The Accessibility Problem for Geometric Rough Differential Equations

The theory of rough paths introduced by Lyons [3, 4] (then developed in further directions by Gubinelli [5, 6]) teaches us that the effects of paths on differential systems cannot 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(with some initial condition) requires not only the knowledge of the path \((x_t)_{0 \le t \le T}\) but also of its lift into a signatureWhile 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 \(1\le p <2\)), 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 pathwhere \(a \in \mathbb {R}^2\otimes \mathbb {R}^2\) 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(this now popular example can be found both in [3] and [4].) 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 pathOther 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 [7].

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 \(\gamma >p\ge 1\), \(\xi \in \mathbb {R}^d\), a Lipschitz-\(\gamma \) map \(f:\mathbb {R}^d\rightarrow \mathcal {L}(\mathbb {R}^n, \mathbb {R}^d)\) and a geometric p-rough path \(\textbf{X}\) in \(\mathbb {R}^n\) defined over a compact interval [0, T], let \(\textbf{Y}\) be the solution to the following rough differential equation (RDE)Does there exist an \(\mathbb {R}^n\)-valued path x with bounded variation such that the solution to the classical ODEsatisfies \(y_T=\xi +\textbf{Y}_{0,T}^1\)? We will call this the accessibility problem.

By the continuity of the Itô-Lyons solution map associated to (3) and the very definition of a geometric rough path, given a geometric p-rough path \(\textbf{X}\) in \(\mathbb {R}^n\), one can approximate, in the p-rough path topology and up to an arbitrarily small error, its solution \(\textbf{Y}\) 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 \((\xi +\textbf{Y}_{0,t}^1)_{0\le t\le T}\) in the p-variation and the uniform convergence topologies by the path \((y_t)_{0\le t\le T}\). The question we ask here is slightly different: we are only interested in the terminal value \(\xi +\textbf{Y}_{0,T}^1\) 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 \((\xi +\textbf{Y}_{0,t}^1)_{0\le t\le T}\) has only finite p-variation while solutions to the ODE (4) have finite 1-variations. Therefore, it is not always possible to replace \(\textbf{X}\) by a path of finite length and replicate the solution \((\xi +\textbf{Y}_{0,t}^1)_{0\le t\le T}\) at all times.

Let us now consider a simple case. Let N be a strictly positive integer and denote by \(T^{(N)}(\mathbb {R}^n)\) the truncated tensor algebra of \(\mathbb {R}^n\) of order N defined byDefine the (linear) map \(f: T^{(N)}(\mathbb {R}^n) \rightarrow \mathcal {L} (\mathbb {R}^n,T^{(N)}(\mathbb {R}^n))\) byThen, given a path \(x:[0,T]\rightarrow \mathbb {R}^n\) of bounded variation, the ODEadmits as a unique solution the order-N truncated signature path \(\left( S_{N}(x)_{(0,t)}\right) _{0\le t\le T}\) of x [4, 8]. It is also known (by using for example the shuffle product property [4, 8]) that the order-N truncated signatures live in the free nilpotent group \(G^{(N)}(\mathbb {R}^n)\) of order N [9]. 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 [4]). 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.

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 [4, 9]. Denote by \((e_i)_{1\le i\le n}\) the canonical basis of \(\mathbb {R}^n\) and, for every integer \(i\in [1,n]\), by \(f^i\) the linear homomorphism defined on \(T^{(N)}(\mathbb {R}^n)\) byIt 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 any integer \(i\in [1,n]\), an integral curve \(\gamma \) of \(f^i\) is also a solution to the ODEwhere \(x^{(i)}: t\mapsto t e_i\) 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:In Sect. 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 Sect. 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 followsWe simply denote the null rough path by 0Second, 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 \(f:\mathbb {R}^n\rightarrow \mathbb {R}^p\) and \(x,v \in \mathbb {R}^n\), \(f_*(x)(v)\) is sometimes used instead of writing \(f_*(x)(\textrm{D}_{x,v})\) (\(\textrm{D}_{x,v}\) being the derivation (tangent vector) at x in the direction v), is identified with \(\textrm{d}f(x)(v)\) 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).

We first recall the classical notion of Lipschitz regularity (as presented for example in [4]). Given two Banach spaces E and F and an integer \(k\ge 1\), 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 \(\Vert .\Vert \) to designate a norm on \(E^{\otimes k}\) and the norm on F. Let us state the main three properties required of such norms, which we will assume to hold in the rest of the paper:For further discussion on the properties required from the norms on tensor products of spaces, we refer the interested reader either to the literature on rough paths (e.g. [4]) or the manuscript [13].

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

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 [2].

There exists a natural embedding between spaces of Lipschitz maps.

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.

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 [13]. We reintroduce it here to allow for a certain flexibility in our proofs.

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

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

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).

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

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.

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

One of the central arguments in building the charts and thus the manifold structure in [1] 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 [13]. It can be obtained from the original version by simple arguments of restricting the maps to bounded sets and rescaling.

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 \(\textrm{Lip-}1\) ones using for example Picard-Lindelöf’s theorem.

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

We now recall and build on the definition of Lipschitz manifolds as laid out in [2].

Finite dimensional vector spaces are naturally endowed with Lipschitz structures [2]. In the remaining of this paper, we will explicitly refer to the structure defined by the following proposition.

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

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.

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

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

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

Like in the Euclidean setting (Whitney’s extension theorems, e.g. [14]), Lipschitz maps defined on compact subsets of Lipschitz manifolds and with values in a finite dimensional space can be extended.

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

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.

Next, we show that the \(\textrm{Lip-}\gamma '\) regularity of connections is stable by \(\textrm{Lip-}\gamma \) diffeomorphisms for \(\gamma '\le \gamma -1\). 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.

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 [1] and will restrict the exposition to the minimum necessary to obtain our results.

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

In the sequel, we will use the following notations.

A key idea in Sussmann’s work [1] was to identify the “candidate” tangent space to orbits. This is best carried through the notion of distributions.

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.

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

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 [1]. 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.

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

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 [2].

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

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

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 \(\textbf{X}\) 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 \(\mathbb {R}^d\)Conversely, the functional \(\mathbb {X}\) completely determines the classical rough path \(\textbf{X}\). The proof relies on all three axioms of Definition 5.1 and can be found in [2].

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

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 [2]. Moreover, the integrals of a rough path along two one-forms that agree on an open set containing its support are the same.

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.

In this subsection, we will recall the most important results of the theory of rough differential equations on manifolds as developed in [2].

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

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

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.

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).

We get now the main theorem of this paper.