An extension theorem to rough paths

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www.elsevier.com/locate/anihpc

An extension theorem to rough paths

Terry Lyons, Nicolas Victoir

^∗

Mathematical Institute, 24–29 St Giles’, Oxford, OX1 3LB, UK

Received 22 April 2004; received in revised form 6 May 2006; accepted 12 July 2006 Available online 22 December 2006

Abstract

We show that any continuous path of finitep-variation can be lifted to a geometricq-rough path, whereq > p.

Résumé

Nous montrons que tout chemin continu dep-variation finie peut être relevé en un « geometricq-rough path », pourq > p.

Keywords:Differential equations; Rough paths

1. Introduction

Let

x:[0,1] →Rⁿ, t→

x1(t ), . . . , xn(t )

be a continuous function of bounded variation, andV₁, . . . , V_nsome smooth functions fromR^dinto itself. Then there exists a (unique) solution to the control differential equation

dy(t )=n

i=1Vi(y(t ))dxi(t ),

y(0)=y₀. (1)

But without the smoothness assumption onx (which is for example almost surely not satisfied by Brownian motion), classical theory fails to give a meaning to the above equation. Rough paths theory [12,13,11] gives a meaning to Eq. (1), wheneverxis a continuous path of finitep-variation lifted to a “geometricp-rough path”.

To understand what a geometricp-rough path is, consider a smooth pathx:[0,1] →Rⁿ, and define S(x)_0,t=1+

0<s₁<t

dxs₁+

s<si1<···<si[p]<t

dxs_i₁ ⊗ · · · ⊗dxs_i_[_p_].

* Corresponding author.

E-mail address:[email protected] (N. Victoir).

doi:10.1016/j.anihpc.2006.07.004

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t→S(x)0,ttakes its values inG^[^p^](Rⁿ), the free nilpotent group of step[p]overRⁿ, thought of a manifold immersed in the tensor algebra_[p]

k=0(Rⁿ)^⊗^k [12,14]. For a givenp1, the set of geometric rough paths is the closure under a givenp-variation metric of the set{S(x), xsmooth} (see definition (12)). A weak geometricp-rough path is a G^[^p^](Rⁿ)-valued path of finitep-variation, where thep-variation is computed using a homogeneous metric associated to the group. The distinction between these two spaces was glossed over in the paper [12] by the first name author. It is obvious that a geometricp-rough path is a weak geometricp-rough path and that a weak geometricp-rough path is a geometricq-rough path for anyq > p. There are examples of weak geometricp-rough paths that are not geometric p-rough paths [6]. The difference between weak geometric rough paths and geometric rough paths is a bit like the difference between Lipschitz functions andC¹functions.

Ifxis a (weak) geometric p-rough path, xprojects onto a path x with values in Rⁿ. The solution to Eq. (1) is uniquely defined for anyx, and the solution is also a (weak) geometricp rough pathy. Moreover, the mapx→y is continuous in an appropriate topology. In the classical setting wherex is smooth, p=1. There is a functional relationship between x and the solution of the differential equation (1). Forp2, there will be infinitely many choices forx projecting ontox. The corresponding solution yand its projection y will in general depend on this choice.

There is often a “canonical” choice for the liftxofx (for example, ifx is smooth,S(x)is a canonical lift ofx to a geometricp-rough path, for anyp1). “Canonical” lifts have been constructed for Brownian motion [13,10], fractional Brownian motion with Hurst parameter greater than 1/4 [3,13], free Brownian motion [2,18], and a large class of random paths on fractals [1,8].

1.1. Our goal

Consider the following natural question: can every continuous path of finitep-variation inV (a Banach space) be lifted to a weak geometricp-rough path (aG^[^p^](V )-valued path of finitep-variation)? We will see, that provided thatp is not an integer number greater than or equal to 2, the answer is affirmative. This is optimal, as a counter example forp=2 was provided in [18]. In particular, any path of finitep-variation inV can be lifted to a geometric q-rough path, for anyq > p. The theorem we prove is actually stronger.

Theorem 1.We fixp∈ [1,+∞). LetV be a Banach space andKa closed normal subgroup ofG⁽^[^p^]⁾(V ). Ifxis a (G⁽^[^p^]⁾(V )/K, · G^([p])(V )/K)continuous path of finitep-variation, withp /∈N\{0,1}, then one can liftx to a weak geometricp-rough path.

Consider a pathx of finite p-variation with values in(G⁽^[^p^]⁾(V )/K, · G⁽^[^p^]⁾(V )/K), where K is as above. x projects to aRⁿ-valued path, and so one can consider again the differential equation (1). This one only makes sense once we liftx to a geometricq-rough pathx, forq > p. The solution depends in general on the choice of the liftx.

We will identify conditions on the Lie algebra generated by the vector fields(Vi)1inin (1) so that the projectiony of the rough path solutionyof Eq. (1) does not depend on the lift ofx. In general,yandywill depend on the liftx ofx.

To help the comprehension of the paper, we start by presenting the main theorem forp∈(2,3)andV =R², where no algebra is necessary and result are quite intuitive.

2. A simple case

We start with a non-surprising technical lemma, whose proof is inspired from the Kolmogorov–Centsov criteria [9].

Lemma 2.Lety be a map from

n0

2ⁿ

k=0{k2⁻ⁿ}into(E, d), a metric space, such that for alln, k∈ {0, . . . ,2ⁿ}, d(yk

2n, yk+1

2n )C2⁻^n/p. (2)

Then, there exists a unique continuous pathy˜:[0,1] →(E, d)which coincides withy on

n0

₂ⁿ

k=0{k2⁻ⁿ}. More- over,y˜is1/p-Hölder.

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Proof. We fixr∈N, and show by induction onmthat for alls, t∈Dmsuch that 0< t−s <2⁻^r, d(y_s, y_t)2C

m

k=r+1

2⁻^k/p. (3)

Whenm=r+1, necessarily,(s, t )is of the form(₂^km,^k₂⁺m¹),k∈ {0, . . . ,2^m−1}, and so (3) is exactly formula (2).

Suppose now that formula (3) is valid for m=r+1, . . . , M−1. Takes, t∈D_M such that 0< t−s <2⁻^r, and considert1=max{u∈DM−1;ut}ands1=max{u∈DM−1;us}. Notice thatd(ys, ys₁)andd(yt₁, yt)are both bounded byC2⁻^M/p, and, by the induction assumption, that

d(ys₁, yt₁)2C

M−1

k=r+1

2⁻^k/p. Therefore,

d(ys, yt)2C2⁻^M/p+2C

M−1

k=r+1

2⁻^k/p

=2C

M

k=r+1

2⁻^k/p, which concludes the induction.

Now let us consider(s, t )∈

m0Dm, and letrbe the natural number such that 2⁻^(r⁺¹⁾< t−s <2⁻^r. From the induction, we obtain

d(ys, yt)2C

∞ k=r+1

2⁻^k/pCp2⁻^(r⁺^1)/p

Cp|t−s|^1/p. (4)

We finally definey˜t for 0t1 by

˜ yt= lim

r→∞y_[2r t] 2r .

From (4), the limit exists andy˜satisfiesd(y˜_s,y˜_t)C|t−s|z^1/p. 2

Letx be aR²-valued path, which is 1/p-Hölder withp∈(2,3). We want to prove that we can liftx to a 1/p- Hölder path with values in the Heisenberg groupH¹equipped with its Carnot–Caratheodory metric. Let us first recall a few fact about this group and its metric. The Heisenberg groupH¹is equal toR³equipped with the product

(x1, y1, z1)×(x2, y2, z2)=

x1+x2, y1+y2, z1+z2+1

2(y1x2−y2x1)

.

The Carnot–Caratheodory distance will be introduced later, the only property we need for this preliminary chapter is that there exists positive constantsc, Csuch that

cmax

|x₁−x₂|,|y₁−y₂|,

z₁−z₂+1

2(y₁z₂−y₂z₁) ^1/2

d

(x₁, y₁, z₁), (x₂, y₂, z₂) , and

d

(x₁, y₁, z₁), (x₂, y₂, z₂)

Cmax

|x₁−x₂|,|y₁−y₂|,

z₁−z₂+1

2(y₁z₂−y₂z₁) ^1/2

.

It is easy to see that to liftx to a 1/p-HölderH¹-valued path, we need to construct the Levy area ofx, i.e. a map A:{0s < t1} →Rsuch that

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• for alls < t < u∈ [0,1], A_s,u=A_s,t+A_t,u+1

2

x_s,t¹ x_t,u² −x_s,t² x_t,u¹

; (5)

• for some constantC, for alls, t∈ [0,1],

|A_s,t|C|t−s|^2/p. (6)

Of course, ifxis of bounded variation,As,t=¹₂s

s<v₁<v₂<t(dx¹_v1dx_v²₂−dx_v2¹ dx_v²₁)satisfies the above condition.

As we do not assumexsmooth, we cannot use this area.

Proposition 3.Letxbe aR²-valued path, which is1/p-Hölder withp=2. Then, one can liftxto a1/p-Hölder path with values in the Heisenberg groupH¹equipped with its Carnot–Caratheodory metric.

In rough paths language, using the fact that path of finitep-variation are 1/p-Hölder after a time change, this means that we can lift anyR²-valued path of finitep-variation to a geometricp-rough path, wheneverp∈(2,3).

Proof. Ifp <2, the result is just a easy consequence of Theorem 1 in [12], or just properties of Young integrals. We therefore assumep >2.

LetC_x be the Hölder constant ofx. We construct inductively the area of x between dyadic times, Ak 2n,^k⁺¹

2n for k=0, . . . ,2ⁿ. We also define inductively

a_n=2^2n/p max

0k2ⁿ|Ak 2n,^k⁺¹

2n |.

First, we setA_0,1=0, and therefore we havea₀=0. Assume then, for a fixedn, that we have constructedAk 2n,^k₂⁺_n¹

fork=0, . . . ,2ⁿ. We defineA 2k 2n+1,^2k+1

2n+1 andA2k+1 2n+1,^2k+2

2n+1 so that they are both equal. Eq. (5) therefore forces them to be equal to

1 2Ak

2n,^k₂⁺n¹ −1 4

x¹_k

2n,^2k⁺¹

2n+1

x²_2k₊₁

2n+1,^k₂⁺_n¹−x²_k

2n,^2k⁺¹

2n+1

x¹_2k₊₁

2n+1,^k₂⁺_n¹

.

In particular

an+12⁻²⁽ⁿ⁺^1)/p2⁻^2n/pa_n 2 +C_x²

2 2⁻²⁽ⁿ⁺^1)/p, i.e.

an+12^2/p⁻¹an+1 2C_x².

It is easy to see by induction that, ifp >2, the sequencea_n is bounded. Transferring this information in terms of the pathx=(x, A), we see that we have constructed elementsxk2−n of the metric space (H¹, d) such that for all n, k∈ {0, . . . ,2ⁿ},

d(xk 2n,xk+1

2n )M_p2⁻^n/p. (7)

We conclude the proof with Lemma 2. 2

The above construction and idea will be the main argument of the proof of the main theorem. To be able to explain it, we need to introduce a few algebraic and geometric notions.

3. Algebraic preliminaries

3.1. Carnot groups

IfGis a simply connected nilpotent Lie group with Lie algebraG, then the Lie group exponential map exp :G→G is a diffeomorphism [15,17]. In this case we let ln :G→Gdenote the inverse of the exponential function. We start with a couple definitions.

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Definition 4.A Carnot group¹is a connected nilpotent Lie groupG, such that its Lie algebraGcan be written as G=W₁⊕ · · · ⊕W_n,

where for alli, W_i₊₁= [W₁, W_i]. For an elementg=exp(w1+ · · · +w_n)∈G, withw_i∈W_i, we let, fort∈R, δtg=exp

t w1+ · · · +tⁿwn

. δis called the dilation operator.

Definition 5.A (symmetric sub-additive) homogeneous norm [5] on a Carnot groupGis a function · G:G→R⁺ such that

(i) gG=0 if and only ifgis the neutral element of the group, 1=exp(0), (ii) δ_tgG= |t|gG,

(iii) for allg, h∈G,g⊗hGgG+ hG, (iv) for allg,gG= g⁻¹G.

Such a norm define a left invariant distance on the group byd_G(g, h)= h⁻¹⊗gG. We will say that(G, · G)is a normed Carnot group.

IfGis a fixed Carnot group with finite dimensional Lie algebra, all homogeneous norms onGare equivalent. The Carnot–Caratheodory norm is an example of a homogeneous norm on a Carnot group [7]. Any homogeneous norms · GonGleads to a left invariant distancedG(x, y)= y⁻¹x(in particular, the Carnot–Caratheodory norm leads to the Carnot–Caratheodory distance). LetGbe a normed Carnot group with Lie algebraG,Ka Lie subgroup ofG,with Lie algebraK. IfKis a closed normal Lie subgroup ofG, or equivalently ifKis closed ideal ofG, thenG/Kis then a Carnot group with Lie algebraG/K[15]. IfGis equipped with a homogeneous norm · G, then we equipG/K with the quotient homogeneous norm onG/K

· G/K:G/K→R, gK→ inf

k∈Kg⊗kG.

We will denote byπ_G,G/K the canonical homomorphism fromGontoG/K. Sometimes, it will be more convenient to writegKforπ_G,G/K(g).

Proposition 6.Let(G, · G)be a normed Carnot group, K a closed normal Lie subgroup ofG. There exists an injectioniG/K,G:G/K→Gsuch that

(i) πG,G/K◦iG/K,Gis the identity map ofG/K,

(ii) for allt∈R⁺,gK∈G/K,δt(iG/K,G(gK))=iG/K,G(δt(gK)), (iii) for allgK∈G/K,gKG/KiG/K,G(gK)G2gKG/K.

Proof. By definition of the homogeneous norm onG/K, for allg∈Gsuch thatgKG/K=1, the set M_g=

g⊗ksuch thatk∈Kand 1g⊗kG2 is non-empty. We definei_G/K,Gon the set of elements

gK, such thatgKG/K=1

to be any function which atgK associates an element of

m∈π_G,G/K⁻¹ (gK)M_m; such function exists by the axiom of choice. We then extendi_G/K,GtoG/Kwith the help of the formulai_G/K,G(δ_tgK)=δ_ti_G/K,G(gK). 2

1 In most definitions of a Carnot group,Gis assumed to be finite dimensional. We do not make such an assumption here.

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3.2. Free nilpotent groups

We know introduce a fundamental example of a Carnot group.

We fix (for the rest of the paper) a normed vector space(V , · 1). We letT (V )=_∞

n=0V^⊗ⁿbe the tensor algebra overV.T (V )equipped with standard addition+, tensor multiplication⊗and scalar product is an associative algebra.

T⁽ⁿ⁾(V ),the quotient algebra ofT (V )by the ideal_∞

m=n+1V^⊗^m,inherits this algebraic structure. One can define onT⁽ⁿ⁾(V )a Lie bracket by the formula

[a, b] =a⊗b−b⊗a,

which makesT⁽ⁿ⁾(V )into a Lie algebra. We letG⁽ⁿ⁾(V )be the Lie subalgebra of T⁽ⁿ⁾(V )generated by elements inV. Note that

G⁽ⁿ⁾(V ) n

i=1

V_i, where

V₁=V and V_i₊₁= [V , V_i]. (8)

G⁽ⁿ⁾(V )is the free nilpotent Lie algebra of stepn[12–14]. The exponential, logarithm and inverse function are defined onT⁽ⁿ⁾(V )by mean of their power series. We denote byG⁽ⁿ⁾(V )=exp(G⁽ⁿ⁾(V )). By the Baker–Campbell–Hausdorff formula,(G⁽ⁿ⁾(V ),⊗)is a connected nilpotent Lie group, called the free nilpotent Lie group of stepnoverV. By construction,(G⁽ⁿ⁾(V ),⊗)is a Carnot group, with Lie algebraG⁽ⁿ⁾(V ).

We are now going to equipG⁽ⁿ⁾(V )with a homogeneous norm. We first let · i be some norms onV^⊗ⁱ such that for all(a_i, a_j)∈V^⊗ⁱ×V^⊗^j,a_i ⊗a_ji+ja_ii + a_jj. To simplify notations, we will write · for all these norms. Now define

gG⁽ⁿ⁾(V )= max

i=1,...,n

i!g_i1/ i

+max

i!g⁻¹

i^{1/ i},

whereg=1+g₁+· · ·+g_n,g_i∈V^⊗ⁱis an element of the groupG⁽ⁿ⁾(V ), andg⁻¹is its inverse. The binomial equality quickly shows thatg→maxi=1,...,n(i!g_i)^{1/ i}is a sub-additive homogeneous norm (but a priori not symmetric). That implies that · _G⁽ⁿ⁾_{(V )}defines a homogeneous norm onG⁽ⁿ⁾(V ). We also let

d_G(n)(V )(g, h)=h⁻¹⊗g

G⁽ⁿ⁾(V ).

Proposition 7.Letg=exp(l1+ · · · +ln),withli∈Vi. Then, c_n max

i=1,...,nl_i^{1/ i}g_G⁽ⁿ⁾_{(V )}C_n max

i=1,...,nl_i^{1/ i}, for some constantscnandCnwhich depends only onn.

Proof. Let us fixi∈ {1, . . . , n}and writeg=1+g₁+ · · · +g_n,withg_i ∈V^⊗ⁱ. By definition of the exponential function,

g_k=

k

i=1

1 i! _j_1,...,ji

j1+···+ji=kl_j₁

⊗ · · · ⊗l_j_i.

Hence, k!g_kk

1/ k

_k

i=1

k! i! _j_1,...,ji

j1+···+ji=k

l_j₁ · · · l_j_i 1/ k

k!(expk−1)1/ k

i=max1,...,nl_i^{1/ i}.

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Applying this tog⁻¹,we obtain k!g⁻¹

k

1/ k

k!(expk−1)1/ k

i=max1,...,n −li^{1/ i}=

k!(expk−1)1/ k

i=max1,...,n −li^{1/ i}. That gives us the upper bound. For the lower bound, observe that by definition of the logarithm function,

lk=

k

i=1

(−1)ⁱ i j1,...,ji

j1+···+ji=k

gj₁⊗ · · · ⊗gj_i,

which, when applied to bothgand its inverse, gives that for all 1kn, lk^{1/ k}c⁻_n¹gG⁽ⁿ⁾(V )

for a constantc_n>0. 2

Corollary 8.LetK=exp(K)be a closed normal subgroup ofG⁽ⁿ⁾(V ). Then, ifg=exp(l1+ · · · +l_n)withl_i∈V_i,

cn gK_G⁽ⁿ⁾_{(V )/K}

maxi=1,...,n(infk_i∈K∩V_il_i+k_i)^{1/ i} Cn.

Corollary 9. LetC(G⁽ⁿ⁾(V ))be the centre ofG⁽ⁿ⁾(V ) andθ the canonical isomorphism betweenG⁽ⁿ⁻¹⁾(V )and G⁽ⁿ⁾(V )/C(G⁽ⁿ⁾(V )). Then the homogeneous norm · G⁽ⁿ⁻¹⁾(V )andθ (·)G⁽ⁿ⁾(V )/C(G⁽ⁿ⁾(V ))are equivalent. We will therefore not distinguish between them.

4. Rough paths

In this paper, byE-valued path, we mean a function from[0,1]intoE.

4.1. Onp-variation

Definition 10.Let(E, d)be a metric space. A(E, d)-valued pathxis said to have finitep-variation if sup

D

#D−1

i=1

d(x_t_i, x_t_i₊₁)^p<∞,

where the supremum runs over all subdivisionsD=(0t1· · ·t#D1)of the interval[0,1].

Note thatx is continuous and of finite regularp-variation if and only if for allst,d(x_s, x_t)ω(s, t ),where (i) ω:{(s, t ),0st1} → R⁺is continuous.

(ii) ωis super-additive, i.e.∀s < t < u,ω(s, t )+ω(t, u)ω(t, u). (9) (iii) ω(t, t )=0 for allt∈ [0,1].

We will say in such case thatx has finitep-variation controlled byω.

We are going to show that a continuous(E, d)-valued path of finitep-variation is, up to reparametrisation of time, 1/p-Hölder continuous. Ifωsatisfies (9), then

(s, t )→ω(0,1)

ω(0, t )

ω(0,1)−ω(0, s) ω(0,1)

is a continuous additive map, equal to zero on the diagonal, andω(0, t )−ω(0, s)ω(s, t )(by the super-additivity ofω). Therefore, a path is of finitep-variation if and only if there exists an non-decreasing and continuous surjection γ from[0,1]onto[0,1]and a positive constantCsuch that

for allst, d(x_s, x_t)^pCγ (t )−γ (s).

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For such aγ, we define γ⁻¹:[0,1] → [0,1],

t→inf

u, γ (u)=t . The following is straightforward to check.

Lemma 11.Let x be a continuous(E, d)-valued path of finitep-variation controlled by(s, t )→C|γ (t )−γ (s)|, whereγis a continuous increasing surjection from[0,1]onto[0,1]. Define

y:[0,1] →E, t→x_γ₋1(t ).

Then,y is a1/p-Hölder(E, d)-valued path.

Reciprocally, ifyis a1/p-Hölder(E, d)-valued path then, it is a continuous path of finitep-variation controlled by(s, t )→C|t−s|.

4.2. Geometricp-rough paths

Definition 12.A weak geometricp-rough path is a(G⁽^[^p^]⁾(V ),·G^([p])(V ))-valued path which has finitep-variation.² Whenx is a path with values in a group(G,⊗), we will writexs,t=x_s⁻¹⊗xt.

5. The extension theorem

We first need an important lemma.

Lemma 13.Let(G, · G)be a normed Carnot group with graded Lie algebra G=W1⊕W2⊕ · · · ⊕Wn.

DefineK to be a closed subgroup ofexp(Wn), which gives us a normed Carnot group(G/K, · G/K). Letx be a 1/p-Hölder(G/K, · G/K)-valued path. Then, ifp > n, there exists a1/p-Hölder(G, · G)-valued pathx˜ such thatπ_G,G/K(x)˜ =x.

Proof. As exp(Wn)is in the center ofG,K is a subgroup of the center ofG. In particular,K is a closed normal subgroup ofG.

To construct our pathx, we are first going to construct its increments˜ x˜_s,twhens, t∈D_m= {₂^km, k∈ {0, . . . ,2^m⁻¹}}

witht−s=2⁻^m, doing this for allm.x˜_s,t will be constructed in such a way that ˜x_s,tGC|t−s|^1/p for a given C <∞. Multiplying the increments, we will then have definedxon all dyadics, and the proof will be finished thanks to Lemma 2.

So we define recursively onmsome elementsy k

2m,^k₂⁺_m¹ ∈K,k∈ {0, . . . ,2^m−1},m∈N, in the aim of defining the elementsx˜ k

2m,^k₂⁺_m¹ with the formula

˜ x k

2m,^k₂⁺_m¹ =i_G/K,G(x k

2m,^k₂⁺_m¹)⊗y k 2m,^k₂⁺_m¹

wherei_G/K,G is the injection of Proposition 6. This will ensure thatπ_G,G/K(x)˜ =x. First, we let,y_0,1=exp(0).

Then, we assume thaty k

2m,^k+1₂m (and hencex˜ k

2m,^k+1₂m) has been constructed for all 0k2^m−1 and a fixedm, and we define the two elementsy 2k

2m+1,^2k⁺¹

2m+1 andy2k+1 2m+1,^2k⁺²

2m+1 to be both equal, and equal to the inverse of δ₂₋1/n

i_G/K,G(x 2k 2m+1,^2k⁺¹

2m+1

)⊗i_G/K,G(x2k+1 2m+1,^2k⁺²

2m+1

)⊗ ˜x⁻_k¹

2m,^k₂⁺_m¹

.

2 The definition of a geometricp-rough path is presented quite differently in [12], as the notion of a homogeneous norm was not mentioned there.

Nonetheless, the difference is easily seen to be only notational.

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We easily check thatπG,G/K(y 2k 2m+1,^2k⁺¹

2m+1)=exp(0), i.e. that y 2k 2m+1,^2k⁺¹

2m+1 =y2k+1 2m+1,^2k⁺²

2m+1 ∈K. As elements of K commute with elements ofG, and with the help of the formulaδ₂1/n(y)=y^⊗²fory∈K, we check that this choice fory 2k

2m+1,^2k+1

2m+1 andy2k+1 2m+1,^2k+2

2m+1 gives

˜ x k

2m,^k⁺¹

2m = ˜x 2k 2m+1,^2k⁺¹

2m+1 ⊗ ˜x2k+1 2m+1,^2k⁺²

2m+1. We then defineam=2^m/psup_k_∈{_0,...,2^m−1}y k

2m,^k⁺¹

2mG. By the assumption thatx is 1/p-Hölder and by the definition ofi_G/K,G,

iG/K,G(x k 2m,^k₂⁺_m¹)

G2x k

2m,^k₂⁺_m¹G/K2C2⁻^m/p. Hence, from the previous inequality, we obtain that

2^1/n2⁻^(m⁺^1)/pa_m₊₁a_m2⁻^m/p+2⁻^(m⁺^1)/p2²⁺^1/pC, i.e.

am+12^1/p⁻^1/nam+2²⁺^1/p⁻^1/nC.

Asn < p, we have 2^1/p⁻^1/n<1 which forces the sequence a_m to be bounded. So we have constructed for every m0,k∈ {0, . . . ,2^m−1}some elementsx˜ k

2m,^k₂⁺_m¹ =i_G/K,G(x k

2m,^k₂⁺_m¹)⊗y k

2m,^k₂⁺_m¹ ∈G, such that ˜x k

2m,^k⁺¹

2mG2x k

2m,^k⁺¹

2m G/K+ y k 2m,^k⁺¹

2mG

2C+sup

m

am

2⁻^m/p=Cp,n2⁻^m/p. Remember also that for all dyadic ₂^km,

˜ x k

2m,^k₂⁺_m¹ = ˜x 2k 2m+1,^2k⁺¹

2m+1 ⊗ ˜x2k+1 2m+1,^2k⁺²

2m+1. That allows us to define

˜ x k

2m,₂^lm =

l−1

j=k

˜ x j

2m,^j₂⁺m¹.

We have proved that for alln, k∈ {0, . . . ,2ⁿ}, ˜x k

2m,^k₂⁺m¹GC2⁻^m/p. (10)

The proof is therefore finished using Lemma 2. 2 We are now ready for our main theorem.

Theorem 14. We fix p∈ [1,+∞). Let K be a closed normal subgroup of G⁽^[^p^]⁾(V ). If x is a (G⁽^[^p^]⁾(V )/K, · G^([p])(V )/K)continuous path of finitep-variation, withp /∈N\{0,1}, then there exists a continuous(G⁽^[^p^]⁾(V ), · G⁽^[^p^]⁾(V ))-valued pathx˜of finitep-variation such that

π_G([p])(V ),G⁽^[^p^]⁾(V )/K(x)˜ =x.

Proof. As noticed in Section 4.1, we assume without loss of generalities thatx is 1/p-Hölder. We denote byK1⊕

· · · ⊕K_[p]the Lie algebra ofK, withKi⊂V_i. We define fork=1, . . . , n, H^(k)=G^(k)(V )/exp(Kk)

and

M^(k)=G^(k)(V )/exp(K1⊕ · · · ⊕Kk) H^(k)/exp(K1⊕ · · · ⊕Kk−1).

(10)

We are going to construct recursively some H^(k)-valued paths y^(k) and G^(k)(V )-valued paths x^(k) which are 1/p-Hölder and such that the projections ofx,x^(k), andy^(k)ontoM^(k)are equal, i.e. such that

π_H(k),M^(k)

y^(k)

=π_M([p]),M^(k)(x), (11)

π_G(k)(V ),M^(k)

x^(k)

=π_M([p]),M^(k)(x). (12)

Using Lemma 13,x^(k)is easily constructed fromy^(k), so we only need to construct the pathsy^(k). Constructing those paths are not very difficult intuitively, as to constructy^(k), we just need to “paste” togetherx^(k⁻¹⁾andx.

Fork=1 y⁽¹⁾=π_M([p]),M⁽¹⁾(x)is aH⁽¹⁾=G⁽¹⁾(V )/exp(K1)valued path, which is 1/p-Hölder.

We now assume that we have aG^(k)(V )-valued pathx^(k) which is 1/p-Hölder and which satisfies equality (12), and we aim to construct a 1/p-HölderH^(k⁺¹⁾-valued pathy^(k⁺¹⁾.

The setZ^(k⁺¹⁾defined by

(g, m)∈G^(k)(V )×M^(k⁺¹⁾such thatπ_G(k)(V ),M^(k)(g)=π_M(k+1),M^(k)(m) equipped with the product

(g₁, m₁)⊗(g₂, m₂)=(g₁⊗g₂, m₁⊗m₂)

is a group, and the applicationΨ:Z^(k⁺¹⁾→H^(k⁺¹⁾defined by the formula: if_k∈V₁⊕ · · · ⊕V_k andl^k⁺¹∈V_k₊₁, Ψ

exp(k),exp

k+l^k⁺¹ exp

K1⊕ · · · ⊕Kk+1

=exp(k)exp l^k⁺¹

exp(Kk+1), is easy seen to be an isomorphism by Baker–Campbell–Hausdorff formula.

Using Proposition 7 and Corollary 8, we see that there exists a constantC_k₊₁such that for all(g, m)∈Z^(k⁺¹⁾, Ψ (g, m)

H^(k+1)C_k₊₁

g_G^(k)_{(V )}+ m_M^(k+1)

.

For allt∈ [0,1],(x_t^(k), π_M([p]),M^(k+1)(xt))∈Z^(k⁺¹⁾, hence we can define y_t^(k⁺¹⁾=Ψ

x_t^(k), π_M([p]),M^(k+1)(x_t) .

Note first thaty^(k⁺¹⁾satisfies the equality (11). BecauseΨ is an isomorphism,y_s,t^(k⁺¹⁾=Ψ (x_s,t^(k), π_M([p]),M^(k⁺¹⁾(x_s,t)) and hence

y_s,t^(k⁺¹⁾

H^(k⁺¹⁾C_k₊1x^(k)_s,t

G^(k)(V )+π_M([p]),M^(k+1)(x_s,t)

M^(k⁺¹⁾

C_k₊₁x^(k)_s,t

G^(k)(V )+ x_s,tM^([p])

. By hypothesis and induction hypothesis,

x_s,t^(k)

G^(k)(V )+ xs,tM⁽^[^p^]⁾(C+C_k)|t−s|^1/p, hencey_s,t^(k⁺¹⁾H^(k+1)C_k₊₁|t−s|^1/p.

Using the induction step until we reach the level[p], we obtain aG⁽^[^p^]⁾(V )-valued pathx⁽^[^p^]⁾which is 1/p-Hölder and such that

π_G([p])(V ),M⁽^[^p^]⁾

x⁽^[^p^]⁾

=π_M([p]),M⁽^[^p^]⁾(x)=x. 2 We ought to make a couple comments on our main theorem.

Remark 15.Note that we could have considered a continuous path of finitep-variation with values in a quotient space ofG⁽ⁿ⁾(V ),withn >[p]. IfK⁽ⁿ⁾is a closed normal subgroup ofG⁽ⁿ⁾(V )andx is a 1/p-Hölder path with values in (G⁽ⁿ⁾(V )/K⁽ⁿ⁾, · G⁽ⁿ⁾(V )/K⁽ⁿ⁾), withp /∈N\{0,1}, then there exists a 1/p-HölderG⁽ⁿ⁾(V )-valued pathx˜such that

π_G(n)(V ),G⁽ⁿ⁾(V )/K⁽ⁿ⁾(x)˜ =x.

To prove this, first letK⁽^[^p^]⁾=π_G(n),G⁽[p])(K⁽ⁿ⁾). The canonical projection ofxintoG^[^p^](V )/K⁽^[^p^]⁾is a 1/p-Hölder path. Hence, by the previous theorem, there exists a 1/p-HölderG^[^p^](V )-valued pathx⁽^[^p^]⁾such that

π_G([p])(V ),G^[^p^](V )/K⁽^[^p^]⁾

x⁽^[^p^]⁾

=π_G(n)(V )/K⁽ⁿ⁾,G^[^p^](V )/K⁽^[^p^]⁾(x).