S T Éphane

Xiaochuan Yang
email: xiaochuan.yang@u-pec.fr

MULTIFRACTAL ANALYSIS FOR THE OCCUPATION MEASURE OF STABLE-LIKE PROCESSES

In this article, we investigate the local behaviors of the occupation measure µ of a class of real-valued Markov processes M, defined via a SDE. This (random) measure describes the time spent in each set A ⊂ R by the sample paths of M. We compute the multifractal spectrum of µ, which turns out to be random, depending on the trajectory. This remarkable property is in sharp contrast with the results previously obtained on occupation measures of other processes (such as Lévy processes), since the multifractal spectrum is usually deterministic, almost surely. In addition, the shape of this multifractal spectrum is very original, reflecting the richness and variety of the local behaviors. The proof is based on new methods, which lead for instance to fine estimates on Hausdorff dimensions of certain jump configurations in Poisson point processes.

Introduction

The occupation measure of a R d -valued stochastic process (X t ) t≥0 describes the time spent by X in any borelian set A ⊂ R d . It is the natural measure supported on the range of the process X, and plays an important role in describing the different fractal dimensions of the range of X. Local regularity results for the occupation measure and its density when it exists (often called local times if X is Markovian) yield considerable information about the path regularity of the process itself, see the survey article by Geman and Horowitz [START_REF] Geman | Occupation densities[END_REF] on this subject.

We describe the local behaviors of this occupation measure via its multifractal analysis. Multifractal analysis is now identified as a fruitful approach to provide organized information on the fluctuation of the local regularity of functions and measures, see for instance [START_REF] Jaffard | Wavelet techniques in multifractal analysis[END_REF][START_REF] Falconer | Fractal geometry[END_REF]. Its use in the study of pointwise regularity of stochastic processes and random measures has attracted much attention in recent years, e.g. (time changed) Lévy processes [START_REF] Jaffard | The multifractal nature of Lévy processes[END_REF][START_REF] Barral | The singularity spectrum of Lévy processes in multifractal time[END_REF][START_REF] Durand | Singularity sets of Lévy processes[END_REF][START_REF] Durand | Multifractal analysis of Lévy fields[END_REF][START_REF] Balança | Fine regularity of Lévy processes and linear (multi)fractional stable motion[END_REF], stochastic differential equations with jumps [START_REF] Barral | A pure jump Markov process with a random singularity spectrum[END_REF][START_REF] Yang | Multifractality of jump diffusion processes[END_REF][START_REF] Xu | The multifractal nature of boltzmann processes[END_REF], the branching measure on the boundary of a Galton-Watson tree [START_REF] Mörters | Thin and thick points for branching measure on a Galton-Watson tree[END_REF][START_REF] Mörters | On the multifractal spectrum of the branching measure on a Galton-Watson tree[END_REF], local times of a continuous random tree [START_REF] Berestycki | Beta-coalescents and continuous stable random trees[END_REF][START_REF] Balança | Uniform multifractal structure of stable trees[END_REF], SPDE [START_REF] Perkins | The multifractal structure of super-Brownian motion[END_REF][START_REF] Mytnik | Multifractal analysis of superprocesses with stable branching in dimension one[END_REF][START_REF] Khoshnevisan | Intermittency and multifractality: A case study via parabolic stochastic pdes[END_REF], Brownian and stable occupation measure [START_REF] Hu | The multifractal structure of stable occupation measure[END_REF][START_REF] Shieh | Logarithmic multifractal spectrum of stable occupation measure[END_REF][START_REF] Marsalle | Slow points and fast points of local times[END_REF][START_REF] Hu | Multifractal structure of a general subordinator[END_REF][START_REF] Dembo | Thick points for spatial Brownian motion: multifractal analysis of occupation measure[END_REF], amongst many other references.

The first author was partially supported by the ANR project MUTADIS, ANR-11-JS01-0009. The second author was supported by grants from Région Ile-de-France.

In this article, we obtain the almost-sure multifractal spectrum of the occupation measure of stable-like jump diffusions, which turns out to be random, depending on the trajectory. This remarkable property is in sharp contrast with the results previously obtained on occupation measures of other processes (such as Lévy processes), since the multifractal spectrum is usually deterministic, almost surely. In addition, the shape of this multifractal spectrum is very original, reflecting the richness and variety of the local behaviors. The proof is based on new methods, which lead for instance to fine estimates on Hausdorff dimensions of certain jump configurations in Poisson point processes.

We first introduce the class of processes we focus on. Definition 1. Let ε 0 > 0, and β : R → [ε 0 , 1 -ε 0 ] be a nowhere constant non-decreasing, Lipschitz continuous map. The stable-like processes M are pure jump Markov processes whose generator can be written as

Lf (x) = 1 0 (f (x + u) -f (x))β(x)u -1-β(x) du. (1)
Introduced by Bass [START_REF] Bass | Uniqueness in law for pure jump Markov processes[END_REF] in the late 80's by solving a martingale problem, this class of processes has sample paths whose characteristics change as time passes, which is a relevant feature when modeling real data (e.g. financial, geographical data). Roughly speaking, the stable-like processes behave locally like a stable process, but the stability parameter evolves following the current position of the process, see [START_REF] Barral | A pure jump Markov process with a random singularity spectrum[END_REF] or [START_REF] Yang | Multifractality of jump diffusion processes[END_REF] for an explanation from the tangent processes point of view.

Let M = {M t , t ∈ [0, 1]} be a stable-like process associated with a given function x → β(x) as in Definition 1. Our purpose is to describe the local behaviors of the occupation measure of M defined as

µ(A) = 1 0 1 A (M t )dt. ( 2
)
It depicts how long M stays in any Borel set A ⊂ R. We investigate the possible local dimensions for µ, as well as its multifractal spectrum. Let us recall these notions. Definition 2. Let ν be a positive measure on R, whose support is Supp (ν) := {x ∈ R : ∀ r > 0, ν(B(x, r)) > 0}. The upper local dimension of ν at the point x ∈ Supp (ν) is defined by dim(ν, x) = lim sup r↓0 log ν(B(x, r)) log r .

Similarly, the lower local dimension of ν at x is dim(ν, x) = lim inf r↓0 log ν(B(x, r)) log r .

When dim(ν, x) and dim(ν, x) coincide at x, the common value is denoted by dim(ν, x), the local dimension of ν at x. Our aim is to investigate two multifractal spectra of the occupation measure µ associated with stable-like processes, related to these local dimensions. Let dim H stand for the Hausdorff dimension in R, with the convention that dim H (∅) = -∞. The first multifractal spectrum (in space) is defined as follows.

Definition 3. Let O be an open set in R, and ν a Borel measure on R. Consider the level sets

E ν (O, h) = {x ∈ O ∩ Supp (ν) : dim(ν, x) = h} E ν (O, h) = {x ∈ O ∩ Supp (ν) : dim(ν, x) = h}.
The upper and lower multifractal spectrum of ν are the mappings

d ν (O, •) : h → dim H E ν (O, h), d ν (O, •) : h → dim H E ν (O, h).
The famous paper by Hu and Taylor [START_REF] Hu | The multifractal structure of stable occupation measure[END_REF] states that for every α-stable subordinator L α whose occupation measure is denoted by µ α , almost surely for all x ∈ Supp µ α , [START_REF] Balança | Uniform multifractal structure of stable trees[END_REF] dim(µ α , x) = α and dim(µ α , x) ∈ [α, 2α].

It is a classical result [START_REF] Blumenthal | Some theorems on stable processes[END_REF] that when α ∈ (0, 1), the image of any interval I by L α has Hausdorff dimension α, almost surely. This implies that the support of µ α has Hausdorff dimension α, almost surely.

With all this in mind, the lower spectrum of µ α is trivial: for each open interval O intersecting Supp (µ), one has d µα (O, h) = α when h = α, -∞ when h = α. Hu and Taylor also prove that the upper spectrum is much more interesting (see Figure 1): Almost surely, for each open interval O that intersects Supp (µ α ), [START_REF] Barral | A pure jump Markov process with a random singularity spectrum[END_REF]

d µα (O, h) = g α (h) := α 2α h -1 when h ∈ [α, 2α], -∞ otherwise.
Our first result gives the possible values for the local dimensions of the occupation measure µ associated with a stable-like process M.

Theorem 1. Consider a stable-like process M associated to a non-decreasing mapping β, as in Definition 1, and the associated occupation measure µ. With probability 1, for every x ∈ Supp (µ), dim(µ, x) = β(x) and dim(µ, x) ∈ [β(x), 2β(x)].

Hence, the support of the lower spectrum d µ is random, depending on the trajectory of M.

The space lower multifractal spectrum is then quite easy to understand, since the level set E µ (O, h) contains either one point or is empty, depending on whether h belongs to the closure of the range of the index process {β(M t ) : M t ∈ O} or not. Theorem 1 indicates that the spectrum related to the upper local dimension dim(µ, •) should be more interesting. This is indeed the case, as resumed in Theorem 2. Set

ĝα (h) := α 2α h -1 when h ∈ [α, 2α), -∞ otherwise.
Note that the only difference between g and ĝ is at the value h = 2α.

Definition 4. For every monotone càdlàg function Υ : R + → R, we denote by S(Υ) the set of jumps of Υ.

Theorem 2. Set the (at most countable) sets of real numbers

E 1 = {β(M t ) : t ∈ S(M) and β(M t ) ≥ 2β(M t-)}, E 2 = {2β(M t-) : t ∈ S(M) and β(M t ) ≥ 2β(M t-)}, E = E 1 ∪ E 2 . ( 5
)
With probability 1, for every non-trivial open interval O ⊂ R, one has

d µ (O, h) = 0 if h ∈ {β(M t ) : M t ∈ O}, -∞ otherwise, (6)
and for every h ∈ R + \ E,

d µ (O, h) = sup ĝα (h) : α ∈ {β(M t ) : M t ∈ O} . (7) Remark 1. If the range of β(•) is included in, for example, [1/2, 9/10], then the set of exceptional values E = ∅ a.s.
First, one shall notice that both spectra are random, depending on the trajectory and on the interval O. In this sense, d µ (O, •) and d µ (O, •) are inhomogeneous, contrarily to what happens for the occupation measure µ α of α-stable subordinators (the spectra do not depend on O).

One shall interpret the space upper spectrum as the supremum of an infinite number of space multifractal spectra of "locally α-stable processes" for all values α ∈ {β(M t ) : M t ∈ O}. This formula finds its origin in the fact that locally, M behaves around each continuous time t as an α-stable process with α = β(M t ). The allure of a typical space upper multifractal spectrum is depicted in Figure 2. This shape is very unusual in the literature.

h d O,µ (h) 0 β(M t ) Figure 2.
First, observe that, since β and M are increasing maps, when t 0 ∈ S(M) is a jump time for M, then the "local" index of M jumps at t 0 from β(M t 0 -) to β(M t 0 ), and for t ≥ t 0 , the only possibility to have d µ (O, M t ) = β(M t 0 ) is when t = t 0 . Similarly, when t < t 0 , it is not possible to have

d µ (O, M t ) = 2β(M t 0 -).
In particular there may be a "hole" in the support of d µ (O, •). Indeed, a quick analysis of the functions g α (•) shows that this happens when there is a time t 0 such that β(M t ) > 2β(M t-), which occurs with positive probability for functions β(•) satisfying 2ε 0 < 1 -ε 0 .

All this explains the set of exceptional points E in Theorem 2. We deal with these exceptional points in the following theorem, whose statement is rather long but whose proof follows directly from a careful analysis of the previous results.

1) h = β(M t ) > 2β(M t-). If M t / ∈ O, d µ (O, h) = -∞, otherwise one has d µ (O, h) = 0 if dim(µ, M t ) = h, -∞ if dim(µ, M t ) > h. (2) h = 2β(M t-) < β(M t ). If M t-/ ∈ O, d µ (O, h) = -∞, otherwise one has d µ (O, h) = 0 if dim(µ, M t-) = h, -∞ if dim(µ, M t-) < h. (3) h = β(M t ) = 2β(M t-). If {M t , M t-} ∈ O, one has d µ (O, h) = 0 if dim(µ, M t-) = h or dim(µ, M t ) = h, -∞ if dim(µ, M t-) < h and dim(µ, M t ) > h.
If only one of M t and M t-belongs to O (say, M t-), one has

d µ (O, h) = 0 if dim(µ, M t-) = h, -∞ if dim(µ, M t-) < h. If neither M t nor M t-belongs to O, one has d µ (O, h) = -∞.
It is also interesting to consider the time multifractal spectra related to L α and M, which describes the variation along time of the Hausdorff dimension of the set of times t such that µ α (or µ) has a local dimension h at x = L α t (or M t ). By abuse of language, one says in this case that µ (or µ α ) has a dimension h at t. For this let us introduce other level sets.

Definition 5. For every open set

O ⊂ [0, 1], set E t µ α (O, h) = {t ∈ O : dim(µ α , L α t ) = h}, E t µ (O, h) = {t ∈ O : dim(µ α , M t ) = h},
and the similar quantities for lower local dimensions E t µ α (O, h) and E t µ (O, h). The corresponding time multifractal spectra of µ α and µ are

d t µ α (O, •) : h → dim H E t µ α (O, h), d t µ (O, •) : h → dim H E t µ (O, h),
and the similar quantities for lower local dimensions.

In the case of a stable subordinator, Hu and Taylor prove that a.s., [START_REF] Bertoin | Lévy processes[END_REF]

d t µ α (O, h) = g α (h) α = 2α/h -1 when h ∈ [α, 2α], -∞ otherwise.
This time upper multifractal spectrum is homogeneous, in the sense that it does not depend on the choice of O. In this article, we also compute the time multifractal spectra of µ.

Theorem 4. Set the (at most countable) set of real numbers

E = {β(M t ) : t ∈ S(M) and β(M t ) ≥ 2β(M t-)}.
d t µ (O, h) = 0 if h ∈ {β(M t ) : t ∈ O}, -∞ otherwise, (9)
and for every h ∈ E ,

d t µ (O, h) = sup ĝα (h) α : α ∈ {β(M t ) : t ∈ O} . ( 10
) Remark 2. Note that E ⊂ E. If, for example, the range of β(•) is included in [1/2, 9/10], then almost surely E = ∅.
The first part is trivial. Observe that there is a subtle difference between ( 9) and ( 6), since at each jump time t for M, there is no s ∈ R such that

M s = M t-.
As for the space multifractal spectrum, there is some uncertainty about the value of d t µ (O, M t ) at the jump times t ∈ S(M), which is dealt with in the following theorem.

Theorem 5. With probability 1, for every non-trivial open interval

O ⊂ R, for every h ∈ {β(M t ) : t ∈ S(M)}, writing h = β(M t ) , one has d t µ (O, h) = d µ (O, h) β(M t-)
.

The correspondance between (4) and (8) follows from the fact that for every α-stable subordinator, almost surely for each measurable set

E ⊂ [0, 1], dim H (L α (E)) = α • dim H (E). ( 11
)
Up to a countable number of points, writing O = (g, d), one has the equality

L α E t µ α (O, h) = E µ α (L α g , L α d ), h .
The method developed by Hu and Taylor consists first in proving [START_REF] Bertoin | Lévy processes[END_REF], and then in applying [START_REF] Durand | Singularity sets of Lévy processes[END_REF] to get [START_REF] Barral | A pure jump Markov process with a random singularity spectrum[END_REF].

Following these lines, we start by proving Theorem 4. The original methods by Hu and Taylor do not extend here, and an alternative way to compute the time multifractal spectrum of µ is needed. For this, some scenario leading to the fact that µ has exactly an upper local dimension equal to h at x = M t is identified. More precisely, it will be proved that d µ (O, M t ) = h when t is infinitely many times very closely surrounded by two "large" jump times for the Poisson point process involved in the construction of M. Using this property, we build in Section 6 a (random) Cantor set of such times t with the suitable Hausdorff dimension. The difficulty lies in the fact that the expected Hausdorff dimension is random and depends on the interval we are working on.

Further, it is natural to look for a uniform dimension formula such as [START_REF] Durand | Singularity sets of Lévy processes[END_REF] for the stable-like jump diffusion process M. As is pointed out by Hu and Taylor [START_REF] Hu | Multifractal structure of a general subordinator[END_REF] (see also page 94 of [START_REF] Bertoin | Lévy processes[END_REF]), as long as the Laplace exponent of a general subordinator oscillates at infinity, i.e. it exhibits different power laws at infinity, one can never expect such an identity to hold. Nevertheless, using an argument based on coupling and time change, we find upper and lower dimension bounds for images of sets by stable-like processes. Theorem 6. Almost surely, for every mesurable set E ⊂ [0, 1], one has

(12) dim H M(E) ∈ dim H (E) • inf t∈E β(M(t)), sup t∈E β(M(t-)) .
Moreover, if the set E satisfies that for every non-trivial subinterval O ⊂

[0, 1], dim H (E) = dim H (E ∩ O), then dim H M(E) = dim H (E) • sup t∈E β(M(t-)).
These results are fine enough for us to deduce Theorem 2 from Theorem 4, even when the stability parameter of these processes varies as time passes. Also, this theorem solves partially a question left open in [START_REF] Yang | Hausdorff dimension of the range and the graph of stable-like processes[END_REF].

Let us end this introduction with a proposition describing the typical behavior of the occupation measure µ. We skip the proof since it can be deduced by adapting the arguments in the proof of our Theorem 7.

Proposition 1 (Typical Behavior). With probability 1, one has:

• for Lebesgue-almost every time t

∈ [0, 1], dim(µ, M t ) = β(M t ). • for µ-almost every point x ∈ R, dim(µ, x) = β(x).
The rest of this paper is organized as follows. We start by recalling basic properties of the stable-like processes in Section 2. The local dimensions of the occupation measure (Theorem 1) are studied in Section 3. The time spectrum (Theorems 4 and 5) is obtained in Section 4 using a general result (Theorem 7), whose proof is given in Sections 5 and 6. Finally, the space spectrum (Theorem 2) is dealt with in Section 7, together with the dimension of images of arbitrary sets by stable-like processes (Theorem 6).

Preliminaries

First of all, stable-like processes admit a Poisson representation which was regularly used to study path properties of such processes, see for instance [START_REF] Barral | A pure jump Markov process with a random singularity spectrum[END_REF][START_REF] Yang | Multifractality of jump diffusion processes[END_REF][START_REF] Yang | Hausdorff dimension of the range and the graph of stable-like processes[END_REF]. Let us recall this representation and a coupling associated with it which will be useful for our purposes.

Let N (dt, dz) be a Poisson measure on R + × R with intensity dt ⊗ dz/z 2 . Such a measure can be constructed from a Poisson point process which is the set of jumps of a Lévy process with triplet (0, 0, dz/z 2 ), see for instance Chapter 2 of [START_REF] Applebaum | Lévy processes and stochastic calculus[END_REF]. We denote

F t = σ ({N (A) : A ∈ B([0, t] × [0, +∞))}).
Recall the definition of a stable-like process and formula [START_REF] Applebaum | Lévy processes and stochastic calculus[END_REF]. The existence and uniqueness of such jump diffusion processes is classical and recalled in the next proposition. Observe that by the substitution u = z 1/β(x) (for each fixed x), the generator of a stable-like process is rewritten as

Lf (x) = 1 0 f (x + z 1 β(x) ) -f (x) dz z 2 . ( 13
)
Proposition 2. Let N be as in the last paragraph.

(1) There exists a unique càdlàg (F t ) t∈[0,1] -adapted solution to

(14) M t = t 0 1 0 z 1 β(M u-) N (du, dz).
Furthermore, M is an increasing strong Markov process with generator L given by (13). (2) For every α ∈ (0, 1), we define

L α t = t 0 1 0 z 1 α N (du, dz). ( 15
)
Then for all α ∈ (0, 1), {L α t , t ∈ [0, 1]} is an α-stable subordinator whose jumps larger than 1 are truncated.

Classical arguments based on Gronwall inequality and Picard iteration yield the first item. For a proof, see Proposition 13 of [START_REF] Barral | A pure jump Markov process with a random singularity spectrum[END_REF] or Proposition 2.1-2.3 of [START_REF] Yang | Multifractality of jump diffusion processes[END_REF] with some slight modifications. The second item is standard, see for instance Section 2.3 of [START_REF] Applebaum | Lévy processes and stochastic calculus[END_REF].

Remark 3. Recall that S(Υ) is the set of jumps of a monotone càdlàg function Υ : [0, 1] → R + .
Observe that by construction, almost surely, the processes M and the family of Lévy processes (L α ) α∈(0,1) are purely discontinuous, increasing, with finite variation, and that they jump at the same times, i.e. S(M) = S(L α ).

Next observation is key for the study of the local dimensions of µ. Proposition 3. Consider the process M and L α for all α ∈ (0, 1) introduced in Proposition 2. Almost surely, for all 0 ≤ s ≤ t ≤ 1,

0 ≤ L β(Ms) t -L β(Ms) s ≤ M t -M s ≤ L β(M t-) t -L β(M t-) s .
This is intuitively true because we construct simultaneously M and L α such that they jump at the same times, and the jump size of L α is always larger than L α whenever α > α . See Proposition 14 of [START_REF] Barral | A pure jump Markov process with a random singularity spectrum[END_REF] for a proof.

Local dimensions of µ : Proof of Theorem 1

Observe that almost surely, for all α ∈ Q ∩ (0, 1), formula (3) is true. This, together with Proposition 3, leads to the local dimension of µ. Proof of Theorem 1 : Three cases may occur.

(1) x = M t with t a continuous time of M. Due to the coupling in their construction (Proposition 2), almost surely, for every α, the process L α is also continuous at t. By continuity, for arbitrary rational numbers α, α ∈ (0, 1) satisfying α < β(M t ) < α , there exists a small δ > 0 such that for all s ∈ (t -δ, t + δ), α < β(M s ) < α . Using the occupation measure µ α of the process L α , and applying Proposition 3 to L α and L α , one gets when r is small

µ α (L α t -r, L α t + r) ≤ µ (M t -r, M t + r) ≤ µ α (L α t -r, L α t + r) (16)
By formula (3) for the lower and upper local dimensions of µ α , for all small ε > 0, almost surely, one has for r small enough that

α -ε ≤ log µ α (L α t -r, L α t + r) log(r) ≤ 2α + ε,
and the same for α . Hence log µ

(M t -r, M t + r) log(r) ≥ log µ α (L α t -r, L α t + r) log(r) ≥ α -ε. and log µ (M t -r, M t + r) log(r) ≤ log µ α (L α t -r, L α t + r) log(r) ≤ 2α + ε. Therefore, α -ε ≤ dim(µ, x) ≤ dim(µ, x) ≤ 2α + ε.
On the other hand, still by formula (3), dim(µ α , x) = α , so there exists a sequence (r n ) converging to 0 such that

α -ε ≤ log µ α (L α t -r n , L α t + r n ) log(r n ) ≤ α + ε, so α -ε ≤ dim(µ, x) ≤ α + ε.
Letting ε tend to zero and α, α tend to β(M t ) with rational values yields

β(M t ) = dim(µ, x) ≤ dim(µ, x) ≤ 2β(M t ).
(2) x = M t with t a jump time for M. Observe that in this case µ (x -r, x + r) = µ (M t , M t + r) for r > 0 small enough. For arbitrary rational numbers α < β(M t ) < α , the inequality ( 16) is straightforward using Proposition 3. We follow the same lines in the first case to obtain the desired result.

(

) x = M t-with t a jump time for M. Now, µ (x -r, x + r) = µ (M t--r, M t-) 3
for r > 0 small. Then the proof goes like the previous items.

Let us end this Section with the proof of the easier part of Theorem 2 : space lower multifractal spectrum of µ. Proof of Formula (6) of Theorem 2 : As noticed in Remark 3, t → β(M t ) is increasing due to the monotonicity of M and β. Hence each level set of t → β(M t ) contains at most one point. This means that for each open interval O that intersects Supp (µ),

E µ (O, h) = {x ∈ Supp (µ) ∩ O : β(x) = h} =      {β(M t )} if h = β(M t ) for some t with M t ∈ O, {β(M t-)} if h = β(M t-) for some t with M t ∈ O, ∅ if h ∈ {β(M t ) : M t ∈ O},
which completes the proof.

A general result to get the time spectrum (Theorems 4 and 5)

Let us present a general result, proved in Sections 5 and 6. This theorem gives the dimension of the random set of times t where the local dimension mapping s → dim(µ, M s ) coincides with a given function. The remarkable feature of this theorem is that it allows to determine these dimensions for all the monotone càdlàg function simultaneously, with probability one.

Theorem 7. For every non-increasing càdlàg function

Υ : [0, 1] → [1, 2]
and every open interval O ⊂ [0, 1], let us define Υ min = inf u∈O Υ(u) and the sets

E t µ (O, Υ) = t ∈ O : dim(µ, M t ) = Υ(t)β(M t ) , E t,≥ µ (O, Υ) = t ∈ O : dim(µ, M t ) ≥ Υ(t)β(M t ) . With probability 1, for every non-increasing càdlàg function Υ : [0, 1] → [1, 2] and every open interval O ⊂ [0, 1], we have (17) dim H E t µ (O, Υ) = dim H E t,≥ µ (O, Υ) = dim H E t,≥ µ (O, Υ min ) = 2 Υ min -1.
The notation E t,≥ µ (O, Υ min ) means that we consider the constant function Υ ≡ Υ min . 4.1. Proof for the time upper multifractal spectrum. Let us explain why Theorems 1 and 7 together imply Theorems 4 and 5. One wants to prove formula [START_REF] Dembo | Thick points for spatial Brownian motion: multifractal analysis of occupation measure[END_REF], which can be rewritten as [START_REF] Jaffard | Wavelet techniques in multifractal analysis[END_REF]

d t µ (O, h) = sup 2α h -1 ∈ [0, 1) : α ∈ {β(M t ) : t ∈ O} .
One combines Theorem 1 and Theorem 7 with the family of functions {Υ h (t) = h/β(M t ) : h ≥ 0}. With probability one, these functions are all càdlàg decreasing. Observe that for every open interval O,

dim H E t µ (O, Υ h ) = dim H t ∈ O : dim(µ, M t ) = Υ h (t)β(M t ) = dim H t ∈ O : dim(µ, M t ) = h = d t µ (O, h).
We prove now that formula [START_REF] Jaffard | The multifractal nature of Lévy processes[END_REF] applied to the family {Υ h : h ≥ 0} implies formula [START_REF] Jaffard | Wavelet techniques in multifractal analysis[END_REF]. Several cases may occur according to the value of h. [START_REF] Balança | Fine regularity of Lévy processes and linear (multi)fractional stable motion[END_REF]. By Theorems 1 and 7, one gets

• If {1} ⊂ Υ h (O): (1) First case: For all t ∈ O, Υ h (t) > 1. • If inf t∈O Υ h (t) ≥ 2, then for all t ∈ O, Υ h (t) > 2. Theorem 1 entails E t µ (O, Υ h ) = ∅. So d t µ (O, h) = dim H E t µ (O, Υ h ) = -∞, which coincides with (18). • If inf t∈O Υ h (t) < 2, consider the entrance time in (1, 2) by Υ h τ = inf{t ∈ O : Υ h (t) < 2}. By construction, ∀ t ∈ (τ, ∞) ∩ O, Υ h (t) ∈ (1,
d t µ (O, h) = d t µ ((τ, ∞) ∩ O, h) = 2 inf t∈(τ,∞)∩O Υ h (t) -1 = 2 sup t∈(τ,∞)∩O β(M t ) h -1,
which coincides with ( 18).

(2) Second case: There exists t ∈ O, such that Υ h (t) < 1. Define the passage time of (-∞, 1) by Υ h as [START_REF] Jaffard | Wavelet techniques in multifractal analysis[END_REF]. • If {1} ⊂ Υ h (O): Let t 0 ∈ O be the unique time such that Υ h (t 0 ) = 1, i.e. h = β(M t 0 ). One distinguishes different cases according to the behavior of t → M t at t 0 .

σ = inf{t ∈ O : Υ h (t) < 1}. • If σ is the left endpoint of O, then for all t ∈ O, Υ h (t) < 1. Theorem 1 yields E t µ (O, Υ h ) = ∅. Again, this gives d t µ (O, h) = -∞, which coincides with
(1) If M is continuous at t 0 : β(M • ) is also continuous at t 0 . By definition, the entrance time τ satisfies τ < t 0 . Theorem 7 entails

d t µ (O, h) = d t µ ((τ, t 0 ) ∩ O, h) = 2 inf t∈(τ,t 0 )∩O Υ h (t) -1 = 1,
which coincides with ( 18).

(

) If t 0 is a jump time for M and β(M t 0 -) < h = β(M t 0 ) < 2β(M t 0 -): Then, using that Υ h (t 0 ) = 1, one deduces that 0 < Υ h (t 0 -) -Υ h (t 0 ) < 1, which implies inf t∈(τ,t 0 )∩O Υ h (t) < 2. 2
The same computation as in item 1. with O replaced by (τ, t 0 ) ∩ O yields formula [START_REF] Jaffard | Wavelet techniques in multifractal analysis[END_REF].

(

) If t 0 is a jump time for M and h = β(M t 0 ) ≥ 2β(M t 0 -): Then Υ h (t 0 -) ≥ 2, thus τ = t 0 . One has E t (O, h) = {t 0 } if dim(µ, M t 0 ) = h, ∅ otherwise. 3
This last formula coincides with the one claimed by Theorem 5.

4.2.

Reduction of the problem. Observing that we have the obvious inclusion

E t µ (O, Υ) ⊂ E t,≥ µ (O, Υ) ⊂ E t,≥
µ (O, Υ min ), we proceed in two parts: • first, in Section 5, we show that [START_REF] Khoshnevisan | Intermittency and multifractality: A case study via parabolic stochastic pdes[END_REF] dim

H E t,≥ µ (O, Υ min ) ≤ 2 Υ min -1
simultaneously for all Υ and O.

In order to get [START_REF] Khoshnevisan | Intermittency and multifractality: A case study via parabolic stochastic pdes[END_REF], it is equivalent to show that, almost surely,

for each γ ∈ [1, 2] and open interval (a, b) ⊂ [0, 1] with rational endpoints, dim H E t,≥ µ (a, b), γ ≤ 2 γ -1.
We will actually prove that for γ ∈ (1, 2), almost surely, [START_REF] Marsalle | Slow points and fast points of local times[END_REF] dim

H E t,≥ µ (0, 1), γ ≤ 2 γ -1.
The extension to arbitrary a, b ∈ [0, 1] ∩ Q and γ ∈ {1, 2} is straightforward.

• second, in Section 6, we complete the result by proving that

(21) dim H E t µ (O, Υ) ≥ 2 Υ min -1
also simultaneously for all Υ and O, almost surely. It is also enough to get the result for O = (0, 1).

Proof of Theorem 7 : upper bound

Our aim is to prove [START_REF] Marsalle | Slow points and fast points of local times[END_REF]. For notational simplicity, we write E t,≥ µ (0, 1), γ = E(γ). Let us first observe that the family of sets {E(γ), γ ∈ [1, 2]} is nonincreasing with respect to γ. Recall that γ ∈ (1, 2) throughout this section.

The strategy is to find a natural limsup set which covers E(γ).

For this, we start by pointing out a property satisfied by all points in E(γ). Heuristically, it says that every t ∈ E(γ) is infinitely many times surrounded very closely by two points which are large jumps of the Poisson point process generating N . Proposition 4. With probability 1, the following holds: for every t ∈ E(γ) and every ε > 0 small, there exists an infinite number of integers

n ≥ 0 such that N ((t -2 -n , t] × [2 -n/(γ-ε) , 1] ≥ 1 and N ((t, t + 2 -n ] × [2 -n/(γ-ε) , 1] ≥ 1.
Proof Let t ∈ E(γ). This implies that [START_REF] Mörters | On the multifractal spectrum of the branching measure on a Galton-Watson tree[END_REF] lim sup

r→0 log µ (M t -r, M t + r) log r ≥ γ • β(M t ).
This equation is interpreted as the fact that the time spent by the process M in the neighborhood of M t cannot be too large. The most likely way for µ to behave like this is that M jumps into this small neighborhood of M t with a larger than normal jump, and quickly jumps out of that neighborhood with another big jump. This heuristic idea is made explicit by the following computations.

Lemma 1. Let 0 < ε < γ -1. If t ∈ E(γ)
, then there exist infinitely many integers n such that

|M t+2 -n -M t | ∧ |M t -M t-2 -n | ≥ 2 - n (γ-ε/4)β(M t ) . ( 23
)
Proof Let us prove that t satisfies [START_REF] Perkins | The multifractal structure of super-Brownian motion[END_REF] lim sup

s→0 + |M t+s -M t | ∧ |M t -M t-s | s 1/(γ-ε/5)β(Mt) ≥ 1.
Assume first that M is continuous at t. Assume toward contradiction that for all s > 0 sufficiently small,

|M t+s -M t | ≤ s 1/((γ-ε/5)β(Mt)) or |M t -M t-s | ≤ s 1/((γ-ε/5)β(Mt)) . If |M t+s -M t | ≤ s 1/((γ-ε/5)β(Mt)) , then setting r = M t+s -M t , (25) µ (M t -r, M t + r) ≥ µ (M t , M t + r) = s ≥ r (γ-ε/5)β(Mt) .
The same holds true when |M t -M t-s | ≤ s 1/((γ-ε/5)β(Mt)) . We have thus proved that (25) holds for every small r by continuity of M at t, this contradicts [START_REF] Mörters | On the multifractal spectrum of the branching measure on a Galton-Watson tree[END_REF].

When t is a jump time for M, the proof goes as above using the two obvious remarks : µ(M t -r, M t + r) = µ(M t , M t + r) for all small r > 0, and M t -M t-s > (M t -M t-)/2 which does not depend on s.

From [START_REF] Perkins | The multifractal structure of super-Brownian motion[END_REF] we deduce [START_REF] Mytnik | Multifractal analysis of superprocesses with stable branching in dimension one[END_REF].

Next technical lemma, proved in [START_REF] Yang | Multifractality of jump diffusion processes[END_REF], shows that when (23) holds, there are necessarily at least two "large" jumps around (and very close to) t. Let us recall this lemma, adapted to our context.

Lemma 2 ([28]

). Let N stand for the compensated Poisson measure associated with the Poisson measure N . There exists a constant C such that for every δ > 1, for all integers n ≥ 1

P   sup 0≤s<t≤1 |s-t|≤2 -n 2 n δ(β(M t+2 -n )+2/n) t s 2 -n δ 0 z 1 β(M u-) N (du, dz) ≥ 6n 2   ≤ Ce -n .
Remark 4. The formula looks easier than the one in [START_REF] Yang | Multifractality of jump diffusion processes[END_REF] because in our context M is increasing. When the function β is constant, the term 2/n in the previous inequality disappears [START_REF] Balança | Fine regularity of Lévy processes and linear (multi)fractional stable motion[END_REF].

Recall formula [START_REF] Geman | Occupation densities[END_REF] of M. Last Lemma allows us to control not exactly the increments of M, but the increments of the "part of M" constitued by the jumps of size less than 2 -n δ . It essentially entails that these "restricted" increments over any interval of size less than 2 -n are uniformly controlled by 2

- n δ(β(M t+2 -n )+2/n)
with large probability. More precisely, Borel-Cantelli Lemma applied to Lemma 2 with δ = γ -ε yields that for all integers n greater than some n γ-ε ,

t+2 -n t 2 -n γ-ε 0 z 1 β(M u-) N (du, dz) ≤ 6n 2 • 2 - n (γ-ε)(β(M t+2 -n+1 )+2/n) ≤ 2 - n (γ-ε/2)β(M t+2 -n+1 ) .
On the other hand, for all integers n greater than some other n γ-ε , a direct computation gives

t+2 -n t 2 -n γ-ε 0 z 1 β(M u-) du dz z 2 ≤ C2 -n 2 -n γ-ε 1 β(M t+2 -n ) -1 ≤ 2 - n (γ-ε)β(M t+2 -n ) .
Therefore, for all large n,

t+2 -n t 2 -n γ-ε 0 z 1 β(M u-) N (du, dz) ≤ t+2 -n t 2 -n γ-ε 0 z 1 β(M u-) N (du, dz) + t+2 -n t 2 -n γ-ε 0 z 1 β(M u-) du dz z 2 ≤ 2 - n (γ-ε/3)β(M t+2 -n+1 ) . (26)
Similarly, one establishes that

t t-2 -n 2 -n γ-4ε 0 z 1 β(M u-) N (du, dz) ≤ 2 - n (γ-ε/3)β(M t+2 -n+1 ) . ( 27
)
Let us introduce, for every integer n ≥ 1, the process

M n t = t 0 1 2 -n γ-ε z 1/β(M u-) N (du, dz), so that M t = M n t + t 0 2 -n γ-ε 0 z 1 β(M u-) N (du, dz).
A direct estimate shows that by right-continuity of M, when n becomes large, one has

3 • 2 - n (γ-ε/3)β(M t+2 -n+1 ) < 2 - n (γ-ε/4)β(M t )
Recalling formula [START_REF] Geman | Occupation densities[END_REF], the three inequalities [START_REF] Mytnik | Multifractal analysis of superprocesses with stable branching in dimension one[END_REF], [START_REF] Xu | The multifractal nature of boltzmann processes[END_REF] and [START_REF] Yang | Hausdorff dimension of the range and the graph of stable-like processes[END_REF] imply that for an infinite number of integers n

| M n t+2 -n -M n t | ∧ | M n t -M n t-2 -n | ≥ 2 - n (γ-ε/3)β(M t+2 -n+1 ) ≥ 2 - n (γ-ε/2)β(M t ) . ( 28
)
Since M n (and M) are purely discontinuous and right continuous, this last inequality proves the existence of at least one time t n 1 ∈ (t -2 -n , t] and another time t n 2 ∈ (t, t + 2 -n ] such that M n (and M) has a jump. The desired property on the Poisson measure N follows, and Proposition 4 is proved.

Further, in order to find an upper bound for the dimension of E(γ), one constructs a suitable covering of it. For n ∈ N * and k = 0, . . . , 2 n -1, set

I n,k = [k2 -n , (k + 1)2 -n ) and I n,k = k+1 =k-1 I n, .
One introduces the collection of sets

E n (γ, ε) = I n,k : N I n,k × 2 -n γ-ε , 1 ≥ 2, k = 0, . . . , 2 n -1 ,
which is constituted by the intervals I n,k containing at least two jumps for N of size greater than 2 -n γ-ε . Finally, one considers the limsup set

E(γ, ε) = lim sup n→+∞ I∈En(γ,ε) I. ( 29
)
Proposition 4 states exactly that E(γ) ⊂ E(γ, ε). So it is enough to find an upper bound for the Hausdorff dimension of E(γ, ε). Next lemma estimates the number of intervals contained in E n (γ, ε). Lemma 3. With probability 1, there exists a constant C such that for all n ≥ 1,

#E n (γ, ε) ≤ Cn 2 2 n( 2 γ-ε -1) .
Proof For a fixed "enlarged" dyadic interval I n,k , the inclusion I n,k ∈ E n (γ, ε) corresponds to the event that a Poisson random variable with parameter q n = 3 • 2 -n • 2 n γ-ε is larger than 2. Since q n → 0 exponentially fast, one has

p n := P( I n,k ∈ E n (γ, ε)) = C n 2 -n(2-2 γ-ε ) ,
where C n is a constant depending on n which stays bounded away from 0 and infinity.

The events { I n,3k ∈ E n (γ, ε)} k≥0 being independent, the random variable #{k ∈ {1, . . . , 2 n /3 : I n,3k ∈ E n (γ, ε)} is a Binomial random variable with parameters ( 2 n /3 , p n ). An application of Markov inequality yields

P # k : I n,3k ∈ E n (γ, ε) ≥ n 2 2 n /3 p n ≤ n -2 .
Further, Borel-Cantelli Lemma gives that almost surely, for n sufficiently large,

# k : I n,3k ∈ E n (γ, ε) ≤ n 2 2 n /3 p n .
The same holds for # k :

I n,3k+1 ∈ E n (γ, ε) and # k : I n,3k+2 ∈ E n (γ, ε) . One concludes that for all n ∈ N sufficiently large, #E n (γ, ε) ≤ 3n 2 2 n /3 p n ,
which proves the claim. Now we are in position to prove the upper bound for the Hausdorff dimension of E(γ).

Proof of (20) : Let n 0 be so large that the previous inequalities hold true for all integers n ≥ n 0 . Recalling (29), one knows that for every n 1 ≥ n 0 , the union n≥n 1 I∈En(γ,ε) I forms a covering of E(γ, ε), thus a covering of E(γ).

Let s > 2 γ-ε -1. Fix η > 0 and n 1 so large that all intervals I ∈ E n 1 (γ, ε) have a diameter less than η. Using the covering just above, one sees that the s-Hausdorff measure of E(γ) is bounded above by

H s η (E(γ)) ≤ n≥n 1 I∈En(γ,ε) | I| s ≤ n≥n 1 Cn 2 2 n( 2 γ-ε -1) |3 • 2 -n | s
which is a convergent series. Therefore, H s η (E(γ)) = 0 as n 1 can be chosen arbitrarily large. This leads to H s (E(γ)) = 0 for every s > 2 γ-ε -1. We have thus proved almost surely,

dim H (E(γ)) ≤ 2 γ -ε -1.
Letting ε → 0 yields the desired upper bound.

Proof of Theorem 7 : lower bound

The aim of this section is to get that with probability one, [START_REF] Mörters | Thin and thick points for branching measure on a Galton-Watson tree[END_REF] holds with O = (0, 1) for all non-increasing càdlàg function Υ :

[0, 1] → [1, 2].
Recalling the notations in Theorem 7, for simplicity, we write

F (Υ) = E t µ (
0, 1), Υ . Let ε > 0 and 0 < b < ε be fixed until the end of Section 6.7. We construct simultaneously for all Υ with 1 + 2ε ≤ Υ min ≤ 2 -2ε and ε > 0, a random Cantor set C(Υ, ε ) ⊂ F (Υ) with Hausdorff dimension larger than 2/(Υ min + ε ) -1. The lower bound for the Hausdorff dimension of F (Υ) follows.

We explain how to extend the proof to the functions Υ satisfying Υ min ∈ [1, 2] \ [1 + 2ε, 2 -2ε] in subsection 6.8.

6.1.

The time scales, and some notations. We aim at constructing Cantor sets inside F (Υ). Recalling Proposition 4, some configurations for the jump points are key in this problem. More precisely, one knows that every point in F (Υ) is infinitely often located in the middle of two large jumps which are really close to each other. So the Cantor set we are going to construct will focus on these behaviors.

Let us define a (deterministic) sequence of rapidly decreasing positive real numbers. First,

η 1,0 = 10 -10 , η 1, = η 1+ε 1, -1 for 1 ≤ ≤ 1 := min{ ≥ 1 : η 1, ≤ e -η -1 1,0 }.
By induction one defines the sequence {η n, : n ∈ N * , 0 ≤ ≤ n } as

η n,0 = η n-1, n-1 , η n, = η 1+ε n, -1 for 1 ≤ ≤ n := min{ ≥ 1 : η n, ≤ e -η -1 n,0 },
which are our time scales. One also sets

η n, n+1 = η n+1,1 η n = η n,0 .
The natural partition of [0, 1] induced by this sequence is denoted by

J n, = J n, ,k = [kη n, , (k + 1)η n, ) : k = 0, . . . , 1 η n,
.

By convention, J n, ,-1 = J n, ,-2 = J n, , 1 ηn +1 = J n, , 1 ηn +2 = ∅. One needs the enlarged intervals

J n, ,k = k+1 i=k-1 J n, ,i .
6.2. Zero jump and double jumps configuration. Two types of jump configuration along the scales are of particular interest, since they are the key properties used to build relevant Cantor sets. Recall that the Poisson random measure N has intensity dt ⊗ dz/z 2 . Definition 6. For any

n ∈ N * , 1 ≤ ≤ n and γ ∈ [1 + 2ε, 2 -2ε], define J z n, (γ) = J n, ,k ∈ J n, : N J n, ,k × [η 1/γ n, +1 , η 1/γ n, ) = 0 (30) J d n, (γ) =    J n, ,k ∈ J n, :    N J n, ,k-2 × [η 1/γ n, /2, η 1/γ n, ) = 1 N J n, ,k+2 × [η 1/γ n, /2, η 1/γ n, ) = 1    . ( 31
)
Remark 5. The superscript "z" refers to "zero jump" while "d" refers to "double jump".

Let us start with straightforward observations:

• for (n, ) = (n , ), the composition (number and position of the intervals) of J z n, (γ) and J z n , (γ) are independent thanks to the Poissonian nature of the measure N .

• The same holds true for the double jump configuration.

• Fixing (n, ), for |k -k | ≥ 3, the events J n, ,k ∈ J z n, (γ) and J n, ,k ∈ J z n, (γ) are independent. • The same holds for J d n, (γ) if one assumes that |k -k | ≥ 5. • For fixed (n, , k), the events J n, ,k ∈ J z n, (γ) and J n, ,k ∈ J d n, (γ) are independent. Next probability estimate is fundamental in the sequel.

Lemma 4. For all n ∈ N * , 1 ≤ ≤ n , γ ∈ [1 + 2ε, 2 -2ε] and J ∈ J n, , p n, ,γ = P J ∈ J z n, (γ) = exp -C n, η (1-1+ε γ ) n, (32) q n, ,γ = P J ∈ J d n, (γ) = C n, η 2-2 γ n, (33)
where C n, , C n, are constants uniformly (with respect to n, and γ) bounded away from 0 and infinity.

Proof The value of p n, ,γ corresponds the probability that a Poisson random variable with parameter p = 3η n, η

1/γ n, +1 -η 1/γ n,
equals zero, thus p n, ,γ = e -p . On the other hand, each condition in (31) relies on the probability that a Poisson variable with parameter q = η 1-1/γ n, equals one. Hence, by independence, q n, ,γ = (e -q • q) 2 . The result follows.

6.3. Random trees induced by the zero jump intervals and estimates of the number of their leaves. In this section, one constructs for a fixed integer n ∈ N * a nested collection of intervals, indexed by 0 ≤ ≤ n . These intervals induce naturally a random tree with height n + 1.

One starts with any interval J n ∈ J n = J n,0 , which is the root of the tree, denoted by T n,0 = {J n }. Define by induction, for 1 ≤ ≤ n , T n, = {J ∈ J n, : J ∈ J z n, (γ) and J ⊂ J for some J ∈ T n, -1 }. One focuses on the J n -rooted random tree T n,γ (J n ) = (T n,0 , . . . , T n, n ). The number of leaves of T n,γ (J n ), denoted by |T n,γ (J n )|, is the cardinality of T n, n .

Fact: Every point belonging to the intervals indexed by the leaves of the tree have the remarkable property that "they do not see" large jump points between the scales η n and η n+1 . This observation is made explicit in Lemma 9.

Remark 6. Observe that we dropped the index γ in the definition of T n, to ease the notations, since these sets will not re-appear in the following sections.

Our goal is to prove the following estimate on the number of leaves of T n,γ (J n ). Proposition 5. With probability one, for every integer n large, for every

J n ∈ J n,0 and γ ∈ [1 + 2ε, 2 -2ε], (34) |T n,γ (J n )| ≥ η n 2η n+1
.

The estimate of |T n,γ (J n )| is divided into several short lemmas.

Lemma 5. For all n ∈ N * , J n ∈ J n,0 and γ ∈ [1 + 2ε, 2 -2ε], one has

P #T n,1 ≥ 1 -log(1/η n,0 ) -2 η n,0 η n,1 p n,1,γ ≥ 1 -3 exp -log(1/η n,0 ) -4 η n,0 3η n,1 p n,1,γ /2
Proof For any (n, ) and for i ∈ {0, 1, 2}, set

T i n, = {J ∈ T n, : J = J n, ,3k+i ∈ J n, : k ∈ N}.
By independence (see the observations before Lemma 4), for each i ∈ {0, 1, 2}, the number of vertices in T i n,1 is a binomial random variable with parameter ( η n,0 /(3η n,1 ) , p n,1,γ ).

By Chernoff inequality, for every binomial random variable X with parameter (n, p), for any δ ∈ (0, 1), one has

P(X ≤ (1 -δ)np) ≤ exp(-δ 2 np/2). ( 35
)
The result follows applying (35) with δ = log(1/η n,0 ) -2 for every i.

Lemma 6. Set a(n, ) = (1 -log(1/η n, -1 ) -2 ) η n, -1 η n, p n, ,γ .
For all n ∈ N * , J n ∈ J n,0 , γ ∈ [1 -2ε, 2 -2ε] and 2 ≤ ≤ n , a.s.

P #T n, ≥ a(n, )#T n, -1 ≥ 1 -3 exp -log(1/η n, -1 ) -4 #T n, -1 η n, -1 3η n, p n, ,γ /2 .
Proof Using again the remarks before Lemma 4, for every i ∈ {0, 1, 2}, the law of the random variable #T i n, conditioning on #T n, -1 is binomial with parameter (#T n, -1

η n, -1 3η n,
, p n, ,γ ). Applying (35) gives the estimate.

Lemma 7. Set b(n, ) = 1 -3 exp - log(1/η n, -1 ) -4 6(1 -log(1/η n, -1 ) -2 ) k=1 a(n, k) .
For all n ∈ N * , J n ∈ J n,0 and γ ∈ [1 -2ε, 2 -2ε], one has

P |T n,γ (J n )| ≥ n =1 a(n, ) ≥ n =1 b(n, ).
Proof One has

P |T n,γ (J n )| ≥ n =1 a(n, ) ≥ P |T n,γ (J n )| ≥ n =1 a(n, ), #T n, n-1 ≥ n-1 =1 a(n, ) ≥ P |T n,γ (J n )| ≥ a(n, n )#T n, n-1 , #T n, n-1 ≥ n-1 =1 a(n, ) .
Conditioning on #T n, n-1 , and using Lemma 6 with = n , this probability is greater than

E E 1 -3 exp -log(1/η n, n-1 ) -4 #T n, n-1 η n, n-1 3η n, n p n, n,γ /2 #T n, n-1 × 1 #T n, n-1 ≥ n-1 =1 a(n, ) ≥ E E 1 -3 exp - log(1/η n, n-1 ) -4 6(1 -log(1/η n, n-1 ) -2 ) a(n, n )#T n, n-1 #T n, n-1 × 1 #T n, n-1 ≥ n-1 =1 a(n, ) ≥ b(n, n ) P #T n, n-1 ≥ n-1 =1 a(n, )
Iterating this computation yields the desired inequality.

We are now in position to prove Proposition 5.

Proof We are going to prove the following lemma:

Lemma 8. For some constant c 1 , for all n ∈ N * large enough, for every J n ∈ J n,0 and γ ∈ [1 + 2ε, 2 -2ε], one has

P |T n,γ (J n )| ≥ η n 2η n+1 ≥ exp -c 1 exp -η -ε/2 n . ( 36
)
Proof One combines the estimates in Lemma 4 and Lemma 7. Let us first estimate =1 a(n, ) for 2 ≤ ≤ n . Observe that for large n, one has

n = log log(1/η n, n ) log(1/η n,0 ) / log(1 + ε) ≤ log log(1/η n, n )/ log(1 + ε) ≤ log 1 + ε η n,0 / log(1 + ε) ≤ 2 log(1/η n,0 ).
Thus for n large enough,

=1 (1 -log(1/η n, -1 ) -2 ) = exp =1 log 1 -log(1/η n, -1 ) -2 ≥ exp -2 =1 log(1/η n, -1 ) -2 ≥ exp -2 log(1/η n,0 ) -2 ≥ exp -4 log(1/η n,0 ) -1 ≥ 1/ √ 2. ( 37
)
Using the rapid decay of (η n, ) to zero and the uniform boundedness of C n, , one can find a constant c 0 > 0 such that for all n large enough, (

) =1 p n, ,γ = exp - =1 C n, η 1-1+ε γ n, ≥ exp -c 0 η 1-1+ε γ n,1 ≥ 1/ √ 2. 38
Combing (37) and (38), one concludes that for all large n (39) =1 a(n, ) ≥ η n,0 2η n, , and in particular,

n =1 a(n, ) ≥ η n 2η n+1
.

Now we estimate the other product n =1 b(n, ). Using (37), ( 38) and (39), there exists a constant c 1 > 0 such that for every large

n n =1 b(n, ) ≥ n =1 1 -3 exp - log(1/η n, -1 ) -4 12
η n,0 η n, = exp n =1 log 1 -3 exp - log(1/η n, -1 ) -4 12 η n,0 η n, ≥ exp -6 n =1 exp - log(1/η n, -1 ) -4 12
η n,0 η n, ≥ exp -c 1 exp - log(1/η n,0 ) -4 12
η n,0 η n,1 ≥ exp -c 1 exp -η -ε/2 n,0 = exp -c 1 exp -η -ε/2 n .
where the fast decay rate of (η n, ) to zero has been used for the third inequality. These last equations prove exactly (36). Finally, to prove Proposition 5, since the cardinality of J n is less than η -1 n ,

P ∃J n ∈ J n : |T n,γ (J n )| < η n 2η n+1 ≤ η -1 n 1 -exp -c 1 exp -η -ε/2 n .
Using the fast decay of η n to zero, this is the general term of a convergent series, and the Borel-Cantelli lemma gives the result.

Remark 7. Essentially, one needs to keep in mind that the number of leaves of the random tree T n,γ (J n ) is the total number of intervals of J n+1 inside J n , up to a constant factor 1/2.

One finishes this section by proving that every point belonging to a leaf of T n,γ (J) "is not close" to large jumps.

Lemma 9. Let J ∈ J n and r ∈ [η n+1 , η n ). Assume that T n,γ (J) is not empty. Then for each t ∈ T n,γ (J), N (B(t, r) × [r 1/Υ n J n,0 (t) , η 1/Υ n J n,0 (t) n ]) = 0.
Proof For each t ∈ T n,γ (J), denote by J n, (t) the unique interval such that t ∈ J n, (t) for all 0 ≤ ≤ n . Denote by 0 the unique integer such that η n, 0 +1 ≤ r < η n, 0 . By construction of the random tree T n,γ (J), one has

N (B(t, r)×[r 1/Υ n J n,0 (t) , η 1/Υ n J n,0 (t) n, 0 ]) ≤ N ( J n, 0 (t)×[η 1/Υ n J n,0 (t) n, 0 +1 , η 1/Υ n J n,0 (t) n, 0 ]) = 0.
Further, all ancestor interval of J n, 0 (t) has no large jumps around, in particular,

N ( J n, 0 (t) × [η 1/Υ n J n,0 (t) n, 0 , η 1/Υ n J n,0 (t) n ]) = 0
Combining these estimates yields the result. 6.4. Double jumps configuration around the leaves, and key lemma.

In the previous section, we have seen that the "zero jump" configuration is quite frequent. The aim here is to estimate the number of intervals with "double jumps" amongst the leaves of the trees. To this end, we introduce further some notations. Set

M n (γ) = η 1-2/(γ+3•2 -n-1 ) n+1 η 3 n .
Definition 7. Let J 0 ∈ J n,0 and T n,γ (J 0 ) be the random tree defined in last subsection. Consider its leaves that we denote {J i } i=1,...,|Tn,γ (J 0 )| , which are intervals of length η n+1 . The families {F(J 0 , γ, m)} m=1,..., Mn(γ)/2 are defined as the following disjoint subfamilies of {J i } i=1,...,|Tn,γ (J 0 )| :

F(J 0 , γ, m) = J 5k+2m |Tn,γ (J 0 )| Mn(γ) : k ∈ 0, ..., |T n,γ (J 0 )| 5M n (γ) .
Hence, two families F(J 0 , γ, m) and F(J 0 , γ, m ) are disjoint and separated by a distance equivalent to |T n,γ (J 0 )|/M n (γ) η n+1 , and the intervals belonging to the same F(J 0 , γ, m) are separated by the distance at least 4η n+1 .

Finally, denote by

D n = {k2 -n : k ∈ Z, n ∈ N * }
the n-th generation dyadic numbers. One is ready to prove the key lemma.

Lemma 10. The following holds with probability 1: there exists a (random) integer n 0 such that for all n ≥ n 0 , for every

J ∈ J n , every γ ∈ D n ∩ [1 + 2ε, 2 -2ε]
, every a ∈ {0, 1, 2, 3}, each family {F(J, γ, m)} m=1,..., Mn(γ)/2 contains at least one interval belonging to

J d n+1 (γ + a • 2 -(n+1) ).
Remark that the intervals belonging to F(J, γ, m) come also from the tree T n,γ (J) associated with J, so they also enjoy the "zero jump" property.

Proof Fix n a positive integer, J ∈ J n , γ ∈ D n ∩[1+2ε, 2-2ε], a ∈ {0, 1, 2, 3}.
Recall that J n = J n,0 and J n+1 = J n, n with the notations of the previous sections.

By Lemma 4 and the observations made before this Lemma, there exists a positive finite constant c 2 such that for all n large

P ∃m, ∀J ∈ F(J, γ + a2 -n-1 , m), J ∈ J d n, n (γ + a2 -n-1 ) |T n,γ+a2 -n-1 (J)| ≥ η n 2η n+1 ≤ M n (γ) 2 1 -q n, n,γ+a•2 -n-1 1
Mn(γ) • ηn 10η n+1 ≤ η 1- 2 γ+3•2 -n-1 n+1 η 3 n 1 -c 2 η 2- 2 γ+a2 -n-1 n+1 ηn 10η 2- 2 γ+3•2 -n-1 n+1 η 3 n ≤ η 1- 2 γ+3•2 -n-1 n+1 η 3 n exp -c 2 η -2 n /10 Remark that η n+1 ≤ e -η -1 n ≤ η n, n-1 implies log(1/η n+1 ) ≤ (1 + ε)η -1 n .
The above probability is thus bounded by above by

η 3 n exp 2 γ + 3 • 2 -n-1 -1 (1 + ε)η -1 n - c 2 10 η -2 n ≤ η 3 n .
On the other hand, by Lemma 8 one has

P |T n,γ+a2 -n-1 (J)| ≤ η n 2η n+1 ≤ 1 -exp -c 1 exp(-η -ε/2 n ) ≤ 2c 1 exp(-η -ε/2 n ).
Thus, P ∃m, ∀J ∈ F(J, γ, m), J ∈ J d n+1 (γ + a2 -n-1 ) ≤ 2η 3 n . One deduces that

P ∃J ∈ J n , ∃ m, ∀J ∈ F(J, γ, m), J / ∈ J d n+1 (γ + a2 -n-1 ) ≤ η -1 n • 2η 3 n = 2η 2 n .
There are less than 2 n possible choices for γ, and 4 choices for a. Hence,

P ∃ γ, ∃ a, ∃ J ∈ J n , ∃ m, ∀J ∈ F(J, γ, m), J / ∈ J d n+1 (γ + a2 -n-1 ) ≤ 2 n+3 η 2
n , which is the general term of a convergent series. An application of Borel-Cantelli Lemma entails the result. Step 1 (localization). For each Υ as above and ε > 0, there exist t ε ∈ (0, 1), r ε > 0 such that ∀ t ∈ [t ε -r ε , t ε +r ε ], we have Υ(t) < Υ min +ε .

Let n 0 be the random integer obtained in Lemma 10. We assume that n 0 is so large that the conclusions (34) of Proposition 5 hold, and also that

2ε /η n 0 > K Υ • 2 n 0 , where K Υ = |Υ(1-) -Υ(0)| < +∞.
For every interval J, let Osc Υ (J) = sup t∈J Υ(t) -inf t∈J Υ(t) be the oscillation of Υ over J. By the monotonicity of Υ, for each n ≥ n 0 one has

(40) #{J ∈ J n : J ⊂ [t ε -r ε , t ε + r ε ] and Osc Υ (J) ≥ 2 -n } ≤ K Υ • 2 n
Step 2 (Initialization of the Cantor set). One chooses arbitrarily one interval

J n 0 ∈ J n 0 contained in [t ε -r ε , t ε + r ε ] such that (41) Osc Υ (J n 0 ) < 2 -n 0 .
Set the generation "zero" of the Cantor set as C n 0 (Υ, ε ) = J n 0 . Simultaneously, we build a measure ν n 0 by setting ν n 0 (J n 0 ) = 1, and ν n 0 is uniformly distributed on J n 0 .

Step 3 (Next generation of the Cantor set). Let us introduce the following notation: for each n ∈ N * and J ∈ J n , set

Υ n J = max D n ∩ [1, inf t∈J Υ(t)] .
We explain how to get the second generation of intervals C n 0 +1 (Υ, ε ) of the Cantor set.

The oscillation restriction (41) for Υ on J n 0 implies that for every J ∈ J n 0 +1 contained in J n 0 , the quantity Υ n 0 +1 J takes necessarily one of the four values {Υ n 0 J n 0 + a2 -n 0 -1 : a = 0, 1, 2, 3}. Moreover, applying Lemma 10 to J n 0 , one obtains that for each a ∈ {0, 1, 2, 3}, each subfamily {F(J n 0 , Υ n 0 J n 0 , m)} m=1,..., Mn 0 (Υ n 0 J n 0 )/2 contains at least one interval J belonging to J d n 0 +1 (Υ n 0 J n 0 + a2 -n 0 -1 ). Recalling that Υ is non-increasing, the quantities Υ n 0 +1 J are also nonincreasing when J ranges from left to right. Since there are M n 0 (Υ n 0 J n 0 )/2 disjoint families {F(J 0 , Υ n 0 J n 0 , m)} which are organized in increasing order, one deduces that there is a ∈ {0, 1, 2, 3} such that there exist

M n 0 (Υ n 0 J n 0 )/8 different integers m ∈ {1, ..., M n 0 (Υ n 0 J n 0 )/2 } for which the family F(J 0 , Υ n 0 J n 0 , m) contains (at least) one interval J satisfying Υ n 0 +1 J = Υ n 0 J n 0 + a2 -n 0 -1 and J ∈ J d n 0 +1 (Υ n 0 +1 J ). Remark that 2 n 0 +1 M n 0 (Υ n 0 J n 0 )/8 /2 where we used that Υ min < 2 -2ε and J n 0 ⊂ [t ε -r ε , t ε + r ε ].
Then, applying (40) for n = n 0 + 1, one can choose the first M n 0 (Υ n 0 J n 0 )/16 intervals J which satisfy Osc Υ (J) < 2 -n 0 -1 among those already selected in the last paragraph.

Finally, C n 0 (Υ, ε ) is the union of these intervals, which are called the basic intervals of generation n 0 + 1. Observe that these intervals are separated by a distance larger than η n 0 /(2M n 0 (Υ n 0 J n 0 )) (thanks to Borel-Cantelli applied to Lemma 5), and they all have their length equal to η n 0 +1 . Simultaneously, one defines a refinement ν n 0 +1 of the measure ν 0 by setting for every J n 0 +1 basic interval of C n 0 +1 (Υ, ε )

ν n 0 +1 (J n 0 +1 ) = ν n 0 (J n 0 ) 1 M n 0 (Υ n 0 J n 0 )/16
, and by saying that ν n 0 +1 is uniformly distributed inside each J n 0 +1 .

Step 4: Induction of the construction of the Cantor set: Assume that for every i = n 0 , n 0 + 1, ..., n 0 + n, the generation C i (Υ, ε ) has been constructed and satisfies the following:

(1) C i (Υ, ε ) is constituted by a finite number of basic disjoint intervals J i belonging to J i , (2) for every i = n 0 + 1, ..., n 0 + n, each basic interval

J i ∈ C i (Υ, ε ) is included in a unique basic interval J i-1 ∈ C i-1 (Υ, ε ). (3) for every i = n 0 , n 0 + 1, ..., n 0 + n -1, each basic interval J i ∈ C i (Υ, ε ) contains M i (Υ i J i )/16 intervals J i+1 ∈ C i+1 (Υ, ε
). These intervals are separated by a distance at least equal to

η i /(2M i (Υ i J i )). Moreover, each J i+1 belongs to J d i+1 (Υ i+1 J i+1 ). (4) Each basic interval J i of C i (Υ, ε ) satisfies Osc Υ (J i ) ≤ 2 -i .
(5) for every i = n 0 + 1, ..., n 0 + n, ν i is a measure supported by the basic intervals J i of C i (Υ, ε ), and if J i-1 is the unique interval in

C i-1 (Υ, ε ) such that J i ⊂ J i-1 , then (42)
ν i (J i ) = ν i-1 (J i-1 ) 1 M i-1 (Υ i-1
J i-1 )/16 and ν i is uniformly distributed inside each J i . We are now able to complete the induction. For any basic interval J n ∈ C n (Υ, ε ), applying the same method as in step 3, one finds

M n (Υ n J n )/16 intervals J n+1 ∈ J d n+1 (Υ n+1 J n+1 ), also satisfying Osc Υ (J n+1 ) < 2 -(n+1) .
Then, C n+1 (Υ, ε ) is the union of these intervals, which constitue the basic intervals of generation n + 1. By construction, these basic intervals J n+1 are separated by at least η n /(2M n (Υ n J n )), (where J n is the "parent" interval of J n+1 , i.e. the unique basic interval in C n (Υ, ε ) such that J n+1 ⊂ J n ), and they all have their length equal to η n+1 .

Simultaneously, the refinement ν n+1 of the measure ν n is defined by setting, for every J n is the "parent" interval of J n+1 ,

ν n+1 (J n+1 ) = ν n (J n ) 1 M n (Υ n J n )/16
, and by saying that ν n+1 is uniformly distributed inside each J n+1 .

If B ∩ C(Υ, ε ) = ∅, let n 1 be the largest integer such that B intersects C n 1 (Υ, ε ) in exactly one basic interval, denoted by J n 1 .

Denote by δ n 1 +1 (Υ, ε , J n 1 ) the minimal distance between any two intervals of

C n 1 +1 (Υ, ε ) which are contained in J n 1 . Then |B| contains at most min M n 1 (Υ n 1 J n 1 ), |B|/δ n 1 +1 (Υ, ε , J n 1 ) intervals of generation n 1 + 1.
In addition, by construction, one has

(44) δ n 1 +1 (Υ, ε , J n 1 ) ≥ η n 1 2M n 1 (Υ n 1 J n 1 )
.

Hence by ( 42), since all the intervals J n 1 +1 of generation n 1 + 1 within J n 1 have the same ν-mass, one has (using (44))

ν Υ,ε (B) ≤ min M n 1 (Υ n 1 J n 1 ), |B|/δ n 1 +1 (Υ, ε , J n 1 ) • ν n 1 +1 (J n 1 +1 ) ≤ min M n 1 (Υ n 1 J n 1 ), |B| 2M n 1 (Υ n 1 J n 1 ) η n 1 ,0 •   n 1 -1 k=n 0 1 M k (Υ min + ε )   • 1 M n 1 (Υ n 1 J n 1 ) ≤ 2   n 1 -1 k=n 0 1 M k (Υ min + ε )   • η -1 n 1 • min (η n 1 , |B|) .
Due to our choices for the sequence (η n ) n≥1 , when n 0 is large,

n 1 -1 k=n 0 1 M k (Υ min + ε ) ≤ (M n 1 -1 (Υ min + ε )) -1 ≤ η 2 Υ min +2ε -1 n 1
, so applying the inequality x ∧ y ≤ x s y 1-s for s ∈ (0, 1) and 0 < x, y < 1 yields

ν Υ,ε (B) ≤ 2η 2 Υ min +2ε -1 n 1 • η -1 n 1 • η 2- 2 Υ min +2ε n 1 • |B| 2 Υ min +2ε -1 = 2|B| 2 Υ min +2ε -1 .
Finally the mass distribution principle applied to the measure ν Υ,ε , which is supported by the Cantor set C(Υ, ε ), allows one to conclude that

dim H C(Υ, ε ) ≥ 2 Υ min + 2ε -1.
6.8. Extension to Υ min ∈ {1, 2}. Letting ε → 0 along a countable sequence yields that almost surely, for all Υ with Υ min ∈ (1, 2),

dim H F (Υ) ≥ 2 Υ min -1. ( 45
)
It remains to treat the extreme cases. First case : Υ min = 1. For each ε 2 > 0, there exists an open interval O ∈ (0, 1) such that every t ∈ O satisfies Υ(t)

≥ 1 + ε 2 > 1. Applying (45) yields that dim H F (Υ) ≥ 2 1+ε 2 -1 . Letting ε 2 → 0 establishes that dim H F (Υ) = 1.
Second case : Υ min = 2, i.e. Υ ≡ 2. In order to prove dim H F (Υ) ≥ 0, it suffices to show that there exists almost surely t ∈ (0, 1) such that dim(µ, M t ) = 2β(M t ), i.e. F (Υ) = ∅. To this end, some changes are needed for the construction of the Cantor set. We only sketch the proof since it is essentially the same as the one in the precedent sections (with simplification). Set

ρ 0 = 1/2 and ρ n = exp(-ρ -1 n-1 ) for all n ≥ 1, η n = ρ n log(1/ρ n ) -1 for all n ≥ 0. Let J n (2) be the set composed of intervals J n,k = [kη n , (k+1)η n ) that satisfy                N [J n,k-2 × ρ 1/2 n log 1 ρn -3 , 1 = 1, N J n,k+2 × ρ 1/2 n log 1 ρn -3 , 1 = 1, N J n,k × ρ 1/2 n log 1 ρn -3 , 1 = 0, (46)
It is easy to check that any point t covered by the collection J n (2) infinitely often satisfy dim(µ, M t ) ≥ 2β(M t ) (necessarily, one has equality thanks to Theorem 1). We construct as before, by induction, the collection C n (Υ ≡ 2) of basic intervals and the Cantor set C(Υ ≡ 2) = n C n (Υ ≡ 2) contained in F (Υ). The same arguments as in Lemma 4 give a constant C n uniformly bounded below and above by 0 and +∞ such that for any fixed J n,k ,

P(J n,k ∈ J n (2)) = C n • ρ n log 1 ρ n 4 .
Thus one bounds from above the probability that there exists J n,k such that none of the intervals J n+1,k contained in J n,k belongs to J n+1 (2) by

1 η n 1 -C n+1 • ρ n+1 log 1 ρ n+1 4 ηn η n+1
.

Observe that

η n η n+1 = C n+1 ρ -3 n-1 • (C n+1 ) -1 ρ -1 n+1 log 1 ρ n+1 -3 with C n+1 ρ -3 n-1 1.
So the probability in question is less than

η -1 n e -C n+1 ρ -3 n-1 ≤ η -1 n e -3ρ -1 n-1 = η -1 n ρ 3 n ≤ ρ n .
Borel-Cantelli Lemma implies the existence of a sequence of embedded interval with length tending to 0 that satisfy (46). This justifies that F (Υ) = ∅.

7. Space spectrum : proof of Theorem 2 7.1. A first theorem on dimensions, and the space spectrum. Throughout this section, we set ε = ε 0 , which is defined in [START_REF] Applebaum | Lévy processes and stochastic calculus[END_REF]. We are going to prove the following theorem. ) α∈(ε,1-ε) . Also, for every non decreasing càdlàg function f : [0, 1] → [ε, 1 -ε], consider the process

(47) L f t = t 0 1 0 z 1 f (t-) N (ds, dz).
Almost surely, for every set E ⊂ [0, 1], for every function f

: [0, 1] → [ε, 1 -ε], if α < inf{f (t) : t ∈ E} and β > sup{f (t) : t ∈ E}, then dim H (L α (E)) ≤ dim H L f (E) ≤ dim H (L β (E)) = β α dim H (L α (E)).
Before proving Theorem 8 next subsection, let us explain how we deduce the space spectrum of the occupation measure.

As mentioned in the introduction, almost surely, for every set E ⊂ R, the image of E by an α-stable process L α has Hausdorff dimension α dim H (E). Applying Theorem 8 to the function f (t) = β(M t ), which is almost surely càdlàg, one is now ready to prove Theorem 6.

Proof of Theorem 6 : The first part (the formula [START_REF] Durand | Multifractal analysis of Lévy fields[END_REF]) is immediate.

For the second part, let E ⊂ [0, 1] be such that for every non-trivial subinterval

I ⊂ [0, 1], dim H (E) = dim H (E ∩ I).
For every η > 0, there exists an interval I of length less than η such that inf This follows from the càdlàg regularity of t → β(M(t)). Hence, applying [START_REF] Durand | Multifractal analysis of Lévy fields[END_REF] to

E ∩ I gives dim H M(E ∩ I) ∈ dim H (E ∩ I) • sup t∈E β(M(t-)) -η, sup t∈E β(M(t-)) .
Since dim H M(E ∩ I) ≤ dim H M(E) and dim H (E) = dim H (E ∩ I), the result follows by letting η tend to zero.

One deduces a corollary from Theorem 6, which will be used in the proof of Theorem 2.

(48) dim H M E t µ (I, h) = dim H E t µ (I 0 , h) • sup t∈I 0 \d(I 0 ) β(M t ).
with the convention that 0 × (-∞) = 0 and (-∞) × (-∞) = -∞.

Proof If E t µ (I, h) is empty or a singleton, there is nothing to prove. One thus assumes that E t µ (I, h) is neither empty nor a singleton, so I 0 is a nontrivial interval. One could check the analysis in the proof of Theorem 4 for a construction of I 0 . Observe that the left-hand side of (48) is smaller than the right-hand side due to Theorem 6. The converse inequality follows by minimality of I 0 and a localization procedure as in the proof of Theorem 6.

Proof of Theorem 2 and 3 : To deduce the space spectrum, one needs some additional analysis other than the time spectrum. This is due to the following basic observation : for all t ∈ S(M), M t-is not in the range of M, but in the support of µ.

When O does not intersect the range of M, the level set E µ (O, h) = ∅, as is given in Theorem 2 and 3.

When O intersects the range of M, by the càdlàg property of M, there is a non-trivial interval O such that

E µ (O, h) = {x ∈ Supp µ ∩ O : dim(µ, x) = h} = {M t ∈ O : dim(µ, M t ) = h} ∪ {M t-∈ O : t ∈ S(M) and dim(µ, M t-) = h} = M(E t µ ( O, h)) ∪ {M t : t ∈ S(M) ∩ O and dim(µ, M t-) = h}
Since the Hausdorff dimension of the second set in the last union is at most 0, one distinguishes two types of situations according to the value of h. ∈ O is trivial. We thus assume that every M t 0 below belongs to O.

d µ (O, h) = d t µ ( O 0 , h) • sup t∈ O 0 \d( O 0 ) β(M t ) = sup ĝα (h)/α : α ∈ {β(M t ) : t ∈ O 0 } • sup t∈ O 0 β(M t ) = sup ĝα (h) : α ∈ {β(M t ) : t ∈ O 0 } = sup ĝα (h) : α ∈ {β(M t ) : t ∈ O} = sup {ĝ α (h) : α ∈ {β(M t ) : M t ∈ O}} ,
(1) (49) is due to 2β(M t 0 -) < h < β(M t 0 ) with t 0 ∈ S(M). For all t > t 0 , dim(µ,

M t-) ≥ β(M t-) > β(M t 0 ) > h. For all t ≤ t 0 , dim(µ, M t-) ≤ 2β(M t-) ≤ 2β(M t 0 -) < h. So H = ∅ and d µ (O, h) = -∞ as desired. (2) (49) is due to the fact that 2β(M t 0 -) < β(M t 0 ) = h with t 0 ∈ S(M)
and dim(µ, M t 0 ) = β(M t 0 ). As in the last item, H = ∅ as desired.

(3) (49) is due to h = 2β(M t 0 -) < β(M t 0 ) or h = 2β(M t 0 -) = β(M t 0 ).
As before for all t = t 0 , one has dim(µ,

M t-) = h. If dim(µ, M t 0 -) = 2β(M t 0 -), H = {t 0 }, otherwise H = ∅. This coincides with Theorem 3. (4) (49) is due to h ≥ 2β(M b-). For all t < b, dim(µ, M t-) ≤ 2β(M t 0 -) < 2β(M b-) ≤ h, hence H = ∅, as desired. (5) (49) is due to h ≤ β(M a ). Recall first that β(M a ) / ∈ O.
Further, for all t > a, dim(µ, M t ) > h. Hence, H = ∅, as desired. 7.2. Proof of Theorem 8. We start with a Lemma describing the distribution properties of the Poisson point process P = {(T n , Z n )} n≥1 .

Lemma 12. For every j ≥ 1, let P j = {n : Z n ∈ [2 -j-1 , 2 -j )}. Almost surely, there exist two positive decreasing sequences (ε j ) j≥1 and (η j ) j≥1 converging to zero such that for every integer J large enough, one has:

(1) 2 J(1-ε J ) ≤ #P J ≤ 2 J(1+ε J ) ,

(2) for every interval I ⊂ [0, 1] with length 2 -J ,

1 ≤ # j≤J(1+η J ) {n ∈ P j : T n ∈ I} ≤ 2 Jε J , (
) for every interval I ⊂ [0, 1] with length 2 -J , 0 ≤ # j≤J/3 {n ∈ P j : T n ∈ I} ≤ 1, 3
) for every interval I ⊂ [0, 1] with length 2 -J , for every j ≥ J(1

+ η J ), #{n ∈ P j : T n ∈ I} ≤ 2 j(1+ε j ) 2 -J .
Routine computations as in Lemma 4 entail Lemma 12.

Let E ⊂ [0, 1], α = inf{f (t) : t ∈ E} and β = sup{f (t) : t ∈ E}.

Call E α (resp. E f , E β ) the image of E by L α (resp. L f , L β ).
Lemma 13. Almost surely, the following holds. With each interval B α such that B α ∩ E α = ∅, one can associate an interval of the form

B α = L α ([T m , T n )) such that |B α | ≤ 2| B α | and possibly a singleton of the form {L α (T n )}, such that E α ∩ (B α ∪ {L α (T n )}) = E α ∩ B α .
The same holds true for every interval B f such that E j ∩ B f = ∅, which can be replaced by

B f = L f ([T m , T n ))
and possibly a singleton.

Proof

Almost surely all the processes L α , L β and L f are strictly increasing and càdlàg.

Let B α = [x α , y α ] be an interval satisfying B α ∩ E α = ∅. If x α is not of the form L α (T m ), then two cases occur:

• when x α / ∈ L α (E): B α can be replaced by [x α , y α ], where x α = inf( B α ∩ E α ), without altering the covering R α . Since L α is increasing and càdlàg, x α is necessarily the image of some jump point T m by L α .

• when x α ∈ L α (E): x α can be written as L α (t), for some t which is a continuous time for L α . Using the density of the jump points, there exists (T m , Z m ) such that T m < t and

L α (t) -L α (T m ) < | B α |/2. We then choose x α = L α (T m ).
In all cases, B α is replaced by

B α = [x α , y α ], where |B α | ≤ 3/2|B α |.
Similarly, if y α is not of the form L α (T n -) (i.e. the left limit of L α at T n for some jump point T n ), then :

• when y α / ∈ L α (E): B α can be replaced by B α = [x α , y α ], where y α = sup(B α ∩ E α ), without altering the covering R α . Since L α is increasing and càdlàg, y α is of the form L α (T n -) for some jump point T n .

• when y α = L α (T n ) for some jump time T n : Then B α can be replaced by {L α (T n )} ∪ B α , where B α = [x α , L α (T n -)]. Indeed, there is no point of E α between L α (T n -) and L α (T n ). • when y = L α (t) for some t which is a continuous time for L α . Using the same argument as above, there exists (T n , Z n ) such that T n > t and L α (T n ) -L α (t) < |B α |/2. We then choose y α = L α (T n -).

This proves the claim.

Observe that the previous Lemma holds almost surely, for every interval B α , for all α, since the randomness is only located in the distribution of the Point Poisson process and the strictly increasing and càdlàg properties of the processes, which hold simultaneously almost surely.

Next Lemma establishes that the increment of the process in an interval I is approximately the same order as the size of the largest jump in I, uniformly for all I and all the parameters. Lemma 14. With probability one, there exists a non-decreasing function g : [0, 1] → R + with g(0) = 0, continuous at 0, such that the following holds. Let (T m , Z m ) and

(T n , Z n ) (with T m < T n ) be two couples of the point Poisson process. Let B α = [L α (T m ), L α (T n -)], B β = [L β (T m ), L β (T n -)] and B f = [L f (T m ), L f (T n -)]. Then when B α = [L α (T m ), L α (T n -)] is small enough, one has (50) |B α | α/β+g(|B α |) ≤ |B β | ≤ |B α | α/β-g(|B α |) and (51) |B α | ≤ |B f | ≤ |B β |.

Proof

The three processes L α , L f and L β are almost surely pure jump processes with finite variations. One deduces that .

Then (51) follows immediately since f is monotone and α ≤ f (t) ≤ β.

We write B = [T m , T n ), and consider J the unique integer such that 2

-(J+1) < |T n -T m | ≤ 2 -J . We assume that J is so large that ε J ≤ (1/(1 -ε) -1)/4 ≤ (1/α -1)/4.
We now make use of Lemma 12.

Let (T N , Z N ) be the point Poisson process in the above sum (52) with largest jump Z N . We write

Z N = 2 -J N . Then one decomposes |B α | into (53) |B α | = Z 1/α N +
j≤J(1+η J ):Tp∈B and p∈P j Z 1/α p + j>J(1+η J ):Tp∈B and p∈P j Z 1/α p .

Call S 1 and S 2 the two above sums.

Assume that J N < J/3.

Observe that since B strictly contains an interval of length 2 -J-1 , the left inequality part (2) of Lemma 12 yields that J N ≤ (J + 1)(1 + η J+1 ).

Since B is contained in an interval of length 2 -J , one knows that all the jumps other than (T N , Z N ) appearing in formula (53) are smaller than 2 -J/(3α) . Hence, the right inequality in part (2) of Lemma 12 yields S 1 ≤ 2 Jε J 2 -J/(3α) .

Similarly, applying part (4) of Lemma 12, S 2 is bounded by

S 2 ≤ j≥J(1+η J )
2 j(1+ε j ) 2 -J 2 -j/α ≤ 2 J(1+η J )(1+ε J -1/α)-1 2 3/4(1/α-1) -1

≤ C ε 2 -J/α 2 J(ε J +η J (1+ε J -1/α)) , (54)
where C ε := 1 2 3/4(1/(1-ε)-1) -1 . Recalling that J N ≤ J/3, one gets |B α | ≤ (Z N ) 1/α + 2 -J(1/(3α)-ε J ) + C ε 2 -J(1/α-ε J -η J (1+ε J -1/α)) ≤ (Z N ) 1/α + (Z N ) 1/(α)-ε J N /3 + (Z N ) 3(1/α-ε J N -η J N (1+ε J N -1/(1-ε))) .

One concludes that (Z N ) 1/α ≤ |B α | ≤ (Z N ) 1/α-ε J N , for some ε J N which depends only on ε J N and η J N (not on α), is decreasing as a function of ε J N and η J N , and which tends to zero when J N tends to infinity. In addition, the fact that (Z N ) 1/α ≤ |B α | implies that J N ≥ -α log 2 |B α | ≥ -ε log 2 |B α | . Hence ε J N ≤ g 1 ( -ε log 2 |B α | ), where g 1 (r) = ε -ε log 2 r . One can write finally (55)

(Z N ) 1/α ≤ |B α | ≤ (Z N ) 1/α-g 1 (|B α |) .
By construction, this mapping g 1 is non decreasing with r, and tends to 0 when r tends to 0.

Observe that since ε J N is small (uniformly in α), one also has 2α) .

(56) (Z N ) 1/α ≤ |B α | ≤ (Z N ) 1/(
Assume now that J N ≥ J/3. All the jumps other than (T N , Z N ) involved in formula (53) are smaller than 2 -J N /α . Hence, part (2) of Lemma 12 yields S 1 ≤ 2 Jε J 2 -J N /α ≤ 2 -J N (1/α-3ε J N ) .

The sum S 2 is still bounded by above by (54) with J replaced by 3J N . One deduces that |B α | ≤ (Z N ) 1/α + (Z N ) 1/(α)-3ε J N + (Z N ) 3(1/α-ε J N -η J N (1+ε J N -1/(1-ε))) .

One concludes that |B α | ≤ (Z N ) 1/α-ε J N for some ε J N which depends only on J N (not on α), and which tends to zero when J N tends to infinity. For the same reasons as above, equation (55) holds true.

Using that (55) holds true with β instead of α (but with the same mapping ε), one sees that

|B β | ≤ (Z N ) 1/β-g 1 (|B β |) ≤ |B α | α/β-αg 1 (|B β |) .
In addition, using (56) with β instead of α, one has |B β | ≤ (Z N ) 1/2β ≤ |B α | α/(2β) . We deduce that αg 1 (|B β |) ≤ (1 -ε)g 1 (|B α | α/(2β) ) := g 2 (|B α |), hence (57)

|B β | ≤ |B α | α/β-(1-ε)g 2 (|B α |) .
Similarly, recalling that |B α | and |B β | are small quantities, (58)

|B β | ≥ (Z N ) 1/β ≥ |B α | 1/(β(1/α-g 1 (|B α |)) ≥ |B α | α/β+2βg 1 (|B α |) ≥ |B α | α/β+g 3 (|B α |) .
where g 3 (r) = 2(1 -ε) ε(r). Finally, (57) and (58) gives the result, with g(r) = max(g 2 (r), g 3 (r)).

Observe that one can also write for some mapping g which enjoys the same properties as g.

One can now prove Theorem 8.

The following holds almost surely, since it depends only on Lemmas 13 and 14.

Let us denote by

d α = dim H E α , d β = dim H E β , d f = dim H E f .
Let s > d α β/α, and let s = sα/β -(sα/β -d α )/2. One has d α < s < sα/β. By definition of d α , there exists η > 0 such that H s η/2 (E α ) ≤ 4 -s . Hence, for some η/2-covering R α of E α , one has

B α ∈R α | B α | s ≤ 2 -s .
First, using 13, by slightly modifying the intervals B α ∈ R α , one can replace these intervals with intervals of the form B α = L α ([T m , T n )) (plus at most a countable number of singletons), satisfying |B α | ≤ 2| B α |, whose union is still covering E α . Hence, the initial η/2-covering R α can be replaced by an η-covering R α , such that one has

B α ∈R α |B α | s ≤ 1.
Let us choose η so small that g(η α/β-g(η) ) < sα/β-dα 2s . Each ball B α is written L α (B), where B = [T m , T n ). As above, we write B β = L β (B) and B f = L f (B), and (50) and (51) hold true.

Since the balls (B α ) form an η-covering of E α , the balls (B β ) form a η := η α/β-g(η) -covering of E β , and the balls (B f ) also form a η-covering of E f . We denote by R β and R f these two coverings. One has

B f ∈R f |B f | s ≤ B β ∈R β |B β | s ≤ B α ∈R α |B α | s(α/β-g(|B α |)) ≤ B α ∈R α |B α | sα/β-(sα/β-dα)/2 = B α ∈R α |B α | s ≤ 1.
Since R β is an η-covering of E β , the s-pre-Hausdorff measure of E β , H s η (E β ) is less than 1. The same holds for H s η (E f ). This remains true for any sufficiently small η > 0, we conclude that both H s (E f There exists an η-covering R β of E β by intervals of the form B β = L β ([T m , T n )), such that

B β ∈R β |B β | s ≤ 1.
One considers the associated intervals (B α ) and (B f ), and the natural coverings R α and R f of E α and E f provided by these intervals.

Let η be so small that g(|B β |) ≤ g(η) <

s-d β α/β 4s
, where g is given by (59). One has

B α ∈R α |B α | s ≤ B β ∈R β |B β | s/(α/β+ g(|B β |)) ≤ B β ∈R β |B β | sβ/α-2s g(|B β |)β/α ≤ B β ∈R β |B β | sβ/α-(sβ/α-d β )/2 = B β ∈R β |B α | s ≤ 1.
This holds true for any η > 0 small enough, so that H s (E α ) < +∞, hence d α ≤ s. Since this holds true for any s > d β α/β, one gets that d α ≤ d β α/β.

One concludes that d α ≤ d f ≤ d β = d α β/α. Remark 8. It is certainly possible to short-cut the end of the proof, since one knows that α being fixed, almost surely, for every set E ⊂ [0, 1], dim H L α (E) = α dim H (E).

But since we consider all α's, it was easier to prove all inequalities at once.

hdFigure 1 .

 1 Figure 1. Space upper multifractal spectrum of µ α .

Theorem 3 .

 3 With probability 1, for every non-trivial open interval O ⊂ R, when h ∈ E, three cases may occur.

(

Figure 3 .

 3 Figure 3. Time upper multifractal spectrum of µ on an open set O. The spectrum (in thick) is obtained as the supremum of a random countable number of functions of the form gα(h) α (drawn using dotted graphs), for the values α ∈ β(M t ), t ∈ O. With probability 1, for every non-trivial open interval O ⊂ [0, 1],

•

  If σ belongs to the open interval O, the proof goes along the same lines as in item 1. replacing O by (-∞, σ) ∩ O.

6. 5 . 2 Υ

 52 Construction of the Cantor sets. We are ready to construct the families of Cantor sets {C(Υ, ε ) associated with càdlàg non-increasing functions Υ :[0, 1] → [1 + 2ε, 2 -2ε],where ε is any positive rational parameter. These sets are constituted by points which only see those double jump configurations studied before, and their Hausdorff dimension satisfies dim H C(Υ, ε ) ≥ min +2ε -1.

Theorem 8 .

 8 Let ε > 0. Denote by P = {(T n , Z n )} n≥1 a Poisson point process that generates the Poisson measure N (dt, dz) with intensity dt ⊗ dz/z 2 . Consider the family (15) of stable processes (L α .

  t∈E∩I β(M(t)), sup t∈E∩I β(M(t-)) ⊂ sup t∈E β(M(t-)) -η, sup t∈E β(M(t-)) .

Corollary 1 .

 1 For every open interval I ⊂ [0, 1] and h ≥ 0, consider the smallest interval I 0 ⊂ I (it may be not open or reduced to a point) such that E t µ (I, h) = E t µ (I 0 , h). Denote by d(I 0 ) the right endpoint of I 0 Almost surely, for every open interval I ⊂ [0, 1] and h ≥ 0, one has

M

  ((0, 1)) ∩ O = M( O). The set O is an open set (a, b) if M enters O continuously, or is a semiopen interval [a, b) if M enters O with a jump. In any case, M a-/ ∈ O and M b / ∈ O because O is open. Observe that

•

  Type A. The time level set E t µ ( O, h) = ∅, so one ignores the second set in the last union when computing the Hausdorff dimension of E µ (O, h). If #E t µ ( O, h) = 1 (necessarily h = β(M t ) for some t ∈ S(M) with β(M t ) ≥ 2β(M t-)), one has d µ (O, h) = 0 which coincide with the formula in Theorem 3. Otherwise Corollary 1 applied to h and O entails the existence of a minimal O 0 (that we can and will suppose open) such that

  as desired. • Type B. The time level set (49) E t µ ( O, h) = ∅ so one has to consider the (at most) countable set H = {M t : t ∈ S(M) ∩ O and dim(µ, M t-) = h}. Compared with the time spectrum, several cases may occur according to the value of h. Recall that O = [a, b) (when M jumps into O) or (a, b) (when M enters O continuously). In the following analysis, the case M t 0 /

  holds true for |B β | by replacing 1/α by 1/β. Similarly, |B f | = p∈N:Tp∈[Tm,Tn) Z 1/f (Tp-) p

( 59 )

 59 |B α | α/β+ g(|B β |) ≤ |B β | ≤ |B α | α/β-g(|B β |)

  ) and H s (E β ) less than 1, hence d f and d β are smaller than s. Since this holds for anys > d α β/α, one gets that max(d f , d β ) ≤ d α β/α.Next, starting with a η-covering of E f by balls B f , one associates with every ballB f = L f ([T m , T n )) the ball B β = L β ([T m , T n )),the same lines of computation (simply using that |B f | ≤ |B β |) yields that d f ≤ d β . The same argument shows that d α ≤ d f . It remains us to prove the last inequality d α ≤ d β α/β. The proof follows exactly the same lines, we write it without details. Let s > d β α/β, and let s = sβ/α -(sβ/α -d β )/2. One has d β < s < s β/α.

Proposition 6. The Cantor set C(Υ, ε ) is defined as

There exists a unique Borel probability measure ν Υ,ε supported exactly by C(Υ, ε ) such that for all n ≥ n 0 , the measure ν Υ,ε restricted to the σ-algebra generated by {J k : n 0 ≤ k ≤ n} is ν n .

The proof is immediate, since the step 4. of the construction ensures that the measure is a well-defined additive set function with total mass 1 on the algebra {J n : n ≥ n 0 } which generates the Borel σ-algebra, thus extends to a unique probability measure on Borel sets.

Observe that the construction of the family of Cantor sets depends only on Lemma 10, which holds with probability one simultaneously for all functions Υ, as desired. 6.6. Properties of the Cantor sets. The following proposition is key, since it shows that our construction guarantees that we have built points in F (Υ).

Proposition 7. Almost surely, for every non-increasing càdlàg function Υ : [0, 1] → [1 + 2ε, 2 -2ε] and for every small ε > 0,

Proof Suppose that t ∈ [0, 1] ∩ C(Υ, ε ) is a continuous time for M and Υ. One wants to prove that dim(µ, M t ) = Υ(t)β(M t ).

We start by bounding dim(µ, M t ) from below. By construction, for every n ≥ n 0 , t is covered by an interval

) entails that t ∈ J n is surrounded by two jumps of the Poisson point process located at t 1 n and t 2 n whose size belong to ∈ [η

and whose mutual distance is at least 3η n , and at most 5η n .

The process M jumps at t 1 n and t 2 n , with jump size η

, and

Hence, µ (B (M t , r n )) ≤ 5η n . Applying this when n becomes large, one gets

where we used the continuity of M and Υ at t.

The rest of the proof is dedicated to prove the converse inequality, i.e. dim(µ, M t ) ≤ Υ(t) • β(M t ), which is more delicate.

Let ε 1 > 0 be small. Thanks to the continuity of M at t, there exists r 0 > 0 such that r 0 ≤ η n 0 and

where J n 0 is the unique interval of C n 0 (Υ, ε ) that contains t. Now for any 0 < r < r 0 /3, there exists a unique integer n ≥ n 0 such that η n+1 ≤ r < η n . Let us call J n (t) and J n+1 (t) the unique intervals of J n and J n+1 that contain t.

By construction of the random tree T n (J n (t)), there is no large jump around t. More precisely, by Lemma 9,

Applying same argument as in Lemma 9 to scales between n 0 and n, together with the fact that the sequence n → Υ n+1 J n+1 (t) is increasing, yields that

One deduces that the increment of M between t -r and t + r has the form

Denote by m the unique integer such that 2 -m-1 ≤ 2r < 2 -m . One has

Applying Lemma 2 entails

Borel-Cantelli Lemma yields that when m becomes larger than some m 0 , for every γ

Assume that r 0 is so small that 2r 0 ≤ 2 -m 0 . By our choices for n and m, one has m > n, so Υ n+1 J n+1 (t) ∈ D m . By choosing γ = Υ n Jn(t) , u = t + r and v = t -r, one gets

In addition, by continuity of M at t, when r 0 is small enough, one has

On the other hand, recalling the constant ε 0 > 0 in Definition 1, an immediate computation shows that

as soon as r 0 is small enough.

Combining these estimates, one obtains that for all r ≤ r 0 , M t+r -M t-r ≤ r 1 (Υ(t)+3ε 1 )(β(M t )+ε 1 ) , which entails for all 0 < r < M t+r 0 -M t-r 0 , µ(B(M t , r)) ≥ r (Υ(t)+3ε 1 )(β(Mt)+ε 1 ) .

One concludes that dim(µ, M t ) ≤ (Υ(t) + 3ε 1 )(β(M t ) + ε 1 ).

Letting ε 1 → 0 yields the desired upper bound for dim(µ, M t ).