getdocb2f1. 237KB Jun 04 2011 12:04:52 AM
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Vol. 13 (2008), Paper no. 38, pages 1120–1139.
Journal URL
http://www.math.washington.edu/~ejpecp/
Self-similarity and fractional Brownian motions on
Lie groups
Fabrice Baudoin
Laure Coutin
Institut de Math´ematiques de Toulouse
Universit´e Paul Sabatier
118 Route de Narbonne, Toulouse, France
fbaudoin@cict.fr
Laboratoire MAP5
Universit´e Ren´e Descartes
45, rue des Saints P`eres, Paris, France
Laure.Coutin@math-info.univ-paris5.fr
Abstract
The goal of this paper is to define and study a notion of fractional Brownian motion on a
Lie group. We define it as at the solution of a stochastic differential equation driven by a
linear fractional Brownian motion. We show that this process has stationary increments and
satisfies a local self-similar property. Furthermore the Lie groups for which this self-similar
property is global are characterized .
Key words: Fractional Brownian motion, Lie group.
AMS 2000 Subject Classification: Primary 60G15, 60G18.
Submitted to EJP on March 14, 2007, final version accepted July 8, 2008.
1120
1
Introduction
Since the seminal works of Itˆ
o [13], Hunt [12], and Yosida [29], it is well known that the (left)
Brownian motion on a Lie group G appears as the solution of a stochastic differential equation
in the Stratonovitch sense
dXt =
d
X
Vi (Xt ) ◦ dBti , t ≥ 0,
(1.1)
i=1
where V1 , ..., Vd are left-invariant vector fields on G and where (Bt )t≥0 is a Brownian motion;
for further details on this, we also refer to [26]. In this paper we investigate the properties of
the solution of an equation of the type (1.1) when the driving Brownian motion is replaced by
a fractional Brownian motion with parameter H. When H 6= 1/2, fractional Brownian motion
is neither a semimartingale neither a Markov process. Nevertheless, there have been numerous
attempts to define a notion of differential equations driven by fractional Brownian motion. One
dimensional differential equations can be solved using a Doss-Sussmann aproach for any values
of the parameter H as in the work of Nourdin, [22]. The situation is quite different in the
multidimensional case. When the Hurst parameter is greater than 1/2 existence and uniqueness
of the solution are obtained by Z¨
ahle in [31] or Nualart-Rascˆanu in [24].
In the case H < 12 , since fractional Brownian motion with Hurst index H has a modification
with sample paths α H¨
older continuous for any α < H, it falls into the framework of rough paths
theory. Rough paths theory was introduced by Lyons in [19] and further developed in [20], see
also [17] and the references therein. Let us give few words on it. Let x be a C 1 by steps path on
[0, 1] with values in Rδ . The geometric smooth functional over x of order p is X = (1, ..., X p )
where
Z
i
dxs1 ⊗ · · · ⊗ dxsi , 0 ≤ s < t ≤ 1, i = 1, ..., p.
Xs,t
=
s H1 , taking its values in MD (R). It is denoted by X = (1, X, X 2 , X 3 ).
Secondly, let B g,m be the sequence of linear interpolation of B g along the dyadic subdivision of
−m for i = 0, ..., 2m ; then for t ∈ [tm , tm ),
mesh m; that is if tm
i = i2
i
i+1
Btg,m = Btgim +
t − tim
(Btgm − Btgm ).
i+1
i
− tm
i
tm
i+1
Let us now denote X m the solution of (2.2) where B g is replaced by B g,m , that is
dXtm =
d
X
Xtm dBtk,m Vk .
k=1
m
m
It is easily seen that for t ∈ [tm
n−1 , tn ), n = 0, ..., 2 − 1,
!
!
d
d
X
X
k
k
k
k
m
m
m
(Btm
− Btm
)Vk .
(Btm
− Btm
)Vk · · · exp
Xt = exp 2 (t − tn−1 )
1
0
n−1
n−1
(2.3)
k=1
k=1
Pd
Since for k = 1, ..., d, Vk ∈ g we have k=1 (Btkm − Btkm )Vk ∈ g, i = 1, ...., 2m and therefore
i
i−1
P
d
k − B k )V
m takes its values in G. Now, from [8], Xm , the
(B
exp
∈
G.
Thus,
X
m
m
k
t
t
k=1
i
i−1
geometric functional build on X m , converges to X for the distance of 1/p H¨older in RD
2
:
1
m,i
i ki
kXs,t
− Xs,t
i
lim sup
= 0,
α
m→∞ 0≤s 1/H, according to Theorem 4 of [8], 1, ( ∆k [s,t] dB m,I )I,|I|≤3
converges in the
0≤s≤t≤1
R
distance of the p variation to 1, ( ∆k [s,t] dB m,I )I,|I|≤3
. That means that, almost surely,
Btg,m = Btgm +
tm
i+1
0≤s≤t≤1
p/k
Z
3
X Z
XX
◦dB I ,
dB m,I −
sup
∆k [tj−1 ,tj ]
k
Π
∆ [tj−1 ,tj ]
j
k=0 I,|I|=k
converges to 0 when m goes to infinity, where supπ runs over all finite subdivisions of [0, 1].
According to Proposition 3.3.3 of [20] page 51, almost surely there exists a subsequence (ml )
and a control ω such that for all 0 ≤ s ≤ t ≤ 1, I word of length |I| ≤ 3, k ∈ N,
Z
1
dB ml ,I ≤
ω(s, t)k/p ,
∆k [s,t]
β(k : p)!
Z
1
◦dB I ≤
ω(s, t)k/p ,
β(k : p)!
∆k [s,t]
Z
Z
1
ml ,I
I
2−l ω(s, t)k/p ,
dB
−
◦dB ≤
∆k [s,t]
β(k
:
p)!
k
∆ [s,t]
P
2 ([p]+1)/p
). Recall that ω is a control if and
where k : p = [k/p] and β > p2 (1 + ∞
r=3 ( r−2 )
only if ω is continuous, super-additive on {0 ≤ s ≤ t ≤ 1} taking its values in [0, +∞[ and
ω(t, t) = 0, t ∈ [0, 1]. Moreover according Theorem 3.1.3, for all 0 ≤ s ≤ t ≤ 1, I word, k ∈ N,
Z
1
dB ml ,I ≤
ω(s, t)k/p ,
β(k : p)!
∆k [s,t]
Z
1
◦dB I ≤
ω(s, t)k/p ,
(2.6)
β(k : p)!
∆k [s,t]
Z
Z
1
2−l ω(s, t)k/p .
◦dB I ≤
dB ml ,I −
∆k [s,t]
β(k
:
p)!
k
∆ [s,t]
Then the processes Y l , l ∈ N, and Y ,
Yt = 1 +
∞
X
X
Z
X
Z
k=1 I=(i1 ,...,ik )
Ytl
=1+
∞
X
k=1 I=(i1 ,...,ik )
∆k [0,t]
∆k [0,t]
◦dB I Vi1 ....Vik ,
dB ml ,I Vi1 ....Vik ,
1129
t ∈ [0, 1],
are well defined. First, since B g,m is C 1 by piece, Y l is the solution of
dXt = Xt dB g,ml , X0 = 1,
and therefore Y l = X ml , l ∈ N. Secondly, thanks to Theorem 5 of [8], X ml = Y l converges to
X in the distance of p variation, for p > 1/H. We conclude that Y = X, almost surely.
✷
3
Self-similarity of a fractional Brownian motion on a Lie group
Recall that for the Euclidean fractional Brownian motion, we have
1
d
(Bct
, ..., Bct
)t≥0 =law (cH Bt1 , ..., cH Btd )t≥0
This property is called the scaling property of the fractional Brownian motion. In this section,
we are going to study scaling properties of fractional Brownian motions on a Lie group.
As in the previous section, let G be a connected Lie and closed group (of matrices) with Lie
algebra g. Let V1 , ..., Vd be a basis of g and denote by (Xt )t≥0 the solution of the equation
!
d
X
i
dXt = Xt
Vi ◦ dBt , X0 = 1G ,
i=1
where (Bt )t≥0 is a d-dimensional fractional Brownian with Hurst parameter H > 14 .
3.1
Local self-similarity
First, we notice that for (Xt )t≥0 we always have an asymptotic scaling property in the following
sense:
Proposition 3.1. Under
the assumption of Theorem 2.1, when c → 0, c > 0, the sequence of
processes 1c (Xct − 1G ) 0≤t converges in law to β a one dimensional Brownian motion. MoreP
over, if f : G → R is a C 2 map such that di=1 (Vi f )(1G )2 6= 0; then, when c → 0, c > 0, the
sequence of processes c1H (f (Xct ) − f (1G )) 0≤t≤1 converges in law to (aβt )0≤t≤1 where (βt )t≥0
is a one-dimensional fractional Brownian motion and
v
u d
uX
a = t (Vi f )(1G )2 .
i=1
Proof. >From Proposition 2.11,
1 g
1
Xct − 1G
= H Bct
+ H R(ct) t ≥ 0,
H
c
c
c
where the remainder term R is given by
R(t) =
∞
X
X
k=2 I∈{1,...,d}k
Z
∆k [0,t]
1130
◦dB I Vi1 ...Vik ∈ g,
Let us observe that the convergence of the above series stems from (2.6). Using the scaling
g −H
property of fractional Brownian motion we get that (c−H Bct
, c R(ct))t≥0 has the same law as
Z
∞
X
X
Btg, cH
dB I Vi1 ...Vik .
c(k−2)H
k
I=(i1 ,...,ik ) ∆ [0,t]
k=2
t≤0
Again, from (2.6), almost surely,
lim sup kc
c→0 t∈[0,1]
H
∞
X
k=2
c
(k−2)H
Z
X
k
I=(i1 ,...,ik ) ∆ [0,t]
dB I Vi1 ...Vik kg = 0.
g −H
G
Thus, (c−H Bct
, c R(ct))t≥0 converges in law to (Btg, 0)t≥0 when c goes to 0 and ( Xctc−1
)t≥0
H
g
converges in law to (Bt )t≥0 when c goes to 0.
Now, by using the Taylor expansion of f between Xt and 1G , two points of M(RD ) seen as
2
R⊗D ,
Z 1
D
D
X
X
∂f
∂f 2
(Xct − 1)i,j
f (Xct ) − f (1G )
−2H i,j
k,l
c
R
(ct)R
(ct)
=
(1
)
+
(θXs )dθ.
G
cH
∂xi,j
cH
0 ∂xk,l
i,j=1
i,j,k,l=1
2
g −H
∂ f
Since, (c−H Bct
, c R(ct))t≥0 converges in law to (Btg, 0)t≥0 when c goes to 0, and ∂xi,j
∂xk,l
are continuous, we deduce that when c → 0, c > 0, the sequence of processes
1
(f (Xct ) − f (1G )) 0≤t≤1 converges in law to (aβt )0≤t≤1 where (βt )t≥0 is a one-dimensional
cH
fractional Brownian motion and
v
u d
uX
a = t (Vi f )(1G )2 .
i=1
✷
Remark 3.2. Slightly more generally, by using the Taylor expansion proved in (2.6), we obtain
in the same way: Let f : G → R be a smooth map such that there exist k ≥ 1 and (i1 , ..., ik ) ∈
{1, ..., d}k that satisfy
(Vi1 · · · Vik f )(1G ) 6= 0.
Denote n the smallest k that satisfies
the above property. Then, when c → 0, c > 0, the sequence
1
of processes cnH (f (Xct ) − f (1G )) 0≤t≤1 converges in law to (βt )0≤t≤1 where (βt )t≥0 is a process
such that
(βct )t≥0 =law (cnH βt )t≥0 .
3.2
Global self-similarity
Despite the local self-similar property, as we will see, in general there is no global scaling property
for the fractional Brownian motion on a Lie group. Let us first briefly discuss what should be a
good notion of scaling in a Lie group (see also [14] and [15]). If we can find a family a map ∆c ,
c > 0, such that
(Xct )t≥0 =law (∆c Xt )t≥0 ,
1131
first of all it is natural to require that the map c → ∆c is continuous and limc→0 ∆c = 1G .
Then by looking at (Xc1 c2 t )t≥0 , we will also naturally ask that ∆c1 c2 = ∆c1 ◦ ∆c2 . Finally, since
(Xs−1 Xt+s )t≥0 =law (Xt )t≥0 , we will also ask that ∆c is a Lie group automorphism.
The following theorem shows that the existence of such a family ∆c on G only holds if the group
is (Rd , +). This is partly due to the following lemma of Lie group theory that says that the
existence of a dilation on G imposes strong topological and algebraic restrictions:
Lemma 3.3. (See [16]) Assume that there exists a Lie group automorphism Ψ : G → G such
that dΨ1G (differential of Ψ at 1G ) has all its eigenvalues of modulus > 1, then G is a simply
connected nilpotent Lie group.
We can now show:
Theorem 3.4. There exists a family of Lie group automorphisms ∆t : G → G, t > 0, such
that:
1. The map t → ∆t is continuous and limt→0 ∆t = 1G ;
2. For t1 , t2 ≥ 0, ∆t1 t2 = ∆t1 ◦ ∆t2 ;
3. (Xct )t≥0 =law (∆c Xt )t≥0 ;
if and only if the group G is isomorphic to (Rd , +).
Proof. If G is isomorphic to (Rd , +), (Xt )t≥0 is a Euclidean fractional Brownian motion and
the result is trivial.
We prove now the converse statement. Let us first show that the existence of the family (∆t )t>0
implies that G is a simply connected nilpotent Lie group.
Let us denote
δc = d∆c (1G ),
the differential map of ∆c at 1G and observe that δc is a Lie algebra automorphism g → g, that
satisfies ∆c (exp M ) = exp δc (M ), M ∈ g. The map f : t → δet is a map from R onto the set of
linear maps g → g that is continuous. We have furthermore the property
f (t + s) = f (t)f (s).
Consequently there exists a linear map φ : g → g such that
δc = Exp(φ ln c), c > 0,
where Exp denotes here the exponential of linear maps (and not the exponential map g → G
which is denoted exp). Since δc is a linear application on the finite dimensional vector space g,
for c∗ > 1 close enough from 1,
φ ln c∗ = Ln(IL(g) + δc∗ − IL(g) ),
where L(g) is the set of linear functions from g into itself, IL(g) is the identity map from g
P
(−1)n+1 ◦(n)
into itself, and for ϕ ∈ L(g) in a neighbour of 0 Ln(IL(g) + ϕ) = ∞
ϕ
. If α is
n=1
n
1132
an eigenvalue of δc∗ , then ln α is an eigenvalue of φ ln c∗ . Let us furthermore observe that if
λ ∈ Sp(φ) is an eigenvalue of φ, then eλ ln c is an eigenvalue of δc , therefore ℜλ > 0 because
limc→0 ∆c = 1G which implies limc→0 δc = 0. Therefore, |α| > 1 from Lemma 3.3 with Ψ = δc∗ ,
G has to be a simply connected nilpotent Lie group.
We deduce from Proposition 2.7 that
+∞
X
Xt = exp
X
k=1 I∈{1,...,d}k
where the above sum is actually finite and
ΛI (B)t =
X
σ∈Sk
ΛI (B)t VI ,
(−1)e(σ)
k−1
k2
e(σ)
Z
t ≥ 0,
◦dB σ
−1 ·I
.
(3.7)
∆k [0,t]
Due to the assumption that
(Xt )t≥0 =law (∆c X t )t≥0 ,
c
we deduce that
+∞
X
exp
X
k=1 I∈{1,...,d}k
=law
ΛI (B)t VI
t≥0
+∞
X
exp
X
k=1 I∈{1,...,d}k
ΛI (B) t (δc V )I
c
.
t≥0
But since the group G is nilpotent and simply connected the exponential map is a diffeomorphism, therefore
+∞
+∞
X
X
X
X
ΛI (B)t VI
=law
ΛI (B) t (δc V )I .
k=1 I∈{1,...,d}k
t≥0
k=1 I∈{1,...,d}k
c
t≥0
Let us now observe that due to the scaling property of the fractional Brownian motion
+∞
+∞
X
X
X
X
1
ΛI (B) t (δc V )I
=law
ΛI (B)t H|I| δc VI ,
c
c
k
k
k=1
k=1
I∈{1,...,d}
I∈{1,...,d}
t≥0
where | I | is the length of the word I. Thus, for every c > 0,
+∞
+∞
X
X
X
X
1
δ
ΛI (B)t VI
=law
c
ckH
k
k=1 I∈{1,...,d}
t≥0
k=1
I∈{1,...,d}k
t≥0
ΛI (B)t VI
t≥0
Let us now observe that V1 , ..., Vd is a basis of g, therefore all commutators are linear combinations of the Vi ’s and
VI =
d
X
αIi Vi , I ∈ {1, ..., d}k , k ∈ N∗ ,
i=1
δc (Vj ) =
d
X
δci,j Vi , j = 1, ..., d.
i=1
1133
The projections on Vj , j = 1, ..., d, are continuous linear maps (since g has a finite dimension),
therefore
N
X
X
ΛI (B)t αj
I
k=1 I∈{1,...,d}k
t≥0,j=1,...,d
has the same law as
N
X
k=1
X
ckH
ΛI (B)t
d
X
i=1
I∈{1,...,d}k
αIj δci,j
,
t≥0
where N is the degreePof nilpotence
P of G. We take now expectation on both sides and observe
2H ,
kH
that the maps t 7→ E( N
c
I=(i1 ,...,ik ), i1 ≤...≤ik ΛI (B)t ), j = 1, ..., d are polynomial in t
k=1
with coefficient associated to t2H given by
1 = c−2H
d
X
(δci,j )2 , j = 1, ..., d.
i=1
Since the family of linear applications (c−H δc )c>0 is bounded, there exists a subsequence (cl )l∈N
and a matrix (˜
αij )i,j=1,..,d such that liml→∞ cl = ∞, and for i, j = 1, ..., d, liml→∞ c−H
δci,jl = α
˜ ij .
l
Then,
N
d
X
X
X
ckH
ΛI (B)t
αIj δci,jl
,
l
k=1
i=1
I∈{1,...,d}k
t≥0, j=1,...,d
P
converges in law to the Gaussian process ( di=1 Bti α
˜ ij )t≥0,j=1,...,d . For j = 1, ..., d, by using the
scaling property of fractional Brownian motion, we observe that the map
2
N
X
X
E
ΛI (B)t αIj
k=1 I∈{1,...,d}k
is a polynomial with degree 1 in t2H . The leading coefficient of t4H is given by
2
X
E
ΛI (B)1 αIj .
I=(i1 ,i2 )
We conclude therefore
E
X
I=(i1 ,i2 )
2
ΛI (B)1 αIj = 0.
From the support theorem of [7] (Proposition 3), we conclude that for all I = (i1 , i2 ), , j = 1, ..., d,
αIj = 0, that is VI = [Vi1 , Vi2 ] = 0. Therefore all the brackets have to be 0, that is G is
commutative. We can now conclude that G is isomorphic to (Rd , +), because so are all simply
connected and commutative Lie groups.
✷
1134
Remark 3.5. The previous theorem in particular applies to the case of a Brownian motion, that
corresponds to H = 12 .
If we relax the assumption that the family (V1 , ..., Vd ) forms a basis of the Lie algebra g, we can
have a global scaling property in slightly more general groups than the commutative ones. Let
us first look at one example.
The Heisenberg group H is the set of 3 × 3
1
x
0
1
0
0
matrices:
z
y , x, y, z ∈ R.
1
The Lie algebra of H is spanned by the matrices
0
0
0
1
0
0
0
0
0
0
D1 =
, D2 =
0
0
0
0
0
for which the following equalities hold
0
0
0
1
and D3 =
0
0
0
0
0
1
0 ,
0
[D1 , D2 ] = D3 , [D1 , D3 ] = [D2 , D3 ] = 0.
Consider now the solution of the equation
dXt = Xt (D1 ◦ dBt1 + D2 ◦ dBt2 ),
X0 = 1,
where (Bt1 , Bt2 ) is a two-dimensional fractional Brownian motion with Hurst parameter H > 14 .
It is easily seen that
R
1
1 B 2 + t B 1 ◦ dB 2 − B 2 ◦ dB 1
1
Bt1
B
s
s
t t
s
2
0 s
Xt = 0
.
1
Bt2
0
0
1
Therefore (Xct )t≥0 =law (∆c Xt )t≥0 , where ∆c is defined by
1
cH x
1
x
z
1
1
y = 0
∆c 0
0
0
0
0
1
c2H z
cH y .
1
In that case, we thus have a global scaling property whereas H is of course not commutative but
step-two nilpotent. Actually, we shall have a global scaling property in the Lie groups that are
called the Carnot groups. Let us recall the definition of a Carnot group.
Definition 3.6. A Carnot group of step (or depth) N is a simply connected Lie group G whose
Lie algebra can be written
V1 ⊕ ... ⊕ VN ,
where
[Vi , Vj ] = Vi+j
and
Vs = 0, for s > N.
1135
Example 3.7. Consider the set Hn = R2n × R endowed with the group law
1
(x, α) ⋆ (y, β) = x + y, α + β + ω(x, y) ,
2
where ω is the standard symplectic form on R2n , that is
0 −In
ω(x, y) = xt
y.
In
0
On hn the Lie bracket is given by
[(x, α), (y, β)] = (0, ω(x, y)) ,
and it is easily seen that
hn = V1 ⊕ V2 ,
where V1 = R2n × {0} and V2 = {0} × R. Therefore Hn is a Carnot group of depth 2 and observe
that H1 is isomorphic to the Heisenberg group.
Notice that the vector space V1 , which is called the basis of G, Lie generates g, where g denotes
the Lie algebra of G. Since G is step N nilpotent and simply connected, the exponential map
is a diffeomorphism. On g we can consider the family of linear operators δt : g → g, t ≥ 0
which act by scalar multiplication ti on Vi . These operators are Lie algebra automorphisms due
to the grading. The maps δt induce Lie group automorphisms ∆t : G → G which are called
the canonical dilations of G. Let us now take a basis U1 , ..., Ud of the vector space V1 . The
vectors Ui ’s can be seen as left invariant vector fields on G so that we can consider the following
stochastic differential equation on G:
dYt =
d Z
X
i=1
t
0
Ui (Ys ) ◦ dBsi , t ≥ 0,
(3.8)
which is easily seen to have a unique solution associated with the initial condition Y0 = 1G . We
have then the following global scaling property:
Proposition 3.8.
(Yct )t≥0 =law (∆cH Yt )t≥0 .
Proof. We keep the notations introduced before the proof of Theorem 2.7. From Theorem 2.7,
we have
N
X
X
Yt = exp
ΛI (B)t UI , t ≥ 0.
k=1 I∈{1,...,d}k
Therefore,
(Yct )t≥0 =law exp
N
X
k=1
cH|I|
X
I∈{1,...,d}k
1136
ΛI (B)t UI
.
t≥0
Since
exp
N
X
cH|I|
k=1
X
I∈{1,...,d}k
ΛI (B)t UI = exp
N
X
X
k=1 I∈{1,...,d}k
N
X
= ∆cH exp
ΛI (B)t (δcH UI )
X
k=1 I∈{1,...,d}k
we conclude
ΛI (B)t UI ,
(Yct )t≥0 =law (∆cH Yt )t≥0 .
✷
The previous proposition admits a counterpart.
Theorem 3.9. Let G be a connected Lie group (of matrices) with Lie algebra g. Let V1 , ..., Vd
be a family of g. Consider now the solution of the equation
!
d
X
dXt = Xt
Vi ◦ dBti , X0 = 1G .
i=1
Assume that there exists a family of Lie group automorphisms ∆t : G → G, t > 0, such that:
1. The map t → ∆t is continuous and limt→0 ∆t = 1G ;
2. For t1 , t2 ≥ 0, ∆t1 t2 = ∆t1 ◦ ∆t2 ;
3. (Xct )t≥0 =law (∆c Xt )t≥0 ;
Then the Lie subgroup H that is generated by eV1 , ..., eVd is a Carnot group.
Proof.
We can readily follow the lines of the proof of Theorem 3.4, so that we do not enter into details.
First we obtain that H has to be a simply connected nilpotent group and that for every c > 0,
+∞
+∞
X
X
X
X
1
ΛI (B)t VI ,
δc
ΛI (B)t VI
=law
ckH
k=1 I=(i1 ,...,ik ), i1 ≤...≤ik
t≥0
k=1
I=(i1 ,...,ik ), i1 ≤...≤ik
t≥0
where δc is the differential map of ∆c at 1G . For k ≥ 1, we denote Vk the linear space generated
by the set of commutators:
{VI , | I |= k} .
By letting c → +∞ and c → 0 and by using the support theorem of [7], we obtain that c−kH δc is
bounded on Vk for c → +∞ and c → 0. Since c−kH δc = Exp((φ − kHId) ln c), for some matrix
φ, we conclude
+∞
M
Vk ,
h=
k=1
where h is the Lie algebra of H. This proves that H is a Carnot group.
1137
✷
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1139
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c
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o
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ob
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ity
Vol. 13 (2008), Paper no. 38, pages 1120–1139.
Journal URL
http://www.math.washington.edu/~ejpecp/
Self-similarity and fractional Brownian motions on
Lie groups
Fabrice Baudoin
Laure Coutin
Institut de Math´ematiques de Toulouse
Universit´e Paul Sabatier
118 Route de Narbonne, Toulouse, France
fbaudoin@cict.fr
Laboratoire MAP5
Universit´e Ren´e Descartes
45, rue des Saints P`eres, Paris, France
Laure.Coutin@math-info.univ-paris5.fr
Abstract
The goal of this paper is to define and study a notion of fractional Brownian motion on a
Lie group. We define it as at the solution of a stochastic differential equation driven by a
linear fractional Brownian motion. We show that this process has stationary increments and
satisfies a local self-similar property. Furthermore the Lie groups for which this self-similar
property is global are characterized .
Key words: Fractional Brownian motion, Lie group.
AMS 2000 Subject Classification: Primary 60G15, 60G18.
Submitted to EJP on March 14, 2007, final version accepted July 8, 2008.
1120
1
Introduction
Since the seminal works of Itˆ
o [13], Hunt [12], and Yosida [29], it is well known that the (left)
Brownian motion on a Lie group G appears as the solution of a stochastic differential equation
in the Stratonovitch sense
dXt =
d
X
Vi (Xt ) ◦ dBti , t ≥ 0,
(1.1)
i=1
where V1 , ..., Vd are left-invariant vector fields on G and where (Bt )t≥0 is a Brownian motion;
for further details on this, we also refer to [26]. In this paper we investigate the properties of
the solution of an equation of the type (1.1) when the driving Brownian motion is replaced by
a fractional Brownian motion with parameter H. When H 6= 1/2, fractional Brownian motion
is neither a semimartingale neither a Markov process. Nevertheless, there have been numerous
attempts to define a notion of differential equations driven by fractional Brownian motion. One
dimensional differential equations can be solved using a Doss-Sussmann aproach for any values
of the parameter H as in the work of Nourdin, [22]. The situation is quite different in the
multidimensional case. When the Hurst parameter is greater than 1/2 existence and uniqueness
of the solution are obtained by Z¨
ahle in [31] or Nualart-Rascˆanu in [24].
In the case H < 12 , since fractional Brownian motion with Hurst index H has a modification
with sample paths α H¨
older continuous for any α < H, it falls into the framework of rough paths
theory. Rough paths theory was introduced by Lyons in [19] and further developed in [20], see
also [17] and the references therein. Let us give few words on it. Let x be a C 1 by steps path on
[0, 1] with values in Rδ . The geometric smooth functional over x of order p is X = (1, ..., X p )
where
Z
i
dxs1 ⊗ · · · ⊗ dxsi , 0 ≤ s < t ≤ 1, i = 1, ..., p.
Xs,t
=
s H1 , taking its values in MD (R). It is denoted by X = (1, X, X 2 , X 3 ).
Secondly, let B g,m be the sequence of linear interpolation of B g along the dyadic subdivision of
−m for i = 0, ..., 2m ; then for t ∈ [tm , tm ),
mesh m; that is if tm
i = i2
i
i+1
Btg,m = Btgim +
t − tim
(Btgm − Btgm ).
i+1
i
− tm
i
tm
i+1
Let us now denote X m the solution of (2.2) where B g is replaced by B g,m , that is
dXtm =
d
X
Xtm dBtk,m Vk .
k=1
m
m
It is easily seen that for t ∈ [tm
n−1 , tn ), n = 0, ..., 2 − 1,
!
!
d
d
X
X
k
k
k
k
m
m
m
(Btm
− Btm
)Vk .
(Btm
− Btm
)Vk · · · exp
Xt = exp 2 (t − tn−1 )
1
0
n−1
n−1
(2.3)
k=1
k=1
Pd
Since for k = 1, ..., d, Vk ∈ g we have k=1 (Btkm − Btkm )Vk ∈ g, i = 1, ...., 2m and therefore
i
i−1
P
d
k − B k )V
m takes its values in G. Now, from [8], Xm , the
(B
exp
∈
G.
Thus,
X
m
m
k
t
t
k=1
i
i−1
geometric functional build on X m , converges to X for the distance of 1/p H¨older in RD
2
:
1
m,i
i ki
kXs,t
− Xs,t
i
lim sup
= 0,
α
m→∞ 0≤s 1/H, according to Theorem 4 of [8], 1, ( ∆k [s,t] dB m,I )I,|I|≤3
converges in the
0≤s≤t≤1
R
distance of the p variation to 1, ( ∆k [s,t] dB m,I )I,|I|≤3
. That means that, almost surely,
Btg,m = Btgm +
tm
i+1
0≤s≤t≤1
p/k
Z
3
X Z
XX
◦dB I ,
dB m,I −
sup
∆k [tj−1 ,tj ]
k
Π
∆ [tj−1 ,tj ]
j
k=0 I,|I|=k
converges to 0 when m goes to infinity, where supπ runs over all finite subdivisions of [0, 1].
According to Proposition 3.3.3 of [20] page 51, almost surely there exists a subsequence (ml )
and a control ω such that for all 0 ≤ s ≤ t ≤ 1, I word of length |I| ≤ 3, k ∈ N,
Z
1
dB ml ,I ≤
ω(s, t)k/p ,
∆k [s,t]
β(k : p)!
Z
1
◦dB I ≤
ω(s, t)k/p ,
β(k : p)!
∆k [s,t]
Z
Z
1
ml ,I
I
2−l ω(s, t)k/p ,
dB
−
◦dB ≤
∆k [s,t]
β(k
:
p)!
k
∆ [s,t]
P
2 ([p]+1)/p
). Recall that ω is a control if and
where k : p = [k/p] and β > p2 (1 + ∞
r=3 ( r−2 )
only if ω is continuous, super-additive on {0 ≤ s ≤ t ≤ 1} taking its values in [0, +∞[ and
ω(t, t) = 0, t ∈ [0, 1]. Moreover according Theorem 3.1.3, for all 0 ≤ s ≤ t ≤ 1, I word, k ∈ N,
Z
1
dB ml ,I ≤
ω(s, t)k/p ,
β(k : p)!
∆k [s,t]
Z
1
◦dB I ≤
ω(s, t)k/p ,
(2.6)
β(k : p)!
∆k [s,t]
Z
Z
1
2−l ω(s, t)k/p .
◦dB I ≤
dB ml ,I −
∆k [s,t]
β(k
:
p)!
k
∆ [s,t]
Then the processes Y l , l ∈ N, and Y ,
Yt = 1 +
∞
X
X
Z
X
Z
k=1 I=(i1 ,...,ik )
Ytl
=1+
∞
X
k=1 I=(i1 ,...,ik )
∆k [0,t]
∆k [0,t]
◦dB I Vi1 ....Vik ,
dB ml ,I Vi1 ....Vik ,
1129
t ∈ [0, 1],
are well defined. First, since B g,m is C 1 by piece, Y l is the solution of
dXt = Xt dB g,ml , X0 = 1,
and therefore Y l = X ml , l ∈ N. Secondly, thanks to Theorem 5 of [8], X ml = Y l converges to
X in the distance of p variation, for p > 1/H. We conclude that Y = X, almost surely.
✷
3
Self-similarity of a fractional Brownian motion on a Lie group
Recall that for the Euclidean fractional Brownian motion, we have
1
d
(Bct
, ..., Bct
)t≥0 =law (cH Bt1 , ..., cH Btd )t≥0
This property is called the scaling property of the fractional Brownian motion. In this section,
we are going to study scaling properties of fractional Brownian motions on a Lie group.
As in the previous section, let G be a connected Lie and closed group (of matrices) with Lie
algebra g. Let V1 , ..., Vd be a basis of g and denote by (Xt )t≥0 the solution of the equation
!
d
X
i
dXt = Xt
Vi ◦ dBt , X0 = 1G ,
i=1
where (Bt )t≥0 is a d-dimensional fractional Brownian with Hurst parameter H > 14 .
3.1
Local self-similarity
First, we notice that for (Xt )t≥0 we always have an asymptotic scaling property in the following
sense:
Proposition 3.1. Under
the assumption of Theorem 2.1, when c → 0, c > 0, the sequence of
processes 1c (Xct − 1G ) 0≤t converges in law to β a one dimensional Brownian motion. MoreP
over, if f : G → R is a C 2 map such that di=1 (Vi f )(1G )2 6= 0; then, when c → 0, c > 0, the
sequence of processes c1H (f (Xct ) − f (1G )) 0≤t≤1 converges in law to (aβt )0≤t≤1 where (βt )t≥0
is a one-dimensional fractional Brownian motion and
v
u d
uX
a = t (Vi f )(1G )2 .
i=1
Proof. >From Proposition 2.11,
1 g
1
Xct − 1G
= H Bct
+ H R(ct) t ≥ 0,
H
c
c
c
where the remainder term R is given by
R(t) =
∞
X
X
k=2 I∈{1,...,d}k
Z
∆k [0,t]
1130
◦dB I Vi1 ...Vik ∈ g,
Let us observe that the convergence of the above series stems from (2.6). Using the scaling
g −H
property of fractional Brownian motion we get that (c−H Bct
, c R(ct))t≥0 has the same law as
Z
∞
X
X
Btg, cH
dB I Vi1 ...Vik .
c(k−2)H
k
I=(i1 ,...,ik ) ∆ [0,t]
k=2
t≤0
Again, from (2.6), almost surely,
lim sup kc
c→0 t∈[0,1]
H
∞
X
k=2
c
(k−2)H
Z
X
k
I=(i1 ,...,ik ) ∆ [0,t]
dB I Vi1 ...Vik kg = 0.
g −H
G
Thus, (c−H Bct
, c R(ct))t≥0 converges in law to (Btg, 0)t≥0 when c goes to 0 and ( Xctc−1
)t≥0
H
g
converges in law to (Bt )t≥0 when c goes to 0.
Now, by using the Taylor expansion of f between Xt and 1G , two points of M(RD ) seen as
2
R⊗D ,
Z 1
D
D
X
X
∂f
∂f 2
(Xct − 1)i,j
f (Xct ) − f (1G )
−2H i,j
k,l
c
R
(ct)R
(ct)
=
(1
)
+
(θXs )dθ.
G
cH
∂xi,j
cH
0 ∂xk,l
i,j=1
i,j,k,l=1
2
g −H
∂ f
Since, (c−H Bct
, c R(ct))t≥0 converges in law to (Btg, 0)t≥0 when c goes to 0, and ∂xi,j
∂xk,l
are continuous, we deduce that when c → 0, c > 0, the sequence of processes
1
(f (Xct ) − f (1G )) 0≤t≤1 converges in law to (aβt )0≤t≤1 where (βt )t≥0 is a one-dimensional
cH
fractional Brownian motion and
v
u d
uX
a = t (Vi f )(1G )2 .
i=1
✷
Remark 3.2. Slightly more generally, by using the Taylor expansion proved in (2.6), we obtain
in the same way: Let f : G → R be a smooth map such that there exist k ≥ 1 and (i1 , ..., ik ) ∈
{1, ..., d}k that satisfy
(Vi1 · · · Vik f )(1G ) 6= 0.
Denote n the smallest k that satisfies
the above property. Then, when c → 0, c > 0, the sequence
1
of processes cnH (f (Xct ) − f (1G )) 0≤t≤1 converges in law to (βt )0≤t≤1 where (βt )t≥0 is a process
such that
(βct )t≥0 =law (cnH βt )t≥0 .
3.2
Global self-similarity
Despite the local self-similar property, as we will see, in general there is no global scaling property
for the fractional Brownian motion on a Lie group. Let us first briefly discuss what should be a
good notion of scaling in a Lie group (see also [14] and [15]). If we can find a family a map ∆c ,
c > 0, such that
(Xct )t≥0 =law (∆c Xt )t≥0 ,
1131
first of all it is natural to require that the map c → ∆c is continuous and limc→0 ∆c = 1G .
Then by looking at (Xc1 c2 t )t≥0 , we will also naturally ask that ∆c1 c2 = ∆c1 ◦ ∆c2 . Finally, since
(Xs−1 Xt+s )t≥0 =law (Xt )t≥0 , we will also ask that ∆c is a Lie group automorphism.
The following theorem shows that the existence of such a family ∆c on G only holds if the group
is (Rd , +). This is partly due to the following lemma of Lie group theory that says that the
existence of a dilation on G imposes strong topological and algebraic restrictions:
Lemma 3.3. (See [16]) Assume that there exists a Lie group automorphism Ψ : G → G such
that dΨ1G (differential of Ψ at 1G ) has all its eigenvalues of modulus > 1, then G is a simply
connected nilpotent Lie group.
We can now show:
Theorem 3.4. There exists a family of Lie group automorphisms ∆t : G → G, t > 0, such
that:
1. The map t → ∆t is continuous and limt→0 ∆t = 1G ;
2. For t1 , t2 ≥ 0, ∆t1 t2 = ∆t1 ◦ ∆t2 ;
3. (Xct )t≥0 =law (∆c Xt )t≥0 ;
if and only if the group G is isomorphic to (Rd , +).
Proof. If G is isomorphic to (Rd , +), (Xt )t≥0 is a Euclidean fractional Brownian motion and
the result is trivial.
We prove now the converse statement. Let us first show that the existence of the family (∆t )t>0
implies that G is a simply connected nilpotent Lie group.
Let us denote
δc = d∆c (1G ),
the differential map of ∆c at 1G and observe that δc is a Lie algebra automorphism g → g, that
satisfies ∆c (exp M ) = exp δc (M ), M ∈ g. The map f : t → δet is a map from R onto the set of
linear maps g → g that is continuous. We have furthermore the property
f (t + s) = f (t)f (s).
Consequently there exists a linear map φ : g → g such that
δc = Exp(φ ln c), c > 0,
where Exp denotes here the exponential of linear maps (and not the exponential map g → G
which is denoted exp). Since δc is a linear application on the finite dimensional vector space g,
for c∗ > 1 close enough from 1,
φ ln c∗ = Ln(IL(g) + δc∗ − IL(g) ),
where L(g) is the set of linear functions from g into itself, IL(g) is the identity map from g
P
(−1)n+1 ◦(n)
into itself, and for ϕ ∈ L(g) in a neighbour of 0 Ln(IL(g) + ϕ) = ∞
ϕ
. If α is
n=1
n
1132
an eigenvalue of δc∗ , then ln α is an eigenvalue of φ ln c∗ . Let us furthermore observe that if
λ ∈ Sp(φ) is an eigenvalue of φ, then eλ ln c is an eigenvalue of δc , therefore ℜλ > 0 because
limc→0 ∆c = 1G which implies limc→0 δc = 0. Therefore, |α| > 1 from Lemma 3.3 with Ψ = δc∗ ,
G has to be a simply connected nilpotent Lie group.
We deduce from Proposition 2.7 that
+∞
X
Xt = exp
X
k=1 I∈{1,...,d}k
where the above sum is actually finite and
ΛI (B)t =
X
σ∈Sk
ΛI (B)t VI ,
(−1)e(σ)
k−1
k2
e(σ)
Z
t ≥ 0,
◦dB σ
−1 ·I
.
(3.7)
∆k [0,t]
Due to the assumption that
(Xt )t≥0 =law (∆c X t )t≥0 ,
c
we deduce that
+∞
X
exp
X
k=1 I∈{1,...,d}k
=law
ΛI (B)t VI
t≥0
+∞
X
exp
X
k=1 I∈{1,...,d}k
ΛI (B) t (δc V )I
c
.
t≥0
But since the group G is nilpotent and simply connected the exponential map is a diffeomorphism, therefore
+∞
+∞
X
X
X
X
ΛI (B)t VI
=law
ΛI (B) t (δc V )I .
k=1 I∈{1,...,d}k
t≥0
k=1 I∈{1,...,d}k
c
t≥0
Let us now observe that due to the scaling property of the fractional Brownian motion
+∞
+∞
X
X
X
X
1
ΛI (B) t (δc V )I
=law
ΛI (B)t H|I| δc VI ,
c
c
k
k
k=1
k=1
I∈{1,...,d}
I∈{1,...,d}
t≥0
where | I | is the length of the word I. Thus, for every c > 0,
+∞
+∞
X
X
X
X
1
δ
ΛI (B)t VI
=law
c
ckH
k
k=1 I∈{1,...,d}
t≥0
k=1
I∈{1,...,d}k
t≥0
ΛI (B)t VI
t≥0
Let us now observe that V1 , ..., Vd is a basis of g, therefore all commutators are linear combinations of the Vi ’s and
VI =
d
X
αIi Vi , I ∈ {1, ..., d}k , k ∈ N∗ ,
i=1
δc (Vj ) =
d
X
δci,j Vi , j = 1, ..., d.
i=1
1133
The projections on Vj , j = 1, ..., d, are continuous linear maps (since g has a finite dimension),
therefore
N
X
X
ΛI (B)t αj
I
k=1 I∈{1,...,d}k
t≥0,j=1,...,d
has the same law as
N
X
k=1
X
ckH
ΛI (B)t
d
X
i=1
I∈{1,...,d}k
αIj δci,j
,
t≥0
where N is the degreePof nilpotence
P of G. We take now expectation on both sides and observe
2H ,
kH
that the maps t 7→ E( N
c
I=(i1 ,...,ik ), i1 ≤...≤ik ΛI (B)t ), j = 1, ..., d are polynomial in t
k=1
with coefficient associated to t2H given by
1 = c−2H
d
X
(δci,j )2 , j = 1, ..., d.
i=1
Since the family of linear applications (c−H δc )c>0 is bounded, there exists a subsequence (cl )l∈N
and a matrix (˜
αij )i,j=1,..,d such that liml→∞ cl = ∞, and for i, j = 1, ..., d, liml→∞ c−H
δci,jl = α
˜ ij .
l
Then,
N
d
X
X
X
ckH
ΛI (B)t
αIj δci,jl
,
l
k=1
i=1
I∈{1,...,d}k
t≥0, j=1,...,d
P
converges in law to the Gaussian process ( di=1 Bti α
˜ ij )t≥0,j=1,...,d . For j = 1, ..., d, by using the
scaling property of fractional Brownian motion, we observe that the map
2
N
X
X
E
ΛI (B)t αIj
k=1 I∈{1,...,d}k
is a polynomial with degree 1 in t2H . The leading coefficient of t4H is given by
2
X
E
ΛI (B)1 αIj .
I=(i1 ,i2 )
We conclude therefore
E
X
I=(i1 ,i2 )
2
ΛI (B)1 αIj = 0.
From the support theorem of [7] (Proposition 3), we conclude that for all I = (i1 , i2 ), , j = 1, ..., d,
αIj = 0, that is VI = [Vi1 , Vi2 ] = 0. Therefore all the brackets have to be 0, that is G is
commutative. We can now conclude that G is isomorphic to (Rd , +), because so are all simply
connected and commutative Lie groups.
✷
1134
Remark 3.5. The previous theorem in particular applies to the case of a Brownian motion, that
corresponds to H = 12 .
If we relax the assumption that the family (V1 , ..., Vd ) forms a basis of the Lie algebra g, we can
have a global scaling property in slightly more general groups than the commutative ones. Let
us first look at one example.
The Heisenberg group H is the set of 3 × 3
1
x
0
1
0
0
matrices:
z
y , x, y, z ∈ R.
1
The Lie algebra of H is spanned by the matrices
0
0
0
1
0
0
0
0
0
0
D1 =
, D2 =
0
0
0
0
0
for which the following equalities hold
0
0
0
1
and D3 =
0
0
0
0
0
1
0 ,
0
[D1 , D2 ] = D3 , [D1 , D3 ] = [D2 , D3 ] = 0.
Consider now the solution of the equation
dXt = Xt (D1 ◦ dBt1 + D2 ◦ dBt2 ),
X0 = 1,
where (Bt1 , Bt2 ) is a two-dimensional fractional Brownian motion with Hurst parameter H > 14 .
It is easily seen that
R
1
1 B 2 + t B 1 ◦ dB 2 − B 2 ◦ dB 1
1
Bt1
B
s
s
t t
s
2
0 s
Xt = 0
.
1
Bt2
0
0
1
Therefore (Xct )t≥0 =law (∆c Xt )t≥0 , where ∆c is defined by
1
cH x
1
x
z
1
1
y = 0
∆c 0
0
0
0
0
1
c2H z
cH y .
1
In that case, we thus have a global scaling property whereas H is of course not commutative but
step-two nilpotent. Actually, we shall have a global scaling property in the Lie groups that are
called the Carnot groups. Let us recall the definition of a Carnot group.
Definition 3.6. A Carnot group of step (or depth) N is a simply connected Lie group G whose
Lie algebra can be written
V1 ⊕ ... ⊕ VN ,
where
[Vi , Vj ] = Vi+j
and
Vs = 0, for s > N.
1135
Example 3.7. Consider the set Hn = R2n × R endowed with the group law
1
(x, α) ⋆ (y, β) = x + y, α + β + ω(x, y) ,
2
where ω is the standard symplectic form on R2n , that is
0 −In
ω(x, y) = xt
y.
In
0
On hn the Lie bracket is given by
[(x, α), (y, β)] = (0, ω(x, y)) ,
and it is easily seen that
hn = V1 ⊕ V2 ,
where V1 = R2n × {0} and V2 = {0} × R. Therefore Hn is a Carnot group of depth 2 and observe
that H1 is isomorphic to the Heisenberg group.
Notice that the vector space V1 , which is called the basis of G, Lie generates g, where g denotes
the Lie algebra of G. Since G is step N nilpotent and simply connected, the exponential map
is a diffeomorphism. On g we can consider the family of linear operators δt : g → g, t ≥ 0
which act by scalar multiplication ti on Vi . These operators are Lie algebra automorphisms due
to the grading. The maps δt induce Lie group automorphisms ∆t : G → G which are called
the canonical dilations of G. Let us now take a basis U1 , ..., Ud of the vector space V1 . The
vectors Ui ’s can be seen as left invariant vector fields on G so that we can consider the following
stochastic differential equation on G:
dYt =
d Z
X
i=1
t
0
Ui (Ys ) ◦ dBsi , t ≥ 0,
(3.8)
which is easily seen to have a unique solution associated with the initial condition Y0 = 1G . We
have then the following global scaling property:
Proposition 3.8.
(Yct )t≥0 =law (∆cH Yt )t≥0 .
Proof. We keep the notations introduced before the proof of Theorem 2.7. From Theorem 2.7,
we have
N
X
X
Yt = exp
ΛI (B)t UI , t ≥ 0.
k=1 I∈{1,...,d}k
Therefore,
(Yct )t≥0 =law exp
N
X
k=1
cH|I|
X
I∈{1,...,d}k
1136
ΛI (B)t UI
.
t≥0
Since
exp
N
X
cH|I|
k=1
X
I∈{1,...,d}k
ΛI (B)t UI = exp
N
X
X
k=1 I∈{1,...,d}k
N
X
= ∆cH exp
ΛI (B)t (δcH UI )
X
k=1 I∈{1,...,d}k
we conclude
ΛI (B)t UI ,
(Yct )t≥0 =law (∆cH Yt )t≥0 .
✷
The previous proposition admits a counterpart.
Theorem 3.9. Let G be a connected Lie group (of matrices) with Lie algebra g. Let V1 , ..., Vd
be a family of g. Consider now the solution of the equation
!
d
X
dXt = Xt
Vi ◦ dBti , X0 = 1G .
i=1
Assume that there exists a family of Lie group automorphisms ∆t : G → G, t > 0, such that:
1. The map t → ∆t is continuous and limt→0 ∆t = 1G ;
2. For t1 , t2 ≥ 0, ∆t1 t2 = ∆t1 ◦ ∆t2 ;
3. (Xct )t≥0 =law (∆c Xt )t≥0 ;
Then the Lie subgroup H that is generated by eV1 , ..., eVd is a Carnot group.
Proof.
We can readily follow the lines of the proof of Theorem 3.4, so that we do not enter into details.
First we obtain that H has to be a simply connected nilpotent group and that for every c > 0,
+∞
+∞
X
X
X
X
1
ΛI (B)t VI ,
δc
ΛI (B)t VI
=law
ckH
k=1 I=(i1 ,...,ik ), i1 ≤...≤ik
t≥0
k=1
I=(i1 ,...,ik ), i1 ≤...≤ik
t≥0
where δc is the differential map of ∆c at 1G . For k ≥ 1, we denote Vk the linear space generated
by the set of commutators:
{VI , | I |= k} .
By letting c → +∞ and c → 0 and by using the support theorem of [7], we obtain that c−kH δc is
bounded on Vk for c → +∞ and c → 0. Since c−kH δc = Exp((φ − kHId) ln c), for some matrix
φ, we conclude
+∞
M
Vk ,
h=
k=1
where h is the Lie algebra of H. This proves that H is a Carnot group.
1137
✷
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