Lorenzo Pucci

Pietro Gravino

Vito D P Servedio

Modeling the emergence of a new language: Naming Game with hybridization

In recent times, the research field of language dynamics has focused on the investigation of language evolution, dividing the work in three evolutive steps, according to the level of complexity: lexicon, categories and grammar. The Naming Game is a simple model capable of accounting for the emergence of a lexicon, intended as the set of words through which objects are named. We introduce a stochastic modification of the Naming Game model with the aim of characterizing the emergence of a new language as the result of the interaction of agents. We fix the initial phase by splitting the population in two sets speaking either language A or B. Whenever the result of the interaction of two individuals results in an agent able to speak both A and B, we introduce a finite probability that this state turns into a new idiom C, so to mimic a sort of hybridization process. We study the system in the space of parameters defining the interaction, and show that the proposed model displays a rich variety of behaviours, despite the simple mean field topology of interactions.

Emergence of a lexicon as a language

. In this framework, the emergence of a particular communication system is not due to an external coordination, or a common psychological background, but it simply occurs as a convergence effect in the dynamical processes that start from an initial condition with no existing words (agents having to invent them), or with no agreement.

 we recall the NG basic pairwise interaction scheme. A fundamental assumption of NG is that vocabulary evolution associated to every single object is considered independent. This lets us simplify the evolution of the whole lexicon as the evolution of the set of words associated to a single object, equally perceived in the sensorial sphere by all agents.

Fig. 1. Naming Game interaction scheme. A speaker and a hearer are picked up randomly. The speaker utters a word chosen randomly in his vocabulary. Failure: the hearer does not know the uttered word and he simply adds it to his vocabulary. Success: the hearer knows the uttered word and both agents by agreement delete all the words in their vocabularies except the uttered word one.

The simplicity of NG in describing the emergence of a lexicon also relies on the fact that competing words in an individual vocabulary are not weighted, so that they can be easily stored or deleted. In fact it turns out that for convergence to a single word consensus state, the weights are not necessary as it was supposed by the seminal work in this research field [START_REF] Steels | A self-organizing spatial vocabulary[END_REF]. Every agent is a box that could potentially contain an infinite number of words, so the number of states that can define the agent is limited only by the number of words diffused in the population at the beginning of the process (so that anyone can speak with at least one word).

In this work we aimed not only at the aspect of competition of a language form with other ones, but also at introducing interactions between them, with the possibility of producing new ones. We investigate conditions for the success of a new idiom, as product of a synthesis and competition of preexisting idioms. To this purpose, we introduce a stochastic interaction scheme in the basic Naming Game accounting for this synthesis, and show in a very simple case that the success of the new spoken form at expense of the old ones depends both on the stochastic parameters and the fractions of the different idioms spoken by populations at the beginning of the process. We have simulated this process starting from an initial condition where a fraction n A of the population speaks with A and the remaining n B with B. It turns out that when the different-speaking fractions are roughly of the same size the new form, which we call C (therefore we shall refer to our model as the "ABC model" in the following), created from the synthesis of A and B, establishes and supplants the other two forms. Instead, when n A > n B (or symmetrically n B > n A ), above a threshold depending on the chosen stochastic parameters, the term A establishes (or symmetrically B), namely one of the starting idioms prevails and settles in the population.

Previous attempts to model the emergence of a Creole language, i.e. an idiom originating by a sort of hybridization and fusion of languages, can be found in literature [START_REF] Satterfield | Toward a Sociogenetic Solution: Examining Language Formation Processes Through SWARM Modeling[END_REF][START_REF] Nakamura | Language Dynamics[END_REF][START_REF] Strimling | Modelling the evolution of creoles[END_REF].

The ABC Model

The model we propose here is based on a mean field topology involving N agents, i.e. any two agents picked up randomly can interact. Despite this pretty simple topology of interactions, the proposed model will show a richness of behaviors. In the basic Naming Game, the initial condition is fixed with an empty word list for each agent. If an agent chosen as speaker still owns an empty list, he invents a new word and utters it. In our proposed new model all agents are initially assigned a given word, either A or B, so that there is no possibility to invent a brand new word, unless an agent with both A and B in his list is selected. In that case, we introduce a finite probability γ that his list containing A and B turns into a new entry C (Fig. 2). We interpret γ as a measure of the need of agents to obtain a sort of hybridization through which they understand each other, or a measure of the natural tendency of two different language forms to be synthesized together. In the latter case, different values of γ would depend on the language forms considered in a real multi language scenario.

The stochastic rule AB → C may be applied either in the initial interaction phase by changing the dictionary of the speaker, or at the final stage by changing the state of the hearer after the interaction with the speaker. In this paper we show the results obtained by running the trasformation AB → C (and also ABC → C when a speaker has got an ABC vocabulary) before the interaction. Introducing the trasformation after the interaction changes the results from a qualitative point of view, producing only a shift of transition lines between the final states in the space of parameters defining the stochastic process.

A further stochastic modification of the basic Naming Game interaction, firstly introduced in [START_REF] Baronchelli | Nonequilibrium phase transition in negotiation dynamics[END_REF], has also been adopted here. It gives account for the emergence or persistance of a multilingual final state, where more than one word is associated to a single object. This is done by mimicking a sort of confidence degree among agents: in case of a successful interaction, namely when the hearer shares the word uttered by the speaker, the update trasformation of the two involved vocabularies takes place with probability β (the case β = 1 obviously reduces to the basic NG). Baronchelli et al. [START_REF] Baronchelli | Nonequilibrium phase transition in negotiation dynamics[END_REF] showed in their model (which corresponds to our model at γ = 0) that a transition occurs around β = 1/3. For β > 1/3 the system converges to the usual one word consensus state (in this case only A or B). For β < 1/3 the system converges to a mixed state, where more than one word remains, so that there exist single and multi word vocabularies at the end of the process (namely A, B and AB). A linear stability analysis of the mean field master equations of this model (describing the evolution of vocabulary frequencies n A , n B and n AB ) shows that the steady state n A = 1, n B = n AB = 0 (or symmetrically n B = 1, n A = n AB = 0), which is stable for β > 1/3, turns unstable if β < 1/3, where viceversa the steady state n A = n B = b (β), n AB = 1 -2b (β) emerges as a stable state, with b(β) being a simple algebraic expression of the parameter β. In our work, as shown next, we found that the transition order-disorder (i.e. single word vs. multi word final state) at β = 1/3 remains for all the values of γ.

The numerical stochastic simulation of the process for selected values of (β, γ) ∈ [0, 1] × [0, 1], indicates that the system presents a varied final phase space as shown in Fig. 3 and4, left panel. The transition line at β = 1/3 remains: for β > 1/3 the system converges to a one-word final state, with a single word among A, B and C, while for β < 1/3 it converges to a multi-word state with one or more than one word spoken by each agent. This result is confirmed by the integration of the mean field master equation of the model, describing the temporal evolution of the fractions of all vocabulary species n A , n B , n C , n AB , n AC , n BC , n ABC , present in the system. The results of such integration, involving a fourth order Runge-Kutta algorithm, display the same convergence behaviour of the stochastic model (Fig. 3 and4, right panel), though they are obviously characterized by less noise.

High confidence β > 1/3

By looking at Fig. 3 at the region with β > 1/3, we note a transition interval between the final state composed only of either A or B (red region) and a final state with only C (orange region). The fuzziness of the border dividing these two domains, evident in the left panel of the figure, can be ascribed to finite size effects, for the separation line gets sharper by enlarging the number of agents N , eventually collapsing towards the strict line obtained by the integration of the mean field master equation (right panel of the figure), which we report in the Appendix section. The linear stability analysis of the cumbersome mean field master equation reveals that these two phases are both locally stable, and turn unstable when β < 1/3. The convergence to one or the other phase depends on the initial conditions, i.e. whether the system enters the respective actraction basins during its dynamical evolution. To demonstrate this, we studied the behaviour of the system by fixing β and varying both γ and the initial conditions n A = α, n B = 1 -α, with α ∈ [0, 1]. The result reported in Fig. 4 clearly shows a dependence on the initial conditions. In particular, if the initial condition |n A -n B | = 1 -2α is sufficiently large, the convergence to the C phase disappears. The corresponding threshold value of α decreases slightly by decreasing the value of parameter β, going from α = 0.34 for β = 1 to α = 0.24 for β = 0.34 (i.e. slightly above the transition signalled by β = 1/3).

By numerically solving the mean field master equation, we analyzed the evolution of the system in proximity of the transition between the (orange) region characterized by the convergence to the C state and the (red) region where the convergence is towards either the state A or B, being these latter states discriminated by the initial conditions n A > 1/2 or n A < 1/2 respectively. We show the result in Fig. 5 obtained by fixing β = 0.8 and α = 0.49. Initially, both the fractions of n A (black curve) and n B (red curve) decrease in favour of n AB (blue curve). Thereafter, also n AB starts to decrease since the mixed AB and ABC states can be turned directly into C, causing an increase of n C (green curve). While the n AB , n ABC , n A , n AC fractions vanish quite soon, mainly because fewer agents have the A state in their vocabulary, the states involving B and C survive, reaching a meta-stable situation in which n C = n B ≈ 0.37 and n BC ≈ 0.26. This meta-stable state of the system is clearly visible in the mid panel of Fig. 5. The life time of the meta-stable state diverges by approaching the corresponding set of parameters (γ = γ c ≈ 0.12234742 in the Figure ; γ c depends on β), with the result that the overall convergence time diverges as well. The stochastic simulation would differ from the solution of the master equation right at this point: a small random fluctuation in the fraction of either B or C will cause the convergence of the stochastic model towards one or the other state, while the deterministic integration algorithm (disregarding computer numerical Another interesting area in the γ, β phase space of the model is the boundary between the region around β = 1/3, where one switches from a convergence to a single state (β > 1/3) to a situation with the coexistence of multiple states (β < 1/3). As β → 1/3 + the time of convergence towards the C consensus phase, which is the absorbing state whenever γ > 0, diverges following a powerlaw with the same exponent as in the case of γ = 0, where we recover the results of [START_REF] Baronchelli | Nonequilibrium phase transition in negotiation dynamics[END_REF], i.e. t conv (β -1/3) -1 (Fig. 6). Of course, in the case γ = 0 there is no C state involved anymore and the competition is only between the A and B states. Moreover, as we note from Fig. 6, the convergence time to the one word consensus state is the highest when γ = 0 and decreases by increasing the value of γ. This result is somewhat counter intuitive since we expect that the presence of three states A, B, C would slow down the convergence with respect Fig. 6. Convergence time in the ABC model for β → 1/3 + : Time of convergence of the system to the one word consensus state as obtained with the mean field master equation as a function of (β -1/3). The initial population densities were set to nA = 0.49 and nB = 0.51. Importantly, if γ > 0 the system converges to the hybrid state C, while for γ = 0 the state B prevails (orange curve). The dotted line refers to a power-law behaviour of exponent -1. We used a fourth order Runge-Kutta algorithm with temporal step dt = 0.1.

to a situation with only two states A and B, but actually in the first case another supplementary channel is yielding the stable C case, i.e. the AB -→ C channel (neglecting of course the rare ABC -→ C) thus accelerating the convergence.

The linearization of the mean field master equation around the absorbing points with β > 1/3 delivers six negative eigenvalues, confirming that the points in the orange and red region of Fig. 3 are locally stable. Moreover, it comes out that those eigenvalues do not depend on γ showing that the choice of the initial conditions on n A and n B is crucial in entering the two different actraction basins. As a consequence of this independence on γ, the equation of the line dividing the orange and red regions cannot be calculated easily.

Low confidence β < 1/3

In the case β < 1/3 we get multi-word final states. The green color in Fig. 3 stands for an asymptotic situation where n B = n C and n BC = 1 -n B -n C (and of course a symmetric situation with B replaced by A when the initial conditions favour A rather than B). The dependence of the asymptotic fractions n B and n C on β is the same of that occurring for γ = 0 and presented in [START_REF] Baronchelli | Nonequilibrium phase transition in negotiation dynamics[END_REF].

Instead, the blue color of Fig. 3 represents an asymptotic state where all vocabulary typologies are present A, B, C, AB, BC, AC and ABC, with n A = n B and n AC = n BC . In this case the vocabulary fractions depend both on β and γ. White dots in the left panel of Fig. 3, which tend to disappear enlarging the population size N , are points where the system has not shown a clear stable state after the chosen simulation time. In fact, they disappear in the final phase space described by the mean field master equation. Contrary to the case of β > 1/3, the final state does not depend on the particular initial conditions provided that initially n A + n B = 1. By fixing β = 0.1 and varing γ and initial conditions α ∈ [0, 1], we get the steady behavior shown in Fig. 7.

The linearization of the mean field master equation around the absorbing points with β < 1/3 in the green region of Fig. 3 reveals that those are actractive points (six negative eigenvalues) irrespective of the initial condition provided that n A + n B = 1. The equation of the transition line that divides the blue and green region can be inferred numerically, again with the linearization of the master equation. In particular the transition point at β = 0 can be found analytically to be at γ = 1/4. The independence on the initial conditions makes the region β < 1/3 substantially different from the complementary region β > 1/3.

We modeled the emergence of a new language C as a result of the mutual interaction of two different populations of agents initially speaking different idioms A and B and interacting each other without restriction (mean field). Such tight connections between individuals speaking two different idioms is certainly unrealistic, but the same reasoning can be extended to accomplish the birth of single hybrid words resulting from the interaction of two pronunciation variants of the same object (eg. the english word rowel, perhaps an hybridization of the latin rota and the english wheel ).

Three parameters govern the time evolution of the model and characterize the final asymptotic state: β the measure of the tendency of a hearer to adopt the shared word used by the speaker (confidence), γ the probability that two forms A and B are synthesized into the form C, and α the initial condition in the space n A + n B = 1. It turns out that:

for β < 1/3 the system converges to multi-word states, all containing a fraction of the state C, and that do not depend on the initial conditions provided that n A + n B = 1. for β > 1/3 the system converges to a consensus state where all agents end up with the same state, either A, B or C. The transition line γ (β) separating the A or B convergence state from C, which are all locally stable independently from γ, depends on the initial distribution of n A and n B , with n A + n B = 1. Moreover, the invention of C produces a reduction of the time of convergence to the consensus state (all agents speaking with C) when starting with an equal fraction of A and B in the population.

Interestingly the modern point of view of linguists links the birth and continuous development of all languages as product of local interaction between the varied language pools of individuals who continuously give rise to processes of competition and exchange between different forms, but also creation of new forms in order to get an arrangement with the other speakers [START_REF] Mufwene | The ecology of language evolution[END_REF]. In this view of a language as mainly a social product it seems that the use of the Naming Game is particularly fit, in spite of the old conception of pure languages as product of an innate psychological background of individuals [START_REF] Steels | Modeling the cultural evolution of language[END_REF]. It would be interesting to apply our model in the study of real language phenomena where a sort of hybridization of two or more languages in a contact language ecology takes ground. There are many examples of this in the history, as for example the formation of the modern romance European languages from the contact of local Celtic populations with the colonial language of Romans, Latin. A more recent example of this is the emergence of Creole languages in colonial regions where European colonialists and African slaves came into contact [START_REF] Mufwene | Population movements and contacts in language evolution[END_REF].

The starting point for a comparison of our model with this kind of phenomena would be retrieving demographic data of the different ethnic groups at the moment they joined in the same territory and observing if a new language established. Our point of view would be obviously not to understand how particular speaking forms emerged, but to understand whether there is a correlation between the success of the new language forms and the initial language demography. In this case, a more refined modeling would take into account also the temporal evolution of the population due to reproduction and displacements, and the particular topologies related to the effective interaction scheme acting in the population.

Fig. 2 .

 2 Fig. 2. ABC model: Example of interaction with the introduction of the trasformation of AB → C with probability γ on the speaker vocabulary. The trasformation is carried before the usual speaker-hearer NG interaction, which includes the stochastic update of the vocabularies in case of a success depending on β.

Fig. 3 .

 3 Fig. 3. Phase diagram γ, β of the ABC model: The new language C is created at the beginning of the interaction by updating the speaker's vocabulary. We have 0 ≤ β ≤ 1 and 0 ≤ γ ≤ 1 on the horizontal and vertical axis, respectively. Left: Results of the stochastic model where the number of agents involved is N = 1000 and the final time is set to 10 6 . The initial condition on populations is fixed as nA nB and nAB = 0 at t = 0. Right: Results of the mean field master equation numerically integrated till a final time t = 1000. The initial condition where chosen as nA = 0.49 and nB = 0.51 (and symmetrically nA = 0.51 and nB = 0.49). We employed a fourth order Runge-Kutta algorithm with temporal step dt = 0.1.

Fig. 4 .

 4 Fig. 4. Phase diagram γ, α of the ABC model: Final phase diagram depending on γ and initial condition nA = 1 -α and nB = α with α ∈ [0, 1], with β = 0.7 fixed. Left: diagram obtained after 10 6 pair interactions of the stochastic process. Right: diagram obtained by integrating up to t = 1000 the mean field master equation with a fourth order Runge-Kutta algorithm employing a temporal step dt = 0.1. For α < 0.3 (or symmetrically α > 0.7) the system converges to the nA = 1 (or symmetrically nB = 1) for every considered value of γ.

Fig. 5 .

 5 Fig. 5. Temporal evolution of the ABC model: Temporal evolution of all vocabulary species occurring in the system calculated by integrating the mean field master equation. The initial population was chosen as nB = 0.51 and nA = 0.49, while we fixed β = 0.8. For γ = 0.1223 the system converges to the B consensus state, while for γ = 0.1224 the system converges to the C consensus state. The initial trivial transient with the formation of the AB mixed states has been removed. By approaching the transition point the convergence time increases.

Fig. 7 .

 7 Fig. 7. Phase diagram γ, α of the ABC model: Phase diagram depending on γ and initial condition nA = 1 -α and nB = α with α ∈ [0, 1], with β = 0.1 fixed. Left: diagram obtained after t = 10 6 steps of the stochastic process. Right: diagram obtained by integrating up to t = 1000 the mean field master equation with a fourth order Runge-Kutta algorithm employing a temporal step dt = 0.1.

Acknowledgements

The authors wish to thank V. Loreto and X. Castelló for useful discussions. The present work is partly supported by the EveryAware european project grant nr. 265432 under FP7-ICT-2009-C.

Appendix: mean field master equation

The mean field master equation of the ABC model, in the case in which the speaker changes her vocabulary with the rule {AB, ABC} γ -→ {C} before the interaction, is the following:


Hermine Bierm

C Éline Lacaux

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

H ÖLDER REGULARITY FOR OPERATOR SCALING STABLE RANDOM FIELDS

Keywords: 2000 Mathematics Subject Classification. Primary: 60617, 60G18; Secondary: 60G60, 60G52, 60G15 Operator scaling random fields, Stable and Gaussian laws, Hölder regularity, Hausdorff dimension

We investigate the sample paths regularity of operator scaling α-stable random fields. Such fields were introduced in [6] as anisotropic generalizations of self-similar fields and satisfy the

In the case of harmonizable operator scaling random fields, the sample paths are locally Hölderian and their Hölder regularity is characterized by the eigen decomposition of R d with respect to E. In particular, the directional Hölder regularity may vary and is given by the eigenvalues of E. In the case of moving average operator scaling random α-stable random fields, with α ∈ (0, 2) and d ≥ 2, the sample paths are almost surely discontinous.

Introduction

In this paper we consider operator scaling stable random fields as introduced in [START_REF] Biermé | Operator scaling stable random fields[END_REF]. More precisely, for E a real d × d matrix with positive real parts of the eigenvalues, a scalar valued random field (X(x)) x∈R d is called operator scaling for E and H > 0 if for every c > 0 (1)

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

(f dd)
= means equality of finite dimensional distributions and as usual c E = exp(E log c) with exp

(A) = ∞ k=0 A k
k! is the matrix exponential. Let us remark that up to consider the matrix E/H, we may assume, without loss of generality, that H = 1.

These fields can be seen as anisotropic generalizations of self-similar random fields. Let us recall that a scalar valued random field (X(x)) x∈R d is said to be H-self-similar with H ∈ R if

{X(cx); x ∈ R d } (f dd)
= {c H X(x); x ∈ R d } for every c > 0. Then a H-self-similar field is also an operator scaling field for the matrix E = I d /H, where I d is the identity matrix of size d×d. Self-similar random fields are used in various applications such as internet traffic modelling [START_REF] Willinger | Self-similarity and heavy tails: Structural modeling of network traffic[END_REF], ground water modelling and mathematical finance, just to mention a few. Various examples can be found for instance in the books [START_REF] Véhel | Fractals in engineering: from theory to industrial applications[END_REF][START_REF] Abry | Lois d'échelle, fractales et ondelettes[END_REF][START_REF] Samorodnitsky | Stable non-Gaussian random processes[END_REF]. A very important class of such fields are given by Gaussian random fields and especially by the fractional Brownian field B H , where H ∈ (0, 1) is the so-called Hurst parameter. The random field B H is H-self-similar and has stationary increments, e.g.

{B H (x + h) -B H (h); x ∈ R d } (f dd) = {B H (x); x ∈ R d } for any h ∈ R d .
It is an isotropic generalization of the famous fractional Brownian motion, implicitely introduced in [START_REF] Kolmogorov | Wienersche Spiralen und einige andere interessante Kurven in Hilbertsche Raum[END_REF] and defined in [START_REF] Mandelbrot | Fractional Brownian motion, fractionnal noises and applications[END_REF].

Nevertheless, Gaussian random fields are not convenient for some heavy tails phenomena modelling. For this purpose, α-stable random fields have been introduced. A scalar valued random field {X(x); x ∈ R d } is said to be symmetric α-stable (SαS), for α ∈ (0, 2), if any linear combination n k=1 a k X(x k ) is SαS. We address to the book [START_REF] Samorodnitsky | Stable non-Gaussian random processes[END_REF] for a well understanding of such fields. The selfsimilar α-stable fields with stationary increments have been extensively used to propose alternative to Gaussian modelling (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) and are isotropic.

However, certain applications (see, e.g., [START_REF] Benson | Aquifer operator-scaling and the effect on solute mixing and dispersion[END_REF][START_REF] Bonami | Anisotropic analysis of some Gaussian models[END_REF] and references therein) require that the random field is anisotropic and satisfies a scaling relation. This scaling relation should have different Hurst indices in different directions and these directions should not necessarily be orthogonal. To reach this goal the authors of [START_REF] Biermé | Operator scaling stable random fields[END_REF] propose a new class of random fields with an anisotropic behavior driven by a d × d matrix E. More precisely, they introduce α-stable random fields which have stationary increments and satisfy the operator scaling property [START_REF] Abry | Lois d'échelle, fractales et ondelettes[END_REF]. Two different classes of such fields are defined and analyzed, using a moving average representation as well as an harmonizable one. In the Gaussian case α = 2, according to [START_REF] Biermé | Operator scaling stable random fields[END_REF] there exist modifications of these fields which are almost surely Hölder-continuous of certain indices. We give similar results here in the stable case α ∈ (0, 2) for harmonizable operator scaling stable random fields. Actually, we obtain their critical global and directional Hölder exponents, which are given by the eigenvalues of E. In general, such fields are anisotropic and their sample paths properties varies with the direction. In particular, in the case where E is diagonalizable, for any eigenvector θ j associated with the real eigenvalue λ j , harmonizable operator scaling stable random field admits H j = 1/λ j as critical Hölder exponent in direction θ j . Let us point out that we establish an accurrate upper bound for the modulus of continuity. Such upper bound has already been given in the case of real harmonizable fractional stable motions, which are self-similar, in [START_REF] Kôno | Hölder continuity of sample paths of some self-similar stable processes[END_REF] and in the case of some Gaussian random processes in [START_REF] Kôno | On the modulus of continuity of sample functions of gaussian processes[END_REF]. Then, in this paper, we generalize these results to any dimension d and any harmonizable operator scaling stable fields. From this upper bound, we can deduce the Hölder sample paths regularity. Let us point out that we also obtain this upper bound in the case of Gaussian operator scaling random field, which improves the sample paths properties establishes in [START_REF] Biermé | Operator scaling stable random fields[END_REF].

Furthermore, whereas in the Gaussian case α = 2, moving average and harmonizable fields have the same kind of regularity properties, this is no more true in the case α ∈ (0, 2). In particular, we show that for d ≥ 2, a moving average operator scaling stable random field does not admit any continuous modification. Remark that if d = 1, the sample paths regularity properties are already known since the processes studied are self-similar moving average stable processes, see for example [START_REF] Samorodnitsky | Stable non-Gaussian random processes[END_REF][START_REF] Kôno | Hölder continuity of sample paths of some self-similar stable processes[END_REF][START_REF] Takashima | Sample path properties of ergodic self-similar processes[END_REF].

One of the main tool for the study of sample paths of operator scaling random fields is the change of polar coordinates with respect to the matrix E introduced in [START_REF] Meerschaert | Limit distributions for sums of independent random vectors[END_REF] and used in [START_REF] Biermé | Operator scaling stable random fields[END_REF]. For X a Gaussian operator scaling random field with stationary increments, using (1), we can write the variogramme of X as

v 2 (x) = E X 2 (x) = τ E (x) 2H E X 2 ( E (x)) ,
where τ E (x) is the radial part of x with respect to E and E (x) is its polar part. Therefore, in the Gaussian case, the sample paths regularity depends on the behavior of the polar coordinates (τ E (x), E (x)) around x = 0. Such property also holds in the stable case α ∈ (0, 2). The Hölder regularity of the sample paths follows from estimates of τ E (x) compared to x . These estimates are given in Section 3 and their proof are postponed to the Appendix.

Furthermore, to study the sample paths in the stable case, the other main tool we use is a series representation of harmonizable operator scaling α-stable random fields. Such representations in series of infinitely divisible laws have been studied in [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][START_REF] Rosiński | On series representations of infinitely divisible random vectors[END_REF][START_REF] Rosiński | On path properties of certain infinitely divisible processes[END_REF]. As in [START_REF] Kôno | Hölder continuity of sample paths of some self-similar stable processes[END_REF], our study is based on a LePage series representation. Actually, the main idea is to choose a representation which is a conditionnally Gaussian series.

In Section 2, we recall the definition of harmonizable operator scaling random fields. Then, Sections 3 and 4 are devoted to the main tools we need for the study of the sample paths of these fields. More precisely, Section 3 deals with the polar coordinates with respect to a matrix E and Section 4 gives the LePage series representation. In Section 5, the sample paths properties and the Hausdorff dimension of the graph are given. Section 6 is concerned with moving average operator scaling random fields.

Harmonizable representation

Let us recall the definition of harmonizable operator scaling stable random fields, given by [START_REF] Biermé | Operator scaling stable random fields[END_REF].

Let E be a real d × d matrix. Let λ 1 , . . . , λ d be the complex eigenvalues of E and a j = (λ j ) for each j = 1, . . . , d. We assume that [START_REF] Adler | The Geometry of Random Field[END_REF] min

1≤j≤d a j > 1.
Let ψ : R d → [0, ∞) be a continuous, E t -homogeneous function, which means according to Definition 2.6 of [START_REF] Biermé | Operator scaling stable random fields[END_REF] that ψ(c E t x) = cψ(x) for all c > 0. We assume moreover that ψ(x) = 0 for x = 0. Such functions were studied in detail in [START_REF] Meerschaert | Limit distributions for sums of independent random vectors[END_REF], Chapter 5 and various examples are given in Theorem 2.11 and Corollary 2.12 of [START_REF] Biermé | Operator scaling stable random fields[END_REF].

Let 0 < α ≤ 2 and W α (dξ) be a complex isotropic α-stable random measure on R d with Lebesgue control measure (see [START_REF] Samorodnitsky | Stable non-Gaussian random processes[END_REF] p.281). If α = 2, W α (dξ) is a complex isotropic Gaussian random measure. Let q = trace (E). Definition 2.1. Since (2) is fulfilled, the random field

(3) X α (x) = R d e i x,ξ -1 ψ(ξ) -1-q/α W α (dξ) , x ∈ R d ,
is well defined and called harmonizable operator scaling stable random field.

From Theorem 4.1 and Corollary 4.2 of [START_REF] Biermé | Operator scaling stable random fields[END_REF], X α is stochastically continuous, has stationary increments and satisfies the following operator scaling property

(4) ∀c > 0, X α (c E x); x ∈ R d (f dd) = cX α (x); x ∈ R d .
For notational sake of simplicity we denote the kernel function by 3) is a real harmonizable stable random field (see [START_REF] Samorodnitsky | Stable non-Gaussian random processes[END_REF] for details on such fields). In this case, X α is self-similar with exponent H, i.e.

(5) f (x, ξ) = e i x,ξ -1 ψ(ξ) -1-q/α . Let us recall that f (x, •) ∈ L α (R d ) for any x ∈ R d ,
∀c > 0, {X α (cx); x ∈ R d } (f dd) = {c H X α (x); x ∈ R d }.
Let us quote that, if α = 2, X α is a fractional Brownian field and its critical Hölder exponent is given by its Hurst index H (see Theorem 8.3.2 of [START_REF] Adler | The Geometry of Random Field[END_REF] for instance).

Example 2.2. Assume that E is diagonalizable. Then, all the eigenvalues of E are real given by (a j ) 1≤j≤d . Let (θ j ) 1≤j≤d be a basis of some corresponding eigenvectors and consider the function ψ defined by

ψ(x) =   d j=1 | x, θ j | 2/a j   1/2 , x ∈ R d .
The function ψ is clearly continuous and non negative on R d . Moreover, since c E t x, θ j = x, c E θ j = c a j x, θ j , it is also E t -homogeneous. Finally, since (θ j ) 1≤j≤d is a basis of R d we have that ψ(x) = 0 if and only if x = 0. Then we can define X α by (3) and in this case the operator scaling property (4) implies that ∀j = 1, . . . , d, ∀c > 0, {X α (ctθ j ); t ∈ R}

(f dd) = c 1/a j X α (tθ j ); t ∈ R .
The random field X α is self-similar with Hurst index H j = 1/a j in the direction θ j . In particular, in the Gaussian case (α = 2), (X 2 (tθ j )) t∈R is a fractional Brownian motion with Hurst index H j . Then, in this case, its critical Hölder exponent is equal to H j .

The main tool in the study of operator scaling random fields is the change of coordinates in a kind of polar coordinates with respect to the matrix E. Then, before we study the sample paths of X α , we recall in the next section the definition of these coordinates and give some estimates of the radial part.

Polar coordinates

According to Chapter 6 of [START_REF] Meerschaert | Limit distributions for sums of independent random vectors[END_REF], since E is a real d × d matrix with positive real parts of the eigenvalues there exists a norm • E on R d such that for the unit sphere

S E = {x ∈ R d : x E = 1} the map Ψ E : (0, ∞) × S E -→ R d \ {0} (r, θ) -→ r E θ
is a homeomorphism. Hence we can write any x ∈ R d \{0} uniquely as x = τ E (x) E E (x) for some radial part τ E (x) > 0 and some direction E (x) ∈ S E such that x → τ E (x) and x → E (x) are continuous. Observe that

S E = {x ∈ R d : τ E (x) = 1} is compact and set (6) m E = min S E
x and M E = max

S E
x .

We know that τ E (x) → ∞ as x → ∞ and τ E (x) → 0 as x → 0. Hence we can extend τ E (•) continuously by setting τ E (0) = 0. Let us 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 E on S E such that for all f ∈ L 1 (R d , dx) we have

R d f (x) dx = ∞ 0 S E f (r E θ) σ E (dθ) r q-1 dr.
In the Gaussian case (α = 2), the variogramme of X 2 can be rewritten as follows

v 2 (x) = τ E (x) 2 E X 2 ( E (x)) 2 ,
using the operator scaling property (4). Then, the Hölder regularity of X 2 follows from estimates of τ E (x) compared to x around x = 0, e.g. the Hölder regularity of τ E around 0, see [START_REF] Biermé | Operator scaling stable random fields[END_REF]. Then, in order to get an upper bound for the modulus of continuity (for any α), we need precise estimates of τ E (x). As done in [START_REF] Mason | Sample path properties of operator-self-similar Gaussian random fields[END_REF] for the study of operator-self-similar Gaussian random fields we use the Jordan decomposition of the matrix E to get estimates of τ E . From the Jordan decomposition's theorem (see [START_REF] Hirsch | Differential equations, dynamical systems, and linear algebra[END_REF] p. 129 for instance), there exists a real invertible d × d matrix P such that D = P -1 EP is of the real canonical form, which means that D is composed of diagonal blocks which are either Jordan cell matrix of the form

     λ 1 . . . 0 0 λ . . . . . . . . . . . . . . . 1 0 . . . 0 λ     
with λ a real eigenvalue of E or blocks of the form

(7)         Λ I 2 0 . . . 0 0 Λ I 2 . . . . . . . . . . . . . . . . . . 0 . . . . . . . . . . . . I 2 0 . . . . . . 0 Λ         with Λ = a -b b a and I 2 = 1 0 0 1 ,
where the complex numbers a ± ib (b = 0) are complex conjugated eigenvalues of E.

Let us denote by • the subordinated norm of the Euclidean norm on the matrix space. Precise estimates of τ E follow from the next lemma. Lemma 3.2. Let J be either a Jordan cell matrix of size l or a block of the form (7) of size 2l associated with the eigenvalue λ = a + ib. Then, for any t ∈ (0, e -1 ] ∪ [e, +∞)

t a ≤ t J ≤ √ 2l e t a |log t| l-1 .
Proof. see the Appendix.

Let us be more precise on the Jordan decomposition of E. Let us recall that the eigenvalues of E are denoted by λ j , j = 1 . . . d and that a j = (λ j ) > 1 for j = 1, . . . , d. There exist J 1 , . . . , J m , where each J j is either a Jordan cell matrix or a block of the form [START_REF] Bonami | Anisotropic analysis of some Gaussian models[END_REF], and P a real d × d invertible matrix such that

E = P      J 1 0 . . . 0 0 J 2 0 . . . . . . . . . . . . 0 0 . . . 0 J m      P -1 .
We can assume that each J j is associated with the eigenvalue λ j of E and that

1 < a 1 ≤ • • • ≤ a m .
We also set H j = a -1 j and have [START_REF] Clausel | Gaussian fields with a group self-similarity[END_REF] 0

< H m ≤ • • • ≤ H 1 < 1.
If λ j ∈ R, J j is a Jordan cell matrix of size lj = l j ∈ N\{0}. If λ j ∈ C\R, J j is a block of the form [START_REF] Bonami | Anisotropic analysis of some Gaussian models[END_REF] of size lj = 2l j ∈ 2N\{0}. Then for any t > 0,

t E = P       t J 1 0 . . . 0 0 t J 2 0 . . . . . . . . . . . . 0 0 . . . 0 t Jm       P -1
We denote by (e 1 , . . . , e d ) the canonical basis of R d and set f j = P e j for every j = 1, . . . , d.

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

W j = Vect f k ; j-1 i=1 li + 1 ≤ k ≤ j i=1 lj . (9)
Then, each W j is a E-invariant set and E = m j=1 W j .

The following result gives bounds on the growth rate of τ E (x) in terms of the real parts of the eigenvalues of E. Proposition 3.3. For any r ∈ (0, 1) there exist c 1 , c 2 > 0 such that for every

1 ≤ j 0 ≤ j ≤ m, c 1 x H j 0 |log x | -(p j 0 ,j -1)H j 0 ≤ τ E (x) ≤ c 2 x H j |log x | (p j 0 ,j -1)H j holds for any x ∈ ⊕ j k=j 0 W k with x ≤ r and p j 0 ,j = max j 0 ≤k≤j l k .
Proof. See the Appendix.

Then, we easily deduce the following corollary.

Corollary 3.4. For any r ∈ (0, 1) there exist c 1 , c 2 > 0 such that for any x ∈ W j with x ≤ r

(10) c 1 x H j |log x | -(l j -1)H j ≤ τ E (x) ≤ c 2 x H j |log x | (l j -1)H j and for any x ∈ R d with x ≤ r (11) c 1 x H 1 |log x | -(l-1)H 1 ≤ τ E (x) ≤ c 2 x Hm |log x | (l-1)Hm
where l = max 1≤j≤m l j .

Therefore we have precise estimates for the Hölder regularity of the radial part. Let us remark that we improve the first statement of Lemma 2.1 of [START_REF] Biermé | Operator scaling stable random fields[END_REF] and that the second one can also be improved in a similar way. From these estimates we deduce the Hölder regularity of X α in Section 5. As already mentionned, the study of the sample paths is based on a series representation. Then, before we state regularity properties, it remains to give the LePage series representation of harmonizable operator scaling random fields X α defined by (3).

Representation as a LePage series

An overview on series representations of infinitely divisible random variable without Gaussian part can be found for example in [START_REF] Rosiński | On path properties of certain infinitely divisible processes[END_REF][START_REF] Rosiński | Series representations of Lévy processes from the perspective of point processes[END_REF] and references therein. In particular, LePage series representation ( [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]) have been used in [START_REF] Kôno | Self-similar stable processes with stationary increments[END_REF][START_REF] Kôno | Hölder continuity of sample paths of some self-similar stable processes[END_REF] to study the sample paths regularity of some self-similar α-stable random motions with α ∈ (0, 2). Here, this representation is also the main representation we use in the case α ∈ (0, 2). Actually, in the Gaussian case α = 2, such representation does not hold.

Let us now introduce some notations that will be used throughout the paper. Let µ be an arbitrary probability measure equivalent to the Lebesgue measure on R d and let m be its Radon-Nikodym derivative that is µ(dξ) = m(ξ)dξ. Notation 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. isotropic complex random variables so that g n (d) = e iθ g n for any θ ∈ R. We also assume that 0

< E(|g n | α ) < +∞. • (ξ n ) n≥1 is a sequence of i.i.d. random variables with common law µ(dξ) = m(ξ)dξ.
According to Chapter 3 and Chapter 4 of [START_REF] Samorodnitsky | Stable non-Gaussian random processes[END_REF], stochastic integrals with respect to an α-stable random measure Λ can be represented as a LePage series as soon as the control measure of Λ is a finite measure. The next proposition generalizes this to W α whose control measure is the Lebesgue measure. It is a consequence of Lemma 4.1 of [START_REF] Kôno | Self-similar stable processes with stationary increments[END_REF], which is a correction of Lemma 1.4 of [START_REF] Marcus | Characterizations of almost surely continuous p-stable random Fourier series and strongly stationary processes[END_REF]. This proposition can also be deduced from [START_REF] Rosiński | On path properties of certain infinitely divisible processes[END_REF][START_REF] Rosiński | On series representations of infinitely divisible random vectors[END_REF], concerned with series representations of stochastic integrals with respect to infinitely divisible random measures. Proposition 4.1. Assume that α ∈ (0, 2). Then, for every complex valued function h ∈ L α R d , the series

(12) Y h = +∞ n=1 T -1/α n m(ξ n ) -1/α h(ξ n )g n converges almost surely. Furthermore, C α Y h (d) = R d h(ξ)W α (dξ) with (13) C α = E(| (g 1 )| α ) -1/α 1 2π π 0 |cos (x)| α dx 1/α +∞ 0 sin (x) x α dx -1/α . Remark 4.1.
According to Proposition 4.1, taking α ∈ (0, 2), the random measure

Λ α (dξ) = C α +∞ n=1 T -1/α n m(ξ n ) -1/α g n δ ξn (dξ)
is a complex isotropic α-stable random measure.

Proof.

Let V n = m(ξ n ) -1/α h(ξ n )g n . Then, V n ,
σ α = E(| (V 1 )| α ) +∞ 0 sin (x)
x α dx. Since g 1 is invariant by rotation and independent with ξ 1 ,

E(| (V 1 )| α ) = E m(ξ 1 ) -1 |h(ξ 1 )| α E(| (g 1 )| α ) = E(| (g 1 )| α ) R d |h(ξ)| α dξ.
Moreover, by definition of an isotropic α-stable random stable measure (see [START_REF] Samorodnitsky | Stable non-Gaussian random processes[END_REF]),

∀z ∈ C, E exp i z R d h(ξ)M (dξ) = exp (-c α α (h)|z| α ) with c α α (h) = 1 2π π 0 |cos (x)| α dx R d |h(ξ)| α dξ. Therefore, we have C α Y h (d) = R d h(ξ)W α (dξ)
where C α is defined by [START_REF] Kôno | On the modulus of continuity of sample functions of gaussian processes[END_REF], which concludes the proof.

From the previous proposition, we deduce the following statement which is the main series representation we use in our investigation. Proposition 4.2. Let α ∈ (0, 2). For every x ∈ R d , the series

(14) Y α (x) = C α +∞ n=1 T -1/α n m(ξ n ) -1/α f (x, ξ n )g n ,
where C α is defined by [START_REF] Kôno | On the modulus of continuity of sample functions of gaussian processes[END_REF], converges almost surely. Furthermore,

Y α (x) : x ∈ R d (f dd) = X α (x) : x ∈ R d .
Proof. From Proposition 4.1, for any x ∈ R d , the convergence of the series follows from the fact that f (x, •) ∈ L α R d . The equality in distribution between X α and Y α is obtained by linearity of both fields.

Using LePage representation ( 14) of X α and the estimates given in Section 3, we give an upper bound for the modulus of continuity of X α and obtain the critical Hölder regularity of its sample paths in the next section.

Hölder regularity and Hausdorff dimension

Throughout this section we fix K a compact set of R d and investigate the Hölder regularity of the sample paths of X α on K, with X α the harmonizable operator scaling stable random field defined by [START_REF] Ayache | A Fourier formulation of the Frostman criterion for random graphs and its applications to wavelet series[END_REF].

Let us recall that for the Gaussian case α = 2, according to Theorem 5.4 of [START_REF] Biermé | Operator scaling stable random fields[END_REF], the Hölder regularity of X 2 depends on the subspaces (W j ) 1≤j≤m defined by [START_REF] Falconer | Fractal geometry[END_REF] and associated to the eigenvalues of E. More precisely, Theorem 5.4 of [START_REF] Biermé | Operator scaling stable random fields[END_REF] implies that, when restricted to the subspace W j , the Gaussian random field {X 2 (x); x ∈ W j } admits H j as critical Hölder exponent. This follows from the fact that the regularity of X 2 is determined by the regularity of τ E around 0, which is given by H j according to [START_REF] Hirsch | Differential equations, dynamical systems, and linear algebra[END_REF]. Here, we give an upper bound for the modulus of continuity of X α in the general case α ∈ (0, 2]. Then we prove that the critical Hölder exponents are the same than in the Gaussian case α = 2. Let us state our main result.

Theorem 5.1. Let α ∈ (0, 2). There exists a modification X * α of X α on K such that (15) lim δ↓0 sup

x,y∈K

x-y ≤δ

|X * α (x) -X * α (y)| τ E (x -y)|log τ E (x -y)| 1/α+1/2+ε = 0 a.s.
for any ε > 0.

This result was proved in the case of harmonizable self-similar processes in [START_REF] Kôno | Hölder continuity of sample paths of some self-similar stable processes[END_REF], e.g. in the case of Example 2.1 with d = 1. The main idea is to use the LePage series representation [START_REF] Kôno | Hölder continuity of sample paths of some self-similar stable processes[END_REF] where g n , n ≥ 1, are Gaussian complex isotropic random variables. Furthermore, it remains to choose the density distribution m of the ξ n . In [START_REF] Kôno | Hölder continuity of sample paths of some self-similar stable processes[END_REF], the authors choose

m(ξ) = c η |ξ||log |ξ|| 1+η , ξ ∈ R\{0} where c η > 0. A straightforward generalization in higher dimension d leads to choose m(ξ) = c η ξ d |log ξ | 1+η , ξ ∈ R d \{0}.
Remark that in this case (e.g. Example 2.1) the matrix E = I d /H = E t and that we can choose

• E t = • .
Then, using classical polar coordinates, we obtain that for x = 0,

τ E t (x) = x H and E t (x) = x x and therefore m(ξ) = c η τ E t (ξ) q |log τ E t (ξ)| 1+η since q = trace(E) = d/H.
Then we choose this density in the general case. The advantage is that m only depends on the radial part τ E t .

Proof of Theorem 5.1. We can assume without loss of generality that

K = [0, 1] d . According to Proposition 4.2 for every x ∈ R d Y α (x) = C α +∞ n=1 T -1/α n m(ξ n ) -1/α f (x, ξ n )g n converges almost surely and Y α (f dd) = X α .
As already mentionned, we assume that g n , n ≥ 1 are Gaussian complex isotropic random variables and that the density distribution of ξ n is defined by

m(ξ) = c η τ E t (ξ) q |log τ E t (ξ)| 1+η , ξ ∈ R d \{0},
where η > 0 and c η is chosen such that R d m(ξ)dξ = 1. In particular, conditionally to (T n , ξ n ) n , Y α (x) is a real-valued Gaussian random variable.

As in the proof of Kolmogorov-Centsov Theorem (see [START_REF] Karatzas | Brownian Motion and Stochastic Calculus[END_REF]), we first prove that almost surely for τ E (x k -x k ) small enough,

|Y α (x k ) -Y α (x k )| ≤ Cτ E (x k -x k )|log τ E (x k -x k )| 1/α+1/2+ε .
where (x k ) k is some countable dense sequence of K. Then, X α satisfies the same property. Finally, we give a modification X * α of X α for which [START_REF] Kôno | Self-similar stable processes with stationary increments[END_REF] holds. In the first step, we construct the sequence (x k ) k we use.

Step 1. For k ∈ N\{0} let us choose ν k ∈ N\{0} the smallest integer such that

c 2 d Hm/2 2 -ν k Hm |ν k log 2| (l-1)Hm ≤ 2 -k ,
where c 2 and l are given by Corollary 3.4 for r = 1/2. Remark that by definition, ν k ≤ ν k+1 . Up to change c 2 in Proposition 3.4, we can assume that c 2 d Hm/2 2 -Hm |log 2| (l-1)Hm > 1, which implies that ν k > 1 for every k. Then, since 1

≤ ν k -1 ≤ ν k , c 2 d Hm/2 2 -(ν k -1)Hm |(ν k -1) log 2| (l-1)Hm > 2 -k and we have (16) 2 -k 2 √ d -Hm c -1 2 < 2 -ν k (ν k log 2) l-1 Hm ≤ 2 -k √ d -Hm c -1 2 .
Let us remark that (ν k ) k≥1 is an increasing sequence and then that ν k ≥ k for every k. Furthermore, taking the logarithm of ( 16), one easily proves that lim

k→+∞ k ν k = H m , which implies ν k = O(k).
For every k ∈ N\{0} and j = (j 1 , . . . , j d ) ∈ Z d we set

x k,j = j 2 ν k , D k = x k,j : j ∈ Z d ∩ [0, 2 ν k ] d and N k = card D k = (2 ν k + 1) d .
Then D k is a 2 -k net of K for τ E in the sense that for any x ∈ K one can find x k,j ∈ D k such that τ E (x-x k,j ) ≤ 2 -k . Actually, by Proposition 3.4, it is sufficient to choose j such that

j i ≤ 2 ν k x i < j i +1 for 1 ≤ i ≤ d.
Let us remark that the sequence (D k ) k is increasing and set D = +∞ k=1 D k .

Step 2. Almost surely, for any x, y ∈ D

Y α (x) -Y α (y) = C α +∞ n=1 T -1/α n m(ξ n ) -1/α (f (x, ξ n ) -f (y, ξ n ))g n ,
where C α is defined by [START_REF] Kôno | On the modulus of continuity of sample functions of gaussian processes[END_REF]. Since the sequences (T n ) n , (ξ n ) n and (g n ) n are independent and (g n ) n is a sequence of i.i.d. Gaussian complex isotropic random variables

R(x, y) = +∞ n=1 T -1/α n m(ξ n ) -1/α (f (x, ξ n ) -f (y, ξ n ))g n is a Gaussian isotropic complex random variable conditionally to (T n , ξ n ) n . Remark that Y α (x) - Y α (y) = C α (R(x, y)) almost surely. Therefore, conditionally to (T n , ξ n ) n , Y α (x) -Y α (y) is a real centered Gaussian random variable with variance v 2 ((x, y) | (T n , ξ n ) n ) = C 2 α 2 E |R(x, y)| 2 | (T n , ξ n ) n = C 2 α 2 E |g 1 | 2 +∞ n=1 T -2/α n m(ξ n ) -2/α |f (x -y, ξ n )| 2 , since |f (x, ξ n ) -f (y, ξ n )| = |f (x -y, ξ n )|.
We consider the set

E k i,j = {ω : |Y α (x k,i ) -Y α (x k,j )| > v((x k,i , x k,j ) | (T n , ξ n ) n ) ϕ(τ E (x k,i -x k,j
))} , where, as in [START_REF] Kôno | On the modulus of continuity of sample functions of gaussian processes[END_REF],

ϕ(t) = 2C ϕ d log 1 t , t > 0
and C ϕ > 0 will be chosen later. Then, for every (k, i, j),

P E k i,j = E E 1 E k i,j | (T n , ξ n ) n .
We give in the following an upper bound of this probability for well chosen (k, i, j). Note that if Z is a real centered Gaussian random variable with variance 1, we have

E 1 E k i,j | (T n , ξ n ) n = P(|Z| > ϕ(τ E (x k,i -x k,j ))) almost surely.
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 ∩ [0, 2 ν k ] d : τ E (x k,i -x k,j ) ≤ δ k .
For every (i, j) ∈ I k , since ϕ is a decreasing function

P(|Z| > ϕ(τ E (x k,i -x k,j ))) ≤ P(|Z| > ϕ(δ k )).
Then, we recall that

∀u ≥ 0, P(Z > u) ≤ e -u 2 /2 √ 2πu .
Therefore, for every k ∈ N\{0} and (i, j) ∈ I k ,

P E k i,j ≤ 2 π e -ϕ 2 (δ k )/2 ϕ(δ k ) = 2 -(1-δ)kCϕd 2C ϕ d(1 -δ)k log 2 . Hence, ∞ k=1 (i,j)∈I k P E k i,j ≤ 1 2C ϕ d(1 -δ) log 2 +∞ k=1 2 -(1-δ)kCϕd card I k .
Since ν k = O(k), the lower bounds of ( 16) and Corollary 3.4 leads to

card j ∈ Z d ∩ [0, 2 ν k ] d : (i, j) ∈ I k = O δ d/H 1 k 2 dk/Hm k 2d(l-1) , for any i ∈ Z d ∩ [0, 2 ν k ] d . Then one can find a finite constant C > 0 such that ∞ k=1 (i,j)∈I k P E k i,j ≤ C C ϕ (1 -δ) ∞ k=1 k 3d(l-1) 2 -kd -2 Hm + 1-δ H 1 +(1-δ)Cϕ ,
which is finite for C ϕ > 2 Hm -1 H 1 and δ small enough. By the Borel-Cantelli Lemma, almost surely there exists an integer k * (ω) such that for every k ≥ k * (ω),

|Y α (x) -Y α (y)| ≤ v((x, y) | (T n , ξ n ) n ) ϕ(τ E (x -y))
as soon as x, y ∈ D k with τ E (x -y) ≤ δ k .

Step 3. As in [START_REF] Kôno | Hölder continuity of sample paths of some self-similar stable processes[END_REF] let us give an upper bound of

v 2 ((x, y) | (T n , ξ n ) n ) = C 2 α 2 E |g 1 | 2 +∞ n=1 T -2/α n m(ξ n ) -2/α |f (x -y, ξ n )| 2 with respect to τ E (x -y). By definition of f v 2 ((x, y) | (T n , ξ n ) n ) ≤ C 2 α 2 E |g 1 | 2 σ 2 (τ E (x -y)) ,
where for all h > 0

σ 2 (h) = +∞ n=1 T -2/α n m(ξ n ) -2/α min M E h E t ξ n , 2 2 ψ(ξ) -2-2q/α ,
with M E given by [START_REF] Biermé | Operator scaling stable random fields[END_REF]. For the sake of clearness we postpone the proof of the control of σ 2 (h) in Appendix and state it in the following lemma.

Lemma 5.2. For any γ ∈ (0, 1) there exists a finite constant c > 0 such that 1) almost surely as soon as h ≤ 1 -γ.

E σ 2 (h) | (T n ) n ≤ c +∞ n=1 T -2/α n h 2 |log h| (1+η)(2/α-
Following [START_REF] Kôno | Hölder continuity of sample paths of some self-similar stable processes[END_REF] let us denote b(h) = h| log h| (1+η)/α . Then by Lemma 5.2,

E +∞ k=1 σ 2 (2 -k ) b 2 (2 -k ) (T n ) n < +∞.
Therefore by independence of (T n ) n and (ξ n ) n , almost surely

lim k→+∞ σ 2 (2 -k ) b 2 (2 -k ) = 0.
Up to change the Euclidean norm • by the equivalent norm • E t defined in Lemma 6.1.5 of [START_REF] Meerschaert | Limit distributions for sums of independent random vectors[END_REF] the map h → h E t ξ is increasing and so is σ 2 . Also, one can conclude, as in [START_REF] Kôno | Hölder continuity of sample paths of some self-similar stable processes[END_REF], that almost surely

lim h→0 σ 2 (h) b 2 (h) = 0.
Therefore, up to change k * one can assume that for every k ≥ k * (ω), for every x, y

∈ D k , ( 17
) |Y α (x) -Y α (y)| ≤ 2dC ϕ τ E (x -y)|log τ E (x -y)| (1+η)/α+1/2 .
as soon as

τ E (x -y) ≤ δ k . Let Ω * = +∞ n=1 k≥n x,y∈D k τ E (x-y)≤δ k |X α (x) -X α (y)| ≤ 2dC ϕ τ E (x -y)|log τ E (x -y)| (1+η)/α+1/2
Since X α and Y α have the same finite dimensional margins P(Ω * ) = 1.

Step 4. Let ω ∈ Ω * . By Step 3 there exists k * (ω) ≥ 1 such that X α satisfies (17

) for k ≥ k * (ω), x, y ∈ D k and τ E (x -y) ≤ δ k .
Let us recall that by Lemma 2.2 of [START_REF] Biermé | Operator scaling stable random fields[END_REF], there exists K E ≥ 1 such that for all x, y ∈ R d

τ E (x + y) ≤ K E (τ E (x) + τ E (y)) .

Let us denote F (h) =

2dC ϕ h|log h| (1+η)/α+1/2 and choose k 0 ∈ N such that 2

k 0 δ k 0 +1 > 3K 2 E and F is increasing on (0, δ k 0 ]. Up to change k * (ω), we can assume that k * (ω) ≥ k 0 . Let x, y ∈ D with x = y such that 3K 2 E τ E (x -y) ≤ δ k * (ω)
. Then there exists a unique k ≥ k * (ω) such that δ k+1 < 3K 2 E τ E (x -y) ≤ δ k . Since x, y ∈ D, there exists n ≥ k + 1 such that x, y ∈ D n . Moreover, by Step 1, for j = k, . . . , n -1, we can choose x (j) , y (j) ∈ D j such that

τ E x -x (j) ≤ 2 -j and τ E y -y (j) ≤ 2 -j . By construction τ E x (k) -y (k) ≤ K 2 E τ E (x -y) + 22 -k . Let us point out that since k ≥ k 0 , 2 k δ k+1 ≥ 2 k 0 δ k 0 +1 > 3K 2
E . Therefore, one easily sees that

τ E x (k) -y (k) ≤ 3K 2 E τ E (x -y).
Since 3K 2 E τ E (x -y) ≤ δ k we obtain by Step 3 on the one hand that

X α x (k) -X α y (k) ≤ F τ E x (k) -y (k) .
On the other hand we can write

X α (x) -X α x (k) = n-1 j=k X α x (j+1) -X α x (j)
with τ E x (j+1) -x (j) ≤ 3K 2 E 2 -(j+1) ≤ δ j+1 since j ≥ k 0 . Moreover, note that x (j) ∈ D j ⊂ D j+1 and then by Step 3

X α (x) -X α x (k) ≤ n-1 j=k F τ E x (j+1) -x (j) ≤ CF (δ k+1 ),
where C = +∞ j=0 (j + 1) (1+η)/α+1/2 δ j < +∞. Using similar computations for X α (y) -X α y (k) , we obtain that

|X α (x) -X α (y)| ≤ F τ E x (k) -y (k) + 2CF (δ k+1 ) ≤ (1 + 2C)F 3K 2 E τ E (x -y) .
Then one can find a constant C > 0 such that for 3K 2

E τ E (x -y) ≤ δ k * (ω) (18) |X α (x) -X α (y)| ≤ Cτ E (x -y)|log τ E (x -y)| (1+η)/α+1/2 .
We now give a modification of X α . For x ∈ D, we set

X * α (x)(ω) = X α (x)(ω). For x ∈ K let x (n) ∈ D such that lim n→+∞ x (n) = x.
In view of ( 18), X * α x (n) (ω) n is a Cauchy sequence and then converges. We set

X * α (x)(ω) = lim n→+∞ X * α x (n) (ω).
Remark that this limit does not depend on the choice of x (n) . Moreover, since X α is stochastically continuous, X * α is a modification of X α . By continuity of τ E , we easily see that

X * α (x)(ω) -X * α x (ω) ≤ Cτ E (x -y)|log τ E (x -y)| (1+η)/α+1/2
as soon as 3K 2 E τ E (x -y) < δ k * (ω) , which concludes the proof.

Following the same lines as the proof of Theorem 5.1 we obtain a similar result in the Gaussian case (α = 2) for more general fields. Let us remark that Y α is not defined for α = 2. However, in Step 2 of the proof, let us replace Y α by X a centered Gaussian random field and v 2 ((x, y)

| (T n , ξ n ) n ) by the variance of X(x) -X(y) v 2 ((x, y)) = E (X(x) -X(y)) 2 .

Furthermore let us replace

Step 3 by the assumption that for some β ∈ R and δ > 0 there exists a finite constant C > 0 such that for x, y

∈ K with τ E (x -y) ≤ δ (19) E (X(x) -X(y)) 2 ≤ Cτ E (x -y) 2 |log τ E (x -y)| β .
Then Step 1, Step 2 and Step 4 yields the following proposition.

Proposition 5.3. Let X be a centered Gaussian random field satisfying [START_REF] Mandelbrot | Fractional Brownian motion, fractionnal noises and applications[END_REF]. There exists a modification X * of X on K such that

(20) lim δ↓0 sup x,y∈K x-y ≤δ |X * (x) -X * (y)| τ E (x -y)|log τ E (x -y)| 1/2+β+ε = 0 a.s.
for any ε > 0.

Let us point out that if X 2 is an operator scaling Gaussian random field as defined in [START_REF] Biermé | Operator scaling stable random fields[END_REF], then

E (X 2 (x) -X 2 (y)) 2 = τ E (x -y) 2 E X 2 ( E (x -y)) 2 ,
and X 2 satisfies ( 19) with β = 0 by (5.2) of [START_REF] Biermé | Operator scaling stable random fields[END_REF]. Therefore this result is more precise than one could expect from the Theorem 5.1, replacing α by 2.

Let us also mention that Marianne Clausel gives a different proof of a similar result for some Gaussian operator scaling random fields with stationary increments in [START_REF] Clausel | Gaussian fields with a group self-similarity[END_REF]. Now, as in [START_REF] Biermé | Operator scaling stable random fields[END_REF], we are looking for global and directional Hölder critical exponents of the harmonizable stable random field X α . These exponents have been introduced in [START_REF] Bonami | Anisotropic analysis of some Gaussian models[END_REF] in the Gaussian realm but can be defined for any random field. Following Definition 5.1 of [START_REF] Biermé | Operator scaling stable random fields[END_REF], H ∈ (0, 1) is said to be the Hölder critical exponent of a random field (X(x)) x∈R d if there exists a modification X * of X such that for any s ∈ (0, H), the sample paths of X * satisfy almost surely a uniform Hölder condition of order s on K, that is there exists a finite positive random variable A such that almost surely [START_REF] Mason | Sample path properties of operator-self-similar Gaussian random fields[END_REF] |X * (x) -X * (y)| ≤ A x -y s for all x, y ∈ K while for any s ∈ (H, 1), almost surely ( 21) fails. Note that the Hölder critical exponent, if exists, is well defined since any continuous modification of X and X * are indistinguishable. Moreover, according to Definition 5.3 of [START_REF] Biermé | Operator scaling stable random fields[END_REF] we say that X admits H(u) as directional regularity in direction u ∈ S d-1 , with S d-1 the Euclidean unit sphere, if the process (X(tu)) t∈R admits H(u) as Hölder critical exponent on K ∩ Ru.

For all j = 1, . . . , m we set

K j = K ∩ j k=1 W k . Let us remark that K m = K. Corollary 5.4. Let α ∈ (0, 2]. There exists a modification X * α of X α on K such that for all j = 1, . . . , m lim δ↓0 sup x,y∈K j x-y ≤δ |X * α (x) -X * α (y)| x -y H j |log x -y | H j (p j -1)+β+1/2+ε = 0 a.s.
for any ε > 0, where p j = max Proposition 5.6. The random field X α admits H m as Hölder critical exponent. Moreover, for any j = 1, . . . , m, for any direction u ∈ W j , the field X α admits H j as directional regularity in the direction u.

Proof. For Z a real SαS random variable we let

Z α = (-log (E(exp (iZ)))) 1/α . Then, for any x, y ∈ R d , X * α (x) -X * α (y) α = C α ( E (x -y))τ E (x -y) where for all θ ∈ S E C α (θ) = c α R d e i θ,ξ -1 α ψ(ξ) -α-q dξ 1/α and c α = 1 2π π 0 |cos(t)| α dt.
From Lebesgue's Theorem, the function C α is continuous on the compact set S E with positive values. Let us denote m α = min

θ∈S E C α (θ) > 0.
According to [START_REF] Hirsch | Differential equations, dynamical systems, and linear algebra[END_REF] in Corollary 3.4, for any j = 1, . . . , m and u a direction in W j ,

X * α (tu) -X * α (su) α ≥ m α c 1 |t -s| H j |log |t -s|| -(l j -1)H j .
Therefore, for any s > H j , it implies that X * α (tu)-X * α (su) |t-s| s is almost surely unbounded as |t -s| ↓ 0 so [START_REF] Mason | Sample path properties of operator-self-similar Gaussian random fields[END_REF] fails almost surely on K ∩ Ru. Moreover, Corollary 5.4 implies that (X * α (tu)) t∈R satisfies (21) on K ∩Ru for any s < H j and thus H j is the directional regularity of X α in the direction u. Moreover, one can find a direction u ∈ S d-1 in which almost surely (X * α (tu)) t∈R does not satisfy (21) on K ∩ Ru for any s > H m . Therefore, almost surely (X * α (x)) x∈R d can not satisfy (21) on K for any s > H m . Then, by Corollary 5.5 X α admits H m as Hölder critical exponent.

Remark 5.1. In the diagonalizable case (see Example 2.2), the W j are the eigenspaces associated with the eigenvalues of E. In particular, for θ j an eigenvector associated with the eigenvalue λ j = a j , the critical Hölder exponent in direction θ j is H j = 1/a j . Proposition 5.6 compared to Theorem 5.4 of [START_REF] Biermé | Operator scaling stable random fields[END_REF] shows that the operator scaling stable field, defined through an harmonizable representation share the same sample paths properties as the Gaussian ones. Therefore it is natural to have also the same result of Theorem 5.6 [START_REF] Biermé | Operator scaling stable random fields[END_REF] for the box-and the Hausdorffdimensions of their graphs on a compact set. We also refer to Falconer [START_REF] Falconer | Fractal geometry[END_REF] for the definitions and properties of box-and the Hausdorff-dimension and keep the notations of [START_REF] Biermé | Operator scaling stable random fields[END_REF]. More precisely, we fix a compact set K ⊂ R d and consider G(X * α )(ω) = {(x, X * α (x)(ω)); x ∈ K} the graph of a realization of the field X * α over the compact K. We denote dim H G(X * α ), resp dim B G(X * α ), the Hausdorff-dimension and the box-dimension of G(X * α ), respectively. Proposition 5.7. Almost surely

dim H G(X * α ) = dim B G(X * α ) = d + 1 -H m . Proof.
The proof is very similar to those of Theorem 5.6 [START_REF] Biermé | Operator scaling stable random fields[END_REF]. It also uses same kinds of arguments as in [START_REF] Benassi | Local self-similarity and the Hausdorff dimension[END_REF]. For sake of completeness we recall the main ideas. Corollary 5.5 allows as usual to state the

upper bound dim H G(X * α ) ≤ dim B G(X * α ) ≤ d + 1 -H m , a.
s. where dim B denotes the upper box-dimension. The lower bound will also follows from Frostman criterion (Theorem 4.13 (a) in [START_REF] Falconer | Fractal geometry[END_REF]). One has to prove that the integral I s

I s = K×K E (X * α (x) -X * α (y)) 2 + x -y 2 -s/2 dx dy,
is finite to get that almost surely dim H G(X * α ) ≥ s. In our case, the fundamental lemma of [START_REF] Ayache | A Fourier formulation of the Frostman criterion for random graphs and its applications to wavelet series[END_REF] allows us to write this integral using the characteristic function of the SαS field X * α . Actually, when one remarks that, using Fourier-inversion, (ξ 2 + 1) -s/2 = 1 2π R e iξt f s (t)dt, where f s ∈ L ∞ (R) ∩ L 1 (R), one gets

I s = K×K 1 2π x -y -s R e -|t| α X * α (x)-X * α (y) α α x-y α f s (t)dt dx dy.
By a change of variables, as f s ∈ L ∞ (R), one can find C > 0 such that

I s ≤ C K×K x -y 1-s X * α (x) -X * α (y) -1 α dx dy ≤ Cm -1 α K×K x -y 1-s τ E (x -y) -1 dx dy,
where m α = min

θ∈S E c α R d e i θ,ξ -1 α ψ(ξ) -α-q dξ 1/α
as introduced in the proof of Proposition 5.6.

Since K×K x -y 1-s τ E (x -y) -1 dx dy is proved to be finite as soon as s < d + 1 -H m in [START_REF] Biermé | Operator scaling stable random fields[END_REF],

dim B G(X * α ) ≥ dim H G(X * α ) ≥ d + 1 -H m a.
s. and the proof is complete.

Harmonizable operator scaling stable random fields share many properties with Gaussian operator random fields. In particular, they have locally Hölder sample paths and critical directional Hölder exponent depending on the directions. In the Gaussian case (α = 2), [START_REF] Biermé | Operator scaling stable random fields[END_REF] establishes such properties in the framework of harmonizable and moving average Gaussian operator scaling random field. However, for stable laws, harmonizable and moving average representations do not have the same behavior as we see in the next section.

Moving average representation

Let us recall the definition of moving average operator scaling stable random fields introduced in [START_REF] Biermé | Operator scaling stable random fields[END_REF]. Let 0 < α ≤ 2. We consider M α (dy) an independently scattered SαS random measure on R d with Lebesgue control measure, see [START_REF] Samorodnitsky | Stable non-Gaussian random processes[END_REF] for details on such random measures. Let us recall that, as before, q = trace(E). Let ϕ : R d → [0, ∞) be a continuous E-homogeneous function. We assume moreover that there exists β > 1 such that ϕ is (β, E)-admissible. According to Definition 2.7 of [START_REF] Biermé | Operator scaling stable random fields[END_REF] it means that ϕ(x) = 0 for x = 0 and that for any 0 < A < B there exists a constant C > 0 such that, for A ≤ y ≤ B, [START_REF] Meerschaert | Limit distributions for sums of independent random vectors[END_REF] |ϕ(x + y) -ϕ(y)| ≤ Cτ E (x) β holds for any τ E (x) ≤ 1.

Definition 6.1. Since β > 1, the α-stable random field

(23) Z α (x) = R d ϕ(x -y) 1-q/α -ϕ(-y) 1-q/α M α (dy) , x ∈ R d .
is well defined and called moving average operator scaling stable random field.

From Theorem 3.1 and Corollary 3.2 of [START_REF] Biermé | Operator scaling stable random fields[END_REF], it is stochastically continuous, has stationary increments and satisfies the following operator scaling property

∀c > 0, Z α (c E x); x ∈ R d (f dd) = cZ α (x); x ∈ R d ,
as the harmonizable field X α .

In the Gaussian case (α = 2), the variogramme of Z 2 is similar to the one of the harmonizable field X 2 . Then, [START_REF] Biermé | Operator scaling stable random fields[END_REF] proves that Z 2 and X 2 admit the same critical Hölder sample paths properties. However, when α ∈ (0, 2), let us recall that moving average self-similar α-stable random motions does not have in general continuous sample paths (see [START_REF] Samorodnitsky | Stable non-Gaussian random processes[END_REF]). The next proposition states the same property for Z α .

* = U ∩ Q d is dense in U , for any y ∈ U f * (U * , y) = sup x∈U * ϕ(x -y) 1-q/α -ϕ(-y) 1-q/α = +∞.
Then R d f * (U * , y) α dy = +∞ and the necessary condition for sample boundedness (10.2.14) of Theorem 10.2.3 p.450 of [START_REF] Samorodnitsky | Stable non-Gaussian random processes[END_REF] fails. We conclude the proof by Corollary 10.2.4 of [START_REF] Samorodnitsky | Stable non-Gaussian random processes[END_REF].

Appendix

Proof of lemma 3.2. The lower bound is straightforward. Actually, for any t > 0, t λ is an eigenvalue of the matrix t J and therefore t a = t λ ≤ t J .

Let us prove the upper bound. First, let us assume that J is a Jordan cell matrix of size l. In this case λ = a ∈ R and

t J = t a       1 0 . . . 0 log t 1 0 . . . . . . . . . . . . 0 (log t) l-1 (l-1)! . . . log t 1       .
Let us recall that the norm defined for a matrix A = (a ij ) 1≤i,j,≤d by A ∞ = max

1≤i≤d d j=1 |a ij | is the subordinated norm of the infinite norm x ∞ = max 1≤i≤d |x i | for x ∈ R d . By definition, we can deduce that t J ∞ = t a l-1 j=0 |log t| j j! . Then, t J ≤ √ l t J ∞ ≤ √ lt a |log (t)| l-1 l-1 j=0 1 j!
for any t ∈ (0, e -1 ] ∪ [e, +∞). Therefore, for any t ∈ (0, e -1 ] ∪ [e, +∞), we have

t J ≤ √ l e t a |log t| l-1 .
In the second case, let us assume that J is a block of the form (7) of size 2l associated with the eigenvalue λ = a + ib for b = 0. Then t J = t a R(t)N (t) where

R(t) =      R b (t) 0 . . . 0 0 R b (t) 0 . . . . . . . . . . . . 0 0 . . . 0 R b (t)      with R b (t) = cos(b log t) -sin(b log t) sin(b log t) cos(b log t) , and
N (t) =      I 2 0 . . . 0 N 1 (t) I 2 0 . . . . . . . . . . . . 0 N l-1 (t) . . . N 1 (t) I 2      with N j (t) = |log t| j j! 0 0 |log t| j j! . Hence, t J ≤ t α R(t) N (t) .
Since R(t) is an orthogonal matrix, R(t) = 1. Furthermore, N (t) is a (2l) × (2l) matrix and

N (t) ≤ √ 2l N (t) ∞ = √ 2l l-1 j=0 |log t| j j!
Therefore, we also obtain that t

J ≤ √ 2l et a |log t| l-1
for any t ∈ (0, e -1 ] ∪ [e, +∞).

Proof of Proposition 3.4. Let r ∈ (0, 1). One can find r E ∈ (0, r) such that for any x ≤ r E we have τ

E (x) ≤ e -1 . Let x ∈ j k=j 0 W k with x ≤ r E . Then x = τ E (x) E E (x) and l E (x) ∈ j k=j 0 W k .
We first establish the lower bound of Proposition 3.4. Let us write l E (x) = m k=1 l k (x) where each l k (x) ∈ W k . Let L k be the coordinates of l k (x) in the basis f k-1 i=1 li +1 , . . . , f k i=1 li of W k . Hence, by definition of P ,

P -1 l E (x) =    L 1 . . . L m    and x = τ E (x) E l E (x) = P    τ E (x) J 1 L 1 . . . τ E (x) Jm L m    . Since l E (x) ∈ j k=j 0 W k , L k = 0 for k / ∈ {j 0 , . . . , j}, x ≤ P   j k=j 0 τ E (x) J k L k 2   1/2 ≤ P   j k=j 0 τ E (x) J k 2 L k 2   1/2 By Lemma 3.2, x ≤ √ 2e P   j k=j 0 l k τ E (x) 2a k |log τ E (x)| 2(l k -1) L k 2   1/2
since τ E (x) ≤ 1/e. Hence, since τ E (x) ≤ 1, |log τ E (x)| ≥ 1, a k ≥ a j 0 and l k ≤ p j 0 ,j = max

j 0 ≤i≤j l i ≤ d,
x ≤ √ 2de P τ E (x) a j 0 |log τ E (x)| (pj 0 ,j -1)

j k=j 0 L k 2 1/2
≤ √ 2de P τ E (x) a j 0 |log τ E (x)| (pj 0 ,j -1) P -1 l E (x)

Then, [START_REF] Rosiński | On path properties of certain infinitely divisible processes[END_REF] x ≤ √ 2deM E P P -1 τ E (x) a j 0 |log τ E (x)| (pj 0 ,j -1) where M E is defined by [START_REF] Biermé | Operator scaling stable random fields[END_REF]. Let us take the logarithm of this equation. Choosing r 0 < min(1, r E ) small enough, one can find C > 0 such that [START_REF] Rosiński | On series representations of infinitely divisible random vectors[END_REF] | log τ E (x)| ≤ C| log x | for x < r 0 .

Using in [START_REF] Rosiński | On path properties of certain infinitely divisible processes[END_REF], we obtain that there exists C > 0 such that for x ≤ r 0

x H j 0 |log x | -H j 0 (pj 0 ,j -1) ≤ Cτ E (x).

By continuity of the map,

x → x H j 0 |log x | -H j 0 (pj 0 ,j -1) τ E (x) -1 on x ∈ R d / 0 < x < 1 , up to change C, the previous inequality holds for x ≤ r, which gives the lower bound in [START_REF] Karatzas | Brownian Motion and Stochastic Calculus[END_REF].

Let us now establish the upper bound in [START_REF] Karatzas | Brownian Motion and Stochastic Calculus[END_REF]. We write x = m k=1 x k with each x k ∈ W k . We then denote by X k the coordinates of x k in the basis f k-1 i=1 li +1 , . . . , f k i=1 li of W k . Since x ∈ j k=j 0 W k , X k = 0 for every k / ∈ {j 0 , . . . , j}. Hence, by definition of P , since τ E (x) -1 ≥ e. Hence, since τ E (x) -1 ≥ e > 1, a k ≤ a j and l k ≤ p j 0 ,j , 0 < m E ≤ √ 2de P P -1 τ E (x) -a j |log τ E (x)| p j 0 ,j -1 j k=j 0 X k 2 ≤ √ 2de P P -1 τ E (x) -a j |log τ E (x)| p j 0 ,j -1 P -1 x .

P -1 x =    X 1 . . .
Then, using (25) and P -1 x ≤ P -1 x , there exists a constant C > 0 such that τ E (x) < C x H j |log x | H j (pj 0 ,j -1) for x ≤ r 0 . By continuity of the map

x → τ E (x)
x H j |log x | H j (pj 0 ,j -1) on x ∈ R d / 0 < x < 1 , up to change C, the previous inequality holds for x ≤ r, which gives the upper bound in [START_REF] Karatzas | Brownian Motion and Stochastic Calculus[END_REF] and concludes the proof.

Proof of Lemma 5.2. It is sufficient to consider

I(h) = E m(ξ n ) -2/α min M E h E t ξ n , 2 2 ψ(ξ n ) -2-2q/α
By definition,

I(h) = R d m(ξ) 1-2/α ψ(ξ) -2-2q/α min M E h E t ξ , 2 2 dξ.
Using the formula of integration in polar coordinates with respect to E t , see Proposition 3.1,

I(h) = S E t +∞ 0 m r E t θ 1-2/α ψ r E t θ -2-2q/α
min M E (hr) E t θ , 2 2 r q-1 drσ E t (dθ).

Since ψ is E t -homogeneous, By the change of variable ρ = hr, I(h) is equal to

I(h) = S E t
c 1-2/α η h 2 S E t +∞ 0 ψ(θ) -2-2q/α min M E ρ E t θ , 2 2 ρ -3 log ρ h (1+η)(2/α-1)
drσ E t (dθ).

For any γ ∈ (0, 1), there exists A γ such that for every ρ > 0 and every h ≤ 1 -γ,

I 2 = M 2 E M 2 E t S E t ψ(θ) -2-2q/α σ E t (dθ) 1 0 ρ E t 2 ρ -3 ||log (ρ)| + 1| (1+η)(2/α-1) dρ,
where M E and M E t are defined by [START_REF] Biermé | Operator scaling stable random fields[END_REF]. Since ψ is continuous with positive value on the compact set S E t , S E t ψ(θ) -2-2q/α σ(dθ) < +∞.

Hence I 1 < +∞. It follows from Proposition 3.4 that for any δ ∈ (0, 1), there exists a constant c δ > 0 such that

ρ E t ≤ Cρ a 1 |log |ρ|| l-1
for all ρ ≤ δ . Hence, since a 1 > 1, 1 0 ρ E t 2 ρ -3 ||log (ρ)| + 1| (1+η)(2/α-1) dρ < +∞ and I 2 < +∞, which concludes the proof.

Corollary 5 . 5 .

 55 1≤k≤j l k , β = 1/α if α = 2 and β = 0 if α = 2. Proof. It follows from Theorem 5.1 and Corollary 3.3, since a j ≤ a d for any j = 1 . . . d. Let α ∈ (0, 2]. The random field X * α has locally H-Hölder sample paths on R d for every H ∈ (0, H m ). Now let us give the directional and global Hölder critical exponents of X α .

Proposition 6 . 1 .

 61 Assume α ∈ (0, 2) and d ≥ 2. Then, the random field Z α is almost surely unbounded on every open ball. Proof. Let us remark that ϕ(0) = 0 by continuity and E-homogeneity and q = d j=1 a j > d > α, as soon as d ≥ 2. Then, for any U open set, since U

l

  l E (x) = τ E (x) -E x = P    τ E (x) -J 1 X 1 . . . τ E (x) -Jm X m   . Then, l E (x) ≤ P   j k=j 0 τ E (x) -J k X k k τ E (x) -2a k |log τ E (x)| 2(l k -1)

2 r 2 r

 22 ) -2-2q/α min M E (hr) E t θ , 2 -2+q-1-2q/α drσ E t (dθ) ) -2-2q/α min M E (hr) E t θ , 2 -3 |log(r)| (1+η)(2/α-1) drσ E t (dθ).

1 ρ

 1 (ρ) -log (h)| ≤ A γ ||log(ρ)| + 1||log (h)|. Since 2/α > 1, I(h) ≤ A 2/α-1 γ c 1-2/α η h 2 |log (h)| (1+η)(2/α-1) (I 1 + I 2 )withI 1 = 4 S E t ψ(θ) -2-2q/α σ E t (dθ) +∞ -3 ||log (ρ)| + 1| (1+η)(2/α-1) dρ and

  which is a necessary and sufficient condition for X α to be defined.

	Now, let us give some examples of such random fields.
	Example 2.1. Let I d be the identity matrix of size d × d, E = I d /H (with 0 < H < 1) and ψ(x) = x H with • the Euclidean norm. Then the random field defined by (

  n ≥ 1, are i.i.d. isotropic complex random variables. By Lemma 4.1 in [15], Y h converges almost surely and ∀z ∈ C, E exp i zY h = exp (-σ α |z| α ) with

Acknowledgments

We would like to warmly thank Yimin Xiao for the careful reading of our manuscript and his remarks leading to the improvement of our results.