Annales Mathématiques Blaise Pascal

Gilles Carron

ANNALES MATHÉMATIQUES
BLAISE PASCAL
Gilles Carron
Harmonic functions on Manifolds whose large spheres are
small.
Volume 23, no 2 (2016), p. 249-261.
<http://ambp.cedram.org/item?id=AMBP_2016__23_2_249_0>
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Annales mathématiques Blaise Pascal 23, 249-261 (2016)
Harmonic functions on Manifolds whose large
spheres are small.
Gilles Carron
Abstract
We study the growth of harmonic functions on complete Riemannian manifolds
where the extrinsic diameter of geodesic spheres is sublinear. It is an generalization
of a result of A. Kasue. Our estimates also yields a result on the boundedness of
the Riesz transform.
Résumé
On étudie la croissance des fonctions harmoniques sur les variétés rieman-
niennes complètes dont le diamètre des grandes sphères géodésiques croît sous
linéairement. Il s’agit d’une généralisation de travaux de A. Kasue. Nous obtenons
aussi un résultat de continuité pour la transformée de Riesz
1. Introduction
When (M, g) is a complete Riemannian manifold with non negative Ricci
curvature, S-Y. Cheng and S-T. Yau have proven that any harmonic func-
tion h : M → R satisfies the gradient estimate [4]:
sup
z∈B(x,R)
|dh|(z) ≤ C(n)
R
sup
z∈B(x,2R)
|h(z)|.
This result implies that such a manifold can not carry non constant har-
monic function h : M → R with sublinear growth:
|h(x)| = o
(
d(o, x)
)
, d(o, x)→ +∞ .
A celebrated conjecture of S-T. Yau predicted the finite dimensionality of
the space of harmonic functions with polynomial growth on a complete
I’m partially supported by the grants ACG: ANR-10-BLAN 0105 and GTO: ANR-12-
BS01-0004.
Keywords: Poincaré inequality, harmonic function, Riesz transform.
249
Gilles Carron
Riemannian manifold with non negative Ricci curvature:
Hν(M, g) =
{
h ∈ C2(M) , ∆gh = 0, |h(x)| = O
(
dν(o, x)
)}
.
This conjecture has been proven by T. Colding and B. Minicozzi in a much
more general setting.
We say that a complete Riemannian manifold (Mn, g) satisfies the dou-
bling condition if there is a constant ϑ such that for any x ∈M and radius
R > 0:
volB(x, 2R) ≤ ϑ volB(x,R).
If B ⊂ M is a geodesic ball, we will use the notation r(B) for the radius
of B and κB for the ball concentric to B and with radius κr(B). And if
f is an integrable function on a subset Ω ⊂ M , we will note fΩ its mean
over Ω:
fΩ =
1
vol Ω
∫
Ω
f.
We say that a complete Riemannian manifold (Mn, g) satisfies the
scaled (L2) Poincaré inequality if there is a constant µ such that for any
ball B ⊂M and any function ϕ ∈ C1(2B):
‖ϕ− ϕB‖2L2(B) ≤ µ r
2(B)‖dϕ‖2L2(2B) .
Theorem ([5]). If (M, g) is a complete Riemannian manifold that is dou-
bling and that satisfies the scaled Poincaré inequality then for any ν, the
space of harmonic function of polynomial growth of order ν has finite di-
mension:
dimHν(M, g) < +∞.
It is well known that a complete Riemannian manifold with non negative
Ricci curvature is doubling and satisfies the scaled Poincaré inequality,
hence the Yau’s conjecture is true.
The proof is quantitative and gives a precise estimation of the dimension
of the space of harmonic functions with polynomial growth of order ν. In
fact, the condition on the Poincaré inequality can be weakened and the
result holds on a doubling manifold (M, g) that satisfies the mean value
estimation [6, 11]: for any harmonic function h defined over a geodesic
ball 3B:
sup
x∈B
|h(x)| ≤ Cvol 2B
∫
2B
|h|.
An example of Riemannian manifold satisfying the above condition are
Riemannian surfaces of revolution (R2, gγ) where γ ∈ (0, 1] and such that
250
Harmonic functions on Manifolds
on R2 \ {0} ' (0,∞)× S1 we have gγ = (dr)2 + fγ(r)2(dθ)2, where for all
r > 1: fγ(r) = rγ (see [8, Proposition 4.10]). Using the new variable
ρ(r) = exp
(∫ r
1
ds
fγ(s)
)
,
we see that this metric is conformal to the Euclidean metric (dρ)2+ρ2(dθ)2
on R2. In dimension 2, the Laplace equation is conformally invariant hence
harmonic functions on (R2, gγ) are harmonic functions on the Euclidean
space. We know that any harmonic function h on R2 such that for h =
O (ρα) for some α < 1 is necessary constant. Hence we see that when
γ ∈ (0, 1), any harmonic function h on (R2, gγ) satisfying for some ε > 0:
h(x) = O
(
eCr
1−γ−ε)
is necessary constant. In particular, a harmonic function with polynomial
growth is constant.
In [9, 10], A. Kasue has shown that this was a general result for manifold
whose Ricci curvature satisfies a quadratic decay lower bound and whose
geodesic spheres have sublinear growth (see also [13] for a related results):
Theorem 1.1. If (M, g) is complete Riemannian manifold with a based
point o whose Ricci curvature satisfies a quadratic decay lower bound:
Ricci ≥ − κ
2
d2(o, x)g ,
and whose geodeosic sphere have sublinear growth:
diam ∂B(o,R) = o(R) , R→ +∞,
then any harmonic function with polynomial growth is constant.
Following A. Grigor’yan and L. Saloff-Coste [8], we say that a ball
B(x, r) is remote (from a fixed point o) if
3r ≤ d(o, x).
Our first main result is a refinement of A. Kasue’s result when the hypoth-
esis of the Ricci curvature is replaced by a scaled Poincaré inequality for
remote ball: there is a constant µ such that all remote balls B = B(x, r)
satisfy a scaled Poincaré inequality:
∀ϕ ∈ C1(2B) : ‖ϕ− ϕB‖2L2(B) ≤ µr
2‖dϕ‖2L2(2B).
251
Gilles Carron
Theorem 1.2. Let (M, g) be a complete Riemannian manifold whose re-
mote balls satisfy the scaled Poincaré inequality and assume that geodesic
spheres have sublinear growth:
diam ∂B(o,R) = o(R) , R→ +∞.
If h : M → R is a harmonic function such that for IR :=
∫
B(o,R) h
2:
lim
R→+∞
log(IR)
diam ∂B(o,R/4)
R
= 0,
then h is constant.
For instance, on such a manifold, a harmonic function h : M → R sat-
isfying:
|h(x)| ≤ Cd(o, x)ν (volB(o, d(o, x)))−
1
2
is constant.
Moreover, consider (M, g) be a complete Riemannian manifold satisfy-
ing the hypothesis of the Theorem 1.2 and assume that for some γ ∈ (0, 1),
the diameter of geodesic spheres satisfies
diam ∂B(o,R) ≤ CRγ .
If h : M → R is a harmonic function satisfying, for some positive constants
C and ε,
|h(x)| ≤ CeCd(o,x)1−γ−ε volB(o, d(o, x))−
1
2 ,
then h is constant.
Remark 1.3. Our result is a slight improvement of the Theorem 1.1. Indeed
if (M, g) is a complete Riemannian manifold with a based point o whose
Ricci curvature satisfies a quadratic decay lower bound:
Ricci ≥ − κ
2
d2(o, x)g .
On a remote ball B ⊂M , the Ricci curvature is bounded from below
Ricci ≥ − κ
2
4r2(B)g ,
hence according to [3, inequality (4.5)], we have the Poincaré inequality:
∀ϕ ∈ C1(B) : ‖ϕ− ϕB‖2L2(B) ≤ C(n)r
2(B)‖dϕ‖2L2(B).
252
Harmonic functions on Manifolds
Moreover a slight variation of the Bishop–Gromov comparison theorem
(see for instance [12, Lemma 3.1]) implies that (M, g) has polynomial
growth: there is some N > 0 such that for all r > 1:
volB(o, r) ≤ CrN .
A by product of the proof will imply that on the class of manifold
considered by A. Kasue, the doubling condition implies a Cheng–Yau’s
estimate for for the gradient of harmonic function:
Theorem 1.4. Let (Mn, g) be a complete Riemannian manifold that is
doubling and whose Ricci curvature satisfies a quadratic decay lower bound.
Assume that the diameter of geodesic sphere has a sublinear growth
diam ∂B(o,R) = sup
x,y∈∂B(o,R)
d(x, y) = o(R) .
Then there is a constant C such that for any geodesic ball B ⊂ M and
any harmonic function h : 3B → R
sup
x∈B
|dh|2(x) ≤ Cvol 2B
∫
2B
|dh|2.
This result has consequences for the boundness of the Riesz transform.
When (Mn, g) is a complete Riemannian manifold with infinite volume,
the Green formula and the spectral theorem yield the equality:
∀ f ∈ C∞0 (M) ,
∫
M
|df |2g dvolg = 〈∆f, f〉L2 =
∫
M
∣∣∣∆ 12 f ∣∣∣2 dvolg .
Hence the Riesz transform
R := d∆−
1
2 : L2(M)→ L2(T ∗M)
is a bounded operator. It is well known [14] that on a Euclidean space,
the Riesz transform has a bounded extension R : Lp(Rn) → Lp(T ∗Rn)
for every p ∈ (1,+∞). Also according to D. Bakry, the same is true on
manifolds with non-negative Ricci curvature [2]. As it was noticed in [7,
Section 5], in the setting of the Theorem 1.4, the analysis of A. Grigor’yan
and L. Saloff-Coste [8] implies a scaled L1-Poincaré inequality: there is a
constant C such that any balls B = B(x, r) satisfies:
∀ϕ ∈ C1(2B) : ‖ϕ− ϕB‖L1(B) ≤ Cr2‖dϕ‖L1(2B) .
And according to the analysis of P. Auscher and T. Coulhon [1] (see also
the explanations in [7, Section 5]), the Theorem 1.4 implies:
253
Gilles Carron
Corollary 1.5. Under the assumption of the Theorem 1.4, the Riesz
transform is bounded on Lp for every p ∈ (1,+∞).
Acknowledgment
I thank Hans-Joachim Hein: this project had begun by a very fruitful
discussion where we proved together the key Lemma 2.1. I’m also grateful
to the referee for her/his very useful comments who improve the original
manuscript.
2. Absence of harmonic functions
Recall that when (M, g) is a complete Riemannian manifold and o ∈ M ,
we say that a geodesic ball B(x, r) is remote (from o) if
3r ≤ d(o, x).
We define ρ the radius function by ρ(t) = inf
x∈∂B(o,t)
max
y∈∂B(o,t)
d(x, y), we
have
ρ(t) ≤ diam ∂B(o, t) ≤ 2ρ(t).
2.1. An inequality
Lemma 2.1. Let (M, g) be a complete Riemannian manifold whose all
remote balls B = B(x, r) satisfy a scaled Poincaré inequality:
∀ϕ ∈ C1(2B) : ‖ϕ− ϕB‖2L2(B) ≤ µ r
2(B)‖dϕ‖2L2(2B) .
Then there are constants C > 0 and κ ∈ (0, 1) depending only on µ such
that if for some ε ∈ (0, 1/12):
∀ r ∈ [R, 2R] : ρ(r) ≤ εr ,
then ∫
B(o,R)
|dh|2 ≤ C κ
1
ε
∫
B(o,2R)
|dh|2 ,
for any harmonic function h on B(o, 2R).
Proof. Let r ∈ [R + 4εR, 2R − 4εR], our hypothesis implies that there is
some x ∈ ∂B(o, r) such that
B(o, r + εR) \B(o, r) ⊂ B(x, εR+ εr).
254
Harmonic functions on Manifolds
Let h : B(o, 2R) → R be a harmonic function and c ∈ R a real number.
We use the Lipschitz function:
χ(x) =

1 on B(o, r)
r+εR−d(o,x)
εR on B(o, r + εR) \B(o, r)
0 outside B(o, r + εR)
Then integrating by parts and using the fact that h is harmonic we get∫
M
χ2|d(h−c)|2 +2χ(h−c)〈dχ, d(h−c)〉 =
∫
M
〈d((h−c)χ2), d(h−c)〉 = 0.
So that we have:∫
M
|d(χ(h− c))|2
=
∫
M
χ2|d(h− c)|2 + 2χ(h− c)〈dχ, d(h− c)〉+ (h− c)2|dχ|2
=
∫
B(o,r+εR)
(h− c)2|dχ|2 ,
and hence∫
B(o,r)
|dh|2 ≤
∫
B(o,r+εR)
|d(χ(h− c))|2 =
∫
B(o,r+εR)
(h− c)2|dχ|2
≤ 1
ε2R2
∫
B(o,r+εR)\B(o,r)
(h− c)2
≤ 1
ε2R2
∫
B(x,εR+εr)
(h− c)2.
The hypothesis that ε ≤ 1/12 implies that the ball B(x, εR+εr) is remote,
hence if we choose
c = hB(x,ε(R+r)) =
1
volB(x, , ε(R+ r))
∫
B(x,,ε(R+r))
h,
then the Poincaré inequality and the fact that r +R ≤ 3R imply:∫
B(o,r)
|dh|2 ≤ 9µ
∫
B(x,6εR)
|dh|2.
But we have:
B(x, 6εR) ⊂ B(o, r + 6εR) \B(o, r − 6εR) ,
255
Gilles Carron
hence we get∫
B(o,r−6εR)
|dh|2 ≤ 9µ
∫
B(o,r+6εR)\B(o,r−6εR)
|dh|2.
And for all r ∈ [R, 2R− 12εR] we get:∫
B(o,r)
|dh|2 ≤ 9µ1 + 9µ
∫
B(o,r+12εR)
|dh|2.
We iterate this inequality and get∫
B(o,R)
|dh|2 ≤
( 9µ
1 + 9µ
)N ∫
B(o,2R)
|dh|2,
provide that N12εR ≤ R; hence the result with C = 1 + 19µ and
κ =
( 9µ
1 + 9µ
) 1
12
. 
2.2. Harmonic functions with polynomial growth
We can now prove the following extension of Kasue’s results:
Theorem 2.2. Let (M, g) be a complete Riemannian manifold whose all
remote balls B = B(x, r) satisfy a scaled Poincaré inequality:
∀ϕ ∈ C1(2B) : ‖ϕ− ϕB‖2L2(B) ≤ µr
2(B)‖dϕ‖2L2(2B).
Assume that balls anchored at o have polynomial growth:
volB(o,R) ≤ CRµ
and that geodesic spheres have sublinear diameter growth:
lim
t→+∞
ρ(t)
t
= 0.
Then any harmonic function on (M, g) with polynomial growth is constant.
Proof. Let h : M → R be a harmonic function with polynomial growth:
h(x) ≤ C(1 + d(o, x))ν .
We will define
ER =
∫
B(o,R)
|dh|2 and ε(r) = sup
t≥r
ρ(t)
t
.
256
Harmonic functions on Manifolds
We remark first that using the cut off function ξ defined by
ξ(x) =

1 on B(o,R)
2R−d(o,x)
R on B(o, 2R) \B(o,R)
0 outside B(o, 2R).
We obtain
ER ≤
∫
B(o,2R)
|d(ξh)|2 =
∫
B(o,2R)
|h|2|dξ|2 ≤ CR2ν+µ−2. (2.1)
If we iterate the inequality obtained in the Lemma 2.1, we get for all R
such that ε(R) ≤ 1/12:
ER ≤ C`κ
∑`−1
j=0
1
ε(2jR) E2`R .
Using the estimation (2.1), we get
ER ≤ C(R)e
`
(
logκ
`
∑`−1
j=0
1
ε(2jR)
+log(2)(2ν+µ−2)+logC
)
. (2.2)
But the Cesaro theorem convergence implies that:
lim
`→+∞
1
`
`−1∑
j=0
1
ε (2jR) = +∞,
hence if we let `→ +∞ in the inequality (2.2) we get ER = 0 and this for
all sufficiently large R, hence h is constant. 
2.3. An extension
A slight variation of the arguments yields the following extension, which
implies the Theorem 1.2:
Theorem 2.3. Let (M, g) be a complete Riemannian manifold whose all
remote balls B = B(x, r) satisfies a scaled Poincaré inequality:
∀ϕ ∈ C1(2B) : ‖ϕ− ϕB‖2L2(B) ≤ µr
2(B)‖dϕ‖2L2(2B).
Assume that the geodesic spheres have sublinear diameter growth:
lim
t→+∞
ρ(t)
t
= 0 and let ε(r) = sup
t≥r
ρ(t)
t
.
257
Gilles Carron
Let h : M → R be a harmonic function such that IR =
∫
B(o,R)
h2 satisfies
log I(R) = o
(∫ R/4
1
dt
tε(t)
)
,
then h is constant.
Proof. Indeed, the above argumentation shows that if R is large enough
then
ER ≤M(`, R)I(2`+1R) 4−`R−2,
where
log(M(`, R)) = log
(
C`κ
∑`−1
j=0
1
ε(2jR)
)
= ` logC + log κ
`−1∑
j=0
1
ε (2jR)
 .
But
`−1∑
j=0
1
ε (2jR) ≥
1
log 2
`−1∑
j=0
∫ 2jR
2j−1R
dt
tε (t) ≥
1
log 2
∫ 2`−1R
R/2
dt
tε (t) .
Hence we get the inequality:
logER ≤ log I
(
2`+1R
)
− ` log(4) + ` logC + log κlog 2
∫ 2`−1R
R/2
dt
tε (t) − 2 logR.
It is then follows from the above inequality and the Cesaro theorem con-
vergence that h is constant. 
3. Lipschitz regularity of harmonic functions
We are going to prove that a Lipschitz regularity for harmonic function
analogous to the Cheng–Yau gradient inequality:
Theorem 3.1. Let (Mn, g) be a complete Riemannian manifold that sat-
isfy the doubling condition: there is a constant ϑ such that for any x ∈M
and radius R > 0:
volB(x, 2R) ≤ ϑ volB(x,R)
258
Harmonic functions on Manifolds
and assume moreover that the Ricci curvature satisfies a quadratic decay
lower bound
Ricci ≥ − κ
2
r2(x)g ,
where for a fixed point o ∈M : r(x) := d(o, x).
Assume that the diameters of geodesic spheres growth slowly
diam ∂B(o,R) = sup
x,y∈∂B(o,R)
d(x, y) = o(R).
Then there is a constant C such that for any geodesic ball B ⊂ M and
any harmonic function h : 3B → R
sup
x∈B
|dh|2(x) ≤ Cvol 2B
∫
2B
|dh|2.
Proof. According to [7, Proposition 5.3], we need only to show that there
is a constant C such that if R > 0 and if h : B(o, 2R)→ R is a harmonic
function then for any s ≤ σ ≤ R :
1
volB(o, s)
∫
B(o,s)
|dh|2 ≤ CvolB(o, σ)
∫
B(o,σ)
|dh|2. (3.1)
According to the Remark 1.3, we can apply the Lemma 2.1: for all η > 0,
there is a R0 > 0 such that for all R ≥ R0, then∫
B(o,R)
|dh|2 ≤ η
∫
B(o,2R)
|dh|2.
Hence for all R ≥ R0 :
1
volB(o,R)
∫
B(o,R)
|dh|2 ≤ η ϑ 1volB(o, 2R)
∫
B(o,2R)
|dh|2.
Choose η = ϑ−1, then we get that for all R0 ≤ s ≤ σ ≤ R :
1
volB(o, s)
∫
B(o,s)
|dh|2 ≤ ϑvolB(o, σ)
∫
B(o,σ)
|dh|2.
The Ricci curvature being bounded on B(o, 3R0), the Cheng and Yau
gradient estimate yields a constant B such that for all x ∈ B(o,R0):
|dh|2(x) ≤ BvolB(o, 2R0)
∫
B(o,2R0)
|dh|2.
Hence the estimate (3.1) holds with C = max{Bϑ, ϑ}. 
259
Gilles Carron
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Gilles Carron
Laboratoire de Mathématiques Jean
Leray (UMR 6629), Université de Nantes,
2, rue de la Houssinière, B.P. 92208,
44322 Nantes Cedex 3, France
Gilles.Carron@univ-nantes.fr
261


Harmonic functions on Manifolds whose large spheres are small.

Soufiane Aazizi

ANNALES MATHÉMATIQUES
BLAISE PASCAL
Soufiane Aazizi
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Variation of the Subfractional Brownian Motion
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Annales mathématiques Blaise Pascal 23, 141-150 (2016)
A Simple Proof of Berry–Esséen Bounds for the
Quadratic Variation of the Subfractional
Brownian Motion
Soufiane Aazizi
Abstract
We give a simple technic to derive the Berry–Esséen bounds for the quadratic
variation of the subfractional Brownian motion (subfBm). Our approach has two
main ingredients: (i) bounding from above the covariance of quadratic variation of
subfBm by the covariance of the quadratic variation of fractional Brownian motion
(fBm); and (ii) using the existing results on fBm in [1, 2, 4]. As a result, we obtain
simple and direct proof to derive the rate of convergence of quadratic variation of
subfBm. In addition, we also improve this rate of convergence to meet the one of
fractional Brownian motion in [2].
Résumé
Nous donnons une technique simple pour calculer les limites Berry–Esséen pour
la variation quadratique du mouvement Brownien subfractional (subfBm). Notre
approche a deux ingrédients principaux : (i) majorer la covariance des variations
quadratiques de subfBm par la covariance de la variation quadratique du mouve-
ment Brownien fractionnaire (FBM) ; et (ii) utiliser les résultats existants sur fBm
dans [1, 2, 4]. En conséquence, nous obtenons une simple et directe preuve pour
calculer le taux de convergence des variations quadratiques de subfBm. En outre,
nous améliorons aussi ce taux de convergence pour obtenir ceux du mouvement
Brownien fractionnaire dans [2].
1. Introduction and preliminaries
The subfractional Brownian motion (subfBm in short) S = (St, t ≥ 0)
with parameters H ∈ (0, 1), is defined on some probability space (Ω,F , P )
(Here, and everywhere else, we do assume that F is the sigma-field gen-
erated by S). This means that S is a centered Gaussian process with
Keywords: Fractional Brownian motion, Malliavin calculus, Kolmogorov distance, Sub-
fractional Brownian motion, Stein method, Quadratic variation.
141
S. Aazizi
covariance
E[SsSt] = RH(s, t) = s2H + t2H−
1
2
[
(s+ t)2H + |t− s|2H
]
, s, t ≥ 0 (1.1)
The following result, proved in [6], states the convergence of quadratic
variation of subfractional Brownian motion to a centered reduced normal
variable, also provides its rate of convergence. Hereafter, we denote
Zn =
n−1∑
k=0
n2H
[
(S(k+1)/n − Sk/n)2 − V ar
(
S(k+1)/n − Sk/n
)]
, n ≥ 1.
Theorem 1.1 ([6]). Let N be a standard Gaussian random variable (N ∼
N(0, 1)) and suppose that H ∈ (0, 34 ]. Then
Zn
V ar(Zn) converges in distribu-
tion to N and the following Berry–Esséen bounds hold for every n ≥ 1,
dKol
(
Zn√
V ar(Zn)
, N
)
≤ cH ×

n−
1
2 , H ∈
(
0, 12
)
,
n2H−
3
2 , H ∈
[
1
2 ,
3
4
)
,
1√
logn
, H = 34 ,
where cH is a constant depending only on H.
In [6], the proof uses Stein method and Malliavin calculus, based on the
idea developed in [1, 4] for the case of fractional Brownian motion (fBm in
short), which leads to the same rate of convergence. Recently, [2] used the
convolution product of two sequences which improve clearly the rate of
convergence of the fBm. The natural question imposes itself, it is possible
to obtain a rate of convergence of subfBm similar to the one proved by [2]
for the fBm?
The goal of this paper, is to improve the rate of convergence of the
subfBm so that we have at least the same one as the fBm. To perform our
calculation, we will mainly follow the idea taken from [2]. With the proof
of [4] and [2] in hand, we will show how we can retrieve the result of [6],
and how we can improve this result to reach the one of fBm in [2].
For the case of Hurst parameter H > 3/4, we think that it deserves an
entire work, the quadratic variation will converge to a Hermite random
variable similarly to the fractional Brownian Motion [3].
142
A Simple Proof for the Subfractional Brownian Motion
We claim the main result of this paper:
Theorem 1.2. Let N ∼ N (0, 1), there exist a constant cH depending only
on H, such that for every n ≥ 1,
dKol
(
Zn√
V ar(Zn)
, N
)
≤ cH ×

1√
n
, H ∈
(
0, 58
)
,
(logn)3/2√
n
, H = 58 ,
n4H−3, H ∈
(
5
8 ,
3
4
)
,
1
logn, H =
3
4 .
We recall briefly some important tools of Malliavin calculus used
throughout this paper. We mean by H a real separable Hilbert space de-
fined as follows:
(i) denote by E the set of all R-valued functions on [0,∞),
(ii) define H as the Hilbert space obtained by closing E with respect
to the scalar product
〈1[0,s], 1[0,t]〉H = RH(s, t).
For every q ≥ 1, letHq be the qth Wiener chaos ofX, that is, the closed lin-
ear subspace of L2(Ω) generated by the random variables {Hq (X (h)) , h ∈
H, ‖h‖H = 1}, where Hq is the qth Hermite polynomial defined as Hq(x) =
(−1)q e
x2
2
dq
dxq
(e−
x2
2 ). The mapping Iq(h⊗q) = q!Hq (X (h)) provides a lin-
ear isometry between the symmetric tensor product Hq (equipped with
the modified norm ‖ · ‖Hq =
√
q! ‖ · ‖H⊗q) and Hq. Specifically, for all
f, g ∈ Hq and q ≥ 1, one has
E
[
Iq(f)Iq(g)
]
= q! 〈f, g〉H⊗q . (1.2)
Let {ek, k ≥ 1} be a complete orthonormal system in H. Given f ∈ Hp
and g ∈ Hq, for every r = 0, . . . , p ∧ q, the rth contraction of f and g is
the element of H⊗(p+q−2r) defined as
f ⊗r g =
∞∑
i1=1,...,ir=1
〈f, ei1 ⊗ · · · ⊗ eir〉H⊗r ⊗ 〈g, ei1 ⊗ · · · ⊗ eir〉H⊗r .
143
S. Aazizi
In particular, note that f ⊗0 g = f ⊗ g and when p = q, that f ⊗p g =
〈f, g〉H⊗p . Since, in general, the contraction f ⊗r g is not necessarily sym-
metric, we denote its symmetrization by f⊗̃rg ∈ H(p+q−2r). The follow-
ing formula is useful to compute the product of such multiple integrals: if
f ∈ Hp and g ∈ Hq, then
Ip(f) Iq(g) =
p∧q∑
r=0
r!
(
p
r
)(
q
r
)
Ip+q−2r(f⊗̃rg). (1.3)
We will use the notation δk/n = 1[k/n,(k+1)/n], and we send the reader to [5]
for more details on Malliavin calculus.
Now, by self-similarity property of S and (1.1) we deduce for k ≤ l
n2H〈δk/n, δl/n〉H = n2HE
((
S(k+1)/n − Sk/n
) (
S(l+1)/n − Sl/n
))
= E ((Sk+1 − Sk)(Sl+1 − Sl))
= (k + l + 1)2H − 12(k + l + 2)
2H − 12(k + l)
2H
− |l − k|2H + 12 |l − 1− k|
2H + 12 |l + 1− k|
2H
= 12ρ(l − k)−
1
2ρ(l + k + 1),
where ρ(r) = |r + 1|2H + |r − 1|2H − 2|r|2H , r ∈ Z.
On the other hand, we have the relation, for any k, l ∈ N
∣∣∣n2H〈δk/n, δl/n〉H∣∣∣ = 12 |ρ(l − k)− ρ(l + k + 1)|
≤ |ρ(l − k)|. (1.4)
Indeed, we shall prove that the application r ≥ 1 7→ |ρ(r)| is nonincreasing.
We can write the function ρ as
ρ(r) = f(r)− f(r − 1),
where f(r) := (r + 1)2H − r2H . Hence, we have two cases:
144
A Simple Proof for the Subfractional Brownian Motion
Case 1. H > 12 :
f ′(r) = 2H
(
(r + 1)2H−1 − r2H−1
)
> 0 =⇒ f ↗=⇒ ρ(r) > 0,∀r ≥ 1;
f ′′(r) = 2H(2H − 1)
(
(r + 1)2H−2 − r2H−2
)
< 0 =⇒ f ′ ↘=⇒ ρ′ < 0
=⇒ ρ↘ on [1,+∞[.
Hence, |ρ| is nonincreasing on [1,+∞[.
Case 2. H < 12 :
f ′(r) = 2H
(
(r + 1)2H−1 − r2H−1
)
< 0 =⇒ f ↘=⇒ ρ(r) < 0,∀r ≥ 1;
f ′′(r) = 2H(2H − 1)
(
(r + 1)2H−2 − r2H−2
)
> 0 =⇒ f ′ ↗=⇒ ρ′ > 0
=⇒ ρ↗ on [1,+∞[.
Hence, |ρ| is nonincreasing on [1,+∞[.
Moreover, |ρ(0)| ≥ |ρ(1)| for any H ∈ [0, 1]. As consequence, |ρ| is
nonincreasing on [0,+∞[, combining this with the fact that ρ is symmetric,
i.e. ρ(r) = ρ(−r), we have for any l, k ∈ N,
|ρ(l − k)− ρ(l + k + 1)| ≤ |ρ(l − k)|+ |ρ(l + k + 1)|
≤ 2|ρ(l − k)|. (1.5)
With inequality (1.4) in hand, it is now straightforward to obtain Theo-
rem 1.2. Hence, we can write the quadratic variation of S, with respect to
a subdivision πn = {0 < 1n <
2
n < . . . < 1} of [0, 1], as follows
Zn =
n−1∑
k=0
[
n2H
(
S(k+1)/n − Sk/n
)2
− 1 + 12ρ(2k + 1)
]
=
n−1∑
k=0
[
n2H
(
I1(δk/n)
)2
− 1 + 12ρ(2k + 1)
]
= I2

n−1∑
k=0
n2Hδ⊗2k/n︸ ︷︷ ︸
gn
 . (1.6)
145
S. Aazizi
Thus, we can write the correct renormalization of Zn as follows,
Vn =
Zn√
V ar(Zn)
= I2(gn)√
V ar(Zn)
. (1.7)
2. Proof of Theorem 1.2
In the first step, let study the convergence of V ar(Zn)n and
V ar(Zn)
n log n . There-
fore, we have
V ar(Zn)
n
= n−1E[I22 (gn)] = 2‖gn‖2H⊗2
= 2n4H−1
n−1∑
k,l=0
〈δ⊗2k/n, δ
⊗2
l/n〉H⊗2 = 2n
4H−1
n−1∑
k,l=0
〈δk/n, δl/n〉2H
= 12n
n−1∑
k,l=0
|ρ(l − k)− ρ(l + k + 1)|2
= 12n
n−1∑
k,l=0
ρ2(l − k) + 12n
n−1∑
k,l=0
ρ2(l + k + 1)
− 1
n
n−1∑
k,l=0
ρ(l − k) ρ(l + k + 1).
On the other hand, we have
n−1∑
k,l=0
ρ2(l + k + 1) =
n−1∑
l=0
n+l∑
j=l+1
ρ2(j)
=
2n−1∑
j=1
(j−1)∧(n−1)∑
l=0∨(j−n)
ρ2(j)
=
n∑
j=1
ρ2(j)
j−1∑
l=0
+
2n−1∑
j=n+1
ρ2(j)
n−1∑
l=j−n
=
n∑
j=1
jρ2(j) +
2n−1∑
j=n+1
(2n− j)ρ2(j)
= An,1 +An,2. (2.1)
146
A Simple Proof for the Subfractional Brownian Motion
Assume now that H < 34 .
Thus, since
∑
r∈Z
ρ2(r) <∞, we obtain
lim
n→∞
1
n
An,1 = 0.
and
1
n
An,2 ≤
2n−1∑
j=n+1
ρ2(j) −→ 0, as n→∞.
Which leads to
lim
n→∞
1
n
n−1∑
k,l=0
ρ2(l + k + 1) = 0. (2.2)
According to the proof of [2, Theorem 5.6], we have
lim
n→∞
1
n
n−1∑
k,l=0
ρ2(l − k) =
∑
r∈Z
ρ2(r). (2.3)
Finally, by (2.3) and (2.2), together with Cauchy Schwartz inequality
1
n
n−1∑
k,l=0
|ρ(l − k)| |ρ(l + k + 1)| ≤
 1
n
n−1∑
k,l=0
ρ2(l − k)
 12
×
 1
n
n−1∑
k,l=0
ρ2(l + k + 1)
 12
−→ 0, as n→ 0. (2.4)
Combining (2.2), (2.3) and (2.4) we conclude that
lim
n→∞
V ar(Zn)
n
= 12
∑
r∈Z
ρ2(r). (2.5)
Assume now H = 34 . Following similar argument as above we have
V ar(Zn)
n log(n) =
1
n log(n)
n−1∑
k,l=0
ρ2(l − k) + 1
n log(n)
n−1∑
k,l=0
ρ2(l + k + 1)
− 2
n log(n)
n−1∑
k,l=0
ρ(l − k) ρ(l + k + 1).
147
S. Aazizi
Again from the proof of [2, Theorem 5.6], we have
lim
n→∞
1
n log(n)
n−1∑
k,l=0
ρ2(l − k) = 932 . (2.6)
From other side, we have ρ2(r) ∼ 964|r| as |r| → ∞. Implying in turn
from (2.1) that
n−1∑
k,l=0
ρ2(l + k + 1) =
n∑
j=1
jρ2(j) +
2n−1∑
j=n+1
(2n− j)ρ2(j)
≤ 964
 n∑
j=1
1 +
2n∑
j=n+1
(2n− j)1
j

≤ 964n
1 + 2n∑
j=n+1
1
j

≤ 964n
(
1 + n
n+ 1
)
Hence, we have
lim
n→∞
1
n log(n)
n−1∑
k,l=0
ρ2(l + k + 1) = 0. (2.7)
Similarly to (2.4), we obtain by (2.6), (2.7) and Cauchy–Schwartz inequal-
ity
lim
n→∞
1
n log(n)
n−1∑
k,l=0
ρ(l − k) ρ(l + k + 1) = 0. (2.8)
Combining (2.6), (2.7) and (2.8) we deduce that
V ar(Zn)
n log(n) =
9
64 . (2.9)
Let us now derive the explicit bounds. From (1.6), multiplication for-
mula (1.3) and the fact that E‖DZn‖2H = 2V ar(Zn), we obtain
1
2‖DVn‖
2
H − 1 =
2n4H
V ar(Zn)
n−1∑
k,l=0
I2(δk/n⊗̃δl/n) 〈δk/n, δl/n〉H.
148
A Simple Proof for the Subfractional Brownian Motion
It follows by (1.4) that
E
[(1
2 ‖DVn‖
2
H − 1
)2]
= 4n
8H
V ar2(Zn)
E

 n−1∑
k,l=0
I2(δk/n⊗̃δl/n) 〈δk/n, δl/n〉H
2

= 8n
8H
V ar2(Zn)
n−1∑
i,j,k,l=0
〈δi/n, δj/n〉H 〈δk/n, δl/n〉H 〈δi/n⊗̃δj/n, δk/n⊗̃δl/n〉H⊗2
= 4n
8H
V ar2(Zn)
n−1∑
i,j,k,l=0
〈δi/n, δj/n〉H 〈δk/n, δl/n〉H
(
〈δi/n, δk/n〉H 〈δj/n, δl/n〉H
+ 〈δi/n, δl/n〉H〈δj/n, δk/n〉H
)
= 8n
8H
V ar2(Zn)
n−1∑
i,j,k,l=0
〈δi/n, δj/n〉H 〈δi/n, δk/n〉H 〈δk/n, δl/n〉H 〈δj/n, δl/n〉H
≤ 8n
2
V ar2(Zn)
1
n2
n−1∑
i,j,k,l=0
|ρ(i− j)| · |ρ(i− k)| · |ρ(k − l)| · |ρ(j − l)|.
(2.10)
Then, combining the convergence (2.5) and (2.9) together with inequal-
ity (2.10), the rest of the proof is now similar to the one of Theorem 5.6
in [2]. 
Remark 2.1. To retrieve the result of Tudor [6], we start from equality (2.9)
and we follow the same steps as in the proof of Theorem 4.1 in [4].
Acknowledgment
The author would like to express his sincere appreciation to his advisor
Professor Ouknine Youssef and Professor Khalifa Essebay for their impor-
tant remarks and key idea of this paper. The author would like to thank
also the anonymous referee for his valuable comments and suggestions to
improve the quality of the current paper.
149
S. Aazizi
References
[1] J.-C. Breton & I. Nourdin – “Error bounds on the non-normal
approximation of hermite power variations of fractional Brownian mo-
tion”, Electron. Commun. Probab. 13 (2008), p. 482–493.
[2] I. Nourdin – “Lectures on Gaussian approximations with Malliavin
calculus”, in Séminaire de Probabilités XLV, Lecture Notes in Mathe-
matics, vol. 2078, Springer, 2013, p. 3–89.
[3] I. Nourdin, D. Nualart & C. A. Tudor – “Central and non-
central limit theorems for weighted power variations of fractional brow-
nian motion”, Ann. Inst. Henri Poincaré, Probab. Stat. 46 (2010),
no. 4, p. 1055–1079.
[4] I. Nourdin & G. Peccati – “Stein’s method on Wiener chaos”,
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[5] D. Nualart – The Malliavin calculus and related topics, Probability
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[6] C. Tudor – “Berry-Esséen bounds and almost sure clt for the qua-
dratic variation of the sub-fractional Brownian motion”, J. Math. Anal.
Appl. 375 (2011), no. 2, p. 667–676.
Soufiane Aazizi
Department of Mathematics, Faculty of
Sciences Semlalia Cadi Ayyad University,
B.P. 2390 Marrakesh, Morocco
aazizi.soufiane@gmail.com
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