Article4 Sghir Aissa .pdf



Nom original: Article4_Sghir_Aissa.pdf

Ce document au format PDF 1.4 a été généré par 3B2 Total Publishing System 7.51a/W / Acrobat Distiller 5.0.5 (Windows), et a été envoyé sur fichier-pdf.fr le 17/02/2017 à 20:31, depuis l'adresse IP 105.145.x.x. La présente page de téléchargement du fichier a été vue 419 fois.
Taille du document: 166 Ko (19 pages).
Confidentialité: fichier public


Aperçu du document


Portugal. Math. (N.S.)
Vol. 69, Fasc. 4, 2012, 321–339
DOI 10.4171/PM/1920

Portugaliae Mathematica
6 European Mathematical Society

Regularities and limit theorems of some additive
functionals of symmetric stable process in some
anisotropic Besov spaces
Aissa Sghir and Hanae Ouahhabi*
(Communicated by Rui Loja Fernandes)

Abstract. In this paper, we give some regularities and limit theorems of some additive functionals of symmetric stable process of index 1 < a a 2 in some anisotropic Besov spaces.

Mathematics Subject Classification (2010). Primary 60B12; Secondary 60G52.
Keywords. Anisotropic Besov space, symmetric stable process, fractional Brownian motion, fractional derivative, additive functional, slowly varying function.

1. Introduction
Relative compactness in the space of probability measures is a key tool in the
study of weak convergence. A family F of probability measures on the general
metric space S is said to be tight if for each positive e, there is a compact set K
such that PðKÞ > 1 e for all P in F. According to Prohorov’s theorem, tightness is always a su‰cient condition for relative compactness and is also necessary
if S is separable and complete.
The space C½0; 1 of continuous functions is a classical framework for many
regularities and limit theorems in the theory of stochastic processes. The C½0; 1 weak convergence of a sequence of stochastic processes Xn , gives useful results
about the convergence in distribution of continuous functionals of the paths. In
many situations the processes Xn are known to have almost surely paths with at
least some Ho¨lder regularity and the same happens for the limiting process X .
The recent developments in the theory of wavelets and their applications in
probability and statistics show the interest in using more sophisticated function
spaces like the Ho¨lder space C a ½0; 1 , 0 < a < 1, and Besov spaces.
*The authors wish to express their sincere thanks to Prof. M. Ait Ouahra for his suggestions to submit
this work to Portugaliae Mathematica.

(AutoPDF V7 28/1/13 10:00) EMS (170 240mm) Tmath J-2715 PMS, 69:4 PMU: IDP[KN/V] 23/01/2013 pp. 321–339 2715_69-4_04 (p. 321)

322

A. Sghir and H. Ouahhabi

Our aim in this paper is to prove that some additive functionals of local times
of symmetric stable processes of index 1 < a a 2 satisfy certain Ho¨lder conditions
in Lp -norms which are more precise than the classical Ho¨lder conditions in the
uniform norm. We generalize also some limit theorems obtained, in the space of
continuous functions, by Rosen [15] for symmetric stable process of index
1 < a a 2 and Yor [19] for Brownian mtion, a ¼ 2. These will be done by recalling notions of anisotropic Besov spaces, we use a result of Kamont [13] who has
proved the characterization of these spaces in terms of the coe‰cients of the
expansion of a continuous function with respect to a basis which consists of tensor
products of Schauder functions. For the one-parameter Besov spaces, Ciesielski
et al. [12] have shown by using the techniques of constructive approximation of
functions that Besov spaces are isomorphic to spaces of real sequences. These
characterizations make the Besov topology easy to handle, and many applications
have been given in stochastic calculus, such as the regularities of some additive
functionals of local times of symmetric stable process of index 1 < a a 2 (see for
example Ait Ouahra et al. [2] and [4]).
Most of the estimates in this paper contain unspecified positive finite constants.
We use the same symbol Cp for these constants, even when they vary from one line
to the next.
Throughout this paper, X ¼ fXt j t b 0g denotes the real-valued symmetric
stable process of index 1 < a a 2, which is known to have a jointly continuous
local time fLðt; xÞ j t b 0; x a Rg (see Barlow [5] and Boylan [11]).
We have the well known regularity property of the local time and we refer to
Marcus and Rosen [14] for a proof.

Lemma 1.1. For any integer p b 1, there exists a constant 0 < Cp < l such that
for all 0 a t; s a 1 and all x; y a R,
kLðt; xÞ Lðs; xÞk2p a Cp jt sj ða 1Þ=a ;

ð1Þ

kLðt; xÞ Lðt; yÞk2p a Cp tða 1Þ=2a jx yj ða 1Þ=2 ;

ð2Þ

where k k2p ¼ ðEj j 2p Þ 1=2p .
The following lemma gives a regularity property of the local time as a random
function of two variables, its proof can be found in Ait Ouahra and Eddahbi [3].

Lemma 1.2. For any integer p b 1, there exists a constant 0 < Cp < l such that
for all 0 a t; s a 1 and all x; y a R,
kLðt; xÞ Lðs; xÞ Lðt; yÞ þ Lðs; yÞk2p a Cp jt sj ða 1Þ=2a jx yj ða 1Þ=2 :

ð3Þ

(AutoPDF V7 28/1/13 10:00) EMS (170 240mm) Tmath J-2715 PMS, 69:4 PMU: IDP[KN/V] 23/01/2013 pp. 321–339 2715_69-4_04 (p. 322)

Regularities and limit theorems in some anisotropic Besov spaces

323

Remark 1.3. (1) Using (3) and the fact that a.s. Lð0; xÞ ¼ 0, we get the spatial
Ho¨lder regularity of local time:
kLðt; xÞ Lðt; yÞk2p a Cp jx yj ða 1Þ=2 :

ð4Þ

(2) Notice that for a ¼ 2, X is a Brownian motion. Trotter [17] has proved
the existence of an a.s. continuous version of the Brownian local time lðt; xÞ as a
random function of two variables. Moreover, by Boufoussi and Roynette [10], for
each t > 0 fixed, the random function lðt; Þ satisfies the Ho¨lder condition ð4Þ with
exponent 12 , and by Boufoussi [7], the function ðt; xÞ ! lðt; xÞ satisfies the mixed
Ho¨lder condition ð3Þ with exponent 14 in time and exponent 12 in space.
In Sections 2 and 3, we study the Ho¨lder properties of some additive functionals of the local time Lðt; xÞ. We first recall the definition of slowly varying functions and some properties. For more details about slowly varying functions, we
refer the reader to Bingham et al. [6].

Definition 1.4. We say that a measurable function l : R þ ! R þ is slowly varying
at infinity if for all t positive, we have
lim

x!l

lðtxÞ
¼ 1:
lðxÞ

We are interested in the behavior of l at þl, then we can assume for example
that l is bounded on each interval of the form ½0; a , where a > 0.
The following theorem called Potter’s Theorem has played a central role in the
proof of our main results of regularities.

Theorem 1.5. (1) If l is slowly varying function, then for any chosen constants
A > 1 and x > 0, there exists X ¼ X ðA; xÞ such that
( )
lðyÞ
y x y x
a A max
;
ðx b X ; y b X Þ:
lðxÞ
x
x
(2) If moreover l is bounded away from 0 and l on every compact subset of
½0; þl½, then for every x > 0, there exists A 0 ¼ A 0 ðxÞ > 1 such that
( )
lðyÞ
y x y x
ðx > 0; y > 0Þ:
;
a A 0 max
lðxÞ
x
x
We now introduce certain generalized fractional derivative transforms which
play a central role in the sequel.

(AutoPDF V7 28/1/13 10:00) EMS (170 240mm) Tmath J-2715 PMS, 69:4 PMU: IDP[KN/V] 23/01/2013 pp. 321–339 2715_69-4_04 (p. 323)

324

A. Sghir and H. Ouahhabi

For any g a 0; b½ and g a C b B L 1 ðRÞ, we define
Kel; g gðxÞ :¼

1
Gð gÞ

ð þl
lðyÞ
0

gðx e yÞ gðxÞ
dy:
y 1þg

Note that lðxÞ ¼ oðx b Þ asÐ x ! þl for any b > 0 (see Bingham et al. [6], Prop.
þl lð yÞ
1.3.6), so when g > 0, 1 y 1þg < þl. Consequently, if g a C b B L 1 ðRÞ for
some g a 0; b½, then Kel; g gðxÞ defined bounded continuous functions.
But for g ¼ 0, since 1y is not integrable at þl, Kel; 0 is defined by
Kel; 0 gðxÞ :¼

ð þl
lðyÞ
0

gðx e yÞ 1 0; 1½ ðyÞgðxÞ
dy;
y

for any g a C b B L 1 ðRÞ and b > 0.
We put
K l; g :¼ Kþl; g K l; g ;
for the symmetric generalized fractional derivatives.

Remark 1.6. If we take l C 1, we recover the definitions of fractional derivative
g
0
and De
(see Yamada [18], Samko et al. [16]
and Hilbert transform denoted by De
and the references therein).
Following the same arguments used in the proof of Lemma 1 in Ait Ouahra
and Eddahbi [3] in the case of fractional derivatives of local time of symmetric
stable process, we get the following time regularities.
l; g
l; g
Lemma 1.7. (1) Let 0 < g < a 1
Then, for any integer
2 and K a fKe ; K g.

p b 1, there exists a constant 0 < Cp < l such that for all 0 a t; s a 1 and all
x; y a R,
kKLðt; ÞðxÞ KLðs; ÞðxÞk2p a Cp jt sjða 1Þ=a g=a :
(2) In case g ¼ 0 and under the assumption

Ð þl lð yÞ
1

y

dy < l, we get

kKLðt; ÞðxÞ KLðs; ÞðxÞk2p a Cp jt sj x ;
where 0 < x < a 1
a .
Proof. We treat only the case K ¼ Kþl; g , the other cases are similar. Here we
distinguish two cases.

(AutoPDF V7 28/1/13 10:00) EMS (170 240mm) Tmath J-2715 PMS, 69:4 PMU: IDP[KN/V] 23/01/2013 pp. 321–339 2715_69-4_04 (p. 324)

Regularities and limit theorems in some anisotropic Besov spaces

325

(1) Case g > 0. Let b ¼ jt sj 1=a . By the definition of Kþl; g , we have
kKþl; g Lðt; ÞðxÞ Kþl; g Lðs; ÞðxÞk2p
ðb
kLðt; x þ uÞ Lðs; x þ uÞ Lðt; xÞ þ Lðs; xÞk2p
1
a
lðuÞ
du
jGð gÞj 0
u 1þg
ð þl
kLðt; x þ uÞ Lðs; x þ uÞ Lðt; xÞ þ Lðs; xÞk2p
1
þ
lðuÞ
du
jGð gÞj b
u 1þg
:¼ I1 þ I2 :
We estimate I1 and I2 separately.
Estimate of I1 :
Since l is bounded on every compact subset of ½0; þl½; it follows from (3)
that
I1 a Cp jt sj ða 1Þ=2a b ða 1Þ=2 g a Cp jt sj ða 1Þ=a g=a :
Now we return to estimate I2 :
Potter’s Theorem with 0 < x < g implies the existence of AðxÞ > 1 such that
x
u
:
lðuÞ a AðxÞlðbÞ
b
Combining this fact with (1), we obtain
I2 a Cp jt sjða 1Þ=a g=a :
The proof of this case is done.
(2) Case g ¼ 0. By the definition of Kþl; 0 , we have
kKþl; 0 Lðt; ÞðxÞ Kþl; 0 Lðs; ÞðxÞk2p a: J1 þ J2 ;
where
J1 ¼

ð1
lðyÞ
0

kLðt; x þ yÞ Lðt; xÞ Lðs; x þ yÞ þ Lðs; xÞk2p
dy;
y

and
J2 ¼

ð þl
lðyÞ
1

kLðt; x þ yÞ Lðs; x þ yÞk2p
dy:
y

(AutoPDF V7 28/1/13 10:00) EMS (170 240mm) Tmath J-2715 PMS, 69:4 PMU: IDP[KN/V] 23/01/2013 pp. 321–339 2715_69-4_04 (p. 325)

326

A. Sghir and H. Ouahhabi

Now let b ¼ jt sj 2x=ða 1Þ . Using (1), (4) and the fact that l is bounded on every
compact subset of ½0; þl½; we get
ðb

kLðt; x þ yÞ Lðt; xÞk2p þ kLðs; x þ yÞ Lðs; xÞk2p
dy
y
0
ð1
kLðt; x þ yÞ Lðs; x þ yÞk2p þ kLðt; xÞ Lðs; xÞk2p
dy
þ Cp
y
b
h ðb
i
1
a Cp
yða 1Þ=2 1 dy þ jt sj ða 1Þ=a log
:
jt sj 2x=ða 1Þ
0

J1 a Cp

Then
J1 a Cp jt sj

x

2x
j jt sjða 1Þ=a x logjt sjða 1Þ=a x j
a 1

ða 1Þ a x

!

a Cp jt sj x ;
where we have used in the last estimation the elementary inequality: for any
x a 0; 1 , we have jx logðxÞj a e 1 .
Ð þl lð yÞ
We now deal with J2 . Thanks to ð1Þ and the assumption 1
y dy < l, we
obtain
J2 a Cp jt sj ða 1Þ=a :
which gives the desired estimate for g ¼ 0.

r

In the same way we obtain the following space regularities.
l; g
l; g
Lemma 1.8. (1) Let 0 < g < a 1
2 and K a fKe ; K g. Then for any integer p b 1,

there exist a constant 0 < Cp < l such that for any 0 a t a 1, all x; y a ½ M; M ,
kKLðt; ÞðxÞ KLðt; ÞðyÞk2p a Cp t ða 1Þ=2a jx yj ða 1Þ=2 g :
(2) In the case g ¼ 0 and under the assumption

Ð þl lð yÞ
1

y

dy < l, we get

kKLðt; ÞðxÞ KLðt; ÞðyÞk2p a Cp t ða 1Þ=2a jx yjða 1Þ=2 :
M is a finite positive constant.
Proof. We treat only the case K ¼ K l; g , the other cases are similar. Here we distinguish two cases.

(AutoPDF V7 28/1/13 10:00) EMS (170 240mm) Tmath J-2715 PMS, 69:4 PMU: IDP[KN/V] 23/01/2013 pp. 321–339 2715_69-4_04 (p. 326)

Regularities and limit theorems in some anisotropic Besov spaces

327

(1) Case g > 0. Let b ¼ jx yj. By the definition of K l; g , we have
kK l; g Lðt; ÞðxÞ K l; g Lðt; Þð yÞk2p
ðb
1
a
lðuÞ
jGð gÞj 0
kLðt; x þ uÞ Lðt; x uÞk2p þ kLðt; y þ uÞ Lðt; y uÞk2p
du
u 1þg
ð þl
1
þ
lðuÞ
jGð gÞj b



kLðt; x þ uÞ Lðt; x uÞk2p þ kLðt; y þ uÞ Lðt; y uÞk2p
du
u 1þg
:¼ K1 þ K2 :


We estimate K1 and K2 separately.
Estimate of K1 :
Since l is bounded on every compact subset of ½0; þl½; it follows from (2) that,
K1 a Cp t ða 1Þ=2a

ðb
0

a Cp t

ða 1Þ=2a

uða 1Þ=2
du
u 1þg

jx yj ða 1Þ=2 g :

Now we return to estimate K2 :
Potter’s Theorem with 0 < x < g implies the existence of AðxÞ > 1 such that
x
u
lðuÞ a AðxÞlðbÞ
:
b
Combining this fact with (2), we obtain
K2 a Cp t ða 1Þ=2a jx yj ða 1Þ=2 g :
The proof of this case is done.
(2) Case g ¼ 0. Let us give the proof for Kþl; 0 . The other case can be derived
similarly and by linearity. By the definition of Kþl; 0 , we have
kKþl; 0 Lðt; ÞðxÞ Kþl; 0 Lðt; ÞðyÞk2p a: L1 þ L2 ;
where
L1 ¼

ð1
lðuÞ
0

kLðt; x þ uÞ Lðt; xÞ Lðt; y þ uÞ þ Lðt; yÞk2p
du;
u

(AutoPDF V7 28/1/13 10:00) EMS (170 240mm) Tmath J-2715 PMS, 69:4 PMU: IDP[KN/V] 23/01/2013 pp. 321–339 2715_69-4_04 (p. 327)

328

A. Sghir and H. Ouahhabi

and
L2 ¼

ð þl
lðuÞ
1

kLðt; x þ uÞ Lðt; y þ uÞk2p
du:
u

Let us deal with L1 . We have
ðb

L1 a

kLðt; x þ uÞ Lðt; xÞk2p þ kLðt; y þ uÞ Lðt; yÞk2p
du
u
0
ð1
kLðt; x þ uÞ Lðt; xÞk2p þ kLðt; y þ uÞ Lðt; yÞk2p
du:
þ
u
b

We consider the two cases jx yj > 1e and jx yj a 1e .
(a) Case jx yj > 1e . Using (2) and choosing 1e < b < jx yj, we have
L1 a Cp t ða 1Þ=2a jx yjða 1Þ=2 :
(b) Case jx yj a 1e . By choosing 0 < b < jx yj, (2) yields
L1 a Cp t ða 1Þ=2a jx yj ða 1Þ=2 :
Therefore, we deduce that
L1 a Cp t ða 1Þ=2a jx yj ða 1Þ=2 :
Now we deal with L2 . Thanks to ð2Þ and the assumption
obtain

Ð þl lð yÞ
1

y

dy < l, we

L2 a Cp t ða 1Þ=2a jx yj ða 1Þ=2 :
which gives the desired estimate for g ¼ 0.

r

As a consequence of Lemma 1.8 and the Markov property of symmetric stable
processes, we get the following mixed regularities in time and space. (For more
detail about proof, we refer to Ait Ouahra and Eddahbi [3], Theorem 1, for local
time, and Ait Ouahra [1], p. 13, for fractional derivative of local time of symmetric
stable process of index 1 < a a 2.)
l; g
l; g
Lemma 1.9. (1) Let 0 < g < a 1
2 and K a fKe ; K g. For any integer p b 1,

there exists a constant 0 < Cp < l such that, for all 0 a t; s a 1 and all x; y a
½ M; M ,

(AutoPDF V7 28/1/13 10:00) EMS (170 240mm) Tmath J-2715 PMS, 69:4 PMU: IDP[KN/V] 23/01/2013 pp. 321–339 2715_69-4_04 (p. 328)

Regularities and limit theorems in some anisotropic Besov spaces

329

kKLðt; ÞðxÞ KLðt; ÞðyÞ KLðs; ÞðxÞ þ KLðs; ÞðyÞk2p
a Cjt sj ða 1Þ=2a jx yjða 1Þ=2 g :
(2) In the case g ¼ 0 and under the assumption

Ð þl lð yÞ
1

y

dy < l, we get

kKLðt; ÞðxÞ KLðt; ÞðyÞ KLðs; ÞðxÞ þ KLðs; ÞðyÞk2p
a Cjt sj ða 1Þ=2a jx yj ða 1Þ=a :

2. Besov spaces
We will firstly present a brief survey of Besov spaces. For more details, we refer
the reader to Boufoussi [7] and Ciesielski et al. [12].
Let I ¼ ½0; 1 . We denote by L p ðI Þ, 1 a p < þl, the space of Lebesgue integrable real-valued functions defined on I with exponent p. The modulus of continuity of a Borel function f : I ! R in L p ðI Þ norm is defined for all h a R by
op ð f ; tÞ ¼ sup kDh f kp ;
0ahat

where
Dh f ðtÞ ¼ 1½0; 1 h ðtÞ½ f ðt þ hÞ f ðtÞ :
o

Definition 2.1. The Besov space denoted by Bp; m;ln , 1 a p < þl, is a nonseparable Banach space of real-valued continuous functions f on I , endowed
with the norm
op ð f ; tÞ
;
0<ta1 om; n ðtÞ

k f kp;om;ln ¼ k f kp þ sup
where
om; n ðtÞ ¼ t

m

!n
1
1 þ log
;
t

for any 0 < m < 1 and n > 0.
Ciesielski et al. [12] showed by using the techniques of constructive approximation of functions that Besov spaces are isomorphic to spaces of real sequences.
These characterizations allows us to prove in the sequel some results of regularities
of some additive functionals of local times of symmetric stable processes of index
1 < a a 2.

(AutoPDF V7 28/1/13 10:00) EMS (170 240mm) Tmath J-2715 PMS, 69:4 PMU: IDP[KN/V] 23/01/2013 pp. 321–339 2715_69-4_04 (p. 329)

330

A. Sghir and H. Ouahhabi

The Schauder basis on I is defined by
8
>
j1 ðtÞ ¼ t1½0; 1 ðtÞ;
< j0 ðtÞ ¼ 1½0; 1 ðtÞ;
n ¼ 2 j þ k;
j b 0; k ¼ 1; . . . ; 2 j ;
>
: j ðtÞ ¼ j ðtÞ ¼ 2 1 j=2 Fð2 j t kÞ;
j; k
n
where FðuÞ ¼ u1½0; 1=2 ðuÞ þ ð1 uÞ1 1=2; 1 ðuÞ.
In this basis, the decomposition and the coe‰cients of continuous functions f
on I are respectively given by
f ðtÞ ¼

l
X

Cn ð f Þjn ðtÞ:

n¼0

and
8
>
C1 ð f Þ ¼ f ð1Þ f ð0Þ;
< C0 ð f Þ ¼ f ð0Þ;
j b 0; k ¼ 1; . . . ; 2 j ;
n ¼ 2 j þ k;
>
: C ð f Þ ¼ 2 j=2 2f 2k 1 f 2k 2 f 2k :
n
2 jþ1
2 jþ1
2 jþ1
o

We consider the separable Banach subspace of Bp; m;ln , 1 a p < þl, defined by




o ;0
o
Bp; m;ln ¼ f a Bp; m;ln j op ð f ; tÞ ¼ o om; n ðtÞ ðt # 0Þ :
The following characterization theorem is due to Ciesielski et al. [12], Theorem
III.2.
o

;0

m; n
Theorem

2.2. The subspace Bp; l , 1 a p < þl, corresponds to the sequences

Cn ð f Þ

n

such that

2 jþ1
i 1=p
2 jð1=2 mþ1=pÞ h X
lim
jCn ð f Þj p
¼ 0:
n
j!þl
ð1 þ jÞ
n¼2 j þ1

For the proof of our results, we need the following tightness criterion in the
o ;0
subspace Bp; m;ln , 2 a p < þl (see Ait Ouahra et al. [2], Lemma 4.3).

Theorem 2.3. Let fXtn j t a ½0; 1 gnb1 be a sequence of stochastic processes
satisfying:
(1) X0n ¼ 0 for all n b 1.
(2) For all 2 a p < þl, there exists a constant 0 < Cp < þl such that
EjXtn Xsn j p a Cp jt sj pm

for all t; s a ½0; 1 ;

(AutoPDF V7 28/1/13 10:00) EMS (170 240mm) Tmath J-2715 PMS, 69:4 PMU: IDP[KN/V] 23/01/2013 pp. 321–339 2715_69-4_04 (p. 330)

Regularities and limit theorems in some anisotropic Besov spaces

331
o

where 0 < m < 1. Then the sequence fXtn j t a ½0; 1 g is tight in Bp; m;ln
n > 0 and p > maxðm 1 ; n 1 Þ.

;0

for all

Following the same arguments used in the proof of Lemma 9 in Ait Ouahra
et al. [4], we have

Lemma 2.4. (1) Let 0 < g < a 1
and K a fKeg ; K g g.
2

The trajectory t !
o
n; 0
KLðt; ÞðxÞ belongs a.s. to Bp; ða 1Þ=a g=a;
,
1
a
p
<
þl,
for
any n > 1p and all
l
jxj a M.
Ð þl lð yÞ
(2) In the case g ¼ 0 and under the assumption 1
y dy < l, the trajecox; n ; 0
tory t ! KLðt; ÞðxÞ belongs a.s. to Bp; l , 1 a p < þl, for any n > 1p and all
jxj a M, where 0 < x < a 1
a .
(3) Let 0 < g < a 1
,
and
K a fKeg ; K g g. The mapping x ! KLðt; ÞðxÞ belongs
2
oða 1Þ=2 g; n ; 0
, 1 a p < þl, for any n > 1p and all
t a I.
a.s. to Bp; l
Ð þl
lð yÞ
(4) In the case g ¼ 0 and under the assumption 1
y dy < l, the mapping
oða 1Þ=2; n ; 0
, 1 a p < þl, for any n > 1p and all
x ! KLðt; ÞðxÞ belongs a.s. to Bp; l
t a I , where M is a positive finite constant.
Proof. We are going to prove (1) since the other cases follow in the same manner.
By Theorem 2.3, it su‰ces to show that a.s.
2 jþ1
i 1=p
2 jð1=2 ðða 1Þ=a g=aÞþ1=pÞ h X
p
jC
ðKLðt;
ÞðxÞj
¼ 0;
n
j!þl
ð1 þ jÞ n
n¼2 j þ1

lim

where
Cn ðKLðt; ÞðxÞ
¼2

j=2






!
2k 1
2k 2
2k
2KL
; ðxÞ KL
; ðxÞ KL jþ1 ; ðxÞ :
2 jþ1
2 jþ1
2

For any l > 0, we set


2 jþ1
1=p
2 jð1=2 ðða 1Þ=a g=aÞþ1=pÞ X
p
I ¼ P sup
jC
ðKLðt;
ÞðxÞj
>
l
:
n
ð1 þ jÞ n
jb0
n¼2 j þ1
By Tchebychev’s inequality, we have
2 jþ1
1 X 2 jpð1=2 ðða 1Þ=a g=aÞþ1=pÞ X
EjCn ðKLðt; ÞðxÞj p :
Ia p
l jb0
ð1 þ jÞ pn
j
n¼2 þ1

(AutoPDF V7 28/1/13 10:00) EMS (170 240mm) Tmath J-2715 PMS, 69:4 PMU: IDP[KN/V] 23/01/2013 pp. 321–339 2715_69-4_04 (p. 331)

332

A. Sghir and H. Ouahhabi

In view of the definition of Cn ðKLðt; ÞðxÞ and Lemma 1.7, we deduce that
CX
1
<l
p
l jb0 ð1 þ jÞ pn

Ia

for all n >

1
:
p

The result is a simple application of the Borel–Cantelli lemma.

r

In the following we generalize, in Besov spaces, the results obtained in the
space of continuous functions by Rosen [15] in the case of symmetric stable processes of index 1 < a a 2 and by Yor [19] in the case of Brownian motion. The
fractional Brownian motion is obtained as a limit in law of linear local times of
symmetric stable processes. To state this result, let fBtH ðxÞ j t b 0; x a Rg denote
a fractional Brownian sheet with index H a 0; 1½. It the continuous centered
Gaussian process with covariance function


1
E BtH ðxÞ; BsH ð yÞ ¼ ðsbtÞ ðjxj H þ jyj H jx yj H Þ:
2
Let pt ðx; yÞ be the transition probability density for the symmetric stable processes
and write pt ð0; x yÞ ¼ pt ðx yÞ ¼ pt ðjx yjÞ. The a-potential density is defined by
u a ðxÞ ¼

ð þl

e at pt ðxÞ dt:

0

Theorem 2.5. Let z an independent exponential random variable of mean 1. Then
as e ! 0 the sequence of processes

e





Lðz;
exÞ

Lðz;

j
x
a
R
;
ða 1Þ=2
1

converges in law to the process
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
f2 ca u 1 ð0ÞBza 1 ðxÞ j x a Rg
o

;0

n
, 2 a p < þl, for all n > 1p . B a 1 is independent of z,
in the Besov space Bp; ða 1Þ=2;
l
where

!
ð þl
1
ds
ca ¼
p1 ð0Þ p1 1=a
s
s 1=a
0

(AutoPDF V7 28/1/13 10:00) EMS (170 240mm) Tmath J-2715 PMS, 69:4 PMU: IDP[KN/V] 23/01/2013 pp. 321–339 2715_69-4_04 (p. 332)

Regularities and limit theorems in some anisotropic Besov spaces

333

and
ð
1
dp
u ð0Þ ¼
2p 1 þ j pj a
1

is the 1-potential at 0.
Proof. By Theorem 1.3 in Rosen [15], we have the convergence of the finitedimensional distributions. It remains to show tightness.
By virtue of (4), for any 1 a p < þl, we obtain
E½Lðz; exÞ Lðz; e yÞ 2p ¼

ð þl

e s E½Lðs; exÞ Lðs; eyÞ 2p ds

0

a Cp ðejx yjÞ 2pðða 1Þ=2Þ :
This together with Theorem 2.3 completes the proof of Theorem 2.5.

r

3. Anisotropic Besov spaces
Now we denote by L p ðI 2 Þ the space of Lebesgue integrable functions with exponent p ð1 a p < lÞ. For any function f : I 2 ! R, any h a R, the progressive
di¤erence in direction x1 (resp. x2 ), is defined by
Dh; 1 f ðx1 ; x2 Þ ¼ f ðx1 þ h; x2 Þ f ðx1 ; x2 Þ;
Dh; 2 f ðx1 ; x2 Þ ¼ f ðx1 ; x2 þ hÞ f ðx1 ; x2 Þ:
For any ðh1 ; h2 Þ a R 2 , we set
Dh1 ; h2 f ¼ Dh1 ; 1 Dh2 ; 2 f ;
Dh;2 i f ¼ Dh; i Dh; i f ;

i ¼ 1; 2:

For any Borel function f : I 2 ! R such that f a L p ðI 2 Þ, one can measure its
smoothness by its modulus of continuity computed in L p ðI 2 Þ norm.
To this end let us define, for any t a I and ðt1 ; t2 Þ a I 2 ,
oð1; 0Þ p ð f ; t1 Þ ¼ sup kDh1 ; 1 f kp ;
jh1 jat1

oð0; 1Þ p ð f ; t2 Þ ¼ sup kDh2 ; 2 f kp ;
jh2 jat2

oð1; 1Þ p ð f ; t1 ; t2 Þ ¼

sup
jh1 jat1 ; jh2 jat2

kDh1 ; h2 f kp :

(AutoPDF V7 28/1/13 10:00) EMS (170 240mm) Tmath J-2715 PMS, 69:4 PMU: IDP[KN/V] 23/01/2013 pp. 321–339 2715_69-4_04 (p. 333)

334

A. Sghir and H. Ouahhabi

Definition 3.1. Let 0 < a1 ; a2 < 1 and n a R. The anisotropic Besov space, denoted by Lipp ða1 ; a2 ; nÞ, 1 a p < þl, is a non-separable Banach space of realvalued continuous functions f on I 2 , endowed with the norm
a1 ; a2

k f kbon

oð1; 0Þ: p ð f ; t1 Þ
a1 ; a2
ðt1 ; 1Þ
0<t1 a1 on

:¼ k f kp þ sup

oð1; 0Þ: p ð f ; t2 Þ
oð1; 1Þ: p ð f ; t1 ; t2 Þ
þ sup
;
a1 ; a2
a1 ; a2
ð1;
t
Þ
ðt1 ; t2 Þ
o
n
2
0<t2 a1
0<t1 ; t2 a1 on

þ sup

where
ona1 ; a2 ðt1 ; t2 Þ

¼

t1a1 t2a2



1
1 þ log
t 1 t2

!n
:

We consider the separable Banach subspace of Lipp ða1 ; a2 ; nÞ, 1 a p < þl,
defined by



Lipp ða1 ; a2 ; nÞ :¼ f a Lipp ða1 ; a2 ; bÞ j oð1; 0Þ p ð f ; t1 Þ ¼ o ona1 ; a2 ðt1 ; 1Þ as t1 ! 0;


oð0; 1Þ: p ð f ; t2 Þ ¼ o ona1 ; a2 ð1; t2 Þ as t2 ! 0;



oð1; 1Þ: p ð f ; t1 ; t2 Þ ¼ o ona1 ; a2 ðt1 ; t2 Þ as t1 bt2 ! 0 ;
where t1 bt2 :¼ minðt1 ; t2 Þ.
Now, for any continuous functions f on I 2 , we have the decomposition

f ðt1 ; t2 Þ ¼

l
X

X

Cn; n 0 ð f Þjn n jn 0 ðt1 ; t2 Þ;

m¼0 maxðn; n 0 Þ¼m

where Cn; n 0 ð f Þ ¼ Cn1 Cn2 ð f Þ with




Cn1 ð f ÞðtÞ ¼ Cn f ð:; tÞ ;


Cn2 ð f ÞðtÞ ¼ Cn f ðt; Þ :

In order to state our main results, we need the following characterization theorem.
(See Kamont [13], Theorem A.2, who described anisotropic Besov spaces in terms
of the coe‰cients of the expansion of a continuous function with respect to a basis
which consists of tensor products of Schauder functions.)

(AutoPDF V7 28/1/13 10:00) EMS (170 240mm) Tmath J-2715 PMS, 69:4 PMU: IDP[KN/V] 23/01/2013 pp. 321–339 2715_69-4_04 (p. 334)

Regularities and limit theorems in some anisotropic Besov spaces

335


Theorem 3.2. The
subspace Lipp ða1 ; a2 ; nÞ, 1 a p < þl, corresponds to the se-

quences Cn; n 0 ð f Þ such that

2 jþ1
i 1=p
2 jð1=2 a1 þ1=pÞ h X
jCn; l 0 ð f Þj p
¼ 0;
lim
n
j!þl
ð1 þ jÞ
n¼2 j þ1

l 0 ¼ 0; 1;

2 jþ1
i 1=p
2 jð1=2 a2 þ1=pÞ h X
p
lim
jC
ð
f
Þj
¼ 0;
l;
n
j!þl
ð1 þ jÞ n
n¼2 j þ1

l ¼ 0; 1;
0

j þ1
0
2 jþ1
2X
i 1=p
2 jð1=2 a1 þ1=pÞ 2 j ð1=2 a2 þ1=pÞ h X
p
0 ð f Þj
lim
jC
¼ 0:
n;
n
n
j; j 0 !þl
ð1 þ j þ j 0 Þ
j0
n¼2 j þ1 0

n ¼2 þ1

The first result of this section is the following.

Theorem 3.3.
(1) The trajectory ðt; xÞ ! Lðt; xÞ belongs a.s. to anisotropic Besov


a 1
1
space Lipp a 1
2a ; 2 ; n , 1 a p < þl, for any n > p .
g
a 1
g
(2) Let 0 < g < 2 , and K a fKe; K g. The trajectory ðt; xÞ ! KLðt; ÞðxÞ
a 1
belongs a.s. to Lipp a 1
n> 1.
2a ; 2 g; n , 1 a p < þl, for any
Ð þl lð yÞ p
(3) In the case g ¼ 0 and under the assumption
1 y dy < l, the trajec
a 1 a 1
tory ðt; xÞ ! KLðt; ÞðxÞ belongs a.s. to Lipp 2a ; 2 ; n , 1 a p < þl, for any
n > 1p .

Proof. We are going to prove (1) since the other cases follow in the same manner.
Notice that a.s., for all x a R, Lð0; xÞ ¼ 0, thus C0; n ðLÞ ¼ 0. Therefore by Theorem 3.2, it su‰ces to show that

lim

j!þl

2 jþ1
2 jð1=2 ða 1Þ=2þ1=pÞ h X

ð1 þ jÞ b

jC1; n ðLÞj p

i 1=p

¼ 0;

n¼2 j þ1

2 jþ1
i 1=p
2 jð1=2 ða 1Þ=2aþ1=pÞ h X
p
jC
ðLÞj
¼ 0;
lim
n;
0
j!þl
ð1 þ jÞ n
n¼2 j þ1
2 jþ1
i 1=p
2 jð1=2 ða 1Þ=2aþ1=pÞ h X
lim
jCn; 1 ðLÞj p
¼ 0;
n
j!þl
ð1 þ jÞ
n¼2 j þ1
0

j þ1
0
2 jþ1
2X
i 1=p
2 jð1=2 ða 1Þ=2aþ1=pÞ 2 j ð1=2 ða 1Þ=2þ1=pÞ h X
p
0
lim
jC
ðLÞj
¼ 0:
n; n
j; j 0 !þl
ð1 þ j þ j 0 Þ n
j0
n¼2 j þ1 0

n ¼2 þ1

where

(AutoPDF V7 28/1/13 10:00) EMS (170 240mm) Tmath J-2715 PMS, 69:4 PMU: IDP[KN/V] 23/01/2013 pp. 321–339 2715_69-4_04 (p. 335)

336

A. Sghir and H. Ouahhabi







2k 0 1
2k 0 2
2k 0

L
1;

L
1;
;
C1; n ðLÞ ¼ 2 j=2 2L 1;
2 j0
2 j0
2 j0






2k 1
2k 2
2k
j=2
Cn; 0 ðLÞ ¼ 2
2L
;0 L
;0 L j ;0 ;
2j
2j
2






2k 1
2k 2
2k
;
1

L
;
1

L
;
1
:
Cn; 1 ðLÞ ¼ 2 j=2 2L
2j
2j
2j
The first three inequalities follow immediately by the same arguments used in
proof of Lemma 2.4. We will now prove the last inequality. We write
0

2 ðð jþ j Þ=2Þ Cn; n 0 ðLÞ


2k 2 2k 0 2

; j 0 þ1
¼
2 jþ1
2






2k 1 2k 0 1
2k 1 2k 0 2
2k 2k 0 1

2L

2L
;
;
;
¼ 4L
2 jþ1
2 j 0 þ1
2 jþ1
2 j 0 þ1
2 jþ1 2 j 0 þ1






2k 1 2k 0
2k 2 2k 0 1
2k 2k 0 2
2L
2L
;
; j 0 þ1 þ L jþ1 ; j 0 þ1
2 jþ1 2 j 0 þ1
2 jþ1
2
2
2






0
0
0
2k 2k
2k 2 2k
2k 2 2k 2
þ L jþ1 ; j 0 þ1 þ L
þ
f
;
;
2
2
2 jþ1 2 j 0 þ1
2 jþ1
2 j 0 þ1
2
D1=2
jþ1 ; 1

2
D1=2
jþ1 ; 2 L



:¼ 2En; n 0 ðLÞ þ 2Fn; n 0 ðLÞ þ Gn; n 0 ðLÞ;
where





2k 1 2k 0 1
2k 1 2k 0 2

L
;
;
2 jþ1
2 j 0 þ1
2 jþ1
2 j 0 þ1




2k 2k 0 1
2k 2k 0 2
L jþ1 ; j 0 þ1 þ L jþ1 ; j 0 þ1 ;
2
2
2
2




2k 1 2k 0 1
2k 1 2k 0
Fn; n 0 ðLÞ ¼ L

L
;
;
2 jþ1
2 j 0 þ1
2 jþ1 2 j 0 þ1




2k 2 2k 0 1
2k 2 2k 0
;
; j 0 þ1 þ L
;
L
2 jþ1
2
2 jþ1 2 j 0 þ1




2k 2k 0
2k 2k 0 2
Gn; n 0 ðLÞ ¼ L jþ1 ; j 0 þ1 L jþ1 ; j 0 þ1
2
2
2
2




0
2k 2 2k
2k 2 2k 0 2
L
þL
;
; j 0 þ1 :
2 jþ1 2 j 0 þ1
2 jþ1
2
En; n 0 ðLÞ ¼ L

It su‰ces then to show that

(AutoPDF V7 28/1/13 10:00) EMS (170 240mm) Tmath J-2715 PMS, 69:4 PMU: IDP[KN/V] 23/01/2013 pp. 321–339 2715_69-4_04 (p. 336)

Regularities and limit theorems in some anisotropic Besov spaces

337

0

j þ1
0
2 jþ1
2X
i 1=p
2 jð ðða 1Þ=2aÞþ1=pÞ 2 j ð ðða 1Þ=2Þþ1=pÞ h X
p
0 ðLÞj
lim
jE
¼ 0;
n;
n
j; j 0 !þl
ð1 þ j þ j 0 Þ n
j0
n¼2 j þ1 0

n ¼2 þ1
0

j þ1
0
2
2X
i 1=p
2 jð ðða 1Þ=2aÞþ1=pÞ 2 j ð ðða 1Þ=2Þþ1=pÞ h X
p
0 ðLÞj
lim
jF
¼ 0;
n;
n
j; j 0 !þl
ð1 þ j þ j 0 Þ n
j0
n¼2 j þ1 0
jþ1

n ¼2 þ1
0

j þ1
0
2
2X
i 1=p
2 jð ðða 1Þ=2aÞþ1=pÞ 2 j ð ðða 1Þ=2Þþ1=pÞ h X
p
0 ðLÞj
lim
jG
¼ 0:
n;
n
j; j 0 !þl
ð1 þ j þ j 0 Þ n
j0
n¼2 j þ1 0
jþ1

n ¼2 þ1

Let us for example give the proof of the first equality.
For any l > 0, we set

0
2 jð ðða 1Þ=2aÞþ1=pÞ 2 j ð ðða 1Þ=2Þþ1=pÞ
I ¼ P sup sup
ð1 þ j þ j 0 Þ n
jb0 j 0 b0


2 jþ1
h X

0

j þ1
2X

jEn; n 0 ðLÞj p

i 1=p


>l :

n¼2 j þ1 n 0 ¼2 j 0 þ1

By Tchebychev’s inequality, we have
0

j þ1
0
2 jþ1
2X
1 X X 2 jð ðða 1Þ=2aÞþ1=pÞ 2 j ð ðða 1Þ=2Þþ1=pÞ X
Ia p
EjEn; n 0 ðLÞj p :
l jb0 j 0 b0
ð1 þ j þ j 0 Þ n
j0
n¼2 j þ1 0

n ¼2 þ1

In view of the definition of En; n 0 ðLÞ and (3), we deduce that
Ia

C XX
1
<l
p
l jb0 j 0 b0 ð1 þ j þ j 0 Þ pn

for all n >

1
:
p

The result is a simple application of the Borel–Cantelli lemma. This completes the
proof of Theorem 3.3.
r
For the proof of the next theorem, we need the following tightness criterion
in the subspace Lipp ða1 ; a2 ; nÞ, 2 a p < þl (see Boufoussi and Lakhel [9],
Lemma 2.5).

Theorem 3.4. Let fXs;nt j ðs; tÞ a ½0; 1 2 gnb1 be a sequence of random fields
satisfying:
(1) X ;n0 ¼ X0;n : ¼ x for some a R.
(2) For all 2 a p < þl, there exists a constant 0 < Cp < þl such that
EjXs;nt Xsn0 ; t Xs;nt 0 þ Xsn0 ; t 0 j p a Cp js s 0 j a1 p jt t 0 j a2 p

for all t; s a ½0; 1 ;

(AutoPDF V7 28/1/13 10:00) EMS (170 240mm) Tmath J-2715 PMS, 69:4 PMU: IDP[KN/V] 23/01/2013 pp. 321–339 2715_69-4_04 (p. 337)

338

A. Sghir and H. Ouahhabi

where 0 < a1 ; a2 < 1: Then the sequence fX n gnb1 is tight in Lipp ða1 ; a2 ; nÞ for
all n > 2p .
Now we are ready to state and prove our second result.

Theorem 3.5. The sequence of processes

e



Lðt; exÞ Lðt; 0Þ j ðt; xÞ a ½0; 1 2
ða 1Þ=2
1




converges in law as e ! 0 to the process
pffiffiffiffiffi a 1
2
f2 ca BLðt;
0Þ ðxÞ j ðt; xÞ a ½0; 1 g
in Lipp

a 1
2a


2
; a 1
2 ; n , 2 a p < þl, for all n > p .

Proof. By Theorem 1.2 of Rosen [15], we have the convergence of the finitedimensional distributions. The tightness follows from (1) and Theorem 3.4.
r

Remark 3.6. In Section 3, the local time Lðt; xÞ is analyzed in both its variables
through the anisotropic Besov Space, in contrast to Section 2 where one is interested in the regularity in t.

References
[1] M. Ait Ouahra, Weak convergence to fractional Brownian motion in some anisotropic
Besov space. Ann. Math. Blaise Pascal 11 (2004), 1–17. Zbl 1077.60025 MR 2077234
[2] M. Ait Ouahra, B. Boufoussi, and E. Lakhel, The´ore`mes limites pour certaines fonctionnelles associe´es aux processus stables dans une classe d’espaces de Besov. Stoch.
Stoch. Rep. 74 (2002), 411–427. Zbl 1015.60068 MR 1940494
[3] M. Ait Ouahra and M. Eddahbi, The´ore`mes limites pour certaines fonctionnelles
associe´es aux processus stables sur l’espace de Ho¨lder. Publ. Mat. 45 (2001), 371–386.
Zbl 0995.60037 MR 1876912
[4] M. Ait Ouahra, M. Eddahbi, and M. Ouali, Fractional derivatives of local times
of stable Le´vy processes as the limits of the occupation time problem in Besov space.
Probab. Math. Statist. 24 (2004), 263–279. Zbl 1080.60074 MR 2157206
[5] M. T. Barlow, Necessary and su‰cient conditions for the continuity of local time of
Le´vy processes. Ann. Probab. 16 (1988), 1389–1427. Zbl 0666.60072 MR 958195
[6] N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular variation. Encyclopedia
Math. Appl. 27, Cambridge University Press, Cambridge 1987. Zbl 0617.26001
MR 898871

(AutoPDF V7 28/1/13 10:00) EMS (170 240mm) Tmath J-2715 PMS, 69:4 PMU: IDP[KN/V] 23/01/2013 pp. 321–339 2715_69-4_04 (p. 338)

Regularities and limit theorems in some anisotropic Besov spaces

339

[7] B. Boufoussi, Espaces de Besov: Caracte´risations et applications. The`se de l’Universit
de Nancy 1. France, Nancy 1994.
[8] B. Boufoussi and A. Kamont, Temps local brownien et espaces de Besov anisotropiques. Stochastics Stochastics Rep. 61 (1997), 89–105. Zbl 0884.60033 MR 1473915
[9] B. Boufoussi and E. h. Lakhel, Un re´sultat d’approximation d’une EDPS hyperbolique en norme de Besov anisotropique. C. R. Acad. Sci. Paris Se´r. I Math. 330
(2000), 883–888. Zbl 0960.60059 MR 1771952
[10] B. Boufoussi and B. Roynette, Le temps local brownien appartient p.s. a` l’espace de
Besov Bp;1=2l . C. R. Acad. Sci. Paris Se´r. I Math. 316 (1993), 843–848.
Zbl 0788.46035 MR 1218273
[11] E. S. Boylan, Local times for a class of Marko¤ processes. Illinois J. Math. 8 (1964),
19–39. Zbl 0126.33702 MR 0158434
[12] Z. Ciesielski, G. Kerkyacharian, and B. Roynette, Quelques espaces fonctionnels associe´s a` des processus gaussiens. Studia Math. 107 (1993), 171–204. Zbl 0809.60004
MR 1244574
[13] A. Kamont, Isomorphism of some anisotropic Besov and sequence spaces. Studia
Math. 110 (1994), 169–189. Zbl 0810.41010 MR 1279990
[14] M. B. Marcus and J. Rosen, p-variation of the local times of symmetric stable processes and of Gaussian processes with stationary increments. Ann. Probab. 20 (1992),
1685–1713. Zbl 0762.60069 MR 1188038
[15] J. Rosen, Second order limit laws for the local times of stable processes. In Se´minaire
de Probabilite´s, XXV, Lecture Notes in Math. 1485, Springer-Verlag, Berlin 1991,
407–424. Zbl 0758.60078 MR 1187796
[16] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional integrals and derivatives.
Gordon and Breach Science Publishers, Yverdon 1993. Zbl 0818.26003 MR 1347689
[17] H. F. Trotter, A property of Brownian motion paths. Illinois J. Math. 2 (1958),
425–433. Zbl 0117.35502 MR 0096311
[18] T. Yamada, On the fractional derivative of Brownian local times. J. Math. Kyoto
Univ. 25 (1985), 49–58. Zbl 0625.60090 MR 777245
[19] M. Yor, Le drap brownien comme limite en loi de temps locaux line´aires. In Seminar
on probability, XVII, Lecture Notes in Math. 986, Springer-Verlag, Berlin 1983,
89–105. Zbl 0514.60075 MR 770400
Received September 24, 2012
A. Sghir, Laboratoire de Mode´lisation Stochastique et De´terministe et URAC 04, Faculte´
des Sciences, B.P. 717, 6000 Oujda, Maroc
E-mail: semastai@hotmail.fr
H. Ouahhabi, Laboratoire de Mode´lisation Stochastique et De´terministe et URAC 04,
Faculte´ des Sciences, B.P. 717, 6000 Oujda, Maroc
E-mail: ahanae31@gmail.com

(AutoPDF V7 28/1/13 10:00) EMS (170 240mm) Tmath J-2715 PMS, 69:4 PMU: IDP[KN/V] 23/01/2013 pp. 321–339 2715_69-4_04 (p. 339)


Article4_Sghir_Aissa.pdf - page 1/19
 
Article4_Sghir_Aissa.pdf - page 2/19
Article4_Sghir_Aissa.pdf - page 3/19
Article4_Sghir_Aissa.pdf - page 4/19
Article4_Sghir_Aissa.pdf - page 5/19
Article4_Sghir_Aissa.pdf - page 6/19
 




Télécharger le fichier (PDF)


Article4_Sghir_Aissa.pdf (PDF, 166 Ko)

Télécharger
Formats alternatifs: ZIP



Documents similaires


article4 sghir aissa
article14 sghir aissa
article13 sghir aissa
article10 sghir aissa
article9 sghir aissa
article8 sghir aissa

Sur le même sujet..