Multifractal analysis for the occupation measure of stable-like processes

In this article, we investigate the local behaviors of the occupation measure $\mu$ of a class of real-valued Markov processes M, defined via a SDE. This (random) measure describes the time spent in each set A $\subset$ R by the sample paths of M. We compute the multifractal spectrum of $\mu$, which turns out to be random, depending on the trajec-tory. This remarkable property is in sharp contrast with the results previously obtained on occupation measures of other processes (such as L{\'e}vy processes), since the multifractal spectrum is usually determinis-tic, 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 [14] on this subject.
We describe the local behavior 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 [18,13]. 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 [17,5,11,12,2], stochastic differential equations with jumps [4,28,26], the branching measure on the boundary of a Galton-Watson tree [21,22], local times of a continuous random tree [7,3], SPDE [24,23,19], Brownian and stable occupation measure [15,25,20,16,10], amongst many other references.
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), where 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 behavior. 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.

Definitions and notations
We first introduce the class of processes we focus on. Definition 1.1. Let ε 0 > 0, and β : R → [ε 0 , 1−ε 0 ] be a nowhere constant non-decreasing, Lipschitz continuous map. The stable-like process M is the pure jump Markov process whose generator can be written as (1.1) Introduced by Bass [6] 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 [4] or [28] for an explanation from the tangent processes point of view.
Let us comment on the assumptions. The truncation at 1 for the jump measure is not a restriction, since there is almost surely a finite number of large jumps in the unit interval. From a technical standpoint, our actual proof relies heavily on the monotonicity of the index function β. In particular, it is used (see Section 3) for determining the range of the local dimensions of the occupation measure; it is also essential in the transference from time spectrum to space spectrum (see Theorem 7.2). Of course, it would be worth studying the same question with weaker assumptions.
Let M = {M t , t ∈ [0, 1]} be a stable-like process associated with a given function x → β(x) as in Definition 1.1. Our purpose is to describe the local behavior of the occupation measure of M defined as µ(A) =  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. and the similar set for lower local dimensions E t µ (O, h). The corresponding time upper multifractal spectrum of µ is One defines similarly the time lower multifractal spectrum d t µ (O, ·).

Main results
Let us start with known results on stable subordinators. The famous paper by Hu and Taylor [15] states that for every α-stable subordinator {L α t ; t ∈ R + } whose occupation measure is denoted by µ α , almost surely for all x ∈ Supp µ α , dim(µ α , x) = α and dim(µ α , x) ∈ [α, 2α]. (1.3) It is a classical result [9] 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. Hu and Taylor also prove that the upper spectrum is much more interesting (see The method developed by Hu and Taylor consists first in proving (1.5), and then in applying (1.6) to get (1.4). We follow this strategy in our proof.
Our first result gives the possible values for the local dimensions of the occupation measure µ associated with a stable-like process M, which is an analog of (1.3).
Theorem 1.5. Consider a stable-like process M associated to a non-decreasing mapping β, as in Definition 1.1, and the associated occupation measure µ. With probability 1, for every x ∈ Supp (µ), Hence, the supports of the lower spectra d µ and d t µ are random, depending on the trajectory of M.
The time lower multifractal spectrum is then quite easy to understand, since the level considering space lower spectrum. Theorem 1.5 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 1.7 on time spectra. Set Note that the only difference between g andĝ is at the value h = 2α. Definition 1.6. For every monotone càdlàg function Υ : R + → R, we denote by S(Υ) the set of jumps of Υ.
In addition, with probability one, for every h ∈ R + \ E 1 and every non-trivial open interval (1.9) Remark 1.8.
• The first part is trivial. Observe that there is a subtle difference between (1.7) and (1.11) (space lower spectrum), since at each jump time t for M, • If, for instance, the range of β(·) is included in [1/2, 9/10], then almost surely E 1 = ∅.
• The proof of Theorem 1.7 is based on a more general result (Theorem 4.1) for time spectrum.
The space multifractal properties of µ are summarized in the following theorem. Theorem 1.9. Set the (at most countable) sets of real numbers (1.10) where E 1 is defined in Theorem 1.7. With probability 1, for every non-trivial open interval O ⊂ R, one has (1.11) and for every h ∈ R + \ E, 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, t ∈ [0, 1]}. This formula finds its origin in the fact that locally, M behaves around each point of continuity t as an α-stable process with α = β(M t ).
The allure of a typical space upper multifractal spectrum is depicted in Figure 3. This shape is very unusual in the literature.
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 t0− ) to β(M t0 ), and for t ≥ t 0 , the only possibility to have d µ (O, M t ) = β(M t0 ) is when t = t 0 . Similarly, when t < t 0 , it is not possible to have d µ (O, M t ) = 2β(M t0− ).
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 , see Definition 1.1.
All this explains the set of exceptional points E in Theorem 1.9. We deal with these exceptional points in Section 7, Proposition 7.1, whose statement is rather long but whose proof follows directly from a careful analysis of the previous results.
Following these lines, we start by proving Theorem 1.7. 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.
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 1.5) are studied in Section 3. The time spectra (Theorems 1.7) are obtained in Section 4 using a general result (Theorem 4.1), whose proof is given in Sections 5 and 6. Finally, the space spectrum (Theorem 1.9) is dealt with in Section 7, together with the dimension of images of arbitrary sets by stable-like processes (Corollary 7.3).

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 [4,28,27]. 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 [1]. We denote Recall the definition of a stable-like process and formula (1.1). 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   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 [4] or Proposition 2.1-2.3 of [28] with some slight modifications. The second item is standard, see for instance Section 2.3 of [1].

Remark 2.2.
Recall that S(Υ) is the set of jumps of a monotone càdlàg function Υ : 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 2.3. Consider the process M and L α for all α ∈ (0, 1) introduced in Proposition 2.1. Almost surely, for all 0 ≤ s ≤ t ≤ 1, 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 [4] for a proof.

Local dimensions of µ: Proof of Theorem 1.5
Recall that (1.3) holds for a stable subordinator. A straightforward adaptation of Hu and Taylor's argument leads to (1.3) for a stable subordinator without jumps of size larger than 1. Now, observe that almost surely, for all α ∈ Q ∩ (0, 1), formula (1.3) is true. This, together with Proposition 2.3, leads to the local dimension of µ.
Proof of Theorem 1.5. Three cases may occur.
1. x = M t , where t is a continuity point of M. Due to the coupling in their construction (Proposition 2.1), 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 2.3 to L α and L α , one gets when r is small By formula (1.3) for the lower and upper local dimensions of µ α , for all small ε > 0, almost surely, one has for r small enough that ≤ 2α + ε, and the same for α . Hence On the other hand, still by formula (1.3), dim(µ α , x) = α , so there exists a sequence (r n ) converging to 0 such that Letting ε tend to zero and α, α tend to β(M t ) with rational values yields β( 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 (3.1) is straightforward using Proposition 2.3. We follow the same lines in the first case to obtain the desired result.
which completes the proof.

A general result to get the time spectrum (Theorems 1.7)
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.

Proof for the time upper multifractal spectrum
Let us explain why Theorems 1.5 and 4.1 together imply the time upper multifractal spectrum. Theorems 1.7. Let us start with formula (1. This proves (1.9).
One wants to prove formula (1.8) for h / ∈ E 1 , which can be rewritten as We prove now that formula (4.1) applied to the family {Υ h : h ≥ 0} implies formula (4.2). Several cases may occur according to the value of h.

Second case:
This has been considered at the beginning of this section.

Reduction of the problem
we proceed in two parts: • first, in Section 5, we show that simultaneously for all Υ and O.
• second, in Section 6, we complete the result by proving that also simultaneously for all Υ and O, almost surely. It is also enough to get the result for O = (0, 1).
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 5.1. With probability 1, one has: for each t ∈ E(γ) and ε > 0 small, there exists an infinite number of integers 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.
Proof. Let us prove that t satisfies lim sup Assume first that M is continuous at t. Assume toward contradiction that for all s > 0 The same holds true when |M t − M t−s | ≤ s 1/((γ−ε/5)β(Mt)) . We have thus proved that (5.4) holds for every small r by continuity of M at t, this contradicts (5.1).
When t is a jump time for M, the proof goes as above using the two obvious remarks: Next technical lemma, proved in [28], shows that when (5.2) 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 5.3 ([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 Remark 5.4. The formula looks easier than the one in [28] because in our context M is increasing. When the function β is constant, the error term 2 −n in time (on the LHS of the inequality) is hidden and the term 2/n in the previous inequality disappears, see [2].
Recall formula (2.2) 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 N 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 5.3 with δ = γ − ε yields that for all integers n greater than some n γ−ε , (the term 2 −n+1 on the RHS of the inequality is not a typo) On the other hand, for all integers n greater than some other n γ−ε , a direct computation Therefore, for all large n, Let us introduce, for every integer n ≥ 1, the process . A direct estimate shows that by right-continuity of M, when n becomes large, one Recalling formula (2.2), the three inequalities (5.2), (5.5) and (5.6) imply that for an infinite number of integers n 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 5.1 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 One introduces the collection of sets 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 . 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 5.5. With probability 1, there exists a constant C such that for all n ≥ 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 where C n is a constant depending on n which stays bounded away from 0 and infinity.
Let s > 2 γ−ε − 1. Fix η > 0 and n 1 so large that all intervals I ∈ E n1 (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 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, Letting ε → 0 yields the desired upper bound.

Proof of Theorem 4.1: lower bound
The aim of this section is to get that with probability one, (4.5) holds with O = (0, 1) for all non-increasing càdlàg function Υ : Recalling the notations in Theorem 4.1, for simplicity, we write 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.

The time scales, and some notations
We aim at constructing Cantor sets inside F (Υ). Recalling Proposition 5.1, 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 between two large jumps which are really close to each other. So the Cantor set we are going to construct will focus on this behavior.
By convention, J n, ,−1 = J n, ,−2 = J n, , Observe the multi-scale structure of these intervals: the main scales are determined by the sequence {J n,0 ; n ≥ 1} which decays extremely fast, and the intermediate scales are determined by {J n, ; ∈ {0, . . . , n }} for each n, observing that J n, n = J n+1,0 . We use subsequently the notation J n = J n,0 . When it comes to the estimates for the intermediate scales (e.g. Section 6.3), J n,0 will be used, otherwise J n will be used. The same remark is valid for η n and η n,0 .
Finally, one needs the enlarged intervals J n, ,k = k+1 i=k−1 J n, ,i .

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 . Remark 6.2. 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.
• 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.

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,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 6.11.

Remark 6.4.
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.
Iterating this computation yields the desired inequality.
We are now in position to prove Proposition 6.5.
Thus for n large enough, 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, where the fast decay rate of (η n, ) to zero has been used for the third inequality.
Finally, to prove Proposition 6.5, since the cardinality of J n,0 is less than η −1 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 6.10. 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+1,0 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 6.11. Let J ∈ J n,0 and r ∈ [η n+1 , η n ). Assume that T n,γ (J) is not empty. Then for each t ∈ T n,γ (J), 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 Further, all ancestor intervals of J n, 0 (t) in the intermediate scales (for a fixed n), i.e. intervals J n, (t) where 0 ≤ < 0 , satisfy the zero jump configuration by construction, in particular, Combining these estimates yields the result.

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 n . : 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 .
Recall that J n = J n,0 and J n+1,0 = J n, n with the notations of the previous sections.
By Lemma 6.3 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 ) The above probability is thus bounded by above by On the other hand, by Lemma 6.9 one has There are less than 2 n possible choices for γ, and 4 choices for a. Hence, 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), Let n 0 be the random integer obtained in Lemma 6.13. We assume that n 0 is so large that the conclusions (6.5) of Proposition 6.5 hold, and also that 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 #{J ∈ J n : J ⊂ [t ε − r ε , t ε + r ε ] and Osc Υ (J) ≥ 2 −n } ≤ K Υ · 2 n (6.11) (6.13) 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 , , and by saying that ν n+1 is uniformly distributed inside each J n+1 .
Proposition 6.14. 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 6.13, which holds with probability one simultaneously for all functions Υ, as desired.

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).
Borel-Cantelli Lemma yields that when m becomes larger than some m 0 , for every γ ∈ D m ∩ [1 + 2ε, 2 − 2ε] and |v − u| ≤ 2 −m (with u > v), Assume that r 0 is so small that 2r 0 ≤ 2 −m0 . By our choices for n and m, one has m > n, so Υ n+1 Jn+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 1.1, an immediate computation shows that as soon as r 0 is small enough.
In addition, by construction, one has . (6.15) Hence by (6.13), since all the intervals J n1+1 of generation n 1 + 1 within J n1 have the same ν-mass, one has (using (6.15)) Due to our choices for the sequence (η n ) n≥1 , when n 0 is large, , so applying the inequality x ∧ y ≤ x s y 1−s for s ∈ (0, 1) and 0 < x, y < 1 yields 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.
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 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 1.5). 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 6.3 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.
So the probability in question is less than Borel-Cantelli Lemma implies the existence of a sequence of embedded intervals with length tending to 0 that satisfy (6.17). This justifies that F (Υ) = ∅.
7 Space spectrum: proof of Theorem 1.9 Next proposition deals with (at most countable) exceptional values of h. This, combined with Theorem 1.9, completes the statement of the space spectrum.

A first theorem on dimensions, and the space spectrum
Throughout this section, we set ε = ε 0 , which is defined in (1.1). We are going to prove the following theorem.
The following corollary should be understood as a generalization of the uniform dimension result of type (1.6) to stable-like processes.    Proof. The first part (the formula (7.2)) is immediate.
For every η > 0, there exists an interval I of length less than η such that inf t∈E∩I β(M(t)), sup These results are fine enough for us to deduce Theorem 1.9 from Theorem 1.7. Also, the corollary solves partially a question left open in [27].
Before proving Theorem 7.2 next subsection, let us explain how we deduce the space spectrum of the occupation measure. The following lemma which will be used in the proof of Theorem 1.9.  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 non-trivial interval. One could check the analysis in the proof of Theorem 1.7 for a construction of I 0 . Observe that the left-hand side of (7.3) is less than or equal to the right-hand side due to Corollary 7.3.
as desired.
Routine computations as in Lemma 6.3 entail Lemma 7.5.
Lemma 7.6. 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 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 point of continuity 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 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 . 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 point of continuity 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 7.7. 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.
We now make use of Lemma 7.5.
Let (T N , Z N ) be the point Poisson process in the above sum (7.7) with largest jump Z N . We write Z N = 2 −J N . Then one decomposes |B α | into |B α | = Z 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 7.5 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 (7.8) are smaller than 2 −J/(3α) . Hence, the right inequality in part (2) of Lemma 7.5 yields S 1 ≤ 2 Jε J 2 −J/(3α) .
One concludes that 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 (Z N ) 1/α ≤ |B α | ≤ (Z N ) 1/α−g1(|B α |) . (7.10) By construction, this mapping g 1 is non decreasing with r, and tends to 0 when r tends to 0.
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 (7.10) holds true.

Stable-like occupation measure
One can now prove Theorem 7.2.
The following holds almost surely, since it depends only on Lemmas 7.6 and 7.7.
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 First, using 7.6, 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 .
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 (7.5) and (7.6) 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 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 ) and H s (E β ) less than 1, hence d f and d β are smaller than s. Since this holds for any s > 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 ball B 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.
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.