I Bailleul
email: ismael.bailleul@univ-rennes1.fr

R Catellier
email: remi.catellier@unice.fr

Non-explosion criteria for rough differential equations driven by unbounded vector fields

We give in this note a simple treatment of the non-explosion problem for rough differential equations driven by unbounded vector fields and weak geometric rough paths of arbitrary roughness.

-Introduction

Although rough paths theory has now been explored for twenty years, a few elementary questions are still begging for a definite answer. We consider the existence problem for the local time and occupation time of solutions to rough differential equations as the main open problem, in relation with reflection problems. At a more fundamental level, the question of global in time existence of solutions of a rough differential equation dz t " Fpz t q dX t ,

under relaxed boundedness assumptions on the vector fields F " pV 1 , . . . , V ℓ q has not been clarified so far. Given a weak geometric p-rough path X defined on some time interval r0, T s, the preceding equation is known to have a solution defined on the whole of r0, T s if the driving vector fields V i are C γ b , for some regularity exponent γ ą p; see for instance T. Lyons' seminal paper [START_REF] Lyons | Differential equations driven by rough signals[END_REF] or the lecture notes [START_REF] Caruana | Differential equations driven by rough paths[END_REF]. One would ideally like to relax these boundedness assumptions to some linear growth assumption, but the following elementary counter-examples of Gubinelli and Lejay [START_REF] Lejay | Global solutions to rough differential equations with unbounded vector fields[END_REF] shows that this is not sufficient. Consider the dynamics (1.1) on R2 , with F " pV 1 , V 2 q, and vector fields V 1 px, yq " px sinpyq, xq and V 2 px, yq " 0, driven by the non-geometric pure area rough path X t " 1 `tp1 b 1q. Writing z t " px t , y t q, one sees that z is actually the solution of the ordinatry differential equation 9 z t " p 9

x t , 9 y t q " `xt sinpy t q 2 `x2 t cospy t q, x t sinpy t q ˘.

The solution started from an initial condition of the form pa, 0q, with a positive, has constant null y-component and has an exploding x-component since 9

x t " x 2 t . The non-explosion problem was explored in a number of works for differential equations driven by p-rough paths, for 2 ď p ă 3, especially in the works of Davie [START_REF] Davie | Differential equations driven by rough paths: an approach via discrete approximations[END_REF] and Lejay [START_REF] Lejay | On rough differential equations[END_REF][START_REF] Lejay | Global solutions to rough differential equations with unbounded vector fields[END_REF]. Davie provides essentially the sharpest result in the regime 2 ď p ă 3.

' To make it simple, assume F is C 3 and has linear growth: ˇˇFpxq ˇˇÀ |x|. Theorem 6.1 (a) in [START_REF] Davie | Differential equations driven by rough paths: an approach via discrete approximations[END_REF] provides a non-explosion criterion in terms of the growth rate of D 2 F ˇˇD 2 F pxq ˇˇď hp|x|q.

2

There is no explosion if hprq À 1 r , and ż 8 ˆrγ´2 hprq ˙p´1 γ´1 dr r p " 8. Davie's criterion is shown to be sharp in the class of all p-rough paths, 2 ď p ă 3, with an example of a rough differential equation where explosion can happen for some appropriate choice of a non-weak geometric rough path in case the criterion is not satisfied -see Section 6 in [START_REF] Davie | Differential equations driven by rough paths: an approach via discrete approximations[END_REF]. The limit case for Davie's criterion is hprq " Op1q r . We essentially recover that bound. ' Lejay [START_REF] Lejay | On rough differential equations[END_REF] works with Banach space valued weak geometric p-rough paths, with 2 ď p ă 3. In the setting where the vector fields V i are C 3 with bounded derivates and are required to have growth rate ˇˇV i pxq ˇˇÀ gp|x|q, he shows non-explosion of solutions to equation (1.1) under the condition that ř k 1 gpkq p diverges. The limit case is gprq » r 1 p . ' The analysis of Friz and Victoir [START_REF] Friz | Multidimensional stochastic processes as rough paths[END_REF], Exercice 10.56, gives a criterion comparable to ours, with an erronous proof. They use a pattern of proof that is implemented in a linear setting and cannot work in a nonlinear framework as it bears heavily on a scaling argument -see the proof of Theorem 10.53. One can see part of the present work as a correct or alternative proof of their statement.

We identify in the sequel a vector field V on R d with the first order differential operator f Þ Ñ pDf qpV q. For a tuple I " pi 1 , . . . , i k q P t1, . . . , ℓu k , and vector fields V 1 , . . . , V ℓ , we define the differential operators V I :" V i 1 ¨¨¨V i k , and V rIs :"

" V i 1 , . . . , rV i k´1 , V i k s ‰ ı ,
under proper regularity assumptions on the V i . (Note that the operator V rIs is actually of order one, so V rIs is a vector field.) The local increment z t ´zs of a solution z to the rough differential equation (1.1) is known to be well-approximated by the time 1 value of the ordinary differential equation

y 1 r " rps ÿ k"1 ÿ IPt1,¨¨¨,ℓu k Λ k,I ts V rIs `s, y r pxq ˘, (1.2)
where Λ ts :" log X ts , and 0 ď r ď 1 -see [START_REF] Bailleul | Flows driven by rough paths[END_REF] or [START_REF] Boutaib | Dimension-free Euler estimates of rough differential equations[END_REF] for instance. The following simplified version of our main result, Theorem 6, actually gives a non-explosion result in terms of growth assumptions on the vector fields V rIs that appear in the approximate dynamics (1.2). Pick an arbitrary p ą 1 and a weak geometric p-rough path X.

1. Theorem -There is no explosion for the solutions of the rough differential equation (1.1) is the functions V i 1 ¨¨¨V in Id are C 2 with bounded derivatives, for any 1 ď n ď rps and any tuple pi 1 , . . . , i n q P 1, ℓ n .

Theorem 6 is sharper than that statement as it involves the vector fields V rIs -recall Example 3 of [START_REF] Lejay | Global solutions to rough differential equations with unbounded vector fields[END_REF]. In the case where 2 ď p ă 3, our non-explosion criterion becomes

ˇˇD 2 x F ˇˇ_ ˇˇD 3 x F ˇˇÀ 1 1 `|x| ,
for a multiplicative implicit constant independent of x P R d . We mention here that we have been careful on the growth rate of the different quantities but that one can optimize the regularity assumptions that are made on the vector fields V i to get slightly sharper results.

This explains the discrepancy between Davie's optimal criterion in the case 2 ď p ă 3 and our result. These refinements are not needed for the applications [START_REF] Bailleul | Limit theorems for systems of mean field rough differential equations[END_REF]; we leave them to the reader. Note also here that one can replace R d by a Banach space and give versions of the statements involving infinite dimensional rough paths, to the price of using slightly different notations, such as in [START_REF] Bailleul | Flows driven by Banach space walued rough paths[END_REF]. There is no difference between the finite and the infinite dimensional settings for the explosion problem.

Our main result, Theorem 6, holds for dynamics (1.1) with a drift and time-dependent vector fields. It is proved in Section 2 on the basis of some intermediate technical estimates whose proof is given in Section 3. Theorem 6 holds for Hölder p-rough paths. A similar statement holds for more general continuous rough paths, with finite p-variation, such as proved in Section 4 with other corollaries and extensions.

Notations. We gather here a number of notations that are used throughout the paper. ' Given a positive finite time horizon T , we denote by ∆ T the simplex tpt, sq P r0, T s 2 : 0 ď s ď t ď T u. ' We refer the reader to Lyons' seminal article [START_REF] Lyons | Differential equations driven by rough signals[END_REF] or any textbook or lectures notes on rough paths [START_REF] Lyons | System control and rough paths[END_REF][START_REF] Caruana | Differential equations driven by rough paths[END_REF][START_REF] Friz | Multidimensional stochastic processes as rough paths[END_REF][START_REF] Baudoin | Diffusion Processes and Stochastic Calculus[END_REF][START_REF] Bailleul | A flows-based approach to rough differential equations[END_REF] for the basics on rough paths theory and simply mention here that we work throughout with finite dimensional weak geometric Hölder p-rough paths X " 1 ' X 1 ' ¨¨¨' X rps , with values in À rps i"0 pR ℓ q bi say, and norm

}X} :" max 1ďiďrps sup 0ďsătďT ˇˇX i ts ˇˇ1 i |t ´s| 1 p .
Note that if Λ " `0 ' Λ 1 ' ¨¨¨' Λ rps ˘is the logarithm of the rough path X, we have for all 0 ď s ď t ď T , all i P t1, ¨¨¨, rpsu,

ˇˇΛ i ts ˇˇÀ i }X} i |t ´s| i p
' Last, we use the notation a À b to mean that a is smaller than a constant times b, for some universal numerical constant. One can trade in the above assumption some growth condition on the V i against some growth condition on its derivatives; this is the rationale for introducing the notion of α-growth.

-Solution flows to rough differential equations

Assumption -Time regularity and growth.

There exists some regularity exponents κ 1 ě We assume that the derivative in x of the V rIns pt, ¨q ¨¨¨V rI 1 s pt, ¨q also satisfies the previous estimate.

Let X be an R ℓ -valued weak geometric Hölder p-rough path. Set Λ ts :" log X ts , for all 0 ď s ď t ď T , and denote by µ ts the time 1 map of the ordinary differential equation

y 1 r " pt ´sqV 0 `s, y r pxq ˘`rps ÿ k"1 ÿ IPt1,¨¨¨,ℓu k Λ k,I t,s V rIs `s, y r pxq ˘(2.1)
that associates to x the value at time 1 of the solution path to that equation with initial condition x. Note that Assumption 1 ensures that equation (2.1) is well-defined up to time 1 . Following [START_REF] Bailleul | Flows driven by rough paths[END_REF], we define a solution flow to the rough differential equation

dϕ t " V 0 pt, ϕ t qdt `Fpt, ϕ t qdX t , (2.2)
where F :" pV 1 , . . . , V ℓ q, as a flow locally well-approximated by µ. Here, we take advantage in this definition of some variant of the definition of [START_REF] Bailleul | Flows driven by rough paths[END_REF] introduced by Cass and Weidner in [START_REF] Cass | Tree algebras over topological vector spaces in rough path theory[END_REF]. For a parameter a, the notation C a stands for a constant depending only on a.

Definition -

A flow ϕ : ∆ T ˆRd Þ Ñ R d is
whenever |t ´s| ď ε X .
Note that we require the flow to be globally defined in time and space, unlike local flows of possibly exploding ordinary, or rough, differential equations. The latter are only defined on an open set of R `ˆR d depending on X. This definition differs from the corresponding definition in [START_REF] Bailleul | Flows driven by rough paths[END_REF] in the fact that ε R is required to be independent of X. We first state a local in time existence result for the flow, in the spirit of [START_REF] Bailleul | Flows driven by rough paths[END_REF].

5. Theorem -Let the vector fields V 0 and pV 1 , . . . , V ℓ q satisfy Assumption 1 and Assumption 2.

' There exists a positive constant a 1 such that for all R ą 0, and all pt, sq P ∆ T with |t ´s| for some universal positive constant a 2 . One writes ϕpXq to emphasize the dependence of ϕ on X. ' Given a weak geometric rough path X and ps, tq P ∆ T and R such that condition (2.4) holds, then ϕpX 1 q is well-defined on rs, ts ˆBp0, Rq for X 1 sufficiently close to X, and ϕpX 1 q converges to ϕpXq in L 8 `rs, ts ˆBp0, Rq ˘as X 1 tends to X.

One says that ϕ depends continuously on X in the topology of uniform convergence on bounded sets. As you can see from the statement of Theorem 5, the quantity |t ´s| is only required in that case to be smaller than a constant depending on X and R, unlike what is required from a solution defined globally in time. The proof of Theorem 5 mimics the proof of the analogue local in time result proved in [START_REF] Bailleul | Flows driven by rough paths[END_REF]. As the proof of latter contains typos that makes reading it hard, we give in Section 3 a self-contained proof of this result.

6. Theorem -Let V 0 and pV 1 , . . . , V ℓ q satisfy Assumption 1 and Assumption 2. There exists a unique global in time solution flow ϕ to the rough differential equation (2.2).

' One can choose in the defining relation (2.3) for a solution flow

η " 1 `rps p , ε X " c 1 `1 `}X} ˘´p , C R,X " c 2 p1 `Rq α `1 `}X} ˘rps`1 ,
for some universal positive constants c 1 , c

´1'

, if α ă 1 p1 `Rq|t ´s|

1 p e c 4 N |t´s| 1 p , if α " 1.
The non-trivial part of the proof consists in proving that one can patch together the local flows contructed in Theorem 5 and define a globally well-defined flow. As this requires a careful track of a number of quantities, we provide a proof of the technical results in Section 3. Since it is the main contribution of this work, we also give a proof of this theorem using some results of lemmas and propositions of Section 3.

Proof of Theorem 6 -Fix ps, tq P ∆ T . For n ě 0 and 0 ď k ď 2 n set t n k :" k2 ´npt ´sq`s and µ n t,s :"

µ t n 2 n ,t n 2 n ´1 ˝¨¨¨˝µ t n 1 ,t n 0 .
Here is the makor input for the proof of the statement. Proposition 16 below states the existence of universal positive constants c 1 ă 1 and c 2 such that for |t ´s|

› › › L 8 pBp0,Rqq ď R `c p1 `Rq α ,
for all 0 ď k ď 2 n ´1. Let n " npRq be the least integer such that

2 ´n 1 p p1 `Rq α rps`1 |t ´s| 1 p `1 `}X} ˘ď a 1 p1 `2cq α rps`1
. This is the smallest integer such that for all the intervals pt n k , t n k`1 q satisfy the assumption of Theorem 5, with starting point µ t n k t n k´1 ˝¨¨¨˝µ t n 1 t n 0 pxq and x P Bp0, Rq. Then, we have for all m 0 , ¨¨¨, m 2 n ´1 P N, › › ›µ

m 2 n ´1 t n 2 n t n 2 n ´1 ˝¨¨¨˝µ m 0 t n 1 t n 0 ´µts › › › L 8 pBp0,Rqq ď c 1 ˇˇt ´sˇˇ1`rps p p1 `Rq α `1 `}X} ˘rps`1 .
Sending successively m 2 n ´1, . . . , m 0 to 8 and using the continuity of ϕ with respect to its R d -valued argument gives

› › ›ϕ t n 2 n t n 2 n ´1 ˝¨¨¨˝ϕ t n 1 t n 0 ´µts › › › ď c 2 ˇˇt ´sˇˇ1`rps p p1 `Rq α `1 `}X} ˘rps`1 . (2.6)
Set, for x P Bp0, Rq,

ϕ ts pxq :" ϕ t n 2 n t n 2 n ´1 ˝¨¨¨˝ϕ t n 1 t n 0 pxq. Splitting the intervals pt n k , t n k`1
q into dyadic sub-intervals, one shows that for all u P rs, ts of the form u " k2 ´N pt ´sq `s, one has ϕ t,u ˝ϕu,s pxq " ϕ t,s pxq.

Finally, since the map px, s, tq Ñ ϕ t n k`1 ,t n k pxq is a continuous for all 0 ď k ď 2 n ´1, so is ϕ. This proves the first item of Theorem 6.

The second item is a byproduct of the bound of Equation (2.3) and Corollary 13 below. The third item of the statement is straightforward given that ϕ is constructed from patching together local solution flows.

Choose finally a positive constant c 3 big enough such that setting

N :" " c 3 `1 `}X} ˘pȷ ,
one has t´s N ď ε X and `1 `}X} ˘N ´1 p ď 1. Define also

t i :" i N pt ´sq `s,
and R 0 :" 0 and R i :" sup xPBp0,Rq

ˇˇϕ t i s pxq ´xˇˇ,
for 1 ď i ď N . Note that

C |t i`1 ´ti |,}X} " rps ÿ i"1 ˆ1 `}X} N 1 p ˙i |t ´s| i p À |t ´s| 1 p ,
for a universal positive multiplicative factor. We thus have

ϕ t i s pxq ´x " ϕ t i t i´1 `ϕt i´1 s pxq ˘´µ t i t i´1 `ϕt i´1 s pxq μt i t i´1 `ϕt i´1 s pxq ˘´ϕ t i´1 s pxq `ϕt i´1 s pxq ´x,
and there is an absolute positive constant K such that

R i ď R i´1 `Kp1 `R `Ri´1 q α |t ´s| 1 p ;
the bounds on ϕ ts pxq ´x given in the statement follows from that relation. ◃

As a corollary of Theorem 6, one proves in Theorem 24 the differentiability of the solution flow with respect to some parameters. This theorem will be of crucial importance in the forthcoming work [START_REF] Bailleul | Limit theorems for systems of mean field rough differential equations[END_REF]; we state it here in a readily usable form.

Assumption -Let

A be a Banach, parameter space and let U be a bounded open subset of A. Let pV i q 0ďiďℓ be time and parameter-dependent vector fields on R d with the following regularity properties.

' There exists some exponents κ 1 ą 1`rps´p p and κ 2 ą rps p , such that we have for all integers β 1 , β 2 with 0 ď β 1 `β2 ď rps `1,

sup 0ďsďtďT › › ›D β 1 a D β 2 x V 0 pt, ., .q ´Dβ 1 a D β 2 x V 0 ps, ., .q › › › L 8 pR d ˆU q |t ´s| κ 1 ă `8,
' For all 1 ď i ď ℓ, and all integers β 1 , β 2 with 0 ď β 1 `β2 ď rps `2, we have

sup 0ďsďtďT › › ›D β 1 a D β 2 x V i pt, ., .q ´Dβ 1 a D β 2 x V i ps, ., .q › › › L 8 pR d ˆU q |t ´s| κ 2 ă ` 8
Refer to Definition 22 in Section 4.3 for the definition of the local accumulation N β of X. 8. Theorem -Let X be a R ℓ valued weak geometric Hölder p-rough path and suppose that V 0 , V 1 , ¨¨¨, V ℓ satisfy Assumptions 7. Let ϕpa, ¨q stand for all a P U for the solution flow to the equation dϕpa, ¨q " V 0 `t, a, ϕpa, ¨q˘d t `σ`t , a, ϕpa, ¨q˘d X t .

(2.7) Then for all 1 ď s ď t ď T , the function pa, xq Þ Ñ ϕ ts pa, xq is differentiable and ' for |t ´s|

-Complete proof of Theorem 5

The structure of the proof is simple. One first proves C 2 estimates on the time r map of the ordinary differential equation (2.1), this is the content of Lemma 9. Building on a Taylor formula given in Lemma 11, and quantified in Lemma 12 and Corollary 13, one shows in Proposition 15 that the µ's defined what could be called a 'local approximate flow', after [START_REF] Bailleul | Flows driven by rough paths[END_REF]. We then follow the construction recipe of a flow from an approximate flow given in [START_REF] Bailleul | Flows driven by rough paths[END_REF], by patching together the local flows. The crucial global in time existence result is obtained as a consequence of a Grönwall type argument, as can be expected from the fact that, in their simplest form, the growth assumptions of Theorem 6 mean that all the vector fields appearing in the approximate dynamics have α-growth. Readers familiar with [START_REF] Bailleul | Flows driven by rough paths[END_REF] can go directly to Section 4.

Recall the definition of y r as the solution of the ordinary differential equation (2.1) defining µ ts . The first step in the analysis consists in getting some local in space C 2 estimate on y r p¨q ´Id, with y r p¨q seen as a function of the initial condition x in (2.1). Set

C |t´s|,}X} :" |t ´s| `rps ÿ i"1 |t ´s| i p }X} i .
9. Lemma -Assume V 0 and pV 1 , . . . , V ℓ q satisfy the space regularity Assumption 1, and pick ps, tq P ∆ T with |t ´s|

ˇˇD k y r pxq ˇˇÀ C |t´s|,}X} ,
under the assumption that |t ´s|

1 p `1 `}X} ˘ď 1.
The second step of the analysis is an elementary explicit Taylor expansion; see [START_REF] Bailleul | Flows driven by rough paths[END_REF] for the model situation. Given 1 ď n ď rps, set

∆ n 1 :" ! `rn , ¨¨¨, r 1 ˘P r0, 1s n : r n ď r n´1 ď ¨¨¨ď r 1 )
and

I n,rps :" ! pI 1 , ¨¨¨, I n q P t1, ¨¨¨, du k 1 ˆ¨¨¨ˆt1, ¨¨¨, du kn ; n ÿ m"1 k m ď rps ) ;
indices k m above are non-null.

11. Lemma -Assume V 0 and pV 1 , . . . , V ℓ q satisfy the space regularity Assumption 1. For any 1 ď n ď rps and any vector space valued function f on R d of class C n we have the Taylor formula f `µts pxq ˘" f pxq `pt ´sq `V0 ps, ¨qf ˘pxq

`n ÿ i"1 1 i! ÿ I i,rps i ź m"1 Λ km,
`n´1 ÿ i"1 1 i! ÿ I i,rps pt ´sq i ź m"1 Λ km,Im ts ż ∆ i`1 1 `V0 ps, ¨qV rI i s ps, ¨q ¨¨¨V rI 1 s ps, ¨qf ˘`y r i`1 pxq ˘dr `n ÿ i"2 ÿ I i´1,rps k 1 `¨¨¨`k i ěrps`1 i ź m"1 Λ km,Im ts ż ∆ i 1 `VrI i s ps, ¨q ¨¨¨V rI 1 s ps, ¨qf ˘`y r i pxq ˘dr.
Proof -The proof is done by induction, and relies on the following fact. For all u P r0, 1s and all g P C 1 pB; Bq, we have gpy r q ´gpxq " pt ´sq

ż r 0 `V0 ps, ¨qg ˘py u q du `ÿ 1ďkďrps IPt1,¨¨¨,ℓu k Λ k,I t,s ż r 0 `VrIs ps, ¨qg ˘py u q du;
this is step 1 of the induction. For step 2, apply step 1 successively to g " f and u " 1, then g " pV rIs ps, ¨qf ˘and u " r. This gives f `µts pxq ˘´f pxq " pt ´sq `V0 ps, ¨qf ˘pxq

`pt ´sq ż 1 0 ! `V0 ps, ¨qf ˘py r q ´`V 0 ps, ¨qf ˘pxq ) dr `ÿ 1ďkďrps IPt1,¨¨¨,ℓu k Λ k,I ts `VrIs ps, ¨qf ˘pxq `ÿ 1ďkďrps IPt1,¨¨¨,ℓu k pt ´sqΛ k,I ts ż 1 0 ż r 1 0
`V0 ps, ¨qV rIs ps, ¨qf ˘py r 2 q dr 2 dr 1 `ÿ 1ďk 1 ,k 2 ďrps

I 1 Pt1,¨¨¨,ℓu k 1 I 2 Pt1,¨¨¨,ℓu k 2 2 ź m"1 Λ km,Im ts ż 1 0 ż r 1 0
`VrI 2 s ps, ¨qV rI 1 s ps, ¨qf ˘py r 2 q dr 2 dr 1 .

The last term of the right hand side can be decomposed into

1 2 ÿ I 2,rps 2 ź m"1
Λ km,Im ts `VrI2s ps, ¨qV rI1s ps, ¨qf ˘pxq

`ÿ I 2,rps 2 ź m"1 Λ km,Im ts ż ∆ 2 1
! `VrI2s ps, ¨qV rI1s ps, ¨qf ˘py r2 q ´`V rI2s ps, ¨qV rI1s ps, ¨qf ˘pxq ) dr 2 dr 1 `ÿ k1`k2ěrps`1

I 1,rps 2 ź m"1 Λ km,Im ts ż ∆ 2 1
`VrI2s ps, ¨qV rI1s ps, ¨qf ˘py r2 q dr 2 dr 1 ;

this proves step 2 of the induction. The n to pn `1q induction step is done similarly, and left to the reader. ◃

Given f P C n b `Rd , R d ˘, set }f } n :" ˇˇf p0q ˇˇ`sup kPt1,¨¨¨,nu › › D k f › › 8 .
A

ˇˇD x ε n,f ts ˇˇÀ }f } n`1 p1 `Rq nα `1 `}X} ˘rps`1 pt ´sq 1`rps p .
' If finally f satisfies Assumption H, then the previous bound on D x ε n,f ts holds with p1 `Rq α in place of p1 `Rq nα .

Proof -Write dr for dr i . . . dr 1 on ∆ i 1 , and recall that ε n,f ts pxq " pt ´sq ż 1 0 ! `V0 ps, ¨qf ˘py r q ´`V 0 ps, ¨qf ˘pxq ) dr

`n´1 ÿ i"1 1 i! ÿ I i,rps pt ´sq i ź m"1 Λ km,Im ts ż ∆ i`1 1 ! V 0 ps, ¨qV rI i s ps, ¨q ¨¨¨V rI 1 s ps, ¨qf ) py r i`1 q dr `n ÿ i"2 ÿ I i´1,rps k 1 `¨¨¨`k i ěrps`1 i ź m"1 Λ km,Im ts ż δ i 1 !
V rI i s ps, ¨q ¨¨¨V rI 1 s ps, ¨qf

) py r i q dr. (3.1)
Recall also that C |t´s|,}X} À 1 under the assumption of the statement.

' As we have for all positive radius R, and all points x, y P Bp0, Rq, the estimate ˇˇV 0 ps, ¨qf pxq ´V0 ps, ¨qf pyq ˇˇď }f } n `1 One can replace in the previous bounds the first term p1 `Rq pi`1qα by p1 `Rq α and the second term p1 `Rq pi`1qα by 1 if f satisfies Assumption H. So ˇˇˇˇˇn ´1 ÿ i"1

`
1 i! ÿ I i,rps pt ´sq i ź m"1 Λ km,Im ts ż ∆ i`1 1
`V0 ps, ¨qV rI i s ps, ¨q ¨¨¨V rI 1 s ps, ¨qf ˘py r i`1 q dr ˇˇˇÀ }f } n pt ´sq 1`1 p p1 `Rq nα `1 `}X} ˘rps and ˇˇˇˇˇn ´1 ÿ i"1

1 i! ÿ I i,rps pt ´sq i ź m"1 Λ km,Im ts ż∆ i`1 1 D ! V 0 ps, ¨qV rI i s ps, ¨q ¨¨¨V rI 1 s ps, ¨qf ˘`y r i`1 pxq ) Dy r i`1 pxq dr ˇˇˇÀ }f } n`1 pt ´sq 1`1 p p1 `Rq nα `1 `}X} ˘rps .
Once again, if the function f satisfies Assumption H, one can replace p1 `Rq nα by p1 `Rq α in the first bound and p1 `Rq nα by 1 in the second bound.

The analysis of the last term in the right hand side of the decomposition (3.1) for ε n,f ts is a bit trickier since greater powers of }X} can pop out. Indeed, one has

ˇˇˇn ÿ i"2 ÿ I i´1,rps k 1 `¨¨¨`k i ěrps`1 i ź m"1 Λ km,Im ts ż ∆ i 1 !
V rI i s ps, ¨q ¨¨¨V rI 1 s ps, ¨qf

) py r i q dr ˇˇÀ }f } n rps ÿ l"1 `1 `}X} ˘rps`i |t ´s| rps`i p p1 `Rq nα .
But recall that |t ´s| The first estimate of the corollary follows then from the fact that exppΛ ts q " X ts .

The two other estimates are consequences of the fact that the identity map satisfies Assumption 1 and Assumption H. ◃ 14. Remark -As in Remark 10, one can require that V 0 and V rIs are more regular, and ask ' For all 1 ď k 1 , ¨¨¨, k n ď rps, with ř n i"1 k i ď rps, and all I k i P t1, ¨¨¨, ℓu k i , the functions V 0 ps, ¨qV rI n´1 s ps, ¨q ¨¨¨V rI 1 s ps, ¨q and V rIns ps, ¨q ¨¨¨V rI 1 s ps, ¨q are C 2`n b with α-growth, uniformly in time.

Under that stronger regularity assumption, we have for all 2 ď k ď n `1,

sup xPBp0,Rq ˇˇˇD k µ ts pxq´!pt ´sqD k V 0 ps, xq `rps ÿ j"1 ÿ IPt0,¨¨¨,ℓu j X j,I ts D k V I ps, xq ) ˇˇÀ p1 `Rq α `1 `}X} ˘rps`1 |t ´s| rps`1 p .
The next proposition shows that µ satisfies a localized version of an approximate flow; see [START_REF] Bailleul | Flows driven by rough paths[END_REF].

15. Proposition -Given 0 ď s ď u ď t ď T , with pt ´sq The bounds of the statement can be read on that decomposition; we give the details for pµ tu ˝µus qpxq and live the details of the estimate for its derivative to the reader. It follows from Assumption 2 on the time regularity of V 0 and the V rI i s . . . V rI 1 s Id that ˇˇp1q ˇˇ`ˇˇp4q ˇˇÀ p1 `Rq α ´pt ´uqpu ´sq κ 1 ``1 `}X} ˘rps pt ´uq

1 p pu ´sq κ 2 À p1 `Rq α pt ´sq 1`rps p `1 `}X} ˘rps .
Lemma 12 takes care of the remainder terms p6q. By using lemma 9 and the fact that V 0 is Lipschitz continuous in space, uniformly in time, one gets ˇˇp2q ˇˇÀ pt ´uqpu ´sq To estimate the terms p3q and p5q, set gps, ¨q :" V rI i s ps, ¨q ¨¨¨V rI 1 s ps, ¨qId.

We start by doing a Taylor expansion of g `s, µ ts pxq ˘using Lemma 11, to the order n " rps´ř i j"1 k j . As gps, ¨q satisfies Assumption H as a consequence of Assumption 1, one can use Lemma 12 to get the expected bounds, using the fact that X u,s X t,u " X t,s and exppΛq " X. Details of these algebraic computations can be found in the proof of the corresponding statement in [START_REF] Bailleul | Flows driven by rough paths[END_REF]. ◃

Remark -One has similar local bounds for higher derivatives of µ tu ˝µus ´µts in the setting of Remark 14.

Write here part of the conclusion of

2 n t n`1 2 n ´1 ˝¨¨¨˝µ t n`1 1 t n`1 0 ˘pxq ˇď ˇˇ`µ t n`1 2 n t n`1 2 n ´1 ˝¨¨¨˝µ t n`1 1 t n`1 0 ˘pxq ´µus pxq ˇˇ`ˇˇµ us pxq ˇď L 2 ´1`rps p |t ´s| 1`rps p p1 `Rq α `1 `}X} ˘rps`1 `R `C|t´s|,}X} p1 `Rq α ď R `2p1 `Rq α ε 0 .
and sup xPBp0,Rq ˇˇDµ tu pxq ˇˇď 1 `2ε 0 , by Lemma 9. Furthermore we have

µ t n`1 2 n`1 t n`1 2 n`1 ´1 ˝¨¨¨˝µ t n`1 1 t n`1 0 pxq ´µts pxq " ´µt n`1 2 n`1 t n`1 2 n`1 ´1 ˝¨¨¨˝µ t n`1 2 n `1t n`1 2 n ´µtu ¯˝´µ t n`1 2 n t n`1 2 n ´1 ˝¨¨¨˝µ t n`1 1 t n`1 0 ¯pxq `µtu ˝`µ t n`1 2 n t n`1 2 n ´1 ˝¨¨¨˝µ t n`1 1 t n`1 0
˘pxq ´µtu ˝µus pxq `µtu ˝µus pxq ´µts pxq.

We thus have for all x P Bp0, Rq, the estimate

ˇˇµ t n`1 2 n`1 t n`1 2 n`1 ´1 ˝¨¨¨˝µ t n`1 1 t n`1 0 pxq ´µts pxq ˇď L ˇˇˇt ´s 2 ˇˇˇ1 `rps p ´1 `R `2ε 0 p1 `Rq α ¯α`1 `}X} ˘rps`1 `p1 `2ε 0 q L p1 `Rq α ˇˇˇt ´s 2 ˇˇˇ1 `rps p `1 `}X} ˘rps`1 `C0 |t ´s| 1`rps p p1 `Rq α `1 `}X} ˘rps`1 ,
from which the induction step follows given our choice of ε 0 and L. ◃

The same bound for the derivative of the approximate flow requires a bound on |t ´s| that depends on p1 `Rq α , such as described here. 17. Proposition -One can find a positive constant ε 1 ă 1 such that for 0 ď s ď t ď T with

C 0 |t ´s| 1 p p1 `Rq α 1`rps `1 `}X} ˘ď ε 1 ,
we have, for all positive radius R,

sup |x|ďR ˇˇD `µt n 2 n t n 2 n ´1 ˝¨¨¨˝µ t n 1 t n 0 ˘pxq ´D`µ ts ˘pxq ˇˇď L |t ´s| 1`rps p p1 `Rq α `1 `}X} ˘rps`1 .
Proof -The proof is a variation on the theme of the proof of Proposition 16. We provide the details for the reader's convenience, and keep the notation u for s`t 2 . We proceed here as well by induction and loot at the 'n to n `1' induction step of the proof.

D ´µt n`1 2 n`1 t n`1 2 n`1 ´1 ˝¨¨¨˝µ t n`1 1 t n`1 0 ¯pxq ´Dµ ts pxq " ! D `µt n`1 2 n`1 t n`1 2 n`1 ´1 ˝¨¨¨˝µ t n`1 2 n `1t n`1 2 n ˘´Dµ tu ) `µt n`1 2 n t n`1 2 n ´1 ˝¨¨¨˝µ t n`1 1 t n`1 0 ˘pxq ˆD`µ t n`1 2 n t n`1 2 n ´1 ˝¨¨¨˝µ t n`1 1 t n`1 0 ˘pxq `"`D µ tu ˘´`µ t n`1 2 n t n`1 2 n ´1 ˝¨¨¨˝µ t n`1 1 t n`1 0 ˘pxq ¯´Dµ tu `µus pxq ˘* ˆD`µ t n`1 2 n ,t n`1 2 n ´1 ˝¨¨¨˝µ t n`1 1 ,t n`1 0 ˘pxq `Dµ tu `µus pxq ˘´D `µt n`1 2 n t n`1 2 n ´1 ˝¨¨¨˝µ t n`1 1 t n`1 0 ˘pxq ´Dµ us pxq D`µ
tu ˝µus ˘pxq ´Dµ ts pxq. We know from the induction step and the R-dependent assumption on u ´s that ˇˇD ´µt n`1

2 n t n`1 2 n ´1 ˝¨¨¨˝µ t n`1 1 t n`1 0 ¯pxq ´Dµ us pxq ˇˇď ε 1 2 ´1`rps p ,
and ˇˇD ´µt n`1

2 n t n`1 2 n ´1 ˝¨¨¨˝µ t n`1 1 t n`1 0 ¯pxq ˇˇď 1 `2ε 1 .
We also have from Proposition 16 ˇˇ´µ t n`1

2 n t n`1 2 n ´1 ˝¨¨¨˝µ t n`1 1 t n`1 0 ¯pxq ˇˇď `1 `2ε 1 ˘p1 `Rq ´1,
and Lemma 9 gives us a uniform control on the Lipschitz size of the µ ba . We thus have

ˇˇD ´µt n`1 2 n`1 t n`1 2 n`1 ´1 ˝¨¨¨˝µ t n`1 1 t n`1 0 ¯pxq ´Dµ ts pxq ˇď L p1 `2ε 1 q 1`α 2 ´1`rps p |t ´s| 1`rps p `1 `}X} ˘rps`1 p1 `Rq α `L 2 ´1`rps p ε 1 p1 `2ε 1 q |t ´s| 1`rps p `1 `}X} ˘rps`1 p1 `Rq α `L p1 `ε1 q 2 ´1`rps p |t ´s| 1`rps p `1 `}X} ˘rps`1 p1 `Rq α `C0 |t ´s| 1`rps p `1 `}X} ˘rps`1 p1 `Rq α ď ´C0 `b2 ´1`rps p `p1 `2εq α `ε1 p1 `2ε 1 q `p1 `ε1 q ˘¯|t ´s| 1`rps p p1 `Rq α `1 `}X} ˘rps`1 .
An adequate choice of ε 1 closes the induction step, given the definition of L. ◃ 18. Remark -In the improved regularity conditions on the vector fields stated in Remark 14, we have for all 2 ď k ď n `1 and for all

C 0 |t ´s| 1 p p1 `Rq kα 1`rps `1 `}X} ˘ď ε 1 , one have sup |x|ďR ˇˇD k `µt n 2 n t n 2 n ´1 ˝¨¨¨˝µ t n 1 t n 0 ˘pxq ´Dk µ ts pxq ˇˇď L |t ´s| 1`rps p p1 `Rq α `1 `}X} ˘rps`1 .
With all these preliminary results at hand, we are now in a position to give a proof of our local well-posedness result, Theorem 5.

Proof of Theorem 5 -We treat existence and uniqueness one after the other. We keep the above notations, and set, in addition,

µ n ts " µ t n 2 n t n 2 n ´1 ˝¨¨¨˝µ t n 1 t n 0 .
Local in time existence -For all x P Bp0, Rq, µ n`1 ts pxq ´µn ts pxq "

2 n ÿ k"1 ´µt n`1 2 n`1 t n`1 2 n`1 ´1 ˝¨¨¨˝µ t n`1 2k`3 t n`1 2k`2 ˝`µ t n`1 2k`2 t n`1 2k`1 ˝µt n`1 2k`1 t n`1 2k μt n`1 2 n`1 t n`1 2 n`1 ´1 ˝¨¨¨˝µ t n`1 2k`3 t n`1 2k`2 ˝`µ t n`1 k`1 t n`1 2k ˘μ t n k t n k´1 ˝¨¨¨˝µ t n 1 t n 0 pxq. (3.
2) It follows from Proposition 17 that the maps

µ t n`1 2 n`1 t n`1 2 n`1 ´1 ˝¨¨¨˝µ t n`1 2k`3 t n`1 2k`2
are Lipschitz continuous, uniformly in n, with a Lipschitz constant that depends neither on X nor on R. Furthermore, thanks to Proposition

The sequence µ n ts is thus uniformly convergent on the ball Bp0, Rq to a limit, continuous, function denoted by ϕ ts ; it satisfies the estimate sup xPBp0,Rq ˇˇϕ ts pxq ´µts pxq ˇˇÀ |t ´s|

1`rps p p1 `Rq α `1 `}X} ˘rps`1 .
Finally, for all dyadic points a P rs, ts and all x P Bp0, Rq, we have by construction ϕ ta pxq ˝ϕas pxq " ϕ ts pxq.

As X is an Hölder continuous rough path, the function px; s, tq Þ Ñ µ ts pxq, from Bp0, Rq ˆt0 ď s ă t ď T u to R d , is continuous. The continuity of ϕ as a function of px; s, tq follows in a straightforward way; its continuous dependence on X is a consequence of the continuous dependence of µ with respect to X. Note however that ϕ ts is only defined at that stage for s and t close enough.

Uniqueness -Let ψ stand for another solution flow, with associated constants ε X and C R,X , and exponent η ą 1. Take R and ps, tq satisfying the conditions of Proposition 17, with |t ´s| ď C R,X . Then ˇˇϕ ts pxq ´ψts pxq ˇˇď ˇˇµ n ts pxq ´ϕts pxq ˇˇ`ˇˇµ n ts pxq ´ψts pxq ˇď ˇˇµ n ts pxq ´ϕts pxq

ˇ2 n ´1 ÿ k"0 ˇˇˇ´`µ t n 2 n t n 2 n ´1 ˝µt n k`2 t n k`1 ˘˝µ t n k`1 t n k ´`µ t n 2 n t n 2 n ´1 ˝µt n k`2 t n k`1 ˘˝ψ t n k`1 t n k ¯˝ψ t n k t n k´1 ˝¨¨¨˝ψ t n 1 t n 0 pxq ˇˇÀ ˇˇµ n ts pxq ´ϕts pxq ˇˇ`2 ´npη´1q . (3.4)
Local uniqueness follows from that estimate. We have used here the fact that the µ n ts are Lipschitz continuous, uniformly in n, and that

sup xPBp0,Rq ˇˇψ t,s ´µt,s pxq ˇˇÀ C R,X |t ´s| η . ◃

-Corollaries and extensions

We emphasize in the Section 4.1 and Section 4.2 two consequences on solutions to differential equations of the above results/computations. Young and mixed rough/Young equations are considered in Section 4.1, and differentiability of the solution flow with respect to parameters is considered in Section 4.2. The estimates on the derivative flow we get there will be used in the forthcoming work [START_REF] Bailleul | Limit theorems for systems of mean field rough differential equations[END_REF] on limit theorems for systems of mean field rough differential equations. We worked so far in with weak geometric Hölder p-rough paths; one can actually work with general rough paths, controlled by arbitrary controls [START_REF] Lyons | Differential equations driven by rough signals[END_REF]. A non-explosion criterion with quantitative estimates is provided in Section 4.3 in this more general setting.

-Young and mixed rough-Young differential equations

The proofs of theorems 5 and 6 do not use the fact the the drift term is driven only by time. Instead we treat the signal t Ñ t as a Lipschitz path, and deal with it using Young differential calculus techniques. A direct counterpart of this approach is a loss of regularity in the coefficients, either in time and space. A real reward of this approach, which does not modify the proof but requires only more notations, is an extension of the results to a mixed Young-Rough differential equation.

Let V 0 and F " pV 1 , . . . , V ℓ q be given; let another family G :" pW 1 , . . . , W m q of vector fields on B be given. A solution flow to the mixed rough-Young differential equation is defined as in Definition 4, with the 'approximate flow' µ ts defined as the time 1 map of the ordinary differential equation

y 1 r " V 0 `s, y r ˘pt ´sq `m ÿ j"1 Y j ts W j `s, y r ˘`rps ÿ k" ÿ IPt1,¨¨¨,ℓu k Λ k,I ts V X rIs `s, y r ˘.
The constants ε and C that appear in the defining estimate (2.3) are now allowed to depend on R, X and Y .

19. Corollary -Let X be an R ℓ -valued weak geometric Hölder p-rough path and Y be an R m -valued 1 q -Hölder path, with 1 p `1 q ą 1 and p ě 2. Assume pV 0 , Fq and pW i , Fq satisfy Assumption 1 and Assumption 2 for all 1 ď i ď m. Assume furthermore that there exists a positive exponent κ such that κ `1 q ą 1, and

sup xPBp0,Rq sup 0ďsătďT ˇˇW i pt, xq ´Wi ps, xq ˇ|t ´s| κ À p1 `Rq α .
Then the rough differential equation dϕ t " V 0 pt, ϕ t qdt `Gpt, ϕ t q dY t `Fpt, ϕq dX t , has a unique global in time solution flow.

On can choose the constants ε X,Y and C R,X,Y such that pt ´sq

1 q `1 `}Y } ˘`pt ´sq 1 p `1 `}X} ˘À 1, and C R,X,Y » p1 `Rq α `1 `}Y } `}X} ˘`1 `}X} ˘rps and N » max ! " p1 `}Y }q q ‰ , " p1 `}X}q p ‰ ) .
The proof is left to the reader since it is a direct modification of Section 3, with more notations.

-Derivative flow

Rough differential equations dϕ t " V 0 pt, ϕ t qdt `Fpt, ϕ t q d t (4.1) generate flows of diffeomorphisms under appropriate regularity conditions on the driving vector fields. The pair pϕ, Dϕq, made up of ϕ and its differential, also satisfies an equation, with 'triangular' structure dpDϕq " DV 0 pt, ϕ t qDϕ t dt `DFpt, ϕ t qDϕ t dX t .

One can find results on derivative flows in the book [START_REF] Friz | Multidimensional stochastic processes as rough paths[END_REF] of Friz and Victoir, Chapter 11; see also the interesting works [START_REF] Coutin | Sensitivity of rough differential equations: an approach through the Omega lemma[END_REF] and [START_REF] Coutin | Perturbed linear rough differential equations[END_REF] of Coutin and Lejay. One gets another proof of the differentiability of the flow with respect to the initial point as a direct byproduct of the results of Section 2. Pick p ą 2.

20. Assumption -Let V 0 and V 1 , . . . , V ℓ be a set of time dependent vector fields on B such that there exists two exponents with κ 1 ą 1`rps´p p , and κ 2 `1 p ą 1, such that

sup 0ďsătďT › › V 0 pt, ¨q ´V0 ps, ¨q› › C 2`n b |t ´s| κ 1 ă `8
and each V i satisfies the estimate

sup 0ďsătďT › › V i pt, ¨q ´Vi ps, ¨q› › C 3`n b |t ´s| κ 2 ă `8.
21. Theorem -Let X be a weak geometric Hölder p-rough path and pV 0 , V 1 , ¨¨¨, V ℓ q which satisfy Assumption 20. Let ϕ stand for the solution flow to the rough differential equation (4.1). Then each ϕ ts is of class C n , has linear growth and bounded derivatives. Furthermore for a suitable positive constant ε 3 , independent of X, and |t ´s| The previous bound allows us to send n to 8, and to get, as a consequence, that A k ts " D k ϕ ts . The construction of the global in time flow and its derivatives is done by gluing all these local flows, as above.

We now turn to the global bounds. As previously let N be the least integer such that T 1 p N ´1 p `1 `}X} ˘À 1, where the implicit multiplicative constant is chosen such that all the previous bounds hold. Setting t i :" i N pt ´sq `s, one can use the local in time bounds on some time interval of length pt i`1 ´ti q. We have ϕ t i s pxq ´x "ϕ t i t i´1 `ϕt i´1 s pxq ˘´µ t i t i´1 `ϕt i´1 s pxq μt i t i´1 `ϕt i´1 s pxq ˘´ϕ t i´1 s pxq `ϕt i´1 s pxq ´x

Hence, if one sets R 0 i :" sup x ˇˇϕ t i ,s pxq ´xˇˇ, one has

R 0 i ď R 0 i´1 `C|t ´s| 1 p À i|t ´s| 1 p .
Similarly One obtains the bounds for the higher order derivatives using Faà di Bruno formula. ◃

-Finite p-variation rough paths

It is well-known the global bound for the differential of the flow, or the global bound for the flow for vector field with linear-growth, is not not good - [START_REF] Cass | Integrability and tail estimates for Gaussian rough differential equations[END_REF], [START_REF] Friz | Integrability of (non-)linear rough differential equations and integrals[END_REF], [START_REF] Friz | A Course on Rough Paths[END_REF]. Indeed, in the setting of weak geometric Hölder p-rough paths, N " `1`}X} ˘p, and for a Gaussian rough path X, the quantity }X} only has Gaussian tail and E " e c}X} p ‰ " `8 for any p ą 2 and any positive constant c. To derive some moment bounds of solutions of rough differential equations, one need more advanced tools; we recall them here for the reader's convenience.

Definition -A weak geometric continuous rough path with finite p-variation is a continuous rps-level weak geometric rough path such that }X} r0,T s,p´var :"

rps ÿ i"1 sup π partition of r0,T s ¨ÿ pt k ,t k`1 qPπ |X i t,s | p i '1 p ă ` 8
Set wpt, sq :" }X} p rs,ts,p´var . If X is a weak geometric continuous rough path with finite p-variation then w is a control; it is in particular increasing in its two variables, super-additive and continuous on the diagonal. Note also that a weak geometric Hölder p-rough path is always of finite p-variation since wpt, sq ď |t ´s| `1 `}X} ˘p. The advantage of using the p-variation norm instead of the Hölder norm is related to integrability properties for random rough paths.

22. Definition -Given β ą 0 define τ β 0 " 0 and τ β i`1 " inftt P rτ β i , T s : wpτ β i , tq ě βu ^T. The quantity N β :" supti ě 0 : τ β i ă T u is called the local accumulated variation of X. The following result combines results from Friz and Victoir [START_REF] Friz | Multidimensional stochastic processes as rough paths[END_REF] and Cass, Litterer and Lyons [START_REF] Cass | Integrability and tail estimates for Gaussian rough differential equations[END_REF] Theorem -Let β ą 0, p ě 2 and let X be a centered Gaussian process defined over some finite interval r0, T s. Suppose that the covariance function is of finite ρ-two dimensional variation for some ρ P p1, 2q. Then for any p P p2ρ, 4q, X can be lifted as a level-rps weakly geometric continuous finite p-variation rough path, and for β ą 0, the process N In particular, for p P p2ρ, 4q, and for any constant C ą 0, E rexp pCN β qs À 1.

Friz and Riedel gave in [START_REF] Friz | Integrability of (non-)linear rough differential equations and integrals[END_REF] what is now the classical proof of this result, based on Borell's isoperimetric inequality in Gaussian spaces. Cass and Ogrodnik [START_REF] Cass | Tail estimates for Markovian rough paths The[END_REF] use heat kernel estimates as a substitute to isoperimetry to prove a similar result for Markovian rough paths. Compare the following definition to definition 4.

Definition -A flow

ϕ : ∆ T ˆRd Þ Ñ R d is
N 1´1 p β ˙1 1´α ´p1 `Rq ¸, if α ă 1 p1 `Rqwpt, sq 1 p e c 3 N β , if α " 1.
One gets back Theorem 6 when X is an Hölder p-rough path, with N replaced by N β .

Proof -The proof follows exactly the same steps as the proofs of Theorems 5 and Theorem 6. We give here the main changes and leave the computations to the reader. -recall |t ´s| ď wpt, sq. The same replacement is done in Corollary 13 and Remark 14. Finally, using the inequality |t ´s| ď wpt, sq and the fact that the real-valued functions u Ñ wpt, uq and u Ñ wpu, sq are increasing, one can also replace `1 `}X} ˘rps`1 |t ´s| The proofs of Proposition 16 and Proposition 17 are a bit different, but the spirit is the same. The main difference is that one cannot say immediately that w `t, t`s 2 ˘ď 1 2 wpt, sq. But given pt, sq P ∆ T , there exists ũ P ps, tq such that wpt, uq " wpu, sq ď 1 2 wpt, sq. Consider any sequence of embedded partitions pπ n q nPN " ´`t n i ˘iPt0,¨¨¨,nu ¯nPN with mesh going to 0. One proves by induction the existence of constants 0 ă β ď 1 and L ą 0 such that for wpt, sq ď β, one has for all k ď n, Let the integer 0 ď i 0 ď n be such that t n`1 i 0 ď ũ ă t n`1 i 0 `1. One closes the induction and proves the following bound for all n P N by taking u " t n`1 i 0 `1, using the fact that wpt n`1 i 0 `1, ũq `wpũ, t n`1 i 0 q ÝÑ nÑ8 0.

The same trick holds for the proof of Proposition 17, assuming that wpt, sqp1 `Rq For the proof of the local existence, one can proceed as in Lemma 2.1 of [START_REF] Deya | A priori estimates for rough PDEs with application to rough conservation laws[END_REF], and as in the proof of Theorem 5. Let `pt n i q iPt0,¨¨¨,nu ˘nPN be the sequence of dyadic partitions. Remark that since w is superadditive, there exists i such that wpt n i`1 , t p . Here we crucially use the fact that the composition of the flows are globally Lipschitz continuous, uniformly in n. Hence M n converges uniformly to a limit ϕ t,s ´µt,s and The remainder of the proof follows easily from the proof of Theorem 5 and Theorem 6. Indeed, by construction, ϕ is a flow for all dyadic points, and then by continuity for all points, and thanks to the continuity of µ with respect to X, ϕ is continuous with respect to X. Note also that thanks to the superadditivity property of the control, one has

2 n ´1 ÿ k"0
wpt n i`1 , t n i q rps`1 p À max iPt0,¨¨¨,2 n ´1u wpt n i`1 , t n i q rps`1´p p wpt, sq, and since w is continuous on the diagonal, the above sum goes to 0 as n goes to infinity. Local uniqueness of the flow follows -see Equation (3.4). The proof of global existence is similar to the proof of Theorem [START_REF] Boutaib | Dimension-free Euler estimates of rough differential equations[END_REF]. Use the sequence of times pτ β i q iPN from definition 22. We have ϕ τ β i ,s pxq ´x " ϕ τ β

R i ď R i´1 `wpτ β i , τ β i´1 q
rps`1 p p1 `R `Ri q α `Cpτ β i , τ β i´1 , Xqp1 `R `Ri´1 q α . Since wpτ β i , τ β i´1 q ď β, one has Cpτ β i , τ β i´1 , Xq À wpτ β i , τ β i´1 q 1 p ., hence R i ´Ri´1 À p1 `R `Ri´1 q α `wpτ β i , τ β i´1 q rps`1 p `wpτ β i , τ β i´1 q 1 p ˘.

  Pick α P r0, 1s. A finite dimensional-valued function f defined on R d is said to have α-growth if sup xPR d ˇˇf pxq ˇ1 `|x| ˘α ă `8. Let V 0 and V 1 , ¨¨¨, V d : r0, T s ˆRd Ñ R d be time-dependent vector fields on R d . 2. Assumption -Space regularity and growth. For any 1 ď n ď rps and for any tuple I P t1, ¨¨¨, ℓu n , ' the vector fields V 0 ps, ¨q and V rIs ps, ¨q are Lipschitz continuous with α-growth, and their derivatives DV 0 ps, ¨q and DV rIs are C 1 b pB, Bq, uniformly in time, ' for all indices 1 ď k 1 , ¨¨¨, k n ď rps with ř k i ď rps, and all tuples I k i P t1, ¨¨¨, ℓu k i , the functions V 0 ps, ¨qV rI n´1 s ps, ¨q ¨¨¨V rI 1 s ps, ¨qId and V rIns ps, ¨q ¨¨¨V rI 1 s ps, ¨qId are C 2 b with α-growth, uniformly in time.

1 p ` 1 ` 1 p 1 p ` 1 `

 11111 }X} ˘À 1, and a P U , sup xPR d ˇˇD a ϕ ts pa, xq ˇˇÀ |t ´s| `1 `}X} ˘rps ' there exists positive constants β and c such that one has sup xPR d ˇˇD a ϕ ts pa, xq ˇˇÀ |t ´s| }X} ˘ecN β for all 0 ď s ď t ď T .

1 p ` 1

 11 `}X} ˘rps p1 `Rq α À pt ´sq 1`rps p `1 `}X} ˘rps p1 `Rq α .

1 p e c 1 |t´s| 1 p

 11 N .

  Gaussian tails, namely there exists a constant µ ą 0 such that

  s pxq ˇˇď L p1 `Rq α wpt, sq rps`1 p

α 1 `

 1 rps ď β. One can again replace in Proposition 16,Propositino 17 and Remark 18 `

  |ϕ t,s pxq ´µt,s pxq| À p1 `Rq α

i τ β i´1 `ϕτ β i´1 s pxq ˘´µ τ β i τ β i´1 `ϕτ β i´1 s pxq μτ β i τ β i´1 `ϕτ β i´1 s pxq ˘´ϕ τ β i´1 s pxq `ϕτ β i´1 s pxq

  ´x. Define R i :" sup xPBp0,Rq ˇˇϕ τ β i s pxq ´xˇˇa nd R 0 " 0. The fourth item of the statement follows then from the induction relation

  For all 1 ď n ď rps and 1 ď k 1 , ¨¨¨, k n ď rps, with ř n i"1 k i ď rps, for all tuples I i P t1, ¨¨¨, du k i , we have ˇˇV rIns pt, ¨q ¨¨¨V rI 1 s pt, ¨qpxq ´VrIns ps, ¨q ¨¨¨V rI 1 s ps, ¨qpxq ˇ| t ´s| κ 2 À p1 `Rq α .

	1`rps´p p	and κ 2 ě rps p with the following properties.
	' One has	sup xPBp0,Rq	sup 0ďsătďT	ˇ|t ˇˇV 0 pt, xq ´V0 ps, xq ´s| κ 1 À p1 `Rq α ,
	' sup		sup
	xPBp0,Rq	0ďsătďT

said to be a solution flow to the rough differential equation (2.2) if there exists an exponent η ą 1 independent of X, such that one can associate to any positive radius R two positive constants C R,X and ε X such that

	one has	sup	ˇˇϕ ts pxq ´µts pxq ˇˇď C R,X |t ´s| η ,	(2.3)
		xPBp0,Rq

  2 .

	' One has for all f P C	rps`1 b	and all |t ´s| ď ε X the estimate
	sup	ˇˇˇf ˝ϕt,s pxq ´!f pxq `pt ´sqV 0 ps, ¨qf	`rps ÿ	ÿ	X k,I t,s V I ps, ¨qf	)	pxq ˇˇÀ
	xPBp0,Rq					k"1	IPt0,¨¨¨,ℓu k
				b }f } C rps`1	p1 `Rq αprps`1q `1 `}X}	˘rps`1 |t ´s|	rps`1 p	.
	When f " Id, one can replace p1 `Rq αprps`1q by p1 `Rq α and }f } C n b by 1 in the
	previous bound.
	' The map that associates ϕ to X is continuous from the set of weak geometric Hölder
	p-rough paths into the set of continuous flows endowed with the topology of uniform
	convergence on bounded sets.
	' Finally, there exists two positive universal constants c 3 , c 4 such that setting N :" " c 3 `1 `}X} ˘pı ,
	one has for all pt, sq P ∆ T , sup xPBp0,Rq ˇˇϕ s,t pxq ' ' ' ' % ´xˇˇÀ $ ' ' ' ' & p1 `Rq ¨˜1 `c4	p1 `Rq 1´α |t ´s| 1 p N	¸1 1´α

  Apply repeatedly Grönwall lemma. We only prove the estimate for y r pxq ´x and leave the remaining details to the reader. It suffices to write |y r pxq ´x| ďpt ´sq ˇˇV 0 `s, x ˘ˇ`r The derivative equations satisfied by Dy r and D 2 y r are used to get the estimates of the statement on these quantities, using once again that the condition of the statement imposes to C to be of order 1.

							ps ÿ	ÿ	ˇˇΛ k,I t,s	ˇˇˇˇV rIs ps, xq ˇpt
			´sq	ż r	k"1 ˇˇV 0 `s, y u pxq ˘´V 0 ps, xq ˇˇdu IPt1,¨¨¨,ℓu k
					0
			`rps ÿ		ÿ		ˇˇΛ k,I t,s	ˇˇż r	ˇˇV rIs `s, y u pxq ˘´V rIs ps, xq ˇˇdu
			k"1	IPt1,¨¨¨,ℓu k	0
			ÀC `|t ´s|, }X}	˘´`1	`|x| ˘α	`ż r	|y u pxq ´x|du	¯.
							0
	to get the conclusion from Grönwall lemma, using the fact that C |t´s|,}X} À 1, for
	|t ´s|
	Then
	' ˇˇy r pxq	´xˇˇÀ	`1 `|x| ˘αC |t´s|,}X} ,

1 p `1 `}X} ˘ď 1. ' ˇˇDy r pxq ´Id ˇˇÀ C |t´s|,}X} , ' ˇˇD 2 y r pxq ˇˇÀ C |t´s|,}X} . The maps y r p¨q are thus C 1 b , uniformly in r P r0, 1s. Proof -1 p `1 `}X} ˘ď 1. ◃ 10. Remark -Would Assumption 1 require in addition that the vector fields V 0 ps, ¨q and V rIs ps, ¨q were C n`2 b with α-growth, uniformly in 0 ď s ď T , we would then have the estimate sup 2ďkďn`2

  function g P C n `Rd , R d ˘is said to satisfy Assumption H if for all 1 ď k 1 , . . . , k n ď rps with ř p i"1 k i ď rps, and all tuples I k i P t1, . . . , ℓu k i , the functions V 0 ps, ¨qV rI n´1 s ps, ¨q ¨VrI 1 s ps, ¨qg and V rIns ps, ¨q ¨VrI 1 s ps, ¨qg are C 2 b , with α-growth, uniformly in s P r0, T s. 12. Lemma -Assume Assumption 1 holds, and pick a function f P C n b `Rd , R d ˘, for some 2 ď n ď rps. Given ps, tq P ∆ T with |t ´s|

		1 p `1 `}X} ˘ď 1, we have
	sup	ˇˇε n,f ts pxq ˇˇÀ }f } n p1 `Rq nα `1 `}X}	˘rps`1 pt ´sq	1`rps p ,
	xPBp0,Rq
	for all positive radius R.

' If furthermore D n`1 f exists and is a bounded function, then sup xPBp0,Rq

  We estimate the size of the spatial derivative of the first term in the above decompops, ¨qf ˘`y r pxq ˘´D `V0 ps, ¨qf ˘`x ˘¯Dy r pxq ˇˇÀ }f } n |t ´s| 1`1 p p1 `Rq 2α `1 `}X} ˘rps .Once again, one can replace p1 `Rq 2α by p1 `Rq α if f satisfies Assumption H.ˇˇ`V 0 ps, ¨qV rI i s ps, ¨q ¨¨¨V rI 1 s ps, ¨qf ˘pxq ˇˇÀ }f } n p1 `Rq pi`1qα ˇˇD `V0 ps, ¨qV rI i s ps, ¨q ¨¨¨V rI 1 s ps, ¨qf ˘pxq ˇˇÀ }f } n`1 p1 `Rq pi`1qα .

	R˘α |x ´y| uniformly in 0 ď s ď T , it follows from Lemma 9 that ˇˇˇp t ´sq ż 1 ! `V0 ps, ¨qf ˘py r q ´`V 0 ps, ¨qf ˘pxq ) ts writing ˇˇˇp t ´sq ż 1 0 ´D`V 0 ps, ¨qf ˘`y r pxq ˘Dy r pxq ´D`V 0 ps, ¨qf ˘`x ˘¯ˇˇˇˇd r À ˇˇˇp t ´sq ż 1 0 dr ´D`V 0 ps, ¨qf ˘`x ˘¯´D y r pxq ´Id ¯ˇˇp t ´sq ż 1 0 dr ´D`V ' The two other terms in the decomposition (3.1) of ε n,f ts are estimated in the same way. Remark that sup xPBp0,Rq and that sup dr ˇˇˇÀ }f } sition of ε n,f xPBp0,Rq
	0

n pt ´sq C |t´s|,}X} p1 `Rq 2α .

Note that if V 0 ps, ¨qf is globally Lipschitz continuous one can replace p1 `Rq 2α above by p1 `Rq α . 0

  Hence for all i P t1, ¨¨¨, rpsu; this gives the expected upper bound. The same idea is used for the spatial derivatives. Once again, one can replace p1 `Rq nα by p1 `Rq α if the function f satisfies Assumption H.

				1 p `1 `}X} ˘ď 1, so we have `1 `}X}	˘i´1 |t ´s|	i´1 p	À 1, ◃
	13. Corollary -We have
	sup	ˇˇˇf ˝µts pxq ´!f pxq `pt ´sqV 0 ps, ¨qf pxq	`rps ÿ	ÿ	X k,I ts V I ps, ¨qf	)	pxq ˇˇÀ
	xPBp0,Rq							k"1	IPt0,¨¨¨,ℓu k
				}f } rps`1 p1 `Rq αprps`1q `1 `}X}	˘rps`1 |t ´s|	rps`1 p .
	for all f P C	rps`1 b	with α-growth, and 1 ď k ď rps. We also have
	sup		ˇˇˇµ ts pxq ´´x `pt ´sqV 0 ps, xq	`rps ÿ	ÿ	ts V I ps, xq X k,I ¯ˇˇÀ
	xPBp0,Rq			k"1	IPt0,¨¨¨,ℓu k
						p1 `Rq α `1 `}X}	˘rps`1 |t ´s|	rps`1 p .
	and
	sup
	xPBp0,Rq
								)	ˇˇˇÀ
								dr
				p1 `Rq αprps`1q	rps ÿ	|t ´s|	i`rps p	}X} rps`i
						k"1
				À p1 `Rq αprps`1q pt ´sq	1`rps p	`1 `}X}	˘rps`1 .

ˇˇˇD µ ts pxq ´´Id `pt ´sqDV 0 ps, xq `rps ÿ k"1 ÿ IPt0,¨¨¨,ℓu k X k,I ts DV I ps, xq ¯ˇˇÀ p1 `Rq α `1 `}X} ˘rps`1 |t ´s| rps`1 p . Proof -We only have to bound the sum over I n,rps of the terms n ź m"1 Λ km,Im ts ż ∆n dr

´`V rIns ps, ¨q ¨¨¨V rI 1 s ps, ¨qf ˘py rn q ´`V rIns ps, ¨q ¨¨¨V rI 1 s ps, ¨qf ˘pxq for n " rps, thanks to lemmas 11 and 12. We have k 1 " ¨¨¨" k rps " 1, on I rps,rps . As we know that we have ˇˇ`V I ps, ¨qf ˘pxq ´`V I ps, ¨qf ˘pyq ˇˇÀ p1 `Rq αrps }f } rps`1 |x ´y|, for all I P t1, ¨¨¨, du rps , and all x, y P Bp0, Rq, it follows that ˇˇˇˇÿ I 1 ,¨¨¨,I rps Pt1,¨¨¨,ℓu ´Λ1 ts ¯rps ż ∆ n 1 ! `VI rps ps, ¨q ¨¨¨V I 1 ps, ¨qf ˘py rn q

´`V I rps ps, ¨q ¨¨¨, V I 1 ps, ¨qf ˘pxq

  tu ˝µus pxq ´µts pxq " pt ´uq ´V0 `u, µ us pxq ˘´V 0 `s, µ us pxq ˘pt ´uq ´V0 `s, µ us pxq ˘´V 0 `s, x ˘r

	ps ÿ i"1	1 i!	I i,rps ÿ	m"1 ! i ź	Λ km,Im tu	m"1 `i ź	Λ km,Im us	m"1 ´i ź	Λ km,Im ts	) `VrIis ps, ¨q ¨¨¨V rI1s ps, ¨qId ˘pxq
	`rps ÿ i"1	1 i!	I i,rps ÿ	m"1 i ź	Λ km,Im tu	! `VrIis pu, ¨q ¨¨¨V rI1s pu, ¨qId ˘`µ us pxq VrIis
											ps, ¨q ¨¨¨V rI1s ps, ¨qId ˘`µ us pxq ˘)
	`rps ÿ i"1	1 i!	ÿ I i,rps	i ź m"1	Λ km,Im tu	VrIis ! `VrIis ps, ¨q ¨¨¨V rI1s ps, ¨qId ˘`µ u,s pxq ps, ¨q ¨¨¨V rI1s ps, ¨qId ˘pxq )
	`ε tu `µus pxq ˘`ε us	`x˘`ε	ts pxq
	": p1q `¨¨¨`p6q.
							i"1 `µus pxq 1 ÿ i! m"1 i ź I i,rps ˘,	Λ km,Im tu	!	V rI i s pu, ¨q ¨¨¨V rI 1 s pu,	εtu ¨q) `µus pxq
	where εts pxq :" ε rps,Id ts	pxq `ε1 ts pxq and
	ε 1 ts pxq :"	ÿ				rps ź	Λ 1,i k t,s	ż	!	) pV I Idqps, y rn q ´pV I Idqps, xq	dr,
				IPt1,¨¨¨,du rps	m"1	∆ rps
	for any 0 ď a ď b ď T . As we also have
	µ us pxq ´µts pxq " ´pt ´uqV 0 ps, xq
				`rps ÿ i"1	1 i!	I i,rps ÿ	m"1 ! i ź	Λ km,Im us	m"1 ´i ź	Λ km,Im ts	) `VrI i s ps, ¨q ¨¨¨V rI 1 s ps,	¨q˘`xε

1 p `1 `}X} ˘rps ď 1, we have sup xPBp0,Rq ˇˇµ tu ˝µus pxq ´µts pxq ˇˇ_ ˇˇˇD `µtu ˝µus ˘pxq ´D`µ ts ˘pxq ˇˇÀ p1 `Rq α `1 `}X} ˘rps`1 |t ´s| 1`rps p . Proof -First, remark that µ tu ˝µus pxq " µ us pxq `pt ´uqV 0 `u, µ us pxq ȓps ÿ us pxq `ε ts pxq, this gives µ

  The proof is done by induction on n. Note first that we can take L enough to have ε 0 L ď 1 and C |t´s|,}X} À ε 0 . Proposition 15 provides the initialisation of the induction. Assume step n of the induction has been proved and set

	and
			L ą	1	C 0 p ´2´1 `rps´p	p1 `2ε 0 q	.
	16. Proposition -For all 0 ď s ă t ď T with
			L |t ´s|	1 p `1 `}X} ˘ď ε 0 ,
	and all positive radius R, we have
	sup |x|ďR	ˇˇµ t n 2 n t n 2 n ´1 ˝¨¨¨˝µ t n 1 t n 0 pxq ´µts pxq ˇˇď L |t ´s|	1`rps p	p1 `Rq α `1 `}X}	˘rps`1 .
	Proof -u :"	t	`s 2	" t n`1 2
					Proposition 15 under the form
		sup	ˇˇµ t,u ˝µu,s pxq ´µt,s pxq ˇˇď C 0 p1 `Rq α `1 `}X}	˘rps`1 |t ´s|	1`rps p ,
	xPBp0,Rq
	for some positive constant C 0 . Given n ě 1, and 0 ď s ď t ď T , set t n k :" k2 ´npt ´sq `s.
	Pick ε 0 such that
			2 ´1`rps´p p	p1 `2ε 0 q ă 1

n , so the statement of the proposition holds on the intervals ps, uq and pu, tq. We have ˇˇ`µ t n`1

  Finally there exists some positive constants c 1 , ¨¨¨, c n , independent of X, such that for all 0 ď s ď t ď T , and every 1 ď k ď n, we haveˇˇD k ϕ ts pxq ´Dk ϕ s,s pxq ˇˇÀ |t ´s|This implies that for all 0 ď k ď n the function D k t,s is Lipschitz continuous, with Lipschitz constant not greater than a constant multiple of |t ´s| , such that D k µ n ts converges uniformly to A k t,s as n goes to 8. One then needs to prove that the A k are indeed the k-th derivative of ϕ and get the bounds of the statement. Note that the small time bounds are direct consequences of equation (4.2) and Remark 10 once we know that the D k µ n ts converge.

										1 p `1 `}X} ˘ď ε 3 , we
	have				› › ›D k ϕ ts	´Dk µ ts	› › ›	8	À |t ´s|	1`rps p	`1 `}X}	˘rps`1 ,
	for all 0 ď k ď n. sup	ˇˇϕ ts pxq	´xˇˇÀ |t ´s|	1 p `1 `}X}	ȃnd
									xPR d
					sup					1 p e c k |t´s|	1 p N ,
	where N "	"	xPR d c `1 `}X} ˘´p	ı .
	Proof -We work here with α " 0, so we know from the above computations that for |t ´s| 1 p `1 `}X} ˘À 1, and all 0 ď k ď pn `1q, we have
				› › ›D k `µn ts ´µts ˘pxq › › ›	8	À |t ´s|	1`rps p `1 `}X}	˘rps`1 .	(4.2)
										1`rps p `1 `}X}	˘rps`1 . It
	follows from this fact and the proof of Theorems 5 and 6 that there exists some maps
	A k t,s We have
	µ n ts px `hq	´ÿ 0ďjďk	1 j!	D k µ t,s pxq ¨hj "	1 k!	ż 1 0	dλD k`1 µ n ts pλh `xq ¨pp1 ´λqhq k h,
	where h j " ph, ¨¨¨, hq loooomoooon	. Hence, thanks to Remark 18, for all |t ´s|	1 p `1 `}X} ˘À 1, the
				j times
	maps D k`1 µ n ts are bounded, uniformly in n, and
	ˇˇˇˇˇµ n t,s px `hq	´ÿ 0ďjďk	1 j!	D k µ n t,s pxq ¨hj ˇˇˇˇˇÀ	|t ´s|	1`rps p `1 `}X}	˘rps`1 |h| k`1 .

  , we have Dϕ t i s pxq ´Id " ´Dϕ t i t i´1 `ϕt i´1 s pxq ˘´µ t i t i´1 `ϕt i´1 s pxq ˘¯Dϕ t i´1 s pxq `´Dµ t i t i´1 `ϕt i´1 s pxq ˘´Id ¯Dϕ t i´1 s pxq `Dϕ t i´1 s pxq ´Id. Again,given the choice of N , one have can use all the local bounds on ϕ, Dϕ, and µ and Dµ, and setting R 1 i :" sup x ˇˇDϕ t i ,s pxq ´Id ˇˇ, one has

	R 1 i ď C|t ´s|	1 p `R1 i´1 `1 `|t ´s| ˘1 p ,
	and
	sup

x ˇˇDϕ s,t pxq ´Id ˇˇÀ |t ´s|

  said to be a solution flow to the rough differential equation (2.2) if there exists an exponent η ą 1 such that one can associate to any positive radius R two positive constants C R and ε, independent of X, such that one has sup R " c 2 p1 `Rq α , for some positive universal constants c 1 , c 2 , in the defining identity(4.3). Id, one can replace p1 `Rq αprps`1q by p1 `Rq α and }f } C n b by 1 in the previous bound. ' The map that associates ϕ to X is continuous from the set of weak geometric continuous rough paths with finite p-variation into the set of continuous flows endowed with the topology of uniform convergence on bounded sets.' Finally, there exists β ą 0 and c 3 ą 0 such that one has for all pt, sq P ∆ T ,

	' One has for all f P C sup xPBp0,Rq ˇˇˇf ˝ϕt,s pxq ´!f pxq `pt ´sqV 0 ps, ¨qf rps`1 b and all wpt, sq ď ε the estimate `rps ÿ k"1 ÿ IPt0,¨¨¨,ℓu k X k,I t,s V I ps, ¨qf }f } C rps`1 b p1 `Rq αprps`1q wpt, sq ) pxq ˇˇÀ rps`1 p . When f " sup xPBp0,Rq ´xˇˇÀ ' ' % ˇˇϕ ts pxq $ ' ' & 1 ˜ˆp1 `Rq 1´α `c4 wpt, sq p

xPBp0,Rq ˇˇϕ ts pxq ´µts pxq ˇˇď C R wpt, sq η , (4.3)

whenever wpt, sq ď ε.

23. Theorem -Let X be a weak geometric continuous rough path with finite p-variation. Let V 0 and pV 1 , . . . , V ℓ q satisfy Assumption 1 and Assumption 2. There exists a unique global in time solution flow ϕ to the rough differential equation (2.2).

' One can choose η " 1`rps p , ε " c 1 and C

  First, there is no loss of generality in assuming that |t´s| ď wpt, sq; replace if necessary wpt, sq by wpt, sq " |t ´s| `wpt, sq. Set One can replace the constant C |t´s|,}X} by Cpt, s, Xq in Lemma 9 and Remark 10 as soon as wpt, sq ď 1; this ensures that Cpt, s, Xq À 1. Lemma 11 remains the same as it relies only on algebraic manipulations. In Lemma 12, one has to assume that wpt, s, Xq ď 1, and one can replace `1 `}X} ˘rps`1 |t ´s|

		rps ÿ		k
		Cpt, s, Xq :"	wpt, sq	p .
		k"1
				1`rps
				p	in the estimates by
		1`rps
	wpt, sq	p

  n i´1 q ď p2 n ´ 1q ´1wpt, sq. Define the partition p π " ␣ s " t 0 , ă ¨¨¨ă t n i´1 ă t n i`1 ă ¨¨¨ă t n 2 n " tRepeating this operation until we get the trivial partition of rs, ts we see that

					( and
	set M n t,s :" µ n t,s ´µt,s , and
	x M n t,s :" µ t n 2 n ,t n 2 n ´1 ˝¨¨¨˝µ t n i`2 ,t n i`1 ˝µt n i`1 ,t n i´1 ˝µt n i´1 ,t n i´2 ˝µt n 1 ,t n 0 ´µt,s .
	We have x M s,t ´M n t,s "	! µ t n 2 n ,t n 2 n ´1 ˝¨¨¨˝µ t n i`2 ,t n i`1 ˝pµ t n i`1 ,t n i ˝µt n i ,t n i´1 q	)
			´µt n 2 n ,t n 2 n ´1 ˝¨¨¨˝µ t n i`2 ,t n i`1 ˝pµ t n i`1 ,t n i´1 q	˝µt n i´1 ,t n i´2 ˝µt n 1 ,t n 0 pxq.
					α
	The induction hypothesis and the bound wpt, sqp1 `Rq	1`rps ď β, then give
		ˇˇx M n s,t	´M n t,s ˇˇÀ p2 n ´1q ´rps`1 p p1 `Rq α wps, tq	rps`1 p .
			M n t,s "	2 n ÿ	ρ k t,s ,
				k"0
	with	ˇˇρ k t,s pxq ˇˇÀ p1 `Rq α wpt, sq	rps`1 p	p2 n ´kq ´rps`1

I.Bailleul thanks the U.B.O. for their hospitality, part of this work was written there.

R. Catellier acknoledges the support of the Lebesgue Centre.

Acknowledgements

Most of this work was done while R. Catellier was working at University of Rennes 1 (IRMAR -UMR 6625), with the support of the Labex Centre Henri Lebesgue.

When α " 1 one end up with the following bound :

When α ă 1, one ends up with

By using Jensen formula, one finally has the bound

which ends the proof. ◃ 24. Theorem -Let p ą 2 and X be a weak geometric continuous finite p-variation rough path and let pV 0 , ¨¨¨V ℓ q which satisfies Assumption 20. Let ϕ stands for the solution flow to the rough differential equation (4.1). Then each ϕ t,s is of class C n , has linear growth and bounded derivatives. Furthermore for a suitable positive constant ε 3 , independent of X, and wpt, sq ď ε 3 , we have

for all 0 ď k ď n. Finally there exists β ą 0 and some positive constants c 1 , ¨¨¨, c n , independent of X, such that for all 0 ď s ď t ď T , and every 1 ď k ď n, we have ◃


On the definition of a solution to a rough differential equation

-Main result

a) Setting -Rough paths theory was introduced by T. Lyons in [START_REF] Lyons | Differential equations driven by rough signals[END_REF] as a theory of controlled differential equations 9

x t " Fpx t q 9 h t (1.1)

with F P LpR , R d q, and a non-differentiable control h with values in R . The path x takes here its values in R d . Variants of the Cauchy-Lipschitz theorem gives the wellposed character of this equation for an absolutely continuous control, and L.C. Young extended this analysis to controls that are α-Hölder, for α ą 1 2 ; the solution path is then a continuous function of the control h P C α pr0, 1s, R q, under proper regularity and boundedness conditions on F. Nothing better than that can be done in a deterministic setting in the Hölder class, on account of the fact that since the solution path x t is expected to be no better than α-Hölder, so is Fpx t q, so the equation involves making sense of the product Fpx t q 9 h t of an α-Hölder function with an pα´1q-Hölder distribution, either directly or in its integral form. It is known that no such product can be defined as a continuous function of its two arguments when the Hölder regularity exponents add up to a nonpositive real number -see e.g. [START_REF] Bahouri | Fourier analysis and nonlinear partial differential equations[END_REF]. Lyons' deep insight was to realize that what really controls the dynamics in (1.1) is not the path h, but rather the data of the path together with a number of its iterated integrals ż h i s dh j s , ij rďs h i r dh j r dh k s , . . . for all possible indices 1 ď i, j, k, ¨¨¨ď . The rougher the control h is, the more iterated integrals you need. This can be understood from the fact that these iterated integrals are the coefficients that appear in a Taylor expansion of the solution to equation (1.1) when the control is smooth, and this leads to the notion of solution adopted in [START_REF] Davie | Differential equations driven by rough paths: and approach via discrete approximations[END_REF][START_REF] Bailleul | Flows driven by rough paths[END_REF] by Davie and Bailleul. There is no miracle though, and one cannot still make sense of these iterated integrals when the control is not sufficiently regular. Lyons' considerable feat was to extract from these 'non-existing' quantities the analytic and algebraic properties that they should satisfy and to work directly with objects enjoying these properties as new controls. These are the weak geometric Hölder p-rough paths that we encounter below. We refer the reader to the gentle introductions [START_REF] Caruana | Differential equations driven by rough paths[END_REF][START_REF] Friz | A Course on Rough Paths[END_REF][START_REF] Bailleul | A flows-based approach to rough differential equations[END_REF] for different points of view on rough differential equations; one can learn everything from scratch in the first 19 pages of [START_REF] Bailleul | Flows driven by rough paths[END_REF].

b) Weak geometric Hölder p-rough paths and rough differential equations -We refer the reader to [START_REF] Lyons | Differential equations driven by rough signals[END_REF][START_REF] Baudoin | Diffusion Processes and Stochastic Calculus[END_REF][START_REF] Bailleul | Flows driven by rough paths[END_REF] for basics on rough paths and recall all we need in this section. Pick an integer p ě 1. Denote by pe i q i"1.. the canonical basis of R , and by T rps " À i"1 pR q bi the truncated tensor algebra over R ; it is equipped with the Lie bracket operation ra, bs :" ab ´ba, and endowed with a norm }a} :"

rps ÿ i"1 |a i | 1{i ,
for a ": ' rps i"1 a i ; denote by pa I q IP 1, k ,0ďkďrps the coordinates of a generic element a of T rps in the canonical basis of T rps . We sometimes write a " pa i q i"1..rps instead of a " '

}X i ts } ď O `|t ´s| i{p ˘.
The archetype of a weak geometric Hölder p-rough path is given by the canonical lift of a smooth R -valued path h under the form

H t :" ˜1, h t ´h0 , ż t 0 ph u 1 ´h0 q dh u 1 , . . . , ż 0ďu rps 﨨¨ďu 1 ďt dh u rps ¨¨¨dh u 1 ¸; (1.2)
it can be seen, after Chen [START_REF] Chen | Integration of paths, geometric invariants and a generalized Baker-Hausdorff formula[END_REF], that H takes values in G rps and that

H ts :" H ´1 s H t " ˜1, h t ´hs , ż t s ph u 1 ´h0 q dh u 1 , . . . , ż sďu rps 﨨¨ďu 1 ďt dh u rps ¨¨¨dh u 1 ¸.
Denote by 1, the set of integers between 1 and . Define inductively on the size of the tuple J :" ti 1 , Iu P 1, k`1 , the element e rJs " " e i 1 , e rIs ‰ of g rps . Given 0 ď s ď t ď T , set

Λ ts :" log X ts " ÿ IP 1, k 0ďkďrps
Λ I ts e rIs P g rps .

Let F " `V1 , . . . , V ˘stand for a collection of smooth enough vector fields on R d , and X stand for a weak geometric Hölder p-rough path over R . There are several definitions of a solution to the rough differential equation dz t " Fpz t q dX t .

(1.3) Besides Lyons' original definition [START_REF] Lyons | Differential equations driven by rough signals[END_REF] and Gubinelli's formulation in terms of controlled paths [START_REF] Gubinelli | Controlling rough paths[END_REF], Davie gave in [START_REF] Davie | Differential equations driven by rough paths: and approach via discrete approximations[END_REF] a formulation of a solution in terms of Taylor expansion for the time increment of a solution. We identify freely a vector field to a first order differential operator via the expression

V i f " pDf qpV i q
for any C 1 function f on R d . For a tuple I " pi 1 , . . . , i k q P 1, k , we write |I| :" k, and define a k th order differential operator setting

V I " V i 1 ¨¨¨V i k , that is to say V I f " V i 1 `¨¨¨pV i k f q ˘.
With a slight abuse of notation, we write V I pxq for `VI Id ˘pxq. An R d -valued path pz t q 0ďtďT defined on some finite time interval r0, T s is a solution to the rough differential equation (1.3) in the sense of Davie if one has

z t " z s `ÿ I;|I|ďrps X I ts pV I qpz s q `O`| t ´s| a ˘, (1.4)
for any 0 ď s ď t ď T , for some exponent a ą 1. The function Op¨q is allowed to depend on F. Two paths satisfying that condition with different exponents a, a 1 ą 1 coincide if they start from the same point. This definition is a priori not well suited to make sense of a solution to a rough differential taking values in a manifold -an expression like m `V pmq, for a point m of a manifold M and a vector V pmq P T m M , has for instance no intrinsic meaning. Bailleul gave in [START_REF] Bailleul | Flows driven by rough paths[END_REF] a more general notion of solution to a rough differential equation by requiring from a potential solution that it satisfies the estimate f pz t q " f pz s q `ÿ I;|I|ďrps X I ts pV I f qpz s q `Of `|t ´s| a ˘, for any real-valued sufficiently regular function f defined on the state space. The function O f p¨q is allowed to depend on f . The exponent a that appears here may be different from the exponent that appears in equation (1.4); this makes no difference as long as a ą 1, as noted above. This definition makes perfect sense in a manifold setting, and taking f to be a coordinate function in the above equation recovers the notion of solution in the sense of Davie.

c) Main result -Cass and Weidner proved in Theorem 5.3 of their recent work [START_REF] Cass | Tree algebras over topological vector spaces in rough path theory[END_REF] that the two notions are actually equivalent using some deep insights from algebra based on the use of Grossman-Larson and Connes-Kreimer Hopf algebras. That point can be proved by elementary means. We assume as usual that the vector fields V i in (1.3) are C γ , for some γ ą p.

Theorem -An R d -valued path z is a solution to the rough differential equation (1.3) in the sense of Davie if and only if it is a solution of that equation in the sense of Bailleul.

Proof -We only need to prove that solutions to equation (1.3) in Davie' sense are solutions in Bailleul' sense. Given a globally Lipschitz vector field V on R d , we denote by exppV q the time 1 map of the differential equation

9 y u " V py u q,
that associates to x the value at time 1 of the solution to the equation started from x. Also, define inductively on the size of the tuple ti 1 , Iu P 1, k`1 , the vector fields

V rti 1 ,Ius :" " V i 1 , V rIs ‰ ,
starting with V ris :" V i . Given 0 ď s ď t ď T , recall we write Λ ts for log X ts ; this is an element of g rps . It is proved in Proposition 9 of [START_REF] Bailleul | Flows driven by rough paths[END_REF] that one has

› › › › › › exp ¨ÿ I;|I|ďrps Λ I ts V rIs '´!Id `ÿ I;|I|ďrps X I ts pV I q ) › › › › › › L 8 ď O ´|t ´s| γ p ¯(1.5)
for a Op¨q depending only on the vector fields V i . (See also [START_REF] Boutaib | Dimension-free Euler estimates of rough differential equations[END_REF] for a similar statement.) A path z is thus a solution to the rough differential equation (1.3) in the sense of Davie if and only if

z t " exp ¨ÿ I;|I|ďrps Λ I ts V rIs 'pz s q `O`| t ´s| a 1 ˘,
for some exponent 1 ă a 1 ď γ p . Let y stand now for a solution path to the ordinary differential equation 9 y u " ÿ I;|I|ďrps Λ I ts V rIs py u q, started from z s at time 0. We use in the computation below the notation O `|t ´s| a 1 for a function whose value may change from line to line. For a γ-Hölder real-valued function f on R d one has

f pz t q " f py 1 q `O`| t ´s| a 1 " f pz s q `ż 1 0 d du f py u q du `O`| t ´s| a 1 " f pz s q `ÿ |I 1 |ďrps Λ I 1 ts ż 1 0 V I 1 py u 1 q du 1 `O`| t ´s| a 1 " f pz s q `ÿ |I 1 |ďrps Λ I 1 ts V I 1 pz s q `ÿ |I 1 |ďrps,|I 2 |ďrps |I 1 |`|I 2 |ďrps ż 1 0 V I 2 V I 1 py u 2 q du 2 du 1 `O`| t ´s| a 1 ˘,
and, after repeating rps times the same computation giving a function as the integral of its derivative, one eventually has

f pz t q " f pz s q `rps ÿ k"1 1 k! Λ I 1 ts ¨¨¨Λ I k ts `VI k ¨¨¨V I 1 ˘pz s q `O`| t ´s| a 1 " f pz s q `ÿ I;|I|ďrps X I ts pV I f qpz s q `Of `|t ´s| a 1 ˘,
from the fact that Λ ts " logX ts . We repeat that computation in Section 2 in a more general setting.

The following remarks emphasize the robust character of the above proof.

' One can weaken the regularity assumptions on the vector fields V i by only requiring that they are C rγs in the usual sense and the rγs-derivative of V i is pγ ´rγsq-Hölder continuous, without requiring that the V i or their derivatives be bounded. Indeed, given a solution z of (1.3) in the sense of Davie, and a time s, the classical Cauchy Lipschitz theory ensures that exp ´řI;|I|ďrps Λ ts V rIs ¯pz s q is well-defined for t close enough to s, possibly depending on z s .

' For a non-necessarily Hölder weak geometric p-rough path X controlled by a general control wps, tq rather than by |t ´s| 1{p , one replaces pt ´sq 1{p by wps, tq in the definitions of a solution to a rough differential equation driven by X, and requires in both definitions that the remainder is of order wps, tq ap , for some exponent a ą 1. The proof of Theorem 1 remains the same. The equivalence problem for the different definitions of a solution to a rough differential equation is further explored in the very recent work [START_REF] Brault | The nonlinear sewing lemma II: Lipschitz continuous formulation[END_REF] of Brault and Lejay.

' Let E stand for a Banach space. For dynamics driven by an E-valued rough paths, following [START_REF] Cass | On the integration of weakly geometric rough paths[END_REF] and [START_REF] Cass | Tree algebras over topological vector spaces in rough path theory[END_REF], one needs to define T pnq pEq as the completion of the truncated algebraic tensor algebra with respect to a system of cross (semi)norms.

One then defines the n-step free Lie algebra over E as the closure in T pnq pEq of the algebraic free Lie algebra. The n-step free nilpotent Lie group is then indeed the image by the exponential map of the closed Lie algebra. An appropriate formalism where vector fields are indexed by the elements of E is required, as in [START_REF] Bailleul | Flows driven by Banach space valued rough paths[END_REF][START_REF] Boutaib | Dimension-free Euler estimates of rough differential equations[END_REF] or [START_REF] Cass | Tree algebras over topological vector spaces in rough path theory[END_REF]. Identity (1.5) is proved in this setting in [START_REF] Bailleul | Flows driven by Banach space valued rough paths[END_REF] by elementary means similar to their finite dimensional analogues. (We erronously worked in the non-complete versions of the Lie and truncated tensor algebras in [START_REF] Bailleul | Flows driven by Banach space valued rough paths[END_REF]. The complete setting should be adopted, and nothing is changed to the story told in [START_REF] Bailleul | Flows driven by Banach space valued rough paths[END_REF] in this extended setting. Thanks to T. Cass for pointing this out.)

' One of the nice points of Cass and Weidner's work [START_REF] Cass | Tree algebras over topological vector spaces in rough path theory[END_REF] is the fact that they can also handle differential equations driven by a more general notion of rouhg path called branched rough paths. In a finite dimensional setting, one can appeal to Hairer and Kelly's result [START_REF] Hairer | Geometric versus non-geometric rough paths[END_REF] stating that a solution to a rough differential equation driven by a branched rough path over R is also the solution to a differential equation driven by a weak geometric rough path over a larger space R 1 , with driving vector fields built from the initial V i . The equivalence of the two notions of solution for equations driven by weak geometric rough paths implies the equivalence of the two notions of solution for the equation driven by a branched rough path, since the Taylor expansion of f py t q from the branched rough path and from the weak geometric rough path point of views coincide, by Theorem 5.8 in [START_REF] Hairer | Geometric versus non-geometric rough paths[END_REF]. We give a self-contained proof of the equivalence of a Davie-type and Bailleul-type notion of a solution to a rough differential equation driven by a branched rough path in Section 2.

-Branched rough paths and their associated flows in a nutshell

We prove in this section the following analogue of Theorem 1.

Theorem -An R d -valued path z is a solution to a rough differential equation driven by a branched rough path in the sense of Davie if and only if it is a solution of that equation in the sense of Bailleul.

Theorem 2 gives an alternative proof of Theorem 5.3 of [START_REF] Cass | Tree algebras over topological vector spaces in rough path theory[END_REF] in the setting of dynamics driven by branched rough paths. We include in this section a short self-containted introduction to finite dimensional branched rough paths before proving Theorem 2 in Section 2.2. We refer to Gubinelli's original paper [START_REF] Gubinelli | Ramification of rough paths[END_REF] and Hairer and Kelly's paper [START_REF] Hairer | Geometric versus non-geometric rough paths[END_REF] for alternative accounts on branched rough paths in a finite dimensional setting, and to [START_REF] Cass | Tree algebras over topological vector spaces in rough path theory[END_REF] for an account of the theory in an infinite dimensional setting.

What are branched rough paths good for? Together with Equation (1.2) giving the lift of a smooth R -valued path to the rps-step free nilpotent Lie group, the definition of a weak geometric rough path X makes it clear that the higher order levels of X play the role of the non-existing integrals ş X i dX j , ť X i dX j dX k , etc. The archetypal example of a non-smooth weak geometric p-rough path is given by realizations of the Brownian rough path B " pB, Bq, for 2 ă p ă 3, defined by B jk ts :"

ż t s pB u ´Bs q ˝dB u .
Note the use of Stratonovich integration. If one defines instead the iterated integral B with an Itô integral, the corresponding object B no longer takes values in the rps-step free nilpotent Lie group G rps . This is related to the fact that Itô integration procedure does not satisfy the usual chain rule. A different algebraic setting is needed to handle the integration theory of such controls; this is precisely what branched rough paths provide us with. The name "branched rough paths" come from the fact that these objects are indexed by trees.

-Branched rough paths

1. The starting point -Consider an ordinary controlled R d -valued differential equation

9 x r " ÿ i"1 V i px r q 9 h i r
driven by some smooth R -valued path h. Given any tuple I P 1, k , write H I ts for the iterated integral ş sďs 1 﨨¨ďs k ďt dh i 1 s 1 . . . dh i k s k . Branched rough path appears naturally if one expands in the Taylor formula

f px t q " f px s q `ÿ |I|ďk 0 H I ts pV I f qpx s q `O`| t ´s| k 0 `1˘( 2.1)
all the iterated derivatives V I f in terms of the derivatives of its different terms. We have for instance

V 1 `V2 f ˘pxq " pD x f q `pV 1 V 2 qpxq ˘`pD 2 x f q `V1 pxq, V 2 pxq ˘. (2.
2) The best way to represent the resulting sum in (2.1) is to index it by labelled rooted trees. Let T stand for the set of possibly empty rooted labelled trees, with labels in t1, . . . , u. Given some trees τ 1 , . . . , τ k in T and a P t1, . . . , u, denote by rτ 1 . . . τ k s a the element of T obtained by attaching the trees τ 1 , . . . , τ k to a new root with label a. Any element of T can be constructed in this way starting from the empty tree 1. We denote by |τ | the number of vertices of a tree τ , and define its symmetry factor σpτ q recursively

σ `rτ n 1 1 . . . τ n k k s a ˘" n 1 ! . . . n k ! σpτ 1 q n 1 . . . σpτ k q n k
, where τ 1 , . . . , τ k are distinct trees with respective multiplicities n 1 , . . . , n k . This number does not depend on the label a. Denote by xT y the real vector space spanned by T and by xT y ˚its algebraic dual, identified with xT ˚y, for some copy T ˚of T .

Set V p1 ˚q :" 0. and define recursively, for a P 1, and τ 1 , . . . , τ n in T , vector fields on R d indexed by dual trees V p' å q :" V a , V `prτ 1 . . . τ n s a q ˚˘:" 1 σ `rτ 1 . . . τ n s a ˘pD n V a q `V pτ 1 q, . . . , V pτ n q ˘; (2.3)

vector fields on R d are seen as first order differential operators. We define differential operators indexed by dual forests setting

V `τ 1 ¨¨¨τ n ˘f " 1 σ `τ1 . . . τ n ˘`D n f ˘`V pτ 1 q, . . . , V pτ n q ˘, (2.4
f px t q " f px s q `ÿ ϕPF ;|ϕ|ďk 0 H ϕ ts `V pϕ ˚qf ˘px s q `O`| t ´s| k 0 `1˘.
(2.5)

The

H τ i us dh a u ; (2.6)
formulas (2.5) and (2.6) go back to Cayley' seminal work [START_REF] Cayley | On the analytical forms called trees[END_REF]. In the same way as the family of iterated integrals of h lives in an algebraic structure, the free nilpotent Lie group, the family of all H τ also lives in some algebraic structure, the dual of a Hopf algebra. We refer the reader to Sweedler's little book [START_REF] Sweedler | Hopf algebras[END_REF] for the basics on Hopf algebras; all that we need to know is elementary and recalled below.

2. Hopf algebraic structure -Let pF, ¨q stand for the set of commuting real-valued polynomials with indeterminates the elements of T and polynomial multiplication operation; equip the algebraic tensor product F b F with the induced product pa b bqpc b dq = pacq b pbdq. We define a coproduct ∆ on F as follows. Given a labelled rooted tree τ , denote by Subpτ q the set of subtrees of τ with the same root as τ . Given such a subtree s, we obtain a collection τ 1 , . . . , τ n of labelled rooted trees by removing s and all the adjacent edges to s from τ . Write τ zs for the monomial τ 1 . . . τ n . One defines a linear map ∆ : T Ñ F b F by the formula ∆τ " ÿ sPSubpτ q pτ zsq b s, and extend it to F by linearity and by requiring that it is multiplicative ∆pτ 1 . . . τ n q :" ∆pτ 1 q . . . ∆pτ n q. This coproduct is coassociative, p∆ b Idq∆ " pId b ∆q∆, as it should be in any Hopf algebra. Given a real-valued map Y on T , we extend it into an element of the dual space

F ˚of F setting Y τ 1 ...τn :" xY, τ 1 ¨¨¨τ n y :" n ź i"1 Y τ i ,
for a monomial τ 1 ¨¨¨τ n , and by linearity. One defines a convolution product on F ˚setting pY ‹ Xq τ :" pY b Xqp∆τ q "

ÿ sPSubpτ q Y τ zs X s , (2.7)
for a labelled rooted tree τ . The third ingredient needed to define the Hopf algebra structure of F is an antipode S, that is a map S : F Ñ F inverting ∆ in the sense that MpId b Sq∆ " MpS b Idq∆ " Id, where M stands for the multiplication map Mpa b bq " ab P F. An explicit formula for S was first obtained in [START_REF] Connes | Hopf algebras, renormalization and noncommutative geometry[END_REF]; see [START_REF] Chartier | Algebraic structures of B-series[END_REF] for a simple and enlighting proof. From its definition, the inverse a ´1 of any element a of F is given by the formula a ´1 " S ˚a, where S ˚is the dual map in F ˚of the antipode map in F, that is `a´1 , τ ˘" `a, Sτ ˘.

Pick an integer N ě 0. Write F pkq for the vector space spanned by the monomials pτ 1 . . . τ n q ˚, with

ř n i"1 |τ i | " k,

;

(2.10)

we also define a distance dpX, Yq " }X ´Y} on the nonlinear set of Hölder branched p-rough path. Given 0 ď s ď t ď T and a Hölder branched p-rough path X defined on the time interval r0, T s, we define an element of xT ďrps y setting

Λ ts :" log ‹ X ts ;
by definition exp ‹ pΛ ts q " X ts P G rps .

4. Rough differential equations driven by Hölder branched p-rough paths -Let X be a Hölder branched p-rough path over R , and F " pV 1 , . . . , V q be γ-Hölder vector fields on R d , for γ ą p. A path z in R d is said to be a solution path to the rough differential equation dz t " Fpz t q dX t (2.11) in the sense of Davie if one has

x t " x s `ÿ ϕPF ;|ϕ|ďrps `V pX ts qId ˘px s q `O`| t ´s| a ˘.
for some constant a ą 1. (This is actually Gubinelli's definition [START_REF] Gubinelli | Ramification of rough paths[END_REF] rather than Davie's definition.) The path z is a solution to Equation (2.11) in the sense of Bailleul if one has

f px t q " f px s q `ÿ ϕPF ;|ϕ|ďrps `V pX ts qf ˘px s q `O`| t ´s| a ˘,
for some constant a ą 1, for every real-valued function γ-Hölder function f on R d . (The Op¨q term is allowed to depend on f .) Theorem 2 states the equivalence of these two notions of definition. Conditions for the existence of a unique solution are given in Gubinelli' seminal paper [START_REF] Gubinelli | Ramification of rough paths[END_REF].

-Proof of Theorem 2

We only need to prove that a solution in Davie' sense is a solution in Bailleul' sense. We proceed as in Section 1 and prove that the result comes from a Taylor expansion property satisfied by the time 1 map of an approximate dynamics and the morphism property stated in Lemma 3.

Recall we write Λ ts for log ‹ X ts . Given some times 0 ď s ď t ď T , let µ ts stand for the well-defined time 1 map associated with the ordinary differential equation 9 y u " V `Λts ˘py u q.

(2.12) (The assumption that γ ą p and the vector fields V i are γ-Lipschitz ensures that all the vector field V pΛ ts q is globally Lipscthiz continuous.)

5. Lemma -There is a constant a ą 1 with the following property. For any γ-Hölder real-valued function f on R d , one has

f ˝µts " f `V `Xts ˘f `Of `|t ´s| a ˘,
for a remainder term O f `|t ´s| a ˘that may depend on f , and that has a C 1 -norm bounded above by a constant multiple of |t ´s| a .

Proof -The Taylor expansion property follows from the morphism property of the V -map from Lemma 3, writing

f py 1 q " f py 0 q `ż 1 0 V pΛ ts qf ( py s 1 q ds 1 " f py 0 q ` V pΛ ts qf ( pxq `ż 1 0 ż s 1 0 ! V pΛ ts qV `Λts ˘f ) py s 2 q ds 2 ds 1 " f py 0 q ` V pΛ ts qf ( pxq `ż 1 0 ż s 1 0 ! V `Λ‹2
ts ˘f ) py s 2 q ds 2 ds 1 " f py 0 q ` V pΛ ts qf ( pxq `1 2

! V `Λ‹2 ts ˘f ) pxq `ż 1 0 ż s 1 0 ż s 2 0 ! V pΛ ‹3 ts qf
) py s 3 q ds 3 ds 2 ds 1 and, by induction, f py 1 q " f py 0 q `rps ÿ k"1

1 k! V `Λ‹k ts ˘f ( pxq `ż ! V `Λ‹prps`1q
ts ˘f ) py rps`1 q 1 0ďs 1 﨨¨ďs rps`1 ds " V pX ts qf ( py 0 q `Θf `|t ´s| a ˘`ż ! V `Λ‹prps`1q ts ˘f ) py rps`1 q 1 0ďs 1 﨨¨ďs rps`1 ds

The Θ f `|t ´s| a ˘term comes from the fact that Λ P H N and X P H N are built from one another using the restriction of the maps log ‹ and exp ‹ , so X ts P H N coincides with ř N i"0 1 n! Λ ‹n ts up to some terms in HzH N of size O `|t´s| a ˘. The conclusion follows then from that explicit representation of the Taylor remainder.

It follows from Lemma 5 that a path z is a solution in Davie' sense iff z t " exp `V pΛ ts q ˘pz s q `O`| t ´s| a 1 ˘, for some exponent 1 ă a 1 ď γ p . The end of the proof of Theorem 2 is then identical to the proof of Theorem 1. `εId tu ˝µus . One then knows from Theorem 1 in [START_REF] Bailleul | Flows driven by rough paths[END_REF] that the rough differential equation (2.11) has a unique solution flow, associated with the C 1 -approximate flow µ. This provides a direct proof that the rough differential equation (2.11) has a unique solution flow.

Remarks

2.

Here as in Section 1, working with Hölder rough paths or rough paths controlled by some more general control wps, tq does not make any difference. Working with branched rough paths over an infinite dimensional space E does not make any difference either, as long as one takes care of working with a symmetric system of cross seminorms generating the topology of E, and complete accordingly the different algebraic tensor spaces E bk , such as emphasized in [START_REF] Cass | Tree algebras over topological vector spaces in rough path theory[END_REF], Section 3.

  ) with σ `τ1 . . . τ n ˘:" σ `rτ 1 . . . τ n s a ˘, for any 1 ď a ď . Set |τ 1 ¨¨¨τ n | " |τ 1 | `¨¨¨`|τ n |, and denote by ϕ " τ 1 ¨¨¨τ n , a generic forest, and by F the set of these forests. In those terms, Equation (2.1) rewrites

  and denote by π ďN the natural projection from F ˚to the quotient space F ˚z À kěN `1 F pkq . Set π ďN pF ˚q ": F ďN .The image of exp ‹ pxT ˚yq by π ďN is then diffeomorphic to the real vector space xT ďN y spanned by π ďN pT ˚q; it is a Lie group when equipped with the operation π ďN ˝‹, which we still denote by ‹. Write G N Ă F ďN for that Lie group; it plays for branched rough paths the role that the rps-step free nilpotent Lie group plays for weak geometric Hölder p-rough paths.Denote by ' å the element of the dual canonical basis, dual to ' a . It is elementary to check that the F ˚-valued function H ts " pH τ ts q τ PT , |τ |ďN associated with a smooth Rvalued path h, as defined in (2.6), considered as a function of the time parameter t, satisfies in π N pF ˚q the ordinary differential equation this implies that H ts is an element of G pN q for all times t ě s, and thatH ts " H ´1 s H t ,with H r :" H r0 , for all r ě 0.Definition -Let 2 ă p. A Hölder branched p-rough path on r0, T s is a G for all forests ϕ " τ 1 . . . τ k with |ϕ| :" ř k i"1 |τ i | ď rps, and X ts :" X ´1 s ‹ X t .

		dH ts "	ÿ	t H ts ‹ `'å dh a	˘;
				a"1
					rps -valued path
	X such that	sup 0ďsătďT	|t ´s| |X ϕ ts | |ϕ| p	ă 8,
	The norm of X is defined as
	}X} :"	max ϕPF ; |ϕ|ďrps	sup 0ďsătďT	|X ϕ ts | |ϕ| |t ´s| p

  -1. Write ε f ts for the remainder f ˝µts ´V pX ts qf . One can see directly that the µ ts form a C 1 -approximate flow from the identity µ tu ˝µus " V pX tu qId ( ˝µus `εId tu ˝µus " V pX us qV pX tu qId `εV pXtuqId

			us	`εIdtu ˝µus
	" V pX ts qId `εV pXtuqId us	`εId tu ˝µus
	" µ ts	`εId

ts

`εV pXtuqId us

part of this work was written there. Partial support from the ANR-16-CE40-0020-

Denote by θ " τ 1 . . . τ n , with τ i P T , a generic element of the canonical basis B of F. Write pθ ˚qθPB for the dual canonical basis and use bold letters to denote generic elements of F ˚. For an element a of F ˚of the special form a " ř τ 1 PT a τ 1 pτ 1 q ˚, with a τ 1 P R, we have for instance

where the inside sum is over the set of subtrees τ 1 of τ with the same root as τ , and such that τ zτ 1 is a subtree τ 2 of τ . Note in particular that a ‹ a is again of the form ř τ 1 PT b τ 1 pτ 1 q ˚. More generally, we have for such an a P F å‹n "

where the inside sum is over the set of disjoint rooted subtrees τ 1 , . . . , τ n of τ , with respective roots α 1 , . . . , α n such that α i`1 is a descendant of α i in τ , for all 1 ď i ď n ´1, and any node of τ is in one of the subtrees τ i . The convolution product (2.7) is closely related to the identities (2.3) and (2.4) defining the differential operators V pτ ˚q and V pτ 1 ¨¨¨τ n q.

The following statement is proved by induction on n `n1 .

3. Lemma -For any dual forests τ 1 ¨¨¨τ n and σ 1 ¨¨¨σ n1 , one has

3. Hölder branched p-rough paths -We define a Lie bracket on F ˚setting ra, bs ‹ :" a ‹ b ´b ‹ a, for which the real vector space xT ˚y spanned by the dual trees is a Lie algebra. The exponential map exp ‹ : F ˚Ñ F ˚and the logarithm map log ‹ : F ˚Ñ F ˚are defined by the usual series

with the convention a ‹0 " 1 ˚, where 1 ˚denote the dual of the empty tree. It is clear from the above formula for a ‹n that we have

with the same inside sum as in (2.8). Formula (2.8) justifies the convergence of the these sums, in the sense that `exp ‹ paq, θ ˘and `log ‹ pbq, θ ˘are actually finite sums for any monomial θ, provided pa, 1q " 0 -the pairing p¨, ¨q is a pairing between F ˚and F. Setting F 0 :" a P F ˚; pa, 1q " 0 ( and F 1 :" a P F ˚; pa, 1q " 1 ( , one can see that exp ‹ : F 0 Ñ F 1 and log ‹ : F 1 Ñ F 0 are reciprocal bijections; the following result can be seen as a consequence. (See Sweedler's above mentioned book, or Reutenauer's book [START_REF] Reutenauer | Free Lie algebras[END_REF], Theorem 3.2.) 4. Theorem -The pair `exp ‹ pxT ˚yq, ‹ ˘is a group.