Paul Balanc
email: paul.balanca@ecp.fr

Fine regularity of Lévy processes and linear (multi)fractional stable motion

Keywords: AMS 2000 subject classifications: 60G07, 60G17, 60G22, 60G44 2-microlocal analysis, Hölder regularity, multifractal spectrum, oscillating singularities, Lévy processes, linear fractional stable motion

In this work, we investigate the fine regularity of Lévy processes using the 2-microlocal formalism. This framework allows us to refine the multifractal spectrum determined by Jaffard and, in addition, study the oscillating singularities of Lévy processes. The fractal structure of the latter is proved to be more complex than the classic multifractal spectrum and is determined in the case of alpha-stable processes. As a consequence of these fine results and the properties of the 2-microlocal frontier, we are also able to completely characterise the multifractal nature of the linear fractional stable motion (extension of fractional Brownian motion to α-stable measures) in the case of continuous and unbounded sample paths as well. The regularity of its multifractional extension is also presented, indirectly providing an example of a stochastic process with a non-homogeneous and random multifractal spectrum.

Introduction

The study of sample path continuity and Hölder regularity of stochastic processes is a very active field of research in probability theory. The existing literature provides a variety of uniform results on local regularity, especially on the modulus of continuity, for rather general classes of random fields (see e.g. Marcus and Rosen [START_REF] Marcus | Markov processes, Gaussian processes, and local times[END_REF], Adler and Taylor [START_REF] Adler | Random fields and geometry[END_REF] on Gaussian processes and Xiao [START_REF] Xiao | Uniform modulus of continuity of random fields[END_REF] for more recent developments).

On the other hand, the structure of pointwise regularity is generally more complex as the latter often tends to behave erratically as time passes. This type of sample path behaviour was first put into light on Brownian motion by Orey and Taylor [START_REF] Orey | How often on a Brownian path does the law of iterated logarithm fail?[END_REF] and Perkins [START_REF] Perkins | On the Hausdorff dimension of the Brownian slow points[END_REF]. They respectively studied fast and slow points which characterize logarithmic variations of the pointwise modulus of continuity, and proved that the sets of times with a given pointwise regularity have a distinct fractal geometry. Khoshnevisan and Shi [START_REF] Khoshnevisan | Fast sets and points for fractional Brownian motion[END_REF] have recently extended this study of fast points to fractional Brownian motion.

Lévy processes with a jump compound also present an interesting pointwise behaviour. Indeed, Jaffard [START_REF] Jaffard | The multifractal nature of Lévy processes[END_REF] has proved that despite the random variations of the pointwise exponent, the level sets of the latter show a specific fractal structure. This seminal work has been enhanced and extended by Durand [START_REF] Durand | Singularity sets of Lévy processes[END_REF], Durand and Jaffard [START_REF] Durand | Multifractal analysis of Lévy fields[END_REF] and Barral et al. [START_REF] Barral | A pure jump Markov process with a random singularity spectrum[END_REF]. Particularly, the latter have proved that Markov processes have a range of admissible pointwise behaviours wider and richer than Lévy processes. In the aforementioned works, multifractal analysis happens to be the key concept to study and characterise the local fluctuations of the pointwise regularity. In order to be more specific, we recall a few definitions.

Definition 1 (Pointwise exponent). A function f : R → R d belongs to C α t , where t ∈ R and α > 0, if there exist C > 0, ρ > 0 and a polynomial P t of degree less than α such that ∀u ∈ B(t, ρ); f (u) -P t (u) ≤ C|t -u| α .

The pointwise Hölder exponent of f at t is then defined by α f,t = sup{α ≥ 0 : f ∈ C α t }, where by convention sup{∅} = 0.

Multifractal analysis is interested in the fractal geometry of the level sets of the pointwise exponent, which are also called the iso-Hölder sets of f :

E h = t ∈ R : α f,t = h
for every h ∈ R + ∪ {+∞}.

(1.1)

The geometry of the collection (E h ) h∈R+ is then studied through its Hausdorff dimension, defining for that purpose the local spectrum of singularities d f (h, V ) of f :

d f (h, V ) = dim H (E h ∩ V ) for every h ∈ R + ∪ {+∞} and V ∈ O, (1.2)
where O designates the collection of nonempty open sets of R and dim H is the Hausdorff dimension, with by convention dim H (∅) = -∞ (we refer to [START_REF] Falconer | Fractal geometry[END_REF] for the complete definition of the latter).

Even though (E h ) h∈R+ are random sets, stochastic processes such as Lévy processes [START_REF] Jaffard | The multifractal nature of Lévy processes[END_REF], Lévy processes in multifractal time [START_REF] Barral | The singularity spectrum of Lévy processes in multifractal time[END_REF] and fractional Brownian motion have a deterministic multifractal spectrum. Furthermore, these random fields are also said to be homogeneous since the quantity d X (h, V ) is independent of the open set V for any h ∈ R + . In addition, when the pointwise exponent is constant along sample paths, the spectrum is described as degenerate, i.e. its support is reduced to a single point (e.g. the Hurst exponent H in the case of f.B.m.). Nevertheless, note that Barral et al. [START_REF] Barral | A pure jump Markov process with a random singularity spectrum[END_REF] and Durand [START_REF] Durand | Random wavelet series based on a tree-indexed Markov chain[END_REF] have provided examples of respectively Markov jump processes and wavelet random series with a non-homogeneous and random spectrum of singularities.

As outlined in Equations (1.1) and (1.2), multifractal analysis usually focuses on the structure of pointwise regularity. Unfortunately, as presented by Meyer [START_REF] Meyer | Wavelets, vibrations and scalings[END_REF], the pointwise Hölder exponent suffers of a couple of drawbacks: it lacks of stability under the action of pseudo-differential operators and it is not always characterised by the wavelets coefficients. In addition, several simple deterministic examples such as the Chirp function t → |t| α sin |t| -β show that it does not fully capture the local geometry and oscillations of a function.

Several approaches, such as the oscillating, chirp and weak scaling exponents introduced by Arneodo et al. [START_REF] Arneodo | Singularity spectrum of multifractal functions involving oscillating singularities[END_REF] and Meyer [START_REF] Meyer | Wavelets, vibrations and scalings[END_REF], have emerged in the literature to address the limits of the pointwise exponent and supplement the latter by characterising other aspects of the local regularity. Interestingly, the aforementioned concepts are embraced by a single framework called 2-microlocal analysis. It has first been introduced by Bony [START_REF] Bony | Second microlocalization and propagation of singularities for semilinear hyperbolic equations[END_REF] in the deterministic frame to study singularities of generalised solutions of PDEs. Several authors have then investigated in [START_REF] Jaffard | Pointwise smoothness, two-microlocalization and wavelet coefficients[END_REF][START_REF] Jaffard | Wavelet methods for pointwise regularity and local oscillations of functions[END_REF][START_REF] Meyer | Wavelets, vibrations and scalings[END_REF][START_REF] Véhel | The 2-microlocal formalism[END_REF] this framework more deeply, determining in particular the close connection between the 2-microlocal formalism and the previous scaling exponents. More recently, Herbin and Lévy Véhel [START_REF] Herbin | Stochastic 2-microlocal analysis[END_REF] have developed a stochastic approach of this framework to investigate the fine regularity of stochastic processes such as Gaussian processes, martingales and stochastic integrals.

Similarly to the pointwise Hölder exponent, the introduction of this formalism starts with the definition of appropriate functional spaces, named 2-microlocal spaces. We begin with a simpler, but narrower, definition to give an intuition of these concepts. Definition 2. Suppose t ∈ R, s ∈ R and σ ∈ (0, 1) such that σ -s / ∈ N. A function f : R → R d belongs to the 2-microlocal space C σ,s t if there exist C > 0, ρ > 0 and a polynomial P t such that for all u, v ∈ B(t, ρ):

f (u) -P t (u) -f (v) + P t (v) ≤ C|u -v| σ |u -t| + |v -t| -s . (1.3)
In addition, P t is unique if we suppose its degree is smaller than σ -s . In this case, it corresponds to the Taylor polynomial of order σ -s of f at t.

The 2-microlocal spaces are therefore parametrised by a pair (s , σ) of real numbers and we clearly observe on Equation (1.3) that they extend the underlying ideas of the classic Hölder spaces. To define these elements for any σ ∈ R \ Z, we need to slightly complexify the form of the increments considered. Definition 3. Suppose t ∈ R and b < t is fixed. In addition, consider s ∈ R, σ ∈ R \ Z and k ∈ Z such that σ -s / ∈ N and σ + k ∈ (0, 1). A function f : R → R d belongs to the 2-microlocal space C σ,s t if there exist C > 0, ρ > 0 and a polynomial P t,k such that for all u, v ∈ B(t, ρ): I k b+ f (u) -P t,k (u) -I k b+ f (v) + P t,k (v) ≤ C|u -v| σ+k |u -t| + |v -t| -s , (1.4) where I k b+ f designates the derivative of order -k when k ≤ 0 and the iterated integral of order k when k > 0, i.e. I k b+ f (u) := 1/Γ(k -1) u b (u -s) k-1 f (s) ds. The time-domain characterisation (1.3)-(1.4) of 2-microlocal spaces has first been obtained by Kolwankar and Lévy Véhel [START_REF] Kolwankar | A time domain characterization of the fine local regularity of functions[END_REF] in the case σ ∈ (0, 1) and then extended by Seuret and Lévy Véhel [START_REF] Seuret | A time domain characterization of 2-microlocal spaces[END_REF] and Echelard [START_REF] Echelard | Analyse 2-microlocale et application au débruitage[END_REF] to σ ∈ R \ Z. Note that the previous characterisation does not depend on the value of the constant b, since a modification of the latter simply induces an adjustment of the polynomial P t .

Even though, we restrict ourselves in Definitions 2-3 to usual functions, 2-microlocal spaces were originally introduced by Bony [START_REF] Bony | Second microlocalization and propagation of singularities for semilinear hyperbolic equations[END_REF] for tempered distributions S (R). The first definition given by Bony [START_REF] Bony | Second microlocalization and propagation of singularities for semilinear hyperbolic equations[END_REF] relies on the Littlewood-Paley decomposition of distributions, and thereby corresponds to a description in the Fourier space. Another characterisation based on wavelet coefficients has also been presented by Jaffard [START_REF] Jaffard | Pointwise smoothness, two-microlocalization and wavelet coefficients[END_REF]. In addition, note that the previous characterisation is in fact equivalent the localised 2-microlocal spaces which are also defined for distributions in D (R) (we refer to [START_REF] Meyer | Wavelets, vibrations and scalings[END_REF] for a more precise distinction between global and local definitions of the 2-microlocal spaces).

One major property of the 2-microlocal spaces is their stability under the action of pseudodifferential operators. In particular, as proved by Jaffard and Meyer [START_REF] Jaffard | Wavelet methods for pointwise regularity and local oscillations of functions[END_REF]Th 1.1], they satisfy

∀α > 0; f ∈ C σ,s t ⇐⇒ I α + f ∈ C σ+α,s t , ( 1.5)
where the fractional integral of f of order α ≥ 0 is defined by:

I α + f (u) := 1/Γ(α) R (u - s) α-1 + f (s) ds.
Note that the latter definition of the operator I α + coincides with the fractional integral presented in [START_REF] Jaffard | Wavelet methods for pointwise regularity and local oscillations of functions[END_REF] for tempered distributions (we refer to the book of Samko et al. [START_REF] Samko | Fractional integrals and derivatives[END_REF] for an extensive study of the subject).

Similarly to the pointwise Hölder exponent, the introduction of 2-microlocal spaces leads naturally to the definition of a regularity tool named the 2-microlocal frontier:

∀s ∈ R; σ f,t (s ) = sup σ ∈ R : f ∈ C σ,s t .
Due to several inclusion properties of the 2-microlocal spaces, the map s → σ f,t (s ) is well-defined and satisfies:

• σ f,t (•) is a concave non-decreasing function; • σ f,t (•)
has left and right derivatives between 0 and 1.

Furthermore, as a consequence of Equation (1.5), σ f,t (•) is stable under the action of pseudodifferential operators. As a function, the 2-microlocal frontier σ f,t (•) offers a more complete and richer description of the local regularity and cover in particular the usual Hölder exponents:

α f,t = σ f,t (0) and α f,t = -inf{s : σ f,t (s ) ≥ 0},
where the last equality has been proved by Meyer [START_REF] Meyer | Wavelets, vibrations and scalings[END_REF] under the assumption ω(h) = O (1/|log(h)|) on the modulus of continuity of f . Several other scaling exponents previously outlined can also be retrieved from the frontier: the chirp and weak scaling exponents introduced by Meyer [START_REF] Meyer | Wavelets, vibrations and scalings[END_REF] are given by:

β c f,t = dσ f,t ds s →-∞ -1 -1 and β w f,t = lim s →-∞ σ f,t (s ) -s ;
These two elements characterise the asymptotic regularity of a function after a large number of integrations and the latter was been specifically introduced to supplement the pointwise exponent in multifractal analysis. The oscillating exponent defined by Arneodo et al. [START_REF] Arneodo | Singularity spectrum of multifractal functions involving oscillating singularities[END_REF] can also be retrieved from the 2-microlocal frontier:

β o f,t = dσ f,t ds s =-α f,t - -1 -1.
The latter aims to capture the oscillating behaviour by studying the regularity after infinitesimal integrations. Note that the original definition of these exponents are based on Hölder spaces (see [START_REF] Seuret | The local Hölder function of a continuous function[END_REF] for an extensive review).

In the stochastic framework, Brownian motion provides a simple example of 2-microlocal frontier: with probability one and for all t ∈ R

∀s ∈ R; σ B,t (s ) = 1 2 + s ∧ 1 2 . (1.6)
Using the common terminology of Arneodo et al. [START_REF] Arneodo | Oscillating singularities on Cantor sets: a grand-canonical multifractal formalism[END_REF] and Meyer [START_REF] Meyer | Wavelets, vibrations and scalings[END_REF], Brownian motion is said to have cusp singularities as β w B,t = α B,t and β o B,t = 0,. On the other hand, oscillating singularities appear when the slope of the frontier is strictly smaller than 1 at s = -α f,t , or equivalently, when β w B,t > α B,t . This oscillating behaviour is well-illustrated by the chirp function whose frontier and scaling exponents at 0 respectively are equal to σ f,0 (s ) = (α + s

)(1 + β), α f,0 = 0, β c f,0 = β o f,0 = β and β w f,0 = ∞.
In this paper, we combine the 2-microlocal formalism with the classic use of multifractal analysis to obtain a finer and richer description of the regularity of Lévy processes. Following the path of [START_REF] Jaffard | The multifractal nature of Lévy processes[END_REF][START_REF] Durand | Singularity sets of Lévy processes[END_REF][START_REF] Durand | Multifractal analysis of Lévy fields[END_REF], we extend the multifractal description (Section 2) to the aforementioned scaling exponents and the 2-microlocal frontier. We present in particular how this formalism allows to capture and describe the oscillating singularities of Lévy processes. The fractal structure of the latter is determined for a few classes of Lévy processes which include alpha-stable processes.

This finer analysis of the sample path properties of Lévy processes happens to be very useful for the study of another class of processes named linear fractional stable motion (LFSM). The LFSM is a common α-stable self-similar process with stationary increments which can be seen as the extension of the fractional Brownian motion to the non-Gaussian frame. In Section 3, we completely characterize the multifractal nature of the LFSM, unifying the geometrical description of the sample paths independently of their boundedness. In addition, we also extend this analysis to the multifractional generalisation of the LFSM.

Statement of the main results

As it is well known, an R d -valued Lévy process (X t ) t∈R+ has stationary and independent increments. Furthermore, its law is determined by the Lévy-Khintchine formula (see e.g. [START_REF] Sato | Lévy processes and infinitely divisible distributions[END_REF]): for all t ∈ R + and λ ∈ R d , E[e i λ,Xt ] = e tψ (λ) where ψ is given by

∀λ ∈ R d ; ψ(λ) = i a, λ - 1 2 λ, Qλ + R d e i λ,x -1 -i λ, x 1 { x ≤1} π(dx).
In the previous expression, Q is a non-negative symmetric matrix and π is the Lévy measure, i.e. a positive Radon measure on

R d \ {0} such that R d (1 ∧ x 2 ) π(dx) < ∞.
Throughout this paper, it will always be assumed that π(R d ) = +∞ since otherwise, the Lévy process corresponds to the sum of a simple compound Poisson process with drift and a Brownian motion whose regularity is well-known. Sample path properties of Lévy processes are known to depend on the growth of the Lévy measure near the origin. More precisely, Blumenthal and Getoor [START_REF] Blumenthal | Sample functions of stochastic processes with stationary independent increments[END_REF] have defined the following exponents β and β ,

β = inf δ ≥ 0 : R d 1 ∧ x δ π(dx) < ∞ and β = β if Q = 0; 2 if Q = 0. (1.7)
Owing to π's definition, β, β ∈ [0, 2]. Pruitt [START_REF] Pruitt | The growth of random walks and Lévy processes[END_REF] proved that α X,0 a.s.

= 1/β when Q = 0. Note that several other exponents have been introduced in the literature to study the sample path properties of Lévy processes (see e.g. [START_REF] Khoshnevisan | Level sets of additive Lévy processes[END_REF][START_REF] Khoshnevisan | Hausdorff dimension of the contours of symmetric additive Lévy processes[END_REF] for some recent developments).

Jaffard [START_REF] Jaffard | The multifractal nature of Lévy processes[END_REF] has studied the spectrum of singularities of Lévy processes under the following assumption on the measure π,

j∈N 2 -j C j log(1 + C j ) < ∞, where C j = 2 -j-1 < x ≤2 -j π(dx).
(1.8)

Under the Hypothesis (1.8), Theorem 1 in [START_REF] Jaffard | The multifractal nature of Lévy processes[END_REF] states that the multifractal spectrum of a Lévy process X is almost surely equal to

∀V ∈ O; d X (h, V ) =      βh if h ∈ [0, 1/β ); 1 if h = 1/β ; -∞ if h ∈ (1/β , +∞].
(1.9)

Durand [START_REF] Durand | Singularity sets of Lévy processes[END_REF] has extended this result to Hausdorff g-measures, where g is a gauge function, and Durand and Jaffard [START_REF] Durand | Multifractal analysis of Lévy fields[END_REF] have generalized the study to multivariate Lévy fields.

In this work, we first establish in Proposition 2 a new proof of the multifractal spectrum (1.9) which does not require Assumption (1.8). Results obtained by Durand [START_REF] Durand | Singularity sets of Lévy processes[END_REF] on Hausdorff g-measure are also indirectly extended using this method.

In order to refine and extend the spectrum of singularities (1.9) using the 2-microlocal formalism, we are interested the fractal geometry of the collections of sets ( E h ) h∈R+ and ( E h ) h∈R+ respectively defined by

E h = t ∈ E h : ∀s ∈ R; σ X,t (s ) = (h + s ) ∧ 0 and E h = E h \ E h .
The introduction of these two collections corresponds to the natural distinction presented in the literature [START_REF] Arneodo | Oscillating singularities on Cantor sets: a grand-canonical multifractal formalism[END_REF][START_REF] Arneodo | Singularity spectrum of multifractal functions involving oscillating singularities[END_REF][START_REF] Meyer | Wavelets, vibrations and scalings[END_REF] between two types of singularities: the family ( E h ) h∈R+ gathers the cusp singularities of Lévy processes, i.e. times at which the slope of the 2-microlocal frontier is equal to 1, whereas the collection ( E h ) h∈R+ regroups the oscillating singularities of the process, i.e. when β w X,t > α X,t and β o X,t > 0. In our first important result, we provide a general description of the fractal geometry of these singularities. Theorem 1. Suppose X is a Lévy process such that β > 0. Then, with probability one, the cusp singularities ( E h ) h∈R+ of X satisfy

∀V ∈ O; dim H ( E h ∩ V ) =      βh if h ∈ [0, 1/β ); 1 if h = 1/β ; -∞ if h ∈ (1/β , +∞].
(1.10) Furthermore, the oscillating singularities ( E h ) h∈R+ of X are such that

∀V ∈ O; dim H ( E h ∩ V ) ≤ 2βh -1 if h ∈ (1/2β, 1/β ); -∞ if h ∈ [0, 1/2β] ∪ [1/β , +∞], (1.11)
where the 2-microlocal frontier at t ∈ E h verifies σ X,t (s )

≤ h+s 2βh ∧ 1 β + s ∧ 0 for all s ∈ R. Remark 1. Theorem 1 induces that dim H ( E h ) < dim H ( E h ) for every h ∈ [0, 1/β ].
Therefore, in terms of Hausdorff dimension, chirp oscillations that might appear on a Lévy process are always singular compared to the common cusp behaviour.

We also note that even though sample paths of Lévy processes do not satisfy the condition ω(h) = O(1/|log(h)|) outlined in the introduction, Theorem 1 nevertheless ensures that the pointwise Hölder exponent can be retrieved from the 2-microlocal frontier at any t ∈ R + using the formula α X,t = -inf{s : σ X,t (s ) ≥ 0}. As a consequence, the pointwise regularity of Lévy processes can also be characterised by its wavelet coefficients.

The determination of the 2-microlocal regularity of Lévy processes allows to deduce the behaviour of several scaling exponents. In particular, we are interested in the multifractal spectrum of the weak scaling exponent, whose level sets are defined as:

E w h = t ∈ R : β w X,t = h for every h ∈ R + ∪ {+∞}.
Corollary 1. Suppose X is a Lévy process such that β > 0. Then, with probability one

∀V ∈ O; dim H (E w h ∩ V ) =      βh if h ∈ [0, 1/β ); 1 if h = 1/β ; -∞ otherwise.
(1.12) Furthermore, the oscillating exponent is such that β o X,t ≤ max 0, 2βh -1 and

dim H t ∈ E h : β o X,t > 0 ≤ 2βh -1 if h ∈ (1/2β, 1/β ); -∞ otherwise. (1.13)
Finally, the chirp scaling exponent satisfies β c X,t = 0 for all t ∈ R. According to Corollary 1, the multifractal spectrum associated to the weak scaling exponent is the same as the classic one (1.9) despite the oscillating singularities which might exist. We also note that the latter do not influence the chirp scaling exponent, showing that chirp oscillations tend to disappear after multiple integrations.

Following the ideas presented by Meyer [START_REF] Meyer | Wavelets, vibrations and scalings[END_REF], it is also natural to investigate geometrical properties of the sets (E σ,s ) σ,s ∈R defined by

E σ,s = t ∈ R + : ∀u > s ; X• ∈ C σ,u t and ∀u < s ; X• / ∈ C σ,u t .
This collection of sets can be seen as the level sets of the 2-microlocal frontier for a fixed σ.

Corollary 2. Suppose X is a Lévy process such that β > 0. Then, with probability one and for all σ ∈ R -,

∀V ∈ O; dim H (E σ,s ∩ V ) =      βs if s ∈ [0, 1/β ); 1 if s = 1/β ; -∞ otherwise. (1.14)
where s denotes the common 2-microlocal parameter s = σ -s . Furthermore, for all s ∈ R, E 0,s = E -s and E σ,s is empty if σ > 0.

As for the weak scaling exponent, we obtain in Corollary 2 a multifractal spectrum which takes the same form as Equation (1.9) (note that the latter corresponds to the case σ = 0). In addition, the oscillating singularities are also not captured by these scaling exponents and the spectrum associated.

Theorem 1 provides an upper bound of the Hausdorff dimension of the oscillating singularities of a general Lévy process. In Section 2.3, we obtain the exact estimates for some specific classes of Lévy processes, proving in particular that the Blumenthal-Getoor exponent does not entirely characterise the structure of these chirp oscillations. Proposition 1. Suppose π is a Lévy measure on R such that π(R ± ) = 0 and X is a Lévy process with generating triplet (a, Q, π). Then, with probability one,

E h = ∅ for all h ∈ R + , i.e. ∀t ∈ R + , ∀s ∈ R; σ X,t (s ) = α X,t + s ∧ 0.
Note in particular that subordinators do not have oscillating singularities, which is quite understandable because of their monotonicity.

Nevertheless, these singularities might appear as well for rather natural classes of processes such as alpha-stable Lévy processes. Theorem 2. Suppose X is a Lévy process parametrised by (0, 0, π), where the Lévy measure π has the following form

π(dx) = a 1 |x| -1-α1 1 R+ dx + a 2 |x| -1-α2 1 R-dx, (1.15)
and a 1 , a 2 > 0 and α 1 , α 2 ∈ (0, 2).

Then, the Blumenthal-Getoor exponent of π is equal to β = max(α 1 , α 2 ) and with probability one, the oscillating singularities of X satisfy

∀V ∈ O; dim H ( E h ∩ V ) = (α 1 + α 2 )h -1 if h ∈ 1/(α 1 + α 2 ), 1/β ; -∞ otherwise. (1.16)
One of the interesting aspects of the previous result is to show that the Hausdorff dimension of the oscillating singularities of Lévy processes is not necessarily governed by the Blumenthal-Getoor exponent, but also takes into account the symmetrical aspect of the Lévy measure. Furthermore, Theorem 2 proves that the upper bound obtained in Theorem 1 is optimal, since in the case of an alpha-stable process parametrised by (α, β α ), with probability one

∀V ∈ O; dim H ( E h ∩ V ) = 2αh -1 if h ∈ (1/2α, 1/α) and β α ∈ (-1, 1); -∞ otherwise.
(1.17)

Note that owing to Proposition 1, an alpha-stable process whose skewness parameter β α is equal to 1 or -1 does not have oscillating singularities.

The fine 2-microlocal structure presented Theorems 1 and 2 happens to be interesting outside the scope of Lévy processes. More precisely, it allows to characterized the multifractal nature of the linear fractional stable motion (LFSM). The latter is a fractional extension of alpha-stable Lévy processes and is usually defined by the following stochastic integral (see e.g. [START_REF] Samorodnitsky | Stable non-Gaussian random processes[END_REF])

X t = R (t -u) H-1/α + -(-u) H-1/α + M α (du), (1.18)
where M α is an alpha-stable random measure parametrised by α ∈ (0, 2) and

β α ∈ [-1, 1],
and H ∈ (0, 1) is the Hurst exponent. Several regularity properties have been determined in the literature. In particular, sample paths are known to be nowhere bounded [START_REF] Maejima | A self-similar process with nowhere bounded sample paths[END_REF] if H < 1/α and Hölder continuous when H > 1/α. In this latter case, Takashima [START_REF] Takashima | Sample path properties of ergodic self-similar processes[END_REF], Kôno and Maejima [START_REF] Kôno | Hölder continuity of sample paths of some self-similar stable processes[END_REF] proved that the pointwise and local Hölder exponents satisfy almost surely H -1/α ≤ α X,t ≤ H and α X,t = H -1/α. Throughout this paper, we will assume that α ∈ [1, 2), which is required to obtain Hölder continuous sample paths (H > 1/α).

Using an alternative representation of LFSM presented in Proposition 3, we enhance the aforementioned regularity results and obtain a precise description of the multifractal structure of the LFSM. Theorem 3. Suppose X is a linear fractional stable motion parametrized by α ∈ [1, 2), β α ∈ [-1, 1] and H ∈ (0, 1). Then, with probability one and for all σ ≤ H -

1 α ∀V ∈ O; dim H (E σ,s ∩ V ) = α(s -H) + 1 if s ∈ H -1 α , H ; -∞ otherwise. (1.19)
where s = σ -s . When σ > H -1 α , E σ,s is empty for all s ∈ R. In addition, the weak scaling exponent satisfies with probability one

∀V ∈ O; dim H (E w h ∩ V ) = α(h -H) + 1 if h ∈ H -1 α , H ; -∞ otherwise. (1.20)
Finally, the chirp scaling exponent β c X,t is equal to 0 for all t ∈ R. Therefore, we observe that the multifractal structure presented in Theorem 3 corresponds to the spectrum of alpha-stable processes translated by a factor H -1 α . Interestingly, we also note that on the contrary to usual Hölder exponents, the weak scaling exponent and the 2-microlocal formalism allow to describe the multifractal nature of the LFSM independently of the continuity of its sample paths, unifying the continuous (H > 1 α ) and unbounded (H < 1 α ) cases (see Figure 1). In the latter case, the 2-microlocal domain is located strictly below the s -axis, implying that sample paths are nowhere bounded. Nevertheless, the proof of Theorem 3 ensures in this case the existence of a modification of the LFSM such that the sample paths are distributions in D (R) whose 2-microlocal regularity can be studied as well.

In addition, the classic multifractal spectrum can be explicated when sample paths are Hölder continuous.

Corollary 3. Suppose X is a linear fractional stable motion parametrized by

α ∈ [1, 2), β α ∈ [-1, 1] and H ∈ (0, 1), with H > 1/α.
Then, with probability one, the multifractal spectrum of X is given by

∀V ∈ O; d X (h, V ) = α(h -H) + 1 if h ∈ H -1 α , H ; -∞ otherwise. (1.21)
An equivalent multifractal structure is presented in Proposition 4 for a similar class of processes called fractional Lévy processes (see [START_REF] Benassi | On roughness indices for fractional fields[END_REF][START_REF] Marquardt | Fractional Lévy processes with an application to long memory moving average processes[END_REF][START_REF] Cohen | A general framework for simulation of fractional fields[END_REF]).

The LFSM admits a natural multifractional extension which has been introduced and studied in [START_REF] Stoev | Stochastic properties of the linear multifractional stable motion[END_REF][START_REF] Stoev | Path properties of the linear multifractional stable motion[END_REF][START_REF] Dozzi | Real harmonizable multifractional stable process and its local properties[END_REF]. The definition of the linear multifractional stable motion (LMSM) is

s ′ σ 0 -1 -1 2 dim H (E σ,s ′ ) = α(s -H) + 1 1 2 1 2 -H + 1 α H -1 α -H s = σ -s ′ (a) Continuous sample paths (H > 1 α ) s ′ σ 0 -1 -1 2 dim H (E σ,s ′ ) = α(s -H) + 1 1 2 1 2 H -1 α -1 2 -1 α s = σ -s ′ (b) Unbounded sample paths (H < 1 α )
Figure

t ∈ R + , H(t) -1/α ≤ α X,t ≤ H(t)
and α X,t = H(t) -1/α almost surely. Ayache and Hamonier [START_REF] Ayache | Linear Multifractional Stable Motion: fine path properties[END_REF] have also investigated the existence of an optimal local modulus of continuity. Theorem 3 can be generalized to the LMSM in the continuous case. More precisely, we assume that the Hurst function satisfies the following assumption,

H : R → 1 α , 1 is δ-Hölderian, with δ > sup u∈R H(u). (H 0 )
Since the LMSM is clearly a non-homogeneous process, it is natural to focus on the study of the spectrum of singularities localized at t ∈ R + , i.e.

∀t ∈ R + d X (h, t) = lim ρ→0 d X (h, B(t, ρ)) = lim ρ→0 dim H (E h ∩ B(t, ρ)).
Theorem 4. Suppose X is a linear multifractional stable motion parametrized by α ∈ (1, 2),

β α ∈ [-1, 1] and an (H 0 )-Hurst function H.
Then, with probability one, for all t ∈ R and for all σ < H(t)

-1 α , lim ρ→0 dim H E σ,s ∩ B(t, ρ) = α s -H(t) + 1 if s ∈ H(t) -1 α , H(t) ; -∞ otherwise. (1.22)
where s = σ -s . Furthermore, the set E σ,s ∩ B(t, ρ) is empty for any σ > H(t) -1 α and ρ > 0 sufficiently small. Theorem 4 extends the results presented in [START_REF] Stoev | Stochastic properties of the linear multifractional stable motion[END_REF][START_REF] Stoev | Path properties of the linear multifractional stable motion[END_REF], and also ensures that the localized multifractal spectrum is equal to

∀t ∈ R + ; d X (h, t) = α h -H(t) + 1 if h ∈ H(t) -1 α , H(t) ; -∞ otherwise. (1.23)
Moreover, we observe that Proposition 3 and Theorem 4 still hold when the Hurst function H(•) is a continuous random process. Thereby, similarly to the works of Barral et al. [START_REF] Barral | A pure jump Markov process with a random singularity spectrum[END_REF] and Durand [START_REF] Durand | Random wavelet series based on a tree-indexed Markov chain[END_REF], it provides a class stochastic processes whose spectrum of singularities, given by Equation (1.23), is non-homogeneous and random.

Lévy processes

In this section, X will designate a Lévy process parametrized by the generating triplet (a, Q, π). The Lévy-Itō decomposition states that it can represented as the sum of three independent processes B, N and Y , where B is a d-dimensional Brownian motion, N is a compound Poisson process with drift and Y is a Lévy process characterized by 0, 0, π(dx

)1 { x ≤1} .
Without any loss of generality, we restrict the study to the time interval [0, 1]. Furthermore, as outlined in the introduction, we also assume that the Blumenthal-Getoor β is strictly positive. As noted by Jaffard [START_REF] Jaffard | The multifractal nature of Lévy processes[END_REF], the component N does not affect the regularity of X since its trajectories are piecewise linear with a finite number of jumps. Sample path properties of Brownian motion are well-known and therefore, we first focus in this section on the study of the jump process Y .

It is well-known that the process Y can be represented as a compensated integral with respect to a Poisson measure J(dt, dx) of intensity L 1 ⊗ π:

Y t = lim ε→0 [0,t]×D(ε,1)
x J(ds, dx) -t

D(ε,1)
x π(dx) , (

where for all 0 ≤ a < b, D(a, b) := {x ∈ R d : a < x ≤ b}. Moreover, as presented in [START_REF] Sato | Lévy processes and infinitely divisible distributions[END_REF]Th. 19.2], the convergence is almost surely uniform on any bounded interval. In the rest of this section, for any m ∈ R + , Y m will denote the Lévy process:

Y m t = lim ε→0 [0,t]×D(ε,2 -m )
x J(ds, dx) -t

D(ε,2 -m )
x π(dx) .

(2.2)

Finally, in the following proofs, c and C will denote positive constants which can change from a line to another. More specific constants will be written c 1 , c 2 , . . . Furthermore, we will write u n v n when there exists two constants c 1 , c 2 independent of n such that

c 1 v n ≤ u n ≤ c 2 v n for every n ∈ N.

Pointwise exponent

We extend in this section the multifractal spectrum (1.9) to any Lévy process. To begin with, we prove two technical lemmas that will be extensively used in the rest of the article.

Lemma 2.1. For any δ > β, there exists a positive constant c(δ) such that for all

m ∈ R + P sup t≤2 -m Y m/δ t 1 ≥ m2 -m/δ ≤ c(δ)e -m . Proof. Let δ > β. We observe that for any m ∈ R + , sup t≤2 -m Y m/δ t 1 ≥ m2 -m/δ = ε∈{-1,1} d sup t≤2 -m ε, Y m/δ t ≥ m2 -m/δ
Hence, it is sufficient to prove that there exists c(δ) > 0 such that for any

ε ∈ {-1, 1} d , P sup t≤2 -m ε, Y m/δ t ≥ m2 -m/δ ≤ c(δ)e -m .
Let λ = 2 m/δ and M t = e λ ε,Y m/δ t for all t ∈ R + . According to Theorem 25.17 in [START_REF] Sato | Lévy processes and infinitely divisible distributions[END_REF], we have

E[M t ] = exp t D(0,2 -m/δ ) e λ ε,x -1 -λ ε, x π(dx) . Furthermore, we observe that for all s ≤ t ∈ R + , E[ M t | F s ] = M s exp (t -s) D(0,2 -m/δ ) e λ ε,x -1 -λ ε, x π(dx) ≥ M s ,
since for any y ∈ R, e y -1 -y ≥ 0. Hence, M is a positive submartingale, and using Doob's inequality (Theorem 1.7 in [START_REF] Revuz | of Grundlehren der Mathematischen Wissenschaften[END_REF]), we obtain

P sup t≤2 -m ε, Y m/δ t ≥ m2 -m/δ = P sup t≤2 -m M t ≥ e m ≤ e -m E[M 2 -m ]. For all y ∈ [-1, 1], we note that e y -1 -y ≤ y 2 . Thus, for any m ∈ R + , E[M 2 -m ] ≤ exp 2 -m D(0,2 -m/δ ) λ 2 ε, x 2 π(dx) ≤ exp 2 -m D(0,2 -m/δ ) λ 2 x 2 π(dx) . If β < 2, let us set γ > 0 such that β < γ < 2 and γ < δ. Then, 2 -m D(0,2 -m/δ ) λ 2 x 2 π(dx) = 2 -m(1-2/δ) D(0,2 -m/δ ) x γ • x 2-γ π(dx) ≤ 2 -m(1-2/δ) 2 -m/δ(2-γ) D(0,1) x γ π(dx) = 2 -m(1-γ/δ) D(0,1) x γ π(dx) ≤ D(0,1) x γ π(dx), since γ < δ. If β = 2, we simply observe that 2 -m D(0,2 -m/δ ) λ 2 x 2 π(dx) ≤ 2 -m(1-2/δ) D(0,1) x 2 π(dx) ≤ D(0,1) x 2 π(dx), as δ > 2. Therefore, there exists c(δ) > 0 such that for all m ∈ R + , E[M 2 -m ] ≤ c(δ),
concluding the proof of this lemma. Lemma 2.2. Suppose δ > β. Then, with probability one, there exist c 1 > 0 and M (ω

) > 0 such that ∀u, v ∈ [0, 1] : |u -v| ≤ 2 -m ; Y m/δ u -Y m/δ v ≤ c 1 m2 -m/δ (2.3)
for any m ≥ M (ω).

Proof. We first note that for any m ∈ R + and any δ > β,

sup u,v∈[0,1]:|u-v|≤2 -m Y m/δ u -Y m/δ v ≥ 3m2 -m/δ ⊆ 2 m -1 k=0 sup t≤2 -m Y m/δ t+k2 -m -Y m/δ k2 -m ≥ m2 -m/δ .
Therefore, the stationarity of Lévy processes and Lemma 2.1 yield

P sup u,v∈[0,1]:|u-v|≤2 -m Y m/δ u -Y m/δ v ≥ 3m2 -m/δ ≤ 2 m c(δ)e -m = c(δ)e -cm .
Using the latter estimate and Borel-Cantelli lemma, we obtain Equation (2.3).

Let us recall the definition of the collection of random sets (A δ ) δ>0 introduced by Jaffard [START_REF] Jaffard | The multifractal nature of Lévy processes[END_REF]. For every ω ∈ Ω, S(ω) denotes the countable set of jumps of Y•(ω). Moreover, for any

ε > 0, let A ε δ be A ε δ = t∈S(ω) ∆Yt ≤ε t -∆Y t δ , t + ∆Y t δ .
Then, the random set A δ is defined by

A δ = lim sup ε→0 + A ε δ . As noted in [25], if t ∈ A δ , we necessarily have α Y,t ≤ 1 δ .
The other side inequality is obtained in the next statement which extends Proposition 2 from [START_REF] Jaffard | The multifractal nature of Lévy processes[END_REF]. Proposition 2. Suppose δ > β. Then, with probability one, for all t ∈ [0, 1] \ S(ω):

t / ∈ A δ =⇒ α Y,t ≥ 1 δ . Proof. Suppose ω ∈ Ω, t / ∈ A δ , u ∈ [0, 1] and m ∈ N such that 2 -(m+1)δ ≤ |t -u| < 2 -mδ . Since t / ∈ A δ , there exists ε 0 > 0 such that for all ε ≤ ε 0 , t / ∈ A ε δ . The component [t,u]×D(ε0,1)
x J(ds, dx) -(u -t)

D(ε0,1)
x π(dx) is piecewise linear, and therefore does not influence the pointwise exponent α Y,t . Without any loss of generality, we may assume that 2 -m ≤ ε 0 . Then, for any jump ∆Y s such that

∆Y s ∈ [2 -m , ε 0 ], we have ∆Y s δ ≥ 2 -mδ ≥ |t -u|, implying that [t,u]×D(2 -m ,ε0)
x J(ds, dx) = 0.

Furthermore, using Lemma 2.2, we obtain

Y m u -Y m t ≤ c m2 -m ≤ c log |t -u| -1 |t -u| 1/δ ,
assuming that |t -u| is sufficiently small. Therefore, the remaining term to estimate corresponds to -(u -t) D(2 -m ,ε0) x π(dx). To study the latter, we distinguish two different cases, depending on the polynomial component we subtract in Definition 1.

1. If δ ≥ 1, let us set P t ≡ 0. Then, (u -t) D(2 -m ,ε0) x π(dx) ≤ c |t -u| D(2 -m ,ε0) x δ • x 1-δ π(dx) ≤ c |t -u| • 2 -m(1-δ) D(2 -m ,ε0) x δ π(dx) ≤ c |t -u| 1/δ .
2. If δ < 1 (and thus β < 1), we set P t (u) ≡ -(u -t) D(0,ε0) π(dx), which corresponds to the linear drift of the Lévy process. We observe that

-(u-t) D(2 -m ,ε0) x π(dx)-P t (u) = (u -t) D(0,2 -m ) x π(dx).
Then, similarly to the previous case, the latter satisfies (u -t)

D(0,2 -m ) x π(dx) ≤ c |t -u| D(0,2 -m ) x δ • x 1-δ π(dx) ≤ c |t -u| • 2 -m(1-δ) D(0,2 -m ) x δ π(dx) ≤ c |t -u| 1/δ .
Therefore, owing to the previous estimates, we have proved that

Y u -Y t -P t (u) ≤ c 0 log |t -u| -1 |t -u| 1/δ
, where the constant c 0 is independent of u. The latter inequality and Definition 1 prove that α Y,t ≥ 1 δ . Proposition 2 ensures that almost surely

∀h > 0; E h = δ<1/h A δ \ δ>1/h A δ \ S and E 0 = δ>0 A δ ∪ S.
(2.4) Furthermore, since the estimate of the Hausdorff dimension obtained in [START_REF] Jaffard | The multifractal nature of Lévy processes[END_REF] does not rely on Assumption (1.8), the Lévy process Y satisfies with probability one

∀V ∈ O; dim H (E h ∩ V ) = βh if h ∈ [0, 1/β]; -∞ otherwise.

2-microlocal frontier of Lévy processes

We now aim to refine the multifractal spectrum of Lévy processes by studying their 2microlocal structure. Let us begin with a few basics remarks and estimates on their 2microlocal frontier. Firstly, according to [START_REF] Meyer | Wavelets, vibrations and scalings[END_REF]Th. 3.13], with probability one, for all t ∈ [0, 1] and for any -s < α Y,t , the sample path Y•(ω) belongs to the 2-microlocal space C 0,s t . Furthermore, owing to the density of the set of jumps

S(ω) in [0, 1], necessarily Y•(ω) / ∈ C σ,s t
for any σ > 0 and all s ∈ R. Hence, since the 2-microlocal frontier is a concave function with left-and right-derivatives between 0 and 1, with probability one and for all t ∈ [0, 1]:

∀s ∈ R + ; σ Y,t (s ) ≥ (α Y,t + s ) ∧ 0 and σ Y,t (s ) ≤ 0.
Therefore, we are interested in obtaining finer estimates of the negative component of the 2-microlocal frontier of Y . As outlined in the introduction and Definitions 2-3, we need to analyse the following type of increments in the neighbourhood of t:

u b (u -s) k-1 + Y s ds -P t,k (u) - v b (v -s) k-1 + Y s ds + P t,k (v) (2.5)
where b < t is fixed and k ≥ 1. The polynomial component to be subtracted can be estimate using our work on pointwise exponent. Indeed, when k = 0, the P t,0 ≡ P t where the latter has been presented in the proof of Proposition 2,. Then, the consistency of the definition of the 2-microlocal spaces imposes that P t,k-1 must correspond to the derivative of P t,k . This last property shows us that the form of P t,k can be completely deduce from the known polynomial P t .

For the sake of readability, we divide the proof of Theorem 1 and its corollaries in several different technical lemmas. To begin with, we give simple estimates on the jumps of a Lévy process.

Lemma 2.3. For any ε > 0, there exists an increasing sequence (m n ) n∈N such that with probability one, for all t ∈ [0, 1] and for every n ≥ N (ω)

∃u ∈ B(t, 2 -mnα ); ∆Y u ≥ 2 -mn and J B(u, 2 -mnγ ) × D(2 -mn(1+ε) , 1) = 1, where α = β(1 -2ε) and γ = β(1 + 4ε). Proof. Suppose m ∈ N, ε > 0, α = β(1 -2ε) and γ = β(1 + 4ε). Let I be an interval such that I = I 1 ∪ I 2 ∪ I 3
, where I 1 , I 2 , I 3 are three consecutive and disjoint intervals of size 2 -mnγ . Then, we are interested in the following event:

A = J I 1 , D(2 -m , 1) = 0 ∩ J I 3 , D(2 -m , 1) = 0 ∩ J I 2 , D(2 -m , 1) = 1 ∩ J I, D(2 -m(1+ε) , 2 -m ) = 0 ,
Since J is a Poisson measure, A corresponds to the intersection of independent events whose probability is equal to

P(A) = 2 -mγ π(D(2 -m , 1)) • exp -3 • 2 -mγ π(D(2 -m , 1)) + 2 -mγ π(D(2 -m(1+ε) , 2 -m )) .
As described in [START_REF] Blumenthal | Sample functions of stochastic processes with stationary independent increments[END_REF], β can be defined by β = inf δ ≥ 0 : lim sup r→0 r δ π B(r, 1) < ∞ . Therefore, there exists r 0 > 0 such that for all r ∈ (0, r 0 ], π B(r, 1) ≤ r -β(1+ε) . Hence, for any m ∈ N sufficiently large:

P(A) ≥ 2 -mγ π(D(2 -m , 1)) exp -2 -mβε+1 ≥ 2 -mγ-1 π D(2 -m , 1) .
Furthermore, according to the definition of β, there also exists an increasing sequence (m n ) n∈N such that for all n ∈ N, π(D(2 -mn , 1)) ≥ 2 mnβ (1-ε) . Therefore, along this sequence, we obtain P(A) ≥ 2 -mn5βε-1 for every n ∈ N.

Let now consider an interval I of size 2 -mnα . There exist at most 2 -mnα+mnγ disjoint sub-intervals I of size 3 • 2 -mnγ . We designate by B the event where A is not satisfied by all these sub-elements I. Owing to the previous estimate of P(A) and the independence of these different events, we obtain

P(B) = P (A c ) 2 mn (γ-α) ≤ 1 -2 -mn5βε-1 2 mn (γ-α)
.

Note that γ -α = 6βε. Hence, log P(B) ≤ -2 -mn5βε-1 • 2 mn6βε = -2 mnβε-1 and the probability P(B) satisfies P(B) ≤ exp -2 mnβε-1 . Finally, we know there exist at most 2 mnα+1 disjoint intervals J of size 2 -mnα inside [0, 1]. We denote by B n the event where B is satisfied for one of the previous interval I. Since B n is the reunion of events, we obtain

P(B n ) ≤ 2 mnα+1 • exp -2 mnβε-1 ≤ exp -2 mnβε-1 + c m n α .

Therefore,

n∈N P(B n ) < ∞ and owing to Borel-Cantelli lemma, with probability one, there exists N (ω) such that for every n ≥ N (ω), ω ∈ B c n . The latter inclusion means that for every interval I previous defined, there exists a sub-element I such that the event A is satisfied on I, which proves this lemma. Let us set t ∈ [0, 1]. We need to evaluate the size of the increments described in Equation (2.5). Hence, let us first determine the form of the local process Y (u, k) := I k b+ Y (u) -P t,k (u) used. We know that when k = 0, the polynomial component is described in Proposition 2, and thus we define the local process Y (s, 0) in the neighbourhood of t as following:

∀u ∈ R; Y (u, 0) = Y u -Y t -P t (u).
Then, since the polynomial component must correspond to the Taylor development of the process at t, we define the elements Y (•, k) be induction:

∀u ∈ R; Y (u, k) = u t Y (s, k -1) ds.
One can easily verify that the derivative of Hence, we need to show in this proof that for any k ∈ N, the increments of the process Y (•, k) are sufficiently large in the neighbourhood of t. More precisely, we will show by induction that there exist t n,k → n t, ρ n,k > 0 and δ n,k > 0 such that for every k ∈ N and all n ∈ N:

Y (•, k) is Y (•, k -
∀u

∈ [t n,k , t n,k + ρ n,k ); |Y (u, k)| ≥ δ n,k . (2.7)
To initialize the induction with k = 0, we make use of the estimate obtained in Lemma 2.3: there exists an increasing sequence (m n ) n∈N such that with probability one, for all t ∈ [0, 1] and for every n ≥ N (ω)

∃v ∈ B(t, 2 -mnα ); |∆Y v | ≥ 2 -mn and J B(v, 2 -mnγ ) × D(2 -mn(1+ε) , 1) = 1,
where α = β(1 -2ε) and γ = β(1 + 4ε). Since the reasoning which follows is completely symmetric, we may assume without any loss of generality that v ≥ t and ∆Y v ≥ 0. Let us set n ≥ N (ω) and a proper v ≥ t. We know there is no other jump of size greater than 2 -mn(1+ε) inside the ball B(v, 2 -mnγ ). Therefore, for all u ∈ B(v, 2 -mnγ ),

Y u -Y v = -∆Y v 1 {u<v} -(u -v) D(2 -mn(1+ε) ,1)
x π(dx)

+ Y mn(1+ε) u -Y mn(1+ε) v .
Furthermore, according to Lemma 2.2, the norm of the latter increment satisfies:

Y mn(1+ε) u -Y mn(1+ε) v ≤ c 1 m n 2 -mn(1+ε) , as we note that |u -v| ≤ 2 -mnβ(1+4ε) = 2 -mn(1+ε)β(1+4ε)/(1+ε) with β(1 + 4ε)/(1 + ε) > β.
Then, similarly to the proof of Proposition 2, we need to distinguish two different cases.

1. If β ≥ 1, P t ≡ 0 and thus Y (u, 0) = Y u -Y t . Let us first assume that Y (v, 0) ≥ 2 -mn-1
and set t n,0 = v and ρ n,0 = 2 -mnγ . Then, for all u ∈ [t n,0 , t n,0 + ρ n,0 ):

|Y (u, 0)| ≥ |Y (v, 0)| -Y mn(1+ε) u -Y mn(1+ε) v -(u -v) D(2 -mn (1+ε) ,1)
x π(dx) .

Using the estimates presented in Proposition 2, we obtain an upper bound of the last term:

(u -v) D(2 -mn(1+ε) ,1) x π(dx) ≤ c 2 -mnγ • 2 -mn(1+ε)(1-β(1+ε)) ≤ c 2 -mn(1+ε) , since -γ + β(1 + ε) 2 = -β(2ε -ε 2 ) < 0. Hence, |Y (u, 0)| ≥ 2 -mn-1 -c 2 -mn(1+ε) ≥ 2 -mn-2
for any n sufficiently large.

Let now assume that Y (v, 0) ≤ 2 -mn-1 . Since ∆Y v ≥ 2 -mn , we necessarily have Y (v -, 0) ≤ -2 -mn-1 . Then, we set in this case t n,0 = v -2 -mnγ and ρ n,0 = 2 -mnγ , and obtain as well

|Y (u, 0)| ≥ 2 -mn-1 -c 2 -mn(1+ε) ≥ 2 -mn-2 . 2. If β < 1, P t (u) ≡ -(u -t) D(0,1) π(dx).
Similarly to the previous case, we first assume that Y (v, 0) ≥ 2 -mn-1 and set t n,0 = v and ρ n,0 = 2 -mnγ . Then, for all u ∈ [t n,0 , t n,0 + ρ n,0 ]:

|Y (u, 0)| ≥ |Y (v, 0)| -Y mn(1+ε) u -Y mn(1+ε) v -(u -v) D(0,2 -mn (1+ε) )
x π(dx) ,

where the latter element satisfies

(u -v) D(0,2 -mn(1+ε) ) x π(dx) ≤ c 2 -mnγ • 2 -mn(1+ε)(1-β(1+ε)) ≤ c 2 -mn(1+ε) .
Hence, |Y (u, 0)| ≥ 2 -mn-1 -c 2 -mn(1+ε) ≥ 2 -mn-2 for any n sufficiently large. In the case Y (v, 0) ≤ 2 -mn-1 , we observe that Y (v -, 0) ≤ -2 -mn-1 . Therefore, setting t n,0 = v -2 -mnγ and ρ n,0 = 2 -mnγ , we obtain

|Y (u, 0)| ≥ 2 -mn-1 -c 2 -mn(1+ε) ≥ 2 -mn-2 .
Therefore, in both cases, we have proved that

∀u ∈ [t n,0 , t n,0 + ρ n,0 ); |Y (u, 0)| ≥ δ n,0 ,
where ρ n,0 = 2 -mnγ , δ n,0 = 2 -mn-1 and B(t n,0 , ρ n,0 ) ⊂ B(t, 2 -mnα+1 ).

Let now assume that Equation (2.7) is satisfied for k ∈ N. Without any loss of generality, we may suppose that Y (u, k) ≥ δ n,k on the interval [t n,k , t n,k + ρ n,k ) (otherwise, simply consider the process -Y (u, k) in the following reasoning). In this case, the function u → u t Y (s, k) ds is strictly increasing on the previous interval.

Let us first assume that

t n,k +ρ n,k /2 t Y (s, k) ds ≥ 0. In this case, we set t n,k+1 = t n,k + 3/4ρ n,k , ρ n,k+1 = ρ n,k /4 and δ n,k+1 = ρ n,k δ n,k /4. Then, for all u ∈ [t n,k+1 , t n,k+1 + ρ n,k+1 ) Y (u, k + 1) ≥ u t n,k +ρ n,k /2 Y (s, k) ds ≥ (u -t n,k -ρ n,k /2) δ n,k ≥ ρ n,k δ n,k /4 = δ n,k+1 ,
In the other case

t n,k +ρ n,k /2 t Y (s, k) ds ≤ 0, we consider the set of parameters t n,k+1 = t n,k , ρ n,k+1 = ρ n,k /4 and δ n,k+1 = ρ n,k δ n,k /4. Then, for all u ∈ [t n,k+1 , t n,k+1 + ρ n,k+1 ) Y (u, k + 1) ≤ - t n,k +ρ n,k /2 u Y (s, k) ds ≤ -(t n,k + ρ n,k /2 -u) δ n,k ≤ -ρ n,k δ n,k /4 = -δ n,k+1 ,
Therefore, assuming that Equation (2.7) holds for k ∈ N, we have proved that it does too for k + 1 with

ρ n,k+1 = ρ n,k /4, δ n,k+1 = ρ n,k δ n,k /4 and B(t n,k+1 , ρ n,k+1 ) ⊂ B(t n,k , ρ n,k ) ⊂ B(t, 2 -mnα+1 ).
Finally, the lower bound on Y n,k presented in Equation (2.7) will now help us to obtain the expected bound on the 2-microlocal frontier. Owing to the previous definitions, for every k ∈ N, |t n,k -t| ≤ 2 -mnα+1 and there exist c k > 0 independent of n ∈ N such that δ n,k = c k 2 -mn kγ • 2 -mn . Hence, for every n ∈ N,

|Y (t n,k , k) -Y (t, k)| ≥ δ n,k ≥ c k 2 -mn(1+kγ) ≥ c k |t n,k -t| (1+kγ)/α ,
where we recall that α = β(1 -2ε) and γ = β(1 + 4ε). Therefore, this last inequality proves that the pointwise exponent of

Y (•, k) at t satisfies α Y (•,k),t ≤ (1 + kγ)/α → ε→0 1/β + k.
Owing to the Definition 3 of the 2-microlocal spaces, this last inequality induces that with probability one, for any t ∈ [0, 1] and all s ∈ R, σ Y,t (s ) ≤ 1 β + s ∧ 0. As we have obtained a uniform upper bound on the 2-microlocal frontier, we now study more precisely the regularity of Y at times where α Y,t < 1/β. To begin with, we prove a simple lemma related to the number of jumps inside an interval. Lemma 2.5. Suppose δ > β, ε > 0 and k ∈ N are such that δ > β(1 + 2ε)(k + 1)/k. Then, with probability one, there exists M (ω) such that:

∀t ∈ [0, 1]; J B(t, 2 -mδ ), D(2 -m(1+ε) , 1) ≤ k, (2.8)
for every m ≥ M (ω).

Proof. Let m ∈ N and I be an interval of size 2 -mδ+2 . Since J is a Poisson random measure,

P J I, D(2 -m(1+ε) , 1) > k = exp(-λ m ) +∞ =k+1 λ m ! ,
where λ m = 2 -mδ+2 π D(2 -m(1+ε) , 1) ≤ 2 -mδ+2+mβ(1+2ε) → 0. Hence, we obtain the inequality P J I, D(2 -m(1+ε) , 1) > k ≤ c λ k+1 m . Considering a covering of the interval [0, 1] with 2 mδ overlapping sub-elements I of size 2 -mδ+2 , we denote by B m the event where at least one of these intervals has more than k jumps inside. Then,

P(B m ) ≤ c 2 mδ • λ k+1 m ≤ c 2 mδ-mδ(k+1)+mβ(k+1)(1+2ε) .
Since δk > β(1+2ε)(k+1), there exists γ > 0 such P(B m ) ≤ c 2 -mγ . Therefore, m∈N P(B m ) < ∞ and owing to Borel-Cantelli lemma, there exists M (ω) such that for every m ≥ M (ω), J I, D(2 -m(1+ε) , 1) ≤ k. Finally, since we consider intervals I of size 2 -mδ+2 covering [0, 1] and overlapping, we have proved that for all t ∈ [0, 1], J B(t, 2 -mδ ), D(2 -m(1+ε) , 1) ≤ k.

In the next lemma, we start with the study of the 2-microlocal frontier of Y at points t ∈ [0, 1] where α Y,t ∈ [0, 1/2β].

Lemma 2.6. With probability one, for all h ∈ [0, 1/2β], the singularities of Y satisfy E h = E h and E

h = ∅, i.e. for all t ∈ E h ∀s ∈ R; σ Y,t (s ) = (α Y,t + s ) ∧ 0.
Proof. Suppose h ∈ [0, 1/2β] and t ∈ E h \ S(ω) (t is not a jump time). Since we already know that σ Y,t (s ) ≥ (h + s ) ∧ 0, we need to only prove the other side inequality. For that purpose, we will proceed similarly to the proof of Lemma 2.4.

More precisely, let us set ε > 0 and δ > max 2β(1 + 2ε), 1/(h + ε) . Since t ∈ E h and owing to Equation (2.4), there exist two sequences (v n ) n∈N and (m n ) n∈N such that

∀n ∈ N; v n ∈ B t, 2 -mn/(h+ε)
and ∆Y vn ≥ 2 -mn .

Without any loss of generality, we may assume that v n ≥ t. Furthermore, owing to Lemma 2.5, J B(v n , 2 -mnδ ), D(2 -mn(1+ε) , 1) = 1, i.e. there is no other jump larger than 2 -mn(1+ε) in the neighbourhood of v n . Then, similarly to the proof of Lemma 2.4, we need to distinguish two different cases.

Y (u, 0) ≥ Y (v n , 0) -Y mn(1+ε) u -Y mn(1+ε) vn -(u -v n ) D(2 -mn (1+ε) ,1)
x π(dx) .

Still using the same estimates as in the proof of Lemma 2.5, we know that the last two terms are upper bounded by c 2 -mn(1+ε) , proving that Y (u, 0) ≥ 2 -mn-2 for any n sufficiently large. The case Y (v n , 0) ≤ 2 -mn-1 is treated completely similarly, using

t n = v n -2 -mnδ and ρ n = 2 -mnδ . 2. If β < 1, P t (u) ≡ -(u -t) D(0,1) π(dx). Assuming first that Y (v n , 0) ≥ 2 -mn-1 . we still set t n = v n and ρ n = 2 -mnδ . Then, for all u ∈ [t n , t n + ρ n ): Y (u, 0) ≥ Y (v n , 0) -Y mn(1+ε) u -Y mn(1+ε) vn -(u -v n ) D(0,2 -mn(1+ε) )
x π(dx) ,

As previously, the last two terms are upper bounded by c 2 -mn(1+ε) , proving that Y (u, 0) ≥ 2 -mn-2 for any n sufficiently large. The case Y (v n , 0) ≤ 2 -mn-1 is treated similarly using t n = v n -2 -mnδ and ρ n = 2 -mnδ . Therefore, we have proved in both cases that for all u ∈ [t n , t n + ρ n ), with n sufficiently large, Y (u, 0) ≥ 2 -mn-2 . Reproducing the same reasoning as in the proof of Lemma 2.4, there exists s n such that for every n ∈ N, s n ∈ B(t, 2 -mn/(h+ε) ) and

Y (s n , 1) ≥ c 2 -mn • 2 -mnδ ≥ |t -s n | (h+ε)(1+δ) .
Hence, α Y (•,1),t ≤ (h + ε)(1 + δ). Considering the limit ε → 0 and δ → 1/h, we obtain α Y (•,1),t ≤ h + 1. The latter inequality is sufficient to prove that σ Y,t (s ) = (h + s ) ∧ 0 for all s ∈ R.

To conclude this proof, let us consider the case t ∈ S(ω). We observe that for all u ≥ t,

u t Y s ds = (u -t)Y t + u t (Y s -Y t ) ds with u t (Y s -Y t ) ds = o(|t -u|),
as Y is right-continuous. Similarly, for all u ≤ t,

t u Y s ds = (t -u)Y t-+ o(|t -u|).
Therefore, since ∆Y t = Y t -Y t-= 0, there does not exist a polynomial P t which can cancel both terms (u -t)Y t and (t -u)Y t-, proving that σ Y,t (s ) = s ∧ 0 for all s ∈ R.

In the last technical lemma, we focus on the particular case α Y,t ∈ (1/2β, 1/β) and try to distinguish oscillating singularities from the common cusp behaviour.

Lemma 2.7. With probability one, for all

h ∈ (1/2β, 1/β), Y satisfies ∀V ∈ O; dim H ( E h ∩ V ) = βh and dim H ( E h ∩ V ) ≤ 2βh -1 (< βh).
(2.9)

Furthermore, for any t ∈ E h and all s ∈ R, σ Y,t (s ) ≤ (h + s )/2βh.

Proof. On the contrary to the previous lemma, we know that some oscillating singularities might appear at a given time t. Hence, the first step in this proof is to isolate these behaviours and estimate the fractal dimension of the corresponding set of times.

For that purpose, let us set δ ∈ (β, 2β) and ε > 0. We are interested in the doublejump configurations, i.e. when two jumps greater than 2 -m(1+ε) are sufficiently close. More precisely, suppose I is an interval of size 2 -mδ+1 and p m designates the probability of obtaining at least two jumps greater than 2 -m(1+ε) inside I. Then,

c 2 -m2δ • 2 m2β ≤ p m = P J I, D(2 -m(1+ε) , 1) ≥ 2 ≤ C 2 -m2δ • 2 m2β(1+2ε)
where we may assume that δ > β(1 + 2ε). We consider 2 mδ consecutive, but disjoint, intervals of size 2 -mδ+1 which are sufficient to cover [0, 1]. Then, if we denote by N 1 m the number of intervals with the previous configuration, it follows a Binomial distribution of parameters 2 mδ and p m . Moreover, Chernoff's inequality induces that

P N 1 m ≥ c 0 2 -mδ+m2β(1+2ε) ≤ exp -c 2 -mδ+m2β ,
where 2β > δ. Let us consider now the same configurations of intervals translated by 2 -mδ and denote by N 2 m the corresponding Binomial random variable. Owing to the previous estimate and Borel-Cantelli lemma, with probability one, there exists M (ω) such that for every m ≥ M (ω), N 1 m ≤ c 0 2 -mδ+m2β(1+2ε) and N 2 m ≤ c 0 2 -mδ+m2β(1+2ε) . Then, let T m index the previous intervals with a double-jump configuration and F (δ, ε) designate the following set:

F (δ, ε) = lim sup m→∞ I∈Tm c(I) -2 -mδ+2 , c(I) + 2 -mδ+2 ,
where c(I) denotes the center of any interval I ∈ T m . Using a simple covering based on intervals of size 2 -mδ , we can obtain an upper bound of the Hausdorff dimension of F (δ, ε). More precisely, for any m 0 ∈ N, the series

+∞ m=m0 c |T m | • 2 -mδ γ ≤ c +∞ m=m0 2 -m(δ(1+γ)-2β(1+2ε)) , converges when δ(1 + γ) > 2β(1 + 2ε), i.e. γ > 2β(1 + 2ε)/δ -1. Therefore, dim H F (δ, ε) ≤ 2β(1 + 2ε)/δ -1 almost surely.
Let now set h ∈ (1/2β, 1/β). We aim to prove that E h ⊂ F (δ, ε) for any δ < 1/h and ε > 0. For that purpose, we need to show that for every

t ∈ E h \ F (δ, ε), the 2-microlocal frontier at t satisfies σ Y,t (s ) ≤ (h + s ). As t ∈ E h , there exist two sequences (v n ) n∈N and (m n ) n∈N such that ∀n ∈ N; v n ∈ B t, 2 -mn/(h+ε)
and ∆Y vn ≥ 2 -mn .

We may assume that ε is sufficiently small to satisfy 2 -mn/(h+ε) ≤ 2 -mδ , i.e. δ < 1/(h + ε). Furthermore, since t / ∈ F (δ, ε), for every m sufficiently large, there is no double-jump configuration in the neighbourhood of t and v n , meaning that J B(v n , 2 -mnδ ), D(2 -mn(1+ε) , 1) = 1: there does not exist other jump larger than 2 -mn(1+ε) in the neighbourhood of v n . Therefore, we obtain the configuration presented in the proof of Lemma 2.6, and as the latter remains valid, we have

∀s ∈ R; σ Y,t (s ) ≤ (h + s ).
This upper bound shows that E h ⊂ F (δ, ε), and considering the limits δ → 1/h and ε → 0, it induces the inequality dim H E h ≤ 2βh -1. Furthermore, since 2βh -1 < βh and E h = E h ∪ E h , we have also proved that dim H E h = βh.

To conclude this lemma, we obtain an upper bound of the 2-microlocal frontier in the case t ∈ E h . Since the sketch of the proof is similar to Lemmas 2.4 and 2.6, we only present the main elements. Still using the previous two sequences

(v n ) n∈N and (m n ) n∈N , Lemma 2.5 induces that J B(v n , 2 -mn2β(1+3ε) ), D(2 -mn(1+ε) , 1) = 1.
Then, using the methodology presented in Lemma 2.6, there exists (s n ) n∈N such that for every n ∈ N, s n ∈ B(t, 2 -mn/(h+ε) ) and

Y (s n , 1) ≥ c 2 -mn • 2 -mn2β(1+3ε) ≥ |t -s n | (h+ε)(1+2β(1+3ε)) , Hence, α Y (•,1),t ≤ (h + ε)(1 + 2β(1 + 3ε)) → ε→0 h(1 + 2β
), and using the reasoning presented in Lemma 2.6, we obtain σ Y,t ≤ (h + s )/2βh for all s ∈ R.

Before finally proving Theorem 1 and its corollaries, we recall the following result on the increments of a Brownian motion. The proof can be found in [START_REF] Adler | The geometry of random fields[END_REF] (inequality (8.8.26)).

Lemma 2.8. Let B be a d-dimensional

Brownian motion. There exists an event Ω 0 of probability one such that for all ω ∈ Ω 0 , ε > 0, there exists h(ω) > 0 such that for all ρ ≤ h(ω) and t ∈ [0, 1], we have

sup u,v∈B(t,ρ) B u -B v ≥ ρ 1/2+ε .
Proof of Theorem 1. We use the notations introduced at the beginning of the section. As previously said, the compound Poisson process N can be ignored since it does not influence the final regularity. Furthermore, if Q = 0, and therefore B ≡ 0 and β = β, Lemmas 2.4, 2.6 and 2.7 on the component Y yields Theorem 1.

Otherwise, the Lévy process X corresponds to the sum of the Brownian motion B and the jump component Y . Still using Lemmas 2.4, 2.6 and 2.7, it is sufficient to prove that with probability one and for all t ∈ [0, 1], σ X,t = σ B,t ∧σ Y,t . Owing to the definition of 2-microlocal frontier, we already know that σ X,t ≥ σ B,t ∧ σ Y,t . Furthermore, when σ B,t (s ) = σ Y,t (s ), the upper bound is straightforward, and thus σ X,t (s ) = σ B,t (s ) ∧ σ Y,t (s ).

Therefore, to obtain Theorem 1, we have to prove that with probability one, for all t ∈ [0, 1], σ X,t ≤ σ B,t = s → 1/2 + s ∧ 1/2. For that purpose, we distinguish two different cases.

1. If β = β = 2, we only need to slightly modify the proof of Lemma 2.4. More precisely, owing to Lévy's modulus of continuity, the increments of the Brownian motion satisfy:

∀u, v : |u -v| ≤ 2 -mnγ ; B u -B v ≤ c m n 2 -mnγ/2 = c m n 2 -mn(1+4ε) ,
since γ = β(1 + 4ε). Therefore, the term due to the increments of the Brownian motion does not influence the rest of the estimates presented in the proof, ensuring that σ X,t (s ) ≤ (1/2 + s ) ∧ 0, for all s ∈ R.

2. If β < 2, let δ = 2 and ε > 0. According to Lemma 2.5, there exist k ∈ N and M (ω) ∈ N such that for all m ≥ M , there are at most k jumps of size greater than 2 -m(1+3ε) in any interval of size 2 -mδ . Hence, there always exists a sub-interval I of size c 0 2 -δm with no jump greater than 2 -m(1+3ε) inside. Still using Lemma 2.2, we know that for all m ≥ M (ω)

∀u, v ∈ [0, 1] : |u -v| ≤ 2 -δm ; Y m(1+3ε) u -Y m(1+3ε) v ≤ c 1 m2 -m(1+3ε) .
Let us set t ∈ [0, 1] and I be one of the previous interval of size c 0 2 -δm . According to Lemma 2.8, there exist u, v 1+2ε) . Then,

∈ I such that B u -B v ≥ c 0 2 -m(
X u -X v ≥ B u -B v -Y m(1+3ε) u -Y m(1+3ε) v -|u -v| • D(2 -m(1+3ε) ,1)
x π(dx) , 1+3ε) . Hence, we obtain a lower bound of the increments on the interval I, ensuring that the rest of the proof presented in Lemma 2.5 holds similarly.

where |u -v| • D(2 -m(1+3ε) ,1) x π(dx) ≤ c 2 -m(
Proof of Corollary 1. Recall that β w X,t = lim s →-∞ σ X,t (s ) -s . Hence, using the global upper bound on the 2-microlocal frontier proved in Theorem 1, we know that β w X,t ≤ 1/β with probability one. In addition, owing to the geometrical properties of the frontier, we observe that for every h

∈ [0, 1/β ] ∀h ∈ [0, 1/β ); E h ⊆ E w h ⊆ E h ∪ h <h E h .
The first inclusion clearly shows that dim

H (E w h ∩ V ) ≥ dim H ( E h ∩ V ) = βh.
In addition, we also know that for every h < h, dim H E h ≤ 2βh -1 < βh, which proves the other side inequality.

To obtain the upper bound on the oscillating exponent, we only need to note that according to its characterisation using the 2-microlocal frontier,

∀h ∈ [0, 1/β ); t ∈ E h : β o X,t > 0 = E h .
Finally, the chirp exponent is equal to one because of the upper bound σ X,t (s ) ≤ 1/β + s .

Proof of Corollary 2. Owing to upper bound on the 2-microlocal frontier obtained in Theorem 1, the case σ = 0 corresponds to the classic spectrum of singularity. Hence, let us set σ < 0. We recall that s denotes the parameter σ -s . If s ≥ 1/β or s < 0, the result is straight forward using Theorem 1 and properties of the 2-microlocal frontier.

Therefore, we suppose that s ∈ [0, 1/β ) and note that E σ,s = {t ∈ R + : σ X,t (s ) = σ}, since the negative component of the 2-microlocal frontier of X can not be constant. Hence, similarly to the previous corollary, E σ,s satisfies

∀s ∈ [0, 1/β ); E s ⊆ E σ,s ⊆ E s ∪ h<s E h .
These two inclusions lead to the same estimates, and therefore the expected equality on the Hausdorff dimension.

Oscillating singularities of some classes of Lévy processes

In this section, we aim to understand more precisely the oscillating singularities of Lévy processes captured by the collection of sets ( E h ) h∈R+ . Note that to simply our presentation, we assume that d = 1.

Let us begin with the proof of Proposition 1 where we present a class Lévy processes with no chirp oscillations. Recall that in this case, we consider Lévy measures such that π(R ± ) = 0.

Proof of Proposition 1. In order to prove that E h = ∅ for all h ∈ R + , we extend Lemma 2.6 to any h ∈ [0, 1/β). We may assume without any loss of generality that π(R -) = 0. We still consider the two sequences (v n ) n∈N and (m n ) n∈N such that

∀n ∈ N; v n ∈ B t, 2 -mn/(h+ε)
and ∆Y vn ≥ 2 -mn .

where we suppose that v n ≥ t and Y designates the jump component. In addition, we first assume that β ≥ 1 and Y (v n , 0) ≥ 2 -mn-1 , and we set t n = v n and ρ n = 2 -mnδ . Then, since the Lévy process only has positive jumps, for all u ∈ [t n , t n + ρ n ),

|Y (u, 0)| ≥ |Y (v n , 0)| -Y mn(1+ε) u -Y mn(1+ε) vn -(u -v n ) D(2 -mn (1+ε) ,1)
x π(dx) ,

According to the proof presented in Lemma 2.6, this inequality is sufficient to show that σ Y,t (s ) ≤ (α X,t + s ) ∧ 0. The cases Y (v n , 0) ≤ 2 -mn-1 and β ≤ 1 are then treated similarly, proving that the 2-microlocal frontier of the process X is equal to (α X,t + s ) ∧ 0.

Proposition 1 proves in particular that Lévy subordinators, in which case β ≤ 1, only have cusp singularities.

The second important class of Lévy processes we consider are characterised by the following Lévy measure

π(dx) = a 1 |x| -1-α1 1 R+ dx + a 2 |x| -1-α2 1 R-dx,
where a 1 , a 2 > 0 and α 1 , α 2 ∈ (0, 2).

The proof of Theorem 2 is rather technical and will be divided in several parts for the sake of readability. To begin with, we present two simple technical lemmas related to the Binomial distribution. Recall that Chernoff's inequality states that for any ε ∈ (0, 1),

P N ≤ np(1 -ε) ≤ exp -np ε 2 /2 .
(2.10) and

P N ≥ np(1 + ε) ≤ exp -np ε 2 /2 . (2.11)
where N follows a Binomial distribution of parameters n and p.

Lemma 2.9. Suppose N follows a Binomial distribution with parameters n and p. Then, there exists c > 0 such that when n > c and p < 1/c,

P # empty intervals of size ≥ 1/p LL(1/p) ≥ np LL(1/p)h(p) 4 ≥ 1 -exp -np/8 LL(1/p) 2 ,
using the notations: LL(1/p) := log(log(1/p)) and h(p) := exp(-1/ LL(p)).

Proof. To obtain this lower bound, we first estimate the probability of obtaining an empty interval of size of l 0 = 1/p LL(1/p) . Setting p 0 := (1 -p) l0 , we note that

log(p 0 ) ≥ 1 p LL(1/p) log(1 -p) ≥ -2 LL(1/p) .
Therefore, p 0 ≥ h(p) 2 when c is sufficiently large.

Let n 0 denote the number of disjoint sub-intervals of size l 0 and N 0 be a r.v. following a Binomial distribution of parameters n 0 and p 0 . Owing to Chernoff's inequality,

P N 0 ≥ n 0 p 0 h(p) ≥ 1 -exp -n 0 p 0 g(p) 2 /2 ,
where g(p) := 1 -h(p). Note that n 0 ≥ np LL(1/p)h(p) and g(p)

1/ LL(1/p) when c is large enough. Therefore, using the previous estimates,

P N 0 ≥ np LL(1/p)h(p) 4 ≥ 1 -exp -np/8 LL(1/p) 2 ,
proving the lemma. Lemma 2.10. Suppose N follows a Binomial distribution with parameters n and p. Then, there exists c > 0 such that when n > c and p < 1/c

P # successes spaced by ≥ 1/p ≥ np/6 ≥ 1 -exp -np/16 .
Proof. The sketch of the proof is similar to Lemma 2.9. The set {1, . . . , n} can be divided in n 0 intervals of size l 0 = 1/p . The probability p 0 of obtaining at least a success in one of these intervals is equal to:

p 0 = 1 -(1 -p) 1/p -→ p→0 1 -e -1 ≥ 1/2.
Furthermore, only considering one third of the previous intervals, i.e. n 0 /3, we consider the Binomial distribution B(n 0 /3, p 0 ). Still using Chernoff's inequality, we obtain

P N 0 ≥ np/6 ≥ 1 -exp -np/16 , which concludes the proof.
Proof of Theorem 2. As observed in the proofs of Theorem 1 and Proposition 1, chirp singularities appear when a compensation phenomena between jumps exists. Hence, the main goal of the proof is to characterise in more details this particular behaviour in the case of the Lévy measure considered. Firstly, we clearly note the Blumenthal-Getoor exponent β of π is equal to max(α 1 , α 2 ).

Hausdorff dimension (upper-bound).

To obtain a tighter upper bound of the Hausdorff dimension, we need to enhance the estimates presented in Lemma 2.7. We have observed in the proof of Proposition 1 that oscillating singularities do not appear when there are jumps of the same sign. Hence, we are interested in the double-jump configurations with jumps of opposite signs.

Suppose δ ∈ (β, α 1 + α 2 ), ε > 0 and I is an interval of size 2 -jδ+1 . We are interested in the following type of configurations:

J I, (2 -j(1+ε) , 1] ≥ 1 and J I, [-1, -2 -j(1+ε) ) ≥ 1.
The probability p j of such an event satisfies:

p j 2 -jδ 2 jα1(1+ε) • 2 -jδ 2 jα2(1+ε) = 2 -j2δ+j(α1+α2)(1+ε) .
Using this probability, the rest of the proof is rather similar to Lemma 2.7. We consider 2 jδ consecutive, but disjoint, intervals of size 2 -jδ+1 sufficient to cover [0, 1] and we denote by N 1 j the number of intervals with the previous configuration. Owing to Chernoff's inequality,

P N 1 j ≥ c 0 2 -jδ+j(α1+α2)(1+ε) ≤ exp -c 2 -jδ+j(α1+α2) ,
where (α 1 + α 2 ) > δ. Consider now the same configurations of intervals translated by 2 -jδ and denote by N 2 j the corresponding Binomial random variable. Using Borel-Cantelli lemma, with probability one, there exists M (ω) such that for every j ≥ M (ω), N 1 j ≤ c 0 2 -jδ+j(α1+α2)(1+ε) and N 2 j ≤ c 0 2 -jδ+j(α1+α2)(1+ε) . Using the same notation T j , we define

F (δ, ε) = lim sup j→∞ I∈Tj c(I) -2 -jδ+2 , c(I) + 2 -jδ+2 ,
where c(I) denotes the center of any interval I ∈ T j . Then, for any

j 0 ∈ N +∞ j=j0 c |T j | • 2 -jδ γ ≤ c +∞ j=j0 2 -j(δ(1+γ)-(α1+α2)(1+ε)) , is finite when δ(1 + γ) > (α 1 + α 2 )(1 + ε), i.e. γ > (α 1 + α 2 )(1 + ε)/δ -1. Therefore, dim H F (δ, ε) ≤ (α 1 + α 2 )(1 + ε)/δ -1.
The rest of the proof of Lemma 2.7 does not change, proving that for any δ < 1/h and ε > 0, E h ⊂ F (δ, ε). Therefore, with probability one,

∀h ∈ 1/(α 1 + α 2 ), 1/β ; dim H E h ≤ (α 1 + α 2 )h -1.
Finally, when h / ∈ 1/(α 1 + α 2 ), 1/β , the proof of Lemma 2.6 can also be similarly adapted to prove that E h = ∅ with probability one.

Construction (lower-bound).

In order to prove the lower bound of the Hausdorff dimension, we need to construct a proper set of times with singularities.

For our construction procedure, we will need a set of parameters p = (δ, δ , δ , γ, ρ) such that δ < δ < δ ∈ (β,

α 1 + α 2 ), δ < γ ∈ (β, α 1 + α 2 ), δ < √ βδ and δ > √ δ δ .
In addition, we also define the sequence

j n = (δ/δ ) n → ∞.
The first step consists in constructing collections of intervals such that for every t inside, there is no jump of size 2 -j or greater closer than 2 -jδ for all j ≤ j 0 , where j 0 is a given index. More precisely, we define by induction a collection, indexed by the random variables S n , of disjoint intervals of size 2 -jnδ in the following way. Suppose S n is defined such that for every t inside an interval, there is no jump greater than 2 -jn+1 closer than 2 -jnδ of t. In every interval of size 2 -jnδ , we consider consecutive sub-intervals of size 2 -jn+1δ with no jumps greater than 2 -jn+2 inside. Removing the left and right elements of these collections, we obtain the family S n+1 , which corresponds to the offspring of S n . Owing to this construction procedure, we know that the remaining intervals satisfy the expected property, i.e. for any t inside, there is no jump greater than 2 -jn+2 closer than 2 -jn+1δ .

In order to determine the number of this type of intervals, we need to estimate the law of |S n+1 | conditionally to |S n |. For any n ∈ N, let us denote by p n+1 the probability of obtaining at least one jump greater 2 -jn+2 inside an interval of size 2 -jn+1δ . Note that for every n sufficiently large, an interval of size 2 -jnδ can be divided in at least 2 -jnδ /2 -jn+1δ h(p n+1 ) = 2 jn+1(δ-δ ) h(p n+1 ) sub-intervals. Furthermore, let M n+1 designate the following random variable:

M n+1 = # family ≥ 1/p n+1 LL(1/p n+1 ) of consecutive empty intervals of size 2 -jn+1 .
According to Lemma 2.9,

P M n+1 ≥ s 0 2 jn+1δ p n+1 LL(1/p n+1 )h(p n+1 ) 5 |S n | ≥ s 0 2 jnδ ≥ 1 -exp -s 0 2 jn+1δ p n+1 /8 LL(1/p n+1 ) 2 ,
for any s 0 ∈ R + such that s 0 2 jnδ ≥ 1. As previously outlined, for every collection of consecutive empty intervals, we remove the extremal elements to constitute the family S n+1 . Noting that 1/p n+1 LL(1/p n+1 ) -2 ≥ h(p n+1 )/p n+1 LL(1/p n+1 ) for any n sufficiently large, we therefore obtain

P |S n+1 | ≥ s 0 2 jn+1δ h(p n+1 ) 6 |S n | ≥ s 0 2 jnδ ≥ 1 -exp -s 0 2 jn+1δ p n+1 /8 LL(1/p n+1 ) 2 .
Furthermore, the probability of obtaining an empty interval of size 2 -jn+1δ is equal to:

q n+1 := P J [0, 2 -jn+1δ ] × D 2 -jn+2 , 2 -jn+1 = 0 exp -c 2 jn+2β-jn+1δ = exp -c 2 -jn+1δ(1-β/δ ) -→ 1.
Hence, p n+1 = 1 -q n+1 2 -jn+1δ(1-β/δ ) for any n sufficiently large, and there exists c 1 > 0 such that

P |S n+1 | ≥ s 0 2 jn+1δ h(p n+1 ) 6 |S n | ≥ s 0 2 jnδ ≥ 1 -exp -c 1 s 0 2 jn+1βδ/δ /(n + 1) 2 .
Furthermore, note 2 jn+1βδ/δ -jnδ = 2 jn+1(βδ/δ -δ ) . Since we have assumed that δ < √ βδ and s 0 2 jnδ ≥ 1, there exists r > 0 such that

P |S n+1 | ≥ s 0 2 jn+1δ h(p n+1 ) 6 |S n | ≥ s 0 2 jnδ ≥ 1 -exp -2 jn+1r .
Therefore, by induction, the law of |S n+m | satisfies

P |S n+m | ≥ s 0 2 jn+mδ m k=1 h(p n+k ) 6 |S n | ≥ s 0 2 jnδ ≥ m k=1 1 -exp -2 j n+k r .
Finally, we note that

m k=1 h(p n+k ) 6 ≥ exp -c m k=1 1/(n + k) ≥ exp -c log(n + m) , implying that P |S n+m | ≥ s 0 2 jn+mδ-c2 log(n+m) |S n | ≥ s 0 2 jnδ ≥ m k=1 1 -exp -2 j n+k r , (2.12)
where c 2 is a constant independent of n and m.

The previous bound gives us an estimate of the probability of obtaining intervals without any jump in a given neighbourhood. Using this estimate, we will be able to construct our main collection of nested intervals indexed by (T ) ∈N such that a most scales 2 -jn , there is no jump in the neighbourhood, and at specific ones 2 -j n( ) , a particular double-jump configuration appears. To construct this collection, let us first define this sequence n( ) ∈N :

n(0) = 1 and n( + 1) = 2 n( ) ∀ ∈ N.
For every ∈ N, we are interested in the following type of configuration: in an interval of size 2 -j n( ) δ /3, there exist two jumps ∆X u and ∆X v of opposite sign inside the middle third and such that |u -v| ≤ 2

-j n( ) γ , |∆X u |, |∆X v | ∈ [2 -j n( ) -1 , 2 -j n( ) ] and |∆X v -∆X u | ≤ 2 -j n( ) ρ .
Using the independence property on the Poisson measure J, the probability r of the previous event can be lower bounded by

r exp -c 2 -j n( ) δ+j n( )+1 β • 2 -j n( ) (δ-α1) • 2 -j n( ) (γ-α2-1+ρ) 2 -j n( ) (δ+γ-α1-α2-1+ρ) .
The collection of intervals T is constructed by induction. T 0 is initialised with the singleton corresponding to the interval [0, 1]. Then, assuming T is defined, for any I ∈ T , we consider the sub-intervals of size 2 -j n( +1) δ with a double-jump configuration and with no jump in the neighbourhood at all intermediate scales 2 -jn , n( ) < n < n( + 1). Note that if none satisfy the previous conditions, we avoid the extinction of the tree by selecting a single sub-interval of size 2 -j n( +1) δ .

We aim to estimate the size of T +1 conditionally to T . For any I ∈ T , we denote by c(I) the middle point between the two jump times inside I. Then, for every integer k ∈ [j n( ) δ, j n( ) δ ], we want to estimate the number double-jumps configurations inside the interval of size 2 -k :

I k := c(I) -2 -k , c(I) -2 -k-1 ∪ c(I) + 2 -k-1 , c(I) + 2 -k .
(2.13)

We designate by S n( )+1,k the number of sub-intervals of size 2 -j n( )+1 δ inside I k which are empty. Using Chernoff's inequality, the latter satisfies

P |S n( )+1,k | ≥ 2 j n( )+1 δ-k-2 ≥ 1 -exp -2 j n( )+1 δ-k-4 ≥ 1 -exp -2 j n( )+1 r .
The last inequality is due to j n( )+1 δ -k ≥ j n( )+1 δ -j n( ) δ > j n( )+1 r, if r is sufficiently small. Therefore, using the estimates obtained previously,

P |S n( +1)-1,k | ≥ 2 j n( +1)-1 δ-c3n( )-k ≥ n( +1)-1 k=1 1 -exp -2 j n( )+k r ≥ 1 -exp -2 j n( ) r n( +1) ,
where c 3 is a positive constant and we recall that n( + 1) = 2 n( ) . An interval of size 2 -j n( +1)-1 δ can be divided in at least 2 j n( +1) (δ-δ )-1 sub-intervals. Hence, if M ,k denotes the number of double-jump configurations existing among the sub-intervals of S n( +1)-1,k , Lemma 2.10 and the estimate of r induce that

P |M ,k | ≥ 2 -j n( +1) (δ+γ-α1-α2-1+ρ) • 2 j n( +1) δ-c3n( )-k-4 ≥ 1 -exp -2 j n( ) r n( +1)+1 .
We observe that the intervals I k are disjoints for different integers k. Hence, the probability of the intersection of the previous event for every k ∈ [j n( ) δ, j n( ) δ ] satisfies

P j n( ) δ k≥j n( ) δ |M ,k | ≥ 2 j n( +1) (α1+α2-γ+1-ρ)-c4n( )-k ≥ 1 -exp -2 j n( ) r (n( +1)+1)2βj n( ) ,
since we assume that δ ≤ α 1 + α 2 ≤ 2β. The previous construction procedure leads to estimate of size of T +1 . Therefore, conditionally to the event |T | ≥ k 0 2 j n( ) (α1+α2-γ+1-ρ)-2j n( -1) δ , we obtain

P |T +1 | ≥ k 0 2 j n( +1) (α1+α2-γ+1-ρ)-2j n( ) δ |T | ≥ k 0 2 j n( ) (α1+α2-γ+1-ρ)-2j n( -1) δ ≥ 1 -exp -2 j n( ) r (n( +1)+1) 2βj n( ) 2 j n( )c ≥ 1 -exp -2 j n( ) r 2 j n( ) c 5 .
For any 0 ∈ N, we know that the construction ensures that |T 0 | ≥ 1 almost surely. Hence, choosing k 0 = 2 -c6j n( 0 ) , with the proper constant c 6 , we obtain that the following lower bound

P > 0 |T | ≥ 2 j n( ) (α1+α2-γ+1-ρ)-2j n( -1) δ-c6j n( 0 ) ≥ > 0 1 -exp -2 j n( -1) r 2 j n( -1) c 5 .
Considering the logarithm of the right term, we observe

> 0 log 1 -exp -2 j n( -1) r 2 j n( -1) c 5 ≥ - > 0 2 j n( -1) c5 • exp -2 j n( -1) r -→ 0→∞ 0.
Hence, the previous probability converges to 1 for any set of parameters p = (δ, δ , δ , ρ).

Since the family of events considered is increasing with 0 , it implies that almost surely there exists 0 (ω) such that |T | ≥ 2 j n( -1) (α1+α2-γ+1-ρ)-2j n( ) δ-c6j n( 0 ) for all > 0 (ω). Furthermore, owing to the construction procedure described previously, we also know that for any I ∈ T , every interval I k defined in Equation (2.13) contains at least 2 j n( +1) (α1+α2-γ+1-ρ)-c4n( )-k proper double-jump configurations.

Hausdorff dimension (lower-bound). The previous estimates now allow us to study more precisely the oscillating behaviour of the Lévy process. Suppose h ∈ (1/(α 1 + α 2 ), 1/β) and p is a set of parameters such that δ < 1/h < δ and 1/h < γ. Then, let us define the set of interest G(h, p) as following:

G(h, p) = > 0 I∈T c(I) -2 -j n( ) /h+1 , c(I) -2 -j n( ) /h-1 ∪ c(I) + 2 -j n( ) /h-1 , c(I) + 2 -j n( ) /h+1 , ( 2.14)
where c(I) still denotes the middle point of any double-jump interval I ∈ T and 0 (ω) corresponds to the random index previously defined. Owing to the construction of the tree T , we note that G(h, p) corresponds to to the intersection of collections T ,h of nested intervals of size 3 • 2 -j n( ) /h-1 . Furthermore, according to the estimates obtained in the previous paragraph, we know that every I ∈ T ,h contains at least 2 j n( +1) (α1+α2-γ+1-ρ)-c4n( )-j n( ) /h-1 sub-elements separated by 2 j n( +1) (γ-α1-α2-1+ρ) . In order the estimate the Hausdorff dimension of the set G(h, p), we construct by induction a mass measure µ on it. To begin with, µ 0 attributes an equivalent weight on every interval I ∈ T 0,h . Then, similarly to the procedure on Cantor's set, µ +1 is defined on the intervals I ∈ T +1,h such that the weight µ (I), I ∈ T ,h , is equally distributed on its offspring. The measure µ is then defined as the limit of the sequence (µ ) ≥ 0 , which clearly exists since the cumulative distribution functions uniformly converge on [0, 1].

Since every I ∈ T ,h contains at least 2 j n( +1) (α1+α2-γ+1-ρ)-c4n( )-j n( ) /h elements,

∀I ∈ T ,h ; µ(I) ≤ k= 0+1 2 -j n(k) (α1+α2-γ+1-ρ)+c4n(k-1)+j n(k-1) /h ≤ 2 -j n( ) (α1+α2-γ+1-ρ)+c7j n( -1) ,
since we note that n( ) ≤ j n( ) and k=1 j n(k) ≤ c j n( ) for any ∈ N.

As we aim to use the usual mass distribution principle to determine a gauge function g such that the Hausdorff measure H g of G(h, p) is positive, we need to obtain an upper bound of µ B(t, r) for any t ∈ [0, 1] and r > 0 sufficiently small. There exists ∈ N such that 2 -j n( ) /h ≤ r < 2 -j n( -1) /h and without any loss of generality, we may assume that > 0 . Furthermore, as r < 2 -j n( -1) /h , we may also suppose that B(t, r) ⊂ I where I ∈ T -1,h (otherwise, consider the intersection B(t , r ) between B(t, r) and the closest element I). Since the sub-intervals I ∈ T ,h , with I ⊂ I are separated by at least 2 j n( +1) (γ-α1-α2-1+ρ) , we know that the ball B(t, r) intersects with at most r 2 j n( ) (γ-α1-α2-1+ρ)+1 of them. Hence, since µ(I) has the same value for every I ∈ T ,h with I ⊂ I, we obtain µ B(t, r) ≤ r 2 j n( ) (γ-α1-α2-1+ρ)+1 µ(I) ≤ r 2 j n( -1) /h+c4n( -1)+1 µ(I), as µ(I) ≤ µ(I)/2 j n( ) (α1+α2-γ+1-ρ)-c4n( -1)-j n( -1) /h . In addition, we know that µ(I) ≤ 2 -j n( -1) (α1+α2-γ+1-ρ)+c7j n( -2) and j n( -2) ≤ n( -1), inducing µ B(t, r) ≤ r 2 -j n( -1) (α1+α2-γ+1-ρ)+j n( -1) /h+c8n( -1) , Furthermore, as γ > β, 1/h > β and ρ > 1,

α 1 + α 2 -γ -1/h + 1 -ρ < 0, 1 -(α 1 + α 2 - γ + 1 -ρ)h > 0 and r 1-(α1+α2-γ+1-ρ)h ≤ 2 -j n( -1) /h+j n( -1) (α1+α2-γ+1-ρ))
. Finally, since j n( -1) = δ/δ n( -1) ≤ c log 1/r , there exist c 9 , c 10 > 0 such that for all t ∈ [0, 1] and

r > 0 µ B(t, r) ≤ c 9 log(1/r) c10 r (α1+α2-γ+1-ρ)h .
Using the mass distribution principle (see [START_REF] Falconer | The geometry of fractal sets[END_REF] for instance), this inequality proves that G(h, p) has a positive g-Hausdorff measure, where the gauge function g is defined by g(r) = log(1/r) c10 r (α1+α2-γ+1-ρ)h . Therefore, if we restrict ourselves to rational parameters p, we have proved that with probability one, for all h ∈

(1/(α 1 + α 2 ), 1/β), dim H G(h, p) ≥ (α 1 + α 2 -γ + 1 -ρ)h.

2-microlocal frontier (lower-bound).

In this last step of the proof, we aim to show that the 2-microlocal frontier of every t ∈ G(h, p) has a chirp oscillation shape.

Let us set ω ∈ Ω and t ∈ G(h, p). As previously outlined in this work, we know that we may ignore the component of Lévy process which corresponds to the jumps of size greater than 2 -j n( 0 ) . Furthermore, owing to the construction of the set G(h, p), we know that for every ∈ N, the distance between t and the closest jump time s such that

|∆X s | ∈ [2 -j n( ) -1 , 2 -j n( ) ], satisfies 2 -j n( ) /h-2 ≤ |s -t| ≤ 2 -j n( ) /h+1 .
Therefore, owing to the characterisation (2.4) of the set E h , G(h, p) ⊂ E h , i.e. α X,t = h.

We aim to prove that the 2-microlocal frontier of X at t shows a chirp oscillation behaviour: σ X,t (s ) > h + s for all s < -h. Similarly to the proof of Theorem 1, we therefore investigate the regularity of the integral of X. In addition, we assume that β ≥ 1, as the proof in the other case β < 1 is completely similar.

Let us set u ∈ R, ε > 0 and h := h(1 + ε) > h. There exist m > 0 such that 2 -(m+1)/h < |t -u| ≤ 2 -m/h . Furthermore, let ∈ N be the greatest integer such that j n( ) ≤ m. We have to distinguish two different cases depending on the value of m.

Let us first suppose that j n(

) (1 + ε) ≤ m. Since 1/h > β, Lemma 2.2 implies that ∀v ∈ B(t, |u -t|); |X m v -X m t | ≤ c m2 -m ≤ c log |u -t| -1 |u -t| h . Furthermore, we note that |t -u| ≤ 2 -j n( ) (1+ε)/h = 2 -j n( ) /h ,
; |X v -X t | ≤ c log |u -t| -1 |u -t| h .
where H ∈ (0, 1), (a + , a -) ∈ R 2 \ (0, 0) and M α is an α-stable random measure on R with Lebesgue control measure λ and skewness intensity β α (•) ∈ [-1, 1]. Throughout this paper, it is assumed that β α is constant, and equal to zero when α = 1. In this context, for any Borel set A ⊂ R, the characteristic function of M α (A) is given by

E e iθMα(A) = exp -λ(A)|θ| α 1 -iβ α sign(θ) tan(απ/2) if α ∈ (0, 1) ∪ (1, 2); exp -λ(A)|θ| if α = 1.
For the sake of readability, we consider in the rest of the section the particular case (a + , a -) = (1, 0) (even though as stated [START_REF] Samorodnitsky | Stable non-Gaussian random processes[END_REF], the law of the process depends on values (a + , a -) chosen).

To begin with, we present in the next statement an alternative representation for the two-parameter field (t,

H) → X(t, H) = R (t -u) H-1/α + -(-u) H-1/α + M α (du).
In the case H ≥ H/α, the formula has been previously obtained by Takashima [START_REF] Takashima | Sample path properties of ergodic self-similar processes[END_REF]. Proposition 3. For all t ∈ R and H ∈ (0, 1), the random variable X(t, H) satisfies

X(t, H) a.s. =            C H R L u (t -u) H-1/α-1 + -(-u) H-1/α-1 + du if H ∈ 1 α , 1 ; L t if H = 1 α C H R (L u -L t )(t -u) H-1/α-1 + -L u (-u) H-1/α-1 + du if H ∈ 0, 1 α , ( 3.2

)

where C H = H -1/α and L is an α-stable Lévy process defined by

∀t ∈ R + L t = M α ([0, t]) and ∀t ∈ R -L t = -M α ([t, 0]).
Proof. For the sake of readiness, we present in the proof for any H ∈ (0, 1), even though the first case can be found in [START_REF] Takashima | Sample path properties of ergodic self-similar processes[END_REF]. Suppose t ∈ R and H ∈ (0, 1). Since (L t ) t∈R is an α-stable Lévy process, it has càdlàg sample paths. According to [START_REF] Applebaum | Lévy processes and stochastic calculus[END_REF] (chap. 4.3.4), the theory of the stochastic integration based α-stable Lévy processes coincide integrals with respect to α-stable random measure. Therefore, the r.v. X(t, H) is almost surely equal to R (tu)

H-1/α + -(-u)
H-1/α + dL u . Let ε > 0 and b < t. Using a classic integration by parts, we obtain

L t-ε ε H-1/α -L s (t -b) H-1/α = t-ε b (t -u) H-1/α dL u -H - 1 α t-ε b L u (t -u) H-1/α-1 du. (3.3) 1. If H ∈ 1 α , 1 , H -1/α > 0. Hence, t-ε b L u (t -u) H-1/α-1 du almost surely converges to t b L u-(t -u) H-1/α-1 du when ε → 0. Similarly, t-ε b (t -u) H-1/α dL u converges in L α (Ω).
(t -u) H-1/α -(-u) H-1/α dL u = C H t b L u (t -u) H-1/α-1 -(-u) H-1/α-1 du -L b (t -b) H-1/α -(-b) H-1/α . that t ∈ [0, 1]. Then, X t = C H t b L u (t -u) H-1/α-1 + du + C H 0 b L u (-u) H-1/α-1 + du + C H b -∞ L u (t -u) H-1/α-1 + -(-u) H-1/α-1 + du,
where b < 0 is fixed. The second term is simply a constant that does not influence the regularity. Similarly, using the dominated convergence theorem, we note that the third one is a smooth function on the interval [0, 1], and therefore has no impact on the 2-microlocal frontier. Therefore, we only need to focus on the first term. Let us define the process Y u = L u 1 {u≥t} . Since the 2-microlocal spaces and frontier presented in Definitions 2 and 3 are localised at a point t, we necessarily have σ L,t = σ Y,t . Furthermore, we note that

Z t := C H t b L u (t -u) H-1/α-1 + du = C H R Y u (t -u) H-1/α-1 + du = I H-1/α + Y (t)
Owing to the property of stability of 2-microlocal spaces under fractional integration (see see Theorem 1.1 in [START_REF] Jaffard | On the pointwise regularity of functions in critical Besov spaces[END_REF]), we obtain σ Z,t = σ Y,t + H -1/α, and therefore σ X,t = σ L,t + H -1/α.

2.

If H < 1/α, we first observe that according to the multifractal spectrum of alpha-stable processes and H > 0, dim H ({t ∈ R : α L,t ≤ 1/α -H}) < 1. Hence, for almost every ω ∈ Ω, Formula (3.2) is well-defined almost everywhere on R. Anywhere else, we may simply assume that X(t, H) is set to zero. We will explain later why the value 0 at these particular times does not modify the 2-microlocal frontier.

Similarly to the previous case H > 1/α, the regularity of X only depends on the behaviour of the component

Z : t -→ C H t b (L u -L t )(t -u) H-1/α-1 + du.
One might recognize a Marchaud fractional derivative (see e.g. [START_REF] Samko | Fractional integrals and derivatives[END_REF]). Let us modify this expression to exhibit a more classic form of fractional derivative. For almost all s ∈ [0, 1] and ε > 0, we have

s-ε b L u (s -u) H-1/α du = C H s-ε b L u du s u+ε (v -u) H-1/α-1 dv + ε H-1/α s-ε b L u du = C H s-ε b du s u+ε (L u -L v )(v -u) H-1/α-1 dv + ε H-1/α s-ε b L u du + C H s-ε b du s u+ε L v (v -u) H-1/α-1 dv
The last two terms are equal to for almost all t ∈ [0, 1]. Note that the last two formulas ensure that Z• ∈ L 1 loc (R), and thus X• ∈ L 1 loc (R) with probability one. Let us now explain in which sense we investigate the 2-microlocal regularity of X. As previously outlined in the introduction, in the case H < 1/α, sample paths of LFSM are nowhere bounded. As a consequence, it is meaningless to consider the usual Hölder regularity. On the other hand, the 2-microlocal formalism has been introduced in a more general frame which are distributions D (R). Since we have previously proved that X• ∈ L 1 loc (R), with probability one, X• is a distribution whose 2-microlocal frontier is well-defined. We refer to [START_REF] Meyer | Wavelets, vibrations and scalings[END_REF] for a complete presentation of the 2-microlocal spaces for distributions. Also note that in this context, we can modify the values of X t on the negligible set {t ∈ R : α L,t ≤ 1/α -H} without modifying X• in the sense of distributions. Then, let first consider the term Y : t → t b L u (t -u) H-1/α du. Since H -1/α > -1, it is a Riemann-Liouville fractional integral of order H -1/α + 1 > 0. Hence, using the techniques previously presented, we obtain that σ Y,t = σ L,t + H -1/α + 1. Furthermore, the almost everywhere derivative d dt t b L u (t -u) H-1/α du coincide with the derivative in the sense distribution. Still using the stability of 2-microlocal spaces, the 2-microlocal frontier of the latter is therefore equal to σ L,t + H -1/α. In addition, the 2-microlocal frontier of t → L t (t -b) H-1/α is equal to σ L,t (the multiplication with a locally smooth function having no effect). Hence, as σ L,t > σ L,t + H -1/α, we have proved that σ Z,t = σ L,t + H -1/α, and thus σ X,t = σ L,t + H -1/α with probability one. Therefore, in both cases, we have proved that with probability one and for all t ∈ [0, 1], ∀s ∈ R; σ X,t (s ) = σ L,t (s ) + H -1/α. Then, using the same reasoning as in the proof of Corollary 2, we observe that ∀s ∈ [0, 1/α]; E s-H+1/α (L) ⊆ E σ,s (X) ⊆ E s-H+1/α (L) ∪ h<s-H+1/α E h (L).

for any σ < H -1/α. Furthermore, since : is an alpha-stable process, Theorem 1 induces that dim H E s-H+1/α (L) = α s -H + 1/α = α(s -H) + 1

and dim H E h (L) < α(s + H) -1 for every h < s -H + 1/α. These two estimates clearly prove the spectrum presented in Equation (1.19). Finally, the spectrum of singularity for the weak scaling exponent is obtained similarly.

Another class of processes similar to the LFSM has been introduced and studied in [START_REF] Benassi | On roughness indices for fractional fields[END_REF][START_REF] Marquardt | Fractional Lévy processes with an application to long memory moving average processes[END_REF][START_REF] Cohen | A general framework for simulation of fractional fields[END_REF]. Named fractional Lévy processes, it is defined by

X t = 1 Γ(d + 1) R (t -u) d + -(-u) d + L(du),
where d ∈ (0, 1/2) and L is a Lévy process enjoying Q = 0 (no Brownian component), E[L(1)] = 0 and E[L(1) 2 ] < +∞. Owing to this last assumption on L, LFSMs are not fractional Lévy processes. Nevertheless, their multifractal regularity can be determined as well.

Proposition 4. Suppose X is a fractional Lévy process parametrized by d ∈ (0, 1/2). Then, with probability one and for all σ ≤ d,

∀V ∈ O; dim H (E σ,s ∩ V ) = β(s -d) if s ∈ d, d + 1 β ; -∞ otherwise. (3.4)
where β designates the Blumenthal-Getoor exponent of the Lévy process L. Furthermore, for all s ∈ R, E σ,s is empty if σ > d.

Proof. Marquardt [36] has established (Theorem 3.4) a representation of fractional Lévy processes equivalent to Proposition 3:

X t = 1 Γ(d) R L u (t -u) d-1 + -(-u) d-1 + du.
Based on this result, a straightforward adaptation of the proof of Theorem 3 yields Equation (3.4).

Similarly to the LFSM, this statement refines regularity results established in [START_REF] Benassi | On roughness indices for fractional fields[END_REF][START_REF] Marquardt | Fractional Lévy processes with an application to long memory moving average processes[END_REF] and proves that the multifractal spectrum of a fractional Lévy process is equal to

∀V ∈ O; d X (h, V ) = β(h -d) if h ∈ d, d + 1 β ; -∞ otherwise. (3.5)
Let us finally conclude this section with the proof of Theorem 4.

Proof of Theorem 4. Suppose (X t ) t∈R is a linear multifractional stable motion with α ∈ (1, 2) and Hurst function H(•) ∈ (1/α, 1). According to the representation obtained in Proposition 3, X t is almost surely equal to X(t, H(t)).

To begin with, we first use the uniform estimate of the local Hölder exponent obtained by Ayache and Hamonier [6,Th. 8.1] to obtain an upper bound on the 2-microlocal frontier. The latter have proved that with probability one and for all t ∈ R, α X,t = H(t) -1/α. In addition, the 2-microlocal frontier is known to satisfy the inequality σ X,t ≤ lim inf u→t α X,u for any t ∈ R, which proves that σ X,t ≤ H(t) -1/α with probability one.

Let us now set ω ∈ Ω and t ∈ R and decompose X into two parts: X u = X(u, H(t)) + X(u, H(u)) -X(u, H(t)) .

According to the proof of Theorem 3, we already that the 2-microlocal frontier of the first component X( • , H(t)) is equal to σ L,t + H(t) -1/α. As a consequence, we have to prove that the second term Y u := X(u, H(u)) -X(u, H(t)) is negligible in terms of 2-microlocal regularity. For that purpose, we observe that for any u, v ∈ B(t, ρ)

Y u -Y v = X(u, H(u)) -X(u, H(t)) -X(v, H(v)) + X(v, H(t)) = H(u) H(t) ∂ H X(u, h) dh - H(v) H(t) ∂ H X(v, h) dh.
Therefore, since H is δ-Hölderian,

|Y u -Y v | ≤ H(u) H(t) ∂ H X(u, h) -∂ H X(v, h) dh + H(v) H(u) ∂ H X(v, h) dh ≤ c |u -t| δ • |u -v| γ + c |u -v| δ
where γ < inf u∈B(t,ρ) H(u) -1/α and δ > sup u∈R H(u). Using Definition 2 of the 2microlocal spaces, this inequality proves that σ Y,t ≥ δ + s ∧ H -1/α for all s ∈ R. Since δ > H(t), σ X,t ≤ H(t) -1/α and σ L,t (s ) ≤ (1/α + s ) ∧ 0, we therefore obtain with probability one and for all t ∈ R ∀s ∈ R; σ X,t (s ) = σ L,t (s ) + H(t) -1/α. Then, the multifractal structure described in Equation (1.22) is obtained using the same arguments as in the proof of Theorem 3.

Remark 2.

In the case H(•) does not satisfy the assumption δ > sup t∈R H(t), the proof of Theorem 4 can be modified to extend the statement and generalize results obtained in [START_REF] Stoev | Path properties of the linear multifractional stable motion[END_REF]. This complete study is made in [START_REF] Balança | Sample paths properties of irregular multifractional Brownian motion[END_REF] for the multifractional Brownian motion. For the sake of clarity, we prefer to focus in this work on (H 0 )-Hurst functions and the multifractal structure of the LMSM presented in Theorem 4. Remark 3. Even though it is assumed all along this section that H(•) is deterministic, owing to the deterministic representation presented in Proposition 3, Theorems 3 and 4 still hold if H(•) is a continuous random process. Hence, based on these results, a class of random processes with random and non-homogeneous multifractal spectrum can be easily constructed. A similar extension of the multifractional Brownian motion has been introduced and studied by Ayache and Taqqu [START_REF] Ayache | Multifractional processes with random exponent[END_REF].

  1) and Y (t, k) = 0, proving that the Taylor development of Y (•, k) at t is P ≡ 0. Therefore, this construction procedure ensures that the difference between Y (•, k) and I k b+ Y corresponds to the polynomial function appearing in the definition of the 2-microlocal spaces.

  Therefore, using Equation (3.3) with t = 0 and b < 0, we obtain almost surely t b

LL

  v (v -b) H-1/α dv, which converges to s b L v (v -b) H-1/α dv as ε → 0 since H -1/α > -1.Similarly, as we consider times at which α L,s > 1/α -H, the dominated convergence theorem implies that the first term converges toC H s b dv v b (L u -L v )(v -u) H-1/α-1 du. Therefore, s b L u (s-u) H-1/α du = C H s b dv v b (L u -L v )(v-u) H-1/α-1 du+ s b L v (v-b) H-1/α dv.According to classic real analysis results, the previous expression is differentiable almost everywhere on the interval [0, 1], and therefore u (t -u) H-1/α du -L t (t -b) H-1/α ,

  The previous lemma will help us to obtain a uniform upper bound on the 2-microlocal frontier of Y . With probability one, for all t ∈ [0, 1], the 2-microlocal frontier of Y at t satisfies

	Lemma 2.4. ∀s ∈ R; σ Y,t (s ) ≤	1 β	+ s ∧ 0.	(2.6)

Proof. Let us first observe that to obtain an upper bound of the 2-microlocal frontier of the R d -valued Y = (Y 1 , . . . , Y d ) process, it is sufficient to prove this bound holds for one the component Y i . Furthermore, we also know that each of these components is a onedimensional Lévy process and there exists i ∈ {1, . . . , d} such that the Blumenthal-Getoor exponent of Y i is equal to β. Hence, considering these two remarks, we may assume without any loss of generality that we study only one component, and thus d = 1.

1 .

 1 If β ≥ 1, P t ≡ 0 and thus Y (u, 0) = Y u -Y t . Consider n ∈ N and first assume that Y (v n , 0) ≥ 2 -mn-1 . Let us also set t n = v n and ρ n = 2 -mnδ . Then, for all u ∈ [t n , t n + ρ n ):

  implying there is no jump time s such that |∆X s | ≥ 2 -m and s ∈ B(t, |u -t|). Using in addition the estimates on the drift obtained in Proposition 2, we obtain ∀v ∈ B(t, |u -t|)

Therefore, u t (X v -X t ) dv ≤ c log |u -t| -1 |u -t| 1+h , where we recall that h > h.

We now consider the second case j n( ) ≤ m ≤ j n( ) (1 + ε). As previously, we know that for all v ∈ B(t, |u -t|), |X m v -X m t | ≤ c log |u -t| -1 |u -t| h . Nevertheless, in this case, there might exist a double-jump of size 2 -j n( ) inside the interval B(t, |u -t|). Owing to the construction of the set G(h, p), the contribution of the double-jump to

In the previous exponents, we note that ρ > 1 and γ > 1/h, implying that ρh + 1 > h + 1 and h(1 + γ) > h + 1. Hence, we have proved there exists ε 0 > 0 such that

for all u in the neighbourhood of t. This last inequality proves that the regularity at t is singular, as the 2-microlocal frontier must satisfy

Therefore, with probability one, for all h ∈ (1/(α

Linear (multi)fractional stable motion

The linear fractional stable motion (LFSM) is a stochastic process that has been considered by several authors: Maejima [START_REF] Maejima | A self-similar process with nowhere bounded sample paths[END_REF], Takashima [START_REF] Takashima | Sample path properties of ergodic self-similar processes[END_REF], Kôno and Maejima [START_REF] Kôno | Hölder continuity of sample paths of some self-similar stable processes[END_REF], Samorodnitsky and Taqqu [START_REF] Samorodnitsky | Stable non-Gaussian random processes[END_REF], Ayache et al. [START_REF] Ayache | Linear fractional stable sheets: wavelet expansion and sample path properties[END_REF], Ayache and Hamonier [START_REF] Ayache | Linear Multifractional Stable Motion: fine path properties[END_REF]. Its general integral form is defined by

When b → -∞, the left-term clearly converges to X(t, H) in L α (Ω). According to [START_REF] Pruitt | The growth of random walks and Lévy processes[END_REF], we know that almost surely for any ε > 0, lim sup u→-∞ u 1/α+ε |L u | = 0. Furthermore, we also have (t

Therefore, as H < 1 and using the dominated convergence theorem, the right-term almost surely converges to the expected integral.

2. If H ∈ 0, 1 α , we observe that Equation (3.3) can be slightly transformed into

According to [START_REF] Pruitt | The growth of random walks and Lévy processes[END_REF], α Y,t a.s.

= 1/α. Therefore, up to an extracted sequence, the previous expression almost surely converges when ε → 0 and using a similar formula for t = 0, we obtain

The property lim sup u→-∞ u 1/α+ε |L u | = 0 and the previous equivalents finally prove Equation (3.2).

To end this proof, let us consider the integral representation in the particular case H = 1/α. In fact, Equation (3.2) is a slightly misuse since the expression does not exist. Nevertheless, we prove that it converges almost surely to X(t, 1/α) = L t when H → 1/α.

Suppose first that H 1/α and rewrite X(t, H) as

The first component of the expression converges to zero since C H → H→1/α 0 and α Y,t a.s. = 1/α. As the second part simply converges to L t , we get the expected limit. The case H 1/α is treated similarly. Note that Picard [START_REF] Picard | Representation formulae for the fractional Brownian motion[END_REF] has determined a similar representation for fractional Brownian motion.

Proof of Theorem 3. Let us set H ∈ (0, 1) and α ∈ [1, 2). In order to obtain the multifractal structure of the LFSM, we first relate the 2-microlocal frontier of X at t to the frontier of the alpha-stable process L.

1. If H > 1/α, we note that the representation obtained in Proposition 3 is defined almost surely for all t ∈ R. Therefore, let us set ω ∈ Ω and t ∈ R. As previously, we can assume