\pi^{0}$ nuclear modification factor measured by LHCb in $p{\rm
Pb}$ collisions at $\sqrt{s_{\rm NN}}=8.16\,{\rm TeV}$ [13].

##  3 Open charm production

The LHCb collaboration recently measured $D^{0}$ production in $p{\rm Pb}$
collisions at $\sqrt{s_{\rm NN}}=8.16\,{\rm TeV}$ [14]. The higher collision
energy provides access to lower $x$ than the $5.02\,{\rm TeV}$ measurement.
Furthermore, the $8.16\,{\rm TeV}$ dataset is much larger than the $5.02\,{\rm
TeV}$ dataset, allowing for much higher precision and granularity. Results are
shown in Figure 5. The upper row of figures shows results at forward rapidity,
which agree well with both nPDF and CGC predictions. The backward results,
however, show tension with nPDF calculations at high $p_{\rm T}$. This tension
occurs in a kinematic region similar to the charged-particle and $\pi^{0}$
enhancements.

Figure 5: The $D^{0}$ nuclear modification factor measured by LHCb in $p{\rm
Pb}$ collisions at $\sqrt{s_{\rm NN}}=8.16\,{\rm TeV}$ [14].

##  4 Ultraperipheral collisions

In addition to inclusive forward particle production in $p{\rm Pb}$
collisions, the low-$x$ regime can also be probed using ultraperipheral heavy-
ion collisions (UPCs). In UPCs, a photon emitted by a nucleus interacts with
the other colliding nucleus. LHCb has studied vector meson production in UPCs
in ${\rm PbPb}$ collisions at $\sqrt{s_{\rm NN}}=5.02\,{\rm TeV}$ [15]. The
$J/\psi$ and $\psi(2S)$ cross sections measured by the LHCb collaboration are
shown in Figure 6. The vector meson photoproduction cross section is
proportional to the square of the gluon density in the nucleus at leading
order and probes $x\lesssim 10^{-5}$ [16]. Perturbative QCD calculations of
UPC cross sections suffer from large scale uncertainties [17], but the LHCb
studies of $J/\psi$ and $\psi(2S)$ production together constrain the scale
dependence of low-$x$ physics models.

Figure 6: The (left) $J/\psi$ and (right) $\psi(2S)$ photoproduction cross
sections measured in ${\rm PbPb}$ UPCs at $\sqrt{s_{\rm NN}}=5.02\,{\rm TeV}$
[15].

##  5 Conclusions

LHCb data has allowed modern nPDFs to precisely describe the collinear
structure of the nucleon at $x\lesssim 10^{-5}$. Despite this successful
description, a satisfactory understanding of the underlying low-$x$ physics
remains elusive. Even with newfound precision, perturbative QCD calculations
relying on nPDFs and linear QCD still often agree with CGC predictions. As a
result, the task of low-$x$ physics at the LHC is now to overconstrain parton
densities across a wide kinematic range in order to find the onset of
nonlinear QCD effects. Accomplishing this will require understanding final-
state effects in light-hadron production, which will benefit from future LHCb
studies of identified particle production in $p{\rm Pb}$ collisions.
Measurements of light neutral meson production will also pave the way for
studies of low-$p_{\rm T}$ direct photon production at LHCb, which will
directly probe the gluon nPDF at low $x$ and $Q^{2}$. Finally, LHCb is
planning to collect proton-oxygen data during Run 3. These data will help
determine the $A$ dependence of low-$x$ effects and could help reveal the
onset of nonlinear QCD.

## Acknowledgements

This material is based on work supported by the U.S. National Science
Foundation.

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# The accessibility problem for geometric rough differential equations

Youness Boutaib  RWTH Aachen University
Chair for Mathematics of Information Processing
Pontdriesch 10
52062 Aachen, Germany<EMAIL_ADDRESS>
###### Abstract.

We show how to use geometric arguments to prove that the terminal solution to
a rough differential equation driven by a geometric rough path can be obtained
by driving the same equation by a piecewise linear path. For this purpose, we
combine some results of the seminal work of Sussmann on orbits of vector
fields [Sus73] with the rough calculus on manifolds developed by Cass,
Litterer and Lyons in [CLL12].

###### Key words and phrases:

Rough paths, rough analysis on manifolds, accessibility, orbits.

###### 1991 Mathematics Subject Classification:

Primary 60L20, 58J65; Secondary 58A30, 53Cxx

##  1\. Introduction

###  1.1. Formulation of the problem

The theory of rough paths introduced by T. Lyons [Lyo98, LCL07] (then
developed in further directions by M. Gubinelli [Gub04, Gub10]) teaches us
that the effects of paths on differential systems can not always be explained
by the collection of values taken by the path through time. The understanding
of these effects requires, in general, a convention on what is meant by the
sequence of the iterated integrals of the driving path. More specifically,
giving a meaning to the differential equation

(1) $\mathrm{d}y_{t}=f(y_{t})\mathrm{d}x_{t}$

(with some initial condition) requires not only the knowledge of the path
$(x_{t})_{0\leq t\leq T}$ but also of its lift into a signature

(2) $S(x)_{0,t}=\left(1,\int_{0\leq u_{1}\leq
t}\mathrm{d}x_{u_{1}},\int_{0\leq u_{1}\leq u_{2}\leq
t}\mathrm{d}x_{u_{1}}\otimes\mathrm{d}x_{u_{2}},\int_{0\leq u_{1}\leq
u_{2}\leq u_{3}\leq
t}\mathrm{d}x_{u_{1}}\otimes\mathrm{d}x_{u_{2}}\otimes\mathrm{d}x_{u_{3}},\ldots\right).$

While the signature (2) can be given a canonical meaning for any smooth path
$x$ (or more precisely, for any path with finite $p$-variation for $1\leq
p<2$), this is no longer the case for rougher paths like the ones frequently
encountered in stochastic analysis. For example, one may interpret (2) for the
Brownian motion using either Itô’s or Stratonivich’s integration theory and
the solution to the differential equation (1) will also depend in general on
this choice. What could be even less intuitive is that the lift (2) for smooth
paths and the solution to (1) can also be interpreted in non-canonical ways,
thus describing unexpected effects. For example, taking $x$ to be the null
path in $\mathbb{R}^{2}$, the classical theory of ordinary differential
equations (ODEs) tells us that the solution $y$ to (1) is constant (equal to
its initial value $y_{0}$). However, in the theory of rough paths, one may
interpret $x$ as a rough path and choose the polynomial tensor path

$t\longmapsto\left(1,(0,0),at\right),$

where $a\in\mathbb{R}^{2}\otimes\mathbb{R}^{2}$ is any (antisymmetric) second
order tensor in $\mathbb{R}^{2}$, as a definition for the second order
signature of $x$. In general, the solution to (1) driven by this rough path is
not going to be constant. Moreover, the solution to (1) driven by this rough
lift of the null path $x$ can be given a physical meaning, for instance as the
limiting behaviour of systems driven by the increasingly oscillating paths

$x(n):t\longmapsto\left(\frac{\cos(2\pi n^{2}t)}{n},\frac{\sin(2\pi
n^{2}t)}{n}\right)\in\mathbb{R}^{2},$

(this now popular example can be found both in [Lyo98] and [LCL07].) More
explicitly, the sequence $(y(n))_{n}$ of solutions to the differential
equation (1) driven by the paths $(x(n))_{n}$ converges (in some rough path
topology) to the solution to the same equation interpreted in the rough path
sense to be driven by the rough path

$t\longmapsto\left(1,(0,0),\left(\begin{array}[]{cc}0&\pi t\\\ -\pi t&0\\\
\end{array}\right),\ldots\right).$

Other examples of equations driven by the Brownian motion whose limiting
behaviour is to be treated by the theory of rough paths and which can be given
a physical meaning can be found in [FGL15].

In view of the above, it is legitimate to ask the following question: can
rough signals through their non-canonical lifts exhibit terminal output states
that cannot be observed in systems that are driven by (piecewise) smooth paths
(and their canonical lifts)? More formally, for $\gamma>p\geq 1$,
$\xi\in\mathbb{R}^{d}$, a Lipschitz-$\gamma$ map
$f:\mathbb{R}^{d}\to\mathcal{L}(\mathbb{R}^{n},\mathbb{R}^{d})$ and a
geometric $p$-rough path $\mathbf{X}$ in $\mathbb{R}^{n}$ defined over a
compact interval $[0,T]$, let $\mathbf{Y}$ be the solution to the following
rough differential equation (RDE)

(3)
$\left\\{\begin{array}[]{l}\mathrm{d}\mathbf{Y}_{t}=f(\mathbf{Y}_{t})\mathrm{d}\mathbf{X}_{t},\quad\forall\;\;0\leq
t\leq T\\\ \mathbf{Y}_{0}=\xi\\\ \end{array}\right.$

Does there exist an $\mathbb{R}^{n}$-valued path $x$ with bounded variation
such that the solution to the classical ODE

(4)
$\left\\{\begin{array}[]{l}\mathrm{d}y_{t}=f(y_{t})\mathrm{d}x_{t},\quad\forall\;\;0\leq
t\leq T\\\ y_{0}=\xi\\\ \end{array}\right.$

satisfies $y_{T}=\xi+\mathbf{Y}_{0,T}^{1}$? We will call this the
accessibility problem.

###  1.2. Review of the literature and strategy

By the continuity of the Itô solution map associated to (3) and the very
definition of a geometric rough path, given a geometric $p$-rough path
$\mathbf{X}$ in $\mathbb{R}^{n}$, one can approximate, in the $p$-rough path
topology and up to an arbitrarily small error, its solution $\mathbf{Y}$ by
the solution $y$ (identified as its canonical signature $S(y)$) to the RDE
when driven by a path of bounded variation. Consequently, one can approximate
the path $(\xi+\mathbf{Y}_{0,t}^{1})_{0\leq t\leq T}$ in the $p$-variation and
the uniform convergence topologies by the path $(y_{t})_{0\leq t\leq T}$. The
question we ask here is slightly different: we are only interested in the
terminal value $\xi+\mathbf{Y}_{0,T}^{1}$ of the output of the system and, as
a price for this weaker condition, we want to reproduce this same terminal
state by driving the RDE (3) by a path of bounded variation. Indeed, in
general, the path $(\xi+\mathbf{Y}_{0,t}^{1})_{0\leq t\leq T}$ has only finite
$p$-variation while solutions to the ODE (4) have finite $1$-variations.
Therefore, it is not always possible to replace $\mathbf{X}$ by a path of
finite length and replicate the solution $(\xi+\mathbf{Y}_{0,t}^{1})_{0\leq
t\leq T}$ at all times.

Let us now consider a simple case. Let $N\in\mathbb{N}^{*}$ and denote by
$T^{(N)}(\mathbb{R}^{n})$ the truncated tensor algebra of $\mathbb{R}^{n}$ of
order $N$ defined by

$T^{(N)}(\mathbb{R}^{n})=\mathbb{R}\oplus\mathbb{R}^{n}\oplus\left(\mathbb{R}^{n}\right)^{\otimes
2}\oplus\cdots\oplus\left(\mathbb{R}^{n}\right)^{\otimes N}.$

Define the (linear) map
$f:T^{(N)}(\mathbb{R}^{n})\to\mathcal{L}(\mathbb{R}^{n},T^{(N)}(\mathbb{R}^{n}))$
by

$\forall x\in\mathbb{R}^{n},\,\forall(a_{0},\ldots,a_{N})\in
T^{(N)}(\mathbb{R}^{n}):\quad f(a_{0},\ldots,a_{N})(x)=(0,a_{0}\otimes
x,\ldots,a_{N-1}\otimes x).$

Then, given a path $x:[0,T]\to\mathbb{R}^{n}$ of bounded variation, the ODE

(5)
$\left\\{\begin{array}[]{l}\mathrm{d}y_{t}=f(y_{t})\mathrm{d}x_{t},\quad\forall\;\;0\leq
t\leq T\\\ y_{0}=(1,0,\cdots,0)\in T^{(N)}(\mathbb{R}^{n})\\\
\end{array}\right.$

admits as a unique solution the order-$N$ truncated signature path
$\left(S_{N}(x)_{(0,t)}\right)_{0\leq t\leq T}$ of $x$ [LCL07, Che62]. It is
also known (by using for example the shuffle product property [LCL07, Che62])
that the order-$N$ truncated signatures live in the free nilpotent group
$G^{(N)}(\mathbb{R}^{n})$ of order $N$ [Reu93]. As $f$ is smooth, one may also
consider the equation (5) in the rough sense. The traces of solutions to (5)
when driven by (geometric) rough paths also take their values in
$G^{(N)}(\mathbb{R}^{n})$ (for example by the continuity of the shuffle
product property [LCL07]). The answer to the accessibility problem in the case
of (5) is affirmative and the argument relies on one final ingredient; that
the order-$N$ truncated signatures of paths of bounded variation fill the free
nilpotent group $G^{(N)}(\mathbb{R}^{n})$. The most known proof of this fact
is a geometric one based on a theorem by Chow.

######  Theorem 1.1 (Chow-Rashevskii).

[Cho39, Gro96, Ras38] Let $M$ be a connected smooth manifold and let
$\mathcal{D}$ be a finite family of smooth vector fields in $M$ such that
these vector fields and their successive (higher order) Lie brackets span
every tangent space of $M$. Then every two points in $M$ can be joined by a
concatenation of integral curves of the vector fields in $\mathcal{D}$.

First note that $G^{(N)}(\mathbb{R}^{n})$ is a Lie group whose tangent space
is given (through a globally defined logarithm map) by
${\mathcal{L}}^{(N)}(\mathbb{R}^{n})$, the space of Lie polynomials of degree
$N$ [LCL07, Reu93]. Denote by $(e_{i})_{1\leq i\leq n}$ the canonical basis of
$\mathbb{R}^{n}$ and, for every $i\in[\\![1,n]\\!]$, by $f^{i}$ the linear
homomorphism defined on $T^{(N)}(\mathbb{R}^{n})$ by

$f^{i}(y)=f(y)(e_{i}),\quad\textrm{for all }y\in T^{(N)}(\mathbb{R}^{n}).$

It is clear then, by the very definition of
${\mathcal{L}}^{(N)}(\mathbb{R}^{n})$, that the left-invariant vector fields
$f^{1},\ldots,f^{n}$ and their successive Lie brackets span the tangent spaces
of $G^{(N)}(\mathbb{R}^{n})$. Hence, by Chow’s theorem, any point in
$G^{(N)}(\mathbb{R}^{n})$ can be connected to the unit element
$(1,0,\cdots,0)$ by a concatenation of integral curves of the vector fields
$f^{1},\ldots,f^{n}$. For $i\in[\\![1,n]\\!]$, an integral curve $\gamma$ of
$f^{i}$ is also a solution to the ODE

$\mathrm{d}\gamma_{t}=f(\gamma_{t})\mathrm{d}x^{(i)}_{t},$

where $x^{(i)}:t\mapsto te_{i}$ is an axis path. Therefore, a concatenation of
integral curves of the vector fields $f^{1},\ldots,f^{n}$ starting at
$(1,0,\cdots,0)$ corresponds to a solution to the ODE (5) driven by a
concatenation of axis paths, which itself is a path of bounded variation (we
will revisit this argument more in detail in the proof of the main theorem of
this article). Therefore $G^{(N)}(\mathbb{R}^{n})$ is indeed exactly the space
of all signatures of paths of bounded variation and the accessibility problem
is solved in this case.

In the current paper, we will generalise the geometric argument above to the
more generic RDE (3) where the terminal solutions to the corresponding ODE do
not define a smooth manifold with a structure as simple and as well understood
as the Lie group $G^{(N)}(\mathbb{R}^{n})$. There will be three main
ingredients to these generalisations:

  * •

A thorough understanding of Lipschitz vector fields and their flows (Section
2). This section recalls several of the results in [Bou20] and provides new
ones that will be the base of all the proofs of Lipschitz regularity in this
paper.

  * •

A geometric structure on which one can define rough paths and RDEs (Sections 3
and 5). In Section 3, we recall the definition of Lipschitz manifolds in
[CLL12], show how to endow the unit ball (and sets that are diffeomorphic to
it) with such a structure and explain how the Lipschitz regularity of vector
fields translate into the Lipschitz regularity of the transport map they
induce between Lipschitz manifolds. Section 5 recalls the main elements of the
theory of rough paths on manifolds as laid out in [CLL12].

  * •

Sussmann’s seminal work on orbits of vector fields [Sus73], which is a more
flexible and more general version of Chow’s theorem (Section 4). We will adapt
these results to our needs and build local Lipschitz structures on
neighborhoods contained in the same orbit.

In Section 6, we will combine all our constructions in the preceding sections
to prove the general accessibility property for RDEs driven by geometric rough
paths. We will conclude the paper with some final remarks in Section 7.

We end this introduction with a few words on some prerequisites, conventions
and notations. First, and due to it becoming a far more popular and understood
topic nowadays than a few years ago, some knowledge of the classical theory of
(geometric) rough paths is assumed. Classical rough paths in Banach spaces are
denoted in bold letters, with components denoted as follows

$\mathbf{X}_{s,t}=\left(1,\mathbf{X}^{1}_{s,t},\mathbf{X}^{2}_{s,t},\ldots\right).$

We simply denote the null rough path by $0$

$0_{s,t}=\left(1,0,0,\ldots,0,\ldots\right).$

Second, most of the maps that will be considered in this paper are defined
between vector spaces. Consequently, most of the statements below can be
expressed in terms of directional or Fréchet derivatives. However, as we aim
to build a non-flat geometric structure, the results and the intuition are
more straightforward when sometimes expressed in a geometric language (e.g.
pushforwards). The standard characterisations of the tangent spaces to finite
dimensional vector spaces will be heavily used. For example, for a smooth map
$f:\mathbb{R}^{n}\to\mathbb{R}^{p}$ and $x,v\in\mathbb{R}^{n}$, $f_{*}(x)(v)$
is sometimes used instead of writing $f_{*}(x)(\mathrm{D}_{x,v})$
($\mathrm{D}_{x,v}$ being the derivation (tangent vector) at $x$ in the
direction $v$), is identified with $\mathrm{d}f(x)(v)$ and can then be
understood either as a vector in $\mathbb{R}^{p}$ or a tangent vector to
$\mathbb{R}^{p}$ at $f(x)$. Finally, we will heavily rely on the following
notations throughout this article (some of which have already been used in
this introduction).

$\mathcal{L}(E,F)$ | : |  The space of (continuous) linear mappings from a normed vector space $E$ to a normed vector space $F$.
---|---|---
$E^{\otimes n}$ | : |  The space of homogeneous tensors of the vector space $E$ of order $n$ for $n\in\mathbb{N}^{*}$.
$B_{n}(x,\alpha)$ | : |  The open ball centred at $x\in\mathbb{R}^{n}$ of radius $\alpha$, $\mathbb{R}^{n}$ being endowed with the Euclidean norm.
$\pi_{n,p,d}$ | : |  The orthogonal projection defined from $\mathbb{R}^{n}\simeq\mathbb{R}^{p}\oplus\mathbb{R}^{n-p}$ into $\mathbb{R}^{d}\simeq\mathbb{R}^{p}\oplus\mathbb{R}^{d-p}$ whose kernel is $\\{0_{\mathbb{R}^{p}}\\}\oplus\mathbb{R}^{n-p}$ and image is $\mathbb{R}^{p}\oplus\\{0_{\mathbb{R}^{d-p}}\\}$.
$\mathrm{Id}_{U}$ | : |  The identity map on the set $U$.
$T_{x}M$ | : |  The tangent space to a manifold $M$ at $x\in M$.
$\mathcal{\tau}(M)$ | : |  The set of vector fields on a manifold $M$.
$[p]$ | : |  The integer part of a real number $p$, i.e. the only integer such that $0\leq p-[p]<1$.
$\overline{A}$ | : |  The closure of a subset $A$ of a topological space.

##  2\. A review of results on Lipschitz maps and their flows

###  2.1. Lipschitz maps

We first recall the classical notion of Lipschitz regularity (as presented for
example in [LCL07]). Given two Banach spaces $E$ and $F$ and an integer $k\geq
1$, we denote by $\mathcal{L}_{s}(E^{\otimes k},F)$ the space of symmetric
$k$-linear mappings from $E$ to $F$. We will use, without ambiguity, the same
notation $\|.\|$ to designate a norm on $E^{\otimes k}$ and the norm on $F$.
We will also avoid the discussion on the required properties from the norms on
tensor products of spaces and instead refer the interested reader either to
the literature on rough paths (e.g. [LCL07]) or the manuscript [Bou20].

######  Definition 2.1.

Let $n\in\mathbb{N}$ and $0<\varepsilon\leq 1$. Let $E$ and $F$ be two normed
vector spaces and $U$ be a subset of $E$. Consider the maps $f^{0}:U\to E$,
and, for every $k\in[\\![1,n]\\!]$, $f^{k}:U\to\mathcal{L}_{s}(E^{\otimes
k},F)$. For $k\in[\\![0,n]\\!]$, the map $R_{k}:E\times
E\to\mathcal{L}(E^{\otimes k},F)$ defined by

$\forall x,y\in U,\forall v\in E^{\otimes
k}:f^{k}(x)(v)=\sum\limits_{j=k}^{n}f^{j}(y)\left(\frac{v\otimes(x-y)^{\otimes(j-k)}}{(j-k)!}\right)+R_{k}(x,y)(v)$

is called the remainder of order $k$ associated to
$f=(f^{0},f^{1},\ldots,f^{n})$.
The collection $f=(f^{0},f^{1},\ldots,f^{n})$ is said to be Lipschitz of
degree $n+\varepsilon$ on $U$ (or in short a $\textrm{Lip}-(n+\varepsilon)$
map) if there exists a constant $M$ such that for all $k\in[\\![0,n]\\!]$,
$x,y\in U$ and $v_{1},\ldots,v_{k}\in E$:

  1. (1)

$\|f^{k}(x)(v_{1}\otimes\cdots\otimes v_{k})\|\leq
M\|v_{1}\otimes\cdots\otimes v_{k}\|$;

  2. (2)

$\|R_{k}(x,y)(v_{1}\otimes\cdots\otimes v_{k})\|\leq
M\|x-y\|^{n+\varepsilon-k}\|v_{1}\otimes\cdots\otimes v_{k}\|$.

The smallest constant $M$ for which the properties above hold is called the
$\textrm{Lip}-(n+\varepsilon)$-norm of $f$ and is denoted by
$\|f\|_{\textrm{Lip}-(n+\varepsilon)}$.

Note that Definition 2.1 has the advantage of being purely quantitative: no
topological assumptions are made on the domain $U$.

######  Notation.

For $\gamma>0$, we will denote by $\lceil\gamma\rceil$ the unique integer such
that $0<\gamma-\lceil\gamma\rceil\leq 1$.

The following proposition shows that Lipschitz regularity is a uniform local
property.

######  Proposition 2.2.

[Bou20] Let $\gamma>0$. Let $E$ and $F$ be two normed vector spaces and $U$ be
a subset of $E$. We assume that there exists $\delta>0$ and $M\geq 0$ such
that, for every $x\in U$, $f_{|B(x,\delta)\cap U}$ is $\textrm{Lip}-\gamma$
with a norm less than or equal to $M$ (where
$f=(f^{0},\ldots,f^{\lceil\gamma\rceil}))$. Then $f$ is $\textrm{Lip}-\gamma$
and there exists a constant $C_{\delta,\gamma}$ (depending only on $\delta$
and $\gamma$) such that

$\|f\|_{\textrm{Lip}-\gamma}\leq C_{\delta,\gamma}M.$

A very useful characterisation of Lipschitz maps can be obtained in the case
when the domain is assumed to be open. The proof of the following basic result
can be found for example in [CLL12].

######  Theorem 2.3.

Let $n\in\mathbb{N}$, $0<\varepsilon\leq 1$ and $M\geq 0$. Let $E$ and $F$ be
two normed vector spaces and $U$ be a subset of $E$. Let $f^{0}:U\to F$ and,
for every $k\in[\\![1,n]\\!]$, $f^{k}:U\to\mathcal{L}(E^{\otimes k},F)$ be
maps. We consider the two following assertions

(A1):

$(f^{0},f^{1},\ldots,f^{n})$ is $\textrm{Lip}-(n+\varepsilon)$ and
$\|f^{0}\|_{\textrm{Lip}-(n+\varepsilon)}\leq M$.

(A2):

$f^{0}$ is $n$ times differentiable, with $f^{1},\ldots,f^{n}$ being its
successive derivatives. Moreover, $\|f^{0}\|_{\infty}$, $\ldots$,
$\|f^{n}\|_{\infty}$ are upper-bounded by $M$ and for all $x,y\in U$, one has

$\|f^{n}(x)-f^{n}(y)\|\leq M\|x-y\|^{\varepsilon}.$

If $U$ is open then $\mathbf{(A1)}\Rightarrow\mathbf{(A2)}$. If $U$ is open
and convex then $\mathbf{(A1)}\Leftrightarrow\mathbf{(A2)}$.

######  Remark 2.4.

Following Theorem 2.3, the maps $f^{1},\ldots,f^{n}$ are uniquely determined
by $f^{0}$ if the domain of definition is open. While we stress that this is
not the case in general, we will often use the shorthand statement that
“$f^{0}$ is Lipschitz” instead of “$(f^{0},\ldots,f^{n})$ is Lipschitz”,
assuming that there is no confusion as to the knowledge of
$f^{1},\ldots,f^{n}$.

There exists a natural embedding between spaces of Lipschitz maps.

######  Theorem 2.5.

[Bou20] Let $\gamma,\gamma^{\prime}>0$ such that $\gamma^{\prime}\leq\gamma$.
Let $E$ and $F$ be two normed vector spaces and $U$ be a subset of $E$. Let
$f:U\to F$ be a $\textrm{Lip}-\gamma$ map. Then $f$ is
$\textrm{Lip}-\gamma^{\prime}$ and there exists a constant
$M_{\gamma,\gamma^{\prime}}$, depending only on $\gamma$ and
$\gamma^{\prime}$, such that $\|f\|_{\textrm{Lip}-\gamma^{\prime}}\leq
M_{\gamma,\gamma^{\prime}}\|f\|_{\textrm{Lip}-\gamma}$.

As an illustration of the above results, we construct a Lipschitz bijection
from the Euclidean space to the unit ball, which will be of use to us in the
subsequent sections.

######  Lemma 2.6.

Let $d\in\mathbb{N}^{*}$. Let $F:\mathbb{R}^{d}\to B_{d}(0,1)$ be the map
given by

$F:y\longmapsto\frac{y}{\sqrt{1+\|y\|^{2}}}.$

Then $F$ is a homeomorphism. Moreover:

  * •

$F$ is Lipschitz of any degree,

  * •

For every $\delta\in(0,1)$, its inverse $F^{-1}_{|B_{d}(0,\delta)}$ is
Lipschitz of any degree.

###### Proof.

$F$ is clearly a homeomorphism and is classically smooth with bounded
derivatives at any order. For $n\in\mathbb{N}$ and $\varepsilon\in(0,1)$ and
for every $x\in\mathbb{R}^{d}$, as $F^{(n+1)}$ is bounded on $B_{d}(x,1)$ by a
universal constant then $F^{(n)}$ is $\varepsilon$-Hölder on $B_{d}(x,1)$ with
a Hölder constant that is independent of $x$. Hence, by Theorem 2.3, $F$ is
Lip-$(n+\varepsilon)$ on $B_{d}(x,1)$ with a Lipschitz norm that is
independent of $x$. We deduce then by Proposition 2.2 that $F$ is
Lip-$(n+\varepsilon)$.
The inverse of $F$ is given by

$F^{-1}:y\mapsto\frac{y}{\sqrt{1-\|y\|^{2}}}.$

The same previous arguments apply to $F^{-1}$ when restricted to
$B_{d}(0,\delta)$ for $\delta\in(0,1)$ (the Lipschitz constants will explode
as $\delta$ nears 1.) This concludes this proof. ∎

In order to obtain sharper quantitative results or to include important
functions (for example linear maps) for which one of the axioms of Lipschitz
regularity fails, a notion of almost Lipschitz regularity has been introduced
in [Bou20]. We reintroduce it here to allow for a certain flexibility in our
proofs.

######  Definition 2.7.

[Bou20] Let $n\in\mathbb{N}^{*}$, $0<\varepsilon\leq 1$ and
$\delta\in(0,\infty)\cup\\{\infty\\}$. Let $E$ and $F$ be two normed vector
spaces and $U$ be a subset of $E$. Consider the maps $f^{0}:U\to E$, and, for
every $k\in[\\![1,n]\\!]$, $f^{k}:U\to\mathcal{L}_{s}(E^{\otimes k},F)$.
Denote by $R_{0}:U\times U\to E$ the remainder map of order $0$ associated to
$f=(f^{0},f^{1},\ldots,f^{n})$. The collection $f$ is said to be almost
Lipschitz of degree $n+\varepsilon$ on domains of size $\delta$ of $U$ if
$(f^{1},\ldots,f^{n})$ is $\textrm{Lip}-(n+\varepsilon-1)$ and there exists a
non-negative constant $M$, such that

$\forall x,y\in U:\quad\|x-y\|<\delta\Rightarrow\|R_{0}(x,y)\|\leq
M\|x-y\|^{n+\varepsilon}.$

If $\|R_{0}\|_{\infty,\delta}$ denotes the smallest value for such a constant
$M$, we will denote

$\|f\|_{\delta,\textrm{Lip}-(n+\varepsilon)}=\max(\|f^{1}\|_{\textrm{Lip}-(n+\varepsilon-1)},\|R_{0}\|_{\infty,\delta}).$

When $\delta$ is infinite, we will merely say that $f$ is almost
Lip-$(n+\varepsilon)$ on $U$.

On open convex sets, there exists a simple criterion for identifying almost
Lipschitz maps.

######  Lemma 2.8.

[Bou20] Let $n\in\mathbb{N}^{*}$ and $0<\varepsilon\leq 1$. Let $E$ and $F$ be
two normed vector spaces and $U$ be an open convex subset of $E$. Let $f:U\to
F$ be an $n$-times continuously differentiable map with successive derivatives
respectively denoted $f^{1},\ldots,f^{n}$. Then $f$ is almost Lipschitz of
degree $n+\varepsilon$ on $U$ if and only if $f^{1}$ is Lipschitz of degree
$n+\varepsilon-1$ on $U$. In this case

$\|f\|_{\infty,\textrm{Lip}-(n+\varepsilon)}=\|f^{1}\|_{\textrm{Lip}-(n+\varepsilon-1)}.$

An example of a consequence of working with almost Lipschitz maps is the
following general composition theorem.

######  Theorem 2.9.

[Bou20] Let $E$, $F$ and $G$ be three normed vector spaces. Let $U$ be a
subset of $E$ and $V$ be a subset of $F$. Let $\delta>0$ and $\gamma\geq 1$.
Let $f:U\to F$ be an almost $\textrm{Lip}-\gamma$ map on domains of size
$\delta$ of $U$ and $g:V\to G$ be a $\textrm{Lip}-\gamma$ map such that
$f(U)\subseteq V$. Then $g\circ f$ is $\textrm{Lip}-\gamma$ and there exists a
constant $C_{\delta,\gamma}$ (depending only on $\delta$ and $\gamma$) such
that

$\|g\circ f\|_{\textrm{Lip}-\gamma}\leq
C_{\delta,\gamma}\|g\|_{\textrm{Lip}-\gamma}\max(\|f\|_{\delta,\textrm{Lip}-\gamma}^{\gamma},1).$

######  Remark 2.10.

In the above theorem, if $U$ and $V$ are open, then the collection $g\circ f$
is defined by taking $g\circ f$ and its higher derivatives. Otherwise, said
collection will consist of $g\circ f$ and “formal higher derivatives” obtained
by applying the chain rule using the collections $g$ and $f$ (see [Bou20] for
more details.)

######  Remark 2.11.

In the context of Theorem 2.9, if $f$ is assumed to be $\textrm{Lip}-\gamma$,
then there exists a constant $C_{\gamma}$ depending only on $\gamma$ such that

$\|g\circ f\|_{\textrm{Lip}-\gamma}\leq
C_{\gamma}\|g\|_{\textrm{Lip}-\gamma}\max(\|f\|_{\textrm{Lip}-(n+\varepsilon)}^{\gamma},1).$

A simple case of composition is where one of the maps is linear, although,
technically, a continuous linear map defined on the whole space is not
Lipschitz in general (but is almost Lipschitz).

######  Proposition 2.12.

[Bou20] Let $\gamma>0$ and $E$, $F$ and $G$ be three normed vector spaces.

  * •

Let $U$ be a subset of $E$, $f:U\to F$ be a $\textrm{Lip}-\gamma$ map and
$u:F\to G$ a bounded linear map. Then $u\circ f$ is $\textrm{Lip}-\gamma$ and
$\|u\circ f\|_{\textrm{Lip}-\gamma}\leq\|u\|\|f\|_{\textrm{Lip}-\gamma}$.

  * •

Let $v:E\to F$ a bounded linear map and $g:F\to G$ be a $\textrm{Lip}-\gamma$
map. Then $g\circ v$ is $\textrm{Lip}-\gamma$ and $\|g\circ
v\|_{\textrm{Lip}-\gamma}\leq\|g\|_{\textrm{Lip}-\gamma}\max(1,\|v\|^{\gamma})$.

We have similar results for the image of a pair of Lipschitz maps by a
bilinear map.

######  Proposition 2.13.

[Bou20] Let $\gamma>0$. Let $E$, $F$, $G$ and $H$ be normed vector spaces and
$U$ be a subset of $E$. Let $f:U\to F$ and $g:U\to G$ be two
$\textrm{Lip}-\gamma$ maps. Let $B:F\times G\to H$ be a continuous bilinear
map. Then $B(f,g):U\to H$ is $\textrm{Lip}-\gamma$ and there exists a constant
$C_{\gamma}$ such that

$\|B(f,g)\|_{\textrm{Lip}-\gamma}\leq
C_{\gamma}\|B\|\|f\|_{\textrm{Lip}-\gamma}\|g\|_{\textrm{Lip}-\gamma}.$

We will mostly use Proposition 2.13 for composition maps of the type $x\mapsto
g(x)\circ f(x)$, with $f$ and $g$ being Lipschitz maps taking values in
appropriate spaces of linear maps. An example of such result is the following
lemma about the Lipschitz regularity of pullbacks of one-forms, which will be
of use to us in a subsequent section.

######  Lemma 2.14.

Let $\gamma>1$ and $\delta>0$. Let $E$, $F$ and $G$ be normed vector spaces
and $U$ be a subset of $E$. Let $f:U\to F$ be an almost Lip-$\gamma$ map on
domains of size $\delta$ of $U$ and $\alpha:F\to\mathcal{L}(F,G)$ be a
Lip-$(\gamma-1)$ one-form. Then $f^{*}\alpha$ is a Lip-$(\gamma-1)$ one-form
and there exists a constant $C_{\delta,\gamma}$ such that

$\|f^{*}\alpha\|_{\textrm{Lip}-(\gamma-1)}\leq
C_{\delta,\gamma}\|\alpha\|_{\textrm{Lip}-(\gamma-1)}\|f_{*}\|_{\textrm{Lip}-(\gamma-1)}\max(\|f\|_{\delta,\textrm{Lip}-\gamma}^{\gamma-1},1).$

###### Proof.

By the composition theorem 2.9, $\alpha\circ f$ is Lip-$(\gamma-1)$ and there
exists a constant $C_{\delta,\gamma}$ such that

$\|\alpha\circ f\|_{\textrm{Lip}-(\gamma-1)}\leq
C_{\delta,\gamma}\|\alpha\|_{\textrm{Lip}-(\gamma-1)}\max(\|f\|_{\delta,\textrm{Lip}-\gamma}^{\gamma-1},1).$

By the definition of almost Lipschitz regularity, $f_{*}$ is Lip-$(\gamma-1)$.
Considering the bilinear map

$B:(u,v)\in\mathcal{L}(E,F)\times\mathcal{L}(F,G)\longmapsto v\circ
u\in\mathcal{L}(E,G),$

we conclude, by applying Proposition 2.13, that
$f^{*}\alpha=B(f_{*},\alpha\circ f)$ is Lip-$(\gamma-1)$ with the desired
bound on its Lip-$(\gamma-1)$ norm. ∎

In the rest of this paper, Lipschitz diffeomorphisms will be key in
transporting Lipschitz structures.

######  Definition 2.15.

Let $\gamma>0$. Let $E$ and $F$ be two normed vector spaces, $U$ be a subset
of E and $V$ a subset of $F$. A map $f:U\to V$ is said to be a
$\textrm{Lip}-\gamma$ diffeomorphism if $f$ is $\textrm{Lip}-\gamma$ and
bijective and $f^{-1}$ is also $\textrm{Lip}-\gamma$. We define in a similar
way almost $\textrm{Lip}-\gamma$ diffeomorphisms.

One of the central arguments in building the charts and thus the manifold
structure in [Sus73] is the constant rank theorem. To reproduce this argument
in our setting, we will need a Lipschitz version of said theorem. The
following is a simplified statement of the original Lipschitz constant rank
theorem that appeared in [Bou20]. It can be obtained from the original version
by simple arguments of restricting the maps to bounded sets and rescaling.

######  Theorem 2.16 (Constant Rank).

[Bou20] Let $\gamma>1$ and $n,d,p\in\mathbb{N}^{*}$. Let $U$ be an open subset
of $\mathbb{R}^{n}$ and $\varphi:U\to\mathbb{R}^{d}$ be a Lip-$\gamma$ map
(whose derivative at each point is) of rank at most $p$. Let $x_{0}\in U$ such
that $\mathrm{d}\varphi(x_{0})$ is of rank $p$. Then, there exist

  * •

a $\textrm{Lip}-\gamma$ diffeomorphism $f:U_{0}\to B_{n}(0,1)$ defined on an
open subset $U_{0}$ of $\mathbb{R}^{n}$ containing $x_{0}$ and such that
$f(x_{0})=0$ and,

  * •

a $\textrm{Lip}-\gamma$ diffeomorphism $g:W\to B_{d}(0,1)$ defined on an open
subset $W$ of $\mathbb{R}^{d}$ containing $\varphi(x_{0})$ and such that
$g(\varphi(x_{0}))=0$,

such that, for all $(x_{1},\ldots,x_{p})\in B_{n}(0,1)$

$g\circ\varphi\circ
f^{-1}(x_{1},\ldots,x_{n})=(x_{1},\ldots,x_{p},0,\ldots,0)\in B_{d}(0,1),$

i.e., the following diagram commutes

$\begin{array}[]{ccc}U_{0}(\subseteq\mathbb{R}^{n})&\overset{\varphi}{\longrightarrow}&(\varphi(U_{0})\subseteq)W(\subseteq\mathbb{R}^{d})\\\
f\Big{\downarrow}&&\Big{\downarrow}g\\\
B_{n}(0,1)&\overset{\pi_{n,p,d}}{\longrightarrow}&B_{d}(0,1)\\\ \end{array}$

###  2.2. Flows of Lipschitz vector fields

We first recall the notions of integral curves and flows. The existence and
uniqueness results are very classical in the case of $\mathcal{C}^{1}$ vector
fields and trivial for Lip-$1$ ones using for example Picard-Lindelöf’s
theorem.

######  Definition 2.17.

Let $d\in\mathbb{N}^{*}$ and $f$ be a Lip-$1$ vector field on
$\mathbb{R}^{d}$. For $y\in\mathbb{R}^{d}$, we call integral curve of $f$
starting at $y$ the unique solution $\gamma_{y}$ to the differential equation

$\left\\{\begin{array}[]{l}\mathrm{d}\gamma_{y}(t)=f(\gamma_{y}(t))\mathrm{d}t,\quad\forall
t\in\mathbb{R},\\\ \gamma_{y}(0)=y.\\\ \end{array}\right.$

The restriction of an integral curve of $f$ to an arbitrary interval will
still be called an integral curve of $f$. We call flow of the vector field $f$
and denote $\tilde{f}$ the map

$\tilde{f}:(t,y)\in\mathbb{R}\times\mathbb{R}^{d}\longmapsto\gamma_{y}(t)\in\mathbb{R}^{d}.$

We will denote by $\tilde{f}_{t}$, for $t\in\mathbb{R}$, the map
$y\mapsto\tilde{f}(t,y)$; and by $\tilde{f}_{y}$, for $y\in\mathbb{R}^{d}$,
the integral curve $t\mapsto\tilde{f}(t,y)$.

We will need quantitative results on the differentiability and Lipschitz
regularity of flows. The analogous qualitative results for smooth vector
fields are classical.

######  Theorem 2.18.

[Bou20] Let $d\in\mathbb{N}^{*}$ and $\gamma>1$. Let $f$ be a
$\textrm{Lip}-\gamma$ vector field on $\mathbb{R}^{d}$. Then

  1. (1)

$\forall T,r\in\mathbb{R}^{*}_{+},\forall
y\in\mathbb{R}^{d}:\quad\tilde{f}((-T,T)\times B_{d}(y,r))\subseteq
B_{d}(y,r+T\|f\|_{\textrm{Lip}-1})$,

  2. (2)

$\tilde{f}$ is almost $\textrm{Lip}-\gamma$ on $(-T,T)\times\mathbb{R}^{d}$
and there exists a constant $C$ depending only on $T$, $\gamma$ and
$\|f\|_{\textrm{Lip}-\gamma}$ such that
$\|\mathrm{d}\tilde{f}\|_{\textrm{Lip}-(\gamma-1)}\leq C$.

##  3\. Lipschitz geometry

###  3.1. Lipschitz Manifolds

We now recall and build on the definition of Lipschitz manifolds as laid out
in [CLL12].

######  Definition 3.1.

[CLL12] Let $\gamma>0$. Let $n\in\mathbb{N}^{*}$ and let $M$ be an
$n$-topological manifold. Let $I$ be a countable set and, for every $i\in I$,
$U_{i}$ be an open subset of $M$ and $\phi_{i}:M\to\mathbb{R}^{n}$ be a
compactly supported map such that its restriction on $U_{i}$ defines a
homeomorphism. We say that $M$ is a Lipschitz-${\gamma}$ manifold with the
Lipschitz-${\gamma}$ atlas $((U_{i},\phi_{i}))_{i\in I}$ if the following
properties are satisfied

  * •

$(U_{i})_{i\in I}$ is a pre-compact locally finite cover of $M$;

  * •

For every $i\in I$, $\phi_{i}(U_{i})=B_{n}(0,1)$;

  * •

There exists $\delta\in(0,1)$, such that $(U_{i}^{\delta})_{i\in I}$ covers
$M$, where, for every $i\in I$,

$U_{i}^{\delta}=\left({\phi_{i}}_{|U_{i}}\right)^{-1}(B_{n}(0,1-\delta));$

  * •

There exists $L>0$, such that, for every $i,j\in I$,
$\phi_{j}\circ{({\phi_{i}}_{|U_{i}})}^{-1}:B_{n}(0,1)\to\mathbb{R}^{n}$ is
Lipschitz-$\gamma$ and
$\|\phi_{j}\circ{({\phi_{i}}_{|U_{i}})}^{-1}\|_{\textrm{Lip}-\gamma}\leq L$.

Finite dimensional vector spaces are naturally endowed with Lipschitz
structures [CLL12]. In the remaining of this paper, we will explicitly refer
to the structure defined by the following proposition.

######  Proposition 3.2.

Let $\gamma\geq 1$ and $n\in\mathbb{N}^{*}$. Denote by $(e_{1},\ldots,e_{n})$
the canonical basis of $\mathbb{R}^{n}$. Let $\varphi$ be a Lip-$\gamma$
extension on $\mathbb{R}^{n}$ of $\mathrm{Id}_{B_{n}(0,1)}$ with support in
$B_{n}(0,2)$. For $x\in\mathbb{R}^{n}$, let $\varphi_{x}$ be the map
$y\mapsto\varphi(y-x)$. Finally, let $I$ be the countable subset of
$\mathbb{R}^{n}$ defined by

$I=\left\\{\sum_{i=1}^{n}\frac{k_{i}}{\sqrt{n}}e_{i},\;\;k_{1},\ldots,k_{n}\in\mathbb{Z}\right\\}.$

Then $(B_{n}(x,1),\varphi_{x})_{x\in I}$ is a Lip-$\gamma$ atlas on
$\mathbb{R}^{n}$.

######  Remark 3.3.

It is straightforward to endow the Cartesian product of two Lipschitz
manifolds (with the same regularity) with a natural Lipschitz structure (c.f.
[CLL12]).

Lipschitz structures can be canonically transported by homeomorphisms,
irrespective of their regularity.

######  Proposition 3.4.

Let $\gamma\geq 1$. Let $M$ and $N$ be two topological spaces. Let $F:M\to N$
be a homeomorphism. We assume that $M$ is a Lip-$\gamma$ manifold. Then $N$
has a natural Lip-$\gamma$ structure induced by $F$.

###### Proof.

Let $(U_{i},\varphi_{i})_{i\in I}$ be Lip-$\gamma$ atlas on $M$. For each
$i\in I$, we define $V_{i}=F(U_{i})$ and $\psi_{i}=\phi_{i}\circ F^{-1}$. It
is then trivial to show that $(V_{i},\psi_{i})_{i\in I}$ defines a
Lip-$\gamma$ atlas on $N$. ∎

######  Remark 3.5.

In the context of Proposition 3.4, the local representation of the function
$F$ in this induced structure is the identity function, which is obviously
smooth (and Lipschitz when restricted to a bounded domain).

Recall that in the context of Definition 3.1, a single chart does not
constitute a Lipschitz atlas. Consequently, one cannot construct a Lipschitz
structure on the unit ball using only the identity map. In this regard,
Proposition 3.4 provides us with a simple way to show that the unit ball has
indeed a Lipschitz structure. We will see in the next subsection that this
structure is also natural.

######  Corollary 3.6.

For every $\gamma\geq 1$, the unit ball in the Euclidean space has a structure
of a Lip-$\gamma$ manifold.

###### Proof.

It suffices to consider the homeomorphism of Lemma 2.6 in the context of
Proposition 3.4. ∎

###  3.2. Lipschitz regularity on manifolds

We introduce now the notion of Lipschitz regularity of maps and one-forms on
Lipschitz manifolds.

######  Definition 3.7.

[CLL12] Let $\gamma_{0}\geq\gamma>0$ and $n\in\mathbb{N}^{*}$. Let $M$ be a
Lip-$\gamma_{0}$ $n$-dimensional manifold and $((U_{i},\phi_{i}))_{i\in I}$ be
a Lip-$\gamma_{0}$ atlas on $M$. Let $V$ be a Banach space.

  * •

We say that a map $f:M\to V$ is Lip-$\gamma$ if there exists a non-negative
constant $C$, such that, for every $i\in I$,
$f\circ{({\phi_{i}}_{|U_{i}})}^{-1}:B_{n}(0,1)\to V$ is Lip-$\gamma$ with a
norm less than or equal to $C$. The smallest such constant $C$ is then called
the Lip-$\gamma$ norm of $f$ and is denoted by $\|f\|_{\textrm{Lip}-\gamma}$.

  * •

Assume that $\gamma\in(0,\gamma_{0}-1]$. We say that a $V$-valued one-form
$\alpha$ is Lip-$\gamma$ if there exists a non-negative constant $C$, such
that, for every $i\in I$ the pullback

$({\phi_{i}}_{|U_{i}}^{-1})^{*}\alpha:B_{n}(0,1)\to\mathcal{L}(\mathbb{R}^{n},V)$

is a Lip-$\gamma$ one-form with a norm less than or equal to $C$. The smallest
such constant $C$ is then called the Lip-$\gamma$ norm of $\alpha$ and is
denoted by $\|\alpha\|_{\textrm{Lip}-\gamma}$.

######  Remark 3.8.

It is easy to check that the classical and manifold notions of Lipschitz
regularity are the same on finite-dimensional vector spaces (endowed with
their natural Lipschitz structure of Proposition 3.2).

######  Remark 3.9.

It is straightforward to generalise the above definition of Lipschitz
regularity on manifolds to maps and one-forms that are not globally defined.
In this case, one checks only the Lipschitz regularity (of the local
representations) on (images of) the intersections of the domain of definition
with the chart domains (c.f. [CLL12]).

The following lemma is going to be of technical use to us later.

######  Lemma 3.10.

Let $p\in\mathbb{N}^{*}$ and $\gamma\geq 1$. Let $V$ be a subset of a normed
vector space. We assume that there exists a homeomorphism $F:\mathbb{R}^{p}\to
V$ that is Lip-$\gamma$. Then the inverses of the chart maps in the
$F$-induced Lip-$\gamma$ atlas on $V$ are Lip-$\gamma$ in the classical sense
with a uniform upper bound on their Lip-$\gamma$ norms.

###### Proof.

If $(B_{p}(x,1),\phi_{x})_{x\in I}$ denotes a Lip-$\gamma$ atlas on
$\mathbb{R}^{p}$ (Proposition 3.2), then the corresponding atlas on $V$ is
given by $(F(B_{p}(x,1)),\phi_{x}\circ F^{-1})_{x\in I}$. Note that the maps
$(F\circ\phi_{x}^{-1})_{x\in I}$ are the local representations of the
Lip-$\gamma$ map $F$ on $\mathbb{R}^{p}$. Therefore, they are Lip-$\gamma$
with norms bounded uniformly from above. ∎

Next, we show that Lipschitz regularity is preserved on manifolds endowed with
a Lipschitz structure induced by a Lipschitz homeomorphism.

######  Lemma 3.11.

Let $p\in\mathbb{N}^{*}$ and $\gamma\geq 1$. Let $V$ be a subset of a normed
vector space and $f$ be a Banach-space valued Lip-$\gamma$ map defined on $V$.
We assume that there exists a homeomorphism $F:\mathbb{R}^{p}\to V$ that is
Lip-$\gamma$. Then $f$ is Lip-$\gamma$ on $V$ in the manifold sense in the
$F$-induced Lipschitz structure.

###### Proof.

Consider the Lip-$\gamma$ atlas ${(U_{x},\phi_{x})}_{x\in I}$ on
$\mathbb{R}^{p}$ constructed in Proposition 3.2. Since $f\circ F$ is
Lip-$\gamma$ on $\mathbb{R}^{p}$ (in the classical sense and hence also in the
manifold sense), then there exists a constant $C$ such that, for all $x\in I$,
$f\circ F\circ\phi_{x}^{-1}$ is Lip-$\gamma$ with a norm bounded from above by
$C$. This concludes the proof (by the definitions of Lipschitz regularity on
manifolds and the $F$-induced Lipschitz structure on $V$). ∎

######  Remark 3.12.

By applying Lemma 3.11 in the case of the unit ball, a map defined on the unit
ball that is Lipschitz in the classical sense is also Lipschitz in the
manifold sense when the unit ball is endowed with the Lipschitz structure of
Corollary 3.6.

Like in the Euclidean setting (Whitney’s extension theorems, e.g. [Ste70]),
Lipschitz maps defined on compact subsets of Lipschitz manifolds and with
values in a finite dimensional space can be extended.

######  Theorem 3.13.

[CLL12] Let $\gamma_{0}\geq\gamma>0$ such that $\gamma_{0}\geq 1$. Let $M$ be
a Lip-$\gamma_{0}$ manifold and $K$ be a precompact subset of $M$. Let $f$ be
a $V$-valued Lip-$\gamma$ map defined on $K$, where $V$ is a finite
dimensional space. Then we can extend $f$ to a compactly supported
Lip-$\gamma$ map on $M$ with a Lip-$\gamma$ norm depending only on
$\|f\|_{\textrm{Lip}-\gamma}$, $M$, $K$, $\gamma_{0}$ and $\gamma$.

######  Remark 3.14.

In Theorem 3.13, the dependence on $M$ is through its Lipschitz structure (and
the subsequent choice of a partition of unity) while the dependence on $K$ is
through the set of chart domains that intersect $K$.

###  3.3. Lipschitz transports

The notion of transport will be key in translating the definition of solutions
to RDEs from the classical to the manifold setting.

######  Definition 3.15.

Let $M$ and $N$ be two Lip-$\gamma$ manifolds of dimensions $n$ and $d$
respectively. We call transport from $M$ to $N$ a map $g$ defined on $M\times
N$ such that, for all $x\in M$,
$g(x,.)\in\mathcal{L}(T_{x}M,\mathcal{\tau}(N))$. For such a map and for
$(U,\phi)$ and $(V,\psi)$ charts in $M$ and $N$ respectively, we define
$g_{(\phi,\psi)}$ by the following. For all $(x,y)\in\phi(U)\times\psi(V)$,
$v\in\mathbb{R}^{n}$,

$g_{(\phi,\psi)}(x,y)(v)=\psi_{*}(\psi^{-1}(y))\left(g(\phi^{-1}(x),\psi^{-1}(y))(\phi^{-1})_{*}(x)(v)\right)\in\mathbb{R}^{d}.$

We say that $g$ is Lip-$\gamma$ if there exists a constant $C$ such that for
all charts $(U,\phi)$ and $(V,\psi)$ in $M$ and $N$ respectively, the map
$g_{(\phi,\psi)}$ is Lip-$\gamma$ with norm less than $C$.

######  Example.

If $E$ and $F$ are two vector spaces and $f:E\times F\to\mathcal{L}(E,F)$ is a
map, then there exists a natural transport from $E$ to $F$ induced by $f$.

######  Remark 3.16.

We will also sometimes consider the extension of Definition 3.15 to transports
that are not globally defined.

On the one hand, we will see later that RDEs (and by extension ODEs and flow
equations) on a Lipschitz manifold admit a unique global solution when the
vector fields define a Lipschitz transport map (with enough Lipschitz
regularity in the case of RDEs). On the other hand, by Lemma 3.11 and Remark
3.12, vector fields on the unit ball that are Lipschitz in the classical sense
are also Lipschitz in the manifold sense. However, it is an elementary
exercise to construct smooth vector fields so that their flows starting from
the unit ball leave the unit ball in a finite time. The next lemma reconciles
these two arguments and sheds more light on Lipschitz geometry: Lipschitz
vector fields do not necessarily define Lipschitz transports on the entire
unit ball.

######  Lemma 3.17.

Let $n$ and $p$ be two positive integers. Let
$f:B_{p}(0,1)\to\mathcal{L}(\mathbb{R}^{n},\mathbb{R}^{p})$ be a Lip-$\gamma$
map. Define a transport $g$ from $\mathbb{R}^{n}$ to $B_{p}(0,1)$ by

$g(x,y):v_{x}\in\mathbb{R}^{n}\longmapsto
f(y)(v_{x})\in\mathbb{R}^{p},\;\;\;\textrm{ for all
}(x,y)\in\mathbb{R}^{n}\times B_{p}(0,1).$

Then :

  1. (1)

$g$ is Lip-$\gamma$ on $\mathbb{R}^{n}\times B_{p}(0,\delta)$, for every
$\delta\in(0,1)$.

  2. (2)

if $f$ is compactly supported, then $g$ is Lip-$\gamma$ on
$\mathbb{R}^{n}\times B_{p}(0,1)$.

###### Proof.

Consider the Lip-$(\gamma+1)$ atlases ${(U_{x},\phi_{x})}_{x\in I}$ and
${(\widetilde{U}_{z},\varphi_{z})}_{z\in J}$ on $\mathbb{R}^{n}$ and
$\mathbb{R}^{p}$ respectively (as in Proposition 3.2) and the Lipschitz
bijection $F:\mathbb{R}^{p}\to B_{p}(0,1)$ of Lemma 2.6. For $z\in J$, we
define $y_{z}=F(z)$, $V_{y_{z}}=F(\widetilde{U}_{z})$ and
$\psi_{y_{z}}=\varphi_{z}\circ F^{-1}$. Let $\delta\in(0,1)$. As the
collection $\\{B_{p}(z,2)\\}_{z\in J}$ is locally finite and that
$F^{-1}\left(B_{p}(0,\delta)\right)$ is precompact, then there exists a finite
subset $J_{\delta}\subset J$ such that if $z\notin J_{\delta}$ then
$B_{p}(z,2)\cap F^{-1}\left(B_{p}(0,\delta)\right)=\varnothing$ and
consequently $V_{y_{z}}\cap B_{p}(0,\delta)=\varnothing$. Therefore, we
restrict ourselves to the case $z\in J_{\delta}$. Let us finally note that as
$\cup_{z\in J_{\delta}}F(\widetilde{U}_{z})$ is precompact, then there exists
$M_{\delta}\in(0,1)$ such that $\cup_{z\in
J_{\delta}}F(\widetilde{U}_{z})\subset B(0,M_{\delta})$. Let $x\in I$ and
$z\in J_{\delta}$. By Definition 3.15, we need to study the regularity of the
map

$\begin{array}[]{rccl}g_{x,{y_{z}}}:&B_{n}(0,1)\times\psi_{y_{z}}\left(B_{p}(0,\delta)\cap
V_{y_{z}}\right)&\longrightarrow&\mathcal{L}(\mathbb{R}^{n},\mathbb{R}^{p})\\\
&(u,v)&\longmapsto&(\psi_{y_{z}})_{*}(\psi_{y_{z}}^{-1}(v))\circ
f(\psi_{y_{z}}^{-1}(v))\circ(\phi_{x}^{-1})_{*}(u)\\\ \end{array}$

In anticipation of the second part of the proof, we will study instead the
regularity of the (unrestricted) map

$g_{x,{y_{z}}}:(u,v)\in B_{n}(0,1)\times
B_{p}(0,1)\longmapsto(\psi_{y_{z}})_{*}(\psi_{y_{z}}^{-1}(v))\circ
f(\psi_{y_{z}}^{-1}(v))\circ(\phi_{x}^{-1})_{*}(u)\in\mathcal{L}(\mathbb{R}^{n},\mathbb{R}^{p}).$

By Lemma 3.11, $f\circ\psi_{y_{z}}^{-1}$ is Lip-$\gamma$ with a constant that
can be bounded from above independently of ${y_{z}}$. Therefore, by
Proposition 2.13, the map

$(u,v)\in B_{n}(0,1)\times B_{p}(0,1)\longmapsto
f(\psi_{y_{z}}^{-1}(v))\circ(\phi_{x}^{-1})_{*}(u)\in\mathcal{L}(\mathbb{R}^{n},\mathbb{R}^{p})$

is Lip-$\gamma$ with a norm that is independent of both $x$ and ${y_{z}}$. By
definition of the chart map $\psi_{y_{z}}$, one has

$\forall v\in
B_{p}(0,1):\quad(\psi_{y_{z}})_{*}(\psi_{y_{z}}^{-1}(v))=\mathrm{d}(\varphi_{z}\circ
F^{-1})\circ(F\circ\varphi_{z}^{-1})(v).$

Since $F\circ\varphi_{z}^{-1}:B_{p}(0,1)\to B_{p}(0,M_{\delta})$ is
Lip-$\gamma$ and $\varphi_{z}\circ
F^{-1}:B_{p}(0,M_{\delta})\to\mathbb{R}^{p}$ is Lip-$(\gamma+1)$ (with
Lipschitz constant depending on $\delta$) then
$v\mapsto(\psi_{y_{z}})_{*}(\psi_{y_{z}}^{-1}(v))$ is Lip-$\gamma$. Using
again Proposition 2.13, this implies that $g_{x,{y_{z}}}$ is indeed
Lip-$\gamma$ (with Lipschitz norm bounded form above independently of $x$ and
${y_{z}}$). This concludes the first part of the proof.
Assume now that $f$ is compactly supported. Let $\delta\in(0,1)$ such that $f$
is supported in $B_{p}(0,\delta)$. Let $x\in I$. By the above, $g_{x,y_{z}}$
is Lip-$\gamma$ for all $z\in J_{\delta}$ with a Lipschitz norm bounded from
above by a constant that only depends on $\delta$. For $z\notin J_{\delta}$,
$g_{x,y_{z}}$ is the null connection, therefore also Lip-$\gamma$. Hence $g$
is Lip-$\gamma$. ∎

Next, we show that the Lip-$\gamma^{\prime}$ regularity of connections is
stable by Lip-$\gamma$ diffeomorphisms for $\gamma^{\prime}\leq\gamma-1$. To
reduce the assumptions and make the construction more intuitive, we will
restrict ourselves to the following specific situation that is relevant to our
main result.

######  Corollary 3.18.

Let $n$ and $d$ be two positive integers and $\gamma>1$. Let $E$ be a normed
vector space and $V$ be a subset of $E$. Let $F:B_{p}(0,1)\to V$ be a
Lip-$\gamma$ diffeomorphism and endow $V$ with the Lip-$\gamma$ structure
induced by $F$. Let $f:V\to\mathcal{L}(\mathbb{R}^{n},E)$ be a compactly
supported Lip-$\gamma$ map such that for all $v\in\mathbb{R}^{n}$,
$f(.)(v)\in\tau(V)$. Then the transport $g$ from $\mathbb{R}^{n}$ to $V$
induced by $f$ is Lip-$(\gamma-1)$.

###### Proof.

Consider the usual Lip-$(\gamma+1)$ atlas ${(U_{x},\phi_{x})}_{x\in I}$ and
${(\widetilde{U}_{z},\varphi_{z})}_{z\in J}$ on $\mathbb{R}^{n}$ and
$B_{p}(0,1)$ respectively. For $z\in J$, we define $y_{z}=F(z)$,
$V_{y_{z}}=F(\widetilde{U}_{z})$ and $\psi_{y_{z}}=\varphi_{z}\circ F^{-1}$.
Let $x\in I$ and $z\in J$. By Definition 3.15, we need to study the Lipschitz
regularity of the map

$\begin{array}[]{rccl}g_{x,{y_{z}}}:&B_{n}(0,1)\times
B_{p}(0,1)&\longrightarrow&\mathcal{L}(\mathbb{R}^{n},\mathbb{R}^{p})\\\
&(u,v)&\longmapsto&(\psi_{y_{z}})_{*}(\psi_{y_{z}}^{-1}(v))\circ
f(\psi_{y_{z}}^{-1}(v))\circ(\phi_{x}^{-1})_{*}(u)\\\ \end{array}$

Define the following compactly supported Lip-$(\gamma-1)$ on $B_{p}(0,1)$

$\widehat{f}:z\in B_{p}(0,1)\longmapsto\left(v\mapsto(F^{-1})_{*}(F(z))(f\circ
F)(z)(v)\right)\in\mathcal{L}(\mathbb{R}^{n},\mathbb{R}^{p}),$

and denote by $\widehat{g}$ its associated transport from $\mathbb{R}^{n}$ to
$B_{p}(0,1)$. Note then that $g_{x,{y_{z}}}=\widehat{g}_{x,z}$. By Lemma 3.17,
we conclude that $g$ is Lip-$(\gamma-1)$. ∎

##  4\. Local Lipschitz parametrisation of orbits of Lipschitz vector fields

###  4.1. Orbits and distributions

In this subsection, we will introduce the notion of orbits and their
“candidate” tangent spaces. We will in most cases follow the terminology used
in [Sus73] and will restrict the exposition to the minimum necessary to obtain
our results.

######  Definition 4.1.

Let $d\in\mathbb{N}^{*}$. Let $\mathcal{D}$ be a family of Lip-$1$ vector
fields on $\mathbb{R}^{d}$. Let $n\in\mathbb{N}^{*}$. Let
$f^{1},\ldots,f^{n}\in\mathcal{D}$ and $t_{1},\ldots,t_{n}\in\mathbb{R}$. The
$\mathbb{R}^{d}$-valued map on $\mathbb{R}^{d}$ defined by
$\tilde{f}^{n}_{t_{n}}\circ\cdots\circ\tilde{f}^{1}_{t_{1}}$ is called a
$\mathcal{D}$-diffeomorphism.

Orbits of vector fields will be a central notion in the rest of this paper.

######  Definition 4.2.

Let $d\in\mathbb{N}^{*}$. Let $\mathcal{D}$ be a family of Lip-$1$ vector
fields on $\mathbb{R}^{d}$ and $y\in\mathbb{R}^{d}$. We call
$\mathcal{D}$-orbit at $y$ and denote by $\mathbb{L}_{y}(\mathcal{D})$ the set
of images of $y$ by all $\mathcal{D}$-diffeomorphisms.

In the sequel, we will use the following notations.

######  Notation.

Let $d\in\mathbb{N}^{*}$ and $\mathcal{D}$ be a family of Lip-$1$ vector
fields on $\mathbb{R}^{d}$. Following similar notational conventions as
before, for $n\in\mathbb{N}^{*}$ and
$\xi=(f^{1},\ldots,f^{n})\in\mathcal{D}^{n}$,

  * •

we denote by $\tilde{\xi}$ the map

$\tilde{\xi}:\mathbb{R}^{n}\times\mathbb{R}^{d}\longrightarrow\mathbb{R}^{d}\quad,\quad(t_{1},\ldots,t_{n},x)\longmapsto\tilde{f}^{n}_{t_{n}}\circ\cdots\circ\tilde{f}^{1}_{t_{1}}(x),$

  * •

for $y\in\mathbb{R}^{d}$, $\tilde{\xi}_{y}$ denotes the map
$(t_{1},\ldots,t_{n})\mapsto\tilde{\xi}(t_{1},\ldots,t_{n},y)$,

  * •

for $s=(s_{1},\ldots,s_{n})$ we denote by $\tilde{\xi}_{s}$ the
$\mathcal{D}$-diffeomorphism $x\mapsto\tilde{\xi}(s_{1},\ldots,s_{n},x)$.

More generally, for $p\in\mathbb{N}^{*}$,
$n_{1},\ldots,n_{p}\in\mathbb{N}^{*}$,
$(\xi^{1},\ldots,\xi^{p})\in\mathcal{D}^{n_{1}}\times\cdots\times\mathcal{D}^{n_{p}}$
and, if $\eta$ denotes the ordered collection $(\xi^{1},\ldots,\xi^{p})$, then
$\tilde{\eta}$ is the map given by the identity

$\forall(t_{1},\ldots,t_{p})\in\mathbb{R}^{n_{1}}\times\cdots\times\mathbb{R}^{n_{p}},\forall
x\in\mathbb{R}^{d}:\quad\tilde{\eta}(t_{1},\ldots,t_{p},x)=\tilde{\xi}^{p}_{t_{p}}\circ\cdots\circ\tilde{\xi}^{1}_{{t_{1}}}(x).$

For $y\in\mathbb{R}^{d}$, $\tilde{\eta}_{y}$ is the map $\tilde{\eta}(.,y)$
and for $s=({s_{1}},\ldots,{s_{p}})$, $\tilde{\eta}_{{s}}$ is the map
$\tilde{\xi}^{p}_{{s_{p}}}\circ\cdots\circ\tilde{\xi}^{1}_{{s_{1}}}$.

A key idea in Sussmann’s work [Sus73] was to identify the “candidate” tangent
space to orbits. This is best carried through the notion of distributions.

######  Definition 4.3.

Let $M$ be a $\mathcal{C}^{1}$-manifold. A distribution $\Delta$ on $M$ is a
mapping that associates to every point $p\in M$ a linear subspace of $T_{p}M$
which is denoted $\Delta(p)$.

######  Definition 4.4.

Let $d\in\mathbb{N}^{*}$ and $\gamma\geq 1$. Let $\mathcal{D}$ be a family of
Lip-$\gamma$ vector fields on $\mathbb{R}^{d}$. Let $\Delta$ and $\Gamma$ be
two distributions on $\mathbb{R}^{d}$.

  * •

We say that $\mathcal{D}$ spans $\Delta$ if, for every $p\in\mathbb{R}^{d}$,
one has

$\Delta(p)=\textrm{span}\\{f(p),\;f\in\mathcal{D}\\}.$

  * •

If $\gamma>1$, we say that $\Delta$ is $\mathcal{D}$-invariant if, for every
$\mathcal{D}$-diffeomorphism $g$ and $y\in\mathbb{R}^{d}$, $g_{*}(y)$ induces
a diffeomorphism from $\Delta(y)$ onto $\Delta(g(y))$.

  * •

We say that $\Delta$ contains $\Gamma$ and write $\Gamma\subseteq\Delta$ if,
for every $y\in\mathbb{R}^{d}$, $\Gamma(y)\subseteq\Delta(y)$.

One is then able to identify the candidate tangent space to a
$\mathcal{D}$-orbit as the collection of tangent vectors spanned by the vector
fields of $\mathcal{D}$ and the transport of these via
$\mathcal{D}$-diffeomorphisms.

######  Theorem 4.5.

[Sus73] Let $d\in\mathbb{N}^{*}$ and $\gamma\geq 1$. Let $\mathcal{D}$ be a
family of Lip-$\gamma$ vector fields on $\mathbb{R}^{d}$.

  1. (1)

There exists a unique natural distribution spanned by $\mathcal{D}$. This will
be denoted by $\mathcal{L}(\mathcal{D})$.

  2. (2)

Assume that $\gamma>1$. Then there exists a unique smallest distribution
$P_{\mathcal{D}}$ containing $\mathcal{L}(\mathcal{D})$ that is
$\mathcal{D}$-invariant. More explicitly, for every $y\in\mathbb{R}^{d}$,
$P_{\mathcal{D}}(y)$ is given by

$P_{\mathcal{D}}(y):=\left\\{g_{*}(v)\;:\;g\;\;\mathcal{D}-\textrm{diffeomorphism},v\in\mathcal{L}(\mathcal{D})(g^{-1}(y))\right\\}.$

As a direct consequence of the definition of invariance, one has the
following.

######  Corollary 4.6.

[Sus73] Let $d\in\mathbb{N}^{*}$ and $\gamma>1$. Let $\mathcal{D}$ be a family
of Lip-$\gamma$ vector fields on $\mathbb{R}^{d}$. Then the dimension of
$P_{\mathcal{D}}$ is constant on $\mathcal{D}$-orbits, i.e. if $x$ and $y$
belong to the same $\mathcal{D}$-orbit then
$\mathrm{dim}\left(P_{\mathcal{D}}(x)\right)=\mathrm{dim}\left(P_{\mathcal{D}}(y)\right)$.

While $P_{\mathcal{D}}$ is, by construction, rich enough to contain all the
tangent vectors generated by integral curves of the vector fields in
$\mathcal{D}$ and their compositions, it can still be precisely generated by
these. This is the content of the theorem below, which is the summary of the
three main lemmas in [Sus73]. The only difference with the original statements
is the Lipschitz regularity condition instead of the $\mathcal{C}^{\infty}$
assumption, which does not impact the validity of the original proofs.

######  Theorem 4.7.

[Sus73] Let $d\in\mathbb{N}^{*}$ and $\gamma>1$. Let $\mathcal{D}$ be a family
of Lip-$\gamma$ vector fields on $\mathbb{R}^{d}$.

  1. (1)

For $n\in\mathbb{N}^{*}$, $\xi\in\mathcal{D}^{n}$, $x,y\in\mathbb{R}^{d}$ and
$T\in\mathbb{R}^{n}$ such that $\tilde{\xi}(T,x)=y$, one has

$(\tilde{\xi}_{x})_{*}(T)(\mathbb{R}^{n})\subseteq P_{\mathcal{D}}(y).$

  2. (2)

Conversely, for all $y\in\mathbb{R}^{d}$, there exist $x\in\mathbb{R}^{d}$,
$n\in\mathbb{N}^{*}$, $\xi\in\mathcal{D}^{n}$ and $T\in\mathbb{R}^{n}$ such
that

$\tilde{\xi}(T,x)=y\textrm{ and
}(\tilde{\xi}_{x})_{*}(T)(\mathbb{R}^{n})=P_{\mathcal{D}}(y).$

###  4.2. A local Lipschitz structure on orbits

We obtain now one of the main results in this paper by endowing neighbourhoods
of points in their orbits with Lipschitz structures.

######  Theorem 4.8.

Let $d\in\mathbb{N}^{*}$ and $\gamma>1$. Let $\mathcal{D}$ be a family of
Lip-$\gamma$ vector fields on $\mathbb{R}^{d}$. Let $y\in\mathbb{R}^{d}$. Then
there exist a subset $V_{y}$ of $\mathbb{R}^{d}$ containing $y$ and contained
in $\mathbb{L}_{y}(\mathcal{D})$ and a Lip-$\gamma$ diffeomorphism
$\Psi_{y}:V_{y}\to B_{p}(0,1)$ centred at $y$, where
$p=\textrm{dim}(P_{\mathcal{D}}(y))$. Moreover, $V_{y}$ has a Lip-$\gamma$
structure induced by $\Psi_{y}^{-1}$ with its tangent space given precisely by
the distribution $P_{\mathcal{D}}$.

###### Proof.

By Theorem 4.7, let $n\in\mathbb{N}^{*}$, $\xi\in\mathcal{D}^{n}$,
$T\in\mathbb{R}^{n}$ and $x\in\mathbb{R}^{d}$ such that $\tilde{\xi}(T,x)=y$
and $(\tilde{\xi}_{x})_{*}(T)(\mathbb{R}^{n})=P_{\mathcal{D}}(y)$. By the same
theorem, for every $t\in\mathbb{R}^{n}$,

$(\tilde{\xi}_{x})_{*}({t})(\mathbb{R}^{n})\subseteq
P_{\mathcal{D}}(\tilde{\xi}_{x}({t})).$

Since $\mathrm{dim}(P_{\mathcal{D}}(\tilde{\xi}_{x}({t})))=p$, then
$\mathrm{rank}(\tilde{\xi}_{x}({t}))_{*}\leq p$. Therefore the map
$(\tilde{\xi}_{x})_{*}$ is of maximal rank $p$ at $T$. Moreover, by Theorems
2.18 and 2.9, $\tilde{\xi}_{x}$ defines a Lip-$\gamma$ map on every open
bounded set of $\mathbb{R}^{n}$. By Theorem 2.16, there exist then an open
subset $U$ of $\mathbb{R}^{n}$ containing $T$, an open subset $V$ of
$\mathbb{R}^{d}$ containing $y$, Lip-$\gamma$ diffeomorphisms $f:U\to
B_{n}(0,1)$ and $g:V\to B_{d}(0,1)$ (centred at $T$ and $y$ respectively) such
that the following diagram commutes

$\begin{array}[]{ccc}U(\subseteq\mathbb{R}^{n})&\overset{\tilde{\xi}_{x}}{\longrightarrow}&V(\subseteq\mathbb{R}^{d})\\\
f\Big{\downarrow}&&\Big{\downarrow}g\\\
B_{n}(0,1)&\overset{\pi_{n,p,d}}{\longrightarrow}&B_{d}(0,1)\\\ \end{array}$

In particular, for all $t\in U$, $({\tilde{\xi}_{x}})_{*}(t)$ is of rank $p$.
Define $V_{y}=\tilde{\xi}_{x}(U)$ and the map $\Psi_{y}:V_{y}\to B_{p}(0,1)$
as the composition $\Psi_{y}:=\pi_{d,p,p}\circ g$. Then $\Psi_{y}$ is
invertible and $\Psi_{y}^{-1}=g^{-1}\circ\pi_{p,p,d}$. Hence, by Proposition
2.12, $\Psi_{y}$ is a Lip-$\gamma$ diffeomorphism. This induces a natural
$p$-dimensional Lip-$\gamma$ structure on $V_{y}$ by Proposition 3.4 and
Corollary 3.6. For $t\in U$, decomposing the invertible map $\Psi_{y}^{-1}$ as
$\Psi_{y}^{-1}=\tilde{\xi}_{x}\circ f^{-1}\circ\pi_{p,p,n}$, one gets

$T_{\tilde{\xi}(t,x)}V_{y}\subseteq\tilde{\xi}_{x}(t)_{*}(\mathbb{R}^{n})\subseteq
P_{\mathcal{D}}(\tilde{\xi}(t,x)).$

As $\tilde{\xi}(t,x)\in\mathbb{L}_{y}(\mathcal{D})$, then
$P_{\mathcal{D}}(\tilde{\xi}(t,x))$ is of the same dimension as
$P_{\mathcal{D}}(y)$, i.e. $p$, the same as $T_{\tilde{\xi}(t,x)}V_{y}$.
Therefore, we conclude that
$T_{\tilde{\xi}(t,x)}V_{y}=P_{\mathcal{D}}(\tilde{\xi}(t,x))$. ∎

######  Remark 4.9.

Using Theorem 4.8, one may build a locally Lip-$\gamma$ structure on
$\mathbb{L}_{y}(\mathcal{D})$ as defined in [BL15]. However, as we didn’t show
the existence of solutions to rough differential equations in such setting, we
will restrict ourselves to local arguments using the above proposition.

##  5\. Rough Calculus on manifolds

###  5.1. Rough paths on manifolds

In order to carry out our main argument, we need to generalise the notion of
rough paths to manifolds. As rough calculus on manifolds is less known among
rough paths specialists, and in view of the multiple manifold approaches
available in the literature, we will briefly recall in this section the main
definitions and results that we will subsequently use, most of which are due
to [CLL12].

We fix a notation for the integral of a rough path, when endowed with a
starting point, along a one-form.

######  Notation.

Let $\gamma>p\geq 1$ and $T\geq 0$. Let $E$ and $F$ be two Banach spaces. Let
$x\in E$ and $\mathbf{X}$ be a geometric $p$-rough path on $E$ defined over
the interval $[0,T]$. Let $\alpha:E\to\mathcal{L}(E,F)$ be a Lip-$(\gamma-1)$
one-form. We denote by $\int\alpha(x,\mathbf{X})\mathrm{d}\mathbf{X}$ the
integral of the rough path $\mathbf{X}$ along $\alpha$ when endowed with $x$
as a starting point, i.e. the trace of $\mathbf{X}$ is understood to be given
by the path $t\mapsto x+\mathbf{X}^{1}_{0,t}$.

In the absence of the linear structure, rough paths on Lipschitz manifolds can
be defined using a functional approach.

######  Definition 5.1.

[CLL12] Let $\gamma_{0},p\in\mathbb{R}$ such that $\gamma_{0}>p\geq 1$ and
$T\geq 0$. Let $M$ be a $\textrm{Lip}-\gamma_{0}$ manifold and $x\in M$.
$\mathbb{X}$ is a (geometric) $p$-rough path over $[0,T]$ on $M$ starting at
$x$ if, for every $\gamma\in\mathbb{R}$ such that $\gamma_{0}\geq\gamma>p$ and
every Banach space $E$, the following conditions are satisfied:

  1. (1)

$\mathbb{X}$ maps compactly supported $\textrm{Lip}-(\gamma-1)$ $E$-valued
one-forms on $M$ to $E$-valued geometric $p$-rough paths (in the classical
sense).

  2. (2)

There exists a control $\omega$ such that for every compactly supported
$E$-valued $\textrm{Lip}-(\gamma-1)$ one-form $\alpha$ on $M$,
$\mathbb{X}(\alpha)$ is controlled by
$\|\alpha\|_{\textrm{Lip}-(\gamma-1)}\omega$, i.e.

$\forall(s,t)\in\Delta_{[0,T]},\forall
i\in[\\![1,[p]]\\!]:\quad\|\mathbb{X}(\alpha)_{(s,t)}^{i}\|\leq\frac{(\|\alpha\|_{\textrm{Lip}-(\gamma-1)}\omega(s,t))^{i/p}}{\beta_{p}(i/p)!}.$

  3. (3)

(Chain rule) For every Banach space $F$ and every compactly supported
$\textrm{Lip}-\gamma$ map $\psi:M\to F$ and every $E$-valued
$\textrm{Lip}-(\gamma-1)$ one-form $\alpha$ on $F$ we have

$\mathbb{X}(\psi^{*}\alpha)=\int{\alpha(\psi(x),\mathbb{X}(\psi_{*}))\mathrm{d}\mathbb{X}(\psi_{*})}.$

Naturally, the classical and manifold notions of rough paths on finite-
dimensional vector spaces (endowed with their natural Lipschitz structures)
are the same. More explicitly, given a geometric $p$-rough path $\mathbf{X}$
with a starting point $x$ in $\mathbb{R}^{d}$, one can define a $p$-rough path
$\mathbb{X}$ on $\mathbb{R}^{d}$ in the manifold sense in the following way.
For every compactly supported Banach space-valued $\textrm{Lip}-(\gamma-1)$
one-form $\alpha$ on $\mathbb{R}^{d}$

$\mathbb{X}(\alpha):=\int\alpha(x,\mathbf{X})\mathrm{d}\mathbf{X}.$

Conversely, the functional $\mathbb{X}$ completely determines the classical
rough path $\mathbf{X}$. The proof relies on all three axioms of Definition
5.1 and can be found in [CLL12].

Next, we adapt the notion of support of rough paths.

######  Definition 5.2.

[CLL12] Let $\gamma_{0}>p\geq 1$ and $M$ be a $\textrm{Lip}-\gamma_{0}$
manifold. Let $\mathbb{X}$ be a $p$-rough path on $M$ with starting point $x$.

  1. (1)

Given an open subset $U$ of $M$, we say that $\mathbb{X}$ misses $U$ if, for
every $\gamma\in(p,\gamma_{0}]$ and every Banach space-valued compactly
supported Lip-$(\gamma-1)$ one-form $\alpha$ on $M$ with support in $U$, we
have $\mathbb{X}(\alpha)=0$.

  2. (2)

We define the support of the rough path $\mathbb{X}$ as follows

$\mathrm{supp}(\mathbb{X}):=\\{x\\}\bigcup\left(M-\bigcup_{\mathbb{X}\textrm{
misses }U}U\right).$

Once again, the classical and manifold notions of support of a rough path
coincide. In this case, the manifold support of a classical rough path (with a
starting point) viewed as a manifold rough path is given by the image of its
trace.
More generally, this manifold notion of support generalises several key
properties of the classical supports of rough paths. For instance, the support
of a rough path (on a manifold) is a compact set [CLL12]. Moreover, the
integrals of a rough path along two one-forms that agree on an open set
containing its support are the same.

######  Proposition 5.3.

[CLL12] Let $\gamma_{0}\geq\gamma>p\geq 1$. Let $M$ be a Lip-$\gamma_{0}$
manifold, $U$ be an open subset of $M$ and $E$ be a Banach space. Let $\alpha$
and $\beta$ be two $E$-valued compactly supported Lip-$(\gamma-1)$ one-forms
such that $\alpha_{|U}=\beta_{|U}$. Let $\mathbb{X}$ be a $p$-rough path in
$M$ such that $\mathrm{supp}(\mathbb{X})\subset U$, then
$\mathbb{X}(\alpha)=\mathbb{X}(\beta)$.

The rigid nature of the axioms in Definition 3.1 has the benefit of ensuring
that the restrictions of a rough path to shorter intervals are supported in
the domains of the Lipschitz charts; the lengths of these intervals depending
only on the control of the rough path.

######  Theorem 5.4.

[CLL12] Let $\gamma_{0}>p\geq 1$ and $\omega$ be a control. Let $M$ be a
$\textrm{Lip}-\gamma_{0}$ manifold. Then there exists $t_{0}>0$ such that, for
every $p$-rough path $\mathbb{X}$ on $M$ controlled by $\omega$, there exists
a chart domain $U$ such that $\mathrm{supp}(\mathbb{X}_{|[0,t_{0}]})\subset
U$.

######  Remark 5.5.

With the notations of Definition 3.1 and Theorem 5.4, the chart $(\phi,U)$ can
be chosen to be any such chart such that $B(\phi(x),\delta)\subset B(0,1)$,
$x$ being the starting point of $\mathbb{X}$. The time $t_{0}$ depends only
the constants involved in the definition of the Lipschitz atlas on $M$ and the
control $\omega$.

###  5.2. RDEs on manifolds

In this subsection, we will recall the most important results of the theory of
rough differential equations on manifolds as developed in [CLL12].

######  Notation.

Let $M$ and $N$ be two smooth manifolds and $g$ be a transport map from $M$ to
$N$. If $\alpha$ is a Banach space-valued one form on $M\times N$, we denote
by $\alpha^{g}$ the one-form defined on $M\times N$ as follows. For all $z\in
M\times N$ and $v=(v_{x},v_{y})\in T_{z}(M\times N)$,

$\alpha^{g}(z)(v):=\alpha(z)(v_{x},g(z)(v_{x})).$

We can now introduce a functional definition of solutions to RDEs on
manifolds.

######  Definition 5.6.

[CLL12] Let $\gamma>p\geq 1$. Let $M$ and $N$ be two Lip-$\gamma$ manifolds,
$x_{0}\in M$ and $y_{0}\in N$. Let $\mathbb{X}$ be a $p$-rough path in $M$
with starting point $x_{0}$. Let $g$ be a transport map from $M$ to $N$. We
say that a rough path $\mathbb{Y}$ in $N$ with starting point $y_{0}$ is a
solution to the RDE

(6)
$\left\\{\begin{array}[]{rcll}\mathrm{d}\mathbb{Y}_{t}&=&g(\mathbb{X}_{t},\mathbb{Y}_{t})\mathrm{d}\mathbb{X}_{t},&\quad\forall
t\leq T,\\\ \mathbb{Y}_{0}&=&y_{0}\\\ \end{array}\right.$

if there exists a $p$-rough path $\mathbb{Z}$ in $M\times N$ with stating
point $(x_{0},y_{0})$ such that $\pi_{M}(\mathbb{Z})=\mathbb{X}$,
$\pi_{N}(\mathbb{Z})=\mathbb{Y}$ and for all compactly supported Banach-space
valued Lip-$(\gamma-1)$ one-forms $\alpha$ on $M\times N$, one has
$\mathbb{Z}(\alpha)=\mathbb{Z}(\alpha^{g})$.

######  Theorem 5.7.

[CLL12] In the setting of Definition 5.6, if $g$ is Lip-$\gamma$ (in the sense
of Definition 3.15), then there exists a unique solution to the RDE (6).

##  6\. The accessibility property of geometric RDEs

###  6.1. Local solutions on orbits

We now show that one may make sense of and locally solve RDEs on an orbit of
Lipschitz vector fields.

######  Proposition 6.1.

Let $p\geq 1$ and $\gamma>p+1$. Let $n,d\in\mathbb{N}^{*}$ and $T>0$. Let
$\mathcal{D}=\\{f^{1},\ldots,f^{n}\\}$ be a family of Lip-$\gamma$ vector
fields on $\mathbb{R}^{d}$ and define the map
$f:\mathbb{R}^{d}\to\mathcal{L}(\mathbb{R}^{n},\mathbb{R}^{d})$ by

$\forall y\in\mathbb{R}^{d},\forall
x=(x_{1},\ldots,x_{n})\in\mathbb{R}^{n}:\quad
f(y)(x)=\sum_{i=1}^{n}x_{i}f^{i}(y).$

Let $y\in\mathbb{R}^{d}$. Let $\mathbb{X}$ be a geometric $p$-rough path on
$\mathbb{R}^{n}$. Then there exists $T_{0}\in(0,T]$ such that the rough
differential equation

(7)
$\left\\{\begin{array}[]{l}\mathrm{d}\mathbb{Y}_{t}=f(\mathbb{Y}_{t})\mathrm{d}\mathbb{X}_{t}\\\
\mathbb{Y}_{0}=y\\\ \end{array}\right.$

admits a solution $\mathbb{Y}$ as a $p$-rough path in $V_{y}$ (defined in
Theorem 4.8) defined over $[0,T_{0}]$.

###### Proof.

First note that since for all $v\in\mathbb{R}^{n}$ and $u\in V_{y}$,

$f(u)(v)\in\mathcal{L}\left(\mathcal{D}\right)(u)\subseteq
P_{\mathcal{D}}(u),$

the differential equation $(\ref{eq:RDESubmanifold})$ is well defined on
$V_{y}$. Let $p=\mathrm{dim}P_{\mathcal{D}}(y)$. By Lemma 3.11, $f$ is
Lipschitz on $V_{y}$ in the manifold sense. Let $\widehat{f}$ be a compactly
supported Lip-$\gamma$ map with support in
$\Psi_{y}^{-1}\left(B_{p}\left(0,\frac{1+\sqrt{2}}{2\sqrt{2}}\right)\right)$
and such that it agrees with $f$ on
$\Psi_{y}^{-1}\left(\overline{B_{p}\left(0,1/\sqrt{2}\right)}\right)$111The
choice of the value $1/\sqrt{2}$ is merely due to the fact that
$B_{p}\left(0,1/\sqrt{2}\right)$ is a Lipschitz chart domain in
$B_{p}\left(0,1\right)$ in the Lipschitz structure endowed by $F$ in Lemma
2.6.. Then, by Corollary 3.18, the corresponding transport $\widehat{g}$ is
Lip-$(\gamma-1)$. By Theorem 5.7 (and since $\gamma-1>p$), let
$\widehat{\mathbb{Z}}=(\mathbb{X},\widehat{\mathbb{Y}})$ be a solution to the
rough differential equation defined on $\mathbb{R}^{n}\times V_{y}$ by

$\left\\{\begin{array}[]{l}\mathrm{d}\widehat{\mathbb{Y}}_{t}=\widehat{f}(\widehat{\mathbb{Y}}_{t})\mathrm{d}\mathbb{X}_{t}\\\
\widehat{\mathbb{Y}}_{0}=y\\\ \end{array}\right.$

By Theorem 5.4 (and the subsequent remark), let $T_{0}\in(0,T]$ such that
$\mathbb{Z}:=\widehat{\mathbb{Z}}_{[0,T_{0}]}$ is supported in the chart
domain
$B_{n}(0,1)\times\Psi_{y}^{-1}\left(B_{p}\left(0,1/\sqrt{2}\right)\right)$
(where its starting point also lives). Let $\alpha$ be a compactly supported
Lip-$(\gamma_{0}-1)$ Banach space valued one form on $\mathbb{R}^{n}\times
V_{y}$, with $\gamma_{0}\in(p,\gamma]$. By the definition of the solution to
an RDE on a manifold, one has
$\mathbb{Z}(\alpha)=\mathbb{Z}(\alpha^{\widehat{f}})$ (where $\widehat{f}$ is
identified with its induced transport map). For $(x,u)\in\mathbb{R}^{n}\times
V_{y}$ and $(v_{x},v_{u})\in\mathbb{R}^{n}\times T_{u}V_{y}$, one has by
definition

$\alpha^{\widehat{f}}(x,u)(v_{x},v_{u})=\alpha(x,u)(v_{x},\widehat{f}(u)(v_{x})).$

From the above expression, one sees that $\alpha^{\widehat{f}}$ and
$\alpha^{f}$ agree in the chart domain
$B_{n}(0,1)\times\Psi_{y}^{-1}\left(B_{p}\left(0,1/\sqrt{2}\right)\right)$
which contains the support of $\mathbb{Z}$. Therefore, by Proposition 5.3,
$\mathbb{Z}(\alpha^{\widehat{f}})=\mathbb{Z}(\alpha^{f})$ and consequently

$\mathbb{Z}(\alpha)=\mathbb{Z}(\alpha^{\widehat{f}})=\mathbb{Z}(\alpha^{f}).$

Therefore $\mathbb{Z}$ is a solution to $(\ref{eq:RDESubmanifold})$ on
$[0,T_{0}]$.∎

######  Remark 6.2.

The careful reader may have noticed the stronger than usual condition on the
regularity of the vector fields $\gamma>p+1$ (instead of $\gamma>p$) in the
statement of Proposition 6.1. Let us say a few words about this:

  * •

While one can theoretically assume first order objects on manifolds (such as
vector fields, one-forms and transport maps) to be of any regularity, it is
not geometrically sensible to assume these objects to be of at least the same
regularity as the manifold itself: for example, one speaks only of
$\mathcal{C}^{n-1}$ vector fields on $\mathcal{C}^{n}$ manifolds. There are
two interlinked reasons for this. First, whenever two chart
$(U_{1},\varphi_{1})$ and $(U_{2},\varphi_{2})$ have intersecting domains, one
usually would want the regularity of such objects on the domain $U_{1}\cap
U_{2}$ when pushed forward by $\varphi_{2}$ to be deduced from its regularity
when pushed forward by $\varphi_{1}$. In this case, one encounters the
pushforward of the transition map $\varphi_{1}\circ\varphi_{2}^{-1}$, which is
one degree less regular than the regularity of the manifold. Second, and in
most practical and non-trivial examples, one cannot prove such strong
regularity if derivatives are involved, e.g. Corollary 3.18. Thus, Theorem 5.7
holds mainly theoretically if we only assume $\gamma>p$.

  * •

It will become clear later that, on the manifold $V_{y}$, one only needs an
existence result for solutions to RDEs. It will be enough then to identify
these manifold solutions with the classical ones on $\mathbb{R}^{d}$. The
proof of Theorem 5.7 is entirely based on the similar result (called the
universal limit theorem) holding on Banach spaces (e.g. [Lyo98, LCL07]): the
solution is first constructed on the unit balls before being pushed forward
onto the manifold. This existence and uniqueness result strongly relies on the
assumption of the transports being of regularity $\gamma$ with $\gamma>p$. To
relax this regularity assumption to $\gamma-1$, one may attempt to build on
the very known existence (only) result by Davie [Dav08] then Friz and Victoir
[FV08, FV10]. However, we will restrict ourselves here to the framework of the
theory given in [CLL12] as our aim is to illustrate the geometric argument
used to answer the accessibility question.

###  6.2. Identification with classical solutions

The next two lemmas allow us to “build our way back” from local solutions to
RDEs on the orbits to classical solutions on the Euclidean space.

######  Lemma 6.3.

Let $n,d\in\mathbb{N}^{*}$ and $\gamma>1$. Let $\mathcal{D}$ be a family of
Lip-$\gamma$ vector fields on $\mathbb{R}^{d}$. Let $y\in\mathbb{R}^{d}$. Then
for every Banach space-valued Lip-$(\gamma-1)$ one-form $\alpha$ on
$\mathbb{R}^{n}\times\mathbb{R}^{d}$, $\alpha_{|\mathbb{R}^{n}\times V_{y}}$
is a Lip-$(\gamma-1)$ one-form on $\mathbb{R}^{n}\times V_{y}$, and there
exists a constant $c_{\gamma,y}$ depending only on $\gamma$ and $\Psi_{y}$
such that

$\|\alpha_{|\mathbb{R}^{n}\times V_{y}}\|_{\textrm{Lip-}(\gamma-1)}\leq
c_{\gamma,y}\|\alpha\|_{\textrm{Lip-}(\gamma-1)}.$

###### Proof.

We denote $p=\textrm{dim}(P_{\mathcal{D}}(y))$. Let ${(U_{x},\phi_{x})}_{x\in
I}$ and ${(\widetilde{U}_{z},\varphi_{z})}_{z\in J}$ be the Lip-$\gamma$
atlases on $\mathbb{R}^{n}$ and $V_{y}$ respectively. For $(x,z)\in I\times
J$, we denote

$\xi_{(x,z)}:(u,v)\in\mathbb{R}^{n}\times
V\longmapsto(\phi_{x}(u),\varphi_{z}(v))\in B_{n}(0,1)\times B_{p}(0,1).$

To prove the claim, we need to show that the one-forms
$((\xi^{-1}_{(x,z)})_{*}\alpha)_{(x,z)\in I\times J}$ are Lip-$\gamma$ with
uniformly bounded norms. By the definition of $\phi_{x}$ and Lemma 3.10, the
maps $(\xi^{-1}_{(x,z)})_{(x,z)\in I\times J}$ are almost Lip-$\gamma$ with
uniformly bounded constants. Hence the result by Lemma 2.14. ∎

######  Lemma 6.4.

Let $\gamma,p\in\mathbb{R}$ such that $\gamma>p\geq 1$. Let
$n,d\in\mathbb{N}^{*}$. Let $\mathcal{D}$ be a family of Lip-$\gamma$ vector
fields on $\mathbb{R}^{d}$. Let $y\in\mathbb{R}^{d}$. Then every $p$-rough
path $\mathbb{Z}$ in $\mathbb{R}^{n}\times V_{y}$ with starting point $z$
defines a $p$-rough path $\widehat{\mathbb{Z}}$ in
$\mathbb{R}^{n}\times\mathbb{R}^{d}$ with the same starting point and such
that, for every compactly supported Banach space-valued Lip-$(\gamma-1)$ one-
form $\alpha$ on $\mathbb{R}^{n}\times\mathbb{R}^{d}$

$\widehat{\mathbb{Z}}(\alpha)=\mathbb{Z}(\alpha_{|\mathbb{R}^{n}\times
V_{y}}).$

###### Proof.

Let $\mathbb{Z}$ be a $p$-rough path in $\mathbb{R}^{n}\times V_{y}$ with
starting point $z$ and with $p$-variation controlled by a control function
$\omega$. We define the functional $\widehat{\mathbb{Z}}$ acting on any
compactly supported Banach space-valued Lip-$(\gamma_{0}-1)$ one-form (with
$\gamma\geq\gamma_{0}>p$) $\alpha$ on $\mathbb{R}^{n}\times\mathbb{R}^{d}$ by
$\widehat{\mathbb{Z}}(\alpha)=\mathbb{Z}(\alpha_{|\mathbb{R}^{n}\times
V_{y}})$. Let $\alpha$ be such a one-form. Then by the definition of a rough
path on a manifold and Lemma 6.3 , $\widehat{\mathbb{Z}}(\alpha)$ is a
$p$-rough path. Moreover, for every $0\leq s\leq t\leq T$ and
$i\in[\\![1,p]\\!]$, one has (using the bound obtained in Lemma 6.3)

$\left\|\widehat{\mathbb{Z}}(\alpha)^{i}_{s,t}\right\|=\left\|\mathbb{Z}(\alpha_{|\mathbb{R}^{n}\times
V_{y}})^{i}_{s,t}\right\|\leq\frac{\left(c_{\gamma,y}\|\alpha\|_{\textrm{Lip-}(\gamma-1)}\omega(s,t)\right)^{i/p}}{\beta_{p}\left(\frac{i}{p}\right)!}.$

Note that $c_{\gamma,y}\omega$ is also a control function. Finally, given a
compactly supported Lip-$\gamma_{0}$ map $g$ defined on
$\mathbb{R}^{n}\times\mathbb{R}^{d}$ with values in a Banach space $W$ and a
Banach space-valued Lip-$(\gamma_{0}-1)$ one-form $\alpha$ on $W$, it suffices
to note that

$(g^{*}\alpha)_{|\mathbb{R}^{n}\times V_{y}}=(g_{|\mathbb{R}^{n}\times
V_{y}})^{*}\alpha$

to prove the chain rule property for $\widehat{\mathbb{Z}}$ using that of
$\mathbb{Z}$. Hence $\widehat{\mathbb{Z}}$ is indeed a $p$-rough path on
$\mathbb{R}^{n}\times\mathbb{R}^{d}$. ∎

We show now that, for a short time, the (traces of the) solutions to a
classical RDE remain in the same orbit (associated to its vector fields).

######  Lemma 6.5.

Let $p\geq 1$ and $\gamma>p+1$. Let $n,d\in\mathbb{N}^{*}$. Let
$\mathcal{D}=\\{f^{1},\ldots,f^{n}\\}$ be a family of Lip-$\gamma$ vector
fields on $\mathbb{R}^{d}$ and define the map
$f:\mathbb{R}^{d}\to\mathcal{L}(\mathbb{R}^{n},\mathbb{R}^{d})$ by

$\forall y\in\mathbb{R}^{d},\forall
x=(x_{1},\ldots,x_{n})\in\mathbb{R}^{n}:\quad
f(y)(x)=\sum_{i=1}^{n}x_{i}f^{i}(y).$

Let $y\in\mathbb{R}^{d}$. Let $\mathbf{X}$ be a geometric $p$-rough path on
$\mathbb{R}^{n}$. Consider the following classical RDE on $\mathbb{R}^{d}$

(8)
$\left\\{\begin{array}[]{l}\mathrm{d}\mathbf{Y}_{t}=f(\mathbf{Y}_{t})\mathrm{d}\mathbf{X}_{t}\\\
\mathbf{Y}_{0}=y\\\ \end{array}\right.$

Then there exists $T_{0}>0$ such that the trace of the solution to (8) on the
time interval $[0,T_{0}]$ lives in $V_{y}$, i.e. for all $t\in[0,T_{0}]$ one
has $y+\mathbf{Y}^{1}_{0,t}\in V_{y}$.

###### Proof.

Let $\mathbb{Z}$ be a solution to the RDE (8) considered in the manifold sense
on $V_{y}$ defined over a non-trivial interval $[0,T_{0}]$ (Proposition 6.1).
By Lemma 6.4, $\mathbb{Z}$ defines a rough path $\widehat{\mathbb{Z}}$ on
$\mathbb{R}^{n}\times\mathbb{R}^{d}$. Let $\alpha$ be a Lip-$(\gamma_{0}-1)$
compactly supported Banach-space-valued one form on
$\mathbb{R}^{n}\times\mathbb{R}^{d}$, with $\gamma_{0}\in(p,\gamma]$. By the
definition of solutions to RDEs on manifolds and the very definition of
$\widehat{\mathbb{Z}}$, one has the following

$\widehat{\mathbb{Z}}(\alpha)=\mathbb{Z}(\alpha_{|\mathbb{R}^{n}\times
V_{y}})=\mathbb{Z}\left(\left(\alpha_{|\mathbb{R}^{n}\times
V_{y}}\right)^{f}\right)=\mathbb{Z}\left(\left(\alpha^{f}\right)_{|\mathbb{R}^{n}\times
V_{y}}\right)=\widehat{\mathbb{Z}}\left(\alpha^{f}\right).$

Therefore $\widehat{\mathbb{Z}}$ is a solution to the RDE (8) on
$\mathbb{R}^{d}$ in the manifold sense; and by the consistency of definitions,
also in the classical sense. By the uniqueness of such solution, we have then
$\mathbf{Y}=\pi_{\mathbb{R}^{d}}(\widehat{\mathbb{Z}})$ (when restricted to
$[0,T_{0}]$). Since $\widehat{\mathbb{Z}}$ is supported in
$\mathbb{R}^{n}\times V_{y}$, we conclude that the trace of $\mathbf{Y}$ lives
indeed in $V_{y}$. ∎

We get now the main theorem of this paper.

######  Theorem 6.6.

Let $p\geq 1$ and $\gamma>p+1$. Let $n,d\in\mathbb{N}^{*}$. Let
$\mathcal{D}=\\{f^{1},\ldots,f^{n}\\}$ be a family of Lip-$\gamma$ vector
fields on $\mathbb{R}^{d}$ and define the map
$f:\mathbb{R}^{d}\to\mathcal{L}(\mathbb{R}^{n},\mathbb{R}^{d})$ by

$\forall y\in\mathbb{R}^{d},\forall
x=(x_{1},\ldots,x_{n})\in\mathbb{R}^{n}:\quad
f(y)(x)=\sum_{i=1}^{n}x_{i}f^{i}(y).$

Let $y_{0}\in\mathbb{R}^{d}$. Let $\mathbf{X}$ be a geometric $p$-rough path
on $\mathbb{R}^{n}$. Then the solution to the rough differential equation

(9)
$\left\\{\begin{array}[]{l}\mathrm{d}\mathbf{Y}_{t}=f(\mathbf{Y}_{t})\mathrm{d}\mathbf{X}_{t}\\\
\mathbf{Y}_{0}=y_{0}\\\ \end{array}\right.$

takes values in $\mathbb{L}_{y_{0}}(\mathcal{D})$, the $\mathcal{D}$-orbit at
$y_{0}$. In particular, for every $T\geq 0$, there exists a piecewise linear
path $x$ in $\mathbb{R}^{d}$ such that
$y_{0}+\mathbf{Y}^{1}_{0,T}=\gamma(y_{0},x,T)$, where
$(\gamma(y_{0},x,t))_{0\leq t\leq T}$ is the solution to the ODE

(10)
$\left\\{\begin{array}[]{l}\mathrm{d}\gamma_{t}=f(\gamma_{t})\mathrm{d}x_{t}\\\
\gamma_{0}=y_{0}\\\ \end{array}\right.$

###### Proof.

First note that for $z_{1},z_{2}\in\mathbb{R}^{d}$, if $V_{z_{1}}\cap
V_{z_{2}}\neq\varnothing$, then $z_{1}$ and $z_{2}$ lie on the same $D$-orbit.
Combining this trivial observation with Lemma 6.5 and a compactness argument
yields the first part of the theorem. To prove the second part, we make the
observation that all points of $\mathbb{L}_{y_{0}}(\mathcal{D})$ are solutions
to ODEs of the type (10) driven by piecewise linear paths. Indeed, let
$y\in\mathbb{R}^{d}$, $i\in[\\![1,n]\\!]$, $t\in\mathbb{R}$ and
$z=\tilde{f}^{i}_{t}(y)$. If $t\geq 0$ (resp. $t\leq 0$), then $z$ is the
value at time $t$ of the solution to (10) when driven by the linear path
$x:\;s\mapsto s\cdot e_{i}$ (resp. $x:\;s\mapsto-s\cdot e_{i}$), with
$(e_{1},\ldots,e_{n})$ denoting the canonical basis of $\mathbb{R}^{n}$. In
other words, $z=\gamma(y,x,|t|)$. Note also that the time $|t|$ of the arrival
of such solution to $z$ can be arbitrarily chosen by rescaling (time-
reparametrising) $x$. More generally, let $z$ be a point in
$\mathbb{L}_{y_{0}}(\mathcal{D})$, $p$ be a positive integer,
$f^{1},\ldots,f^{p}$ be vector fields in $\mathcal{D}$ and
$t_{1},\ldots,t_{p}$ be real numbers such that

$z=\tilde{f}^{p}_{t_{p}}\circ\cdots\circ\tilde{f}^{1}_{t_{1}}(y_{0}).$

By the argument above, let $x_{1}$ be a linear path such that

$\tilde{f}^{1}_{t_{1}}(y_{0})=\gamma\left(y_{0},x_{1},\frac{|t_{1}|}{|t_{1}|+\ldots+|t_{p}|}T\right).$

Inductively, for every $i\in[\\![2,p]\\!]$, let $x_{i}$ be a linear path such
that

$\tilde{f}^{i}_{t_{i}}\circ\cdots\circ\tilde{f}^{1}_{t_{1}}(y_{0})=\gamma\left(\tilde{f}^{i-1}_{t_{i-1}}\circ\cdots\circ\tilde{f}^{1}_{t_{1}}(y_{0}),x_{i},\frac{|t_{i}|}{|t_{1}|+\ldots+|t_{p}|}T\right).$

Then

$z=\gamma\left(y_{0},x_{p}*\cdots*x_{1},T\right),$

where $x_{p}*\cdots*x_{1}$ is the concatenation of $x_{1},\ldots,x_{p}$ (in
this order), which is a piecewise linear path. This concludes the proof. ∎

##  7\. Concluding remarks

In [Sus73], Sussmann constructs a smooth structure on orbits by “patching up”
the smooth structures of the local neighborhoods (denoted $V_{y}$ in this
article, see Theorem 4.8). However, we are not able to build such a global
structure in the Lipschitz case due to the rigidity of the current definition
of Lipschitz structures. Indeed, these are specifically tailored to allow for
global solutions to RDEs on manifolds to exist, but bridges to and from
classical smooth structures are currently lacking (the only non-trivial
exception being the case of smooth compact manifolds, see [CLL12]). In the
case of orbits, patching up the local neighborhoods yields a locally Lipschitz
structure (as defined in [BL15]) which allows only for local solutions to RDEs
(this result was not proven in [BL15] but can be carried using a local
argument). It is difficult however to imagine that general RDEs do not admit
global solutions on orbits of vector fields since (the traces of) these
solutions when considered in the ambient Euclidean space lie in the orbits.
But whether the orbits admit a Lipschitz structure or a “weaker” structures
that still allows for the global solution to RDEs remains an open question to
our knowledge.

A more important problem -at least from an analytical point of view- is to
quantify the necessary length of a piecewise linear path to yield the same
terminal solution as a given geometric rough path. If we try and exploit our
geometric argument by breaking down the original signal so that the solution
is written as a concatenation of rough paths that live in individual charts,
then the question boils down to solving the same problem when restricted to
the charts $V_{y}$. However, the understanding of these charts $V_{y}$
requires a deep knowledge of the whole orbit and on how to transport new
tangent directions from points that are possibly very far away from $y$.
Consequently, it is more judicious to reserve such argument for special cases
where such knowledge is readily available. An interesting example thereof is
the case of the free nilpotent group and the inversion of signatures (e.g.
[LX18, LX17]).

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