The non-linear sewing lemma I: weak formulation

We introduce a new framework to deal with rough differential equations based on flows and their approximations. Our main result is to prove that measurable flows exist under weak conditions, even solutions to the corresponding rough differential equations are not unique. We show that under additional conditions of the approximation, there exists a unique Lipschitz flow. Then, a perturbation formula is given. Finally, we link our approach to the additive, multiplicative sewing lemmas and the rough Euler scheme.


Motivations
The theory of rough paths allows one to define the solution to a differential equation of type for a path x which is irregular, say α-Hölder continuous.Such an equation is then called a Rough Differential Equation (RDE) [19,20,24].The key point of this theory is to show that such a solution can be defined provided that x is extended to a path x, called a rough path, living in a larger space that depends on the integer part of 1/α.When α > 1/2, no such extension is needed.This case is referred as the Young case, as the integrals are constructed in the sense given by L.C. Young [23,27].Provided that one considers a rough path, integrals and differential equations are natural extensions of ordinary ones.
The first proof of existence of a solution to (1) from T. Lyons relied on a fixed point [24,25,26].It was quickly shown that RDE shares the same properties as ordinary differential equations, including the flow property.In [15], A.M. Davie gives an alternative proof that used an Euler type approximation and gave counter-example to uniqueness.More recently, I. Bailleul gave a direct construction through the flow property [3,4,6].
A flow in a metric space V is a family {ψ t,s } 0≤s≤t≤T of maps from V to V such that ψ t,s • ψ s,r = ψ t,r for any 0 ≤ r ≤ s ≤ t ≤ T .When y t (s, a) is a family of solutions to differential equations with y s (s, a) = a, the element ψ t,s (a) can be seen as a map which carries a to y t (s, a).A flow requires some compatibilities on the family of solutions so that solutions may exist although no flow may hold.Flow are related to dynamical systems.There are a slightly different objects than solutions.One of their interest lies in their characterization as lipeomorphims (Lipschitz functions with a Lipschitz inverse), diffeomorphisms...
In this article, we develop a generic framework to construct flows from approximations.We do not focus on a particular form of the solutions, so that our construction is a non-linear sewing lemma, modelled after the additive and multiplicative sewing lemmas [18].
In this first part, we study flows under weak conditions and prove existence of a measurable flow even when the solutions of RDE are not necessarily unique.This is based on a selection theorem [10] due to J.E. Cardona and L. Kapitanski.Such a result is new in the literature where existence of flows were only proved under stronger regularity conditions (the many approaches are summarized in [14]).Besides, our approach also contains the additive and multiplicative sewing lemmas [12,18].The rough equivalent of the Duhamel formula for solving linear RDE [13] with a perturbative terms follows directly from our construction.
Our starting point in the world of classical analysis is the product formula which relates how the iterated product of an approximation of a flow converge to the flow [1,11].It is important both from the theoretical and numerical point of view.
On a Banach space V, let us consider a family ( , a) ∈ R + × V → Φ( , a), called an algorithm of class C 1 in ( , a) such that Φ(0, a) = a.The parameter is related to the quality of the approximation.
For a vector field f , let also ψ be a flow on V, by which we mean that map which associate to each a in V the solution ψ t (a) := y t solution to the ODE ẏt = f (y t ) for t ≥ 0 with y 0 = a.
The Euler scheme for solving ODE is the prototypical example of such behavior.Set Φ( , a) = a + f (a) so that (2) holds.In this case, (3) expresses nothing more than the convergence of the Euler scheme.
For example, one recovers easily the proof of Lie's theorem on matrices.For this, if A is a matrix, then exp(tA) is the solution to Ẏt = AY t with Y 0 = Id.The exponential exp(tA) is easily recovered as For two matrices A and B, exp(t(A + B)) is given by For this, we have only to consider Φ( , a) = exp( A) exp( B)a and to verify that ∂ Φ(0, a) = (A + B)a for any matrix a.
For unbounded operators, it is also related to Chernoff and Trotter's results on the approximation of semi-groups [17,28].The product formula also justifies the construction of some splitting schemes [8].
In this article, we consider as an algorithm a family {φ t,s } 0≤s≤t≤T of functions from V to V which is close to the identity map in short time and such that φ t,s • φ s,r is close to φ t,r for any time s ≤ r ≤ t.For a path x of finite p-variation, 1 ≤ p < 2 with values in R d and a smooth enough function f : R m → L(R d , R m ), such an example is given by φ t,s (a) = a + f (a)x s,t .
We then study the behavior of the composition φ π of the φ t i+1 ,t i along a partition π = {t i } i=0,...,n .Clearly, as the mesh of the partition π goes to 0, the limit, when it exists is a candidate to be a flow.In the example given above, it will be the flow associated to the family of Young differential equations y t (a) = a + t 0 f (y s (a)) dx s .After having introduced some necessary notations (Section 1.2) including the central notion of galaxies (Section 1.2.3) which partition the space of families {φ t,s } 0≤s≤t≤T of functions from a Banach space V to itself according to their proximity, we present the main results of our articles in 1.3

Notations, definitions and concepts
The following notations and hypotheses will be constantly used throughout all this article.

Hölder and Lipschitz continuous functions
Let V and W be two metric spaces.
The space of continuous functions from V to W is denoted by C(V, W).
Let d (resp.d ) be a distance on V (resp.W).For γ ∈ (0, 1], we say that a function f : V → W is γ-Hölder if whenever this quantity is finite.If γ = 1 we say that f is Lipschitz.We then set f Lip := f 1 . For any integer r ≥ 0 and γ ∈ (0, 1], we denote by C r+γ (V, W) the subspace of functions from V to W whose derivatives d k f of order k ≤ r are continuous and such that d r f is γ-Hölder.
We denote by C r+γ b (V, W) the subset of C r+γ (V, W) of bounded functions with bounded derivatives up to order r.

Controls and remainders
From now, V is a topological, complete metric space with a distance d.A distinguished point of V is denoted by 0.
Let us fix a time horizon T .We write T := [0, T ] as well as A control ω : which is continuous close to its diagonal with ω s,s = 0 for all s ∈ T. A typical example of a control is ω s,t = C|t − s| where C is a non-negative constant.
For p ≥ 1, we say that a path x ∈ C(T, V) is a path of finite p-variation controlled by ω if A remainder is a function : R + → R + be a continuous, increasing function such that for some 0 < κ < 1, A typical example for a remainder is (δ) = δ θ for any θ > 1.
We consider that δ : R + → R + is a non-decreasing function with lim T →0 δ T = 0.

Galaxies
Notation 1.We denote by F(V) the set of functions {φ t,s } (s,t)∈T + 2 from V to V which are continuous in (s, t), i.e. for any a ∈ V, the map (s, t) ∈ T + 2 → φ t,s (a) is continuous.
Notation 2 (Iterated products).For any φ ∈ F(V), any partition π of T and any (s, t) ∈ T + 2 , we write where [t i , t j ] is the biggest interval of such kind contained in [s, t] ⊂ T. If no such interval exists, then φ π t,s = φ t,s .
Clearly, for any partition, φ π ∈ F(V).A trivial but important remark is that from the very construction, In particular, {φ π t,s } (s,t)∈π 2 + enjoys a (semi-)flow property when the times are restricted to the elements of π.
The article is mainly devoted to study the possible limits of φ π as the mesh of π decreases to 0. Notation 3. From a distance d on V, we define the distance ∆ Nγ on the space of functions from V to V by where N γ is defined in Section 1.2.
This distance is extended on F(V) by where ω, are defined in Section 1.2.The distance ∆ Nγ , may take infinite values.
If d is a distance for which (V, d) is complete, then ( C(V, V), ∆ N ) and (F(V), ∆ N, ) are complete.
Definition 1 (Galaxy).We define the equivalence relation ∼ on F(V) by φ ∼ ψ if and only if there exists a constant C such that In other words, φ ∼ ψ if and only if ∆ Nγ , (φ, ψ) < +∞.Each quotient class of F(V)/ ∼ is called a galaxy, which contains elements of F(V) which are at finite distance from each others.

Summary of the main results
The galaxies partition the space F(V).Each galaxy may contain two classes of elements on which we focus on this article: 1.The flows, that is the families ψ which satisfy for any (r, s, t) ∈ T + 3 , or equivalently, ψ π = ψ (See ( 7)) for any partition π. 2. The almost flows which we see as time-inhomogeneous algorithms.Besides some conditions on the continuity and the growth given in Definition 2 below, an almost flow φ is close to a flow with the difference that for a suitable function N γ : V → [1, +∞).Along with an almost flow φ comes the notion of manifold of solutions, that is a family {y t (a)} t≥0, a∈V of paths such that Each path y(a) is a solution which we called solution in the sense of Davie as our definition expands naturally the one introduced by A.M. Davie in [15].
Clearly, a manifold of solutions associated to an almost flow φ is also associated to any almost flow in the same galaxy as φ.Besides, if a flow ψ exists in the same galaxy as φ, then z t (a) = ψ t,0 (a) defines a manifold of solutions.Flows are more constrained object than solutions as (9) implies some compatibility conditions, while it is possible to create new paths of solutions from two different ones.As it will be shown in [9], uniqueness of manifold of solutions is strongly related to existence of a flow.
Given an almost flow φ, it is natural to study the limit of the net {φ π } π as the mesh of π decreases to 0. Any limit will be a good candidate to be a flow.
Our first main result (Theorem 1) asserts that if φ is an almost flow in a galaxy G, then any iterated map φ π belongs to G whatever the partition π although the map φ π is not necessarily an almost flow.More precisely, a control is given on ∆ Nγ , (φ π , φ) which is uniform in the partition π.
An immediate corollary is that any possible limit of {φ π } π also belongs to G as the mesh of the partition decreases to 0.
Our second main result (Theorem 2) is that when the underlying Banach space V is finite-dimensional for compactness reasons, there exists a least one measurable flow in a galaxy containing an almost flow, even when several manifold of solutions may exists.Our proof uses a recent result of J.E. Cardona and L. Kapitanski [10] on selection theorems.
Our third results is to give conditions ensuring the existence of at most one flow in a galaxy.If the sufficient for this that a galaxy G contains a Lipschitz flow ψ.
In this case, {φ π } π converges to ψ whatever the almost flow φ in G.The rate of convergence is also quantified.
Finally, we apply our results to the additive, multiplicative sewing lemmas [18] as well as to the algorithms proposed by A.M. Davie in [15] to show existence of measurable flows even without uniqueness.In the sequel [9], we study in details the properties of Lipschitz flows and give some conditions on almost Lipschitz flows to generate them.In addition, we also apply our results to other approximations of RDE, namely the one proposed by P. Friz & N. Victoir [20] and the one proposed by I. Bailleul [3] by using perturbation arguments.

Outline
We show in Section 2 that a uniform control of the iterated product of approximation of flows with respect to the subdivision.In Section 3, we prove our main result: the existence of a measurable flow under weak conditions of regularity.Then, in Section 4 we show the existence and uniqueness of a Lipschitz flow under stronger assumptions.Moreover, we give a rate of the convergence of the iterated product to the flow.In Section 5, we show that the convergence of the iterated product of an approximation of flow is preserved under the action of a additive perturbation.Finally, we recover in Section 6 the additive [25], multiplicative [13,18] and Davie's sewing lemmas [15].

A uniform control over almost flows
In this section, we define almost flows which serves as approximations.The properties of an almost flow φ are weaker than the minimal condition to obtain the convergence of the iterated products φ π as the mesh of the partitions π decreases to 0. However, we prove in Theorem 1 that we can control φ π uniformly over the partitions π.This justifies our definition.

Definition of almost flows
Definition 2 (Almost flow).An element φ ∈ F(V) is an almost flow if for any (r, s, t) Remark 1.A family φ ∈ F(V) satisfying condition (12) with γ = 1 is said to be quasi-contractive.This notion plays an important role in the fixed point theory [7].
Definition 3 (Iterated almost flow).For a partition π and an almost flow φ, we call φ π an iterated almost flows, where φ π is the iterated product defined in (7).
for any a ∈ V and (r, s, t) ∈ T + 3 .

A uniform control on iterated almost flows
Before proving our main result in Section 3, we prove an important uniform control over φ π .
Theorem 1.Let φ be an almost flow.Then there exists a time horizon T small enough and a constant L T ≥ 1 as well as a constant K T ≥ 1 that decreases to 1 as T decreases to 0 such that for any (s, t) ∈ T 2 + , a ∈ V and any partition π of T.
Remark 2. The distance d may be replaced by a pseudo-distance in the statement of Theorem 1.
The proof of Theorem 1 is inspired by the one of the Claim in the proof of Lemma 2.4 in [15].With respect to the one of A.M. Davie, we consider obtaining a uniform control over a family of elements indiced by (s, t) ∈ T + 2 which are also parametrized by points in V.
Definition 5 (Successive points / distance between two points).Let π be a partition of [0, T ].Two points s and t of π are said to be at distance k if there are exactly k − 1 points between them in π.We write d π (t, s) = k.Points at distance 1 are called successive points in π.
We now restrict ourselves to the case (r, t) ∈ π + 2 .To control U r,t (a) in a way that does not depend on π, we use an induction in the distance between r and t.
Our induction hypothesis is that there exist constants L T ≥ 0 and K T ≥ 1 independent from the partition π such that for any (r, t) with K T decreases to 1 at T goes to 0.
If r and t are successive points, φ π t,r = φ t,r so that U r,t (a) = 0. Thus, ( 16) is true for m = 1.With (11) and since N γ (a) γ ≤ N γ (a) as N γ (a) ≥ 1 by hypothesis, For m = 2, it is also true with L T = 1 as U r,t (a) = d(φ t,s • φ s,r (a), φ t,r (a)) where s is the point in the middle of r and t.This proves (16).
Assume that that ( 16)- (17) when the distance between r and t is smaller than m for some m ≥ 2.
For s ∈ π, r ≤ s ≤ t, with (8), ).With ( 12) and ( 13), a) and U s,s (a) = 0.With ( 16)-( 17) and our hypothesis on η, again since N γ ≥ 1 and s, s are at distance less than m provided that s is at distance at most ) This inequality also holds true for r = s or s = t.
We proceed as in [15] to split π in "essentially" two parts.We set Hence, s and s are successive points with r ≤ s < s ≤ t and ω r,s ≤ ω r,t /2.Besides, since ω is super-additive, ω r,s + ω s ,t ≤ ω r,t .Therefore, With such a choice, (20) becomes with (5) and since If T is small enough so that ) when r and t are at distance m.Condition (17) follows from (18) and (16).
The choice of T , K T and L T does not depend on π.In particular, d(φ π t,s (a), φ t,s (a)) ≤ L T N γ (a) (ω s,t ) becomes true for any (s, t) ∈ T + 2 , is it is sufficient to add the points s and t to π.
Corollary 1.Let φ be an almost flow and π be a partition of T. Then φ π ∼ φ (we have not proved that φ π is itself an almost flow).When V is a finite dimensional space with the norm |•|, S φ (s, a) = ∅.We start with a lemma which will be useful to prove some equi-continuity.We denote B(0, R) the closed ball center in 0 of radius R > 0.
Proposition 1. Assume that V is finite-dimensional and that φ is an almost flow.
Then S φ (r, a) is not empty for any (r, a) ∈ T × V.
Proof.Let us show for an almost flow φ, {φ π •,r (a)} π is equi-continuous and bounded.The result is then a direct consequence of the Ascoli-Arzelà theorem.
We differentiate two cases.If m ≤ n, then π m ⊂ π n , which implies that φ πn t,r = φ πn t,s • φ πn s,r .From Theorem 1 and Lemma 1, for all a ∈ B(0, R), for all |t − s| < δ By continuity of t → φ πn t,r (a), and density of m∈N π m in T, we obtain (24) for all (s, t) ∈ T + 2 with r ≤ s and |t − s| < δ.This proves that {t → φ πn t,r (a)} n is uniformly equicontinuous for all a ∈ V. We conclude the proof with the Ascoli-Arzelà theorem.

The non-linear sewing lemma
After showing a uniform control of φ π with respect to π when φ is an almost flow in the previous section, we prove our main result in Theorem 2. We show that in the finite dimensional case, we can built a flow from φ π using a selection principle due to [10].
We restate the notion of solution in the sense introduced by A.M. Davie [15] in the context of a metric space and a general almost flow.
Definition 6 (Solution in the sense of Davie).For an almost flow φ, a solution in the sense of Davie associated to φ is a path y ∈ C(S, Remark 3. Our definition of solution is a natural extension of that of Davie in [15] for a metric space V and a general almost flow φ. Remark 4. If φ is only an almost flow, it is not guarantee that there exists a solution and even less that the uniqueness of the solution holds.When S φ (r, a) = ∅, we prove below in Lemma 3 that a solution in the sense of Davie exists.Even when S φ (r, a) = ∅ for any r ∈ T and any a ∈ V does not proves that there exists a choice of families of solutions {ψ •,r (a)} r∈T,a∈V which satisfies the flow property.
Notation 5. We denote Ω(r, a) the set of continuous path such that y ∈ C(S, V) verifying y r = a.We denote by G K φ (r, a) the set of paths in Ω(r, a) verifying (25) for the constant K.
We restate here, the definition of a family of abstract integral local funnels (Definition 2 in [10]), which leads to the existence of a measurable flow.Definition 8.A family F (r, a) ⊂ C(T, V) with r ∈ [0, +∞), a ∈ V, will be called a family of abstract local integral funnels with terminal times T (r, a) if F (r, a) satisfies H1 Every set F (r, a) is a non-empty compact in C(T, V) and every path y(r, a) ∈ F (r, a) is a continuous map from T = [T 0 , T 0 + T (r, a)] to V, and y r (r, a) = a.
H2 For all r ≥ 0, the map a ∈ V → S(r, a) is measurable in the sense that for any closed subset A flow is constructed by assigning to each point of the space of particular choice of a solution in the sense of Davie in a sense which is compatible.
Hypothesis 1.Let V be a finite dimensional vector space.Let φ be a Lipschitz almost flow (Definition 9) with N bounded.We fix a time horizon T > 0 such that κ(1 + δ T ) < 1.
Remark 5.In the case N bounded, we can choose where K T and L T are the constants of Theorem 1.
The main theorem of this paper is the following one.
Theorem 2 (Non-linear sewing lemma, weak formulation).Under Hypothesis 1, there exists a family ψ ∈ F(V) in the same galaxy as φ such that ψ t,s is Borel measurable for any (s, t) ∈ T 2 + and ψ satisfies the flow property.Remark 6. Proving such a result with a general Banach space V is false as even existence of solutions to ordinary differential equations may fail [16,22].Remark 7. To prove Theorem 2, we show that (G L T (r, a)) r∈[0,+∞),a∈V is a family of abstract local integral funnels in the sense of Definition 8.Then, we use Theorem 2 of [10].
Lemmas 2-7 prove that G L T (r, a) is a family of local abstract funnels in the sense of the Definition 2 in [10].Then we apply Theorem 2 in [10] to obtain the following theorem.
Lemma 2. Under the Hypothesis 1, the terminal times T (r, a) is independent of the starting time r and the starting point a.We denote T = T (r, a).In particular, H0 holds for F = G L T φ .
Proof.Indeed, κ and δ T does not depend neither of a ∈ V and T 0 .
We recall that S φ (r, a) is defined in Notation 4 Our first result is that when S φ (r, a) = ∅, then there exists at least one solution in the sense of Davie in G L T φ (r, a).
Lemma 3. Assume that K ≥ K T L T in Definition 6, where K T and L T are constants in Theorem 1.For any (r, a) ∈ T × V, S φ (r, a) ⊂ G K φ (r, a) for an almost flow φ (note that S φ (r, a) may be empty).
Proof.If y ∈ S φ (r, a) when S φ (r, a) = ∅, then there exists a sequence {π k } k∈N of partitions such that y t = lim φ π k t,r (a) uniformly in t ∈ [r, T ].We note that y r = a.For k ∈ N and s k ∈ π k , with (8) and Theorem 1, And moreover, Choosing {s k } k∈N so that s k decreases to s and passing to the limit, we obtain with ( 27) that φ t,s k • φ π k s k ,r (a) converges uniformly to φ t,s • y s (a).Thus, when k → +∞, (26) shows that y is a solution in the sense of Davie.a) then {y k } k is equi-continuous with the same argument as in proof of Proposition 1.And the subsequence of {y k } k converges in G K φ (r, a) because of the continuity of a ∈ V → φ t,s (a) for any (s, t) ∈ T + 2 .
Let us denote GL T φ (r, a), the set of paths in Proof.Let {a k } k∈N be a convergent sequence of S (r).For each k ∈ N, we choose a path ỹk ∈ GL T φ (r, a k ) K (which is not empty by definition).Then, for every where t := r + t(T − r) and s := r + s(T − r).Since t − s goes to zero when t − s → 0, it follows that {t ∈ [0, 1] → ỹk t } k∈N is equi-continuous.The sequence {a k } k∈N converges, so it is bounded by a constant A ≥ 0. Applying (28) between s = 0 and t, we get By Ascoli-Arzelà theorem, there is a convergent subsequence to a path y.This path belongs to K since K is closed.Because φ t,s is continuous, ỹ ∈ GL T φ (r, a).Hence S (r) is closed and then Borel.
The proof of the next lemma is an immediate consequence of the definition of a solution in the sense of Davie.
Lemma 7. We assume that Hypothesis 1 hold.For r ≥ 0, if y ∈ G L T φ (r, a) and z ∈ G L T φ (r + r , y r+r (r, a)), then y r+r z ∈ G L T φ (r, a).It shows that H4 holds for F := G L T φ Proof.Let us write x := y s z where s := r + r and U τ,t := d(x t , φ t,τ (x τ )) for τ ≤ t.On the one hand, for any r ≤ τ ≤ s ≤ t with ( 25), ( 12) and ( 13), On another hand, for s ≤ τ ≤ t or τ ≤ t ≤ s with ( 25) Thus, combining (29) and (30), for any r ≤ τ ≤ t ≤ T , Besides, for any r ≤ τ ≤ u ≤ t ≤ T with ( 12) and ( 13), Let λ ∈ (0, 1) such that λ satisfies (5) with For any, two successive points τ, t of a subdivision π, where Let us show by induction over the distance m between points τ and t in π ∩[0, where .
When m = 0, U τ,τ = 0 so that (33) holds.For m = 1, τ and t are successive points then (33) holds with (32).Now, we assume that (33) holds for any two points at distance m.Let τ and t be two points at distance m Since ω is super-additive, one may choose two successive points s and s in π with τ < s < s < t such that ω τ,s ≤ ω τ,t /2 and ω s ,t ≤ ω τ,t /2, as in the proof of Theorem 1.Then, by applying (31) between (τ, s, s ) and (s, s , t) we obtain, , with our choice of A L T (π, λ).This concludes the induction, so (33) holds for any Clearly, D L T (π, λ) → 0 when the mesh of π goes to zero.Then, )) when the mesh of π goes to zero.By continuity of (τ, t) → U τ,t , considering a finer and finer partitions leads to U τ,t ≤ A(λ) λ (ω τ,t ) for any r ≤ τ ≤ t ≤ T (λ).
Finally, choosing T (λ) so that T (λ) increases to T defined in Hypothesis 1 when λ goes to 1, we conclude that for any r ≤ τ ≤ t ≤ T , where L T is defined in Hypothesis 1.This proves that z ∈ G L T φ (r, a).
Proof of Theorem 2. Lemma 2-7 prove that conditions H0-H4 of Definition 8 hold for F = G L T φ .This means that G L T (r, a) is a family of abstract local integral funnels.We apply Theorem 1 in [10].For any (r, a) ∈ T × V, there exists a measurable map a → (t → ψ t,r (a)) respect to the Borel subsets of C 0 (T, V) with the property that ψ r,r (a) = a and ψ t,s • ψ s,r (a) = ψ t,r (a), t ≥ r.

Lipschitz flows
A Lipschitz almost flow which has the flow property is said to be a Lipschitz flow.We recast the definition.
Definition 10 (Lipschitz flow).A flow ψ ∈ F(V) is said to be a Lipschitz flow is for any (s, t) ∈ T + 2 , ψ t,s is Lipschitz in space with ψ t,s Lip ≤ 1 + δ T .
In this section, we consider galaxies that contain a Lipschitz flow.
We prove that such a Lipschitz flow ψ is the only possible flow in the galaxy (Theorem 5), and that the iterated almost flow φ π of any almost flow φ converges to ψ (Theorem 3).We also characterize the rate of convergence (Theorem 4).
where is defined by (5).Hence, the galaxies remain the same when is changed to λ .We define Theorem 3. Let φ be an almost flow such that φ π Lip ≤ 1 + δ T whatever the partition π, we say that φ satisfies the uniform Lipschitz (UL) condition.Then there exists a Lipschitz flow ζ ∈ F(V) with ζ s,t Lip ≤ 1 + δ T such that {φ π } converges to ζ as |π| → 0. Theorem 4. Let φ be an almost flow and ψ be a Lipschitz flow with ψ ∼ φ.Then there exists a constant K that depends only on λ, ∆ N, (φ, ψ), κ and T (assumed to be small enough) so that In particular, {φ π } π converges to ψ as |π| → 0.
Remark 8.In Hypothesis 2, the role of ψ and χ are not exchangeable: ψ is assumed to be Lipschitz, there is no such requirement on χ.The reason of this dissymmetry lies in (41).
Remark 9.If ψ is a Lipschitz almost flow and χ is an almost flow, then (ψ, χ) satisfies Hypothesis 2 for any partition π.The condition (37) is a particular case of (15).
We choose λ and T so that We define (recall that Θ(π) is given by ( 35)), and L(π Here, Θ(π) and thus γ(π) converge to zero when the mesh of π tends to zero.
Assume that the induction hypothesis is true at some level m ≥ 1.Let (r, s, t) ∈ π + 3 with r < s < t and d π (r, t) = m + 1.Let s be such that s and s are successive points in π (possibly, s = t).Clearly, d π (r, s) ≤ m and d π (s , t) ≤ m.Using (42) to decompose F s,t using s and using (44), With the induction hypothesis, since r and s (resp.s and t) are at distance at most m, Choosing s and s to satisfy (21), our choice of L(π) and ( 5) imply that The induction hypothesis (43) is then true at level m + 1, and then whatever the distance between the points of the partition.
We may then rewrite (39) as Since γ(π) decreases to 0 as |π| decreases to 0 and |π ∪ {r, t}| ≤ |π|, it is easily shown that {φ π t,s } π forms a Cauchy net with respect to the nested partitions.Then, it does converges to a limit ζ s,t (a).By Theorem 8 and the continuity of N , Moreover ζ does not depend on the subdivision π.Indeed, if π is another subdivision, we obtain with (45), that {φ π} π converges to ζ when |π| → 0.
Corollary 2. Let ψ and χ be two almost flows with ψ ∼ χ and ψ be Lipschitz.Then for T small enough (in function of some λ < 1, κ and δ) .
Proof.With Remark 9, (ψ, χ) satisfies Hypothesis 2. Letting the mesh of the partition decreasing to 0 as the in proof of Theorem 5, and then letting λ increasing to 1 leads to the result.

Perturbations
In this section, we consider the construction of almost flow by perturbations of existing ones.We assume that V is a Banach space.
Let φ ∈ F(V) be an almost rough path with respect to a function N γ such that N γ (a) ≥ N γ (0) ≥ 1.
Notation 6.For φ ∈ F(V) when V is a Banach space, we write .
for some λ ≥ 0. We say that is a perturbation.
We control the first term with (13), |I r,s,t | ≤ N γ (a) (ω r,t ).For the second one, we use ( 6), ( 12) and (47), for γ ∈ (0, 1).With (47) and (46), we obtain for the third term, where the last line comes from (6).And for the last term, we use ( 15) and (47), Thus, combining estimations for each four terms of (49), we obtain (13) which proves that ψ is an almost flow.Example 1 (Lyons extension theorem).With the tensor product ⊗ as product and a suitable norm, for any integer k, the tensor algebra is a Banach algebra.Chen series of iterated integrals (and then rough paths) take their values in some space T k (X).The Lyons extension theorem states that any rough path x of finite p-variation with values in T k (X) for some k ≥ p is uniquely extended to a rough path with values in T (X) for any ≥ k, which leads to the concept of signature [24,25].This follows a → a ⊗ x s,t as an almost flow which satisfies the UL condition (see also [18] and also [13]).

Rough differential equation
Now, we show that our construction is related to the one of A.M. Davie [15].The main idea is to define an almost flow as the truncated Taylor expansion of the solution of (1).The number of term in the Taylor expansion depends directly on the regularity of x.In the Young case one term is needed whereas in the rough case two terms are required.
Here U and V are two Banach spaces, where we use the same notation |•| to denote their norms which can differ.We denote by L(U, V) the continuous linear maps from U to V. Let f be a map from V to L(U, V).If f is regular, we denote its Fréchet derivative in a ∈ V, df (a) ∈ L(V, L(U, V )).

Almost flow in the Young case
Let x : T → U be a path of finite p-variation controlled by ω with 1 ≤ p < 2.
We define a family (φ t,s ) (s,t)∈T + 2 in F(V) such that for all a ∈ V and (s, t) ∈ T + 2 , φ t,s (a) := a + f (a)x s,t , where x s,t := x t − x s .
Proof.We check that assumptions of Definition 2 hold.Let (r, s, t) be in T + 3 and let a, b be in V. First, φ t,t (a) = a because x t,t = 0. Second, Let x = (1, x 1 , x 2 ) be a rough path with values in T 2 (U) of finite p-variation, 2 ≤ p < 3, controlled by ω (See e.g., [19,24] for a complete definition).
We define a family (φ t,s ) (s,t)∈T + 2 in F(V) such that for all a ∈ V and (s, t) ∈ T + 2 , Proof.We check that the assumptions of Definition 2 hold.The proofs of ( 10), (11) and ( 12) are very similar to the ones in the proof of Proposition 5.The computation to show ( 13) is a bit more involved.Indeed, for any a ∈ V, (r, s, t) ∈ T

Notation 4 .
For an almost flow φ, let us denote by S φ (s, a) the set of all the possible limits of the net {φ π •,s (a)} π in ( C([s, T ], V), • ∞ ) for nested partitions.

Definition 7 (
Splicing of paths).For r ≤ s, let us consider y(r, a) ∈ C a ([r, T ], V) and z ∈ C b ([s, T ], V) with b = y s (r, a).Their splicing is

Lemma 4 .
Under Hyposthesis 1, G L T φ (r, a) is non-empty compact subset of the path y ∈ C(S, V) such that y r = a for any r ∈ T and a ∈ V.It shows that H1 holds for F := G L T φ .Proof.It follows directly from Proposition 1 and Lemma 3 (with K T = 1 and K Besides, |ψ t,s (a) − φ t,s (a)| = | t,s (a)| ≤ λN γ (a) (ω s,t ),which proves that ψ ∼ φ and concludes the proof.
|φ t,s (a) − a| ≤ |f (a)| • |x s,t | ≤ |f (a)| • x p ω 1/p s,t , 6.4.2 Almost flow in the rough case When the regularity of x is weaker than in the Young case, we need more terms in the Taylor expansion to obtain an almost flow.Let T 2 (U) := R ⊕ U ⊕ (U ⊗ U) be the truncated tensor algebra (with addition + and tensor product ⊗).A distance is defined on the subset of elements of T 2 (U) on the form a = 1 + a 1 + a 2 with a i ∈ U ⊗i by d(a, b) = |a −1 ⊗ b| where |•| is a norm on T 2 (U) such that |a ⊗ b| ≤ |a| • |b| for any a, b ∈ U.