Hermine Bierm É

Éline Lacaux
email: celine.lacaux@iecn.u-nancy.fr

Hans-Peter Scheffler
email: scheffler@mathematik.uni-siegen.de

MAP Hermine Biermé
email: hermine.bierme@mi.parisdescartes.fr

MULTI-OPERATOR SCALING RANDOM FIELDS

Keywords: 2010 Mathematics Subject Classification. Primary: 60G17 60G60 60G15 60G52 ; Secondary: 60F05 60G22 60G18 July 4, 2011 Gaussian and stable random fields, local operator scaling property, Hölder regularity

In this paper, we define and study a new class of random fields called harmonizable multi-operator scaling stable random fields. These fields satisfy a local asymptotic operator scaling property which generalizes both the local asymptotic self-similarity property and the operator scaling property. Actually, they locally look like operator scaling random fields whose order is allowed to vary along the sample paths. We also give an upper bound of their modulus of continuity. Their pointwise Hölder exponents may also vary with the position x and their anisotropic behavior is driven by a matrix which may also depend on x.

Introduction

Self-similar random processes and fields are required to model numerous natural phenomena, e.g. in internet traffic, hydrology, geophysics or financial markets, see for instance [START_REF] Willinger | Self-similarity and heavy tails: Structural modeling of network traffic[END_REF][START_REF] Véhel | Fractals in engineering: from theory to industrial applications[END_REF][START_REF] Abry | Lois d'échelle, fractales et ondelettes[END_REF].

A very important class of such fields is given by fractional stable random fields (see [START_REF] Samorodnitsky | Stable non-Gaussian random processes[END_REF]). In particular, the well-known fractional Brownian field B H is a Gaussian H-self-similar random field with stationary increments. It is an isotropic generalization of the famous fractional Brownian motion ( [START_REF] Mandelbrot | Fractional Brownian motion, fractional noises and applications[END_REF][START_REF] Kolmogorov | Wienersche Spiralen und einige andere interessante Kurven in Hilbertsche Raum[END_REF]). Self-similar isotropic α-stable fields have been extensively used to propose an alternative to Gaussian modeling (see [START_REF] Mikosch | Is network traffic approximated by stable Lévy motion or fractional Brownian motion?[END_REF][START_REF] Willinger | Self-similarity and heavy tails: Structural modeling of network traffic[END_REF] for instance) to mimic heavy-tailed persistent phenomena.

However, isotropy property is a serious drawback for many applications in medicine [START_REF] Brunet-Imbault | A new anisotropy index on trabecular bone radiographic images using the fast Fourier transform[END_REF], in geophysics [START_REF] Pecknold | The simulation of universal multifractals[END_REF][START_REF] Candela | Characterization of fault roughness at various scales: Implications of three-dimensional high resolution topography measurements[END_REF] and in hydrology [START_REF] Benson | Aquifer operator-scaling and the effect on solute mixing and dispersion[END_REF], just to mention a few. Recently, an important class of anisotropic random fields has been studied in [START_REF] Biermé | Operator scaling stable random fields[END_REF]. These fields are anisotropic generalizations of self-similar stable random fields. They satisfy an operator scaling property which generalizes the classical self-similarity property. More precisely, for E a real d × d matrix whose eigenvalues have positive real parts, a scalar valued random field (X(x)) x∈R d is called operator scaling of order E and H > 0 if, for every c > 0,

{X(c E x); x ∈ R d } (f dd) = {c H X(x); x ∈ R d }, (1)
where

(f dd)
= means equality of finite dimensional distributions and as usual c E = exp(E log c). Let us recall that the self-similarity property corresponds to the case where E is the identity matrix.

Let us also remark that up to consider the matrix E/H, we may and will assume, without loss of generality, that H = 1. The anisotropic behavior of operator scaling random fields with stationary increments is then driven by a matrix. In particular, when θ j is an eigenvector of E associated with the eigenvalue λ j , any operator scaling random field for E is 1/λ j -self-similar in direction θ j . Furthermore, the critical global and directional Hölder exponents of harmonizable operator scaling stable random fields are given by the eigenvalues of E (see [START_REF] Biermé | Operator scaling stable random fields[END_REF][START_REF] Biermé | Hölder regularity for operator scaling stable random fields[END_REF]). Let us emphasize that these exponents and the directions of self-similarity do not vary according to the position.

Moreover, the self-similarity is a global property which can be too restrictive for applications.

Actually, numerous phenomena exhibit scale invariance that may vary according to the scale or to the position and are usually called multifractal (see [START_REF] Chainais | Virtual super resolution of scale invariant textured images using multifractal stochastic processes[END_REF][START_REF] Riedi | Multifractal processes. In Theory and applications of long-range dependence[END_REF][START_REF] Pecknold | The simulation of universal multifractals[END_REF] for examples). To allow more flexibility, [START_REF] Benassi | Gaussian processes and Pseudodifferential Elliptic operators[END_REF] has introduced the local asymptotic self-similarity property. This property characterizes random fields that locally seem self-similar but whose local regularity properties evolve. Since then, many examples of locally asymptotically self-similar random fields have been introduced and studied, e.g. in [START_REF] Benassi | Gaussian processes and Pseudodifferential Elliptic operators[END_REF][START_REF] Peltier | Multifractional Brownian motion: definition and preliminary results[END_REF][START_REF] Benassi | Identification of Filtered White Noises[END_REF][START_REF] Ayache | The generalized multifractional Brownian motion[END_REF][START_REF] Lacaux | Real harmonizable multifractional Lévy motions[END_REF][START_REF] Stoev | Path properties of the linear multifractional stable motion[END_REF].

In this paper, we introduce the local asymptotic operator scaling property which generalizes both the local asymptotic self-similarity property and the operator scaling property. A scalar valued random field X is locally asymptotically operator scaling at point x of order A(x) if

lim ε→0 + X(x + ε A(x) u) -X(x) ε u∈R d (f dd) = (Z x (u)) u∈R d , (2)
with Z x a non degenerate random field. Let us first remark that the local asymptotic selfsimilarity property of exponent h(x) corresponds to the local asymptotic operator self-similarity of order A(x) = I d /h(x) with I d the identity matrix of order d. Moreover, operator scaling random fields of order E are locally asymptotically operator scaling at point 0 of order E. Of course, if they have also stationary increments, they are locally asymptotically operator scaling at any point x. In addition, if (2) is fulfilled, the random field Z x is operator scaling of order A(x). In other words, a local asymptotic multi-operator random field locally looks like an operator scaling random field whose order is allowed to vary along the sample paths.

Then, we focus on harmonizable multi-operator scaling stable random fields, which generalize harmonizable operator scaling stable random fields. A harmonizable multi-operator scaling stable random field X satisfies the local asymptotic self-similarity property [START_REF] Ayache | The generalized multifractional Brownian motion[END_REF] with Z x a harmonizable operator scaling stable random field of order A(x). Moreover, its local sample path properties at point x are the same as those of Z x which have been established in [START_REF] Biermé | Operator scaling stable random fields[END_REF][START_REF] Biermé | Hölder regularity for operator scaling stable random fields[END_REF]. Hence, its local regularity varies with the position x and its anisotropic behavior is driven by a matrix that depends on x. In particular, for any eigenvector θ j (x) of A(x) associated with the real eigenvalue λ j (x), the random field X admits H j (x) = 1/λ j (x) as pointwise Hölder exponent in direction θ j (x) at point x. Let us point out that we establish an accurrate upper bound for the modulus of continuity. Such upper bound has already been given for real harmonizable fractional stable motions in [START_REF] Kôno | Hölder continuity of sample paths of some self-similar stable processes[END_REF], for some Gaussian random processes in [START_REF] Kôno | On the modulus of continuity of sample functions of gaussian processes[END_REF] and for harmonizable operator scaling stable random fields in [START_REF] Biermé | Hölder regularity for operator scaling stable random fields[END_REF]. Then, in this paper, we generalize these results to harmonizable multi-operator scaling stable random fields. To study the sample paths in the case of α-stable random fields with α ∈ (0, 2), we use a LePage series representation (see [START_REF] Lepage | Conditional moments for coordinates of stable vectors[END_REF][START_REF] Lepage | Multidimensional infinitely divisible variables and processes. II. In Probability in Banach spaces[END_REF] for details on such series) which is chosen to be conditionnally Gaussian as in [START_REF] Kôno | Hölder continuity of sample paths of some self-similar stable processes[END_REF][START_REF] Biermé | Hölder regularity for operator scaling stable random fields[END_REF].

Harmonizable multi-operator scaling stable random fields are defined in Section 2. In this section, we also state all the assumptions we will need and present many examples that fulfill them. Section 3 is devoted to the properties of the polar coordinates: these coordinates are one of the main tools we use to study the sample paths as in [START_REF] Biermé | Operator scaling stable random fields[END_REF][START_REF] Biermé | Hölder regularity for operator scaling stable random fields[END_REF]. In Section 4, we state the sample path properties of the class of random fields under study (modulus of continuity and pointwise directional Hölder exponents). Section 5 is devoted to the local asymptotic operator self-similar property. Some technical proofs are postponed to the Appendix.

Throughout this paper, B(x, γ) denotes the closed Euclidean ball of center x and radius γ.

Harmonizable representation

Harmonizable stable random fields are defined as stochastic integrals of deterministic kernels with respect to a stable random measure. In this paper we will always assume that the following assumption holds: Assumption 1. Let α ∈ (0, 2] and W α be a complex isotropic α-stable random measure with Lebesgue control measure (see [START_REF] Samorodnitsky | Stable non-Gaussian random processes[END_REF] p.281 for details on such measures). Note that W 2 is an isotropic complex Gaussian random measure.

Let us recall (see [START_REF] Samorodnitsky | Stable non-Gaussian random processes[END_REF]) that the stochastic integral

W α (f ) := R d f (ξ)W α (dξ) is well-defined if and only if f ∈ L α R d . Furthermore, for f ∈ L α R d , W α (f ) is a stable complex-valued random variable whose characteristic function is given by ∀z ∈ C, E(exp (i Re (zW α (f )))) = exp (-s α W α (f ) α α |z| α ) where W α (f ) α = R d |f (ξ)| α dξ 1/α and s α = 1 2π 2π 0 |cos (ξ)| α dξ.
Note that if α = 2, for each square integrable function f , the stochastic integral W 2 (f ) is a centered Gaussian random variable.

According to [START_REF] Biermé | Operator scaling stable random fields[END_REF], a harmonizable operator scaling stable random field X = (X(x)) x∈R d is defined by

X(x) = Re R d e i x,ξ -1 ψ(ξ) -1-trace(E 0 )/α W α (dξ), (3)
with E 0 a d × d real matrix whose eigenvalues have real parts greater than 1 and ψ : R d → [0, ∞)

a continuous E 0 -homogeneous function, that is ψ(c E 0 ξ) = cψ(ξ) for all c > 0 and x ∈ R d , such that ∀ξ = 0, ψ(ξ) = 0.
In order to obtain a field whose local behavior is given by a harmonizable operator scaling stable random field, we replace in (3) the matrix E 0 (respectively the function ψ) by a matrix E(x)

(respectively a function ψ x ) which depends on the position x. In this approach, the function ψ x is E(x)-homogeneous. This leads us to consider

X α,ψ (x) = Re R d e i x,ξ -1 ψ x (ξ) -1-trace(E(x))/α W α (dξ).
This approach has already been used to define the multifractional Brownian field in [START_REF] Benassi | Gaussian processes and Pseudodifferential Elliptic operators[END_REF][START_REF] Peltier | Multifractional Brownian motion: definition and preliminary results[END_REF]. To ensure that the field X α,ψ is well-defined, we only have to assume that (E(x), ψ x ) satisfies the assumptions of [START_REF] Biermé | Operator scaling stable random fields[END_REF] for all x. Before we state these assumptions, let us introduce several notations we will use throughout the paper.

Notation. We denote by M >0 R d the space of all d × d real matrices whose eigenvalues have positive real parts. In the following, for any

x ∈ R d , E(x) ∈ M >0 R d . The eigenvalues of E(x)
are denoted by λ 1 (x), . . . , λ d (x). For each j = 1, . . . , d and each x ∈ R d , we set

a j (x) = Re (λ j (x)), H j (x) = 1 a j (x) , H(x) = max 1≤i≤d H i (x) and H(x) = min 1≤i≤d H i (x). ( 4
)
The multi-operator scaling random field X α,ψ is well-defined as soon as the two following assumptions are fulfilled. These assumptions come from [START_REF] Biermé | Operator scaling stable random fields[END_REF] when E and ψ do not vary with the position x.

Assumption 2. Assume that

∀x ∈ R d , min 1≤j≤d a j (x) > 1
with a j defined by (4).

Assumption 3. For every x ∈ R d , let ψ x : R d → [0, +∞) be a continuous function, E(x)-
homogeneous which means, according to Definition 2.6 of [START_REF] Biermé | Operator scaling stable random fields[END_REF], that

ψ x (c E(x) ξ) = cψ x (ξ) for all c > 0 and ξ ∈ R d .
Let us also assume that ψ x (ξ) = 0 for ξ = 0.

Following ideas of [START_REF] Ayache | The generalized multifractional Brownian motion[END_REF], let us now define generalized multi-operator scaling stable random fields.

These fields will be useful in the study the sample paths of harmonizable multi-operator scaling stable random fields.

Theorem 2.1. Assume that Assumptions 1, 2 and 3 are fulfilled. Then, the random field

Y α,ψ (x, y, z) = Re R d e i x,ξ -1 ψ y (ξ) -βα(z) W α (dξ), x, y, z ∈ R d , (5)
where

β α (z) = 1 + q(z) α with q(z) = trace(E(z)) (6)
is well-defined on the non empty set

U = (x, y, z) ∈ R 3d : 0 < 1 + (q(z) -q(y))/α < min 1≤j≤d Re (λ j (y)) = 1 H(y)
.

The random field Y α,ψ is called generalized multi-operator scaling stable random field.

Proof. Let x, y, z ∈ R d and H = 1 + (q(z) -q(y))/α. Since β α (z) = H + q(y)/α, according to Theorem 4.1 of [START_REF] Biermé | Operator scaling stable random fields[END_REF] (applied with ψ = ψ y ), the random variable Y α,ψ (x, y, z) is well-defined as soon as

0 < H < min 1≤j≤d Re (λ j (y)) = 1 H(y) ,
which holds for any (x, y, z) ∈ U .

We now introduce the class of harmonizable multi-operator scaling random fields which will study in this paper.

Definition 2.1. Assume that Assumptions 1, 2 and 3 are fulfilled. Then, the random field

X α,ψ (x) = Y α,ψ (x, x, x) = Re R d e i x,ξ -1 ψ x (ξ) -βα(x) W α (dξ) , x ∈ R d , (7)
with β α defined by [START_REF] Biermé | Hölder regularity for operator scaling stable random fields[END_REF], is well-defined and is called harmonizable multi-operator scaling stable random field.

Remark 2.1. If α = 2, X α,ψ is a real-valued centered Gaussian random field.

Let us emphasize that to study the sample paths of X α,ψ , we need the functions ψ and E to be sufficiently regular. We introduce now all the assumptions we will use in sequel.

Assumption 4. Let T = d i=1 [b i , d i ] with b i < d i for 1 ≤ i ≤ d.
Let us assume that the function

(x, ξ) → ψ x (ξ) is locally Lipschitz on T × R d \{0}, that is for every compact set K ⊂ T × R d \{0},
there exists a finite positive constant c 2,1 = c 2,1 (K) such that

|ψ x 1 (ξ 1 ) -ψ x 2 (ξ 2 )| ≤ c 2,1 ( x 1 -x 2 + ξ 1 -ξ 2 )
for every (x 1 , ξ 1 ), (x 2 , ξ 2 ) ∈ K.

Assumption 5. Let T = d i=1 [b i , d i ] with b i < d i for 1 ≤ i ≤ d.
Let us assume that the map

E : x -→ E(x) is a Lipschitz function on T : there exists a finite positive constant c 2,2 = c 2,2 (T ) such that, for x 1 , x 2 ∈ T E(x 1 ) -E(x 2 ) ≤ c 2,2 x 1 -x 2 . Assumption 6. Let T = d i=1 [b i , d i ] with b i < d i for 1 ≤ i ≤ d.
Let us assume that for any

x, y ∈ T , E(x) and E(y) are commuting matrices:

E(x)E(y) = E(y)E(x).
We now conclude this section by several examples. We first give two straightforward classes of examples. The first one is given by harmonizable operator scaling stable random fields. The second one includes the classical multifractional Brownian field as defined in [START_REF] Benassi | Gaussian processes and Pseudodifferential Elliptic operators[END_REF]. For all x, ξ ∈ R d , let

E(x) = E 0 and ψ x = ψ.
Then, Assumptions 2-6 are fulfilled and under Assumption 1, X α,ψ is a harmonizable operator scaling stable random field for E t 0 with stationary increments, see [START_REF] Biermé | Operator scaling stable random fields[END_REF]. In particular, X α,ψ satisfies the operator-scaling property (1) for E t 0 (and H = 1).

Example 2.2 (Multifractional operator scaling random fields). Let E 0 and ψ be as in Example 2.1 and let h : R d -→ (0, 1) be a locally Lipschitz function. For all x ∈ R d , let us define

E(x) = 1 h(x) E 0 and ψ x = ψ h(x) .
Then, Assumptions 2-6 are fulfilled and under Assumption 1, the random field X α,ψ given by [START_REF] Biermé | Operator scaling stable random fields[END_REF] is well-defined. In particular, if E 0 = I d is the identity matrix and if ψ = • is the Euclidean norm on R d , then X α,ψ is a multifractional harmonizable stable random field, called multifractional Brownian field if α = 2 (see [START_REF] Benassi | Gaussian processes and Pseudodifferential Elliptic operators[END_REF]).

Remark 2.2. Let us focus on the special case d = 1. If we assume that ψ x is an even function for any x ∈ R d , Assumption 3 implies that there exists a positive function c such that

ψ x (ξ) = c(x)|ξ| h(x) ,
x → E(x) = a(x) cos(θ(x)) sin(θ(x)) -sin(θ(x)) cos(θ(x))
where a and θ are locally Lipschitz functions on R d . Assume that

∀x ∈ R d , a(x) cos (θ(x)) > 1.
For every x ∈ R d and ξ ∈ R d , let ψ x (ξ) = ξ 1/(a(x) cos(θ(x))) .

Then Assumptions 2-6 are fulfilled such that, under Assumption 1, the random field X α,ψ given by (7) is well-defined.

Example 2.6. Let E i : R d → M >0 (R d ) satisfying Assumption 2 for i ∈ {1, 2} and let ψ (i) satisfying Assumption 3 with respect to E i for i ∈ {1, 2}. Consider the map

E = E 1 1 [0,1] d + E 2 1 R d [0,1] d .
and for any x ∈ R d , the function

ψ x (ξ) = ψ (1) x (ξ)1 [0,1] d (x) + ψ (2) x (ξ)1 R d [0,1] d (x)
. Then ψ satisfies Assumption 3 with respect to E. The random fields X α,ψ (1) , X α,ψ (2) and X α,ψ are well-defined by [START_REF] Biermé | Operator scaling stable random fields[END_REF] and

X α,ψ = X α,ψ (1) 1 [0,1] d + X α,ψ (2) 1 R d [0,1] d .
The approach proposed in this example allows to define harmonizable stable random fields which are piecewise operator scaling.

In the next section we recall one of the main tools needed to study operator scaling random fields, in particular a change of variables formula with respect to adapted polar coordinates.

Polar coordinates

Let us recall the main properties of polar coordinates adapted to a single matrix as introduced in [START_REF] Meerschaert | Limit distributions for sums of independent random vectors[END_REF]. Let M ∈ M >0 R d . As in Chapter 6 of [START_REF] Meerschaert | Limit distributions for sums of independent random vectors[END_REF], let us consider the norm • M defined by

x M = 1 0 t M x dt t , ∀x ∈ R d ( 8
)
where • is the Euclidean norm on R d . Then, according to Chapter 6 of [START_REF] Meerschaert | Limit distributions for sums of independent random vectors[END_REF],

• M is a norm on R d such that the map

Ψ M : (0, +∞) × S M -→ R d \ {0} (r, θ) -→ r M θ
is a homeomorphism, where

S M = {ξ ∈ R d : ξ M = 1} (9)
is the unit sphere for • M . Hence we can write any ξ ∈ R d \{0} uniquely as

ξ = τ M (ξ) M M (ξ) (10)
with τ M (ξ) > 0 and M (ξ) ∈ S M . Here, for any ξ ∈ R d \{0}, τ M (ξ) should be interpreted as the radial part of ξ with respect to M and M (ξ) ∈ S M as its directional part with respect to M .

Let us now recall the formula of integration in polar coordinates established in [START_REF] Biermé | Operator scaling stable random fields[END_REF].

Proposition 3.1. There exists a unique finite Radon measure σ M on the unit sphere S M defined by [START_REF] Candela | Characterization of fault roughness at various scales: Implications of three-dimensional high resolution topography measurements[END_REF] such that for all f ∈ L 1 (R d , dξ),

R d f (ξ) dξ = +∞ 0 S M f (r M θ) σ M (dθ) r trace(M )-1 dr.
The main difficulty in our setting is that we do not consider a single matrix but a family (E(x)) x∈R d of matrices. Hence we need uniform controls on the polar coordinates. These will follow from the next lemmas.

Lemma 3.2. Let T = d i=1 [b i , d i ] with b i < d i for 1 ≤ i ≤ d. Assume that E : T → M >0 (R d ) is
continuous on T and satisfies Assumption 6 on T . Then the map

P : [0, +∞) × T -→ M R d (t, x) -→ t E(x)
is continuous on [0, +∞) × T (with convention 0 E(x) = 0).

Proof. According to Proposition 2.2.11 of [START_REF] Meerschaert | Limit distributions for sums of independent random vectors[END_REF], since E : T → M >0 R d is continuous on T , the map P is continuous on (0, +∞) × T . Therefore, the main problem is to prove that P is continuous at (0, x) for any x ∈ T .

Let us fix x ∈ T . Then, let δ > 0 such that the real parts of all the eigenvalues of E(x) are greater than 2δ. It follows from Theorem 2.2.4 of [START_REF] Meerschaert | Limit distributions for sums of independent random vectors[END_REF] that

sup θ =1 t 2δ t -E(x) θ ----→ t→+∞ 0.
Then, by continuity of t → t -E(x) on [1, +∞[, one can find a finite positive constant c δ such that

∀t ∈ [1, +∞), t -E(x) := sup θ =1 t -E(x) θ ≤ c δ t -2δ .
Now, since E is continuous on T , there exists r δ ∈ (0, +∞) such that

∀y ∈ B(x, r δ ) ∩ T, E(x) -E(y) ≤ δ,
where B(x, r δ ) is the closed Euclidean ball centered at point x with radius r δ . Therefore for any s ∈ (0, 1] and any y ∈ B(x, r δ ) ∩ T , according to Assumption 6

s E(y) = s E(y)-E(x) s E(x) ≤ s E(y)-E(x) s E(x) ≤ c δ s -E(y)-E(x) s 2δ .
Hence, for any s ∈ (0, 1] and any y ∈ B(x, r δ ) ∩ T ,

s E(y) ≤ c δ s δ ,
which also holds for s = 0 by convention and concludes the proof.

Let us remark that one can establish the continuity of P on [0, +∞)×T without Assumption 6. However, without Assumption 6, the proof is very long and this assumption will be needed in sequel.

Lemma 3.2 leads to an uniform control of t E(x) with respect to the eigenvalues of E(x), stated in the next lemma, whose proof is postponed to the Appendix.

Lemma 3.3. Let T = d i=1 [b i , d i ] with b i < d i for 1 ≤ i ≤ d.
Let H and H be defined by (4).

Assume that E : T → M >0 (R d ) is continuous on T and satisfies Assumption 6. Then, for any δ > 0 and r 0 > 0, there exist some finite constants c 3,1 = c 3,1 (T, δ, r 0 ) > 0 and c 3,2 = c 3,2 (T, δ, r 0 ) such that for any x ∈ T , (i) for all t ∈ [0, r 0 ],

t 1/H(x) ≤ t E(x) ≤ c 3,1 t 1/H(x)-δ ; (ii) for all t ∈ [r 0 , +∞), t 1/H(x) ≤ t E(x) ≤ c 3,2 t 1/H(x)+δ .
Moreover, Lemma 3.2 leads also to an uniform control of • E(x) with respect to the Euclidean norm, stated in the next lemma, whose proof is again postponed to the Appendix.

Lemma 3.4. Let T = d i=1 [b i , d i ] with b i < d i for 1 ≤ i ≤ d. Assume that E : T → M >0 (R d )
is continuous on T and satisfies Assumption 6. Then there exist two finite positive constants

c 3,3 = c 3,3 (T ) and c 3,4 = c 3,4 (T ) such that ∀x ∈ T, ∀ξ ∈ R d , c 3,3 ξ E(x) ≤ ξ ≤ c 3,4 ξ E(x)
and such that

∀x ∈ T, c 3,3 ≤ σ E(x) S E(x) ≤ c 3,4
with σ E(x) the measure introduced in Proposition 3.1.

Using Lemmas 3.3 and 3.4 we can compare uniformly the radial parts with the Euclidean norm. The following proposition, whose proof is postponed to the Appendix, is one of the main tools to obtain Hölder regularity of multi-operator scaling stable random fields.

Proposition 3.5. Let T = d i=1 [b i , d i ] with b i < d i for 1 ≤ i ≤ d.
Let H and H be defined by (4).

Assume that E : T → M >0 (R d ) is continuous on T and satisfies Assumption 6. Then, for any δ ∈ (0, min x∈T H(x)), there exist two finite positive constants c 3,5 = c 3,5 (T, δ) and c 3,6 = c 3,6 (T, δ) such that for all x ∈ T and ξ ≤ 1,

c 3,5 ξ H(x)+δ ≤ τ E(x) (ξ) ≤ c 3,6 ξ H(x)-δ , (11)
and, for all ξ ≥ 1,

c 3,5 ξ H(x)-δ ≤ τ E(x) (ξ) ≤ c 3,6 ξ H(x)+δ . (12)
Let us mention that for any fixed x ∈ R d , the inequality [START_REF] Herbin | Stochastic 2-microlocal analysis[END_REF], respectively [START_REF] Kolmogorov | Wienersche Spiralen und einige andere interessante Kurven in Hilbertsche Raum[END_REF], holds true with | log( ξ )| d instead of ξ -δ , respectively instead of ξ δ , with constants c 3,5 , c 3,6 depending on x (see [START_REF] Biermé | Hölder regularity for operator scaling stable random fields[END_REF] for a proof).

We end this section by comparing the radial parts τ E(x) (ξ) and τ E(y) (ξ), uniformly in ξ, if x and y are closed enough. This result will be useful to obtain an upper bound for the modulus of continuity of multi-operator scaling stable random fields. Its proof is postponed to the Appendix.

Proposition 3.6. Let T = d i=1 [b i , d i ] with b i < d i for 1 ≤ i ≤ d. Assume that E : T → M >0 (R d )
is continuous on T and satisfies Assumption 6. Then, for any ε ∈ (0, 1), there exists γ > 0 such that for all x, y ∈ T with x -y ≤ γ,

c 3,7 τ E(y) (ξ) 1+ε ≤ τ E(x) (ξ) ≤ c 3,8 τ E(y) (ξ) 1-ε , ∀ ξ ≤ 1 (13)
and,

c 3,7 τ E(y) (ξ) 1-ε ≤ τ E(x) (ξ) ≤ c 3,8 τ E(y) (ξ) 1+ε , ∀ ξ ≥ 1 ( 14
)
where c 3,7 = c 3,7 (T ) and c 3,8 = c 3,8 (T ) are two finite positive constants that only depend on T .

Let us emphasize that all these results depend only on the eigenvalues of the matrices. Therefore they also hold when the map E is replaced by

E t : x → E(x) t
, where E(x) t is the transpose matrix of E(x).

Sample paths Regularity of multi-operator scaling stable random fields

4.1. Preliminary result on the scale parameter. In order to study the regularity of the sample paths of X α,ψ defined by ( 7), we consider the increments

X α,ψ (u) -X α,ψ (v) = Y α,ψ (u, u, u) -Y α,ψ (v, v, v), ∀u, v ∈ R d
with Y α,ψ defined by [START_REF] Benson | Aquifer operator-scaling and the effect on solute mixing and dispersion[END_REF]. Observe that

X α,ψ (u) -X α,ψ (v) = Y 1,α,u (u, v) + Y 2,α,u (u, v) + Y 3,α,v (u, v), with    Y 1,α,x (u, v) = Y α,ψ (x, u, u) -Y α,ψ (x, u, v), Y 2,α,x (u, v) = Y α,ψ (x, u, v) -Y α,ψ (x, v, v), Y 3,α,x (u, v) = Y α,ψ (u, x, x) -Y α,ψ (v, x, x).
By Theorem 2.1, the random variables Y 1,α,x (u, v) and Y 2,α,x (u, v) are well-defined as soon as

x ∈ R d and |q(v) -q(u)| < α min 1/H(u) -1, 1/H(v) -1, 1 . (15)
Note that for every x, u, v ∈ R d , Y 3,α,x (u, v) is also well-defined and is an increment of a harmonizable operator scaling stable random field with exponent E = E(x) t and kernel function ψ(ξ) = ψ x (ξ) (see [START_REF] Biermé | Operator scaling stable random fields[END_REF]).

In this section, we compare the scale parameter

X α,ψ (u) -X α,ψ (v) α ,
with τ E(v) t (u-v) uniformly in u, v. In order to obtain our estimates, we study the scale parameters of Y 1,α,x (u, v), Y 2,α,x (u, v) and Y 3,α,x (u, v). The controls of these parameters are stated in the three following lemmas, whose proofs are postponed to the Appendix. Lemma 4.1. Assume that Assumptions 1-6 are fulfilled and let K ⊂ R d be a compact set of R d .

Then, for γ > 0 small enough, there exists c 4,1 = c 4,1 (K, T, γ) a finite positive constant such that,

for every x ∈ K, u, v ∈ T with u -v ≤ γ, Y 1,α,x (u, v) is well-defined and Y 1,α,x (u, v) α α ≤ c 4,1 u -v α .
Lemma 4.2. Assume that Assumptions 1-6 are fulfilled and let

K ⊂ R d be a compact set of R d .
Then, for γ > 0 small enough, there exists c 4,2 = c 4,2 (K, T, γ) a finite positive constant such that,

for every x ∈ K, u, v ∈ T with u -v ≤ γ, Y 2,α,x (u, v) is well-defined and Y 2,α,x (u, v) α α ≤ c 4,2 u -v α .
Lemma 4.3. Assume that Assumptions 1-6 are fulfilled and let K ⊂ R d be a compact set of R d .

Then, there exist two finite positive constants c 4,3 = c 4,3 (K) and c 4,4 = c 4,4 (K) such that for every

x ∈ K and every u, v ∈ R d , c 4,3 τ E(x) t (u -v) α ≤ Y 3,α,x (u, v) α α ≤ c 4,4 τ E(x) t (u -v) α .
From the three previous lemmas, an uniform control of the scale parameter of

X α,ψ (u) -X α,ψ (v)
can be stated. The local behavior of this scale parameter is closely linked to the Hölder regularity of the sample paths of the multi-operator stable random field X α,ψ . Actually, in the Gaussian case the Hölder regularity is characterized by the local behavior of this scale parameter; in the α-stable case (α < 2), the next theorem leads to an upper bound for the Hölder regularity.

X α,ψ (u) -X α,ψ (v) α α    ≥ c 4,5 max τ E(v) t (u -v), τ E(u) t (u -v) α ≤ c 4,6 min τ E(v) t (u -v), τ E(u) t (u -v) α , for every u, v ∈ T such that u -v ≤ γ.
Proof of Theorem 4.4. Let u, v ∈ T such that u -v ≤ γ with γ ∈ (0, 1). Then, for γ small enough, by Lemmas 4.1 and 4.2,

Y 1,α,u (u, v) and Y 2,α,u (u, v) are well-defined. Note that Y 3,α,v (u, v)
is also well-defined. Then, we can write

X α,ψ (u) -X α,ψ (v) = Y 1,α,u (u, v) + Y 2,α,u (u, v) + Y 3,α,v (u, v).
Hence, for γ small enough,

X α,ψ (u) -X α,ψ (v) α α ≥ 2 -2α Y 3,α,v (u, v) α α -Y 1,α,u (u, v) α α -Y 2,α,u (u, v) α α ≤ 2 2α Y 1,α,u (u, v) α α + Y 2,α,u (u, v) α α + Y 3,α,v (u, v) α α
By applying Lemmas 4.1, 4.2 and 4.3, for γ small enough,

X α,ψ (u) -X α,ψ (v) α α ≥ 2 -2α c 4,3 τ E(v) t (u -v) α -(c 4,1 + c 4,2 ) u -v α ≤ 2 2α c 4,4 τ E(v) t (u -v) α + (c 4,1 + c 4,2 ) u -v α . Since max z∈T H(z) < 1, we can choose δ ∈ (0, min z∈T H(z)) such that ∀y ∈ T, H(y) + δ ≤ max z∈T H(z) + δ < 1.
By Proposition 3.5, there exists a finite constant c 3,5 = c 3,5 (T, δ), such that

u -v α ≤ c -α 3,5 u -v α(1-max z∈T H(z)-δ) τ E(v) t (u -v) α . Then, one can choose γ small enough such that (c 4,1 + c 4,2 ) u -v α ≤ 2 -2α-1 c 4,3 τ E(v) t (u -v) α for every u, v ∈ T such that u -v ≤ γ. Therefore we can choose c 4,5 = 2 -2α-1 c 4,3 and c 4,6 = 2 2α c 4,4 + 2 -2α-1 c 4,3 .
From the previous theorem, we easily deduce the stochastic continuity of a harmonizable multioperator scaling stable random field. Corollary 4.5. Assume that Assumptions 1-6 are fulfilled. Then the harmonizable multi-operator scaling stable random field X α,ψ defined by [START_REF] Biermé | Operator scaling stable random fields[END_REF] is stochastically continuous on T .

Proof. By Theorem 4.4, there exists γ ∈ (0, 1) and a finite positive constant c 4,6 such that

X α,ψ (u) -X α,ψ (v) α α ≤ c 4,6 τ E(v) t (u -v) α for any u, v ∈ T satisfying u -v ≤ γ.
Let δ ∈ (0, min x∈T H(x)). By Proposition 3.5, there exists a finite positive constant c 3,6 = c 3,6 (T, δ)

such that X α,ψ (u) -X α,ψ (v) α α ≤ c 4,6 c α 3,6 u -v α(H(v)-δ) for any u, v ∈ T satisfying u -v ≤ γ. In particular, since α(H(v) -δ) > 0, ∀v ∈ T, lim u→v X α,ψ (u) -X α,ψ (v) α
α = 0, which implies the stochastic continuity of X α,ψ on T (see Proposition 3.5.1 of [START_REF] Samorodnitsky | Stable non-Gaussian random processes[END_REF]).

Let us also mention that in a special case, when the field X α,ψ has stationary increments, Yimin Xiao proves in Theorem 3.6 of [START_REF] Xiao | Properties of strong local nondeterminism and local times of stable random fields[END_REF] a strong local non-determinism property that enables him to study their local times.

Modulus of continuity.

In this section, we give an upper bound for the modulus of continuity of a harmonizable multi-operator scaling stable random field X α,ψ around the position x.

Let us emphasize that we control the behavior of an increment

X α,ψ (x + u) -X α,ψ (x + v)
using the polar coordinate τ E(x) t with respect to the matrix E(x) t , which takes into account the anisotropic behavior of X α,ψ around x. As in [START_REF] Kôno | Hölder continuity of sample paths of some self-similar stable processes[END_REF][START_REF] Biermé | Hölder regularity for operator scaling stable random fields[END_REF], one of the main tools we use is a LePage series representation (see [START_REF] Lepage | Conditional moments for coordinates of stable vectors[END_REF][START_REF] Lepage | Multidimensional infinitely divisible variables and processes. II. In Probability in Banach spaces[END_REF] for details on such series) which is a conditionally Gaussian series. Since E may vary with the position x, the main difference to [START_REF] Biermé | Hölder regularity for operator scaling stable random fields[END_REF] is that we need some uniform control of the polar coordinates and an uniform comparison of the radial parts with respect to E(x) t and E(y) t (see Section 3). This leads to an upper bound less accurate than the upper bound given in [START_REF] Biermé | Hölder regularity for operator scaling stable random fields[END_REF] in the case of operator scaling harmonizable stable random fields. The difference is a log term but our upper bound is sufficient to obtain the pointwise Hölder exponents.

Let us also point out that our modulus of continuity is local and not uniform in contrast to [START_REF] Xiao | Uniform modulus of continuity of random fields[END_REF].

Theorem 4.6. Assume that Assumptions 1-6 are fulfilled on T . There exists a modification X * α,ψ of X α,ψ on T such that for all x ∈ T , for all ε > 0,

lim γ↓0 sup u ≤γ, v ≤γ x+u,x+v∈T X * α,ψ (x + u) -X * α,ψ (x + v) τ E(x) t (u -v) 1-ε = 0.
Proof. For every k ∈ N\{0} and j = (j 1 , . . . , j d ) ∈ Z d we set

x k,j = j 2 k and D k = x k,j : j ∈ Z d ∩ 2 k T .
Let us remark that the sequence (D k ) k is increasing and set D = ∞ k=1 D k , which is dense in T .

First

Step: In this step, we assume that α ∈ (0, 2).

Let us fix x 0 ∈ T ∩ D. Following [START_REF] Kôno | Hölder continuity of sample paths of some self-similar stable processes[END_REF][START_REF] Biermé | Hölder regularity for operator scaling stable random fields[END_REF], we consider a Lepage series representation of X α,ψ which is a conditionally Gaussian series. This series depends on the position x 0 we have fixed.

Let (T n ) n≥1 , (g n ) n≥1 and (ξ n ) n≥1 be independent sequences of random variables.

• T n is the nth arrival time of a Poisson process with intensity 1.

• (g n ) n≥1 is a sequence of i.i.d. Gaussian complex isotropic random variables so that g n

(d)
= e iθ g n for any θ ∈ R.

• (ξ n ) n≥1 is a sequence of i.i.d. random variables whose common law is µ x 0 (dξ) = m x 0 (ξ)dξ with

m x 0 (ξ) = c a,x 0 τ E(x 0 ) (ξ) q(x 0 ) log τ E(x 0 ) (ξ) 1+a ,
where a > 0, q is defined by ( 6) and c a,x 0 > 0 is chosen such that

R d m x 0 (ξ)dξ = 1.
Let

d α = E(|Re (g 1 )| α ) -1/α 1 2π 2π 0 |cos (x)| α dx 1/α +∞ 0 sin (x) x α dx -1/α and f α (x, ξ) = e i x,ξ -1 ψ x (ξ) -1-q(x)/α , ∀x, ξ ∈ R d . ( 16
)
According to Proposition 4.1 of [START_REF] Biermé | Hölder regularity for operator scaling stable random fields[END_REF], for every

x ∈ R d Z α (x) = d α Re +∞ n=1 T -1/α n m x 0 (ξ n ) -1/α f α (x, ξ n )g n ,
converges almost surely and Z α

(f dd) = X α,ψ . Then, conditionally to (T n , ξ n ) n , Z α (u) -Z α (v) is a real centered Gaussian random variable with variance v 2 α ((u, v) | (T n , ξ n ) n ) = d 2 α 2 E |g 1 | 2 +∞ n=1 T -2/α n m x 0 (ξ n ) -2/α |f α (u, ξ n ) -f α (v, ξ n )| 2 . ( 17
)
Second

Step: Let us now assume that α ∈ (0, 2] and set Z 2 = X 2,ψ . Following the idea of [START_REF] Biermé | Hölder regularity for operator scaling stable random fields[END_REF],

let us consider (ν k ) k≥1 an increasing sequence of integers such that for k large enough and any

x ∈ T , D ν k is a 2 -k net of T for τ E(x) t
, that is such that for any y ∈ T , there exists u ∈ D ν k satisfying τ E(x) t (y -u) ≤ 2 -k . Here, using Proposition 3.5, we can choose ν k = [k/a 1 ] where a 1 ∈ (0, min z∈T H(z)) and [t] denotes the integer part of t ∈ R.

For k ∈ N\{0} and (i, j) ∈ Z d , we consider the set

E k i,j =    ω : |Z α (x ν k ,i ) -Z α (x ν k ,j )| > v α ((x ν k ,i , x ν k ,j ) | (T n , ξ n ) n ) ϕ τ E(x 0 ) t (x ν k ,i -x ν k ,j ) if α ∈ (0, 2) ω : |Z 2 (x ν k ,i ) -Z 2 (x ν k ,j )| > Z 2 (x ν k ,i ) -Z 2 (x ν k ,j ) 2 ϕ τ E(x 0 ) t (x ν k ,i -x ν k ,j ) if α = 2
where, as in [START_REF] Kôno | On the modulus of continuity of sample functions of gaussian processes[END_REF],

ϕ(t) = 2Ad log 1 t , t ∈ (0, 1]
with A > 0 chosen latter. Then, for every (k, i, j),

P E k i,j = P |N | > ϕ τ E(x 0 ) t (x ν k ,i -x ν k ,j
) , where N is a real centered Gaussian random variable with variance 1. Let us choose δ ∈ (0, 1) and set for k ∈ N\{0}, δ k = 2 -(1-δ)k and

I k = (i, j) ∈ Z d ∩ 2 ν k T 2 : τ E(x 0 ) t (x ν k ,i -x ν k ,j ) ≤ δ k . For every (i, j) ∈ I k , since ϕ is a decreasing function P E k i,j ≤ P(|N | > ϕ(δ k )) ≤ 2 π e -ϕ 2 (δ k )/2 ϕ(δ k ) = 2 -A(1-δ)kd Aπd(1 -δ)k log 2 .
Let us fix a 2 > min z∈T H. Then, using Proposition 3.5, one checks that card

I k ≤ c T,a 2 2 kd(2/a 1 -(1-δ)/a 2 )
with c T a finite positive constant which only depends on T and a 2 . Hence, choosing A > 2/a 1 -1/a 2 and δ small enough,

+∞ k=1 (i,j)∈I k P E k i,j ≤ c T,a 2 Aπd(1 -δ) log 2 +∞ k=1 2 -kd((A+1/a 2 )(1-δ)-2/a 1 ) < +∞.
Therefore, by the Borel-Cantelli Lemma, almost surely there exists an integer k * (ω) such that for

every k ≥ k * (ω), |Z α (u) -Z α (v)| ≤ v α ((u, v) | (T n , ξ n ) n ) ϕ τ E(x 0 ) t (u -v) (18)
as soon as u, v ∈ D ν k with τ E(x 0 ) t (u -v) ≤ δ k .

Third

Step: We now give an upper bound of the conditional variance v α when α ∈ (0, 2) and of the variogram of Z 2 . For the sake of clearness, the proof of the following lemma is postponed to the Appendix.

Lemma 4.7. Let ∈ (0, 1).

(1) If α ∈ (0, 2), almost surely, there exists r x 0 = r x 0 ( , ω) > 0, such that for all u, v ∈

B(x 0 , r x 0 ) ∩ T , v α ((u, v) | (T n , ξ n ) n ) ≤ τ E(x 0 ) t (u -v) 1-. (2) If α = 2, there exists r x 0 = r x 0 ( ) > 0 such that for all u, v ∈ B(x 0 , r x 0 ) ∩ T , Z 2 (u) -Z 2 (v) 2 ≤ τ E(x 0 ) t (u -v) 1-.
Let ε ∈ (0, 1). Combining the previous lemma applied with = ε/2 and (18), almost surely, there exists r x 0 = r x 0 (ε, ω) ∈ (0, 1) and k * (ω) such that for all k ≥ k * (ω),

|Z α (u) -Z α (v)| ≤ τ E(x 0 ) t (u -v) 1-ε/2 ϕ τ E(x 0 ) t (u -v) as soon as u, v ∈ D ν k ∩ B(x 0 , r x 0 ) with τ E(x 0 ) t (u -v) ≤ δ k . Let us now choose t ε such that ∀t ∈ (0, t ε ], t 1-ε/2 ϕ(t) ≤ t 1-ε and k * (ω) = k * (ω, ε) such that δ k * (ω) ≤ t ε and τ E(x 0 ) t (ξ) ≤ δ k * (ω) ⇒ ξ ≤ r x 0 . Then, for k ≥ k * (ω), |Z α (u) -Z α (v)| ≤ τ E(x 0 ) t (u -v) 1-ε for u, v ∈ D ν k satisfying max τ E(x 0 ) t (u -x 0 ), τ E(x 0 ) t (v -x 0 ) ≤ δ k * (ω) and τ E(x 0 ) t (u -v) ≤ δ k . Then, let Ω * x 0 = ε∈Q∩(0,1) +∞ n=1 k≥n u,v∈D k,n |X α,ψ (u) -X α,ψ (v)| ≤ τ E(x 0 ) t (u -v) 1-ε with D k,n = u, v ∈ D ν k , τ E(x 0 ) t (u -v) ≤ δ k , max τ E(x 0 ) t (u -x 0 ), τ E(x 0 ) t (v -x 0 ) ≤ δ n .
Since X α,ψ and Z α have the same finite dimensional margins, we have proved that

P Ω * x 0 = 1. Therefore, P(Ω * ) = 1 with Ω * = x 0 ∈D Ω * x 0 .
Since D = k≥1 D ν k , similar arguments as in Step 4 of [START_REF] Biermé | Hölder regularity for operator scaling stable random fields[END_REF] and Proposition 3.5 lead to the existence of a finite positive constant C = C(T ) > 0 such that for ω ∈ Ω * , for all x 0 ∈ D, for all ε ∈ Q ∩ (0, 1), there exists γ x 0 = γ x 0 (ω, ε) > 0 such that for all u, v ∈ D ∩ B(x 0 , γ x 0 ),

|X α,ψ (u) -X α,ψ (v)| ≤ Cτ E(x 0 ) t (u -v) 1-ε . ( 19
)
Fourth Step: We now define a modification of X α,ψ . First, if ω ∈ Ω * , we set X * α,ψ (y)(ω) = 0 for all y ∈ T . Let us now fix ω ∈ Ω * . Then, we set

X * α,ψ (y)(ω) = X α,ψ (y)(ω), ∀y ∈ D ∩ T.
Let y ∈ T and ε ∈ Q ∩ (0, 1). Then, there exists x 0 ∈ D such that y ∈ B(x 0 , γ x 0 /2) and y (n) ∈ D such that lim n→+∞ y (n) = y. In view of [START_REF] Mandelbrot | Fractional Brownian motion, fractional noises and applications[END_REF], for all n, m such that y (n) , y (m) ∈ B(x 0 , γ x 0 ),

X * α,ψ y (n) (ω) -X * α,ψ y (m) (ω) ≤ Cτ E(x 0 ) t y (n) -y (m) 1-ε , such that X * α,ψ y (n) (ω)
n is a Cauchy sequence and hence converges. We set

X * α,ψ (y)(ω) = lim n→+∞ X * α,ψ y (n) (ω).
Remark that this limit does not depend on the choice of y (n) , nor of the choice of x 0 ∈ D and nor of the choice of ε ∈ Q ∩ (0, 1). Observe also that X * α,ψ (•)(ω) is then well-defined on T . Moreover, by [START_REF] Mandelbrot | Fractional Brownian motion, fractional noises and applications[END_REF] and continuity of τ

E(x 0 ) t , ∀u, v ∈ B(x 0 , γ x 0 /2), X * α,ψ (u)(ω) -X * α,ψ (v)(ω) ≤ Cτ E(x 0 ) t (u -v) 1-ε . ( 20
)
Let us now fix x ∈ T. Then, there exists x 0 ∈ D and γ x = γ x (ε, x) ∈ (0, 1) such that B(x, γ x /2) ⊂ B(x 0 , γ x 0 /2).

Hence by Equation ( 20) and by Proposition 3.6, up to change γ x ,

∀u, v ∈ B(x, γ x /2), X * α,ψ (u)(ω) -X * α,ψ (v)(ω) ≤ Cc 3,8 τ E(x) t (u -v) 1-2ε
. where c 3,8 does not depend on (u, v). This also holds for ω ∈ Ω * (for any choice of γ x (ε, ω)).

To conclude the proof, let us emphasize that X * α,ψ is a modification of X α,ψ since X α,ψ is stochastically continuous (by Corollary 4.5).

Hölder exponents.

In this section, we are interested in the global and directional Hölder regularity of the sample paths of a harmonizable multi-operator scaling stable random field X α,ψ . We first prove that X α,ψ admits a modification whose sample paths are "globally" Hölder on T . This is a consequence of Theorem 4.6 and of the comparison of the radial part τ E(x) t with the Euclidean norm.

Corollary 4.8. Assume that Assumptions 1-6 are fulfilled and let X α,ψ be the harmonizable multioperator scaling stable random field defined by [START_REF] Biermé | Operator scaling stable random fields[END_REF]. Then, there exists a modification of X α,ψ which has H-Hölder sample paths on the compact set T for any H ∈ (0, min y∈T H(y)).

Proof. Let us consider the modification X * α,ψ introduced in Theorem 4.6. Let us now fix ω ∈ Ω and ε ∈ (0, 1). By Theorem 4.6, for any x ∈ T , there exists γ x = γ(x, ε, ω) ∈ (0, 1/2) such that

X * α,ψ (u)(ω) -X * α,ψ (v)(ω) ≤ τ E(x) t (u -v) 1-ε for any u, v ∈ T such that x -u ≤ γ x and x -v ≤ γ x .
Then, by Proposition 3.5, for any δ ∈ (0, min v∈T H(v)), there exists a finite positive constant c 3,6 = c 3,6 (T, δ) such that

X * α,ψ (u)(ω) -X * α,ψ (v)(ω) ≤ c 3,6 u -v (H(x)-δ)(1-ε) for any u, v ∈ T such that x -u ≤ γ x and x -v ≤ γ x . Therefore, for any u, v ∈ T such that x -u ≤ γ x and x -v ≤ γ x , X * α,ψ (u)(ω) -X * α,ψ (v)(ω) ≤ c 3,6 u -v (min y∈T H(y)-δ)(1-ε)
since u -v < 1. Since this holds for any x ∈ T , the function z → X * α,ψ (z)(ω) is hölderian of order (min y∈T H(y) -δ)(1 -ε) on the compact set T . This leads to the conclusion.

As already mentioned, the Hölder sample paths regularity of a continuous modification of X α,ψ may vary both with the position and with the direction. At position x, the dependence on the directions is characterized by the Jordan decomposition of E(x).

Notation Let x ∈ R d . Let us consider the Jordan decomposition of E(x) as in [START_REF] Biermé | Hölder regularity for operator scaling stable random fields[END_REF]. Hence,

E(x) = P (x) -1      J 1 (x) 0 . . . 0 0 J 2 (x) 0 . . . . . . . . . . . . 0 0 . . . 0 J px (x)      P (x). (21)
We can assume that each J j (x) is associated with a j (x) = 1/H j (x), the real part of the eigenvalue λ j (x). Observe that

H(x) = min 1≤j≤px H j (x) and H(x) = max 1≤j≤px H j (x).
We denote by (e 1 , . . . , e d ) the canonical basis of R d and set f j (x) = P (x) -1 e j for every j = 1, . . . , d.

Hence, (f 1 (x), . . . , f d (x)) is a basis of R d . For all j = 1, . . . , p x , let

W j (x) = span f k (x) ; j-1 i=1 d i + 1 ≤ k ≤ j i=1 d i (22)
where d i is the size of

J i (x). Then, R d = px j=1 W j (x). Moreover each W j (x) is an E(y)-invariant set when y ∈ R d is such that E(x)E(y) = E(y)E(x).
When v varies in W j (x), [START_REF] Biermé | Hölder regularity for operator scaling stable random fields[END_REF] proved that the behavior of the radial part τ E(x) t (v) around v = 0 is characterized by H j (x). Then, if we only consider X * α,ψ on a straight line driven by u ∈ W j (x), Corollary 4.8 can be strenghtened. Corollary 4.9. Let x ∈ R d . Assume that Assumptions 1-6 are fulfilled with T = [x -η, x + η] = d j=1 [x j -η, x j + η] for η > 0. Let u ∈ W j (x)\{0} where W j (x) is defined by (22) and 1 ≤ j ≤ p x . Then, there exists a modification X * α,ψ of X α,ψ on T such that the random process X * α,ψ (x + tu) t∈R has H-Hölder sample paths in a (deterministic) neighborhood of t = 0 for any H ∈ (0, H j (x)).

Proof. Let us consider the modification X * α,ψ introduced in Theorem 4.6. Let us choose γ ∈ (0, η) such that Equation ( 13) holds (see Proposition 3.6). Let us now fix t 0 ∈ (-γ/ u , γ/ u ) and ω ∈ Ω. It is sufficient to prove that, for H ∈ (0, H j (x)), the function t → X * α,ψ (x + t 0 u + tu) is H Hölder on (-r t 0 (ω), r t 0 (ω)) for some r t 0 (ω) > 0. Let ε ∈ (0, 1). By Theorem 4.6 and Proposition 3.6, there exists γ t 0 = γ(t 0 , x, ε, ω) ∈ (0, 1/4) such that

X * α,ψ (x + t 0 u + tu) -X * α,ψ (x + t 0 u + su) ≤ τ E(x) t ((t -s)u) 1-2ε
for any t, s ∈ (-γ t 0 / u , γ t 0 / u ).

Then by Corollary 3.4 of [START_REF] Biermé | Hölder regularity for operator scaling stable random fields[END_REF], applied to E(x) t and r = 1/2, there exists a finite positive constant c = c(x) and l j = l j (x) ∈ N * such that

X * α,ψ (x + tu) -X * α,ψ (x + su) ≤ c u (1-2ε)H j (x) |t -s| (1-2ε)H j (x) |log (|t -s| u )| (l j -1)(1-2ε)H j (x)
for any s, t ∈ R such that |t -t 0 | ≤ γ t 0 / u and |s -t 0 | ≤ γ t 0 / u . Then, for δ > 0 small enough, X * α,ψ (x + tu) t∈R has (H j (x) -2εH j (x) -δ)-Hölder sample paths on (-γ/ u , γ/ u ), which concludes the proof.

We now focus on Hölder directional and global pointwise exponents. Let us first define these exponents.

Definition 4.1. Let x ∈ R d , (X(y)) y∈R d be a real-valued random field and S d-1 be the Euclidean unit sphere of R d . Assume that X * is a modification of X which has continuous sample paths in a neighborhood of x.

(1) The Hölder pointwise exponent of X at point x is

H X (x) = sup H > 0, lim y→0 X * (x + y) -X * (x) y H = 0 .
(2) Moreover, the directional Hölder pointwise exponent H X (x, u) of the random field X at point x in direction u ∈ S d-1 is the Hölder pointwise exponent at point 0 of the process

(X(x + tu)) t∈R , that is H X (x, u) = sup H > 0, lim t→0 X * (x + tu) -X * (x) |t| H = 0 .
Note that Corollaries 4.8 and 4.9 give lower bounds of these exponents. Moreover, since the harmonizable random field X α,ψ is a stable random field, an upper bound can be deduced from the behavior of the scale parameter

X α,ψ (u) -X α,ψ (
= [x -η, x + η] = d j=1 [x j -η, x j + η] for η > 0. Let us consider X * α,ψ a continuous modification of X α,ψ on T . (1) Let u ∈ W j (x) ∩ S d-1
where W j (x) is defined by [START_REF] Pecknold | The simulation of universal multifractals[END_REF], 1 ≤ j ≤ p x . Then the directional pointwise Hölder exponent of the random field X α,ψ at point x in direction u is almost surely H j (x), that is

H X α,ψ (x, u) = H j (x) almost surely.
(2) Moreover, the pointwise Hölder exponent of the random field X α,ψ at point x is almost surely H(x), that is

H X α,ψ (x) = H(x) = min 1≤j≤d H j (x) almost surely.
Proof. Let X * α,ψ be a modification of X α,ψ which has continuous sample paths on T (see Theorem 4.6).

(1) Let x ∈ T and u ∈ W j (x) ∩ S d-1 . By Corollary 4.9, it is clear that

H X α,ψ (x, u) ≥ H j (x).
By Theorem 4.4, there exists γ ∈ (0, 1) and c 4,1 = c 4,1 (T, γ) a finite positive constant such that

c 4,1 τ E(x) t (tu) α ≤ X α,ψ (x + tu) -X α,ψ (x) α α = X * α,ψ (x + tu) -X * α,ψ (x) α α
for any t ∈ R such that |t| ≤ γ. Hence, by Corollary 3.4 of [START_REF] Biermé | Hölder regularity for operator scaling stable random fields[END_REF], for any H > H j (x), there exists a finite positive constant c such that

c|t| αH ≤ X * α,ψ (x + tu) -X * α,ψ (x) α α for any t ∈ R such that |t| ≤ γ. Therefore, for any H > H j (x), lim t→0 X * α,ψ (x + tu) -X * α,ψ (x) |t| H α α = +∞, which implies that X * α,ψ (x+tu)-X * α,ψ (x) |t| H
is almost surely unbounded as t → 0 since X * α,ψ is an α-stable random field. This leads to

H X α,ψ (x, u) = H j (x) almost surely.
(2) Let x ∈ T . By Corollary 4.8 and continuity of H, it is clear that

H X α,ψ (x) ≥ H(x) = min 1≤j≤d H j (x).

Moreover, by definition of the directional exponents

H X α,ψ (x, u), u ∈ S d-1 , H X α,ψ (x) ≤ inf u∈S d-1 H X α,ψ (x, u).
Then, since for any 1 ≤ j ≤ p x , W j ∩ S d-1 = ∅, assertion (1) implies

H X α,ψ (x) ≤ min 1≤j≤px H j (x) = H(x)
almost surely, which concludes the proof.

Let us now illustrate the previous results. We denote by

H (0) 1 , . . . , H (0)
d the inverse of the real parts of the eigenvalues of E 0 . Let W 1 , . . . , W p the subspaces associated to the Jordan's decomposition of E 0 as [START_REF] Pecknold | The simulation of universal multifractals[END_REF] and let h, E and ψ be as in Example 2.2. Observe that for any x ∈ R d ,

W j (x) = W j since E(x) = E 0 /h(x). Then, according to Corollary 4.10, if u ∈ W j ∩ S d-1 , for all x ∈ R d , H X α,ψ (x, u) = h(x)H (0) j almost surely. Similarly, for all x ∈ R d , H X α,ψ (x) = h(x) min 1≤j≤d H (0) j almost surely.
Example 4.2. Assume that for every 1 ≤ j ≤ d, H j is a locally Lipschitz function on R d with values in (0, 1). We assume moreover that inf

for all x ∈ R d , H Xα,ϕ (x, f j / f j ) = H j (x) almost surely. Similarly, for all x ∈ R d , almost surely H X * α,ϕ (x) = H(x). Example 4.3. Let d = 2.
Let us consider as in Example 2.5 the map

x → E(x) = a(x) cos(θ(x)) sin(θ(x)) -sin(θ(x)) cos(θ(x))
where a and θ are locally Lipschitz functions on R d such that ∀x ∈ R d , a(x) cos (θ(x)) > 1, and X α,ψ the associated random field. Then, for all x ∈ R d and u ∈ S d-1 , almost surely

H X α,ψ (x, u) = H X α,ψ (x) = 1 a(x) cos (θ(x))
.

Local Operator scaling property

In general, harmonizable multi-operator scaling random fields are not operator scaling: they do not satisfy the global property (1) for any fix matrix E. However, they satisfy a weak property we call local asymptotic operator scaling property, which we introduce in the next definition.

lim ε→0 + X(x + ε A(x) u) -X(x) ε u∈R d (f dd) = (Z x (u)) u∈R d , (23)
with Z x a non degenerate random field. Moreover a random field X which satisfies ( 23) is called multi-operator random field of order A.

As mentioned in the introduction, the local asymptotic operator scaling property generalizes both the operator scaling property and the local asymptotic self-similarity property. On the one hand, an operator scaling random field X with stationary increments is locally asymptotically operator scaling at any point x. On the other hand, a locally asymptotically self-similar random field at point x with order h(x) is locally asymptotically operator scaling at point x of order

I d /h(x).
Note also that the local asymptotic self-similarity property can not capture the operator scaling property since it only reveals local self-similarity which is not sufficient to characterize the anisotropy. Actually, let X be an operator scaling random field of order E 0 . Assume that the Jordan's decomposition of E 0 is given by ( 21) with H = min 1≤j≤p

H j = H 1 such that J 1 = 1 H 1 I d 1 and H 1 > min 2≤j≤p H j .
Let W 1 be the corresponding eigenvector space (see [START_REF] Pecknold | The simulation of universal multifractals[END_REF]).

Then, writing for

u ∈ R d = p j=1 W j , u = u 1 + v with u 1 ∈ W 1 , it is clear that, for any ε > 0, ε 1/H 1 u = ε E 0 u 1 + ε 1/H 1 -E 0 v with ε 1/H 1 -E 0 v -→ ε→0 + 0 since v ∈ p j=2 W j , with min 2≤j≤p H j > H 1 . Then, by operator scaling property, if X is stochastically continuous, lim ε→0 + X(εu) ε H 1 u∈R d (f dd) = (X(π W 1 u))) u∈R d
with π W 1 the projection on W 1 . In other words, if X is non degenerated on W 1 , X is locally asymptotically self-similar of order H 1 at point 0 with tangent field (X(π

W 1 u)) u∈R d .
The following remark gives some properties of the random field Z x , which are immediate consequences of (23).

Remark 5.1. Assume that (23) is fulfilled. Then Z x is operator scaling of order A(x), that is

∀c > 0, Z x c A(x) u u∈R d (f dd) = c (Z x (u)) u∈R d . Moreover, if θ ∈ R d is an eigenvector of A(x) associated with a real eigenvalue λ, then lim ε→0 + X(x + εtθ) -X(x) ε 1/λ t∈R (f dd) = (Z x (tθ)) t∈R .
The main result of this section is stated in the next theorem. As expected, a harmonizable multi-operator scaling stable random field X α,ψ locally looks like a harmonizable operator scaling stable random field.

Theorem 5.1. Let x ∈ R d . Assume that Assumptions 1-6 are fulfilled on T = [x -η, x + η] = d j=1 [x j -η, x j + η].
Then, the random field X α,ψ is locally asymptotically operator scaling at point x of order E(x) t in the sense that

lim ε→0 + X α,ψ (x + ε E(x) t u) -X α,ψ (x) ε u∈R d (f dd) = (X ψx (u)) u∈R d , (24)
where X ψx is a harmonizable α-stable operator scaling field with respect to E(x) t and ψ x in the sense of Theorem 4.1 in [START_REF] Biermé | Operator scaling stable random fields[END_REF].

Remark 5.2. In the case where α = 2, one can prove that (24) holds in distribution on the space of continuous functions endowed with the topology of the uniform convergence on compact sets. Actually, in this case, one can applied the classical criterion of tightness based on second moments of increments. However, if α ∈ (0, 2), proving tightness is much harder and an open problem.

Proof. Let x ∈ R d and u ∈ R d . Then, for ε > 0 small enough, the random variables

Y 1,α,x+ε E(x) t u x + ε E(x) t u, x and Y 2,α,x+ε E(x) t u x + ε E(x) t u,
x are well-defined. Then, for ε > 0 small enough, using the notation of Section 4.1, we get

X α,ψ (x + ε E(x) t u) -X α,ψ (x) = Y 1,α,x+ε E(x) t u x + ε E(x) t u, x + Y 2,α,x+ε E(x) t u x + ε E(x) t u, x +Y 3,α,x x + ε E(x) t u, x . (25)
By Lemma 4.1, for ε > 0 small enough,

Y 1,α,x+ε E(x) t u x + ε E(x) t u, x α α ≤ c 4,1 ε E(x) t u α
where the finite positive constant c 4,1 does not depend on ε. Therefore, by Lemma 3.3, for ε > 0 small enough and δ > 0 small enough

Y 1,α,x+ε E(x) t u x + ε E(x) t u, x α α ≤ c 4,1 c α 3,1 u α ε α/H(x)-αδ
where the finite positive constant c 3,1 does not depend on ε. Since Y 1,α,x+ε E(x) t u x + ε E(x) t u, x is a stable random variable and since H(x) < 1, the previous inequality leads to lim

ε→0 + Y 1,α,x+ε E(x) t u x + ε E(x) t u, x ε = 0 in probability. (26)
Using Lemma 4.2 and Lemma 3.3, the same arguments yield that lim

ε→0 + Y 2,α,x+ε E(x) t u x + ε E(x) t u, x ε = 0 in probability. ( 27
)
Observe that the random field

X ψx (v) v∈R d = (Y α,ψ (v, x, x)) v∈R d
is well-defined and is a harmonizable α-stable operator scaling field with respect to E(x) t and ψ x in the sense of Theorem 4.1 in [START_REF] Biermé | Operator scaling stable random fields[END_REF]. Moreover,

∀v ∈ R d , Y 3,α,x (v, x) = X ψx (v) -X ψx (x).
Then, by stationarity of the increments of X ψx and the operator scaling property (see Corollary

4.2 of [7]), Y 3,α,x x + ε E(x) t v, x v∈R d (f dd) = ε(X ψx (v)) v∈R d . (28)
From Equations ( 25), ( 26), ( 27) and ( 28), one easily deduces that lim

ε→0 + X α,ψ (x + ε E(x) t u) -X α,ψ (x) ε u∈R d (f dd) = (X ψx (u)) u∈R d .
6. Proofs 6.1. Polar coordinates.

Proof of Lemma 3.3. Let δ > 0 and r 0 > 0. Let us recall that for any t ∈ [0, +∞), t λ j (x) , 1 ≤ j ≤ d are eigenvalues of t E(x) . Then, for every j = 1, . . . , d, and for every t ∈ [0, +∞),

t λ j (x) = t Re (λ j (x)) ≤ t E(x) ,
which leads to the lower bounds since 1/H(x) = max 1≤j≤d Re (λ j (x)) and 1/H(x) = min 1≤j≤d Re (λ j (x)). Let us now prove the upper bounds.

(i) Since δ > 0 and since E is continuous on T and satisfies Assumption 6, the map x → E(x) -1/H(x) -δ I d is also continuous on T , satisfies Assumption 6 and takes values in

M >0 R d .
Then, by Lemma 3.2, the function

(t, x) → t E(x)-(1/H(x)-δ)I d
is continuous on [0, +∞) × T and thus bounded on the compact set [0, r 0 ] × T . Therefore, there exists a finite constant c 3,1 = c 3,1 (T, δ, r 0 ) > 0 which only depends on δ, T and r 0 such that

∀(t, x) ∈ [0, r 0 ] × T, t E(x)-(1/H(x)-δ)I d ≤ c 3,1 .
Since for t > 0, t E(x)-(1/H(x)-δ)I d = t δ-1/H(x) t E(x) , the last inequality leads to

∀(t, x) ∈]0, r 0 ] × T, t E(x) ≤ c 3,1 t 1/H(x)-δ .
This inequality is obviously fulfilled for t = 0 since 0 E(x) = 0 by convention.

(ii) Since δ > 0 and since E is continuous on T and satisfies Assumption 6, the map x → -E(x) + (1/H(x) + δ)I d is also continuous on T, satisfies Assumption 6 and takes values in M >0 R d . Then, using the same arguments as in the proof of assertion (i), there exists a finite constant c 3,2 = c 3,2 (T, δ, r 0 ) > 0 which only depends on δ, T and r 0 such that

∀(u, x) ∈ [0, 1/r 0 ] × T, u -E(x)+(1/H(x)+δ)I d ≤ c 3,2 .
Hence,

∀(t, x) ∈ [r 0 , +∞) × T, t E(x)-(1/H(x)+δ)I d ≤ c 3,2 ,
that is

∀(t, x) ∈ [r 0 , +∞) × T, t E(x) ≤ c 3,2 t 1/H(x)+δ .
The proof is then complete.

Proof of Lemma 3.4. Since E is continuous on T and satisfies Assumption 6, one can easily see that the map

N : T × R d -→ [0, +∞) (x, ξ) -→ ξ E(x)
, where • M is defined by [START_REF] Brunet-Imbault | A new anisotropy index on trabecular bone radiographic images using the fast Fourier transform[END_REF], is continuous using Lemma 3.2, Lemma 3.3 and the dominated convergence theorem. Furthermore,

∀x ∈ T, ∀ξ ∈ R d \{0}, ξ E(x) = ξ ξ ξ E(x) = ξ N x, ξ ξ .
Since N is continuous and positive on the compact set T × S d-1 , we have

0 < m T = inf T ×S d-1 N (y, θ) ≤ M T = sup T ×S d-1
N (y, θ) < +∞.

Hence for every x ∈ T and every ξ ∈ R d \{0},

ξ E(x) M T ≤ ξ ≤ ξ E(x) m T . ( 29
)
This inequality is obviously fulfilled for ξ = 0 since 0 = 0 E(x) = 0. Let us now focus on σ E(x) S E(x) . Applying Proposition 3.1, one obtains that

∀x ∈ T, σ E(x) (S E(x) ) = S E(x) σ E(x) (dθ) = q(x) R d 1 τ E(x) (ξ)≤1 dξ
where q is defined by [START_REF] Biermé | Hölder regularity for operator scaling stable random fields[END_REF]. By definition of • E(y) (see [START_REF] Brunet-Imbault | A new anisotropy index on trabecular bone radiographic images using the fast Fourier transform[END_REF]), for any y ∈ T and ξ ∈ R d ,

ξ E(y) ≤ 1 if and only if τ E(y) (ξ) ≤ 1, which leads to ∀x ∈ T, σ E(x) (S E(x) ) = q(x) R d 1 ξ E(x) ≤1 dξ.
Then, using [START_REF] Xiao | Properties of strong local nondeterminism and local times of stable random fields[END_REF] and the continuity of the positive function q on the compact set T , one easily finds two positive finite constants c, C such that Proof of Proposition 3.5. Let r 0 = inf ξ ≥1 inf x∈T τ E(x) (ξ). First, let us prove that τ E(x) (ξ) is uniformly bounded below for x ∈ T and ξ ≥ 1, that is r 0 > 0. Otherwise, for any ε ∈ (0, 1), one could find x ∈ T and ξ ∈ R d such that ξ ≥ 1 and

∀x ∈ T, c ≤ S E(x) σ E(x) (dθ) = σ E(x) S E(x) ≤ C.
τ E(x) (ξ) ≤ ε < 1.
where c 3,4 is a finite positive constant which only depends on T . Note that we can assume that c 3,4 ≥ 1. Let us now choose δ ∈ (0, min v∈T H(v)) such that max v∈T H(v) + δ < 1.

Since c 3,4 τ E(y) (ξ) -ε ≥ 1 and since H(x) -δ ≤ H(x) + δ, using Proposition 3.5, we obtain that

τ E(x) (τ E(y) (ξ) E(y)-E(x) E(y) (ξ)) ≤ c 3,6 c 3,4 τ E(y) (ξ) -ε H(x)+δ
where c 3,6 is a finite positive constant which only depends on T and δ. Then, since H(x) + δ < 1 and since c 3,4 τ E(y) (ξ) -ε ≥ 1,

τ E(x) (τ E(y) (ξ) E(y)-E(x) E(y) (ξ)) ≤ c 3,6 c 3,4 τ E(y) (ξ) -ε .
Hence, by Equation (31),

τ E(x) (ξ) ≤ c 3,6 c 3,4 τ E(y) (ξ) 1-ε .
Let us now assume that τ E(y) (ξ) ≥ 1. Since ξ ≤ 1 and τ E(y) (ξ)

1-ε ≥ 1, τ E(x) (ξ) ≤ c 3,6 ≤ c 3,6 τ E(y) (ξ) 1-ε .
Therefore, for any x, y ∈ T such that x -y ≤ γ, for any ξ ∈ R d such that 0 < ξ ≤ 1,

τ E(x) (ξ) ≤ c 3,8 τ E(y) (ξ) 1-ε
where the finite positive constant c which only depends on T such that

c 3,7 τ E(y) (ξ) 1+ε ≤ τ E(x) (ξ)
for any x, y ∈ T such that x -y ≤ γ, for any ξ ∈ R d such that ξ ≤ 1. The proof of ( 13) is then complete. The proof of ( 14) is similar.

6.2. Results on the scale parameter. This section is devoted to the proof of Lemmas 4.1, 4.2 and 4.3. We begin with two auxiliary lemmas:

Lemma 6.1. Let T = d i=1 [b i , d i ] with b i < d i for 1 ≤ i ≤ d. Assume that E : T → M >0 (R d )
is continuous on T and satisfies Assumptions 2 and 6. Let α ∈ (0, 2]. Then, for all ε ∈ (0, min w∈T (1/H(w)-1)) there exist two finite positive constants γ 1 = γ 1 (T, ε) and c 6,1 = c 6,1 (T, ε)

such that τ E(u) (ξ)≤η min ( ξ α , 1)τ E(u) (ξ) -αβα(u)-αε dξ ≤ c 6,1 η αγ 1
for every η ∈ (0, 1] and u ∈ T .

Proof. Let ε ∈ (0, min w∈T (1/H(w) -1)), u ∈ T and η ∈ (0, 1]. We set

I ε (η, u) = τ E(u) (ξ)≤η min ( ξ α , 1)τ E(u) (ξ) -αβα(u)-αε dξ.
By definition of τ E(u) (see [START_REF] Chainais | Virtual super resolution of scale invariant textured images using multifractal stochastic processes[END_REF]) and of β α (see ( 6)), Proposition 3.1 (applied with M = E(u))

leads to

I ε (η, u) ≤ η 0 S E(u) r E(u) θ α r -α(1+ε)-1 σ E(u) (dθ)dr.
Let δ ∈ (0, min w∈T (1/H(w) -1) -ε). Applying Lemma 3.3 (with r 0 = 1) and Lemma 3.4, one obtains that for any w ∈ T , any r ∈ (0, η] and any θ ∈ S E(w) ,

r E(w) θ ≤ r E(w) θ ≤ c 3,1 c 3,4 θ E(w) r 1/H(w)-δ = c 3,1 c 3,4 r 1/H(w)-δ
, where the finite positive constants c 3,1 = c 3,1 (T, δ) and c 3,4 = c 3,4 (T ) do not depend on (w, r, θ).

Therefore,

I ε (η, u) ≤ c 3,1 c 3,4 α σ E(u) S E(u) η 0 r α(1/H(u)-1-ε-δ)-1 dr.
Since δ < min w∈T (1/H(w) -1) -ε and since u ∈ T , we get 1/H(u) -1 -ε -δ > 0. Then, applying again Lemma 3.4, one easily sees that

I ε (η, u) ≤ c α 3,1 c α+1 3,4 η α(1/H(u)-1-ε-δ) α(1/H(u) -1 -ε -δ) .
Since η ∈ (0, 1],

I ε (η, u) ≤ c 6,1 η αγ 1 with c 6,1 = c α 3,1 c α+1 3,4 α min w∈T (1/H(w) -1 -ε -δ) ∈ (0, +∞)
and

γ 1 = min w∈T (1/H(w) -1 -ε -δ) ∈ (0, +∞).
Note that c 6,1 and γ 1 are well-defined by continuity of H on the compact set T .

Lemma 6.2. Let T = d i=1 [b i , d i ] with b i < d i for 1 ≤ i ≤ d. Assume that E : T → M >0 (R d ) is
continuous on T and satisfies Assumptions 2 and 6. Let α ∈ (0, 2]. Then, for all ε ∈ (0, 1) there exist two finite positive constants γ 2 = γ 2 (ε) and c 6,2 = c 6,2 (T, ε) such that

τ E(u) (ξ)>A min ( ξ α , 1)τ E(u) (ξ) -αβα(u)+αε dξ ≤ c 6,2 A -αγ 2
for every A ≥ 1 and u ∈ T .

Proof. Let A ∈ [1, ∞), u ∈ T , ε ∈ (0, 1) and

I ε (A, u) = τ E(u) (ξ)>A min ( ξ α , 1)τ E(u) (ξ) -αβα(u)+αε dξ.
Let us first observe that

I ε (A, u) ≤ τ E(u) (ξ)>A τ E(u) (ξ) -αβα(u)+αε dξ.
Then, applying as in the proof of Lemma 6.1 Proposition 3.1 with M = E(u) and Lemma 3.4, one obtains that

I ε (A, u) ≤ c 3,4 ∞ A r -α(1-ε)-1 dr
with c 3,4 = c 3,4 (T ) a finite positive constant which only depends on T . Then since ε < 1,

I ε (A, u) ≤ c 3,4 α(1 -ε) A -α(1-ε) ,
which concludes the proof.

Proof of Lemma 4.1. Since Assumption 5 is fulfilled, q and H are uniformly continuous on the compact set T . Then we can consider ε ∈ 0, min min w∈T 1/H(w) -1, 1 and there exists

γ = γ(ε) ∈ (0, 1) such that |q(u) -q(v)| < αε,
for any u, v ∈ T with u -v ≤ γ. Henceforth, by continuity of H on the compact set T , for any

u, v ∈ T with u -v ≤ γ, (15)
holds and then Y 1,α,x (u, v) is well-defined for any x ∈ R d .

Let us now consider x ∈ K and u, v ∈ T such that u -v ≤ γ. Then,

Y 1,α,x (u, v) = Re R d f 1,α,x (u, v, ξ)W α (dξ)
where

f 1,α,x (u, v, ξ) = e i x,ξ -1 ψ u (ξ) -βα(u) -ψ u (ξ) -βα(v) . (32)
Therefore, by definition of

• α , Y 1,α,x (u, v) α α = R d |f 1,α,x (u, v, ξ)| α dξ. (33)
Moreover, for any ξ ∈ R d \{0}, by Assumption 3, ψ u (ξ) = 0 and then by the Mean Value Theorem,

ψ u (ξ) -βα(u) -ψ u (ξ) -βα(v) = ψ u (ξ) -βα(u) |β α (u) -β α (v)|ψ u (ξ) -β ξ,u,v |log ψ u (ξ)| (34) for some |β ξ,u,v | ∈ [0, |β α (v) -β α (u)|]. Furthermore, since β α = 1 + q/α, |β α (w) -β α (w )| < ε
for any w, w ∈ T with w -w ≤ γ. Then, since T is a compact set, one can easily find a finite

positive constant c 1 = c 1 (T, γ(ε)) such that ψ w (ξ) -β ξ,w,w log ψ w (ξ) ≤ c 1 max ψ w (ξ) -1 , ψ w (ξ) ε (35)
for any ξ ∈ R d \{0} and any w, w ∈ T with w -w ≤ γ. Moreover, for any w ∈ T , since ψ w is E(w)-homogeneous,

∀ξ ∈ R d \{0}, ψ w (ξ) = τ E(w) (ξ)ψ w E(w) (ξ) .
By Assumptions 3 and 4, the function ψ is positive and continuous on the compact set (w, θ) ∈ T × R d ; θ E(w) = 1 . Then, there exist two finite positive constants c 2 = c 2 (T ) and c

3 = c 3 (T ) such that ∀w ∈ T, ∀ξ ∈ R d \{0}, c 2 τ E(w) (ξ) ≤ ψ w (ξ) ≤ c 3 τ E(w) (ξ). ( 36
)
Let us also remark that since K is a compact set, there exists a finite positive constant c 4 = c 4 (K) such that

∀y ∈ K, ∀ξ ∈ R d , e i y,ξ -1 ≤ c 4 min( ξ , 1). ( 37
)
Therefore, by (32), (34), ( 35), ( 36) and (37), for any

ξ ∈ R d \{0} |f 1,α,x (u, v, ξ)| ≤ c 5 |β α (u) -β α (v)| min( ξ , 1)τ E(u) (ξ) -βα(u) max(τ E(u) (ξ) -1 , τ E(u) (ξ)) ε (38)
where the finite positive constant c 5 does not depend on (x, u, v, ξ).

Then, by (33),

Y 1,α,x (u, v) α α ≤ c α 5 |β α (u) -β α (v)| α R d min( ξ α , 1)τ E(u) (ξ) -αβα(u) max(τ E(u) (ξ) -1 , τ E(u) (ξ)) αε dξ.
Since ε < min(min w∈T 1/H(w) -1, 1), Lemma 6.1 applied with η = 1 and Lemma 6.2 applied with A = 1 lead to

Y 1,α,x (u, v) α α ≤ c α 5 c 6,1 + c 6,2 |β α (u) -β α (v)
| α where c 6,1 and c 6,2 does not depend on (x, u, v). One easily concludes the proof since by Assumption 5, q and then β α = 1 + q/α is Lipschitz on the compact set T .

Proof of Lemma 4.2. As in the beginning of the proof of Lemma 4.1, we can choose γ small enough such that (15) holds for any u, v ∈ T with u -v ≤ γ. Hence, Y 2,α,x (u, v) is well-defined for any x ∈ R d , and u, v ∈ T with u -v ≤ γ.

Let us now consider x ∈ K and u, v ∈ T with u -v ≤ γ. Then,

Y 2,α,x (u, v) = Re R d f 2,α,x (u, v, ξ)W α (dξ),
where

f 2,α,x (u, v, ξ) = e i x,ξ -1 ψ u (ξ) -βα(v) -ψ v (ξ) -βα(v) . (39)
Therefore, by definition of

• α , Y 2,α,x (u, v) α α = R d |f 2,α,x (u, v, ξ)| α dξ.
Let ξ = 0 and let us split

g α (u, v, ξ) = ψ u (ξ) -βα(v) -ψ v (ξ) -βα(v) = g 1,α,η (u, v, ξ) + g 2,α,η (u, v, ξ) with g 1,α,η (u, v, ξ) = 1 τ E(v) (ξ)<η + 1 τ E(v) (ξ)>1/η g α (u, v, ξ) and g 2,α,η (u, v, ξ) = 1 η≤τ E(v) (ξ)≤1/η g α (u, v, ξ),
where η ∈ (0, 1).

First

Step: Study of g 1,α,η and choice of η.

By Assumption 5, β α = 1 + q/α is continuous on T and we can consider

β α = max w∈T β α (w) ∈ (1, +∞).
Let us choose ε = ε(α, T ) > 0 such that ε < min(min w∈T 1/H(w) -1, 1). Then, according to Proposition 3.6, up to change γ, we can assume γ = γ(ε) ∈ (0, 1) and for all ξ = 0 and w, w ∈ T

such that w -w ≤ γ, τ E(w) (ξ) ≥ c 3,7 τ E(w ) (ξ) min(τ E(w ) (ξ) -1 , τ E(w ) (ξ)) ε/βα
, where the finite positive constant c 3,7 = c 3,7 (T, ε) does not depend on w, w and ξ. Then, by (36) (see the proof of Lemma 4.1) and continuity of β α , there exists a finite positive constant

C 1 = C 1 (T, ε), which does not depend on (x, u, v, ξ), such that g 1,α,η (u, v, ξ) ≤ C 1 1 τ E(v) (ξ)<η + 1 τ E(v) (ξ)>1/η τ E(v) (ξ) -βα(v) max(τ E(v) (ξ) -1 , τ E(v) (ξ)) ε . (40)
Then, combining Equations ( 40) and (37), according to Lemma 6.1 and Lemma 6.2, there exist two finite positive constants ν = ν(T, ε) and C 2 = C 2 (K, T, ε), which do not depend on (u, v, x), such that for all η ∈ (0, 1] one has

I 1,α,η (x, u, v) = R d e i x,ξ -1 α g 1,α,η (u, v, ξ) α dξ ≤ C 2 η αν . Choosing η = η(ε, u, v) = u -v 1/ν , one gets that I 1,α,η (x, u, v) ≤ C 2 u -v α .
Second Step: Study of g 2,α,η . Now let us focus on g 2,α,η for this particular choice of η. By homogeneity of ψ u and ψ v ,

g 2,α,η (u, v, ξ) = 1 η≤τ E(v) (ξ)≤1/η τ E(v) (ξ) -βα(v) ψ u τ E(v) (ξ) -E(u) τ E(v) (ξ) E(v) E(v) (ξ) -βα(v) -ψ v E(v) (ξ) -βα(v)
.

By Lemma 3.4, there exist two finite positive constants c 3,3 = c 3,3 (T ) and c 3,4 = c 3,4 (T ) such that

∀w ∈ T, c 3,3 ≤ E(w) (ξ) ≤ c 3,4 .
Then, since ξ = 0 and v ∈ T , v) . By Assumption 6, E(u)E(v) = E(v)E(u) and then

E(v) (ξ) -τ E(v) (ξ) -E(u) τ E(v) (ξ) E(v) E(v) (ξ) ≤ c 3,4 I -τ E(v) (ξ) -E(u) τ E(v) (ξ) E(
I -τ E(v) (ξ) -E(u) τ E(v) (ξ) E(v) = I -τ E(v) (ξ) E(v)-E(u) .
Therefore,

I -τ E(v) (ξ) E(v)-E(u) ≤ E(v) -E(u) log τ E(v) (ξ) max τ E(v) (ξ) -1 , τ E(v) (ξ) E(v)-E(u) , since e M -e M ≤ M -M e M + M , for any M, M ∈ M(R d ) such that M M = M M . Then, since η ≤ τ E(v) (ξ) ≤ 1/η, I -τ E(v) (ξ) E(v)-E(u) ≤ E(v) -E(u) |log η|η -E(v)-E(u) .
Hence, according to Assumption 5, there exists c 2,2 = c 2,2 (T ) such that

I -τ E(v) (ξ) E(v)-E(u) ≤ c 2,2 u -v | log η|η -c 2,2 u-v since η ≤ 1. Finally, since η = u -v 1/ν , one can choose γ small enough such that I -τ E(v) (ξ) E(v)-E(u) ≤ c 3,3 2c 3,4 ≤ 1 2 , which implies that E(v) (ξ) -τ E(v) (ξ) E(v)-E(u) E(v) (ξ) ≤ c 3,3 2 ≤ c 3,4 2 .
Then,

c 3,3 2 ≤ τ E(v) (ξ) E(v)-E(u) E(v) (ξ) ≤ 3c 3,4 2 .
Using the Mean Value Theorem for t → t -βα(v) , the continuity of β α and Assumption 4 with

K = T × y ∈ R d ; c 3,3 2 ≤ y ≤ 3c 3,4 2 ,
one can find two finite positive constants C 3 and C 4 that only depend on T and γ such that

ψ u τ E(v) (ξ) E(v)-E(u) E(v) (ξ) -βα(v) -ψ v E(v) (ξ) -βα(v) ≤ C 3 ψ u τ E(v) (ξ) E(v)-E(u) E(v) (ξ) -ψ v E(v) (ξ) ≤ C 4 u -v 1 + log τ E(v) (ξ) max(τ E(v) (ξ) -1 , τ E(v) (ξ)) c 2,2 u-v .
To conclude let us recall that we have chosen ε ∈ (0, min(min w∈T 1/H(w) -1, 1)). Up to choose γ smaller we may assume that c 2,2 γ < ε. Then, one can find a finite positive constant C 5 = C 5 (T, ε)

such that g 2,α,η (u, v, ξ) ≤ C 5 u -v τ E(v) (ξ) -βα(v) max(τ E(v) (ξ) -1 , τ E(v) (ξ)) ε ,
for all ξ = 0 and u, v ∈ T such that u -v ≤ γ. Then by Lemmas 6.1 and 6.

v 2 α ((u, v) | (T n , ξ n ) n ) = d 2 α 2 E |g 1 | 2 +∞ n=1 T -2/α n m x 0 (ξ n ) -2/α |f α (u, ξ n ) -f α (v, ξ n )| 2
where f α is defined by [START_REF] Lepage | Multidimensional infinitely divisible variables and processes. II. In Probability in Banach spaces[END_REF]. Similarly to the proof of Theorem 4.4 we write

f α (u, ξ n ) -f α (v, ξ n ) = f 1,α,u (u, v, ξ n ) + f 2,α,u (u, v, ξ n ) + f 3,α,v (u, v, ξ n )
where f 1,α,u is defined by (32), f 2,α,u by (39) and f 3,α,y (u, v, ξ) = e i u,ξ -e i v,ξ ψ y (ξ) -βα(y) .

We then denote for j ∈ {1, 2}, Hence,

v 2 j,α ((u, v) | (T n , ξ n ) n ) =
v 2 α ((u, v) | (T n , ξ n ) n ) ≤ 4 3 j=1 v 2 j,α ((u, v) | (T n , ξ n ) n ).
Let ε 1 ∈ (0, min min w∈T 1/H(w) -1, 1 ). ξ 2 1 τ E(x 0 ) (ξ)<η + 1 τ E(x 0 ) (ξ)>1/η τ E(x 0 ) (ξ) -2β 2 (x 0 ) max τ E(x 0 ) (ξ) -1 , τ E(x 0 ) (ξ)

ε 1 ≤ c 2 η 2ν . (42)
Moreover, following the proof of Lemma 4.2 and using Proposition 3.6, choosing γ = γ(ε 1 ) smaller if necessary, one can also find a finite positive constant c 3 = c 3 (T, ε 1 ) such that for u -x 0 ≤ γ/2 and v -x 0 ≤ γ/2,

v 2 2,α ((u, v) | (T n , ξ n ) n ) ≤ c 3 u -v 2 W + σ 2 2 ( u -v ) ,
where W is defined by (41) and for all h ≥ 0,

σ 2 2 (h) = +∞ n=1 T -2/α n m x 0 (ξ n ) -2/α min ξ n 2 , 1 1 τ E(x 0 ) (ξn)<h 1/ν + 1 τ E(x 0 ) (ξn)>h -1/ν τ E(x 0 ) (ξ n ) -2βα(x 0 )
× max(τ E(x 0 ) (ξ) -1 , τ E(x 0 ) (ξ)) 2ε 1 /3 .

Let us recall that the density function of ξ n is m x 0 . Then, using the definition of m x 0 , of β α and β 2 , one can easily find a finite positive constant c 4 = c 4 (T, ε 1 ) such that for h ≥ 0 small enough,

E σ 2 2 (h)|(T n ) n ≤ c 4 ς 2 (h) 2 +∞ n=1 T -2/α n with ς 2 (h) 2 = R d
min ξ 2 , 1 1 τ E(x 0 ) (ξ)<h 1/ν + 1 τ E(x 0 ) (ξ)>h -1/ν τ E(x 0 ) (ξ) -2β 2 (x 0 ) max(τ E(x 0 ) (ξ) -1 , τ E(x 0 ) (ξ)) ε 1 dξ.

Then, (42) leads to the existence of a finite positive constant c 5 = c 5 (ε 1 ) such that almost surely for h ≥ 0 small enough,

E σ 2 2 (h)|(T n ) n ≤ c 5 h 2 +∞ n=1 T -2/α n .
Then, since h → σ 2 2 (h) is monotone, almost surely lim h→0 σ 2 2 (h) h 2-ε = 0 for any ε ∈ (0, 1) (see for instance [START_REF] Biermé | Hölder regularity for operator scaling stable random fields[END_REF]).

Third

Step: Study of v 3,α Using Proposition 3.6, there exist γ = γ(ε 1 ) ∈ (0, 1) and a finite positive constant c 6 = c 6 (T, ε 1 ) such that for any u -x 0 ≤ γ/2 and v -x 0 ≤ γ/2, v 2 3,α ((u, v) | (T n , ξ n ) n ) ≤ c 6 σ 2 3 (τ E(x 0 ) t (u -v)), where, for all h ≥ 0,

σ 2 3 (h) = +∞ n=1 T -2/α n m x 0 (ξ n ) -2/α min h E(x 0 ) ξ n 2 , 1 τ E(x 0 ) (ξ n ) -2βα(x 0 ) max(τ E(x 0 ) (ξ), τ E(x 0 ) (ξ) -1 ) 2ε 1 /3 .
Following the proof of Lemma 5.2 of [START_REF] Biermé | Hölder regularity for operator scaling stable random fields[END_REF], one obtains that

Example 2 . 1 (

 21 Operator scaling random fields). Let E 0 be a d×d real matrix whose eigenvalues have real parts greater than 1. Let us consider a function ψ : R d → [0, ∞) E 0 -homogeneous, locally Lipschitz on R d \{0} and such that ∀ξ = 0, ψ(ξ) = 0.

Theorem 4 . 4 .

 44 Assume that Assumptions 1-6 are fulfilled. Then, for γ > 0 small enough there exist two finite positive constants c 4,5 = c 4,5 (T, γ) and c 4,6 = c 4,6 (T, γ) such that

  v) α when u and v are close to x. More precisely, we use the lower bound stated in Theorem 4.4 and the comparison of the radial part τ E(x) t with the Euclidean norm. Corollary 4.10. Let x ∈ R d . Assume that Assumptions 1-6 are fulfilled with T

Example 4 . 1 .

 41 Let E 0 be a matrix of size d × d whose eigenvalues have real parts greater than 1.

  x∈R d H(x) > 0. Let us considerE = P -1 diag (1/H 1 , . . . , 1/H d ) P,with P ∈ GL d (R) an invertible matrix, and X α,ϕ as in Example 2.4. Let (e j ) 1≤j≤d be the canonical basis of R d and f j = P -1 e j for all 1 ≤ j ≤ d. Then, according to Corollary 4.10, for all 1 ≤ j ≤ d,

Remark 4 . 1 .

 41 One can also be interested in directional and global local Hölder exponents. Then, Corollary 4.10 and the previous examples hold true replacing pointwise Hölder exponents by local ones. Moreover, assumptions of Corollary 3.15 of [11] are satisfied by each example in the Gaussian case (α = 2), such that one can exchange for all x and almost surely. In other words, if α = 2, in the previous examples, there exists an almost sure event Ω * , which does not depend on x, and on which the local Hölder exponents are known.

Definition 5 . 1 .

 51 Let x ∈ R d and A(x) be a d × d real matrix. A random field (X(y)) y∈R d is locally asympotically operator scaling of order A(x) at point x if

Therefore Lemma 3 .

 3 4 holds with c 3,3 = min(1/M T , c) and c 3,4 = max(1/m T , C).

, 8 =

 8 max(c 3,6 c 3,4 , c 3,6 ) = c 3,6 c 3,4 does not depend on x, y ∈ T , nor on ε, γ. Note that the last inequality also holds for ξ = 0 since τ E(x) (0) = τ E(y) (0) = 0. Moreover, by symmetry in τ E(x) (ξ) and τ E(y) (ξ), one can easily find a finite positive constant c 3,7

6 . 3 .

 63 2 and (37), there exists a finite positive constant C 6 = C 6 (T, K, ε) such thatI 2,α,η (x, u, v) = R d e i x,ξ -1 α g 2,α,η (u, v, ξ) α dξ ≤ C 6 u -v α ,for all x ∈ K and all u, v ∈ T such that u -v ≤ γ. The conclusion follows fromY 2,α,x (u, v) α α = I 1,α,η (x, u, v) + I 2,α,η (x, u, v).Proof of Lemma 4.3. Let x ∈ R d . Then, the random fieldX ψx (v) v∈R d = (Y α,ψ (v, x, x)) v∈R dis well-defined and is a harmonizable operator scaling α-stable random field in the sense of Theorem 4.1 of[START_REF] Biermé | Operator scaling stable random fields[END_REF] with respect to E(x) t . Moreover,Y 3,α,x (u, v) = X ψx (u) -X ψx (v).Then, by stationarity of increments of X ψx and the operator scaling property (see Corollary 4.2of [7]), when u = v Y 3,α,x (u, v) α α = τ E(x) t (u -v) α J α x, E(x) t (u -v) where ∀θ ∈ S E(x) t , J α (x, θ) = R d e i θ,ξ -1 α ψ -αβα(x)x (ξ)dξ.Since J α is positive and continuous on the compact set(y, θ) ∈ R d × R d ; y ∈ K,and θ ∈ S E(y) t , there exist c 4,3 = c 4,3 (K) and c 4,4 = c 4,4 (K) two finite positive constants such that ∀y ∈ K, ∀θ ∈ S E(y) t , c 4,3 ≤ J α (x, θ) ≤ c 4,4 , which concludes the proof. Modulus of continuity. Proof of Lemma 4.7. If α = 2, assertion (2) is a direct consequence of Theorem 4.4 and Proposition 3.6.Let us now assume that α ∈ (0, 2). Then, according to[START_REF] Lepage | Conditional moments for coordinates of stable vectors[END_REF]

m 2 α 2 E |g 1 | 2 +∞ n=1 T

 222n=1 x 0 (ξ n ) -2/α |f j,α,u (u, v, ξ n )| 2 and v 2 3,α ((u, v) | (T n , ξ n ) n ) = d -2/α n m x 0 (ξ n ) -2/α |f 3,α,v (u, v, ξ n )| 2 .

First/ 3 .

 3 Step: Study of v 1,α Using (38), Proposition 3.6 and the Lipschitz property of β α on T , one can find γ = γ(ε 1 ) ∈ (0, 1) and a finite positive constant c 1 = c 1 (T, ε 1 ) such that|f 1,α,u (u, v, ξ)| ≤ c 1 u -v min ( ξ , 1) τ E(x 0 ) (ξ) -βα(x 0 ) max(τ E(x 0 ) (ξ) -1 , τ E(x 0 ) (ξ)) ε 1 /3for any ξ ∈ R d \{0} and any u, v ∈ T such that u -x 0 ≤ γ and v -x 0 ≤ γ. Hence, almost surelyv 2 1,α ((u, v) | (T n , ξ n ) n ) ≤ u -v 2 W where W = c 1 +∞ n=1 T -2/α n ζ n (41) with ζ n = m x 0 (ξ n ) -2/α min ξ n 2 , 1 τ E(x 0 ) (ξ n ) -2βα(x 0 ) max τ E(x 0 ) (ξ n ) -1 , τ E(x 0 ) (ξ n ) 2ε 1One easily checks that ζ n , n ∈ N\{0} are i.i.d. integrable random variables and then that W < ∞ almost surely (since T n /n → 1 almost surely and 2/α > 1).Second Step: Study of v 2,αFollowing the proof of Lemma 4.2, one can choose two finite positive constants ν = ν(ε 1 ) and c 2 = c 2 (T, ε 1 ) such that for η small enough,R d

E σ 2 3 1 +∞ n=1 T - 2 /α n ,σ 2 3

 31n=12n2 (h)|(T n ) n ≤ c 7 h 2-εwhere the finite positive constant c 7 = c 7 (ε 1 ) does not depend on h. Therefore, almost surely lim h→0 (h) h 2-ε = 0 for all ε ∈ (0, ε 1 ). Proposition 3.5, Step 1, 2 and 3 lead to the conclusion.

  for any x, ξ ∈ R d ,

	defined on R d with values in the space of diagonal matrices. Let ρ ∈ (0, inf x∈R d	min 1≤j≤d	H j (x)] and
	ψ x (ξ) = |ξ 1 |	H 1 (x) ρ	+ • • • + |ξ d |	H d (x) ρ	ρ	,
	for every x, ξ ∈ R d . Then, Assumptions 2-6 are fulfilled such that, under Assumption 1, the
	random field X α,ψ given by (7) is well-defined.
	Example 2.4. Let E and ψ x as in Example 2.3. Let P ∈ GL d (R) be an invertible matrix. Then
	the map
	x → P -1 E(x)P
	satisfies Assumptions 2, 5 and 6. Moreover, the function
	ϕ : (x, ξ) → ψ x (P ξ)
	satisfies Assumptions 3 and 4. Then, the harmonizable multi-operator scaling stable random field
	X α,ϕ is well-defined by (7).
	Example 2.5. Let d = 2. Let us consider the map
	where h = 1/E. Hence, if d = 1, under Assumptions 1-3, the random process X α,ψ is a multi-
	fractional harmonizable stable random motion, up to a deterministic multiplicative function.
	Example 2.3. For every 1 ≤ j ≤ d, assume that H j is a locally Lipschitz function on R d with
	values in (0, 1). Assume also that
	inf x∈R d	min 1≤j≤d	H j (x) > 0.
	Consider the map
	E = diag (1/H 1 , . . . , 1/H d )

This work has been supported by the grant ANR-09-BLAN-0029-01.

Since E is continuous, H is also continuous and we can choose η > 0 such that 2η < min y∈T 1 H(y) . Then, according to Lemma 3.3 there would exist c 3,1 (T, η, 1) such that

However, this would imply that

where the finite positive constant c 3,4 is given by Lemma 3.4 and only depends on T . Therefore, the lower bound of equation ( 12) holds with c 3,5 = min

where the finite positive constant c 3,3 is given by Lemma 3.4 and only depends on T . Therefore, the upper bound of equation ( 12) holds with c 3,6 = max

.

The proof of equation [START_REF] Herbin | Stochastic 2-microlocal analysis[END_REF], that is the case where ξ ≤ 1, is similar.

Proof of Proposition 3.6. Let ε ∈ (0, 1). Then, since E is continuous on the compact set T , there exists γ = γ(ε) > 0 such that

by definition of τ M (see [START_REF] Chainais | Virtual super resolution of scale invariant textured images using multifractal stochastic processes[END_REF]).

Let us first assume that τ E(y) (ξ) ≤ 1. Then τ E(y) (ξ) E(y)-E(x) E(y) (ξ) ≤ τ E(y) (ξ) -E(y)-E(x) E(y) (ξ) . Hence, by (30) and by Lemma 3.4, τ E(y) (ξ) E(y)-E(x) E(y) (ξ) ≤ c 3,4 τ E(y) (ξ) -ε