Flows driven by rough paths

0.52 MB·English

by  Ismael Bailleul

FLOWS DRIVEN BY ROUGH PATHS
I. BAILLEUL
3
1
0 Abstract. We show in this work how the familiar Taylor formula can be used in a
2 simple way to reprove from scratch the main existence and well-posedness results from

n rough paths theory, in the context of Hölder weak geometric or branched rough paths;
a theexplosionquestion,TaylorexpansionandEulerestimatesarealsodealtwith. Unlike
J other approaches, we work mainly with flows of maps rather than with paths. We

9 illustrateourapproachbyprovingawell-posednessresultforsomemeanfieldstochastic
2 rough differential equation.

]
R
Contents
P
h. 1. Introduction 1
t 2. Flows driven by Hölder weak geometric p-rough paths (2 < p < 3) 3
a
m 2.1. Hölder weak geometric p-rough paths 6
2.2. Approximate rough differential equation 7
[
2.3. Fundamental properties of the approximate flow 9
6
2.4. Flows driven by Hölder weak geometric p-rough paths 12
v
8 3. Flows driven by Hölder weak geometric p-rough paths, general case 16
8 3.1. An algebraic interlude: tensor algebra over Rℓ and free nilpotent Lie group 16
8
3.2. Hölder weak geometric p-rough path, general case 17
0
. 4. Taylor expansion of solution flows to rough differential equations 23
3
5. Flows driven by branched rough paths 25
0
2 5.1. Branched rough paths 26
1 5.2. Flows driven by branched rough paths 29
:
v 6. Paths driven by rough paths 31
i 7. Mean field stochastic rough differential equations 34
X
References 38
r
a
1. Introduction
Since Lyons wrote his groundbreaking article [1] on rough paths, there has been a con-
stantly growing industry in understanding the scope of the theory. Besides providing an
alternative enlighting view on Ito’s theory of stochastic integration and some deep asso-
ciated results, rough paths theory has now invaded the world of numerical simulations,
stochastic and deterministic partial differential equations and finance, to name but a few
areas. Despite Lyons’ Saint Flour lecture notes [2], his book [3] with Qian and the im-
pressive and exhaustive book [4] of Friz and Victoir, rough path still seems to be seen as
Date: January 30, 2013.
This research was partially supported by an ANRgrant "Retour post-doctorant".
1
2 I.BAILLEUL
a somewhat difficult and technical subject where algebra and classical analysis meet in an
intricate way.
We show in this work how the main existence and uniqueness results of the theory can
be proved from scratch using only Taylor formula, in the context of dynamics driven by
Hölder weak geometric or branched rough paths. Contrary to Lyons, Friz-Victoir and
Gubinelli’s approach, we work primarily with maps from the state space E to itself rather
than with E-valued paths. Our dynamics on the space of maps ϕ will have typical form
(1.1) dϕ = Vdt+FdX,
for some driving vector fields V and F = (V ,...,V ) and a (Hölder weak geometric or
1 ℓ
branched p-) rough path X on some time interval [0,T]. Roughly speaking, a rough path
consists of an E-valued non-smooth path X together with a number of objects which
play the role of the missing iterated integrals of X, in the sense that they satisfy the
same algebraic relations as the iterated integrals of any smooth path, as well as some
natural size requirements. Given such a rough path and some vector fields V,V ,...,V ,
1 ℓ
a family (µts)06s6t6T of diffeomorphisms of E is constructed from an ordinary differential
equation. These maps do not form a flow, in the sense that µ is not equal to µ ◦µ for
ts tu us
all06 s 6 u6 t 6 T. However, someconditions onthedrivingvectorfields andtherough
pathensuretheexistenceofauniqueflow ofmaps(ϕts)06s6t6T closeto(µts)06s6t6T. This
is whatwecall a/thesolution totherough differential equation (1.1). This approach mixes
Davie’s original approach [5] and Feyel-de la Pradelle’s picture [6], [7].
All other approaches to rough paths consider paths associated with a point motion as
the fundamental basic object. Lyons, and later Gubinelli, interpret (1.1) as an integral
equation, which requires a suitable notion of rough path integral as a mechanism to attach
toaroughpathY andsomesufficientlyregularoneformg anotherroughpath ·g(Y)dY.
0
Solving equation (1.1) then amounts to find a fixed point to an integral equation of the
form Y = ·g(Y)dY, where Y is some extension of the original rough pathRX. See for
0
instance [1], [2] or [3], and Friz and Hairer’s forthcoming review of Gubinelli’s approach.
Friz andViRctoir’s approach, developed afterDavie’s ideas[5],consider Rd-valueddynamics
driven by a rough path as a limit in some appropriate topology of elementary dynamics
generated by controlled ordinary differential equations in which the control converges in
a rough path sense to some limit rough path. With such a view, no notion of integral is
needed to define adynamics; neither is itthecasein theapproach developped in this work.
Roughness of a rough path is quantified by a real number p > 2. In order to bring
the reader progressivey to the most general results, we deal in section 2 with Hölder weak
geometric p-rough paths, with 2 < p < 3, for which only one substitute of iterated integral
is needed. The case of a general Hölder weak geometric p-rough paths, for any p > 2,
is treated in section 3, while flows driven by branched rough paths are treated in section
5. It comes as a nice feature of our approach that convergence estimates for Euler type
schemes and Taylor expansion for solution flows come almost for free; this is explained
in section 4. Although designed for bounded vector fields, all the machinery works for
vector fields with at most linear growth and proves non-explosion of the dynamics in that
case. The results on flows are applied in section 6 to give a simple proof of the main
existence/uniqueness results for the classical rough differential equations on paths driven
by Hölder weak geometric or branched p-rough paths. They are illustrated differently in
section7whereweproveawell-posednessresultforsomelargeclassofmeanfieldstochastic
rough differential equation.
FLOWS DRIVEN BY ROUGH PATHS 3
Although everything below works word by word in any Banach space, we have chosen to
work in a finite dimensional setting where the definition of a γ-Lipschitz vector field, with
γ non-integer, is well-known, and for which there is no need to pay attention to nice tensor
norms on tensor powers of the Banach space. The reader acquainted with these notions
will have no difficulty in putting herself/himself in that setting while the others will be on
a firm ground.
A few notations will be used thoughout the text, which we gather here.
• Fix T > 0 and a non-integer γ > 1, with integer part [γ] and fractional part
{γ}. We say that a vector field W on Rd is γ-Lipschitz if it is C[γ] with a {γ}-
Hölder continuous differential of order [γ], and all its derivative are bounded. Its
γ-Lipschitz norm kWk is defined as
γ
[γ]
(1.2) kWk = W(r) + W([γ]) < ∞,
γ ∞ {γ}
r=0
X(cid:13) (cid:13) (cid:13) (cid:13)
whereW(r) isthedifferentialo(cid:13)forde(cid:13)rrofW(cid:13) and(cid:13)W([γ]) standsfortheclassical
{γ}
{γ}-Hölder norm of W([γ]).
(cid:13) (cid:13)
• Given any two C1 vector fields V,W on Rd we w(cid:13)rite VW(cid:13) for the vector field x 7→
(D W) V(x) . Given some sufficiently regular vector fields V ,...,V on Rd and
x 1 ℓ
a tuple I = (i ,...,i ) ∈ J1,ℓKr, we identify vector fields with derivation operators
1 r
(cid:0) (cid:1)
and write V for the differential operator f ∈ Cr 7→ V V (...V f) . Writing
I i1 i2 ir
[V,W] for the bracket of two vector fields, this defines a first order differential
(cid:0) (cid:1)
operator, that is a vector field. Set
V = V , V ,...,[V ,V ] ... .
[I] i1 i2 ir−1 ir
• We identify in the sequelhL(R(cid:2)ℓ) = Rℓ ⊗ (Rℓ)∗ an(cid:3)d Riℓ ⊗Rℓ with M (R), via the
ℓ
matrix representation of linear maps in the canonical basis. In these terms, given
any two vectors x,y of Rℓ, the (jk)-th component (x⊗y)jk of x⊗y is xjyk.
• We use the convention that aib stands for the sum aib .
i i i
P
2. Flows driven by Hölder weak geometric p-rough paths (2 < p < 3)
We explain in this opening section the meaning of equation (1.1) and how to solve it in
the simplest case where rough paths are really needed. Recall that a flow on Rd is a family
(ϕts)06s6t6T ofmaps fromRd toitselfsuchthatϕts = ϕtu ◦ϕus,forall0 6 s6 u 6 t 6 T,
and ϕ = Id, for all s. As a guide to our approach, consider the flow of diffeomorphisms
ss
of Rd generated by the controlled ordinary differential equation
N
(2.1) y˙ = f (y )h˙i(u)
u i u
i=1
X
for 0 6 u6 1, for some smooth vector fields f ,··· ,f on Rd, bounded and with bounded
1 N
derivatives, and some smooth RN-valued control h. The map ϕ associates to any x ∈Rd
ts
the value at time t of the solution to (2.1) started at time s from x. This flow can be
constructed directly as follows, without involving any integral equation.
Given 0 6 t 6 T, denote by µ the maps x 7→ x+ N hi(t)−hi(s) V (x) from Rd
ts i=1 i
to iself; they depend continuously on (s,t) in the topology of uniform convergence on Rd.
P (cid:0) (cid:1)
4 I.BAILLEUL
They enjoy the following fundamental property which justifies calling them approximate
solution flows; it is a simple consequence of Taylor formula. The inequality
(2.2) µ ◦µ −µ 6 c |t−s|2
tu us ts ∞ 0
holds for some positive consta(cid:13)nt c0, for all 0 6(cid:13) s 6 u6 t 6 1.
(cid:13) (cid:13)
Theorem 1. There exists a unique flow (ϕts)06s6t61 on Rd to which one can associate
two positive constants c and δ such that
(2.3) ϕ −µ 6 c|t−s|2
ts ts ∞
µholds,wwhheenreevser=t−ss+6i(δt.−Mso)r.eov(cid:13)(cid:13)er, the ϕt(cid:13)(cid:13)s are the uniform limits of µnts := µsnsn−1◦···◦
s1s0 i n
The above flow is the flow associated with the ordinary differential equation (2.1) since
the latter satisfies condition (2.3). We give a complete proof of theorem 1 as it illustrates
perfectly in an elementary setting our approach to flows driven by rough paths.
Proof – 1. Uniform controls. We first show by induction that
(2.4) µn −µ 6 L|t−s|2
ts ts ∞
holds for some positive con(cid:13)stant L, f(cid:13)or all integers n > 1 and all 0 6 s 6 t 6 T
(cid:13) (cid:13)
sufficiently close. Write for that purpose u= s and note that
n+1−2
µn+1−µ 6 µ2 ◦µn−1−µ ◦µn−1 + µ ◦µn−1−µ ◦µ
ts ts ∞ tu us tu us ∞ tu us tu us ∞
(cid:13)(cid:13) (cid:13)(cid:13) (cid:13)(cid:13) (cid:13)(cid:13) +(cid:13)(cid:13)µtu ◦µus−µts ∞. (cid:13)(cid:13)
Usingtheinductionhypothesis,inequality(2.2)andt(cid:13)(cid:13)hefactthatthe(cid:13)(cid:13)maps µts(·) t−s6δ
are uniformly Lipschitz continuous, with a Lipschitz constant no greater than 1 +
(cid:0) (cid:1)
o (1) , we get
δ (cid:0)
(cid:1)µnts+1−µts ∞ 6 µ2tu −µtu ∞+ 1+oδ(1) c(u−s)2+c0(t−s)2
(cid:13) (cid:13) 6 L(cid:13)(t−u)2+(cid:13) 1+(cid:0)oδ(1) c(u(cid:1)−s)2+c0(t−s)2
(cid:13) (cid:13) (cid:13) (cid:13)
2 2 n−1 2
6 (cid:0)+ 1+o (cid:1)(1) L+c (t−s)2.
δ 0
n+1 n+1
(cid:16)n(cid:16) (cid:17) (cid:16) (cid:17) o (cid:17)
To close the induction, it suffices to c(cid:0)hoose δ sm(cid:1) all enough to have the above term
{···} < 1, and to choose L big enough to have {···}L+c 6L.
0
Applyingtheabovereasoningtothe Rd×L(Rd) -valuedmapsx ∈ Rd 7→ µ (x),D µ
ts x ts
associatedwiththeordinarydifferentialequationsatisfiesby(yu)06u61and(Dxyu)06u61,
(cid:0) (cid:1) (cid:0) (cid:1)
where y = x, it follows from the analogue of (2.4) that D µn = (cid:13)n−1D µ re-
0 x ts i=0 x si+1si
mains bounded, so the maps µn ; n >2, t−s 6 δ are uniformly Lipschitz continu-
ts
ous, with a Lipschitz constant no greater than 1+c (t−s), for some positive constant
1
(cid:8) (cid:9)
c .
1
2. The existence and uniqueness proofs of the statement of theorem 1 both rely on
the following elementary identity
(2.5)
m
f ◦···◦f −g ◦···◦g = g ◦···◦g ◦f −g ◦···◦g ◦g ◦f ◦···◦f ,
m 1 m 1 m m−i+1 m−i m m−i+1 m−i m−i−1 1
Xi=1(cid:16) (cid:17)
wheretheg andf aremapsfromRd toitself,andwhereweusetheobviousconvention
i i
concerning the summand for the first and last term of the sum. In particular, if all
FLOWS DRIVEN BY ROUGH PATHS 5
the maps g ◦···◦g are Lipschitz continuous, with a common Lipschitz constant c′,
m k
then
m
(2.6) f ◦···◦f −g ◦···◦g 6 c′ kf −g k .
m 1 m 1 ∞ i i ∞
i=1
(cid:13) (cid:13) X
(cid:13) (cid:13)
a) Existence. Set D := 06 s 6 t 6 T ; t−s6 δ and write D forthe intersection
δ δ
of D with the set of dyadic real numbers. Given s = a2−n0 and t = b2−n0 in D ,
δ δ
(cid:8) (cid:9)
define for n> n
0
(n)
µ = µ ◦···◦µ ,
ts sN(n)sN(n)−1 s1s0
where s =s+i2−n and s = t. Given n > n , write
i N(n) 0
N(n)−1
(n+1)
µ = (cid:13) µ ◦µ
ts si+1si+2−n−1 si+2−n−1si
i=0
(cid:0) (cid:1)
anduse(2.5)withf = µ ◦µ andg = µ andthefactthatall
i si+1si+2−n−1 si+2−n−1si i si+1si
the maps µ ◦···◦µ are Lipschitz continuous with a common
s s s s
N(n) N(n)−1 N(n)−i+1 N(n)−i
Lipschitz constant c′, as established in point 1 above, to get by (2.6) and (2.2)
N(n)−1
µ(n+1)−µ(n) 6 c′ µ ◦µ −µ 6 O(2−n);
ts ts ∞ si+1si+2−n−1 si+2−n−1si si+1si ∞
i=0
(cid:13) (cid:13) X (cid:13) (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13)
so µ(n) converges uniformly on D to some continuous function ϕ. We see that ϕ
δ
satisfies inequality (2.3) on D as a consequence of (2.4), since the µ(n) are of the form
δ
µk(n). As ϕ is a uniformly continuous function of (s,t) ∈ D , by (2.3), it has a unique
δ
continuous extension to D , still denoted by ϕ. To see that it is a flow on D , notice
δ δ
that for dyadic times s 6 u 6 t, we have ϕ(n) = ϕ(n) ◦ ϕ(n), for n big enough; so
ts tu us
ϕ = ϕ ◦ϕ for such triples of times in D , hence for all times since ϕ is continuous.
ts tu us δ
The map ϕ is easily extended as a flow to the whole of {0 6 s 6 t 6 T}.
b) Uniqueness. Let ψ be any flow satisfying contion (2.3). With formulas (2.5) and
(2.6) in mind, rewrite (2.3) under the form ψ = µ + O (t − s)2 , with obvious
ts ts c
notations. Then
(cid:0) (cid:1)
ψts = ψs2ns2n−1 ◦···◦ψs1s0 = µs2ns2n−1 +Oc 2−γpn ◦···◦ µs1s0 +Oc 2−γpn
= µ ◦···◦µ +∆(cid:16) = µ(n)+∆ ,(cid:0) (cid:1)(cid:17) (cid:16) (cid:0) (cid:1)(cid:17)
s2ns2n−1 s1s0 n ts n
where ∆ is of the form of the right hand side of (2.5), sois bounded above by O(2−n)
n
sinceallthemapsµ ◦···◦µ areLipschitzcontinuous withacommon
s2ns2n−1 s2n−ℓ+1s2n−ℓ
Lipschitz constant. Sending n to infinity shows that ψ = ϕ . (cid:3)
ts ts
With the solution flow to ordinary (controlled) differential equations constructed as
above, the reader will easily convince herself/himself that the above proof is sufficiently
robust to justify the following result, which is at the core of our approach to flows driven
by rough paths.
Theorem 2. Let f s,t; · be time dependent regular enough vector fields on
a a=1..N
Rd, uniformly continuous on {0 6 s 6 t 6 T}. Let ν (x,A) = µ (x),M (x,A) be the
(cid:0) (cid:0) (cid:1)(cid:1) ts ts ts
(cid:0) (cid:1)
6 I.BAILLEUL
map from Rd×L(Rd) to itself which associates to (x,A) the time 1 value of the ordinary
differential equation
y˙ = f s,t; y
u a u
a=1..N
X (cid:0) (cid:1)
z˙ = D f (s,t; ·)z
u yu a u
a=1..N
X
started from (x,A). Suppose that there exists α > 1 and a positive constant c such that
0
ν ◦ν −ν 6 c |t−s|α
tu us ts ∞ 0
holds for all 0 6 s 6 t 6 T. T(cid:13)hen there exis(cid:13)ts a unique flow (ϕts)06s6t61 on Rd to which
(cid:13) (cid:13)
one can associate two positive constants c and δ such that
ϕ −µ 6 c|t−s|α
ts ts ∞
holds whenever t−s 6 δ. (cid:13) (cid:13)
(cid:13) (cid:13)
The approach to rough paths driven dynamics developed in this work is a variation on
this result and its proof. Hölder weak geometric p-rough paths (for 2 < p < 3), which
control the rough differential equation (1.1) and play the role of h, are defined in section
2.1. AsRd-valuedobjects,theyarenotregularenoughtoprovidethecrucialproperty (2.2)
and define the solution flow as above. The missing bit of information needed to stabilize
the situation is a substitute of the non-existing iterated integrals tXjdXk, which provide
s r r
a description of what happened to X during any time interval (s,t). The second part of
R
a weak geometric p-rough path plays precisely that role. An approximate solution flows is
defined from ordinary differential equation involving the vector fields V,V ,...,V and the
1 ℓ
driving rough path X, and studied in sections 2.2 and 2.3, where we prove an analogue of
inequality (2.4). A solution flow to equation (1.1) is then defined in section 2.4 as a flow
uniformly well approximated by the approximate solution flow; theorem 14 provides the
main result of this section, under the form of a well-posedness result. The general case
of a Hölder weak geometric p-rough path, for any p > 2, is considered in section 3, while
branched rough paths and their associated dynamics are the subject of section 5.
2.1. Hölderweakgeometricp-roughpaths. GivenasmoothRd-valuedpath(xt)06t6T,
itisanelementary, though important, factthatthemap X : (s,t) 7→ x −x , t rdx ⊗
x t s s s u
dx ∈ Rd× Rd⊗Rd ,takes itsvalues inaparticular subsetofRd× R(cid:16)d⊗Rd sRpeRcifiedby
r
alge(cid:17)braic conditions. It is precisely these conditions which appear in the definition below.
(cid:0) (cid:1) (cid:0) (cid:1)
Definition 3. Let 2 < p < 3 be given. A Hölder weak geometric p-rough path on
the interval [0,T] is a function X : (s,t) 7→ (X ,X ) from the set {0 6 s 6 t 6 T} to
ts ts
Rℓ×(Rℓ⊗Rℓ) satisfying the algebraic requirements
(i) X = X +X , for all 0 6 s 6 u6 t 6 T,
tr us tu
(ii) X = X +X +X ⊗X , for all 0 6 s6 u 6 t 6 T,
ts us tu us tu
(iii) the symmetric part of X is 1X ⊗X , for any 0 6 s 6 t 6 T,
ts 2 ts ts
as well as the analytic conditions
(a) X is 1-Hölder continuous: kXk := sup |Xts| <∞,
p 1 06s<t6T 1
p |t−s|p
(b) kXk := sup |Xts| < ∞.
2 2
p 06s<t6T |t−s|p
FLOWS DRIVEN BY ROUGH PATHS 7
ThealgebraicconditionssatisfiedbyaHölderweakgeometricp-roughpathjustifythink-
ing of its Rd⊗Rd -part as a kind of iterated integral of X against itself. Condition (ii) is
t r u r t u t r
nothing but the analogue of the elementary property = + + , satis-
(cid:0) (cid:1) s s s s u s u u
fied by any reasonable notion of integral on R such that t = u+ t. Note that condition
R Rs Rs R u R R R R
(i) on X implies that X = 0 for all s. We write Xi and Xjk for the components in the
ss ts R tRs R
canonical basis of X and X respectively. Note that the space of p-rough paths is not
ts ts
a vector space as the algebraic conditions on X are not linear. We define a norm on the
space of Hölder weak geometric p-rough paths setting
kXk = kXk ∨kXk ,
1 2
p p
with associated distance function d(X,Y) = kX−Yk, although X−Y is not a Hölder
weak geometric p-rough path. The map X associated with a smooth path is a Hölder
x
weak geometric p-rough path. The term "weak" was coined so as to emphasize the fact
that there are Hölder geometric p-rough paths, as defined above, which are not limit of
some X in the metric topology associated with d – see for instance [8] or [4].
x
Hölder weak geometric p-rough paths appear naturally in a probabilistic context. Let
B be a Brownian motion; the random process B = B −B , tB ⊗◦dB is
t s s r r 06s6t6T
almost-surely a Hölder weak geometric p-rough path,(cid:16)for any 2 < p < 3. It is(cid:17)called the
R
Brownianroughpath. ItiselementarytoprovethatE kBkr isfiniteforallr > 1,using
the scaling properties of Brownian motion – see for instance [4], chap. 13. More generally,
(cid:2) (cid:3)
Brownianmotioncanbereplacedbyanymartingaleenjoyingthesameregularityproperties
as Brownian motion. Notethat usingan Ito integral, BI = B −B , tB ⊗dB
t s s r r 06s6t6T
defines a function satisfying all conditions of definition 3 e(cid:16)xcept (iii)Rsince Bj B(cid:17)k is not
ts ts
equal to tBjdBk + tBkdBj for j = k; it is not a Hölder weak geometric p-rough path.
s r r s r r
2.2. AppRroximate rRough differential equation. Given 2 < p < γ 6 3, let V be a
globally Lipschitz vector field on Rd and V ,...,V be γ-Lipschitz vector fields on Rd. Let
1 ℓ
X be a Hölder weak geometric p-rough path. For 0 6 s 6 t 6 T, denote by µ the map
ts
from Rd to itself which associates to any x ∈ Rd the value at time 1 of the well-defined
and unique solution of the ordinary differential equation
d 1 1 jk
(2.7) y = (t−s)V(y )+Xi V (y )+ X − X ⊗X [V ,V ](y ), 06 u6 1,
du u u ts i u 2 ts 2 ts ts j k u
n o
with y = x. This equation is to be understood as the classical ordinary differential
0
equation version of equation (1.1); the definition of a solution flow to (1.1) given in section
2.4 will make that point clear. The following property of µ is the main reason for its
ts
introduction. Write λ for the Lipschitz constant of V.
Proposition 4. Suppose V is bounded and let M be a common upper bound for the
(γ−1)-Lipschitz norm of V ,...,V and the supremum norm of V. There exists a positive
1 ℓ
constant c, depending only on M,λ,T such that we have for all f ∈ Cγ
(2.8) f◦µts− f+(t−s)(Vf)+Xtis(Vif)+Xjtsk VjVkf 6 c 1+kXkγ kfkCγ |t−s|γp.
∞
(cid:13) n o(cid:13)
Proof(cid:13)– To shorten notations, write a for (t−(cid:0)s), bi (cid:1)for(cid:13)Xi an(cid:0)d cjk for 1(cid:1) X − 1X ⊗
(cid:13) (cid:13) st 2 st 2 st
Xst jk. Let c1 beapositiveconstant such that |b| 6 c1kXk|t−s|1p and |(cid:8)c| 6 c1kXk|t−
(cid:9)
8 I.BAILLEUL
2
s|p. Note that there is a constant c2 depending only on M and T such that we have
for all 0 6 u6 1
(2.9) |yu−x|6 M c2+kXk |t−s|p1.
Setting
(cid:0) (cid:1)
(⋆) := f µ (x) − x+a(Vf)(x)+bi(V f)(x)+cjk [V ,V ]f (x) ,
ts i j k
we have (cid:0) (cid:1) n (cid:0) (cid:1) o
1
(⋆) = a (Vf)(y )−(Vf)(x) +bi (V f)(y )−(V f)(x)
u i u i
Z0 (cid:16)
(cid:8) (cid:9) (cid:8) (cid:9)
+cjk [V ,V ]f (y )− [V ,V ]f (x) du,
j k u j k
which we rewrite und(cid:8)e(cid:0)r the form(cid:1) (cid:0) (cid:1) (cid:9)(cid:17)
1
(⋆) =a (Vf)(y )−(Vf)(x) du +
u
Z0
1(cid:8) u (cid:9)
(2.10) bi a(VV f)(y )+bi′(V V f)(y )+cjk [V ,V ]V f (y ) drdu +
i r i′ i r j k i r
Z0 Z0 n o
1 (cid:0) (cid:1)
cjk [V ,V ]f (y )− [V ,V ]f (x) du.
j k u j k
Z0
By the Lipschitz c(cid:8)o(cid:0)ntinuity o(cid:1)f Vf an(cid:0)d (2.9), t(cid:1)here(cid:9)is a positive constant c depending
3
only on M,λ and T such that the first integral above has absolute value no greater
than c3M 2+kXk kfkC1|t −s|1+p1. As [Vj,Vk] is (γ −1)-Lipschitz, hence (γ − 2)-
Lipschitz, the function [V ,V ]f is also (γ −2)-Lipschitz, so the third integral above
j k
has absolu(cid:0)te value n(cid:1)o greater than 2Mγ c3+kXk γ−2kXkkfkC1|t−s|γp. The second
double integral in (2.10) is of the form
(cid:0) (cid:1)
1 u
bi aA +bi′ (V V f)(y )drdu+cjkC ,
i i′ i r ijk
(cid:16) Z0 Z0 (cid:17)
with |A | 6 2M2kfk ,|C | 6 2M3kfk . Also, by the (γ−2)-Lipschitz character of
i C2 ijk C2
the vector field V V , we have by (2.9)
i′ i
(Vi′Vif)(yr)−(Vi′Vif)(x) 6 M2kfkCγ M c3+kXk |t−s|p1 γ−2.
It follow(cid:12)s that (cid:12) (cid:16) (cid:0) (cid:1) (cid:17)
(cid:12) (cid:12)
bjbk
f µ (x) − f(x)+a(Vf)(x)+bi(V f)(x)+cjk [V ,V ]f (x)+ (V V f)(x)
ts i j k j k
2
(cid:12)(cid:12)(cid:12)is(cid:0)no gre(cid:1)ater(cid:16)than c+kXkγ kfkCγ |t−s|γp for a(cid:0)postive co(cid:1)nstant c depending on(cid:17)ly(cid:12)(cid:12)(cid:12)on
M,λ and T. It remains to use the fact that the symmetric part of X is 1X ⊗X
(cid:0) (cid:1) ts 2 ts ts
to see that cjk[V ,V ]+ bjbkV V = XjkV V . (cid:3)
j k 2 j k ts j k
Remark5. Weonlyuseintheaboveproofthe(γ−2)-Lipschitzcharacter ofthevectorfields
V V rather than their stronger (γ−1)-Lispchitz character. There is no longer uniqueness
j k
forthesolutionof theordinary differentialequation (2.7)iftheV areonly(γ−1)-Lipschitz,
i
or if the vector field V is only supposed to be ρ-Lipschitz, for some positive ρ. However, the
above proof shows that (2.8) holds in that case with the time 1 function associated with any
solution of equation (2.7) in the role of µ ., with the exponent γ replaced by min 1+ρ, γ .
ts p p p
(cid:8) (cid:9)
FLOWS DRIVEN BY ROUGH PATHS 9
IfthevectorfieldV isC1 andV ,...,V areCγ,theclassicalresultsonthedependence of
1 ℓ
solutions to ordinary differential equation with respect to some parameters. For any reals
a,bi,cjk, the map exp aV +biV +cjk[V ,V ] (x) associated with the differential equation
i j k
d (cid:0) (cid:1)
y = aV(y )+biV (y )+cjk[V ,V ](y ), 0 6 u6 1,
u u i u j k u
du
is a continuous function of (a,b,c) and a continuously differentiable function of (b,c). The
following basic fact comes as a consequence of the analytic properties of any Hölder weak
geometricp-roughpathandtheabovetopologyonthesetofHölderweakgeometricp-rough
X
paths. We write µ instead of µ in the proposition below to emphasize its dependence
ts ts
on X.
Proposition 6. • Thediffeomorphisms µX dependcontinuouslyon (s,t),X ,
ts 06s6t6T
in the sense that
(cid:0) (cid:1) (cid:0) (cid:1)
max kµX−µYk , µX −1− µY −1 6 f (s,t),(u,v); X,Y
ts vu ∞ ts vu ∞
for a conntinuous functio(cid:13)(cid:13)n(cid:0) f w(cid:1)ith f(cid:0)(s,t(cid:1)),(s(cid:13)(cid:13),t)o; X,X(cid:0) = 0, for all 0 6(cid:1)s 6 t 6 T,
and any Hölder weak geometric p-rough path X.
(cid:0) (cid:1)
• Also, all the µ are uniformly Lipschitz continuous, with Lipschitz constant no
ts
greater than 1+c|t−s|1/p, for a constant c = c′ 1+kXkγ , with c′ depending only
on M and λ.
(cid:0) (cid:1)
2.3. Fundamentalpropertiesoftheapproximateflow. Thepropertiesof(µts)06s6t6T
described inpropositions 7and10 below willbecrucial to ouranalysis of rough differential
equations made in section 2.4. They correspond to the inequalities (2.2) and (2.4) used in
a crucial way in the proof of theorem 1. As usual, khk stands for the supremum norm of
∞
a map h from Rd to itself.
Proposition 7. There exists a positive constant c depending only on M,λ and T, such
that
µtu ◦µus−µts ∞ 6c 1+kXkγ |t−s|γp
holds for all 0 6 s <u <(cid:13)t 6 T. (cid:13) (cid:0) (cid:1)
(cid:13) (cid:13)
Proof – Fix x ∈ Rd. To shorten notations, write here a for (t −u), bi for Xi , cjk for
tu
Xjk, and primed notations for the same quatities with u and t replaced by s and u
tu
respectively. In those terms,
µ µ (x) = µ (x)+aV µ (x) +biV µ (x) +cjk(V V ) µ (x) +ε (x),
tu us us us i us j k us tu
(cid:0) (cid:1) γ (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
where εtu ∞ 6 c|t−u|p, with the constant c of proposition 4. Using (2.8), (2.9) and
Taylor formula, we obtain the following elementary estimates, where c is a constant
1
(cid:13) (cid:13)
depend(cid:13)ing(cid:13)onlyonM,λandT,andforwhichweonlyusethe(γ−1)-Lipschitzcharacter
of the vector fields V .
i
aV µus(x) −aV(x) 6 c1 1+kXk |u−s|1+p1,
(2.11) (cid:12)(cid:12)biVi(cid:0) µus(x)(cid:1) − biVi +(cid:12)(cid:12) bibi′(cid:0)Vi′Vi (x)(cid:1)6 c1 1+max kXk,kXkγ |u−s|p3,
(cid:12)(cid:12)cjk((cid:0)VjVk) µ(cid:1)us(x(cid:0)) −cjk(VjVk)(x(cid:1)) 6(cid:12)(cid:12) c1 1(cid:16)+kXk kX(cid:0)k|u−s|γp. (cid:1)(cid:17)
(cid:12) (cid:0) (cid:1) (cid:12) (cid:0) (cid:1)
(cid:12) (cid:12)
10 I.BAILLEUL
We thus have
(2.12)
µ µ (x) = x+a′V(x)+b′iV (x)+c′jk(V V )(x)+aV(x)+biV (x)+bi′bi(V V )(x)
tu us i j k i i′ i
(cid:0) (cid:1) +cjk(VjVk)(x)+εtu(x)+ε′us(x),
where
ε′us(x) 6 4c1 1+max kXk,kXkγ |u−s|γp 6 4c1 1+kXkγ |t−s|γp
and (cid:12)(cid:12)εtu(x)(cid:12)(cid:12)6 C|tn−u|γp 6 C(cid:0)|t−s|γp, fo(cid:1)rosome positive c(cid:0)onstant C(cid:1). Noting that we
have
(cid:12) (cid:12)
(cid:12) (cid:12) a′+a = (u−s)+(t−u) = t−s,
b′i+bi = Xi +Xi = Xi ,
us tu ts
c′jk +cjk +bjbk = Xjk +Xjk +Xj Xk = Xjk,
us tu us tu ts
identity (2.12) and estimate (2.8) in proposition 4 provide the awaited inequality:
µtu µus(x) −µts(x) 6 c 1+kXkγ |t−s|γp,
for a positive cons(cid:12)tant(cid:0)c depe(cid:1)nding only(cid:12) on M(cid:0) ,λ and T(cid:1). (cid:3)
(cid:12) (cid:12)
Remark 8. This remark is a sequel to remark 5; as emphasized in the proof, the proof of
the above result uses only the fact that V V is (γ −2)-Lipschitz, for which it suffices that
j k
the V be (γ −1)-Lipschitz, so the conclusion of proposition 7 holds in the case where the
i
V are only supposed to be (γ −1)-Lipschitz and if the vector field V is only ρ-Lipschitz,
i
with an exponent γ in the conclusion replaced by min 1+ρ,γ ; the constant c depends in
p p p
that case of M,ρ and T. From now on, we shall say that such a constant depends on the
(cid:8) (cid:9)
data of the problem.
Given a partition π = {s < s < ··· < s < t} of (s,t), and u ∈ (s,t), denote by
1 n−1
π(s,u) and π(u,t) the partitions of (s,u) and (u,t) induced by π. If u is one of the s then
i
both cardinals of π(s,u) and π(u,t) are strictly smaller than n.
Definition 9. Let ǫ ∈ (0,1) be given. A partition π = {s = s < s < ··· < s <
0 1 n−1
s = t} of (s,t) is said to be of special type ǫ if we have ǫ 6 si−si−1 6 1−ǫ, for all
n si+1−si−1
i = 1...n−1. The trivial partition of any interval into the interval itself is also said to
be of special type ǫ.
A partition of any interval into sub-intervals of equal length if for instance of special
type 1; the dyadic partition {s < ··· < s+k2−n(t−s) < ··· < t} is of special type 1.
3 2
Given a partition π = {s = s < s < ··· < s < s = t} of (s,t) of special type ǫ
0 1 n−1 n
and u ∈ {s ,...,s }, the induced partitions of the intervals (s,u) and (u,t) are also of
1 n−1
specialtype ǫ. Given any partition π(s,t) = {s = s < ··· < s < t = s }of theinterval
0 n−1 n
(s,t), set
µ = µ ◦···◦µ .
π(s,t) snsn−1 s1s0
Proposition 10. Let ǫ ∈ (0,1) be given. There exists somepositive constants c depending
0
on ǫ and the data of the problem, and δ depending on kXk as well, such that for any
0 6 s < t 6 T with t−s 6 δ, and any partition π(s,t) of (s,t) of special type ǫ, we have
(2.13) µπ(s,t)−µts ∞ 6 c0 1+kXkγ |t−s|γp.
(cid:13) (cid:13) (cid:0) (cid:1)
(cid:13) (cid:13)