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El e c t ro n ic

Jo ur

n a l o

f P

r o

b a b i_{l i t y} Vol. 15 (2010), Paper no. 9, pages 241–258.

Journal URL

http://www.math.washington.edu/~ejpecp/

Multidimensional Multifractal Random Measures

Rémi Rhodes and Vincent Vargas

Université Paris-Dauphine, Ceremade, CNRS UMR 7534, F-75016 Paris, France

e-mail:[email protected],

[email protected]

Abstract

We construct and study space homogeneous and isotropic random measures (MMRM) which generalize the so-called MRM measures constructed in[1]. Our measures satisfy an exact scale invariance equation (see equation (1) below) and are therefore natural models in dimension 3 for the dissipation measure in a turbulent flow.

Key words:Random measures, Multifractal processes.

AMS 2000 Subject Classification:Primary 60G57, 28A78, 28A80.

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1 Introduction

The purpose of this paper is to introduce a natural multidimensional generalization (MMRM) of the one dimensional multifractal random measures (MRM) introduced by Bacry and Muzy in[1]. The measures M we introduce are different from zero, homogeneous in space, isotropic and satisfy the following exact scale invariance relation: ifT denotes some cutoff parameter we will define below then the following equality in distribution holds for allλ <1:

(M(λA))_A_⊂_B(0,T) (l aw)

= λdeΩλ₍_M₍_A₎₎

A⊂B(0,T), (1)

whereB(0,T)is the euclidian ball of radiusTinRd andΩ_λis an infinitely divisible random variable independent of(M(A))_A_⊂_B(0,T). In fact, ifνt denotes the law ofΩe−t fort≥0, then it is

straightfor-ward to prove that the laws(νt)t≥0satisfy the following convolution property:

νt+t′=ν_t∗ν_t′ (2)

Therefore, one can find a Levy process(L_t)_t_≥₀such that, for each t,νt is the law ofLt.

Reciprocaly, consider a cutoff parameter T and a family of laws(νt)t≥0 satisfying (2), the normali-sation conditionRexνt(d x) =1 for all t and a technical assumption detailed in section 2. We can

construct a measureM different from zero, homogeneous in space, isotropic and which satisfies the exact scale invariance relation (1) for allλ <1 withΩ_λof lawνln(1/λ).

Let us note that a multi-dimensional generalization of MRM has already been proposed in the lit-terature ([5]). However, this generalization is not exactly scale invariant. Let us stress that to our knowledge equation (1) has never been studied mathematically. In dimension 1, MRM are non trivial (i.e. different from 0) homogeneous solutions to (1) for all λ < 1. In dimension d ≥ 2, MMRM are non trivial isotropic and homogeneous solutions to (1) for allλ < 1. If we consider a non negative random variable Y independent from M than the random measure(Y M(A))_A_⊂Rd is

also solution to (1) for allλ <1. This leads to the following open problem (unicity):

Open problem 1. Let(νt)t≥0 be a fixed family of probability measures which satisfy the convolution property (2), the normalisation conditionRexνt(d x) =1for all t and the non degeneracy condition

stated in proposition 2.9.1 (ifψis the Laplace transform ofν1, the condition is the existence ofε >0 such thatψ(1+ε)<dε). Consider two non trivial (i.e. different from0) homogeneous and isotropic random Radon measures M and M′on the Borelσ-algebraB(Rd). We suppose that there exists some cutoff parameter T such that M and M′satisfy (1) for allλ <1where the law ofΩ_λis given byνln(1/λ). Can one find (in a possibly extended probability space) a non negative and non trivial (i.e. different from0) random variable Y (Y′) independent of M (M′) such that the following equality in law holds:

(Y M(A))_A_⊂_B_(0,_T₎(l aw= () Y′M′(A))_A_⊂_B_(0,_T₎

The above open problem can be solved in the simple case where the family (νt)t≥0 is given by

νt=δ0for allt≥0 (deterministic scale invariance) and where there is no cutoff parameter. This is the content of the following lemma:

Lemma 1.1. Let M be a homogeneous random Radon measure on the Borelσ-algebra_B(Rd₎_{such that}

there existsα >0withE[M(B(0, 1))α]<∞. We suppose that the following equality in distribution holds for allλ <1:

(M(λA))_A_⊂Rd

(l aw)

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Then there exists some (random) constant C _≥ 0such that M = C_L a.s. where_L is the Lebesgue measure.

The proof of the above lemma is to be found in section 4.1.

The study of equation (1) is justified on mathematical and physical grounds. Before reviewing in some detail the physical motivation for constructing the MRMM measures and studying equation (1), let us just mention that equation (1) has recently been used in the probabilistic derivation of the KPZ (Knizhnik-Polyakov-Zamolodchikov) equation introduced initially in[12](see[6],[16]).

1.2 MRMM in dimension 3: a model for the energy dissipation in a turbulent flow

Equations similar to (1) were first proposed in [3] in the context of fully developed turbulence. In[3], the authors conjectured that the following relation should hold between the increments of the longitudinal velocityδv at two scalesl,l′<T, where T is an integral scale characteristic of the turbulent flow: ifPl(δv),Pl′(δv)denote the probability density functions (p.d.f.) of the longitudinal velocity difference between two points separated by a distancel,l′, one has:

P_l(δv) =

G_l,l′(x)Pl′(e−xδv)e−xd x, (3)

whereGl,l′ is a p.d.f. If one makes the assumption that theG_l_,_l′ depend only on the factorλ=l/l′ (scale invariance), then it is easy to show that the G_l_,_l′ are the p.d.f.’s of infinitely divisible laws. Under the assumption of scale invariance, if we take d =1, A= [0,l′], λ=l/l′andΩ_λ₋dln(λ)

of p.d.f. Gl,l′ in relation (1) (valid in fact simultaneously for all A) then equation (3) is the p.d.f. equivalent to (1).

Following the standard conventions in turbulence, we noteεthe (random) energy dissipation mea-sure per unit mass andε_l the mean energy dissipation per unit mass in a ballB(0,l):

εl =

4πl3ε(B(0,l))

We believe the measures we consider in this paper can be used in dimension 3 to model the energy dissipation ε in a turbulent flow. Indeed, it is believed that the velocity field of a stationary (in time) turbulent flow at scales smaller than some integral scale T (characteristic of the turbulent flow) is homogeneous in space and isotropic (see[7]). Therefore, the measureε is homogeneous and isotropic (as a function of the velocity field) and according to Kolmogorov’s refined similarity hypothesis (see for instance[4],[19]for studies of this hypothesis), one has the following relation in law between the longitudinal velocity incrementδvl of two points separated by a distancel and

ε_l:

δv_l(l aw=)U(lεl)1/3,

whereU is a universal negatively skewed random variable independent ofεland of law independent

ofl. Therefore, equations similar to (3) should also hold for the p.d.f. of εl. Finally, let us note

that the statistics of the velocity field and thus also those ofεare believed to be universal at scales smaller thanT in the sense that they only depend on the average mean energy dissipation per unit mass< ε >defined by (note that the quantity below does not depend onl by homogenity):

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where<>denotes an average with respect to the randomness. In particular, the law of _<ε>ε and the p.d.f.’sGl,l′ are completely universal, i.e. are the same for all flows; nevertheless, there is still a big debate in the phyics community on the exact form of the p.d.f.’sG_l,l′. In dimension 3, the measures

M we consider are precisely models for _<ε>ε .

The rest of this paper is organized as follows: in section 2, we set the notations and give the main results. In section 3, we give a few remarks concerning the important lognormal case. In section 4, we gather the proofs of the main theorems of section 2.

2 Notations and main Results

2.1 Independently scattered infinitely divisible random measure.

The characteristic function of an infinitely divisible random variableX can be written asE_[_eiqX_{] =}

eϕ(q), whereϕis characterized by the Lévy-Khintchine formula

ϕ(q) =imq₋1 2σ

2_q2₊

R∗

(eiq x₋1₋iqsin(x))ν(d x)

andν(d x)is the so-called Lévy measure. It satisfiesR_R_∗min(1,x2)ν(d x)<+_∞. LetGbe the unitary group ofRd_{, that is}

G=_{M∈Md(R);M Mt=I}.

Since G is a compact separable topological group, we can consider the unique right translation invariant Haar measure H with mass 1 defined on the Borelσ-algebra _B(G). Let S be the half-space

S=_{(t,y);t∈R_,_y _∈R∗

+} with which we associate the measure (on the Borelσ-algebraB(S))

θ(d t,d y) =y−2d t d y.

Givenϕ, following[1], we consider an independently scattered infinitely divisible random measure

µassociated to(ϕ,H⊗θ)and distributed onG×S(see[14]). More precisely,µsatisfies:

1) For every sequence of disjoint sets(An)n inB(G×S), the random variables (µ(An))n are

inde-pendent and

µ [

A_n

µ(A_n)a.s.,

2) for any measurable set Ain _B(G_×S), µ(A) is an infinitely divisible random variable whose characteristic function is

E(eiqµ(A)) =eϕ(q)H⊗θ(A).

We stress the fact thatµis not necessarily almost surely a signed measure (undoubtedly, the term random measure is misleading). More precisely, it is not always the case that one can consider a version ˜µ ofµ(i.e. for all Ain B(G×S), ˜µ(A) =µ(A) a.s.) such that almost surely A→µ(˜ A) is

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a signed measure. In other words , it is not always the case thatµ(or a version ofµ) satisfies the following strong version 1)’ of 1):

1)’ Almost surely, for every sequence of disjoint sets (A_n)_n in _B(G_×S), the random variables

(µ(An))n are independent and

µ [

µ(An).

Let us additionnally mention that there exists a convex functionψ defined on R _{such that for all}

non empty subsetAofG×S:

1. ψ(q) = +_∞, ifE₍_eqµ(A)_{) = +}_∞_,

2. E(eqµ(A)) =eψ(q)H⊗θ(A)otherwise.

Let qc be defined asqc = sup{q ≥ 0;ψ(q)< +∞}. For any q ∈[0,qc[, ψ(q) <+∞andψ(q) =

ϕ(−iq).

2.2 Multidimensional Multifractal Random Measures (MMRM).

We further assume that the independently scattered infinitely divisible random measureµassociated to(ϕ,H_⊗θ)satisfies:

ψ(2)<+_∞,

andψ(1) =0. The condition onψ(1)is just a normalisation condition. The conditionψ(2)<+_∞

is technical and can probably be relaxed to the condition used in [1]: there exists ε > 0 such thatψ(1+ε)<+_∞. However, in the multidimensional setting, the situation is more complicated because there is no strict decorrelation property similar to the one dimensional setting: in dimension d ≥ 2, there does not exist some distance R such that M(A) and M(B) are independent for two Lebesgue measurable sets A,B _⊂ Rd _{separated by a distance of at least} _R _{(see also lemma 3.2}

below). In[1], the proof of the non-triviality of the MRM deeply relies on a decomposition of the unit interval into independent blocks, which is only possible because such an independent scaleR exists in dimension 1. Nevertheless, the conditionψ(2)<+_∞is enough general to cover the cases considered in turbulence: see[3],[17],[18].

Definition 2.3. Filtration_F_l. LetΩbe the probability space on whichµis defined. _Fl is defined as theσ-algebra generated by_{µ(A_×B);A_⊂G,B_⊂S, dist(B,R2_\_S₎_≥_l_}_.

GivenT, let us now define the function f :R₊_→R_by

f(l) =

l, ifl_≤T T ifl_≥T The cone-like subsetA_l(t)ofS is defined by

A_l(t) =_{(s,y)_∈S;y_≥l,₋f(y)/2_≤s₋t_≤ f(y)/2_}. For forthcoming computations, we stress that fors,t real we have:

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where g_l:R₊_→R_{is given by (with the notation} _x+₌_max₍_x_{, 0}₎_):

g_l(x) =

ln(T/l) +1− x_l, if x ≤l ln+(T/x) if x _≥l

For anyx ∈Rd andm∈G, we denote byx₁mthe first coordinate of the vectormx. The cone product C_l(x)is then defined as

C_l(x) =_{(m,t,y)_∈G_×S;(t,y)_∈A_l(x₁m)_}.

Definition 2.4. ωl(x)process.The processωl(x)is defined asωl(x) =µ(Cl(x)).

Remark 2.5. We make the additional assumption

[−1,1]

|x_|ν(d x)<+_∞,

which ensures the process(t,l) _∈R_×]0,+_∞[_7→ωl(t)is locally bounded. The process l 7→ Ml(A) is

thus a continuous martingale for each measurable subset A. It is not clear how to drop that condition and make the whole machinery work all the same. According to the authors, that condition has been omitted in[1].

Definition 2.6. M_l(dx)measure.For any l>0, we define the measure M_l(d x) =eωl(x)_{d x, that is}

Ml(A) =

eωl(x)_{d x}

for any Lebesgue measurable subset A⊂Rd_.

Theorem 2.7. Multidimensional Multifractal Random Measure (MMRM).

Letϕand T be given as above and letωl and Mlbe defined as above. With probability one, there exists

a limit measure (in the sense of weak convergence of measures) M(d x) = lim

l→0+Ml(d x).

This limit is called the Multidimensional Multifractal Random Measure. The scaling exponent of M is defined by

∀q≥0, ζ(q) =dq−ψ(q). Moreover:

i)_∀x _∈Rd_{, M}₍_{_x_}_{) =}₀

ii) for any bounded subset K ofRd_{, M}₍_K₎_<₊_∞_{and E}_[_M₍_K_)]_{≤ |}_K_|_.

Proposition 2.8. Homogenity and isotropy

1. The measure M is homogeneous in space, i.e. the law of(M(A))_A_⊂Rd coincides with the law of

(M(x+A))_A_⊂Rd for each x∈R d_.

2. The measure M is isotropic, i.e. the law of(M(A))_A_⊂Rd coincide with the law of(M(mA))A⊂Rdfor

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Proposition 2.9. Main properties of the MRM.

1. The measure M is different from0if and only if there existsε >0such thatζ(1+ε)>d; in that case,E₍_M₍_A_{)) =}_|_A_|_.

2. Let q>1and consider the unique n∈N_{such that n}_<_q_≤_n+1. Ifζ(q)>d andψ(n+1)<∞, thenE_[_M₍_A₎q_]_<₊_∞_.

3. For any fixedλ∈]0, 1]and l≤T , the two processes(ωλl(λA))A⊂B(0,T)and(Ωλ+ωl(A))A⊂B(0,T) have the same law, where Ω_λ is an infinitely divisible random variable independent from the process(ωl(A))A⊂B(0,T)and its law is characterized byE[eiqΩλ] =λ−ϕ(q).

4. For anyλ∈]0, 1], the law of(M(λA))_A_⊂_B_(0,_T₎ is equal to the law of (W_λM(A))_A_⊂_B_(0,_T₎, where W_λ =λdeΩλ _and_Ω

λ is an infinitely divisible random variable (independent of(M(A))A⊂B(0,T)) and its characteristic function is

E[eiqΩλ_{] =}_λ−ϕ(q)_.

5. Ifζ(q)₆=_−∞and0<t<T then

M(B(0,t))q

= (t/T)ζ(q)E

M(B(0,T))q

3 The limit lognormal case

In the gaussian case, we haveψ(q) =γ2q2/2 and the conditionζ(1+ε)>dcorresponds toγ2<2d. The approximating measuresM_l is thus defined as:

Ml(A) =

eγXl(x)−

γ2E[_{X l}(x)2]

2 d x

whereX_l is a centered gaussian field (equal to(ωl−E[ωl])/γ) with correlations given by:

E[Xl(x)Xl(y)] =H⊗θ(Cl(x)∩Cl(y))

g_l(_|x₁m− y₁m|)H(d m).

The limit mesures M = lim_l_→₀M_l(d x) we define are in the scope of the theory of gaussian multi-plcative chaos developed by Kahane in[11](see[15]for an introduction to this theory). Formally, the measureM is defined by:

M(A) =

eγX(x)−γ

2E[X(x)2]

2 d x

where X is a centered gaussian field (in fact a random tempered distribution ) with correlations given by:

E_[_X₍_x₎_X₍_y_{)] =}

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Let us suppose thatT =1 for simplicity. Using invariance of the Haar measureH by multiplication, it is plain to see thatE[X(x)X(y)]is of the form F(_|y−x|)where F :R₊ _→R₊_{∪ {∞}}. We have the scaling relation F(a b) =ln(1/a) +F(b)if a≤1 and b≤1 (see also lemma 4.8 below) which entails that for_|x_{| ≤}T:

F(_|x|) =ln(1/|x|)₋

ln(_|em₁|)H(d m).

wheree= (1, 0, . . . , 0)is the first vector of the canonical basis. As a corollary, we get the existence of some constantC (takeC =₋R

Gln(|e m

1|)H(d m)) such that ln(1/|x|) +C is positive definite (as a tempered distribution) in a neighborhood of 0. This easily implies that ln(1/|x_|)is positive definite in a neighborhood of 0: to our knowledge, this result is new. This contrasts with the fact that ln+(1/|x|)is positive definite in dimensiond≤3 but is not positive definite ford≥4 (see[15]). Remark 3.1. It is easy to see that 1− |x|α is positive definite in a neighborhhod of 0 as a function defined in R _if _α _≤ ₂_{. Therefore, one can consider the isotropic and positive definite function in a}

neighborhhood of0inRd _{defined by:}

F(_|x|) =

(1− |x₁m|α)H(d m).

By scaling, one can see that F(_|x_|) =1₋C_|x_|α for some C >0. It is easy to see that this entails that 1− |x|α is positive definite in a neighborhood V of0. Using the main theorem in[13], one can extend 1_{− |}x_|α (defined in V ) in an isotropic and positive definite function defined in Rd_{. This is in contrast}

with the so called Kuttner-Golubov problem which is to determine theα,κ >0such that(1_{− |}x_|α)κ₊is positive definite inRd_{. It is known (see}_[₈_]_{for instance) that for}_{α >}₀_{the function}₍₁_{− |}_x_|α₎₊ _{is not}

positive definite inRd _{if d}_≥₃_.

Finally, let us mention that the fieldX with correlations given by (4) exhibits long range correlations as soon asd_≥2. More precisely, we have:

Lemma 3.2. If d_≥2, we get the following equivalence:

E_[_X₍_x₎_X₍_y_)] _∼

|x−y|→∞

2Γ(d/2) Γ(1/2)Γ((d−1)/2)

|x− y|.

whereΓis the standard gamma function:Γ(x) =R₀∞tx−1e−td t.

4 Proofs

4.1 Proof of lemma 1.1

We will consider the cased =1 for simplicity (the general case is an adaptation of the one dimen-sional case). With no restriction, we can suppose that the α in lemma 1.1 belongs to ]0, 1[. For s,t > 0, we have the following inequalities by using the assumptions on M and the concavity of

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x _7→xα:

E_[_M_[_0,_t₊_s_]α_{] = (}_t₊_s₎αE_[( t

t+s

M[0,t]

t +

s t+s

M[t,t+s]

s )

α_]

≥(t+s)αE_[ t

t+s(

M[0,t]

t )

α₊ s

t+s(

M[t,t+s]

s )

α_]

= (t+s)α( t

t+sE[(

M[0,t]

t )

α_{] +} s

t+sE[(

M[t,t+s]

s )

α_])

= (t+s)αE[M[0, 1]α] =E_[_M_[_0,_t₊_s_]α_]_.

Therefore, the above inequalities are in fact equalities and, since x 7→ xα is strictly concave, this shows that M[0,t]

t and

M[t,t+s]

s are equal almost surely. Let us consider the random non decreasing

function f(t) =M[0,t]. The function f satisfies for alls,t>0: f(t)

t =

f(t+s)₋f(t)

s .

Lettingsgo to 0 in the above equality, we conclude that f has a derivative f′which satisfies: f′(t) = f(t)

t . Therefore, f(t)is of the formC twhere C is some constant.

4.2 Proof of Theorem 2.7

The relationE_[_eωl(x)_{] =}_eψ(1)H⊗θ(Cl(x))₌_{1 (and the fact that for}_l′_<_l_,_ω

l′(x)−ω_l(x)is indepen-dent of_Fl) ensures that for each Borelian subset A_⊂Rd_{, the process} _M_l₍_A₎ _{is a martingale with}

respect to_Fl. Existence of the MRM then results from[10]. Properties i) and ii) result from Fatou’s lemma.

4.3 Characteristic function of

ω

(

x)

As in [1], the crucial point is to compute the characteristic function of ωl(x). We consider

(x1, . . . ,xq)_∈(Rd₎q _and_(λ₁_{, . . . ,}_λ_q₎_∈Rq_{and we have to compute}

φ(λ) =E_eiλ1ωl(x1)+···+iλqωl(xq)_.

Let us denote by_Sq the permutation group of the set_{1; . . . ;q_}. For a generic elementσ∈ Sq, we define

Bσ=_{m∈G;x₁σ(1),m<· · ·<xσ₁(q),m}. Finally, givenx,z∈Rd, we define cone like subset product

C_lσ(x) =Cl(x)∩Bσ={(m,t,y)∈Bσ×S;(t,y)∈Al(x1m)} and

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Lemma 4.4. The characteristic function of the vector(ωl(xi))1≤i≤q exactly matches

exp

iλ1ωl(x1) +· · ·+iλqωl(xq)

=exp X

σ∈Sq

j=1

k=1

ασ(j,k)ρσ_l (xσ(k)−xσ(j))

whereρσ_l (x) =H⊗θ(C_lσ(0,x)), and

ασ(j,k) =ϕ(r_kσ_,_j) +ϕ(r_kσ_+1,_j₋₁)₋ϕ(r_kσ_,_j₋₁)₋ϕ(r_kσ_+1,_j)

r_kσ_,_j=

i=k

λσ(i) (or0if k> j).

Moreover

j=1

k=1

ασ(j,k) =ϕ

k=1

λk

Proof.Without loss of generality, we assumexi₆= xjfori₆= j. We point out that the family(Bσ)σ∈Sd is a partition ofG up to a set of null H-measure. The functionφbreaks down as

φ(λ) =E_eiλ1µ(Cl(x1))+···+iλqµ(Cl(xq))

=E_e

σ∈Sdiλ1µ(Clσ(x

1₎₎₊_···₊_iλ

qµ(Clσ(x q₎₎

= Y

σ∈Sd

E_eiλ1µ(Clσ(x

1₎₎₊_···₊_iλ

qµ(Clσ(x q₎₎

Let us fixσ∈ Sq. We focus on the term

φσ(λ) =E

eiλ1µ(Clσ(x

1₎₎₊_···₊_iλ

qµ(Clσ(x q₎₎₎

eiλσ(1)µ(Clσ(xσ

(1)₎₎₊_···₊_iλ

σ(q)µ(Clσ(xσ

(q)₎₎₎

. Givenσ∈ Sd andp≤q, we further define

φσ(λ,p) =E_eiλσ(1)µ(Clσ(x

σ(1)₎₎₊

···+iλ_σ(p)µ(Clσ(x

σ(p)₎₎₎

From now on, we adapt the proof of[1, Lemma 1]and proceed recursively. We define

Y_qσ=

k=1

λσ(k)µ(Clσ(x σ(k)₎

\Cl(xσ(q))),

which stands for the contribution of the points of the above sum that do not belong to C_l(xσ(q)). Moreover, the points in the setC_l(xσ(q))can be grouped into the disjoint sets

Cl(xσ(k),xσ(q))\Cl(xσ(k−1),xσ(q)).

We stress that the latter assertion is valid since, form∈Bσ, the coordinates are suitably sorted, that is: x₁σ(1),m<x₂σ(2),m<· · ·< x₁σ(q),m. We define

X_kσ_,_q=µ Cl(xσ(k),xσ(q))\Cl(xσ(k−1),xσ(q))

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with the conventionC_l(xσ(k),xσ(0)) =C_l(xσ(0),xσ(k)) =_;, in such a way that one has

λσ(1)µ(Clσ(x

σ(1)_{)) +}

· · ·+λσ(q)µ(Clσ(x

σ(q)_{)) =}_Yσ

q +

k=1

rσ_k_,_qXσ_k_,_q.

Furthermore, since the variable Yq and (Xσ_k_,_q)k are mutually independent, we get the following

decomposition:

φσ(λ) =E

eiYqσ

k=1

ei rkσ,qXσk,q_. ₍₅₎

Similarly, one can prove

φσ(λ,q₋1) =E_eiYqσ

k=1

E_ei rσk,q−1Xkσ,q_. ₍₆₎

Gathering (5) and (6) yields

φσ(λ,q) =φσ(λ,q₋1)

k=1

Eei rσk,qXσk,q

E_ei rkσ,q−1Xσk,q .

For anym_∈Bσ, one has x₁σ(k−1),m<x₁σ(k),m<x₁σ(q),m and therefore

eiαXσk,q₌_eϕ(α)H⊗θ Cl(xσ(k),xσ(q))\Cl(xσ(k−1),xσ(q))

=eϕ(α) H⊗θ(Cl(xσ(k),xσ(q)))−H⊗θ(Cl(xσ(k−1),xσ(q)))

Note that

H_⊗θ(C_l(xσ(i),xσ(j))) =

Bσ

θ A_l(xσ₁(i),m)_∩A_l(x₁σ(j),m)

H(d m)

Bσ

θ Al(0)∩Al(x σ(i),m

1 −x

σ(j),m

1 )

H(d m) =ρσ_l (xσ(i)−xσ(j))

The proof can now be completed recursively. For further details, the reader is referred to[1].

4.5 Homogenity and isotropy

Lemma 4.4 is useful to prove the main properties of the MMRM. For instance, to prove the invariance of the law of the MMRM under translations, it suffices to prove that the law ofω_l is itself invariant. This results from Lemma 4.4 since each termρσ_l (xσ(k)₋xσ(i))is invariant under translations, that is ρσ_l remains unchanged when you replace x1, . . . ,xq by x1+z, . . . ,xq+z for a given z _∈Rd_.

However, Lemma 4.4 may not be adapted to prove the isotropy of the MMRM so that we give a proof by a direct approach:

Lemma 4.6. Proof of the isotropy of the MMRMThe measure M is isotrop, that is:

∀m_∈G, (M(mA))_A_⊂_B(0,T)

l aw

(12)

Proof. Once again, it is sufficient to prove that the characteristic function ofωl is invariant under

G. This time, we compute that characteristic function in a more direct way. We consider x = (x1, . . . ,xq)_∈(Rd)q_,_λ

1, . . . ,λq∈Randm0∈G, and define the function f_x(m,t,y) =

k=1

λk1ICl(xk).

Form_∈G, we definemx as(mx1, . . . ,mxq). We have, using the right translation invariance of the Haar measure and the fact that f_m

0x(m,t,y) = fx(mm0,t,y):

expiλ1ωl(m0x1) +· · ·+iλqωl(m0xq)

expi

f_m

0x(m,t,y)µ(d m,d t,d y)

expi

f_x(mm₀,t,y)µ(d m,d t,d y)i

=exp

ϕ◦f_x(mm₀,t,y)H(d m)θ(d t,d y) =exp

ϕ◦f_x(m,t,y)H(d m)θ(d t,d y) =E

exp

iλ1ωl(x1) +· · ·+iλqωl(xq)

. The isotropy follows.

4.7 Exact scaling and stochastic scale invariance

Lemma 4.8. Exact scaling of Ml(d x). Given∀λ∈]0, 1],∀x1, . . . ,xq∈B(0,T/2), the functionsρ_lσ

satisfy the exact scaling relation

σ∈Sq

q X j=1 j X k=1

ασ(j,k)ρσ_λl(λxσ(k)−λxσ(j)) =₋ln(λ) + X

σ∈Sq

q X j=1 j X k=1

ασ(j,k)ρ_lσ(xσ(k)−xσ(j)). (7)

Proof. We remind that for x real we haveR_A

l(0)∩Al(x)θ

(d t,d y) = gl(|x|). GivenB⊂G andx ∈Rd,

we define:

ρB_l(x) =

1I_B(m)1I_{₍_t_,_y₎_∈_A

l(0)∩Al(x1m)}θ(d t,d y)H(d m).

Then we can compute the functionρB l:

ρB_l(x) =

1I_{₍_t_,_y₎_∈_A

l(0)∩Al(x1m)}H(d m)θ(d t,d y) =

Al(0)∩Al(x1m)

θ(d t,d y)H(d m)

ln(T/l) +1_{− |}xm₁_|/l

1I_|_xm

1|≤l+ln(T/|x

1|)1Il≤|x₁m|≤T

(13)

Givenλ∈]0, 1]andx_∈B(0,T) ρ_λlB(λx) =

ln(T

λl) +1−

|λx₁m_|

λl

1I_|λxm₁|≤λl+ln(

λ|x₁m|)1Iλl≤|λx1m|≤T

H(d m)

ln(T

l ) +1−

|x₁m_| l

1I_|_xm

1|≤l+ln(

|x₁m_|)1Iλl≤|λx1m|≤T

H(d m)

−ln(λ)

1I_|_xm

1|≤l+1Il≤|x

1|≤TH(d m)

=ρ_lB(x)₋ln(λ)H(B)

We therefore obtain

σ∈Sq

q X j=1 j X k=1

ασ(j,k)ρ_λlσ(λxσ(k)−λxσ(j))

= X

σ∈Sq

q X j=1 j X k=1

ασ(j,k)ρ_lσ(xσ(k)−xσ(j))₋ln(λ) X

σ∈Sq

q X j=1 j X k=1

ασ(j,k)H(Bσ)

= X

σ∈Sq

q X j=1 j X k=1

ασ(j,k)ρ_lσ(xσ(k)₋xσ(j))₋ln(λ) X

σ∈Sq

ϕ(

k=1

λk)H(Bσ)

= X

σ∈Sq

q X j=1 j X k=1

ασ(j,k)ρ_lσ(xσ(k)₋xσ(j))₋ln(λ)ϕ(

k=1

λk)

From Lemma 4.4, we deduce that, for any λ∈]0, 1], there exists a random variableC_λ such that

(ωλl(λx))x∈B(0,T/2)

l aw

= (Cλ+ωl(x))x∈B(0,T/2)and such thatCλ is independent of(ωl(x))x∈B(0,T/2) and its characteristic function is given byE_[_eiqCλ_{] =}_λ−ϕ(q)_.

By integrating the previous relation, we obtain the relation

(Mλl(λA))A⊂B(0,T/2)

l aw

= Wλ(Ml(A))A⊂B(0,T/2)

whereWλ=λdeCλis a random variable independent of(Ml(A))A⊂B(0,T/2).

4.9 Non-triviality of the MMRM

Proof of Proposition 2.9 (items 1. and 2.) Suppose we can find a "cube"C_R= [0;R]d andq>1 such that

E_M₍_C_R₎q_<₊_∞_.

Then we can find n_∈N _{such that}_[_{0; 2}−n_R_]d _⊂ _B₍_0,_T_/₂₎_{. We split the cube} _C_R _{into 2}nd _smaller

cubes

Ck,n=

i=1

(14)

wherek= (k1, . . . ,kd)∈N_dn d e f

= Nd_∩[0; 2n−1]d_{. For each fixed value of}_n_{, the cubes}₍_Ck,n₎

k, where

the index kvaries inN_dn form a partition ofC_T. Thus, by using the super-additivity of the function x _7→xq, we have:

E_M₍_C_T₎q₌Eh X

k∈Nn d

M(Ck,n)qi

≥ X

k∈N_dn

E M(Ck,n)q

By using the translation invariance and the scale invariance property of the MMRM, we deduce:

E _M₍_Ck,n₎q₌E _M₍_C0,n₎q₌E _M₍₂−n_C_R₎q₌₂−nζ(q)E _C_R₎q_.

Finally, gathering the previous inequalities yields:

E _C_R₎q_≥₂nd−nζ(q)E _C_R₎q

in such a way that, necessarily,ζ(q)_≥d.

The proof of 1. and 2. is then a consequence of the following lemma:

Lemma 4.10. Let q > 1and consider the unique n_∈N _{such that n}_< _q_≤ _n₊₁_{. If} _ζ(_q₎ _> _{d and}

ψ(n+1)<∞, then we can find a constant C such that: sup

E_M_l([0,T[d)q

≤C.

Proof. The proof is an adaptation of the one in[1](which is itself an adaptation of the correspond-ing result in [2]). Unfortunately, the multi-dimensional setting is bit more complicated because there is no strict decorrelation property similar to the one dimensional setting. With no restriction, we can suppose that T = 1 and d = 2. We consider the following dyadic partition of the cube

[0, 1[2:

[0, 1[2= _∪

0≤i,j≤2m₋₁I (m)

i,j ,

whereI_i(_,m_j)= [ i

2m,

i+1 2m[×[

2m,

j+1

2m [. Let us write the above decompostion in the following form:

[0, 1[2=C₁_∪C₂_∪C₃_∪C₄. where

C1= ∪

i and j evenI

(m)

i,j , C2=_{i and j od d}∪ I

(m)

i,j

and

C₃= _∪

i od d and j evenI

(m)

i,j , C2= ∪

i even and j od dI

(m)

i,j .

Since the measureM_l is homogeneous, we get:

E_M_l_([_{0, 1}_[2₎q_≤₄q−1

i=1

E_M_l₍_C_i₎q

≤4qE_M_l(C1)q

(15)

Now we get the following by subadditivity of x_→xq/(n+1):

E_M_l₍_C₁₎q₌E₍ X

0≤i,j≤2m−1₋₁

M_l(I₂(m_i_,2)_j))q

=E((X

i,j

Ml(I

(m) 2i,2j))

n+1₎q/(n+1)

=E( X

i1,j1,...,in+1,jn+1

k=1 Ml(I

(m) 2ik,2jk))

q/(n+1)

≤E X

i1,j1,...,in+1,jn+1 (

k=1

M_l(I₂(m_i )

k,2jk))

q/(n+1)

=22(m−1)E

M_l(I_0,0(m))q

∗

i1,j1,...,in+1,jn+1

k=1

M_l(I₂(m_i)

k,2jk)

q/(n+1)

≤22(m−1)E

M_l(I_0,0(m))q

∗

i1,j1,...,in+1,jn+1

k=1

M_l(I₂(m_i)

k,2jk)

q/(n+1)

where ∗

i1,j1,...in+1,jn+1 is a sum over indices i1,j1, . . .in+1,jn+1 which are not all equal and the last

inequality is a consequence of Jensen’s inequality. Therefore each term in the above sum is of the form:

r=1

M_l(I₂(m_i )

r,2jr)

nrq/(n+1)

(8) where the sequence of positive integers(n_r)₁_≤_r_≤_k satisfies Pk_r₌₁n_r+1 and the I₂_i

r,2jr are disjoint intervals wich lie at a distance of at least ₂1m. We want to show that each term of the form (8) is bounded by some quantityC_mindependent ofl. We get the following computation using Fubini:

r=1

M_l(I₂(m_i )

r,2jr)

nr₌

In1

2i_1,2j₁×···×I2nk_ik,2_jk

eωl(x1)+···+ωl(xn+1)_{d x}1_{. . .}_{d x}n+1_.

We defineNr=n1+· · ·+nrforrin[1,k]and we introduce the following setAl =Al(x1, . . . ,xn+1): Al=_∪r<r′(∪N

r≤i≤Nr+1−1Cl(x

i₎₎

∩(_∪N_r′_≤_j_≤_N_r′

+1−1Cl(x

j₎₎_.

By construction of_Al, ifxiandxj are in two differentI₂_i

r,2jr thenµ(Cl(x

i₎

\Al)andµ(C_l(xj)_\Al)

are independent. Therefore we get the following factorization:

E_eωl(x1)+···+ωl(xn+1)₌_E_eψ(n+1)H⊗θ(Al)_E_eµ(Cl(x1)\Al)+···+µ(Cl(xn+1)\Al)

=E_eψ(n+1)H⊗θ(Al)

r=1

E_e

Nr≤i_≤Nr₊₁−1µ(Cl(xi)\Al)

= E

eψ(n+1)H⊗θ(Al)

r=1E

eψ(nr)H⊗θ(Al)

r=1

E_e

Nr≤i_≤Nr₊₁−1ωl(xi)

(16)

We have the following inequality: H⊗θ(Al)_≤ X

r<r′

Nr≤i≤Nr+1−1

N_{r′ ≤}j≤N_r′+1−1

H⊗θ(Cl(xi)∩Cl(xj)).

Note that for eachxi,xjin the above sum we have_|xi₋xj_{| ≥} ₂1m and thereforeH⊗θ(Cl(xi)∩Cl(xj))

is bounded by some constant depending onmbut independent ofl. Indeed, using the notation of section 3 forF, we get:

H_⊗θ(C_l(xi)_∩C_l(xj)) =

g_l(_|(xi)m₁ ₋(xj)m₁_|)H(d m)

≤F(_|xi−xj|)

≤F( 1

2m).

In conclusion, we get the existence of some constantCmsuch that:

eωl(x1)+···+ωl(xn+1)_≤_C

m k

r=1

e P

Nr≤i≤Nr+1−1ωl(xi)

Thus, we get by integrating the above relation:

r=1

M_l(I₂(m_i )

r,2jr)

nr_≤_C

m k

r=1

M_l(I₂(m_i )

r,2jr)

nr_.

Since each nr is less or equal to n, we get by induction thatE

Ml(I

(m) 2ir,2jr)

nr_{is bounded} indepen-dently of l and so is the above product. In conclusion, we get the existence of C_m such that we have:

EM_l([0, 1]2)q

≤4q−122mEM_l(I_0,0(m))q

+C_m. Using stochastic scale invariance, we get that:

E_M_l([0, 1]2)q

≤4q−1 2 2m

2mζ(q)E

Ml2m([0, 1]2)q+C_m

≤4q−1 2 2m

2mζ(q)E

M_l([0, 1]2)q

+C_m.

Sinceζ(q)>2, we can choosemsuch that 4q−1 2₂m2ζm(q) <1 and therefore we get:

M_l([0, 1]2)q

≤ Cm

1₋4q−1 22m 2mζ(q)

(17)

4.11 Proof of lemma 3.2

By homogenity and isotropy, we can suppose that y =0 and x =_|x|ewhere eis the first vector of the canonical basis. In this case, we get (see relation (4.2.18) in[9]):

E_[_X₍_x₎_X₍₀_{)] =} Γ(d/2)

2Γ(1/2)Γ((d₋1)/2)

Z 1

ln+(T/(v_|x_|))(1₋v)d/2−3/2_pd v

v whereΓis the standard gamma function: Γ(x) =R∞

0 t

x−1_e−t_{d t}_{. By making the change of variable}

u= v|_Tx| in the above integral, one easily deduces that:

E[X(x)X(0)] _∼

|x|→∞C

|x_|, whereC is given byC= Γ(d/2)

2Γ(1/2)Γ((d−1)/2)

0 ln(1/u)

u. Integrating by parts, we find

0 ln(1/u)

u =

4 and thusC = _Γ(1_/_2)Γ((2Γ(d/2)

d−1)/2).

Acknowledgements: The authors would like to thank E. Bacry, J.F. Muzy and R. Robert for inter-esting discussions.

References

[1] Bacry E., Muzy J.F.: Log-infinitely divisible multifractal processes,Comm. Math. Phys.,236 (2003) no.3, 449-475. MR2021198

[2] Barral, J., Mandelbrot, B.B.: Multifractal products of cylindrical pulses, Probab. Theory Relat. Fields124(2002), 409-430. MR1939653

[3] Castaing B., Gagne Y., Hopfinger E.J.: Velocity probability density-functions of high Reynolds-number turbulence,Physica D46(1990) 2, 177-200.

[4] Castaing B., Gagne Y., Marchand M.: Conditional velocity pdf in 3-D turbulence,J. Phys. II France4(1994), 1-8.

[5] Chainais, P.: Multidimensional infinitely divisible cascades. Application to the modelling of intermittency in turbulence,European Physical Journal B,51no. 2 (2006), pp. 229-243.

[6] Duplantier, B., Sheffield, S.: Liouville Quantum Gravity and KPZ, available on arxiv at the URL http://arxiv.org/abs/0808.1560.

[7] Frisch, U.:Turbulence, Cambridge University Press (1995). MR1428905

[8] Gneiting, T.: Criteria of Polya type for radial positive definite functions, Proceedings of the American Mathematical Society, 129 no. 8 (2001), 2309-2318. MR1823914

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[9] Hiai, F., Petz, D.: The semicircle law, free random variables and Entropy, A.M.S. (2000). MR1746976

[10] Kahane, J.-P.: Positive martingales and random measures,Chi. Ann. Math.,8B(1987), 1-12. MR0886744

[11] Kahane, J.-P.: Sur le chaos multiplicatif,Ann. Sci. Math. Québec, 9no.2 (1985), 105-150. MR0829798

[12] Knizhnik, V.G., Polyakov, A.M., Zamolodchikov, A.B.: Fractal structure of 2D-quantum grav-ity,Modern Phys. Lett A,3(8) (1988), 819-826. MR0947880

[13] Rudin,W.: An extension theorem for positive-definite functions, Duke Math Journal 37 (1970), 49-53. MR0254514

[14] Rajput, B., Rosinski, J.: Spectral representations of infinitely divisible processes, Probab. Theory Relat. Fields82(1989), 451Ð487.

[15] Robert, R., Vargas, V.: Gaussian Multiplicative Chaos revisited, to appear in the Annals of Probability, available on arxiv at the URL http://arxiv.org/abs/0807.1030.

[16] Rhodes, R. Vargas, V.: KPZ formula for log-infinitely divisible multifractal random measures, available on arxiv at the URL http://arxiv.org/abs/0807.1036.

[17] Schmitt, F., Lavallee, D., Schertzer, D., Lovejoy, S.: Empirical determination of universal multifractal exponents in turbulent velocityfields,Phys. Rev. Lett.68(1992), 305-308.

[18] She, Z.S., Leveque, E.: Universal scaling laws in fully developed turbulence,Phys. Rev. Lett. 72(1994), 336-339.

[19] Stolovitzky, G., Kailasnath, P., Sreenivasan, K.R.: Kolmogorov’s Refined Similarity Hypothe-ses,Phys. Rev. Lett.69(8) (1992), 1178-1181.

[20] D.W. Stroock, S.R.S. Varadhan, Multidimensionnal Diffusion Processes, Grundlehren der Mathematischen Wissenschaft 233, Springer, Berlin et al., 1979. MR0532498

(1)

Givenλ∈]0, 1]andx_∈B(0,T) ρ_λB_l(λx) =

ln(T

λl) +1− |λx₁m_|

λl

1I_|λxm₁|≤λl+ln( T

λ|x₁m|)1Iλl≤|λx1m|≤T

H(d m)

ln(T

l ) +1− |x₁m_|

1I_|_xm

1|≤l+ln(

|x₁m_|)1Iλl≤|λx1m|≤T

H(d m)

−ln(λ)

1I_|_xm

1|≤l+1Il≤|x

1|≤TH(d m) =ρ_lB(x)₋ln(λ)H(B)

We therefore obtain

σ∈Sq q X j=1 j X k=1

ασ(j,k)ρ_λσ_l(λxσ(k)−λxσ(j))

= X

σ∈Sq q X j=1 j X k=1

ασ(j,k)ρ_lσ(xσ(k)−xσ(j))₋ln(λ) X

σ∈Sq q X j=1 j X k=1

ασ(j,k)H(Bσ)

= X

σ∈Sq q X j=1 j X k=1

ασ(j,k)ρ_lσ(xσ(k)₋xσ(j))₋ln(λ) X

σ∈Sq

ϕ(

q X

k=1

λk)H(Bσ)

= X

σ∈Sq q X j=1 j X k=1

ασ(j,k)ρ_lσ(xσ(k)₋xσ(j))₋ln(λ)ϕ(

q X

k=1 λk)

From Lemma 4.4, we deduce that, for any λ∈]0, 1], there exists a random variableC_λ such that (ωλl(λx))x∈B(0,T/2)

l aw

= (Cλ+ωl(x))x∈B(0,T/2)and such thatCλ is independent of(ωl(x))x∈B(0,T/2) and its characteristic function is given byE_[_eiqCλ_{] =}_λ−ϕ(q)_.

By integrating the previous relation, we obtain the relation (Mλl(λA))A⊂B(0,T/2)

l aw

= Wλ(Ml(A))A⊂B(0,T/2) whereWλ=λdeCλis a random variable independent of(Ml(A))A⊂B(0,T/2).

4.9 Non-triviality of the MMRM

Proof of Proposition 2.9 (items 1. and 2.) Suppose we can find a "cube"C_R= [0;R]d andq>1 such that

E_M₍_C_R₎q_<₊_∞_.

Then we can find n_∈N _{such that}_{[0; 2}−n_R_]d _⊂ _B_(0,_T_{/2). We split the cube} _C_R _{into 2}nd _smaller

cubes

Ck,n=

d Y

i=1

(2)

wherek= (k1, . . . ,kd)∈N_dn d e f

= Nd_∩[0; 2n₋_1]d_{. For each fixed value of}_n_{, the cubes}₍_Ck,n₎

k, where

the index kvaries inN_dn form a partition ofC_T. Thus, by using the super-additivity of the function

x _7→xq, we have:

E_M₍_C_T₎q₌Eh X

k∈Nn d

M(Ck,n)qi

≥ X

k∈N_dn

E M(Ck,n)q

By using the translation invariance and the scale invariance property of the MMRM, we deduce:

E _M₍_Ck,n₎q₌E _M₍_C0,n₎q₌E _M₍₂−n_C_R₎q₌₂−nζ(q)E _C_R₎q_.

Finally, gathering the previous inequalities yields:

E _C_R₎q_≥₂nd−nζ(q)E _C_R₎q

in such a way that, necessarily,ζ(q)_≥d.

The proof of 1. and 2. is then a consequence of the following lemma:

Lemma 4.10. Let q > 1and consider the unique n_∈N _{such that n}_< _q_≤ _n₊₁_{. If} _ζ(_q₎ _> _{d and}

ψ(n+1)<∞, then we can find a constant C such that:

sup

E_M_l([0,T[d)q ≤C.

[0, 1[2= _∪ 0≤i,j≤2m₋₁I

(m)

i,j ,

whereI_i(_,m_j)= [ i 2m,

i+1 2m[×[

2m, j+1

2m [. Let us write the above decompostion in the following form:

[0, 1[2=C₁_∪C₂_∪C₃_∪C₄. where

C1= ∪

i and j evenI

(m)

i,j , C2=_{i and j od d}∪ I

(m)

i,j

and

C₃= _∪

i od d and j evenI

(m)

i,j , C2= ∪

i even and j od dI

(m)

i,j .

Since the measureM_l is homogeneous, we get:

E_M_l_{([0, 1[}2₎q_≤₄q−1

i=1

E_M_l₍_C_i₎q

≤4qE_M_l(C1)q

(3)

Now we get the following by subadditivity of x_→xq/(n+1):

E_M_l₍_C₁₎q₌E₍ X

0≤i,j≤2m−1₋₁

M_l(I₂(m_i_,2)_j))q

=E((X

i,j Ml(I

(m) 2i,2j))

n+1₎q/(n+1)

=E( X

i1,j1,...,in+1,jn+1

n Y

k=1

Ml(I

(m) 2ik,2jk))

q/(n+1)

≤E X

i1,j1,...,in+1,jn+1 (

n Y

k=1

M_l(I₂(m_i ) k,2jk))

q/(n+1)

=22(m−1)E

M_l(I_0,0(m))q

+ ∗

i1,j1,...,in+1,jn+1

n Y

k=1

M_l(I₂(m_i) k,2jk)

q/(n+1)

≤22(m−1)E

M_l(I_0,0(m))q

+ ∗

i1,j1,...,in+1,jn+1

n Y

k=1

M_l(I₂(m_i) k,2jk)

q/(n+1)

where ∗

i1,j1,...in+1,jn+1 is a sum over indices i1,j1, . . .in+1,jn+1 which are not all equal and the last inequality is a consequence of Jensen’s inequality. Therefore each term in the above sum is of the form:

k Y

r=1

M_l(I₂(m_i ) r,2jr)

nrq/(n+1)

(8) where the sequence of positive integers(n_r)₁_≤_r_≤_k satisfies Pk_r₌₁n_r+1 and the I₂_i

r,2jr are disjoint

intervals wich lie at a distance of at least ₂1m. We want to show that each term of the form (8) is

bounded by some quantityC_mindependent ofl. We get the following computation using Fubini:

k Y

r=1

M_l(I₂(m_i ) r,2jr)

nr₌ Z

In1

2i_1,2j₁×···×I2nk_ik,2_jk

eωl(x1)+···+ωl(xn+1)_{d x}1_{. . .}_{d x}n+1_.

We defineNr=n1+· · ·+nrforrin[1,k]and we introduce the following setAl =Al(x1, . . . ,xn+1): Al=_∪r_<_r′(∪Nr_≤_i_≤_Nr

+1−1Cl(x

i₎₎ ∩(_∪N

r′≤j≤Nr′+1−1Cl(x

j_)).

By construction of_Al, ifxiandxj are in two differentI₂_i

r,2jr thenµ(Cl(x i₎

\Al)andµ(C_l(xj)_\Al) are independent. Therefore we get the following factorization:

E_eωl(x1)+···+ωl(xn+1)₌_E_eψ(n+1)H⊗θ(Al)_E_eµ(Cl(x1)\Al)+···+µ(Cl(xn+1)\Al)

=E_eψ(n+1)H⊗θ(Al) k Y

r=1

E_e P

Nr≤i≤Nr+1−1µ(Cl(x

i₎_\A l)

= E

eψ(n+1)H⊗θ(Al) Qk

r=1E

eψ(nr)H⊗θ(Al) k Y

r=1

E_e P

Nr≤i≤Nr+1−1ωl(x

i₎

(4)

We have the following inequality:

H⊗θ(Al)_≤ X

r<r′

Nr≤i≤Nr+1−1

N_r_{′ ≤}j≤N_r_′₊₁−1

H⊗θ(Cl(xi)∩Cl(xj)).

Note that for eachxi,xjin the above sum we have_|xi₋xj_{| ≥} ₂1m and thereforeH⊗θ(Cl(xi)∩Cl(xj))

is bounded by some constant depending onmbut independent ofl. Indeed, using the notation of section 3 forF, we get:

H_⊗θ(C_l(xi)_∩C_l(xj)) =

g_l(_|(xi)m₁ ₋(xj)m₁_|)H(d m)

≤F(_|xi−xj|)

≤F( 1 2m).

In conclusion, we get the existence of some constantCmsuch that:

eωl(x1)+···+ωl(xn+1)_≤_C m

k Y

r=1

E e

Nr≤i≤Nr+1−1ωl(xi)

. Thus, we get by integrating the above relation:

k Y

r=1

M_l(I₂(m_i ) r,2jr)

nr_≤_C m

k Y

r=1

M_l(I₂(m_i ) r,2jr)

nr_.

Since each nr is less or equal to n, we get by induction thatE

Ml(I

(m) 2ir,2jr)

nr_{is bounded}

indepen-dently of l and so is the above product. In conclusion, we get the existence of C_m such that we have:

EM_l([0, 1]2)q

≤4q−122mEM_l(I_0,0(m))q

+C_m. Using stochastic scale invariance, we get that:

E_M_l([0, 1]2)q

≤4q−1 2 2m

2mζ(q)E

Ml2m([0, 1]2)q+C_m

≤4q−1 2 2m

2mζ(q)E

M_l([0, 1]2)q

+C_m. Sinceζ(q)>2, we can choosemsuch that 4q−1 2₂m2ζm(q) <1 and therefore we get:

M_l([0, 1]2)q

≤ Cm

1₋4q−1 22m 2mζ(q)

(5)

4.11 Proof of lemma 3.2

By homogenity and isotropy, we can suppose that y =0 and x =_|x|ewhere eis the first vector of the canonical basis. In this case, we get (see relation (4.2.18) in[9]):

E_[_X₍_x₎_X_{(0)] =} Γ(d/2)

2Γ(1/2)Γ((d₋1)/2)

Z 1

ln+(T/(v_|x_|))(1₋v)d/2−3/2_pd v

whereΓis the standard gamma function: Γ(x) =R∞ 0 t

x−1_e−t_{d t}_{. By making the change of variable} u= v|_Tx| in the above integral, one easily deduces that:

E[X(x)X(0)] _∼ |x|→∞C

r T |x_|,

whereC is given byC= Γ(d/2) 2Γ(1/2)Γ((d−1)/2)

0 ln(1/u)

u. Integrating by parts, we find R1

0 ln(1/u)

u =

4 and thusC = _Γ(1_/_2)Γ((2Γ(d/2)

d−1)/2).

Acknowledgements: The authors would like to thank E. Bacry, J.F. Muzy and R. Robert for inter-esting discussions.