The singularity spectrum of L´evy processes in multifractal time

The singularity spectrum of L´ evy processes in multifractal time

Julien Barral

INRIA, Domaine de Voluceau Rocquencourt, 78153 Le Chesnay cedex, France.

Email-address: [email protected]

St´ ephane Seuret

Laboratoire d’Analyse et de Math´ematiques Appliqu´ees - Facult´e de Sciences et Technologie - 61, avenue du G´en´eral de Gaulle, 94010 Cr´eteil cedex, France.

Email-address: [email protected]

Key words: Stochastic processes, Hausdorff measures and dimension, Multifractal functions and measures, Self-similar measures

PACS:60Gxx, 28A78, 28A80, 28C15

1 Introduction

The interest for multifractal stochastic processes is mainly motivated by the need for accurate models in the study of the variability of wild signals. These locally irregular signals come from physical phenomena such as fully developed turbulence, TCP Internet traffic, variations of financial prices, or heart beats.

Fractional Brownian Motions (FBM), L´evy processes and multiplicative cas- cades are frequently used when modeling these phenomena. However, these processes are partly satisfactory for different reasons. FBM are monofractal, and thus have the same H¨older exponents at every point. The two other models are multifractal, i.e. the pointwise H¨older exponents take several values, and the level sets of their H¨older exponents are dense fractal sets. Nevertheless the singularity spectra of the L´evy processes have a very specific linear increas- ing shape and, finally, the multifractal multiplicative cascades only generate non-decreasing processes.

Other kinds of multifractal models were thus studied to go beyond these limi- tations. For instance, Gaussian processes with non-constant prescribed H¨older exponents are introduced in [2]. Another approach consists in generating mul- tifractal random wavelet series [23,7].

A third point of view consists in performing a (possibly multifractal) change of time in a given stochastic process (X_t)t≥0. More precisely, given an atomless positive Borel measure µ on R+ supported by an interval of the form [0, T] (T ∈ (0,∞)), then the process X ◦µ([0, t]) is considered. This process shall be viewed as the processX in (again, possibly multifractal) time µ.

The simplest situation lies in taking X equal to a monofractal process, like the FBM (see [32,3,14] and Section 6). In this case, due to the monofractality property, the multifractal nature of X◦µfollows almost straightforward from the one of µ (see Section 6). In the situation when it is assumed that X also has multifractal sample paths, the multifractal time change creates more interesting structures, both from the modeling and mathematical viewpoints (see for instance [37] for preliminary results on this topic, especially concerning large deviation spectra). The fine local study of the sample paths multifractal properties is far more delicate than in the monofractal case. To our knowledge it has never been achieved in a non-trivial case.

This paper deals with the case when X is a L´evy process. We provide con- ditions on the measure µ under which the multifractal nature of the sample paths of the process (Z_t = X ◦ µ([0, t]))t≥0 can be described. Before going further, we detail the reason which led us to consider this problem.

Let b be an integer ≥ 2 and W = (W₀, . . . , Wb−1) a positive random vector.

Then consider in the space of Laplace transforms of probability distributions φ on R+ the equation

φ(u) =E

^b−1 Y

i=0

φ(uW_i)

, ∀ u≥0. (1)

This equation, referred to as the smoothing transformation, is solved in [15,18].

It comes from the modeling of fully developed turbulence [31,30] and of in- teracting particles systems. Subsequently, the problem is then to find all the non-trivial solutions (i.e. 6≡1) of (1). The mapping

ϕ_W :q∈R7→ −log_bE

^b−1 X

i=0

W_i^q

∈R∪ {−∞}. (2) naturally arises in the problem’s solution. Indeed, under the assumption that ϕ_W(p)>−∞ for somep >1, it is proved by Durrett and Liggett in [15] that (1) has non-trivial solutions if and only if there exists β ∈ (0,1] such that ϕ_W(β) = 0 andϕ⁰_W(β)≥0. As a consequence of the concavity of the mapping ϕ_W, such a β is unique and

β = inf{β⁰ ∈[0,1] :ϕ_W(β⁰) = 0}.

It is worth noting that the existence of non-trivial solutions in the general

framework is almost entirely based on the existence of a non-trivial solution in the case β = 1 with ϕ⁰_W(1) > 0. Moreover, in this case, a fundamental non-trivial solution is given by the Laplace transform of the probability distri- bution of kµWk, where µW is an independent multiplicative cascade on [0,1]

generated by the random vector W = (W₀, . . . , Wb−1) used in (1), see [31,26]

and Section 7 for the construction of µ_W. This type of multiplicative cascade measures has been extensively studied in [25,19,16,34,1,4]. Their well-known multifractal properties are closely related toϕ_W and (1).

Therefore, as soon asϕ_W(1) = 0 and ϕ⁰_W(1)>0, it is possible to naturally as- sociate the non-trivial stochastic process (Z_W,t)t∈[0,1] =µ_W([0, t])

t∈[0,1] with (1) such that the Laplace transform of Z_W,1 resolves (1). Moreover, this pro- cess ZW,· is completely characterized by a statistical self-similarity property (see (40) in Section 7).

This raises the problem of finding a natural process satisfying the same prop- erties in the general case β ∈ (0,1]. In the case β ∈ (0,1), ϕ_W(β) = 0 and ϕ⁰_W(β) > 0, we recall how the solution φ of (1) is deduced in [15,18] from the construction of kµ_Wk. First, the random vector W_β = (W₀^β, . . . , W_b−1^β ) is considered. By construction we get thatϕ_Wβ(1) = 0 andϕ⁰_W

β(1) >0, and the situation is reduced to the one described above.

Let φ_β be the Laplace transform of kµ_Wβk. A non-trivial solution of (1) is then given by the mappingφ:u7→φ_β(u^β). Let X_β be a β-stable L´evy subor- dinator independent of µ_Wβ. Remark that the function φ is also the Laplace transform of the random variable Z = X_βkµ_Wβk ([15]). Hence, a method to construct a stochastic process (ZW,t)t∈[0,1] associated with φ and fulfilling the statistical self-similarity property (40) is then the following: Consider the stochastic process

ZW,t=Xβ

µW_β([0, t])=Xβ(ZW_β,t) (t∈[0,1]). (3) This process has the form of a L´evy process in multifractal time, and it pos- sesses the required properties. Indeed, the Laplace transform of Z_W,1 resolves (1), and in addition, since Xβ has by construction independent increments and is independent of µ_Wβ, the increments of Z_W,t also satisfy the statistical self-similarity property (40). Surprizingly enough, stable L´evy subordinators and Mandelbrot multiplicative cascades thus appear as special elements of the same class of processes (obtained by subordinating the integral of a Mandel- brot cascadeµ_W to an independent L´evy subordinator X_β) obeying a certain statistical self-similarity property.

Equation (1) can also be considered in the space of characteristic functions of probability distributions onR. It is shown in [29] that if there exists β ∈(1,2]

such that ϕW(β) = 0 and ϕ⁰_W(β) ≥ 0, then (1) possesses a non-trivial non- positive solution. If ϕ⁰_W(β)>0, we associate naturally with that solution the stochastic process (Z_W,t)t≥0 formally defined as in (3), but with a symmetric

β-stable L´evy process X_β (a Brownian motion without drift if β = 2). Again, the multifractal nature of (Z_W,t)t≥0 appears to be related to ϕ_W.

We now resume the problem we address (i.e. to perform the multifractal anal- ysis of a L´evy process in multifractal time) and our results.

First, the local regularity of a functionf is measured in this paper as follows.

Let d ≥ 1, I a non-trivial subinterval of R+, and f : I → R^d. If x ∈ I, the pointwise H¨older exponenth_f(x) of f atx is defined¹ by

h_f(x) = lim inf

y→x y6=x

log|f(y)−f(x)|

log|y−x| , (4)

where | · |stands for the Euclidean norm, with the convention |log(0)|=∞.

Then the multifractal nature of f is expressed in terms of the size of the levels sets E_h^f of the function h_f(·) defined by E_h^f = {x ∈ I : h_f(x) = h}

(h≥0). This size is measured by the Hausdorff dimension (denoted dim, see Definition 3). Thus we focus on the estimation of the mapping

d_f :h≥07→dimE_h^f,

which is called singularity spectrum or Hausdorff multifractal spectrum of f. A function (resp. a process) is said to be multifractal when its singularity spectrum (resp. the singularity spectrum of its sample paths) is not reduced to a single point (resp.with probability 1).

The singularity spectrum of L´evy processes (X_t)t≥0 – which corresponds in our context to the case where the measureµequals the Lebesgue measure – is performed in [22] (see Theorem 1 below). There is no time change in this case:

L´evy processes without Brownian part have with probability 1 a non-trivial linear multifractal spectrum. This typical shape is explained by the fact that the jump points of L´evy processes satisfy a ubiquity property with respect to the Lebesgue measure (the notion of ubiquity is detailed in Section 3.4).

In our context, when the measureµis not monofractal, that is when the H¨older

1 This exponent does not coincide with the usual pointwise exponent, that we denoteH_f(x), which involves a polynomial ([20]). Ifh_f(x)∈R⁺\N^∗, then h_f(x) = Hf(x) but the two notions may differ if hf(x) ∈ N^∗. Nevertheless hf(x) is the natural notion to be used here. Indeed, the study of (Zt) requires information on the local behavior of t 7→ µ([0,·]), i.e. on the H¨older exponents of the measure µ.

These exponents are in general more tractable by using a definition similar to (4) than with the definition of [20].

exponent function of the measure µ h_µ :t7→lim inf

r→0⁺

logµ(B(t, r))

log(r) (5)

possesses several non-trivial level sets, the situation becomes subtler. We prove that the local behavior of the process (Z_t=X◦µ([0, t]))t≥0 is closely related to some conditioned ubiquityproperties (see Section 3.4), which combine con- ditions on the jump points of (Z_t) with conditions on the local behavior of µ.

Understanding these properties enables us to compute the singularity spec- trumd_Z, under suitable assumptions. These technical assumptions are fulfilled by several classes of statistically self-similar measures µ with a construction based on multiplicative cascade schemes, for instance some R+-martingales (like µ_W above) in the sense of [24,6] or random Gibbs measures (see [9,10]).

Before summarizing our results, we start by recalling precisely the theorem obtained in [22]. Let X = (X_t)_t≥0 be a R^d-valued L´evy process. Recall that X has stationary independent increments and that its characteristic function takes the form E

e^ihλ|Xti=e^−tψ(λ), where ψ(λ) = iha|λi+Q(λ)/2 +

Z

R^d

1−e^ihλ|xi+ihλ|xi1|x|≤1

π(dx)

and where a ∈ R^d, Q is a quadratic form, and π is a Radon measure on R^d\ {0}, called the L´evy measure of X, satisfying

Z

(1∧ |x|²)π(dx)<∞. (6) Define the Blumenthal-Getoor exponent of X as

β = inf

γ ≥0 :

Z

|x|≤1|x|^γπ(dx)<∞

. We always have β ∈[0,2]. Remark that

β = sup0,lim sup

j→+∞

j⁻¹log₂C_j, where C_j =

Z

2^−j−1≤|x|≤2^−jπ(dx) (j ≥1). (7) We focus on the pointwise H¨older exponents of sample paths ofX, thus with- out loss of generality we omit the jump points generated by the compound process with intensity 1_{|x|>1}π(dx). When ^R(1∧ |x|)π(dx) < ∞, there are also several ways to write X as the sum of a Brownian motion B with drift a⁰ ∈ R^d and covariance matrix Q and of a L´evy process X^fof L´evy measure 1_{|x|≤1}π(dx), even when requiring that B and X^fare independent.

For j ≥ 0, let π_j(dx) = 1_{2^−j−1_<|x|≤2^−j_}π(dx). Then let (Y_j)j≥0 be a sequence of independent compound Poisson processes such that the L´evy measure ofY_j

is πj. We then choose X^f. as follows:

Xf_t=^X

j≥0

X_j(t) whereX_j(t) =







Y_j(t) if β <1,

Y_j(t)−^R xπ_j(dx) if β ≥1. (8) Then a general L´evy process (with jumps of norm ≤1) has the form

X =X^f+B(a⁰, Q), (9)

whereB(a⁰, Q) is a Brownian motion with drifta⁰ ∈R^dand covariance matrix Q, independent of X^f(of course if Q= 0 then B is degenerate).

We now state the theorem of [22] using the pointwise H¨older exponent intro- duced above in (4) instead of the classical one.

By convention, dim E =−∞ means that the setE is empty.

Theorem 1 Let X be a L´evy process decomposed in the form X^f+B(a⁰, Q) as in (8) and (9), and consider the associated process X. Suppose that^f β ∈(0,2]

and ^P_j≥12^−jqCjlog(1 +Cj)<+∞ (this holds as soon as β <2).

With probability 1, d

Xe(h) = βh if h∈[0,1/β] and −∞ otherwise.

The influence of B(a⁰, Q) is also studied in [22], and the corresponding result is recalled in Theorem 3.

We now consider a positive Borel measureµwith a support equal to [0,1] and its integral F, i.e. F is the mapping u ∈ [0,1] 7→ µ([0, u]). Let (Z_u)u∈[0,1] be the L´evy process in time F (or µ) given by (Zu =X_F(u))u∈[0,1].

Ifµis a multifractal measure, thenF is a multifractal non-decreasing function.

We are going to assume that µ is atomless, hence F is also continuous on [0,1]. We use the pointwise exponent of µ defined in (5). If h ≥ 0, the level sets E_h^µ of the measure µ are defined as E_h^µ = {u : h_µ(u) = h}. Finally, the singularity spectrum (or Hausdorff multifractal spectrum) ofµis the mapping d_µ:h7→dim E_h^µ.

The so-called scaling functionτµ orL^q-spectrum associated with the measure µis involved in our result. It is classically defined for positive Borel measures µon [0,1] as

τµ :q7→lim inf

j→+∞ −j⁻¹log₂ ^X

0≤k≤2^j−1

µ[k2^−j,(k+ 1)2^−j)^q. (10) The dyadic basis chosen in the definition (10) is not a restriction. Indeed, since supp(µ) = [0,1], a different integer basis b≥2 would give the same value for τ_µ.

The Legendre transform f^∗ of a function f : R⁺ → R∪ {−∞} is defined as f^∗ :h7→infq∈Rhq−f(q).

Roughly speaking, our result yields the singularity spectrumd_Z ofZ when the measureµobeys themultifractal formalismin the sense thatd_µ(h) = τ_µ^∗(h) for all h (for detailed studies of multifractal formalisms for measures, the reader is referred to [13,35]). This property holds for many classes of statistically self-similar measuresµ. These measures also satisfy three technical conditions C1-3 invoked in our statement. For sake of shortness in this introduction, these conditions are specified later in Section 3.4. Among our assumptions, we shall keep this property in mind:

τ_µ⁰(1) exists and is strictly positive. (11) This implies that the lower and upper Hausdorff dimensions ofµcoincide with τ_µ⁰(1) (see [33] for the corresponding definitions).

We shall prove the following result, which includes Theorem 1 as the special case where µis the Lebesgue measure.

Theorem 2 Let X be a L´evy process decomposed in the form X^f+B(a⁰, Q) as in (8) and (9). Suppose that β ∈(0,2], and ^P_j≥12^−jqC_jlog(1 +C_j)<+∞.

Let µ be an atomless positive Borel measure whose support is [0,1], such that (11) and C1hold true.

We introduce the exponents h_µ,β =τ_µ⁰(1)/β and α_max= sup{α:τ_µ^∗(α)≥0}.

Let (Z^e_u)_u∈[0,1] be the stochastic process defined by Z(u) =^e X^f_µ([0,u]) (i.e. the influence of B(a⁰, Q) in the decomposition (9) is not taken into account).

With probability 1:

(1) For every h∈[0, h_µ,β), d

Ze(h)≤βh.

Moreover, if C2(h_µ,β) holds, then for every h∈[0, h_µ,β), d

Ze(h) =βh.

(2) If h∈[h_µ,β, α_max/β], d

Ze(h)≤τ_µ^∗(βh).

Moreover, if C3(βh) holds, then d

Ze(h) =τ_µ^∗(βh).

(3) If h > αmax/β then E

Ze(h) =∅.

The singularity spectrum of Z^e is thus composed of two parts (see Figure 1): First a linear part of slope β, then a concave part which is a dilated and translated version of (a part of) the singularity spectrum of the initial measure µ. This shape reflects the combination of an additive structure (the L´evy process) with a multiplicative structure (the multifractal measure µ). Such a behavior is observed for the heterogeneous sums of Dirac masses studied in [8].

For the sequel, we noteD_µ,β(h) the singularity spectrum obtained in Theorem

hs h

s h /_s !

h /_s ! h h

h

d (h) d (h)

d (h)

hs

! ^{X o F} XoF

!"# Case !$#

Case

Fig. 1. Typical multifractal spectra of Left: a statistically self-similar measure µ, Middle:a L´evy process in multifractal time Xe◦F when β >1, andRight:when β ≤1. Here h_s is the Lebesgue-almost sure exponent, i.e.h_s=τ_µ⁰(0⁺).

2, i.e. it is the mapping

Dµ,β(h) =











βh if h∈[0, h_µ,β) τ_µ^∗(βh) ifh∈[h_µ,β, α_max/β]

−∞ otherwise

(12)

Remark that the singularity spectrum ofZ^e is obtained as the Legendre trans- form of the function

τ_µ,β(q) =







τ_µ(q/β) if q≤β,

0 otherwise

as soon as C2(h_µ,β) and C3(h) hold true for all h∈[τ_µ⁰(1), α_max).

As said above, examples of measures illustrating our result are Gibbs measures and their random counterparts studied in [17,27,9], and of course the indepen- dent random cascades µ_W mentioned above in the study of the fixed points of the smoothing transformation (1). Other examples are the compound Poisson cascades and otherR+-martingales studied in [5,3,6].

We now treat the general case, i.e. the influence of the drift and of the Brow- nian component.

Theorem 3 Under the assumptions of Theorem 2, introduce the exponents

ehµ,β = inf{h ≥0 :βh < τ_µ^∗(h)} if β <1 and hµ,β = inf{h≥0 :βh < τ_µ^∗(2h)}.

We always have h^eµ,β < hµ,β and hµ,β ≤τ_µ⁰(1)/2≤hµ,β. Consider the two mappings (D^f_µ,β is defined if β <1)

d (h)

XoF d (h)

XoF

Q = 0/ Q = 0, <1 and a’=0 ! /

h

_s h /_s ! ^h h /2_s ^h_s ^h Fig. 2. Typical multifractal spectra of Left: a L´evy process in multifractal time Xe◦F when β <1, andRight: when Q6= 0, here withβ = 1.

Df_µ,β(h) =











βh if h∈[0,h^e_µ,β) τ_µ^∗(h) if h∈[h^e_µ,β, α_max]

−∞ otherwise

and D_µ,β(h) =











βh if h∈[0, h_µ,β) τ_µ^∗(2h) if h∈[h_µ,β,^αmax₂ ]

−∞ otherwise.

Let (Z_u)u∈[0,1] be the stochastic process defined by Z(u) = X_µ([0,u]).

(1) Suppose that Q= 0 and (a⁰ = 0 if β < 1). With probability 1, the same conclusions as for Theorem 2 occur here.

(2) Suppose that Q= 0, β <1 and a⁰ 6= 0. With probability 1, (a) d_Z ≤D^f_µ,β.

(b) If C2(h_µ,β) holds, then for every h∈[0,h^e_µ,β), d_Z(h) =D^f_µ,β(h).

(c) If (τ_µ^∗(h^e_µ,β) = β^eh_µ,β andC2(h_µ,β)holds), or if (τ_µ^∗(^eh_µ,β)> β^eh_µ,β and C3(^eh_µ,β) holds), then d_Z(^eh_µ,β) =D^f_µ,β(h^e_µ,β).

(d) If h∈(h^e_µ,β, α_max] and C3(h) holds, then d_Z(h) =D^f_µ,β(h).

(e) If h > α_max then E_Z(h) = ∅.

(3) Suppose that Q6= 0. With probability 1, (a) d_Z ≤D_µ,β.

(b) If C2(h_µ,β) holds, then for every h∈[0, h_µ,β), d_Z(h) =D_µ,β(h).

(c) If (τ_µ^∗(2hµ,β) = βhµ,β and C2(hµ,β) holds), or if (τ_µ^∗(2hµ,β) > βhµ,β

and C3(2h_µ,β) holds), then we have d_Z(h_µ,β) = D_µ,β(h_µ,β).

(d) If h∈(h_µ,β, α_max/2] and C3(2h) holds, then d_Z(h) = D_µ,β(h).

(e) If h > α_max/2 then E_Z(h) =∅.

The conclusions of items (2) and (3) are simple consequences of the fact that respectively a linear drift and a Brownian component are added to the “pure”

L´evy processX. The corresponding spectra are simply obtained as supremum^f of two spectra. This explains their non-concave shapes (see Figure 2).

The paper is organized as follows.

Section 2 recalls some useful properties of measures.

Section 3 introduces the main tools used in the proof of Theorem 2. Proper- ties of Poisson point processes are discussed, and estimates for the increments of X^fobtained in [22] are recalled. Then, results on heterogeneous ubiquitous systems (introduced in [11]) are stated, and conditions C1-3 are defined.

Section 4 is devoted to the proof of Theorem 2 whenB(a⁰, Q)≡0. Sections 5 and 6 complete the proof to yield the general case B(a⁰, Q)6≡0.

Section 7 deals with the validity of conditionC2(h_µ,β)for independent multi- plicative cascades, which play a central role in the fundamental example (3).

2 Local regularity of measures

For every j ≥ 1 and k ∈ [0, . . . ,2^j −1], I_j,k = [k2^−j,(k+ 1)2^−j). I_j,k⁺ and I_j,k⁻ denote the intervals I_j,k+ 2^−j and I_j,k−2^−j.

If u ∈ (0,1), ∀j ≥ 1, Ij(u) denotes the unique dyadic interval of length 2^−j, semi-open to the right, containing u. Then define I_j⁺(u) = I_j(u) + 2^−j and I_j⁻(u) = I_j(u)−2^−j.

The diameter of a set B is denoted by |B|. For the rest of the paper, the convention log(0) =−∞ is adopted.

Definition 1 Let µ be a positive Borel measure on [0,1]. For u₀ ∈(0,1), the lower and upper H¨older exponents of µ at u0 are respectively defined by

α_µ(u₀) = lim inf

j→+∞

logµ(I_j(u₀))

log|Ij(u0)| and α_µ(u₀) = lim sup

j→+∞

logµ(I_j(u₀)) log|Ij(u0)|

When α_µ(u₀) = α_µ(u₀), their common value is denoted α_µ(u₀) and called the H¨older exponent of µ at u0.

The left and right lower and upper H¨older exponents of µ at u₀ are defined by

α⁻_µ(u₀) = lim inf

j→+∞

logµ(I_j⁻(u₀))

log|I_j⁻(u₀)| and α⁺_µ(u₀) = lim inf

j→+∞

logµ(I_j⁺(u₀)) log|I_j⁺(u₀)|

and α⁻_µ(u₀) = lim sup

j→+∞

logµ(I_j⁻(u₀))

log|I_j⁻(u₀)| and α⁺_µ(u₀) = lim sup

j→+∞

logµ(I_j⁺(u₀)) log|I_j⁺(u₀)| . Similarly, when they coincide,α⁻_µ(u₀)andα⁺_µ(u₀)denote their common value.

Finally, we define

h_µ(u₀) = max(α⁻_µ(u₀), α_µ(u₀), α⁺_µ(u₀))

and for h≥0

E^µ_h ={u∈[0,1] :h_µ(u) =h}.

We see that (the exponent h_µ(·) and its level setsE_h^µ are defined in (5))

h_µ(u₀) = min(α⁻_µ(u₀), α_µ(u₀), α⁺_µ(u₀)) = lim inf

r→0⁺

logµ(B(u₀, r)) log|B(u₀, r)| .

Definition 2 If µ is a positive Borel measure on[0,1], and α≥0, denote by Ee_α^µ the set {x:α⁻_µ(x) =α_µ(x) = α⁺_µ(x) =α}.

The following proposition puts together classical results derived from the mul- tifractal formalism for measures (see [13,35]). It provides upper bounds for the Hausdorff dimension of union of level setsE_α^µ,E^e_α^µand E^µ_α. The singularity spectrumd_µ and the scaling function τ_µ were introduced in Section 1. For the reader’s convenience we recall the definition of the Hausdorff dimension.

Definition 3 Let s ≥ 0. The s-dimensional Hausdorff measure of a set E, H^s(E), is defined as

H^s(E) = lim

r&0H^s_r(E), with H^s_r(E) = inf

X

i

|E_i|^s

,

the infimum being taken over all the countable families of sets E_i such that

|E_i| ≤ r and E ⊂ ^S_iE_i. Then, the Hausdorff dimension of E, dim E, is defined as dim E = inf{s≥0 :H^s(E) = 0}= sup{s ≥0 :H^s(E) = +∞}.

Proposition 1 Let µ be a positive Borel measure on [0,1] and let α≥0.

(1) dimE^e_α^µ ≤d_µ(α)≤τ_µ^∗(α).

(2) If α ∈[0, τ_µ⁰(0⁺)], then dim ^S_α⁰_≤αE_α^µ0 ≤τ_µ^∗(α).

(3) If α ≥τ_µ⁰(0⁺), then dim ^S_α⁰_≥αE_α^µ0 ∪E^µ_α⁰≤τ_µ^∗(α).

(4) If τ_µ^∗(α)<0, then E_α^µ =∅.

Next proposition follows from the definition ofτµ and Tchernov inequalities.

Proposition 2 Let µ be a positive Borel measure on [0,1]. For every α≥ 0, C >0 and ε >0, there exists a scale J such that j ≥J implies

log#ⁿk ∈ {0, ..,2^j −1}:µ(Ij,k)≥C2^−j(α+ε)o

log 2^j ≤ sup

α⁰≤α+ε

τ_µ^∗(α⁰) +ε.

3 Tools

In this section, we are given the L´evy process X, decomposed into the sum X =X^f+B(a⁰, Q) described in (9).

3.1 Some notations

We denote byS the Poisson point process with intensity`⊗π associated with the L´evy process X(t), where ` stands for the Lebesgue measure on R+ and π is the L´evy measure.

For every j ≥1, let

G_j ={t : (t, λ)∈S for some λ such that |λ| ∈(2^−j−1,2^−j]}.

Fort ∈G_j,λ_t is the unique elementλ∈R^dsuch that (t, λ)∈S. The jumps of the process Xj(t) are thus exactly located at the points of Gj, and the value of the jump ofX_j at t∈G_j is λ_t.

For every j ≥1 and for every δ >0,A^j_δ is the union of intervals A^j_δ = ^[

t∈Gj

B(t,2^−(j+1)δ).

We clearly have ^S_t∈GjB(t,|λt|^δ) ⊃ A^j_δ. Eventually, for every sequence δ^e = {δ_j}_j of non-negative numbers, we denote

A

eδ = lim sup

j→+∞

A^j_δj = ^\

J≥1

[

j≥J

A^j_δj. (13)

3.2 Coverings and weak redundancy properties associated with Poisson point processes

It is known [38,22] that with probability 1, for every δ < β, if the sequence δeis constantly equal to δ, then A

e^δ =R⁺ (recall (13)). An easy adaptation of the proof of Lemma 3 in [22] yields the following slightly stronger result.

Lemma 1 With probability 1, there exists a non-decreasing non-negative se- quence β^e= (β_j)j≥1 converging to β such that A

βe=R⁺.

Notice that if the L´evy process is stable and if we can write in polar coordinates π(dr, dθ) = αr^−(1+β)drν(dθ) with α ≥ 1/2 and ν a probability measure on

the unit sphere, then the constant sequence (β_j = β)_j can be chosen in the previous statement.

The problem of covering by Poisson intervals is connected with the problem of counting the number of points ofS whose projection on R+ falls in a given dyadic interval Ij,k = [k2^−j,(k+ 1)2^−j). Next Lemmas 2 and 4 are devoted to this question.

Lemma 2 For δ > β and ε_e = {ε_j}j≥1 a sequence of positive numbers, and for every integers j and k, let

K_j,k^δ,eε= #ⁿt∈I_j,k :t∈G_j⁰ for some j⁰ ∈[j/δ, j/(β+ε_j)]^o. (14) There exist two sequences{ε_j}j≥1 and{η_j}j≥1 of positive real numbers converg- ing to 0 such that for every integerT > 0, with probability 1: For everyδ > β, for every j ≥1 large enough (depending onδ), for every k∈ {0, . . . ,2^jT−1}, we have K_j,k^δ,eε ≤2^jηj.

PROOF. By definition of β, there exists a positive non-increasing sequence εe⁽¹⁾ ={ε⁽¹⁾_j }_j converging to zero such that C_j ≤2^{j(β+ε(1)j )}.

LetT be a positive integer. Let δ > β. For every j ≥1 andk ∈ {0, . . . ,2^jT − 1}, the random variable K_j,k^δ,eε(1) is a Poisson variable with intensity C_j^δ,eε(1) =

2^{−j X}

j/δ≤j⁰≤j/(β+ε⁽¹⁾_j )

Cj⁰ ≤ 2^{−j X}

j/δ≤j⁰≤j/(β+ε⁽¹⁾_j )

2^j0(β+ε

(1) j0 )

≤ Mj,δ2^j((β+ε

(1)

[j/δ])/(β+ε⁽¹⁾_j )−1)

, where M_j,δ is equal to 2²

(β+ε(1)

[j/δ])/(β+ε(1) j )

2^β+ε

(1) [j/δ]−1

. In fact it is easily checked that the sequenceM_j,δ can be bounded by a constantM_β independent of δand j since the sequence {ε⁽¹⁾_j } is bounded. Thus C_j^δ,eε(1) ≤ M_β2^j((β+ε

(1)

[j/δ])/(β+ε⁽¹⁾_j )−1)

, for every j and every δ.

Moreover, since the sequence {ε⁽¹⁾_j } is non-increasing and converges to zero as j → +∞, (β+ε⁽¹⁾_[j/δ])/(β+ε⁽¹⁾_j )−1 is bounded by {ε^(δ)_j } = {2ε⁽¹⁾_[j/δ]/β}, which is a non-increasing sequence depending onδ. ThusC_j^δ,eε(1) is bounded by M_β2^jε(δ)j .

We considerε_j =ε⁽¹⁾_[j/log(j+1)] for allj ≥1. For everyδ > β, forj large enough, we have ε_j ≥ ε^(δ)_j , and thus C_j^δ,eε ≤ C_j^δ,eε(1) ≤ 2^jεj (actually, without loss of generality, we can change a little bit the sequence {ε^(δ)_j } so that it takes into account the constant M_β).

We now use the following lemma which is a simple consequence of the Stirling formula

Lemma 3 There exists an integerr >0such that forjlarge enough, for every Poisson random variable N of parameter C > 0, P(N > r(j+C))≤2^−2j. Let P_j^δ = P

∃k ∈ {0, . . . ,2^jT −1} : K_j,k^δ,eε ≥ r(j +C_j^δ,eε). By Lemma 3 for j large enough we have P_j^δ ≤ 2^−2j2^jT so ^P_j≥1P_j^δ < +∞. The Borel-Cantelli lemma implies that for every j large enough, for every k ∈ {0, . . . ,2^jT −1}, K_j,k^δ,eε ≤2^jηj, whereη_j is the positive sequence converging to 0 at infinity defined by 2^jηj =r(j+C_j^δ,eε). This yields the uniform control over k ∈ {0, . . . ,2^jT−1}

of K_j,k^δ,eε for every δ > β with probability 1, and finally with probability 1 for allδ > β since the random functions δ 7→K_j,k^δ,eε are non-decreasing.

We need to introduce the notion ofweakly redundant systeminR+. This notion is later determinant to get upper bounds for the level sets of H¨older exponents.

Definition 4 Let (x_n)n≥0 ∈ R^N+ and (λ_n)n≥0 a positive sequence converging to 0. For every T >0 and j ≥0, we introduce the sets of indices

T_j =ⁿn :x_n∈[0, T], 2^−(j+1) < λ_n≤2^−jo. (15) The family {(x_n, λ_n)}n∈N is said to form a weakly redundant system if for every T > 0 there exists a sequence of integers (NT,j)j≥0 such that

(i) lim_j→∞(log₂N_T,j)/j = 0.

(ii) for every j ≥ 1, T_j can be decomposed into N_T,j pairwise disjoint sub- sets (denoted T_j,1, . . . ,T_{j,NT ,j}) such that for each 1 ≤ i ≤ N_T,j, the family

nB(x_n, λ_n) : n∈ T_j,i^o is composed of disjoint balls.

Lemma 4 Consider the Poisson point process S =^S_j≥0G_j. Let (β_j)j≥0 be a non-decreasing sequence converging to β.

With probability 1, the family ^S_j≥0{(t,|λ_t|^βj) :t∈G_j} forms a weakly redun- dant system.

PROOF. This is a direct consequence of the estimates obtained in the proofs of Lemmas 5 and 8 of [22] for the numbers N_j,k = #{t∈G_j : t ∈[k2^−j,(k+ 1)]2^−j}when β = 1.

3.3 Local regularity of the L´evy process X^f

As a consequence of the work achieved by Jaffard in [22], the increments ofX^f satisfy the following almost-sure properties.

Proposition 3 Let ε >0. With probability 1:

Let t₀ ≥ 0 be not a jump point of X(t), and write^f h_X(t₀) = 1/δ for some δ≥β. For η small enough, there exists ε⁰ >0 such that for allt ≥0,

if |t−t0| ≤η, then ^X

j≥^{log2|t−t0|−1}

β+ε0

|Xj(t)−Xj(t0)| ≤ |t−t0|^1/(β+ε) (16) and |X(t)−X(t₀)| ≤ |t−t₀|^1/(δ+ε). (17) Moreover, still for|t−t0| ≤η, if^P

j<^{log2|t−t0|−1}

β+ε0

Xj(·)has no jump point between t and t₀, we get

X

j<^log2_β+ε^|t−t0^0|−1

|Xj(t)−Xj(t0)| ≤ |t−t0|^1/(β+ε). (18)

Equation (18) implies that when β ≥1 the contribution of the sum of all the drifts associated with the processes X_j(t),j < ^log2_β+ε^|t−t0^0|−1, on a given interval [t0, t], is always less than |t−t0|^1/(β+ε).

3.4 Heterogeneous ubiquity and Hausdorff dimensions of limsup sets

General results of what we call “heterogeneous ubiquity” are obtained in [11]

(see also [12]). Here, a simpler version adapted to our context is stated. It plays a similar role as the geometric Theorem 2 used in [22], but makes it possible to work out problems raised here by considering a multifractal time change. Some additional notations have to be introduced.

Let{u_n}n∈Nbe a sequence of points in [0,1] and{l_n}n∈Na sequence of positive real numbers converging to zero. Let δ >1. For every n ∈N we set

I_n= [u_n−l_n, u_n+l_n], I^e_n⁺ = [u_n+l_n/4], I^e_n⁻ = [u_n−l_n/4, u_n], I_n^δ = [u_n−l_n^δ, u_n+l^δ_n].

In addition, given an integerb ≥2, for u∈[0,1], we set

B_j(u) =ⁿI_n :u∈I_n, l_n ∈(b^−(j+1), b^−j]^o, (19) B^δ_j(u) =ⁿI_j⁰_,k⁰ :∃I_n∈ B_j(u) such that I_j⁰_,k⁰ ⊂I_n^δo. (20) Definition 5 Let {u_n}n∈N be a sequence of points in [0,1], and let{l_n}n∈N be a sequence of positive real numbers converging to zero.

Let µ be a positive Borel measure such that supp(µ) = [0,1] and (11) holds.

The system{(un, ln)}nis said to form an heterogeneous ubiquitous system with respect to (µ, τ_µ⁰(1)) if the following holds true.

(1) There exists a non-increasing sequence (ϕ_j)j≥0 with the properties:

(a) lim_j→∞ϕ_j = 0, (jϕ_j)_j≥0 is non-decreasing at +∞ andlim_j→∞jϕ_j = +∞.

(b) ∀ε >0, (j(ε−ϕ_j))j≥0 is non-decreasing at +∞ , (c) Properties (2), (3) and (4) below hold.

(2) There exist an integer b ≥2 such that

(a) µ-almost every t ∈[0,1] belongs to ^T_N≥0^S_n≥N[u_n−l_n/2, u_n+l_n/2].

(b) For µ-almost every t ∈ [0,1], there exists an integer j(t) such that

∀j ≥j(t), ∀ k such that |k−k_j,t^b | ≤1,

b^{−j(τµ0(1)+ϕj)}≤µ([kb^−j,(k+ 1)b^−j])≤b^{−j(τµ0(1)−ϕj)},

wherek_j,t^b is the unique integerk such thatt∈[kb^−j,(k+1)b^−j). Thus (2)(b) implies for µ-a.e. t ∈ [0,1] a precise control of the µ-mass of the three b-adic intervals around t.

(3) (Self-similarity ofµ) For everyb-adic subintervalLof [0,1], letfL denote the canonical affine mapping from L onto [0,1]. There exists a measure µ^L onL, equivalent to the restriction of µto L, such that property (2)(b) holds for the measure µ^L◦f_L⁻¹ instead of the measure µ.

Let j_L = log_b|L|⁻¹ and for every n≥1, let

U_n^L=











t∈L:











∀ j ≥n+j_L, ∀ k, |k−k_j,t^b | ≤1, µ^L[kb^−j,(k+ 1)b^−j]≤

b^−j

|L|

τ_µ⁰(1)−ϕj−jL











.

The setsU_n^Lclearly form a non-decreasing sequence in[0,1], and by (2)(b) and property (3), ^S_n≥1U_n^L is of full µ^L-measure. Then define

nL = infⁿn≥1 : µ^L(U_n^L)≥ kµ^Lk/2^o. (4) (Control of the growth speed n_L and of the mass kµ^Lk)

There is a dense subset D of (1,∞) such that for every δ ∈ D, for µ-almost every u∈[0,1], one can find an increasing sequence of integers (j_k(u))k≥1 such that for everyk ≥1, there existsL_k ∈B_j^δ

k(u)(u)satisfying

k→∞lim j_Lk

j_k(u) =δ and

n_Lk ≤j_Lk. ϕ_jLk and |L_k|^ϕjLk ≤ kµ^Lkk. (21) The next result is established in [11].

Theorem 4 Let {u_n}n∈N be a sequence of points in [0,1], let {l_n}n∈N be a sequence of positive real numbers converging to zero. Let µbe a positive Borel measure such that supp(µ) = [0,1] and (11) holds.

For every positive sequences εe = (ε_n)n∈N and δ^e = (δ_n)n∈N, define the limsup set

S_µ(δ, τ^e _µ⁰(1),ε) =_e ^\

N≥0

[

n≥N:|ln|^τµ0(1)+εn≤µ(^Ie

+ n),µ(e^I

−

n)≤µ(In)≤|ln|^{τµ0(1)−εn}

I_n^δn.

Suppose that{(u_n, l_n)}_n forms an heterogeneous ubiquitous system with respect to (µ, τ_µ⁰(1)).

There exists a positive sequence ε_e converging to 0 such that for every δ ≥ 1, there exists a non-decreasing sequence δ^e converging to δ as well as a positive Borel measure m_δ such that:

• m_δ(E) = 0 for every Borel set E such that dimE < τ_µ⁰(1)/δ ,

• m_δS_µ(δ, τ^e _µ⁰(1),ε)_e >0.

In particular, dim S_µ(δ, τ^e _µ⁰(1),ε)_e ≥τ_µ⁰(1)/δ.

Moreover, if the system {(u_n, l_n)}n∈N is weakly redundant (see Definition 4), we precisely have dim S_µ(δ, τ^e _µ⁰(1),ε) =_e τ_µ⁰(1)/δ.

The set S_µ(δ, τ^e _µ⁰(1),ε) is constituted by points which are well approximated_e at rate δ >1 by some points u_n, these points being selected according to the behavior ofµaroundu_n. Thus Theorem 4 emphasizes a ubiquity propertycon- ditioned by a measureµ, and shows the existence of exceptional points related simultaneously to the local behavior of the measure µand to the approxima- tion rate by the system{(u_n, l_n)}_n. The condition|l_n|^τµ0(1)+εn ≤µ(I^e_n⁺), µ(I^e_n⁻)≤ µ(I_n)≤ |l_n|^{τµ0(1)−εn} involved in the definition of the set S_µ(δ, τ^e _µ⁰(1),ε) appears_e in the weaker form |ln|^τµ0(1)+εn ≤µ(In)≤ |ln|^{τµ0(1)−εn} in [11], but due to prop- erty (2)(b) the work achieved in [11] makes it possible to add automatically the condition on µ(I^e_n⁺) and µ(I^e_n⁻) and it yields Theorem 4.

Remark 1 For some classes of measures µ, it turns out that property (4) can be simplified in the stronger one: There exists j₀ ≥ 0 such that (21) holds for all b-adic interval L of generation larger than j₀. This is the case for instance for the class of random Gibbs measures described in [9]. Unfortunately independent random cascades do not satisfy this uniform property, and their study required working with the weaker condition (4) (see Sections 1 and 7 as well as [10]).

3.5 Conditions C1-3

Letµ be an atomless positive Borel measure with a support equal to [0,1].

Condition C1

There exist two positive constants γ₁ and γ₂ such that for every small enough sub-interval I of [0,1],|I|^γ1 ≤µ(I)≤ |I|^γ2.

Condition C2(h_µ,β)

Recall that h_µ,β = τ_µ⁰(1)/β. By assumption the function F : t ∈ [0,1] 7→

µ([0, t]) is increasing and continuous on [0,1].

The Poisson point process S can be writtenS ={(t_n, λ_n)}n≥1, with|λ_n| &0.

Let{β_j}_j≥1 be a sequence as found in Lemma 1.

For every (t_n, λ_n) ∈ S such that t_n ∈ G_j, we set u_n = F⁻¹(t_n), and we define the sequence l_n as 2F⁻¹Bt_n,|λ_n|^βj. This ensures that (0,1) ⊂ lim sup_n→∞B(u_n, l_n/2).

Condition C2(h_µ,β) is said to hold when (11) holds and when {(u_n, l_n)}n≥1

forms an heterogeneous ubiquitous system with respect to (µ, τ_µ⁰(1)).

We shall see in Section 7 that this holds under suitable assumptions when µ is an independent multiplicative cascade. Consequently, the assertions of Theorem 2 concerning the linear parts of the spectra apply to the processZ_W defined in (3).

Condition C3(h)

There exists a positive Borel measurem_h on [0,1] such that m_h(E^e_h^µ)>0 and for every Borel set E ⊂[0,1] such that dimE < τ_µ^∗(h), m_h(E) = 0.

Suppose thatµis a independent multiplicative cascade. It is shown in [10] that if the function ϕ_W is everywhere finite, then with probability 1, condition C3(h) holds for all h such that τ_µ^∗(h) > 0. Consequently, the assertions of Theorem 2 concerning the strictly concave parts of the spectra apply to the process Z_W defined in (3).

4 Computation of the Hausdorff spectrum of X^f◦F: Theorem 2

In this section, in order to simplify the notations, we assume that X = X,^f i.e. B(a⁰, Q) = 0 in (9), so thatX^fand Z^e in Theorem 2 are simply denoted X and Z.

By Lemma 1, there exists a non-decreasing sequence of positive real numbers βe={βj}j≥1 converging toβ such that, with probability 1, the setA

eβ (defined in (13)) equals R⁺. Such a sequence is fixed.

4.1 Characterization of the H¨older exponents of Z =X◦F

For every j ≥ 1, for every t ∈ G_j, let l_t = 2|F⁻¹([t− |λ_t|^βj, t+|λ_t|^βj])| and It = [F⁻¹(t)− lt, F⁻¹(t) +lt]. These intervals were considered in condition C2(h_µ,β) in Section 3.5. By construction of the{β_j}_j, we have

[0,1]⊂ ^\

J≥1

[

j≥J

[

t∈G_j

[F⁻¹(t)−l_t/2, F⁻¹(t) +l_t/2].

Definition 6 Let α≥0, δ≥1 and ε >0.

A real number u₀ is said to satisfy the property P(α, δ, ε) if there exist an infinite number of jump points u of Z satisfying

|u−u0| ≤l_F^δ−ε_(u) and l^α+ε_F(u) ≤µIF(u)

≤l^α−ε_F(u). (22) Remark that, by construction, if t=F(u) and t ∈G_j for some integer j ≥1, then under (22) we also have 2^−j ≤l

α−ε β+ε

F(u) if j is large enough.

A real number u₀ is said to satisfy the property P^e(α, δ, ε) if there exist an