Multidimensional Multifractal Random Measures June 12, 2009
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Rémi Rhodes, Vincent Vargas CNRS, UMR 7534, F-75016 Paris, France Université Paris-Dauphine, Ceremade, 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 or phrases: Random measures, Multifractal processes. MSC 2000 subject classifications: 60G57, 28A78, 28A80
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: if T denotes some given cutoff parameter then the following equality in distribution holds for all λ ∈]0, 1]: (1)
(law)
(M(λA))A⊂B(0,T ) = λd eΩλ (M(A))A⊂B(0,T ) ,
1
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where B(0, T ) is the euclidian ball of radius T in Rd and Ωλ is an infinitely divisible random variable independent of (M(A))A⊂B(0,T ) . Let us note that a multi-dimensional generalization of MRM has already been proposed in the litterature ([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 stationary solutions to (1) for all λ ∈]0, 1]. In dimension d ≥ 2, MMRM are isotropic and homogeneous solutions to (1) for all λ ∈]0, 1]. If we consider a non negative random variable Y independent from M then the random measure (Y M(A))A⊂Rd is also solution to (1) for all λ ∈]0, 1]. This leads to the following open problem (unicity): Open problem 1. Consider two homogeneous and isotropic random Radon measures M and M ′ on the Borel σ-algebra B(Rd ). We suppose that there exists some cutoff parameter T such that M and M ′ satisfy (1) for all λ ∈]0, 1]. Can one find (in a possibly extended probability space) a non negative random variable Y (Y ′ ) independent of M (M ′ ) such that the following equality in law holds: (law)
(Y M(A))A⊂B(0,T ) = (Y ′ M ′ (A))A⊂B(0,T ) 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 [11] (see [6], [15]).
1.1 MMRM 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 velocity δv at two scales l, l′ < T , where T is an integral scale characteristic of the turbulent flow: if Pl (δv), Pl′ (δv) denote the probability density functions (p.d.f.) of the longitudinal velocity difference between two points separated by a distance l, l′ , one has: Z (2) Pl (δv) = Gl,l′ (x)Pl′ (e−x δv)e−x dx, R
where Gl,l′ is a p.d.f. If one makes the assumption that the Gl,l′ depend only on the factor λ = l/l′ (scale invariance), then it is easy to show that the Gl,l′ are the p.d.f.’s of infinitely 2
divisible laws. Under the assumption of scale invariance, if we take d = 1, A = [0, l′ ], λ = l/l′ and Ωλ − d ln(λ) of p.d.f. Gl,l′ in relation (1) (valid in fact simultaneously for all A) then equation (2) is the p.d.f. equivalent to (1). Following the standard conventions in turbulence, we note ǫ the (random) energy dissipation measure per unit mass and ǫl the mean energy dissipation per unit mass in a ball B(0, l):
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ǫl =
3 ǫ(B(0, l)) 4πl3
We believe the measures we consider in this paper can be used in dimension 3 to model the energy dissipation ǫ 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 measure ǫ is homogeneous and isotropic (as a function of the velocity field) and according to Kolmogorov’s refined similarity hypothesis (see for instance [4], [18] for studies of this hypothesis), one has the following relation in law between the longitudinal velocity increment δvl of two points separated by a distance l and ǫl : (law)
δvl = U(lǫl )1/3 , where U is a universal negatively skewed random variable independent of ǫl and of law independent of l. Therefore, equations similar to (2) should also hold for the p.d.f. of ǫl . Finally, let us note that the statistics of the velocity field and thus also those of ǫ are believed to be universal at scales smaller than T in the sense that they only depend on the average mean energy dissipation per unit mass < ǫ > defined by (note that the quantity below does not depend on l by homogeneity): < ǫ >=< ǫl > ǫ where denotes an average with respect to the randomness. In particular, the law of and the p.d.f.’s Gl,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.’s Gl,l′ . In ǫ dimension 3, the measures M we consider are precisely models for . 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.
3
2 Notations and main Results 2.1 Independently scattered infinitely divisible random measures. The characteristic function of an infinitely divisible random variable X can be written as E[eiqX ] = eϕ(q) , where ϕ is characterized by the Lévy-Khintchine formula Z 1 2 2 ϕ(q) = imq − σ q + (eiqx − 1 − iq sin(x)) ν(dx) 2 ∗ R R and ν(dx) is the so-called Lévy measure. It satisfies R∗ min(1, x2 ) ν(dx) < +∞. Let G be the unitary group of Rd , that is
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G = {M ∈ Md (R); MM t = 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 Borel σ-algebra B(G). Let S be the half-space S = {(t, y); t ∈ R, y ∈ R∗+ } with which we associate the measure (on the Borel σ-algebra B(S)) θ(dt, dy) = y −2dt dy. We consider an independently scattered infinitely divisible random measure µ associated to (ϕ, H ⊗ θ) and distributed on G × S (see [13]). More precisely, µ satisfies: 1) For every sequence of disjoint sets (An )n in B(G+ ×S), the random variables (µ(An ))n are independent and [ X µ µ(An ) a.s., An = n
n
2) for any measurable set A in 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 that µ is not necessarily a random signed measure. Let us additionnally mention that there exists a convex function ψ defined on R such that for all non empty subsets A of G × S: 1. ψ(q) = +∞, if E(eqµ(A) ) = +∞, 2. E(eqµ(A) ) = eψ(q)H⊗θ(A) otherwise. Let qc be defined as qc = sup{q ≥ 0; ψ(q) < +∞}. For any q ∈ [0, qc [, ψ(q) < +∞ and ψ(q) = ϕ(−iq). 4
2.2
Multidimensional Multifractal Random Measures (MMRM).
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We further assume that the independently scattered infinitely divisible random measure µ associated to (ϕ, H ⊗ θ) satisfies: ψ(2) < +∞, and ψ(1) = 0. The condition ψ(1) is just a normalization condition. The condtion ψ(2) < +∞ is technical and can probably be relaxed to the condition used in [1]: there exists ǫ > 0 such that ψ(1 + ǫ) < +∞. 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. Nevertheless, the condition ψ(2) < +∞ is general enough to cover the cases considered in turbulence: see [3], [16], [17]. Definition 2.3. Filtration Fl . Let Ω be the probability space on which µ is defined. Fl is defined as the σ-algebra generated by {µ(A × B); A ⊂ G, B ⊂ S, dist(B, R2 \ S) ≥ l}. Let us now define the function f : R+ → R by: l, if l ≤ T f (l) = T if l ≥ T The cone-like subset Al (t) of S is defined by: Al (t) = {(s, y) ∈ S; y ≥ l, −f (y)/2 ≤ s − t ≤ f (y)/2}. For forthcoming computations, we stress that for s, t real we have: θ(Al (s) ∩ Al (t)) = gl (|t − s|) where gl : R+ → R is given by (with the notation x+ = max(x, 0)): ln(T /l) + 1 − xl , if x ≤ l gl (x) = ln+ (T /x) if x ≥ l For any x ∈ Rd and m ∈ G, we denote by xm 1 the first coordinate of the vector mx. The cone product Cl (x) is then defined as: Cl (x) = {(m, t, y) ∈ G × S; (t, y) ∈ Al (xm 1 )}. Definition 2.4. ωl (x) process. The process ωl (x) is defined as ωl (x) = µ(Cl (x)). 5
Remark 2.5. For computational purposes, it is important that x 7→ ωl (x) possesses a version with a minimum of regularity. In the 1d-case, ωl is cadlag as a sum of Lévy processes (see [1]). In higher dimensions, we will prove in section 4.4 that ωl admits a version with at most a countable sets of discontinuities. Definition 2.6. Ml (dx) measure. For any l > 0, we define the measure Ml (dx) = eωl (x) dx, that is: Z Ml (A) = eωl (x) dx A
d
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for any Borel measurable subset A ⊂ R . Theorem 2.7. Multidimensional Multifractal Random Measure (MMRM). With probability one, there exists a limit measure (in the sense of weak convergence of measures): M(dx) = lim+ Ml (dx). l→0
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) a.s., ∀x ∈ Rd , M({x}) = 0, i.e. M has no atoms, ii) for any bounded subset K of Rd , M(K) < +∞ a.s. and E[M(K)] ≤ |K|. Proposition 2.8. Homogeneity 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 ∈ Rd . 2. The measure M is isotropic, i.e. the law of (M(A))A⊂Rd coincides with the law of (M(mA))A⊂Rd for each m ∈ G. Proposition 2.9. Stochastic scale invariance. 1. For any fixed λ ∈]0, 1] and l ≤ T , the two processes (ωλl (λx))x⊂B(0,T /2) and (Ωλ + ωl (x))x⊂B(0,T /2) have the same law, where Ωλ is an infinitely divisible random variable independent from the process (ωl (x))x⊂B(0,T /2) and its law is characterized by E[eiqΩλ ] = λ−ϕ(q) .
6
2. For any λ ∈]0, 1], the law of (M(λA))A⊂B(0,T /2) is equal to the law of (Wλ M(A))A⊂B(0,T /2) , where Wλ = λd eΩλ and Ωλ is an infinitely divisible random variable (independent of (M(A))A⊂B(0,T /2) ) and its characteristic function is: E[eiqΩλ ] = λ−ϕ(q) . 3. If ζ(q) 6= −∞ and 0 < t < T then: E M(B(0, t))q = (2t/T )ζ(q)E M(B(0, T /2))q .
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Proposition 2.10. Non-triviality of the MMRM. 1. The measure M is different from 0 if and only if there exists ǫ > 0 such that ζ(1+ǫ) > d; in that case, E(M(A)) = |A|. 2. Let q > 1 and consider the unique n ∈ N such that n < q ≤ n + 1. If ζ(q) > d and ψ(n + 1) < ∞, then E[M(A)q ] < +∞.
3 The limit lognormal case In the gaussian case, we have ψ(q) = γ 2 q 2 /2 and the condition ζ(1 + ǫ) > d corresponds to γ 2 < 2d. The approximating measures Ml are thus defined as: Z γ 2 E[Xl (x)2 ] 2 dx Ml (A) = eγXl (x)− A
where Xl 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)) Z m = gl (|xm 1 − y1 |)H(dm). G
The limit mesures M = liml→0 Ml (dx) we define are in the scope of the theory of gaussian multiplicative chaos developed by Kahane in [10] (see [14] for an introduction to this theory). Formally, the measure M is defined by: Z γ 2 E[X(x)2 ] 2 M(A) = eγX(x)− dx A
7
where X is a centered "gaussian field" (in fact a random tempered distribution ) with correlations given by: Z E[X(x)X(y)] =
G
m ln+ (T /|xm 1 − y1 |)H(dm).
Let us suppose that T = 1 for simplicity. Using invariance of the Haar measure H by multiplication, it is plain to see that E[X(x)X(y)] is of the form F (|y − x|) where F : R+ → R+ ∪ {∞}. We have the scaling relation F (ab) = ln(1/a) + F (b) if a ≤ 1 and b ≤ 1 (see also lemma 4.7 below) which entails that for |x| ≤ T : Z ln(|em F (|x|) = ln(1/|x|) − 1 |)H(dm).
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G
where e = (1, 0, . . . , 0) is the first vector of Rthe canonical basis. As a corollary, we get the existence of some constant C (take C = − G ln(|em 1 |)H(dm)) 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 dimension d ≤ 3 but is not positive definite for d ≥ 4 (see [14]). Remark 3.1. It is plain to see that 1 − |x|α is positive definite in a neighborhhod of 0 as a function defined in R if α ≤ 1. Therefore, one can consider the isotropic and positive definite function in a neighborhhood of 0 in Rd defined by: Z α F (|x|) = (1 − |xm 1 | )H(dm). G
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 of 0. Using the main theorem in [12], 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 the α, κ > 0 such that (1 − |x|α )κ+ is positive definite in Rd . It is known (see [8] for instance) that for α > 0 the function (1 − |x|α )+ is not positive definite in Rd if d ≥ 3.
4 Proofs 4.1 Proof of Theorem 2.7 The relation E[eωl (x) ] = eψ(1)H⊗θ(Cl (x)) = 1 ensures that for each Borelian subset A ⊂ Rd , the process Ml (A) is a martingale with respect to Fl . Existence of the MMRM then results from [9]. Properties i) and ii) result from Fatou’s lemma. 8
4.2 Characteristic function of ωl (x) The crucial point is to compute the characteristic function of ωl (x). We consider (x1 , . . . , xq ) ∈ (Rd )q and (λ1 , . . . , λq ) ∈ Rq and we have to compute 1 q φ(λ) = E eiλ1 ωl (x )+···+iλq ωl (x ) .
Let us denote by Sq the permutation group of the set {1, . . . , q}. For a generic element σ ∈ Sq , we define σ(1),m σ(q),m B σ = {m ∈ G; x1 < · · · < x1 }. Finally, given x, z ∈ Rd , we define the cone like subset product:
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Clσ (x) = Cl (x) ∩ B σ = {(m, t, y) ∈ B σ × S; (t, y) ∈ Al (xm 1 )} and m Clσ (x, z) = {(m, t, y) ∈ B σ × S; (t, y) ∈ Al (xm 1 ) ∩ Al (z1 )}.
Lemma 4.3. The characteristic function of the vector (ωl (xi ))1≤i≤q exactly matches: j q h i XX X σ σ σ(k) σ(j) 1 q α (j, k)ρl (x −x ) E exp iλ1 ωl (x ) + · · · + iλq ωl (x ) = exp σ∈Sq j=1 k=1
where ρσl (x) = H ⊗ θ(Clσ (0, x)), and: σ σ σ σ ασ (j, k) = ϕ(rk,j ) + ϕ(rk+1,j−1 ) − ϕ(rk,j−1 ) − ϕ(rk+1,j ) σ rk,j
=
j X
λσ(i)
(or 0 if k > j).
i=k
Moreover, j q X X
ασ (j, k) = ϕ
j=1 k=1
q X k=1
λk .
Proof. Without loss of generality, we assume xi 6= xj for i 6= j. We point out that the family (B σ )σ∈Sd is a partition of G up to a set of null H-measure. The function φ breaks down as: 1 q φ(λ) =E eiλ1 µ(Cl (x ))+···+iλq µ(Cl (x )) P σ q σ 1 =E e σ∈Sd iλ1 µ(Cl (x ))+···+iλq µ(Cl (x )) Y σ q σ 1 E eiλ1 µ(Cl (x ))+···+iλq µ(Cl (x )) = σ∈Sd
9
Let us fix σ ∈ Sq . We focus on the term: σ σ(q) σ σ(1) σ q σ 1 φσ (λ) = E eiλ1 µ(Cl (x ))+···+iλq µ(Cl (x ))) = E eiλσ(1) µ(Cl (x ))+···+iλσ(q) µ(Cl (x ))) .
Given σ ∈ Sd and p ≤ q, we further define: σ σ(p) σ σ(1) φσ (λ, p) = E eiλσ(1) µ(Cl (x ))+···+iλσ(p) µ(Cl (x ))) .
From now on, we adapt the proof of [1, Lemma 1] and proceed recursively. We define: Yqσ
=
q X
λσ(k) µ(Clσ (xσ(k) ) \ Cl (xσ(q) )),
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k=1
which stands for the contribution of the points of the above sum that do not belong to Cl (xσ(q) ). Moreover, the points in the set Cl (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, for m ∈ B σ , the coordinates are suitably σ(1),m σ(2),m σ(q),m sorted, that is: x1 < x2 < · · · < x1 . We define: σ Xk,q = µ Cl (xσ(k) , xσ(q) ) \ Cl (xσ(k−1) , xσ(q) )
with the convention Cl (xσ(k) , xσ(0) ) = Cl (xσ(0) , xσ(k) ) = ∅, in such a way that one has: λσ(1) µ(Clσ (xσ(1) ))
+···+
λσ(q) µ(Clσ (xσ(q) ))
=
Yqσ
+
q X
σ σ rk,q Xk,q .
k=1
σ Furthermore, since the variable Yq and (Xk,q )k are mutually independent, we get the following decomposition: q iY σ Y σ σ q φ (λ) = E e E eirk,q Xk,q . σ
(3)
k=1
Similarly, one can prove: (4)
q iY σ Y σ σ q φ (λ, q − 1) = E e E eirk,q−1 Xk,q . σ
k=1
10
Gathering (3) and (4) yields: irσ X σ q Y E e k,q k,q irσ X σ . φσ (λ, q) = φσ (λ, q − 1) E e k,q−1 k,q k=1 σ(k−1),m
For any m ∈ B σ , one has x1
σ(k),m
< x1
σ E eiαXk,q =eϕ(α)H⊗θ
σ(q),m
< x1
and therefore;
Cl (xσ(k) ,xσ(q) )\Cl (xσ(k−1) ,xσ(q) )
ϕ(α) H⊗θ(Cl (xσ(k) ,xσ(q) ))−H⊗θ(Cl (xσ(k−1) ,xσ(q) ))
=e
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Note that: σ(i)
H ⊗ θ(Cl (x
σ(j)
,x
)) = =
Z
ZB
σ(i),m
σ
θ Al (x1
σ(j),m
) ∩ Al (x1 σ(i),m
θ Al (0) ∩ Al (x1
Bσ σ =ρl (xσ(i)
− xσ(j) )
) H(dm)
σ(j),m
− x1
) H(dm)
The proof can now be completed recursively.
4.4 Regularity of ωl In the previous subsection, we have computed the finite dimensional distributions of the process ωl . To prove that the limiting measure M inherit the properties of the finite dimensional distributions of ωl , we need to establish some regularity properties of ωl , namely that ωl admits a version with at most a countable set of discontinuities. Remind that the function ϕ can be written as Z 1 2 2 (eiqx − 1 − iq sin(x)) ν(dx). imq − σ q + 2 R∗ Even if it means considering another measure µ, we may (and will) assume that µ is the sum of three independent independently scattered infinitely divisible random measures µ1 , µ2 , µ3 respectively associated with: Z 1 2 2 (eiqx − 1 − iq sin(x)) ν(dx). ϕ1 (q) = imq, ϕ2 (q) = − σ q , ϕ3 (q) = 2 R∗
11
Then ωl (x) is the sum of three independent random variable µ1 (Cl (x))+µ2 (Cl (x))+µ3 (Cl (x)). It is plain to see that for p > 1: E |µ1 (Cl (x))−µ1 (Cl (y))|p ≤ C(l, T )|x−y|p , E |µ2 (Cl (x))−µ2 (Cl (y))|p ≤ C(l, T )|x−y|p/2
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in such a way that the Kolmogorov criterion (see [19]) ensures that µ1 (Cl (x)) and µ2 (Cl (x)) admit a continuous version. Furthermore, the restriction P of µ3 to Fl is a Poisson random Z(ω) measure. Hence, for each realization, µ3 is of the form k=0 δXk , where Z is a random variable taking values in N and (Xk )k is a countable collection of iid random variables taking values in {(m, t, y) ∈ G × S; y ≥ l}. It is therefore plain to see that µ3 (Cl (x)) admits at most a countable set of discontinuities.
4.5 Homogeneity and isotropy Lemma 4.3 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 of ωl is itself invariant. This results from Lemma 4.3 since each term ρσl (xσ(k) − xσ(i) ) is invariant under translations, that is ρσl remains unchanged when you replace x1 , . . . , xq by x1 + z, . . . , xq + z for a given z ∈ Rd . Lemma 4.3 is nevertheless not very convenient to prove the isotropy of the MMRM so that we give a proof by a direct approach. Once again, it is sufficient to prove that the characteristic function of ωl is invariant under G. This time, we compute that characteristic function in a more direct way. We consider x1 , . . . , xq ∈ Rd , λ1 , . . . , λq ∈ R and m0 ∈ G, and define the function q X λk 1ICl (xk ) (m, t, y). f (m, t, y) = k=1
We have, using the right translation invariance of the Haar measure: h i h Z i 1 q E exp iλ1 ωl (m0 x ) + · · · + iλq ωl (m0 x ) =E exp i f (mm0 , t, y) µ(dm, dt, dy) Z = exp ϕ ◦ f (mm0 , t, y)H(dm)θ(dt, dy) Z = exp ϕ ◦ f (m, t, y)H(dm)θ(dt, dy) h i =E exp iλ1 ωl (x1 ) + · · · + iλq ωl (xq ) . The isotropy follows.
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4.6 Exact scaling and stochastic scale invariance Lemma 4.7. Exact scaling of Ml (dx). For all λ ∈ (0, 1] and x1 , . . . , xq ∈ B(0, T /2), the functions ρσl satisfy the exact scaling relation (5) j q j q X XX X XX σ σ σ(k) σ(j) ασ (j, k)ρσl (xσ(k) − xσ(j) ). α (j, k)ρλl (λx − λx ) = − ln(λ) + σ∈Sq j=1 k=1
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σ∈Sq j=1 k=1
R Proof. We remind that for x real we have Al (0)∩Al (x) θ(dt, dy) = gl (|x|). Given B ⊂ G and x ∈ Rd , we define: Z B ρl (x) = 1IB (m)1I{(t,y)∈Al (0)∩Al (xm θ(dt, dy)H(dm). 1 )} Then we can compute the function ρB l : Z Z Z B ρl (x) = 1IB (m)1I{(t,y)∈Al (0)∩Al (xm H(dm)θ(dt, dy) = θ(dt, dy) H(dm) 1 )} B Al (0)∩Al (xm 1 ) Z m m m = ln(T /l) + 1 − |xm |/l 1 I + ln(T /|x |)1 I |x1 |≤l l≤|x1 |≤T H(dm) 1 1 B
Given λ ∈]0, 1] and x ∈ B(0, T ): Z T |λxm T 1 | B m |≤T H(dm) 1I|λxm + ln( )1 I ρλl (λx) = ln( ) + 1 − |≤λl λl≤|λx 1 1 λl λl λ|xm 1 | ZB T |xm | T m |≤T H(dm) + ln( )1 I = ln( ) + 1 − 1 1I|xm |≤l λl≤|λx 1 1 l l |xm B 1 | Z − ln(λ) 1I|xm + 1Il≤|xm H(dm) 1 |≤l 1 |≤T B
=ρB l (x)
− ln(λ)H(B)
13
We therefore obtain: j q X XX
ασ (j, k)ρσλl (λxσ(k) − λxσ(j) )
σ∈Sq j=1 k=1
=
j q X XX
α
σ
(j, k)ρσl (xσ(k)
σ(j)
−x
) − ln(λ)
=
α
σ
(j, k)ρσl (xσ(k)
σ(j)
−x
σ∈Sq j=1 k=1 q
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=
ασ (j, k)H(B σ )
σ∈Sq j=1 k=1
σ∈Sq j=1 k=1
j q X XX
j q X XX
) − ln(λ)
X
ϕ(
σ∈Sq
q X
λk )H(B σ )
k=1
q
j
X XX
ασ (j, k)ρσl (xσ(k) − xσ(j) ) − ln(λ)ϕ(
σ∈Sq j=1 k=1
X
λk )
k=1
From Lemma 4.3, we deduce that, for any λ ∈ (0, 1], there exists a random variable Cλ law such that (ωλl (λx))x∈B(0,T /2) = (Cλ + ωλl (λx))x∈B(0,T /2) and such that Cλ is independent of (ωλl (λx))x∈B(0,T /2) and its characteristic function is given by E[eiqCλ ] = λ−ϕ(q) . By integrating the previous relation, we obtain the relation: law
(Mλl (λA))A⊂B(0,T /2) = Wλ (Ml (A))A⊂B(0,T /2) where Wλ = λd eCλ is a random variable independent of (Ml (A))A⊂B(0,T /2) .
4.8 Non-triviality of the MMRM Suppose we can find a "cube" CR = [0, R]d and q > 1 such that: E M(CR )q < +∞.
Then we can find n ∈ N such that [0, 2−n R]d ⊂ B(0, T /2). We split the cube CR into 2nd smaller cubes d Y k,n [ki 2−n R, (ki + 1)2−n R), C = i=1
def
where k = (k1 , . . . , kd ) ∈ Ndn = Nd ∩ [0, 2n − 1]d . For each fixed value of n, the cubes (C k,n )k , where the index k varies in Ndn form a partition of CT . Thus, by using the super-
14
additivity of the function x 7→ xq , we have: h X q i E M(CT )q = E M(C k,n ) k∈Ndn
≥
X
k∈Ndn
q E M(C k,n )
hal-00394758, version 1 - 12 Jun 2009
By using the translation invariance and the scale invariance property of the MMRM, we deduce: q q q q E M(C k,n ) = E M(C 0,n ) = E M(2−n CR ) = 2−nζ(q) E CR ) . Finally, gathering the previous inequalities yields: q q E CR ) ≥ 2nd−nζ(q) E CR )
in such a way that, necessarily, ζ(q) ≥ d. The proof of 1. and 2. is then a consequence of the following lemma:
Lemma 4.9. Let q > 1 and 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 Ml ([0, T )d)q ≤ C. l
Proof. The proof is an adaptation of the one in [1] (which is itself an adaptation of the corresponding result in [2]). Unfortunately, the multi-dimensional setting is a 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: (m) [0, 1)2 = ∪ m Ii,j , 0≤i,j≤2 −1
(m)
) × [ 2jm , j+1 ). Let us write the above decompostion in the following where Ii,j = [ 2im , i+1 2m 2m form: [0, 1)2 = C1 ∪ C2 ∪ C3 ∪ C4 . where C1 =
∪
i and j even
and C3 =
(m)
∪
i odd and j even
Ii,j , (m)
Ii,j ,
C2 = C2 =
15
∪
i and j odd
∪
(m)
Ii,j
(m)
Ii,j .
i even and j odd
Since the measure Ml is homogeneous, we get: 4 X 2 q q−1 E Ml ([0, 1) ) ≤ 4 E Ml (Ci )q
i=1 ≤ 4 E Ml (C1 )q . q
Now we get the following by subadditivity of x → xq/(n+1) : X (m) E Ml (C1 )q = E ( Ml (I2i,2j ))q 0≤i,j≤2m−1 −1
X (m) = E (( Ml (I2i,2j ))n+1 )q/(n+1)
hal-00394758, version 1 - 12 Jun 2009
i,j
=E ( ≤E
n Y
X
i1 ,j1 ,...,in+1 ,jn+1 k=1 n Y X
(
(m)
(m)
Ml (I2ik ,2jk ))q/(n+1) Ml (I2ik ,2jk ))q/(n+1)
i1 ,j1 ,...,in+1 ,jn+1 k=1
(m) = 22(m−1) E Ml (I0,0 )q +
∗ X
i1 ,j1 ,...,in+1 ,jn+1
n Y (m) Ml (I2ik ,2jk )q/(n+1) E
(m) ≤ 22(m−1) E Ml (I0,0 )q +
∗ X
E
i1 ,j1 ,...,in+1 ,jn+1
k=1 n Y
k=1
(m)
Ml (I2ik ,2jk )
q/(n+1)
∗ P 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 q/(n+1) Y (m) (6) E Ml (I2ir ,2jr )nr r=1
P where the sequence of positive integers (nr )1≤r≤k satisfies kr=1 nr + 1 and the I2ir ,2jr are disjoint intervals wich lie at a distance of at least 21m . We want to show that each term of the form (6) is bounded by some quantity Cm independent of l. We get the following computation using Fubini: Z k Y 1 n+1 (m) nr = E eωl (x )+···+ωl (x ) dx1 . . . dxn+1 . E Ml (I2ir ,2jr ) n
n
r=1
I2i1 ,2j ×···×I2ik ,2j 1
1
k
16
k
We define Nr = n1 + · · · + nr for r in [1, k] and we introduce the following set Al = Al (x1 , . . . , xn+1 ): Al = ∪r
