E Wilson

C Dumont

] M Drelon

I Suisse

C Penverne

M Sauthier

The Palladium Catalyzed Carbonylative Telomerization of Butadiene : One Pot Catalytic and Atom-Economical Synthesis of Unsaturated Esters from Agrobased Alcohols and Polyols

The palladium catalysed carbonylative telomerization reaction of alcohols with butadiene allows an efficient and atomeconomical access to unsaturated alkyl nona-3,8-dienoate esters. The reaction has been studied on the point of view of the nature of the catalyst (phosphine and acid) with ethanol. Commercially available triarylphosphines and carboxylic acids associated to a simple palladium precursor appear as the best combination for the insitu generation of the catalyst. The reaction conditions have been further optimized and the carbonylative telomerization reaction has been efficiently applied to the full transformation of several industrially relevant agrobased monoalcohols and polyols.

Introduction

Among the numerous synthetic methods for esters synthesis, metal-catalyzed carbonylation reactions are of strong interest, being highly atom economical as well as highly efficient. [1] Saturated and unsaturated esters are for example respectively obtained from the hydroesterification of olefins [START_REF] For Reviews | [END_REF] or dienes with an alcohol. [START_REF] Neubert | Recent references for the synthesis of C5 esters : a)[END_REF] The hydroesterification of ethylene, thanks to the development of effective catalytic systems, has found industrial application as an intermediate synthetic step towards methylmethacrylate. [4] The reaction with olefins has also been applied to the conversion of agrobased polyols for the synthesis of plasticizers. [5] The hydroesterification reaction that incorporates butadiene yields C5 unsaturated esters [START_REF] Neubert | Recent references for the synthesis of C5 esters : a)[END_REF] that can be further chemically modified thanks to the versatile reactivity of the double bond. Interestingly, the carbonylative coupling reaction of 2 equivalents of 1,3-butadiene followed by the nucleophilic attack of an alcohol yields alkyl nona-3,8-dienoates (Scheme 1). [START_REF] Faßbach | The reaction is also called carboxytelomerization[END_REF] This carbonylative version of the telomerization reaction [7] encompasses several advantages such as high atomeconomy and the use of industrially accessible reactants (butadiene, CO, alcohols). All of these are already produced on large scale from agro-resources: This further strengthens the attractiveness of such transformation in a context of efficient biomass upgrading. [START_REF] Pomalaza | a) for a review on bio-based butadiene[END_REF] Moreover, the butadiene coupling reactions advantageously lead to the formation of two unsaturations and thus open the way to further functionalization. This herein considered transformation applied with simple alcohols is promoted by palladium-based catalysts. After early works reported by Tsuji, Billups and Knifton, [9] it did not attract significant attention in the literature. The reaction was only recently used for the synthesis of C21 monoesters from βmyrcene under neutral conditions [10] and in an intermediate step for the synthesis of diesters from butadiene with pyridine as solvent. [11] We recently reported that the reaction can be efficiently employed for the functionalization of phenols, natural phenols and a polyphenolic biopolymer such as Kraft lignin. [12] This study revealed that with phenolic nucleophiles, acidic conditions are greatly enhancing the yield of products. Bearing in mind the synthetic potential of this reaction in the field of biomass upgrading, we turned our attention toward the synthesis of unsaturated esters from the catalytic transformation of agrobased aliphatic alcohols and polyols.

Scheme 1

The carbonylative telomerization reaction.

Results and Discussion

The catalytic procedure previously developed for the transformation of phenols was first evaluated with ethanol as a simple biobased aliphatic alcohol. We more particularly focused our attention on the choice of the palladium-based precatalyst and of the acid co-catalyst used as these were key-parameters in the case of phenols conversion. It should be noticed that ethanol is not used as solvent but as a stoichiometric reagent. The very polar solvent dioxane has been selected as the solvent of reaction as it is able to solubilize polyols [5a] 4 equivalents of butadiene were used per ethanol in order to counterbalance the possible butadiene oligomerization and thus favor full conversion of the alcohol. The catalyst was generated in situ from the combination of halide-free Pd(OAc)2 and four equivalents of PPh3. The E isomer was obtained in all cases as the major isomer, the E/Z proportion varied with the reaction conditions from 85%/15% to 70%/30%. [START_REF]Z ratio for the synthesized compounds are given in the ESI[END_REF] In the case of ethanol, both weak and strong acids were evaluated as co-catalysts (Table 1, entries 1-4). Acetic acid, benzoic acid and 2-chlorobenzoic acid showed a high efficiency and yields up to 94 % could be obtained with the latter one. 4-bromobenzoic afforded a very low amount of product. Indeed, in contrast to the C-Cl bond in 2chlorobenzoic acid, the C-Br bond is very reactive toward Pd(0) species. This reactivity is typically used in carbonylation reactions of arylbromide derivatives that lead to the formation of HBr. [START_REF] Martinelli | [END_REF] The strong acid thus contributed to poisoning of the catalyst as exemplified by the catalytic reactions performed with TFA and PTSA as acid cocatalysts (Table 1, entries 5-6). With these two acids, very low yields in product of reaction were obtained. Strong acids are indeed expected to favour the formation of palladium hydride species through the protonation of low valent Pd(0) species. This reactivity prevents the oxidative coupling of two butadiene units, a key step in the catalytic cycle of the carbonylative telomerization reaction.

Table 1. Carbonylative telomerization of butadiene with ethanol: Influence of the acid l [a] Entry Acid pKa Yield [b] 1 The impact of the ligand used to stabilize the catalyst under acidic conditions was also studied. We were more particularly interested in comparing air-stable triarylphosphines with different electronic and steric properties on this catalytic reaction (Table 2). Benzoic acid has been used as co-catalyst for this study.

Triarylphosphines featuring electron-donating groups on the para position of the aryl groups showed higher efficiencies, as higher yields were obtained with P(p-MeOC6H4)3 and P(p-MeC6H4)3 (compare entry 1 with entries 2 and 3). Sterically hindered phosphines led to very low catalytic activities as shown from the catalytic runs involving tris (2,4,6trimethoxyphenyl)phosphine and tris (2,4,6trimethylphenyl) phosphine (entries 4 and5). This contrasts with the results obtained in the case of the (non-carbonylative) telomerization reaction that is efficiently promoted with the sterically hindered P(o-MeOC6H4)3 ligand. [15] This family of ligands doesn't prevent the oxidative coupling step of two butadiene units. However, one can expect that in the case of the carbonylative telomerization reaction, the presence of CO will compete with the coordination of sterically hindered phosphines and thus favour the formation of unstable under-coordinated palladium species. A diphosphine, namely dppb (1,4bis(diphenylphosphino)butane), has been evaluated as ligand, affording the targeted product in low yield, while the product results from butadiene ethoxycarbonylation (ethylpent-3-enoate) has been isolated as main compound. [16] A phosphine-free catalytic reaction didn't show any formation of the products of reaction (entry 8).

Table 2 Carbonylative telomerization of butadiene with ethanol: Influence of the ligand. [a] [a] Conditions: ethanol (0.58 mL, 10 mmol), butadiene (3.5 mL, 40 mmol), Pd(OAc)2 (11.1 mg, 0.05 mmol), ligand (0.2 mmol), benzoic acid (183.2 mg, 1.5 mmol), dioxane (3.0 mL), 30 bar CO, T = 90 o C, t = 20 h; [b] NMR Yield with trimethoxybenzene as internal reference; [c] dppb ligand : 0.1 mmol.

The PPh3/Pd(OAc)2/benzoic acid combination was selected for further applications, as it involves readily available constituants while affording high yields. It was thus used to promote the catalytic conversion of biobased monoalcohols through the carbonylative telomerization reaction (Table 3). Glycerol is a triol that can be readily converted into a monoalcohol by protecting two hydroxyl groups through the formation of an acetal with acetone (as solketal) [17] or carbonate (as glycerol carbonate). [18] Thus, solketal has been reacted and efficiently converted by carbonylative telomerization in the corresponding unsaturated ester with 81 % isolated yield (Table 3, entry 1). On the other hand, the monoester derived from glycerol carbonate has been obtained in very low yield, most probably because of the

Entry

Ligand Yield [b] 1 PPh3 82% 3, entry 2). We then probed the efficiency of the transformation on furfuryl alcohol [19] and tetrahydrofurfuryl [20] alcohol, which are agrobased compounds derived from pentoses issuing from the hemicellulosic feedstock. [21] Simple processes are used for the production of these derivatives at the industrial scale. Interestingly, the carbonylative telomerization of butadiene has been very efficiently applied to the conversion of these two alcohols, the furfuryl and tetrahydrofurfuryl esters have been isolated with respectively 94 % and 81 % isolated yield (entries 3 and 4).

Table 3.Carbonylative telomerization reaction with agrobased monoalcohols.

[a]

[a]Conditions: alcohol (10 mmol), butadiene (3.5 mL, 40 mmol), Pd(OAc)2 The conversion of alcohols by carbonylative telomerization has extended to polyols. With such derivatives, selectivity issues related to substitution degrees are expected. The reaction has been studied in depth with a simple model polyol, ethylene glycol, in order to further optimize the reaction conditions (CO pressure, temperature, see Table 4). The very simple PPh3/Pd(OAc)2/benzoic acid catalytic combination has been used for this study. The ethylene glycol conversion and the selectivity into mono-/diesters have been determined by GC and 1 H NMR analyses performed on the crudes of reaction. The variation of the CO pressure has an impact on the reaction yield. Almost quantitative yields of diester were obtained at 20 and 40 bar of CO (Table 4, entries 1 and 2, respectively). At 50 bar, the reaction is not complete and a mixture of monoester and diester is obtained (entry 3). This is likely to be due to the competition for active sites on the catalyst between the CO and butadiene molecules, favoring CO at higher pressure. Consequently, high CO pressure slows down the butadiene from dimerising. With 0.5% of catalyst in respect to the hydroxyl groups, the reaction can be setup at lower reaction temperatures without loss of yield (entries 4-6) Table 4. Carbonylative telomerization reaction with ethylene glycol. [a] Entry T (°C) CO (bar) EG conv (%) [b] Y2 (%) [b] Y3 (%) [b] The conversion of ethylene glycol and selectivity towards the different products of reaction have been monitored and plotted in function of the reaction time (See Figure 1). The conversion profile gives evidence of an activation period which is likely due to the in situ formation of the actual catalyst.

Entry

Alcohol (R-OH) Isolated Yield [b] 1 81 %

2 13 % 3 94 % 4 81 % OH R R O O CO 2 + + [Pd] H + HO O R O CO 2 + + [Pd] OH HO O R O O R O + 2 3 H + R = O OH O O OH O OH O O O OH

HO OH

After 1 hour, the ethylene glycol starts to be converted into its monoester. The yield in monoester rapidly reaches a maximum of 42 % after 2.5 hours, after which it is converted into diester, reaching full selectivity after 5 hours. At the maximum rate of ethylene glycol conversion, the turn over frequency of the reaction is 40 mol.h -1 . Following this, the scope of the carbonylative telomerization reaction has been extended to polyols derived from agroresources (diols, triols and tetraols, Table 3).

Table 3. Carbonylative telomerization of butadiene with agrobased polyols. [a] Entry Polyol Isolated Yield [b] 1 76 % [a] Conditions: polyol (10mmol), butadiene (3.5 eq per hydroxyl group), Pd(OAc)2 (0.5% mol per hydroxyl group), PPh3 (4 eq/Pd), benzoic acid (0.15 eq per hydroxyl group), 40 bar CO, dioxane (3.0mL), T = 90 o C, t = 20h; [b] Isolated yield of the per-esterified product : diester for entries 1-5, triester for entry 6, tetraester for entry 7.

In addition, as further substrates, 1,3-propanediol, 1,4butanediol and 2,3-butanediol have been selected, as they are accessible from bio-based resources. The first two, 1,3propanediol and 1,4-butanediol, which are primary alcohols, show a similar behaviour with the model substrate ethylene glycol regarding their reactivity towards carbonylative telomerization. The corresponding diesters were obtained with 76 % and 86 % isolated yields (Table 3, entries 1 and 2).

Conversely, 2,3-butanediol bears more sterically hindered secondary alcohols, which should hinder its efficient conversion. However, to our delight, the corresponding diester was obtained with a 81 % yield, comparable to that obtained with 1, 3propanediol and 1,4-butanediol (Table 3, entry 3). This prompted us to study the catalytic conversion of the anhydrohexitols isomannide and isosorbide (secondary alcohols). Isosorbide is particularly relevant as this chiral diol is industrially produced from glucose through a hydrogenation / dehydration two-steps process. [22] Among its various applications, [23] it is used as a precursor for the production of isosorbide-based diesters that are relevant bio-based plasticizers. [24] Isomannide and isosorbide show a very interesting reactivity: they are fully converted into the corresponding diesters that can be isolated in high yield (Table 3, entries 4 and 5). This synthetic pathway is attractive as it allows the synthesis of isosorbide diesters from butadiene and carbon monoxide according to a highly atomeconomical process, a major advance compared to prior methods. Finally, higher polyols were also converted according to the same protocol. Glycerol that bears both primary and secondary alcohols functions was fully converted into the corresponding triester in high isolated yield. Erythritol was also converted in the corresponding tetraester with 36 % isolated yield.

Scheme 2 Hydrogenation of the erythritol tetraester. Reaction conditions :

erythritol tetraester (100 mg, 0.15 mmol), Pd/C (6.4 mg, 0.04 eq.), 30 bar H2, 1:1 DCM: methanol (2 mL), 40 o C, 5 days.

The 1 H-NMR spectrum of this compound showed a high complexity due to the presence of Z/E mixtures. In order to get simpler 1 H NMR spectra and thus better assess the structure of the tetraester, the isolated compound was fully hydrogenated with Pd/C catalyst into the tetranonanoate derivative of erythritol (Scheme 2). The 1 H NMR of this species is fully in line with a symmetric structure (see Supporting Information

Conclusions

In this contribution, we have shown that the carbonylative telomerization reaction is an efficient catalytic transformation for the synthesis of esters from agrobased alcohols, CO and butadiene. The most efficient system in our hand combines palladium acetate, a monophosphine ligand and an acid. Weak acids are particularly efficient co-catalysts: benzoic acid or acetic acid can be used for instance, affording excellent catalytic performances. Regarding the ligand associated to palladium, non-sterically hindered triarylphosphines that bear electrondonating groups on the para-position give superior yields in products of reaction with ethanol as model substrate. In order to provide the synthetic community with a simple catalytic system which comprises readily accessible and economically competitive constituents, we optimized reaction conditions with the PPh3/Pd(OAc)2/benzoic acid combination. With a loading of 0.5%, this has been efficiently used to convert several monoalcohols, diols and higher polyols, which are mostly derived from agroressources. Industrially-relevant alcohols with primary or secondary alcohols such as isosorbide and glycerol have been efficiently converted in the corresponding diester and triester with high isolated yields. On the top of that, glycerol and erythritol can also be converted into their fully esterified analogue. Thus, we have demonstrated that a wide range of alcohols can be efficiently combined with carbon monoxide and butadiene to selectively yield the corresponding esters with C9 side chains. The substrate scope described here comprizes a significant share of biosourced alcohols. Combined with the considered reaction high atom-economical character, it highlights the sustainable character of this approach. This is of high interest, especially when targeting novel products that can find industrial applications, from plasticizers to detergents, with a lesser environmental impact compared to fossil fuel-derived analogues.

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FULL PAPER

The palladium catalysed carbonylative telomerization reaction of agro -based alcohols with butadiene allows an efficient and atom-economical access to the corresponding unsaturated alkyl nona-3,8-dienoate esters.

  [a] Conditions: Ethanol (0.58 mL, 10 mmol), butadiene (3.5 mL, 40 mmol), Pd(OAc)2 (11.1 mg, 0.05 mmol), PPh3 (52.8 mg, 0.2 mmol), acid (1.5 mmol), dioxane (3.0 mL), 30 bar CO, T = 90 o C, t = 20h; [b] NMR Yield with trimethoxybenzene as internal reference.

( 11 .

 11 1 mg, 0.05 mmol), PPh3 (52.8 mg, 0.2 mmol), acid (1.5 mmol), 40 bar CO, dioxane (3.0 mL), T = 90 o C, t = 20 h; [b] purified by silica gel column chromatography.

  Conditions: Ethylene glycol (0.56 mL, 10 mmol), butadiene (6.2 mL, 70 mmol), benzoic acid (366.4 mg, 3 mmol), Pd(OAc)2 (0.1 mmol), PPh3 (0.4 mmol), dioxane (3.0 mL), t = 20 h.[b] Yields were calculated by using both the NMR spectra and the GC chromatograms of the products.

Figure 1 .

 1 Figure 1. Carbonylative telomerization reaction with ethylene glycol. Reaction conditions : Ethylene glycol (1.6 mL, 0.029 mol), butadiene (7 eq. per ethylene glycol), benzoic acid (0.3 eq./ethylene glycol), dioxane (9.0 mL), Pd(OAc)2 (0.01 eq./ethylene glycol), PPh3 (4 eq./Pd), undecane (1.0 mL), 25 bar CO, T = 90 o C.

  ). Beyond the interest in product characterization, this demonstrates how the be further converted, affording additional compounds following the carbonylative telomerization reaction.

Acknowledgements ((optional))

This work was financially supported by the State as part of the "Programme d'Investissement d'Avenir" under reference ANR 15-CE07-0018-01 H2CAT. We acknowledge the Ministère de l'Enseignement supérieur, de la Recherche et de l'Innovation and the CNRS for their financial support. We thank C. Delabre for the GC and GC-MS analysis.


Antoine Ayache
email: antoine.ayache@univ-lille.fr

Lower bound for local oscillations of Hermite processes

Keywords: Wiener chaos, self-similar process, Rosenblatt process, path behavior, nowhere differentiability AMS Subject Classification (2010). Primary: 60G17, 60G18; secondary: 60H05

The most known example of a class of non-Gaussian stochastic processes which belongs to the homogeneous Wiener chaos of an arbitrary order N > 1 are probably Hermite processes of rank N . They generalize fractional Brownian motion (fBm) and Rosenblatt process in a natural way. They were introduced several decades ago. Yet, in contrast with fBm and many other Gaussian and stable stochastic processes and fields related to it, few results on path behavior of Hermite processes are available in the literature. For instance the natural issue of whether or not their paths are nowhere differentiable functions has not yet been solved even in the most simple case of the Rosenblatt process. The goal of our article is to derive a quasi-optimal lower bound for the asymptotic behavior of local oscillations of paths of Hermite processes of any rank N , which, among other things, shows that these paths are nowhere differentiable functions.

Introduction and statement of the main result

Throughout this article the fixed integer N ≥ 1 and positive real number H are arbitrary and such that 1 -1 2N < H < 1 .

(1.1)

The Hermite process of rank N and parameter H is denoted by X N,H = {X N,H (t)} t∈R + .

It is a real-valued stochastic process belonging to the homogeneous Wiener chaos of order N generated by a real-valued Brownian motion B = {B(x)} x∈R on a probability space (Ω, F, P), fixed once and for all. More precisely, {X N,H (t)} t∈R + is defined through the multiple Wiener integral:

X N,H (t) := R N t 0 N p=1 (s -x p )
H-3/2 + ds dB(x 1 ) . . . dB(x N ), for all t ∈ R + , (1.2)

with the convention that, for each (y, α) ∈ R 2 , one has y α + := y α , if y > 0, 0, else.

(1.

3)

The symbol R N in (1.2) denotes integration over R N with diagonals {x p = x p }, p = p excluded. One mentions that two classical books on Wiener chaoses, multiple Wiener integrals and related topics are [START_REF] Janson | Gaussian Hilbert spaces[END_REF][START_REF] Nualart | The Malliavin calculus and related topics[END_REF].

Several decades ago, the well-known papers [START_REF] Taqqu | Weak convergence to fractional Brownian motion and to the Rosenblatt process[END_REF][START_REF] Dobrushin | Non-central limit theorems for non-linear functional of Gaussian fields[END_REF][START_REF] Taqqu | Convergence of integrated processes of arbitrary Hermite rank[END_REF] drew important connections between Hermite processes and Non-Central Limit theorem. The process X 1,H is the very classical fractional Brownian motion (see e.g. [START_REF] Samorodnitsky | Stable non-Gaussian random variables: stochastic models with infinite variance[END_REF][START_REF] Embrechts | Selfsimilar processes[END_REF]). The process X 2,H is the non-Gaussian Rosenblatt process which was first introduced in the pioneering article [START_REF] Rosenblatt | Independence and dependence[END_REF]. Since about fifteen years, there has been significantly increasing interest in the study of Hermite processes and more particularly the Rosenblatt process, we refer to the works [START_REF] Pipiras | Wavelet-type expansion of the Rosenblatt process[END_REF][START_REF] Abry | Wavelet-based synthesis of the Rosenblatt process[END_REF][START_REF] Bardet | A wavelet analysis of the Rosenblatt process: chaos expansion and estimation of the self-similarity parameter[END_REF][START_REF] Maejima | On the distribution of the Rosenblatt process[END_REF][START_REF] Veillette | Properties and numerical evaluation of the Rosenblatt distribution[END_REF][START_REF] Arras | A white noise approach to stochastic integration with respect to the Rosenblatt process[END_REF][START_REF] Bojdecki | From intersection local time to the Rosenblatt process[END_REF][START_REF] Bai | Generalized Hermite processes, discrete chaos and limit theorems[END_REF][START_REF] Bai | Structure of the third moment of the generalized Rosenblatt distribution[END_REF][START_REF] Bai | Convergence of long-memory discrete k-th order Volterra processes[END_REF][START_REF] Bai | Behavior of the generalized Rosenblatt process at extreme critical exponent values[END_REF][START_REF] Bai | The impact of the diagonals of polynomial forms on limit theorems with long memory[END_REF] to cite only a few.

It is well-known that {X N,H (t)} t∈R + satisfies the following two fundamental properties:

(i) It is self-similar with exponent N (H -1) + 1, that is, for each fixed positive real number a, the two processes X N,H (at) t∈R + and a N (H-1)+1 X N,H (t) t∈R + have the same finite-dimensional distributions; observe that one can derive from (1.1) that 1 2 < N (H -1) + 1 < 1.

(1.4) (ii) It has stationary increments, which means that, for every fixed t 0 ∈ R + , the two processes X N,H (t 0 + t) -X N,H (t 0 ) t∈R + and X N,H (t) t∈R + have the same finite-dimensional distributions.

Notice that these two properties imply that E X N,H (t ) -X N,H (t ) 2 = c N,H |t -t | 2N (H-1)+2 , for all (t , t ) ∈ R 2 + , (1.5) where c N,H := E |X N,H (1)| 2 . Thus using (1.5), the inequality 2N (H -1) + 2 > 1 (see (1.4)) and the usual Kolmogorov continuity Theorem, it follows that {X N,H (t)} t∈R + has a modification with continuous paths. Let us emphasize that throughout our article the process X N,H = {X N,H (t)} t∈R + is always identified with this modification of it.

Fine study of paths behavior of stochastic processes is a very classical research topic in probability and harmonic analysis whose roots go back to Wiener's works on Brownian paths in the 20's. Plenty of results on this topic have been derived in the Gaussian and stable frameworks. Yet, so far, only few results (see [START_REF] Mori | The law of the iterated logarithm for self-similar processes represented by multiple Wiener integrals[END_REF][START_REF] Takashima | Sample path properties of ergodic self-similar processes[END_REF][START_REF] Viens | Supremum concentration inequality and modulus of continuity for sub-nth chaos processes[END_REF][START_REF] Arras | On a class of self-similar processes with stationary increments in higher order Wiener chaoses[END_REF]) are known in the framework of non-Gaussian Wiener chaoses. The important article [START_REF] Viens | Supremum concentration inequality and modulus of continuity for sub-nth chaos processes[END_REF] provides a general approach allowing to obtain moduli of continuity for a wide class of sub-nth chaos processes which includes the Hermite process X N,H . An almost sure upper bound of the asymptotic behavior of the local oscillations of the paths of X N,H can be obtained in this way. More precisely, for all ω ∈ Ω, for each fixed point τ ∈ (0, +∞) and for any real number r ∈ (0, τ ], the oscillation on the compact interval [τ -r, τ + r] of the path X N,H (ω) is the positive and finite quantity denoted by Osc X N,H (ω), τ, r and defined as:

Osc X N,H (ω), τ, r := sup X N,H (t , ω) -X N,H (t , ω) : (t , t ) ∈ [τ -r, τ + r] 2 .
(1.6) The modulus of continuity for X N,H (ω) obtained thanks to [START_REF] Viens | Supremum concentration inequality and modulus of continuity for sub-nth chaos processes[END_REF] allows to derive, for almost all ω ∈ Ω, that

lim sup r→0 r -N (H-1)-1 | log 2 r | -N/2 sup τ ∈I Osc X N,H (ω), τ, r < +∞ , (1.7)
where I ⊂ (0, +∞) denotes an arbitrary deterministic compact interval. It seems natural to look for a non-trivial almost sure lower bound for the asymptotic behavior of Osc X N,H (ω), τ, r , as r goes to 0. It is important that such a lower bound be valid on an event of probability 1 not depending on τ , since by this way it would be possible to derive from it nowhere differentiability of paths of Hermite processes.

Even in the most classical case of the Brownian motion such kind of problems are rather difficult to solve. Nowhere differentiability of Brownian paths was first established by Paley, Wiener and Zygmund [START_REF] Paley | Notes on random functions[END_REF]. Later Dvoretzki [START_REF] Dvoretzki | On the oscillation of the Brownian motion process[END_REF] was able to obtain sharp lower bounds for their local oscillations. A general and powerful strategy for dealing with this type of problems on everywhere irregularity of paths was first introduced in the early 70's by Berman [START_REF] Berman | Gaussian processes with stationary increments: local times and sample function properties[END_REF][START_REF] Berman | Gaussian sample functions: uniform dimension and Hölder conditions nowhere[END_REF][START_REF] Berman | Local nondeterminism and local times of Gaussian processes[END_REF] in the Gaussian frame and was later extended by Nolan [START_REF] Nolan | Local nondeterminism and local times for stable processes[END_REF] to the frame of stable distributions. It relies on a very clever intuitive idea called the Berman's principle: "the more regular is a local time in the time variable, uniformly in the space variable, the more irregular is the associated stochastic process"; for instance, when the local time is jointly continuous in the time and space variables then the corresponding process has nowhere differentiable paths. Many important developments of this classical and powerful strategy relying on local times are due to Xiao (see e.g. [START_REF] Xiao | Hölder conditions for the local times and the Hausdorff measure of the level sets of Gaussian random fields[END_REF][START_REF] Xiao | Properties of local-nondeterminism of Gaussian and stable random fields and their applications[END_REF][START_REF] Xiao | Recent developments on fractal properties of Gaussian random fields[END_REF]) who among many other things has introduced strengthened and sectorial versions of the crucial concept of local-nondeterminism. Unfortunately, this strategy can hardly be used in the framework of the Hermite process X N,H since, in contrast with Gaussian and stable processes, there is no explicit and easy exploitable formulas for the characteristic functions of the finite-dimensional distributions of X N,H , even in the most simple case of the Rosenblatt process where N = 2.

An alternative strategy, relying on wavelet-type series representations, which allows to bound from below the asymptotic behavior of the local oscillations of stochastic processes and fields, was first introduced in [START_REF] Ayache | Wavelet construction of generalized multifractional processes[END_REF]. Several variants of it turned out to be efficient (see e.g. [START_REF] Ayache | Asymptotic properties and Hausdorff dimensions of fractional Brownian sheets[END_REF][START_REF] Ayache | Linear fractional stable sheets: wavelet expansion and sample path properties[END_REF][START_REF] Arras | On a class of self-similar processes with stationary increments in higher order Wiener chaoses[END_REF]). Yet, in the case of the Hermite process X N,H , with N ≥ 3, no wavelet-type series representation is available so far. Moreover, in the particular case of the Rosenblatt process X 2,H , the methodology of [START_REF] Ayache | Wavelet construction of generalized multifractional processes[END_REF] seems to be hardly applicable to the wavelet-type series representations of X 2,H which were introduced in [START_REF] Pipiras | Wavelet-type expansion of the Rosenblatt process[END_REF]. Before ending this paragraph, one mentions that the classical strategy due to Berman and the alternative strategy relying on wavelets are presented in detail in the very recent book [START_REF] Ayache | Multifractional stochastic fields : wavelet strategies in multifractional frameworks[END_REF].

Let us now explain in a heuristic way the main ingredient of the new strategy we introduce in the present article. Usually in the literature, fractional Brownian motion, Rosenblatt process and more generally Hermite processes are viewed as stochastic processes whose increments are (rather strongly) correlated, and may even display long range dependence. Our new strategy relies on a different and maybe new way to view increments of these processes in the setting of the study of their path behavior: "many of the increments are independent random variables up to negligible remainders".

This new strategy allows us to prove in the present article the following theorem which provides when ω belongs to Ω * , an universal event of probability 1 not depending on τ , a quasi-optimal lower bound for the asymptotic behavior of Osc X N,H (ω), τ, r , as r goes to 0.

Theorem 1.1 There exist Ω * , an universal event of probability 1 not depending on τ , and c * N,H a (strictly) positive deterministic finite constant only depending (N, H), such that, for all ω ∈ Ω * and for every τ ∈ (0, +∞), one has

lim inf r→0 r -1 | log 2 r | S | log 2 r| N (H-1)+1 Osc X N,H (ω), τ, r ≥ c * N,H > 0 , (1.8)
where log 2 is the logarithm to the base 2, and S is an arbitrary increasing continuous function on R + which satisfies

S(0) ≥ 2 , lim z→+∞ z N 2(1-H) S(z) = 0 , lim z→+∞ z N 2(1-H) +ε S(z) = +∞ , for all fixed ε > 0, ( 1.9)
and

sup z∈R + S z + α log 2 (2 + z) S(z)
< +∞ , for each fixed α > 0.

(1.10)

Remarks 1.2 (i)
The condition S(0) ≥ 2 in (1.9) can be weakened to S(0) > 0, since the inequality (1.8) remains valid when both sides of it are multiplied by any arbitrary positive constant.

(ii) There are many classes of examples of increasing continuous functions S on R + which satisfy the conditions (1.9) and (1.10); a natural one of them is:

S(z) := (2 + z) N 2(1-H) log 2 (3 + z) β , for all z ∈ R + ,
where β is an arbitrarily small fixed positive real number.

(iii) The inequality (1.8) provides, when r goes to 0, a quasi-optimal lower bound for the asymptotic behavior of Osc X N,H (ω), τ, r , since r -1 in it is raised at the same power N (H -1) + 1 as in the reverse inequality (1.7). The issue of finding the optimal lower bound, that is the optimal choice for the function S, is a difficult problem which remains open.

An important straightforward consequence of Theorem 1.1 is that there is no point in (0, +∞) at which a typical path of the Hermite process X N,H satisfy a pointwise Hölder condition of order strictly larger than N (H -1) + 1. More precisely: Corollary 1.3 Let Ω * be the same event of probability 1 as in Theorem 1.1. Let µ be an arbitrary positive real number belonging to the open interval N (H -1) + 1, 1) (recall that one knows from (1.4) that the latter interval is included in (1/2, 1)). Then, for all ω ∈ Ω * and for each τ ∈ (0, +∞), one has that

lim sup t→τ X N,H (t, ω) -X N,H (τ, ω) |t -τ | µ = +∞ .
This clearly implies that, for any ω ∈ Ω * , the path X N,H (ω) is nowhere differentiable on the interval (0, +∞).

2 Proof of Theorem 1.1

First observe that the interval (0, +∞) can be expressed as the countable union of the open, bounded and overlapping intervals 2 -1 q, 1 + 2 -1 q , q ∈ Z + . So, it is enough to prove the theorem for every τ ∈ 2 -1 q, 1 + 2 -1 q , the nonnegative integer q being arbitrary and fixed. For the sake of simplicity, one assumes that q = 0, that is τ ∈ (0, 1); the proof can be done in the same way for any other q. Also one assumes that N ≥ 2, since N = 1 corresponds to the Gaussian case of the fBm which was studied in detail a long time ago. For any integers j ≥ 1 and k ∈ {0, . . . , 2 j -1}, one denotes by ∆(j, k) the increment of the process X N,H such that

∆(j, k) := X N,H (d j,k+1 ) -X N,H (d j,k ) , (2.1)
where d j,k+1 and d j,k are the two dyadic numbers in the interval [0, 1] defined as:

d j,k+1 := (k + 1)/2 j and d j,k := k/2 j . (2.2)
Observe that, in view of (2.1), (1.2) and (1.3), the increment ∆(j, k) can be expressed as:

∆(j, k) := R N d j,k+1 d j,k N p=1 (s -x p ) H-3/2 + ds dB(x 1 ) . . . dB(x N ) (2.3) = R N 1l I j,k (x 1 , . . . , x N ) d j,k+1 d j,k N p=1 (s -x p ) H-3/2 + ds dB(x 1 ) . . . dB(x N ) ,
where 1l I j,k is the indicator function of the unbounded rectangle of R N :

I j,k := (-∞, d j,k+1 ] N . (2.4)
Notice that, in view of the second equality in (1.9), there exists a positive integer J 0 such that, for each integer j ≥ J 0 , one has S(j) ≤ 2 j-4 . From now on, one always assumes that j ≥ J 0 . Let e j be the integer part of S(j) (recall that S(j) ≥ 2), that is

e j := S(j) , (2.5)
and one denotes by L j the non-empty finite set of positive integers defined as:

L j := N ∩ 1, (2 j /e j ) -1 . (2.6)
Observe that the cardinality of L j satisfies, for some positive finite constant c not depending on j, card(L j ) ≤ c 2 j /e j .

(2.7)

For each l ∈ L j , one denotes D j,le j and D j,le j the two non-empty subsets of I j,le j (see (2.4)) defined as: D j,le j := [d j,(l-1)e j +1 , d j,le j +1 ] N and D j,le j := I j,le j \ D j,le j = {x ∈ I j,le j : x / ∈ D j,le j } .

(2.8) One clearly has that D j,le j ∩ D j,le j = ∅ and I j,le j = D j,le j ∪ D j,le j . Thus, using the second equality in (2.3), one gets that ∆(j, le j ) = ∆(j, le j ) + ∆(j, le j ) , (2.11) Roughly speaking, as we have already pointed out in the previous section, the main ingredient of our strategy for proving Theorem 1.1 consists to show that ∆(j, le j ) is in some sense (which will be made more precise in the sequel) negligible with respect to ∆(j, le j ), and that the random variables ∆(j, le j ), l ∈ L j , satisfy the very nice independence property. To this end, some lemmas are needed.

Lemma 2.1 For each fixed integer j ≥ J 0 , the random variables ∆(j, le j ), l ∈ L j , are independent.

Proof

The integrand in (2.10) can be approximated in the sense of the L 2 (R N ) norm by a sequence (φ n ) n∈N of real-valued step functions on R N which vanish on the diagonals {x p = x p }, p = p , and also outside of the cube D j,le j (see (2.8)). Thus, using the "isometry" property of multiple Wiener integral, it turns out that ∆(j, le j ) = lim n→+∞ R N φ n (x 1 , . . . , x N ) dB(x 1 ) . . . dB(x N ), (in L 2 (Ω)).

(2.12)

On the other hand, one knows from the definition of a multiple Wiener integral and from the fact that the step function φ n vanishes outside of D j,le j that the random variable R N φ n (x 1 , . . . , x N ) dB(x 1 ) . . . dB(x N ) can be expressed as a polynomial function in terms of a finite number of increments B(a n i+1 ) -B(a n i ) of the Brownian motion B, where (a n i ) i is a finite increasing sequence of real numbers belonging to the interval [d j,(l-1)e j +1 , d j,le j +1 ]. Thus, it turns out that R N φ n (x 1 , . . . , x N ) dB(x 1 ) . . . dB(x N ) is measurable with respect to the σ-algebra σ B(x) -B(y) : x, y ∈ (d j,(l-1)e j +1 , d j,le j +1 ) . Combining the latter fact with (2.12) it follows that ∆(j, le j ) is measurable with respect to the same σ-algebra.

Finally, observe that the independence of increments property of the Brownian motion B and the fact the intervals (d j,(l-1)e j +1 , d j,le j +1 ), l ∈ L j , are disjoint imply that the σ-algebras σ B(x) -B(y) : x, y ∈ (d j,(l-1)e j +1 , d j,le j +1 ) , l ∈ L j , are independent, which in turn entails that the random variables ∆(j, le j ), l ∈ L j , are independent as well.

Lemma 2.2 Let c be the finite (strictly) positive constant defined as

c := [-1,1] N 1 0 N p=1 (u -y p ) H-3/2 + du 2 dy 1 . . . dy N 1/2 .
(2.13)

Then, for all integer j ≥ J 0 , one has

inf l∈L j ∆(j, le j ) L 2 (Ω) ≥ c 2 -j(N (H-1)+1) .
(2.14)

Proof

Using the fact that the integrand in (2.10) is a symmetric function in the variables x 1 , . . . , x N and the "isometry" property of multiple Wiener integral one gets, for every j ≥ J 0 and l ∈ L j , that ∆(j, le j )

2 L 2 (Ω) := E | ∆(j, le j )| 2 = N ! D j,le j d j,le j +1 d j,le j N p=1 (s-x p ) H-3/2 + ds 2 dx 1 . . . dx N .
Next, using the change of variable s = d j,le j + 2 -j u and (2.2), one obtains that ∆(j, le j )

2 L 2 (Ω) = N ! 2 -2j D j,le j 1 0 N p=1 (d j,le j + 2 -j u -x p ) H-3/2 + du 2 dx 1 . . . dx N .
Then, the first equality in (2.8), the change of variables x p = d j,le j + 2 -j y p , for all p ∈ {1, . . . , N }, the inequality e j ≥ 2, for every j ≥ J 0 , and (2.13) imply that ∆(j, le j )

2 L 2 (Ω) = N ! 2 -(N +2)j [1-e j ,1] N 1 0 N p=1 (2 -j u -2 -j y p ) H-3/2 + du 2 dy 1 . . . dy N = N ! 2 -2j(N (H-1)+1) [1-e j ,1] N 1 0 N p=1 (u -y p ) H-3/2 + du 2 dy 1 . . . dy N ≥ c 2 2 -2j(N (H-1)+1) .
Lemma 2.3 There exists a positive finite constant c such that, for all integer j ≥ J 0 , one has sup l∈L j ∆(j, le j ) L 2 (Ω) ≤ c 2 -j(N (H-1)+1) e H-1 j .

(2.15)

Proof

Using the fact that the integrand in (2.11) is a symmetric function in the variables x 1 , . . . , x N , the "isometry" property of multiple Wiener integral, (2.4), (2.8), the inequality d j,(l-1)e j +1 < d j,le j and the fact that z → z H-3/2 is a decreasing function on (0, +∞), one gets that ∆(j, le j )

2 L 2 (Ω) := E | ∆(j, le j )| 2 = N ! D j,le j d j,le j +1 d j,le j N p=1 (s -x p ) H-3/2 + ds 2 dx 1 . . . dx N ≤ N • N ! d j,(l-1)e j +1 -∞ R N -1 d j,le j +1 d j,le j (s -x N ) H-3/2 N -1 p=1 (s -x p ) H-3/2 + ds 2 dx 1 . . . dx N -1 dx N ≤ N • N ! d j,(l-1)e j +1 -∞ (d j,le j -x N ) 2H-3 dx N (2.16) × R N -1 d j,le j +1 d j,le j N -1 p=1 (s -x p ) H-3/2 + ds 2 dx 1 . . . dx N -1 .
It results from standard computations, (2.2) and the inequality e j -1 ≥ 2 -1 e j that d j,(l-1)e j +1

-∞

(d j,le j -x N ) 2H-3 dx N = (d j,le j -d j,(l-1)e j +1 ) 2H-2 2 -2H = (d j,e j -1 ) 2H-2 2 -2H = (e j -1) 2H-2 2 -2j(H-1) 2 -2H ≤ e 2H-2 j 2 -2(j+1)(H-1) 2 -2H . (2.17)
Moreover, the "isometry" property of multiple Wiener integral, (1.2) and (1.5) imply that

(N -1)! R N -1 d j,le j +1 d j,le j N -1 p=1 (s -x p ) H-3/2 + ds 2 dx 1 . . . dx N -1 (2.18) = E X N -1,H (d j,le j +1 ) -X N -1,H (d j,le j ) 2 = c N -1,H 2 -2j((N -1)(H-1)+1) .
Then combining (2.16), (2.17) and (2.18), one obtains (2.15).

Lemma 2.4 There are Ω an event of probability 1 and C a positive finite random variable such that on Ω, for every integer j ≥ J 0 , one has

sup l∈L j ∆(j, le j ) ≤ C 2 -j(N (H-1)+1) e H-1 j j N/2 . (2.19)
In order to show that Lemma 2.4 holds one needs the following lemma which is in fact Theorem 6.7 on page 82 of the well-known book [START_REF] Janson | Gaussian Hilbert spaces[END_REF].

Lemma 2.5 For any fixed integer N ≥ 1, there exists a (strictly) positive finite universal deterministic constant cN such that, for every random variable χ belonging to the Wiener chaos of order N and for each real number y ≥ 2, one has

P |χ| > y χ L 2 (Ω) ≤ exp -cN y 2/N , (2.20
)

where χ L 2 (Ω) := E |χ| 2 1/2 .
Proof of Lemma 2.4 Let κ ≥ 2 be a finite deterministic constant which will be soon defined more precisely. For all integer j ≥ J 0 , one has

P sup l∈L j ∆(j, le j ) > (κ j) N/2 sup l∈L j ∆(j, le j ) L 2 (Ω) ≤ l∈L j P ∆(j, le j ) > (κ j) N/2 sup l∈L j ∆(j, le j ) L 2 (Ω) ≤ l∈L j P ∆(j, le j ) > (κ j) N/2 ∆(j, le j ) L 2 (Ω) . (2.21)
Moreover, in view of the fact that (κ j) N/2 ≥ 2 (recall that N ≥ 2), one can make use of Lemma 2.5 in order to bound from above the probabilities in (2.21). By this way, for all l ∈ L j , one gets

P ∆(j, le j ) > (κ j) N/2 ∆(j, le j ) L 2 (Ω) ≤ exp -cN κ j , (2.22)
where cN is the same deterministic positive finite constant as in (2.20). Next combining (2.21) and (2.22) with (2.7) and the inequality e j ≥ 2, one obtains that

P sup l∈L j ∆(j, le j ) > (κ j) N/2 sup l∈L j ∆(j, le j ) L 2 (Ω) ≤ c 2 j e -1 j exp -cN κ j = c exp -(c N κ -log 2)j , (2.23)
where c denotes the same constant as in (2.7). One can assume that the finite constant κ is chosen such that κ > (log 2)/c N . Then, one can derive from (2.23) that

+∞ j=J 0 P sup l∈L j ∆(j, le j ) > (κ j) N/2 sup l∈L j ∆(j, le j ) L 2 (Ω) < +∞.
Thus, the Borel-Cantelli Lemma implies that there exist Ω an event of probability 1 and C0 a positive finite random variable, such that, for all ω ∈ Ω and for every integer j ≥ J 0 , one has sup

l∈L j ∆(j, le j , ω) ≤ C0 (ω) j N/2 sup l∈L j ∆(j, le j ) L 2 (Ω) . (2.24)
Finally, (2.19) results from (2.15) and (2.24).

The following lemma is a straightforward consequence of Theorem 6.9 and Remark 6.10 on page 82 of the well-known book [START_REF] Janson | Gaussian Hilbert spaces[END_REF].

Lemma 2.6 For any fixed integer N ≥ 1, there exists an universal deterministic constant

γ N satisfying 0 ≤ γ N < 1 , (2.25)
such that, for each random variable χ belonging to the Wiener chaos of order N one has

P |χ| < 2 -1 χ L 2 (Ω) ≤ γ N , (2.26)
where

χ L 2 (Ω) := E |χ| 2 1/2 .
Remark 2.7 Observe that, in view of the fact that the nonnegative constant γ N is strictly smaller than 1 (see (2.25)), there exists an integer n 0 ≥ 2, only depending on

N , such that 0 ≤ 2γ n 0 N < 1 . (2.27)
Also observe that the second equality in (1.9) and (2.5) imply that there is a positive integer J 1 ≥ J 0 such that, for all integer j ≥ J 1 , one has diam 1, (2 j /e j ) -1 := (2 j /e j ) -1 -1 = (2 j /e j ) -2 ≥ 10 n 0 j .

(2.28)

For each fixed integer j ≥ J 1 , the integer M j ≥ 10 denotes the integer part of (n 0 j) -1 (2 j /e j ) -2 , that is

M j := (n 0 j) -1 (2 j /e j ) -2 . (2.29)
Also, for each fixed integer j ≥ J 1 , one denotes by (U j m ) m∈{0,1,...,M j } the subdivision of the interval 1, (2 j /e j ) -1 by the M j + 1 points defined as: U j M j := (2 j /e j ) -1 and U j m := 1 + m(n 0 j) , for all m ∈ {0, 1, . . . , M j -1}; (2.30) notice that

n 0 j ≤ U j M j -U j M j -1 < 2(n 0 j) . (2.31)
Moreover, for every fixed integer j ≥ J 1 , let ( Λ j m ) m∈{1,...,M j } be the sequence of the nonnegative finite random variables defined, for all m ∈ {1, . . . , M j }, as:

Λ j m := sup ∆(j, le j ) : l ∈ L j m , (2.32)
where ∆(j, le j ) is as in (2.10), and L j m denotes the non-empty finite set of positive integers defined as:

L j m := N ∩ U j m-1 , U j m . (2.33)
Observe that, in view of (2.6), (2.30), (2.31) and (2.33), one has

L j = M j m=1 L j m (2.34)
and card(L j m ) > n 0 j , for all m ∈ {1, . . . , M j }.

(2.35) Lemma 2.8 One denotes by c the same (strictly) positive deterministic constant as in (2.13). Then, there is Ω an event of probability 1 such that on Ω one has

lim inf j→+∞ 2 j(N (H-1)+1) inf 1≤m≤M j Λ j m ≥ 2 -1 c > 0 . (2.

36)

Proof Let Ω be the event defined as:

Ω := +∞ J=J 1 +∞ j=J M j m=1 ω ∈ Ω : Λ j m (ω) ≥ 2 -1 c 2 -j(N (H-1)+1) , (2.37)
where the fixed positive integer J 1 is as in Remark 2.7. In order to show that the lemma holds, it is enough to prove P( Ω) = 1 which is equivalent to prove that

P(Ω \ Ω) = 0 . (2.38)
Notice that, in view of (2.37) the event Ω \ Ω can be expressed as:

Ω \ Ω := +∞ J=J 1 +∞ j=J M j m=1 ω ∈ Ω : Λ j m (ω) < 2 -1 c 2 -j(N (H-1)+1) .
Thus, one knows from the Borel-Cantelli Lemma that in order to derive (2.38) it is enough to prove that

+∞ j=J 1 P M j m=1 ω ∈ Ω : Λ j m (ω) < 2 -1 c 2 -j(N (H-1)+1) < +∞ . ( 2
P M j m=1 ω ∈ Ω : Λ j m (ω) < 2 -1 c 2 -j(N (H-1)+1) ≤ M j m=1 P Λ j m < 2 -1 c 2 -j(N (H-1)+1) = M j m=1 P l∈L j m ω ∈ Ω : ∆(j, le j , ω) < 2 -1 c 2 -j(N (H-1)+1) = M j m=1 l∈L j m P ∆(j, le j ) < 2 -1 c 2 -j(N (H-1)+1) < M j γ n 0 j N < 2γ n 0 N j .
(2.40) Thus, (2.39) follows from (2.27) and (2.40).

Lemma 2.9 Using some of the notations previously introduced in Remark 2.7, for every fixed integer j ≥ J 1 , let (Λ j m ) m∈{1,...,M j } be the sequence of the nonnegative finite random variables defined, for all m ∈ {1, . . . , M j }, as:

Λ j m := sup ∆(j, le j ) : l ∈ L j m , (2.41)
where ∆(j, le j ) is as in (2.9). Moreover, one denotes by Ω * the event of probability 1 defined as Ω * := Ω ∩ Ω (see Lemmas 2.4 and 2.8). Then, one has on Ω * lim inf j→+∞ 2 j(N (H-1)+1) inf

1≤m≤M j Λ j m ≥ 2 -1 c > 0 , (2.42)
where c is the same strictly positive deterministic constant as in (2.13).

Proof Using (2.9), (2.41), the triangle inequality, (2.32), (2.34) and Lemma 2.4, one obtains on the event Ω * , for every j ≥ J 1 , We are now in position to complete the proof of Theorem 1.1. End of the proof of Theorem 1.1 The point τ ∈ (0, 1) is arbitrary and fixed. For any integer j ≥ J 1 , one sets l j (τ ) := 2 j τ /e j ,

2 j(N (H-1)+1) inf 1≤m≤M j Λ j m ≥ 2 j(N (H-1)+1) inf 1≤m≤M j Λ j m -sup l∈L j m ∆(j, le j ) ≥ 2 j(N (H-1)+1) inf 1≤m≤M j Λ j m -sup l∈L j ∆(j, le j ) ≥ 2 j(N (H-1)+1) inf 1≤m≤M j Λ j m -C e H-1 j j N/2 . ( 2
where • denotes as usual the integer part function. One clearly has that 0 ≤ τ -l j (τ )e j 2 j < e j 2 j .

(2.46) Moreover, it easily follows from (2.45), the inequalities 0 < τ < 1, (2.6), (2.5) and the second equality in (1.9) that there exists a positive integer J 2 (τ ) ≥ J 1 , such that, for all integer j ≥ J 2 (τ ), one has l j (τ ) ∈ L j (2.47) and τ -12n 0 jS(j + 1) 2 j , τ + 12n 0 jS(j + 1) 2 j ⊂ (0, 1) ,

where the fixed integer n 0 ≥ 2 is as in (2.27). In view of (2.47) and (2.34), there is m j (τ ) ∈ {1, . . . , M j } such that l j (τ ) ∈ L j m j (τ ) .

(

ω ∈ Ω, Osc X N,H (ω) , τ , 3n 0 jS(j) 2 j ≥ Λ j m j (τ ) (ω) ≥ inf 1≤m≤M j Λ j m (ω) . (2.52)
Thus, assuming that η ∈ 0, 2 -1 c is arbitrary and fixed, one can derive from (2.52) and Lemma 2.9 that, for all ω ∈ Ω * , there exists an integer j 3 = j 3 (τ, η, ω) ≥ J 2 (τ ) such that, for all integer j ≥ j 3 , one has 2 j(N (H-1)+1) Osc X N,H (ω) , τ , 3n 0 jS(j) 2 j ≥ η .

(2.53)

Next, let ρ be an arbitrary positive real number such ρ ≤ 2 -j 3 , one sets j * (ρ) := -log 2 ρ .

(2.54)

One clearly has that j * (ρ) ≥ j 3 and 2 j * (ρ) ≤ ρ -1 < 2 j * (ρ)+1 .

(2.55) Thus, (2.53), (2.54), (2.55), (1.6) and the fact that S is an increasing function imply that ρ -N (H-1)-1 Osc X N,H (ω) , τ , 6n 0 (-log 2 ρ)S(-log 2 ρ)ρ ≥ 2 j * (ρ)(N (H-1)+1) Osc X N,H (ω) , τ , 3n 0 j * (ρ)S(j * (ρ)) 2 j * (ρ) ≥ η .

(2.56)

Next, let r be an arbitrary positive real number such that r ≤ 2 -j 3 . Using the latter inequality, the inequality n 0 ≥ 2, and the fact that the function S is with values in [2, +∞) (see the inequality in (1.9)), one obtains that ν(r) ≤ r ≤ 2 -j 3 , where ν(r) := r 6n 0 (-log 2 r)S(-log 2 r) .

(2.57) Thus, one can take ρ = ν(r) in (2.56). By this way one gets, for all r ∈ (0, 2 - (2.64)

One knows from (2.57) that, for each r > 0 small enough, one has ν(r) ν θ -1 0 r = θ 0 -log 2 (θ -1 0 r) S -log 2 (θ -1 0 r) (-log 2 r)S(-log 2 r) ≤ θ 0 1 + log 2 (θ 0 ) × S -log 2 (r) + log 2 (θ 0 ) S(-log 2 r) .

Thus, using (1.10) one obtains (2.64).

  le j ) = R N 1l D j,le j (x 1 , . . . , x N ) d j,le j +1 d j,le j N p=1 (s -x p ) H-3/2 + ds dB(x 1 ) . . . dB(x N ) (2.10) and ∆(j, le j ) = R N 1l D j,le j (x 1 , . . . , x N ) d j,le j +1 d j,le j N p=1 (s -x p )H-3/2 + ds dB(x 1 ) . . . dB(x N ) .

1

 1 j 3 ] ⊆ (0, 1/2], that ν(r)-N (H-1)-1 Osc X N,H (ω) , τ , -log 2 ν(r) S -log 2 ν(r) (-log 2 r)S(-log 2 r) r ≥ η . (2.58)Let us now show thatθ 0 := sup r∈(0,1/2] -log 2 ν(r) S -log 2 ν(r) (-log 2 r)S(-log 2 r) < +∞ . (2.59)Let us set z = z(r) := -log 2 r, where r ∈ (0, 1/2] is arbitrary. Using (2.57), the inequality z ≥ 1, the second equality in (1.9), and the fact that S is an increasing continuous function, one gets thatS -log 2 ν(r) = S z+log 2 (6n 0 )+log 2 (z)+log 2 (S(z)) ≤ S z+α 0 log 2 (2+z) ,(2.60)where α 0 is a positive finite constant only depending on N and S. Thus, it follows from (2.60) and (1.10) that sup r∈(0,1/2] S -log 2 ν(r) S(-log 2 r) ≤ sup z∈[1,+∞) S z + α 0 log 2 (2 + z) S(z) < +∞ . (2.61) Similarly to (2.60) it can be shown that -log 2 ν(r) ≤ z + α 0 log 2 (2 + z), for all r ∈ (0, 1/2]; thus, one gets that sup r∈(0,1/2]-log 2 ν(r) (-log 2 r) ≤ sup z∈[1,+∞) z + α 0 log 2 (2 + z) z < +∞ . (2.62) Then (2.59) results from (2.61) and (2.62). Next, combining (2.58) and (2.59) with (1.6) and the fact that η ∈ 0, 2 -1 c is arbitrary, it follows that, for all τ ∈ (0, 1) andω ∈ Ω * , lim inf r→0 ν(r) -N (H-1)-1 Osc X N,H (ω) , τ , θ 0 r ≥ 2 Osc X N,H (ω) , τ , r ≥ 2 -1 c . (2.63)Finally, in view of (2.57) and (2.63), in order to derive(1.8) it is enough to show that

  .39) Putting together (2.32), Lemma 2.1, (2.34), Lemma 2.2, Lemma 2.6, (2.35), (2.29) and the inequality e j ≥ 2, one gets, for every j ≥ J 1 ,

Acknowledgements

The author is grateful to the anonymous reviewers for their valuable comments which have led to improvements of the manuscript. This work has been partially supported by the Labex CEMPI (ANR-11-LABX-0007-01) and the GDR 3475 (Analyse Multifractale).


The relationship between fermented food and health

• Increasing amount of data indicate that a diet enriched with fermented food have a beneficial effect on health thanks to a succession of linked effects.

• i.e. the DOMINO effect

Background (1)

+ + + p. 6
The relationship between fermented food and health The relationship between fermented food and health The relationship between fermented food and health Plant-based fermented food can provide sustainability benefits Plant-based fermented food can provide sustainability benefits Many perspectives and many challenges…

• A tiny fraction of the large natural diversity of food microbial strains and consortia are currently harnessed.

• Many technological challenges of plant-based products (anti-nutritional factors, low-solubility of plant proteins, off-flavours, food structure, consumer perception & acceptance).

• Difficulties to extrapolate the effect of long-diet. Can we provide improved protocols for benefit assements of new fermented food & microbial consortia ?

Background (3)

p. 13

From spontaneous fermentations to designed consortia To reconstruct microbial communities, we need microbial ressources. Do we have the good ones ?

Hypothesis: isolation practices filters the diversity of strains and species identified in food although these taxa may represent an important functional role, or participate in certain processes in a transient manner (dynamic succession of populations).

T°S uccession of populations

Metabolic interactions

Master internship: Sybille Tabuteau ( 2022 • A first exemple for connecting microbial ressoures; their genomic data and their associated omics (RNAseq/metabolomic).

• World Institute of Kimchi, Gwangju. South-Korea p. 24

The general idea of the DOMINO project for the curated open FMD and computational tools Implementing Health benefit in the design of consortia

• Human nutritional intervention are the 'golden rule' but long, tedious and expensive;

• It is very likely that long-term diet are necessary for establishing health benefit.

• Setting standard protocols for rapid estimation of health benefit using in-vitro/ex-vivo models ?

p. 27

Implementing Health benefit in the design of consortia

  diversity of microbial solutions (i.e. microbial consortia) 2. Strategies for design & experimental bottlenecks 3. A need for food 'omics' databases & ecological network modelling tools 4. A challenge for implementing health benefits in the design Microbes, fermented foods, health & sustainability… … a complex equation between societal and technological challenges p. 5

Fermented

  discipline to a new era using multi-disciplinary synthetic ecology approaches Build -Try -Learn p. 17 De Ryoy et al., 2014 https:/doi.org/10.1111/1462-2920metagenome-scale modelling M2M: microbiota-scale metabolic complementarity for the identification of key species Belfour et al., 2020 https://doi.org/10.7554/eLife.61968 p. 19 Implementing analyses to map ecological activity on a global scale o RNAseq data analysis tools More relevant at ecosystem level o More complete and complementary food metabolomic analysis = the cornerstone https://doi.org/10.1016/j.tibs.2017.01.004 https://doi.org/10.1093/nar/gky174 p. 20

an omics data resource from microorganisms associated with fermented foods

  MAGs can reveal biases in cultured food bacterial isolates and help identifying new species and strains.

	Strategy : Phylogenomic comparison between Metagenomic Assembled Genomes (MAGs) and isolates with Strategy : Phylogenomic comparison between Metagenomic Assembled Genomes (MAGs) and isolates with
	published genomes published genomes
	~80 food metagenomes -> ~80 MAGS ~80 food metagenomes -> ~80 MAGS
	p. 21 p. 22

): ISME symposium Lausanne August 2022. One target species : Latilactobacillus sakei (80 genomes available, high diversity of strains) One target species : Latilactobacillus sakei (80 genomes available, high diversity of strains) p. 23 2. Omics Food Microbiome databases ODFM, Whon et al., 2021 scientific data. https://doi.org/10.1038/s41597-021-00895-x

• Human nutritional intervention are the 'golden rule' but long, tedious and expensive;

• It is very likely that long-term diet are necessary for establishing health benefit.

• Setting standard protocols for rapid estimation of health benefit using in-vitro/ex-vivo models ? https://doi.org/10.1016/j.jcmgh.2017.12.010 https://doi.org/10.1097/MPG.0000000000001411 https://doi.org/doi:10.1038/nrd2031


Antoine Ayache
email: antoine.ayache@math.univ-lille1.fr

Céline Esser
email: celine.esser@math.univ-lille1.fr

Thomas Kleyntssens
email: tkleyntssens@ulg.ac.be

Different possible behaviors of wavelet leaders of the Brownian motion

Keywords: 60J65, 42C40, Secondary: 26A16 Brownian motion, slow points, fast points, ordinary points, wavelet leaders, modulus of continuity, Hölder spaces

The aim of this paper is to prove that wavelet leaders allow to get very fine properties of the trajectories of the Brownian motion: more precisely, we show that there exist (at least) three different behaviors for the size of wavelet leaders of the Brownian motion. Furthermore, they correspond to the three well-known behaviors of its oscillations, namely to be ordinary, rapid and slow. Some links between the oscillations and the size of wavelet leaders at a given point are also given.

Introduction

The oscillation of a function is a widespread way of quantifying its regularity around a point. This classical notion has been widely used in analysis, for example in the frame of Hölder spaces, and in probability, as for instance in the study of sample path properties of stochastic processes. It is based on increments. More precisely, let f : R → R be a continuous function; for each compact interval K of R, we define the oscillation of f on K by Osc(f, K) = sup

t,t ∈K f (t) -f (t ) . (1)
In order to precisely know how smooth or how rough is f around some fixed point t ∈ R, it is natural to try to determine as sharply as possible the asymptotic behavior of the oscillation Osc(f, I(t, ρ)) when ρ → 0 + . Here I(t, ρ) denotes the closed interval of center t and radius ρ > 0; that is

I(t, ρ) = [t -ρ, t + ρ].
On another hand, wavelet methods have become in the last decades a very powerful tool to finely study regularity properties of functions [START_REF] Daubechies | Ten lectures on wavelets[END_REF][START_REF] Meyer | Wavelets and operators[END_REF][START_REF] Mallat | A Wavelet Tour of Signal Processing[END_REF][START_REF] Jaffard | Wavelet techniques in multifractal analysis, fractal geometry and applications: A jubilee of Benoit Mandelbrot[END_REF]. In this context, wavelet coefficients can be viewed as generalized increments. In order to be more precise, one denotes by ψ : R → R a compactly supported function of L 1 (R) whose first moment vanishes, that is: R ψ(x) dx = 0 .

(2)

Therefore, ψ is called a compactly supported wavelet, and there exists a positive integer N such that the support of ψ is included in [-N, N ]. The wavelet coefficients c J,K , (J, K) ∈ N × Z, of f are defined by

c J,K = 2 J R f (t)ψ(2 J t-K) dt = R f x + K 2 J ψ(x) dx = N -N f x + K 2 J ψ(x) dx . ( 3
)
Observe that ( 2) and (3) imply that

c J,K = R f x + K 2 J -f K 2 J ψ(x) dx (4) = N -N f x + K 2 J -f K 2 J ψ(x) dx . (5)
In what follows, we will use the notation c λ to denote the wavelet coefficient c J,K , where λ is the dyadic interval

λ = λ J,K = K 2 J , K + 1 2 J .
The interval λ J,K provides the location of the wavelet ψ(2 J • -K). We denote by Λ the set of all dyadic intervals of R. Moreover, at an arbitrary given scale J ∈ N, we denote by Λ J the set of all dyadic intervals of size 2 -J . Amplitudes of wavelet coefficients c λ located in some fixed interval λ (that is λ ⊆ λ) can be very fluctuating from one scale to another. In order to avoid such a drawback, the so-called wavelet leaders, which among other things offer the advantage of stability, have been introduced in [START_REF] Jaffard | Wavelet leaders in multifractal analysis[END_REF]. They can be viewed as local suprema of the amplitudes of wavelet coefficients. Thus, a small value of the supremum associated with some interval λ means then that all the wavelet coefficients of the fonction f located in λ have small amplitudes. Roughly speaking this means that f is smooth on λ (see Section 2).

Let us now define the wavelet leaders of f in a precise way. For every (J, K) ∈ N × Z, one denotes by N (λ J,K ) the set

N (λ J,K ) = λ J,K-1 , λ J,K , λ J,K+1 .
Then, the wavelet leader d J,K is defined by

d J,K = max λ∈N (λ J,K ) sup λ ⊆λ c λ . (6)
Again, we use the notation d λ if λ = λ J,K .

Remark 1.1. In a probabilistic framework, the supremum on N (λ J,K ) appearing in [START_REF] Deliège | Mars topography investigated through the wavelet leaders method: A multidimensional study of its fractal structure[END_REF] can create correlation between wavelet leaders, even if it does not exist between wavelet coefficients. It might seem more natural to consider only the dyadic intervals λ ⊆ λ J,K in the definition of the wavelet leaders so that the supremum at a given scale is taken on non-overlapping intervals. These coefficients are called the restricted wavelet leaders [START_REF] Jaffard | Function spaces vs. scaling functions: tools for image classification[END_REF].

Let us point out that the methodology of our article can easily be adapted to their setting.

In fact, we prefer to work with the wavelet leaders defined as in [START_REF] Deliège | Mars topography investigated through the wavelet leaders method: A multidimensional study of its fractal structure[END_REF] since, in contrast with the restricted wavelet leaders, they provide a characterisation of pointwise Hölder spaces (see Section 2).

As wavelet coefficients can viewed as generalized increments, wavelet leaders can be seen as generalized oscillations. It is therefore natural to wonder whether there are some links between the two notions: oscillation (see [START_REF] Abry | Self-similarity and long-range dependence through the wavelet lens. Theory and applications of long-range dependence[END_REF]) and wavelet leader (see [START_REF] Deliège | Mars topography investigated through the wavelet leaders method: A multidimensional study of its fractal structure[END_REF]).

First, observe that for each fixed t ∈ R and J ∈ N, there exists a unique dyadic interval λ ∈ Λ J such that t ∈ λ; this dyadic interval is denoted by λ J (t) and the corresponding wavelet leader by d J (t). If λ j,k ⊆ λ for some λ ∈ N λ J (t) , one has

t - k 2 j ≤ 2 • 2 -J , hence, if x ∈ [-N, N ], t - x + k 2 j ≤ (2 + N )2 -J .
Combining these two inequalities with ( 5), ( 1) and ( 2), one gets that

|c j,k | ≤ N -N f x + k 2 j -f k 2 j |ψ(x)| dx ≤ c 0 Osc f, I t, (2 + N )2 -J ,
where c 0 = ψ L 1 (R) . As a consequence, one has that

d J (t) ≤ c 0 Osc f, I t, (2 + N )2 -J . (7)
Conversely, Jaffard [START_REF] Jaffard | Sur la dimension de boîte des graphes[END_REF] has shown that generally speaking, under the assumption that f belongs to some Hölder space (see Section 2), the reverse inequality holds up to a logarithmic factor:

Osc f, λ J (t) ≤ Cd J (t) log(d J (t)) (8)
for some constant C > 0. It can be derived from ( 7) and ( 8) that there is a priori a loss of information if one considers wavelet leaders instead of oscillations.

The problem we deal with in this article in the following: is this estimation (8) as good as possible, or can one expect an estimation without any logarithmic correction? This question appeared in a general setting in [START_REF] Jaffard | Pointwise smoothness, two-microlocalization and wavelet coefficients[END_REF] where it is proved that this logarithmic correction is needed. In the present paper, we treat the particular case of Brownian motion. Despite the general inequality (8) which leads to believe that there is a loss of information if one works with wavelet leaders instead of oscillations, we show that wavelet leaders are precise enough to reflect very fine properties of the trajectories of Brownian motion, namely the coexistence in them of slow, fast and ordinary points.

The rest of this article is organized in the following way. In Section 2, we make some recalls on Brownian motion and gives some useful properties of its wavelet coefficients, also we present the notions of slow point, fast point and ordinary point. At the end of this section, we state the main result of the article (Theorem 2.5) which shows that the coexistence of slow, fast and ordinary points on Brownian trajectories can be revealed by wavelet leaders. Section 3 is devoted to the proof of Theorem 2.5. Finally, Section 4 gives some links between the behavior of oscillations at a point t and the size of the associated wavelet leaders d J (t).

Brownian motion and statement of the main result

The Brownian motion is an important Gaussian process that models many phenomena (see e.g. [START_REF] Lévy | Processus stochastiques et mouvement brownien[END_REF][START_REF] Kahane | Some Random Series of Functions[END_REF][START_REF] Karatzas | Brownian Motion and Stochastic Calculus[END_REF] and references therein). It is the unique real-valued centered Gaussian process B = {B(t)} t∈R defined on a probability space (Ω, A, P) such that • almost surely, B(0) = 0 and t → B(t) is a continuous function on the real line,

• it has stationary increments, more precisely for every t, s ∈ R, one has B(t) -B(s) ∼ N (0, |t -s|),

• for any

t 1 < t 2 < • • • < t n , n ≥ 1, the increments B(t 2 ) -B(t 1 ), . . . , B(t n ) -B(t n-1 ) are independent.
From now on, if the value of B(t) has to be explicitly associated to an elementary event ω ∈ Ω, the notation B(t, ω) will be used. It is well-known (see e.g. [START_REF] Kahane | Some Random Series of Functions[END_REF]) that there exists an event Ω * ⊆ Ω of probability one satisfying the following property: for each ω ∈ Ω * and for all h ∈ [0, 1/2), there is a constant C(ω) > 0 such that for every t ∈ R, the inequality

Osc B(•, ω), I(t, ρ) ≤ C(ω)ρ h (9)
holds for ρ > 0 small enough. However, this inequality is not valid for h = 1/2. In terms of classical Hölder spaces (see e.g. [START_REF] Krantz | Lipschitz spaces, smoothness of functions, and approximation theory[END_REF]), it means that B belongs to the space C h (R) for every h < 1/2 but fails to belong to C 1/2 (R). This last property can be refined using the notion of generalized Hölder spaces C τ (R). These spaces are defined by replacing the value ρ h in the inequality (9) by τ (ρ), for some modulus of continuity τ (see [START_REF] Clausel | Quelques notions d'irrégularité uniforme et ponctuelle: le point de vue ondelettes[END_REF]). The Brownian motion belongs to the Hölder space C τ (R), where the modulus of continuity τ is given by

τ (ρ) = ρ 1/2 log ρ -1
(see [START_REF] Kahane | Some Random Series of Functions[END_REF]). This means that there exists a constant C (ω) > 0 such that for every t ∈ R, one has

Osc B(•, ω), I(t, ρ) ≤ C (ω)ρ 1/2 log ρ -1 (10)
for ρ > 0 small enough. These previous properties of the Brownian motion are uniform, but its pointwize regularities have also been studied. In this context, the Khintchin law of the iterated logarithm (see e.g. [START_REF] Kahane | Some Random Series of Functions[END_REF]) asserts that for almost every t ∈ R, B satisfies

lim sup ρ→0 |B(t + ρ, ω) -B(t, ω)| ρ 1/2 log log ρ -1 = √ 2. (11)
In particular, for almost every t ∈ R, there exists a constant C (t, ω) > 0 such that

Osc B(•, ω), I(t, ρ) ≤ C (t, ω)ρ 1/2 log log ρ -1
for every ρ > 0 small enough. This is equivalent to the fact that B belongs to the generalized pointwise Hölder spaces C τ (t) introduced recently in [START_REF] Clausel | Quelques notions d'irrégularité uniforme et ponctuelle: le point de vue ondelettes[END_REF][START_REF] Kreit | Generalized pointwise Hölder spaces[END_REF], with τ (ρ) = ρ 1/2 log log ρ -1 .

One can go further and study the regularity properties at the points at which ( 11) is not true. The following theorem [START_REF] Kahane | Some Random Series of Functions[END_REF] summarizes some local properties of the Brownian motion: it shows the existence of three different possible behaviors of the oscillations of the Brownian motion. Note that the last point of this theorem follows from the Khintchin law of the iterated logarithm.

Theorem 2.1. There exists an event Ω * ⊆ Ω of probability 1 such that for every ω ∈ Ω * and every non-empty open interval A of R, there are

t o (ω), t r (ω), t s (ω) ∈ A such that 1. t o (ω) is an ordinary point of B(•, ω), i.e. 0 < lim sup ρ→0 +    Osc B •, ω , I t o (ω), ρ ρ 1/2 log log(ρ -1 )    < +∞ ; 2. t r (ω) is a fast or rapid point of B(•, ω), i.e. 0 < lim sup ρ→0 +    Osc B •, ω , I t r (ω), ρ ρ 1/2 log(ρ -1 )    < +∞ ; 3. t s (ω) is a slow point of B(•, ω), i.e. 0 < lim sup ρ→0 +    Osc B •, ω , I t s (ω), ρ ρ 1/2    < +∞ .
Moreover, for every ω ∈ Ω * , almost every t ∈ R is an ordinary point of B(•, ω).

As we have already mentioned in the introduction, the aim of this article is to study the behavior of the wavelet leaders of the Brownian motion. Wavelets methods appeared to be very useful to study global and pointwise regularities of a given function. Especially, they allow to characterize some functional spaces such as the Hölder spaces [START_REF] Jaffard | Wavelet techniques in multifractal analysis, fractal geometry and applications: A jubilee of Benoit Mandelbrot[END_REF] and gives some numerical methods to study real-life signals (see among others [START_REF] Mallat | Singularity detection and processing with wavelets[END_REF][START_REF] Muzy | Multifractal formalism for fractal signals: The structure function approach versus the wavelet-transform mudulus-maxima method[END_REF][START_REF] Arneodo | The thermodynamics of fractals revisited with wavelets[END_REF][START_REF] Jaffard | Wavelet techniques in multifractal analysis, fractal geometry and applications: A jubilee of Benoit Mandelbrot[END_REF][START_REF] Jaffard | Wavelet leaders in multifractal analysis[END_REF][START_REF] Lashermes | Jaffard Comprehensive multifractal analysis of turbulent velocity using the wavelet leaders[END_REF][START_REF] Wendt | Wavelet leaders and bootstrap for multifractal analysis of images[END_REF][START_REF] Abry | When Van Gogh meets Mandelbrot: Multifractal classification of painting's texture[END_REF][START_REF] Esser | A multifractal formalism for non-concave and non-increasing spectra: The leaders profile method[END_REF][START_REF] Deliège | Mars topography investigated through the wavelet leaders method: A multidimensional study of its fractal structure[END_REF]). More precisely, let c J,K , (J, K) ∈ N × Z, denote the sequence of wavelet coefficients of a locally bounded function f . Then, under some regularity properties on the considered wavelet (see [START_REF] Meyer | Wavelets and operators[END_REF]), f belongs to the Hölder space C h (R) if and only if there exists a constant

C > 0 such that |c J,K | ≤ C2 -hJ ∀J ∈ N, K ∈ Z .
Similarly, the wavelet coefficients give a characterization of the generalized Hölder space C τ (R) (see [START_REF] Kreit | Characterizations of the elements of generalized Hölder-Zygmund spaces by means of their representation[END_REF]). Namely, a function f belongs to C τ (R) if and only if there exists a constant C > 0 such that

|c J,K | ≤ Cτ (2 -J ) ∀J ∈ N, K ∈ Z .
In order to obtain equivalent results in the context of pointwise regularity, one has to work with the wavelet leaders of the function instead of its wavelet coefficients [START_REF] Clausel | Quelques notions d'irrégularité uniforme et ponctuelle: le point de vue ondelettes[END_REF][START_REF] Jaffard | Wavelet leaders in multifractal analysis[END_REF][START_REF] Jaffard | Wavelet techniques in multifractal analysis, fractal geometry and applications: A jubilee of Benoit Mandelbrot[END_REF][START_REF] Kreit | Generalized pointwise Hölder spaces[END_REF]: if f belongs to the generalized pointwise Hölder space C τ (t), then there exists

C > 0 such that d J (t) ≤ C τ (2 -J ) ∀J ∈ N . (12)
Conversely, if f belongs to some Hölder space and if the inequality (12) holds, then

Osc f, I(t, ρ) ≤ C τ (ρ)| log τ (ρ)| , (13)
i.e. f belongs to the generalized pointwise Hölder space C τ (t), where

τ (ρ) = τ (ρ)| log τ (ρ)| .
From now on and in the rest of this paper, we assume that a wavelet whose support is included in [-N, N ] is fixed, and we denote by c J,K (resp. d J,K ), (J, K) ∈ N × Z, the wavelet coefficients (resp. the wavelet leaders) of the Brownian motion B. Let us now present some of the properties of the wavelet coefficients of the Brownian motion, arising directly from the properties of this stochastic process. We refer the reader to [START_REF] Abry | Self-similarity and long-range dependence through the wavelet lens. Theory and applications of long-range dependence[END_REF] for more details.

Remark 2.2. The equivalent definitions (3) and ( 4) of the wavelet coefficients gives that • since B is a centered Gaussian process, {c J,K : (J, K) ∈ N × Z} is a centered Gaussian process as well;

• the stationarity of the increments of B gives that for every fixed J ∈ N, the sequence {c J,K : K ∈ Z} is stationary in K;

• the self-similarity of order 1/2 of B gives that for every J ∈ N, the sequence {c J,K : K ∈ Z} has the same distribution as the sequence {2 -J/2 c 0,K : K ∈ Z}.

Thus, for every (J, K) ∈ N × Z, one has

E c 2 J,K = E c 2 0,0 2 -J and so, c J,K ∼ N (0, 2 -J E c 2 0,0 ).
As a first result concerning the behavior of the wavelet leaders of the Brownian motion, Theorem 2.1 together with the characterization (12) direclty give the following proposition. Proposition 2.3. Let Ω * be the event of probability 1 given in Theorem 2.1. For every ω ∈ Ω * and every t ∈ R,

1. if t is an ordinary point of B(•, ω), then lim sup J→+∞ d J t, ω 2 -J/2 log(J) < +∞ ; 2. if t is a rapid point B(•, ω), then lim sup J→+∞ d J t, ω 2 -J/2 √ J < +∞ ; 3. if t is a slow point of B(•, ω), then lim sup J→+∞ d J t, ω 2 -J/2 < +∞ .
However, the logarithmic correction appearing in the characterization [START_REF] Kahane | Some Random Series of Functions[END_REF] does not allow to get the exact modulus of continuity for the three kinds of behavior of wavelet leaders appearing in Proposition 2.3. A first numerical work [START_REF] Kleyntssens | A refinement of the S ν -based multifractal formalism[END_REF] has led to the idea that the behavior ρ 1/2 log log(ρ -1 ) of the ordinary points seems to be present across the scales of the wavelet coefficients and thus should be transposed in the framework of wavelet leaders. Namely, it leads to the idea that almost surely, for almost every point t, one has

0 < lim sup J→+∞ d J t, ω 2 -J/2 log(J) < +∞ .
This motivates us to consider the following definition.

Definition 2.4. Let ω ∈ Ω and t ∈ R. We say that

1. t is a leader-ordinary point of B(•, ω) if 0 < lim sup J→+∞ d J t, ω 2 -J/2 log(J) < +∞ ; 2. t is a leader-rapid point of B(•, ω) if 0 < lim sup J→+∞ d J t, ω 2 -J/2 √ J < +∞ ; 3. t is a leader-slow point of B(•, ω) if 0 < lim sup J→+∞ d J t, ω 2 -J/2 < +∞ .
The aim of the present paper is to prove the following result, which gives an equivalent of Theorem 2.1 in the context of wavelet leaders.

Proof of Theorem 2.5

This section is devoted to the proof of Theorem 2.5. It will be based on several successive lemma and propositions. Let us start with a straightforward remark.

Remark 3.1. Since the increments of B are independent and that the support of ψ is included in [-N, N ], the equality [START_REF] Daubechies | Ten lectures on wavelets[END_REF] shows that the wavelet coefficients c J 1 ,K 1 , . . . , c Jn,Kn are independent as soon as

K i -N 2 J i , K i + N 2 J i ∩ K l -N 2 J l , K l + N 2 J l = ∅ (14)
for every i = l in {1, . . . , n}. In particular, the coefficients c J,K and c J,K are independent if |K -K| ≥ 2N .

This remark leads us to define the following condition.

Definition 3.2. Let n ≥ 2 and N ∈ N. We say that the dyadic intervals λ J 1 ,K 1 , . . . , λ Jn,Kn satisfy the condition (C N ) if ( 14) is satisfied for every i = l in {1, . . . , n}.

Let us also introduce some notations. Fix (J, K) ∈ N × Z and λ = λ J,K . For any m ∈ N, we denote by S J,K,m or S λ,m the finite set of cardinality 2 m whose elements are the dyadic intervals of scale J + m included in λ J,K ; roughly speaking, "S J,K,m is the set of descendants of λ J,K at the mth generation".

The following lemma allows to obtain a general lower bound for the size of the wavelet leaders of the Brownian motion. Lemma 3.3. Let us denote by ε λ : λ ∈ Λ} an arbitrary sequence of real-valued N (0, 1) Gaussian random variables defined on a probability space Ω and let us fix N ∈ N. Let us assume that for every n ≥ 2 and every dyadic intervals λ 1 , . . . , λ n satisfying the condition (C N ), the variables ε λ 1 , . . . , ε λn are independents. Then, there exists an event Ω * 1 ⊆ Ω of probability 1 such that, for every ω ∈ Ω * 1 and every t ∈ R, one has

lim sup j→+∞      max λ ∈S λ, log 2 (N ) +2 λ∈N (λ j (t)) ε λ (ω)      > 0 . (15)
Proof. Let us fix (J, K) ∈ N × Z. For any m ∈ N and any S ∈ S J,K,m , there is a unique finite sequence (I n ) 0≤n≤m of dyadic intervals which is decreasing in the sense of the inclusion and satisfies I 0 = λ J,K , I m = S and I n ∈ S J,K,n for all n ∈ {1, . . . , m}. Next, we consider the sequence (T n ) 1≤n≤m of dyadic intervals constructed as follows: for every n ∈ {1, . . . , m}, T n is the unique dyadic interval of S J,K,n such that I n-1 = T n ∪ I n . Note that, since the sequence (I n ) 0≤n≤m is decreasing, this construction ensures that the intervals (T n ) 1≤n≤m are pairwise disjoints. Moreover, let us also note that for every n ∈ {1, . . . , m}, one has T n ∈ N I n . For every n ∈ {1, . . . , m}, there is a dyadic interval

T n ∈ S Tn, log 2 (N ) +2 such that k n -N 2 jn , k n + N 2 jn ⊆ T n
where T n = λ jn,kn . Consequently, by assumption, the corresponding Gaussian random variables (ε T n ) 1≤n≤m are independent. In the sequel, the set {T n : 1 ≤ n ≤ m} is denoted by T J,K,m (S).

Let c 0 = 2 -3/2 √ π and denote by p 0 the probability that an arbitrary real-valued N (0, 1) Gaussian random variable belongs to the interval (-c 0 , c 0 ). Elementary calculations allows to obtain that

0 < p 0 < 1 2 . ( 16
)
For all S ∈ S J,K,m , we denote by B J,K,m (S) the Bernoulli random variable defined as

B J,K,m (S) = T ∈T J,K,m (S)
1 {|ε T |<c 0 } . (17)
Notice that, using the definition of p 0 and the independence property of the random variables ε T for T ∈ T J,K,m (S), one has

E B J,K,m (S) = p m 0 . (18)
Next, let G J,K,m be the random variable with values in {0, . . . , 2 m } defined as

G J,K,m = S∈S J,K,m B J,K,m (S) .
Since the cardinality of S J,K,m equals 2 m , using [START_REF] Lévy | Processus stochastiques et mouvement brownien[END_REF], one gets that E G J,K,m = (2p 0 ) m . It follows from Fatou Lemma and ( 16) that

0 ≤ E lim inf m→+∞ G J,K,m ≤ lim m→+∞ E G J,K,m = 0
Hence, the event

Ω * 1,J,K = ω ∈ Ω : lim inf m→+∞ G J,K,m (ω) = 0 (19)
has a probability equal to 1. Since N × Z is a countable set, the probability of the event

Ω * 1 = (J,K)∈N × Z Ω * 1,J,K (20)
is also equal to 1.

Let us now consider ω ∈ Ω * 1 and t ∈ R, and let us prove that ( 15) is satisfied. We fix J ∈ N and K = 2 J t , so that λ J,K = λ J (t). Since for every m ∈ N, G J,K,m takes values in {0, . . . , 2 m }, [START_REF] Kreit | Generalized pointwise Hölder spaces[END_REF] and [START_REF] Kreit | Some characterizations of generalized Hölder spaces[END_REF] imply that there are infinitely many m such that B J,K,m (S) = 0 for every S ∈ S J,K,m , i.e. using [START_REF] Lashermes | Jaffard Comprehensive multifractal analysis of turbulent velocity using the wavelet leaders[END_REF], there exists T ∈ T J,K,m (S) such that

|ε T | ≥ c 0 .
In particular, we have this result for S = λ J+m (t). In this case, T ∈ S λ, log 2 (N ) +2 with λ ∈ N (λ J+m (t)). The conclusion follows. Proposition 3.4. There exists an event Ω * 1 of probability 1 such that for every ω ∈ Ω * 1 and every t ∈ R, one has

lim sup J→+∞ d J (t, ω) 2 -J/2 > 0 .
Proof. For every J ∈ N and every dyadic interval λ ∈ Λ J , let us set

ε λ = 1 2 -J E c 2 0,0 c λ . ( 21
)
Observe that, using the general assumption that the support of the wavelet is included in [-N, N ], Remark 2.2 and Remark 3.1 imply that the assumptions of Lemma 3.3 are fulfilled. Therefore, there exists an event Ω * 1 ⊆ Ω of probability 1 such that, for every ω ∈ Ω * 1 and every t ∈ R, one has

lim sup J→+∞      max λ ∈S λ, log 2 (N ) +2 λ∈N (λ J (t)) ε λ (ω)      > 0 . ( 22
)
Note that using the definition ( 6) of the wavelet leaders and ( 21), one has

d J (t, ω) ≥ max λ ∈S λ, log 2 (N ) +2 λ∈N (λ J (t)) c λ (ω) = 2 -(J+ log 2 (N ) +2) E c 2 0,0 max λ ∈S λ, log 2 (N ) +2 λ∈N (λ J (t)) ε λ (ω) ,
and together with [START_REF] Mallat | A Wavelet Tour of Signal Processing[END_REF], it implies that

lim sup J→+∞ d J (t, ω) 2 -J/2 > 0 .
This gives the conclusion.

As we will see in the proof of Theorem 2.5, this result will allow to get the existence of leader-slow points. Let us now focus on leader-ordinary points. First, let us recall the following classical lemma which provides asymptotic estimates on the tail behavior of a standard Gaussian distribution. Lemma 3.5. Let ε be an arbitrary real-valued N (0, 1) Gaussian random variable. One has

lim x→+∞ P |ε| > x (2π -1 ) 1/2 x -1 e -x 2 /2 = 1 .
Lemma 3.6. Let us denote by ε λ : λ ∈ Λ} an arbitrary sequence of real-valued N (0, 1) Gaussian random variables defined on a probability space Ω and let us fix N ∈ N. Let us assume that for every n ≥ 2 and every dyadic intervals λ 1 , . . . , λ n satisfying the condition (C N ), the variables ε λ 1 , . . . , ε λn are independents. Then, there exists an event Ω * 2 ⊆ Ω of probability 1 such that, for every ω ∈ Ω * 2 and almost every t ∈ R, one has

lim sup j→+∞      1 log(j) max λ ∈S λ, log 2 (N ) +2 λ∈N (λ j (t)) |ε λ |      > 0 .
Proof. Within this proof, we will use the same notations as in the proof of Lemma 3.3. Let us fix t ∈ R. If J ∈ N, we set K = 2 J t . For every m ∈ N, we consider the dyadic interval S = λ J+m (t) ∈ S J,K,m and the associated sequence (T n ) 1≤n≤m of dyadic intervals. Next, we set

E J,m (t) := ω ∈ Ω : max 1≤n≤m |ε T n | ≥ log(2m)
By construction, the Gaussian random variables ε T n , 1 ≤ n ≤ m, are independent. Therefore, one has

P E J,m (t) = 1 - 1≤n≤m P |ε T n | < log(2m) = 1 -1 -P |ε| > log(2m) m
where ε ∼ N (0, 1). Let us set C = 1/2 (2π -1 ) 1/2 > 0. Using Lemma 3.5 and the fact that log(1 -x) ≤ -x if x ∈ (0, 1), there exists M ∈ N such that for any m > M , we have

P E J,m (t) ≥ 1 -1 -C e -1 2 log(2m) log(2m) m ≥ 1 -exp -Cm e -1 2 log(2m) log(2m) ≥ 1 -exp -C m 2 log(2m) ≥ 1 -exp(-m γ )
for γ ∈ (0, 1/2). Consequently, one has in particular

M ∈N P E 2 M ,2 M (t) = +∞ .
In view of the fact that the events E 2 M ,2 M (t), M ∈ N, are independents, it follows from the Borel-Cantelli lemma that

P   M ∈N m≥M E 2 m ,2 m (t)   = 1 .
Therefore, for a fixed t ∈ R, almost surely, there are infinitely many scales j ∈ N such that

max λ ∈S λ, log 2 (N ) +2 λ∈N (λ j (t)) |ε λ | ≥ log j .
Fubini's theorem applied to the function (t, ω) ∈ R ×Ω → χ S(t) (ω), where

S(t) =      ω ∈ Ω : lim sup j→+∞      1 √ log j max λ ∈S λ, log 2 (N ) +2 λ∈N (λ j (t)) |ε λ |      < 1     
, implies then that there is an event Ω * 2 ⊆ Ω of probability 1 on which for almost every t ∈ R,

lim sup j→+∞      1 √ log j max λ ∈S λ, log 2 (N ) +2 λ∈N (λ j (t)) |ε λ |      > 0 .
Proposition 3.7. There exists an event Ω * 2 of probability 1 such that for every ω ∈ Ω * 2 and almost every t ∈ R, one has

lim sup J→+∞ d J (t, ω) 2 -J/2 log(J) > 0 .
Proof. We proceed as in the proof of Proposition 3.4. Using Lemma 3.6, the general assumption that the support of the wavelet is included in [-N, N ], Remark 2.2 and Remark 3.1, we know that there exists an event Ω * 2 ⊆ Ω of probability 1 such that, for every ω ∈ Ω * 2 and almost every t ∈ R, one has

lim sup J→+∞      1 log(J) max λ ∈S λ, log 2 (N ) +2 λ∈N (λ J (t)) |ε λ (ω)|      > 0 , where ε λ = 1 2 -J E c 2 0,0 c λ .
In particular,

max λ ∈S λ, log 2 (N ) +2 λ∈N (λ J (t)) c λ (ω) = 2 -(J+ log 2 (N ) +2) E c 2 0,0 max λ ∈S λ, log 2 (N ) +2 λ∈N (λ J (t)) ε λ (ω) . Consequently, if ω ∈ Ω * 2 , for almost every t ∈ R, one has lim sup J→+∞ d J (t, ω) 2 -J/2 log(J) ≥ lim sup J→+∞ 1 2 -J/2 log(J) max λ ∈S λ, log 2 (N ) +2 λ∈N (λ J (t)) c λ (ω) > 0 ,
and the conclusion follows.

Let us end with a result which will be useful for rapid points. Proof. To avoid making the notations heavier, we suppose that A = (0, 1). The proof can be easily adapted in the general case. The conclusion follows then by covering R with all open intervals with rational endpoints. Let us fix a ∈ (0, 1) and C > 0 such that C 2 < 2a log 2. Let us also consider for every (j, l) ∈ N ×{0, . . . , 2 j(1-a) -1}, the event F j,l := ω ∈ Ω : max k∈{l 2 aj /(2N ) ,...,(l+1)

2 aj /(2N ) -1} |ε j,2kN (ω)| ≥ C j .
Besides, let j 0 be the smallest j such that 2 aj /(2N ) ≥ 1.

Assume for a while that

P     l∈{0,..., 2 j(1-a) -1} F j,l     (23)
is the general term of a convergent series; then the Borel-Cantelli lemma implies that

P   J≥j 0 j≥J l∈{0,..., 2 j(1-a) -1} F j,l   = 1.

Now, let us set

Ω * 3 := J≥j 0 j≥J l∈{0,..., 2 j(1-a) -1} F j,l (24)
and let us consider ω ∈ Ω * 3 . For every j ≥ j 0 , let us set

G j (ω) := k ∈ {0, . . . , 2 j -1} : |ε j,k (ω)| ≥ C j (25)
and

U j (ω) := k∈G j (ω) k 2 j , k + 1 2 j . (26)
Finally, for every n ≥ j 0 , one considers

O n (ω) := j≥n U j (ω).
This last open subset is dense in (0, 1). Indeed, let us consider t ∈ (0, 1), j ≥ j 0 and k such that λ j (t) = λ j,k . Then, either there is l ∈ {0, . . . , 2 j(1-a) -1} such that k ∈ l 2 ja , . . . , (l + 1) 2 ja -1 , or k ∈ { 2 j(1-a) 2 ja , . . . , 2 j -1}.

In the first case, using ( 24) and ( 25), there is k ∈ {l 2 aj /(2N ) , . . . , (l + 1) 2 aj /(2N ) -1} such that 2k N ∈ G j (ω). From ( 26), we get that t is at a distance at most 2•2 j(a-1) of U j (ω).

In the second case, there is k ∈ {( 2 j(1-a) -1) 2 aj /(2N ) , . . . , 2 j(1-a) 2 aj /(2N ) -1} such that 2k N ∈ G j (ω), and similarly, we get that t is at a distance at most c • 2 j(a-1) of U j (ω), for some constant c > 0 depending only on N and a. The the density follows.

Hence, Baire's theorem gives that the set

n≥j 0 O n (ω)
is not empty. If t ∈ n≥j 0 O n (ω), for every n ≥ j 0 , there is j ≥ n such that |ε λ j (t) | ≥ C √ j, and it leads to the conclusion.

It remains then to show that [START_REF] Mallat | Singularity detection and processing with wavelets[END_REF] is the general term of a convergent series. Let us remark that the variables ε j,2kN for k ∈ {l 2 aj /(2N ) , . . . , (l + 1) 2 aj /2N -1} and l ∈ {0, . . . , 2 j(1-a) -1} are independent. Consequently, one has

P     l∈{0,..., 2 j(1-a) -1} F j,l     = 1 - l∈{0,...,[2 j(1-a) ]-1}   1 - k∈{l 2 aj /(2N ) ,...,(l+1) 2 aj /2N -1} P |ε j,2kN | < C j   = 1 -1 -P |ε| < C j 2 aj /(2N ) 2 j(1-a) = 1 -1 -1 -P |ε| ≥ C j 2 aj /(2N ) 2 j(1-a) ≤ 1 -exp 2 j(1-a) log(1 -x j ) (27)
where ε ∼ N (0, 1) and

x j = 1 -P |ε| ≥ C j 2 aj /(2N )
.

Let us remark that x j is always positive and tends to 0 as j → +∞. Indeed, let us set C = (1/2)(2π -1 ) 1/2 . Using Lemma 3.5 and the fact that log(1 -x) ≤ -x if x ∈ (0, 1), there exists J ∈ N such that for any j ≥ J,

0 ≤ x j ≤ (1 -C j exp(-C 2 j/2)) 2 aj /(2N ) ≤ exp - 2 aj 2N C j exp(-C 2 j/2) ≤ exp -C j exp(j(a log 2 -C 2 /2) , (28)
where C is a strictly positive constant depending only on a, N and C. The expression (28) tends to 0 since C 2 < 2a log 2. Moreover, the same argument shows that 2 j(1-a) x j tends to 0. Using the fact that log(1 -x) = -x + o(x) and exp(x) = 1 + x + o(x) as x → 0, we obtain that, for any > 0, the expression ( 27) is upper bounded by

2 j(1-a) ( (x j + x j ) + x j + x j ) (29)
for j large enough. Therefore the expression ( 27) is the general term of a convergent series using the inequality (29) and the inequality (28).

Proposition 3.9. There exists an event Ω * 3 of probability 1 such that for every ω ∈ Ω *

and every non-empty open interval

A of R, there is t(ω) ∈ A such that lim sup J→+∞ d J t(ω), ω 2 -J/2 √ J > 0 .
Proof. We proceed as in the proof of Propositions 3.4 and 3.7, using Lemma 3.8, the assumption that the support of the wavelet is included in [-N, N ], Remark 2.2 and Remark 3.1.

We are now able to prove Theorem 2.5.

Proof of Theorem 2.5. Let us recall (see [START_REF] Jaffard | Wavelet techniques in multifractal analysis, fractal geometry and applications: A jubilee of Benoit Mandelbrot[END_REF]) that there exists an event Ω * 4 ⊆ Ω of probability 1 such that for every ω ∈ Ω * 4 , lim sup

ρ→0 +    Osc B •, ω , I t, ρ ρ 1/2 log(ρ -1 )    < +∞
for every t ∈ R. In view of the inequality [START_REF] Esser | A multifractal formalism for non-concave and non-increasing spectra: The leaders profile method[END_REF],

lim sup J→+∞ d J (t, ω) 2 -J/2 √ J < +∞ (30)
holds for every t ∈ R.

Let us consider the event Let us first show that almost every t ∈ R is a leader-ordinary point of B(•, ω). Using Theorem 2.1, we know that almost every t ∈ R is an ordinary point of B(•, ω). Together with Proposition 2.3 and Proposition 3.7, this implies that for almost every t ∈ R, 0 < lim sup J→+∞ d J (t, ω) 2 -J/2 log(J) < +∞ .

Ω * 0 := Ω * ∩ Ω * 1 ∩ Ω * 2 ∩ Ω * 3 ∩ Ω *
In particular, there exist leader-ordinary points of B(•, ω) in A.

Secondly, Proposition 3.9 shows that there exists t r (ω) ∈ A such that lim sup

J→+∞ d J t r (ω), ω 2 -J/2 √ J > 0.
This result combined with the equation (30) implies that the point t r (ω) is a leader-rapid point of B(•, ω). Finally, Theorem 2.1 and Proposition 2.3 show that there exists t s (ω) ∈ I such that lim sup J→+∞ d J t s (ω), ω 2 -J/2 < +∞ .

Using Proposition 3.4, we see that the point t s (ω) is a leader-slow point of B(•, ω).

4 Some links between the behaviors of the oscillation and the wavelet leaders Theorem 2.5 shows that there are (at least) three different behaviors for the size of the wavelet leaders of the Brownian motion and furthermore, that they correspond to those of the oscillations. A natural question is to determine, for t ∈ R fixed, whether the behavior of Osc B, I t, ρ as ρ → 0 is the same as d J (t) as J → +∞, and conversely. This section gives a partial answer to this question. First, let us note that the inequality (7) direclty leads to the following proposition, which is an equivalent of Proposition 2.3. We can now give some links between the behaviors of the oscillations and the size of the wavelet leaders. 1. For every t ∈ R,

• if t is a leader-rapid point of B(•, ω), then t is a rapid point of B(•, ω),

• if t is a slow point of B(•, ω), then t is a leader-slow point of B(•, ω).

2.

For almost every t ∈ R, t is an ordinary point and a leader-ordinary point of B(•, ω).

Proof. If t is a leader-rapid point, Proposition 4.1 and the equation [START_REF] Jaffard | Wavelet techniques in multifractal analysis, fractal geometry and applications: A jubilee of Benoit Mandelbrot[END_REF] directly imply that t is a rapid point. If t is a slow point, Proposition 2.3 and Proposition 3.4 give that t is a leader-slow point. Finally, the result for the ordinary points can be deduced from the last parts of Theorem 2.1 and 2.5.

Theorem 2 . 5 .

 25 There exists an event Ω * 0 ⊆ Ω of probability 1 such that for every ω ∈ Ω * 0 and every non-empty open interval A of R, there are t o (ω), t r (ω), t s (ω) ∈ A such that t o (ω) is a leader-ordinary point, t r (ω) is a leader-rapid point and t s (ω) is a leader-slow point of B(•, ω). Moreover, for every ω ∈ Ω * 0 , almost every t ∈ R is a leader-ordinary point of B(•, ω).

Lemma 3 . 8 .

 38 Let us denote by ε λ : λ ∈ Λ} an arbitrary sequence of real-valued N (0, 1) Gaussian random variables defined on a probability space Ω and let us fix N ∈ N. Let us assume that for every n ≥ 2 and every dyadic intervals λ 1 , . . . , λ n satisfying the condition (C N ), the variables ε λ 1 , . . . , ε λn are independents. Then, there exists an event Ω * 3 ⊆ Ω of probability 1 such that, for every ω ∈ Ω * 3 and every non-empty open interval A of R, there is t ∈ A such that lim sup j→+∞ |ε λ j (t) | √ j > 0 .

4 of probability 1 ,

 41 where the event Ω * (resp. Ω * 1 , Ω * 2 and Ω * 3 ) is the event of Theorem 2.1 (resp. Proposition 3.4, Proposition 3.7 and Proposition 3.9). Let us fix ω ∈ Ω * 0 and let us consider a non-empty open interval A of R.

Proposition 4 . 1 . 3 .

 413 Let Ω * 0 be the event of probability 1 given in Theorem 2.5. For every ω ∈ Ω * and every t ∈ R,1. if t is a leader-ordinary point of B(•, ω), then lim sup ρ→0 +    Osc B •, ω , I t, ρ ρ 1/2 log log(ρ -1 ) t is a leader-rapid point of B(•, ω), if t is a leader-slow point of B(•, ω), then

Theorem 4 . 2 .

 42 Let Ω * 0 be the event of probability 1 given in Theorem 2.5 and let us fix ω ∈ Ω * 0 .

Acknowledgements

The second author is supported by the the Labex CEMPI (ANR-11-LABX-0007-01).


Francesco Rizzotto
email: francesco.rizzotto@inrae.frwww

Marco Marin

Christine Péchoux

Sandrine Auger

Jasmina Vidic

Colorimetric aptasensor for the detection of Bacillus cytotoxicus spores in milk and ready-to-use food

MicrobAdapt

Micalis Institute

The high incidence of foodborne diseases caused by pathogenic bacteria raises concerns worldwide and imposes considerable public healthcare challenges. This is especially observed with dormant spores of Bacilli, which can often survive treatments used by the food industry to kill growing bacteria. We present a point-of-need colorimetric assay for detection of Bacillus cytotoxicus spores in food. The sensing platform consists of a microtube containing gold nanoparticles (AuNPs), and magnetic particles (MPs), both conjugated with specific aptamer BAS6R that recognize B. cytotoxicus spores. Upon the addition of the sample, spores were determined as present by the enhanced color change of the solution, due to the oxidation of tetramethylbenidine (TMB) with H 2 O 2 . BAS6R@AuNPs aptasensor coupled to BAS6R@MPs proved to be highly sensitive, achieving the naked-eye limit of detection as low as 10 2 cfu/mL in water and milk, and 10 4 cfu/mL in mashed potatoes. Moreover, discrimination between spores of B. cytotoxicus and B. subtilis as well as bacterial vegetative cells was achieved in contaminated food samples, providing a good selectivity. This work provides a promising proof of concept for the development of instrument-free, low-cost and rapid assay for B. cytotoxicus spore detection, which is able to compete in sensitivity with conventional costly and time-consuming laboratory analyses.

ABSTRACT RESULTS

A) UV-Vis absorption spectra of TMB solution containing 10 8 cfu/ml of B. cytotoxicus spores, BAS6R@AuNPs (4 x 10 10 NPs) or their mixture and photographs to illustrate the color change. B) UV-Vis absorption spectra of TMB solution containing Campy@AuNPs (4x10 10 NPs) and Campy@AuNPs ( 4

CONCLUSION

  x 10 10 NPs) with B. cyctotoxicus spores (10 8 cfu/ml). Photographs illustrate no color changes Detection results for B. cytotoxicus spores in three different matrices using BAS6R@AuNPs aptasensor Quantitative evaluation was performed by measuring absorption at 672 nm. Indivudual data points (n=3 indipendent experiments) are shown; error bars represent means ± SD Schematic illustration of B. cytotoxicus spores forming into a sandwich between magnetic particles functionalized with BAS-6R (BAS6R@MN) and BAS6R@AuNPs Detection results for B. cytotoxicus spores in three different matrices using BAS6R@AuNPs aptasensor coupled to magnetic concentration. Quantitative evaluation was performed by measuring absorption at 672 nm. Individual data points (n = 3 independent experiments) are shown, error bars represent means ± SD Specificity study against bacterial vegetative cells and spores in assays performed without (a) and with BAS6R@MN (b). Concentration of B. cytotoxicus spores was 10 7 cfu/mI whereas the other concentrations were 10 8 cfu/mL in tests without spore concentration (a) while concentrations of all strains were 10 4 fu/ mI in tests with concentration (b). Individual data points (n = 3 independent experiments) are shown; error bars represent means ± SD. Inserts are the corresponding images of TMB oxidation: 1, blank; 2, E . coli; 3, S. aureus; 4, B. subtilis cells; 5, B. cytotoxicus cells; 6, B. subtilis spores; and 7, B. cytotoxicus spores.

ACKNOWLEDGEMENTS

We constructed an affordable strategy for the simple and rapid detection of B. cytotoxicus spores in food matrices based on the use of enhanced peroxidase-like activity of AuNPs. Through this rapid colorimetric test, spores of B. cytotoxicus were quantified in lowcost polypropylene microtubes, to a concentration of 10 2 cfu/mL. The detection strategy was demonstrated by absorbance response and color change of the substrate which was catalyzed by AuNPs concentrated on the spores via the specific aptamer. The colorimetric biosensor in this study provides efficient screening method for spores due to lower detection limit, wide linear range and minimal sample processing. It may be used for preliminary monitoring of food products that represent high risk, while confirmation tests and strain identification can be performed only on positive food items. This work provides a promising proof of concept for the development of label-free, instrument-free, low-cost and rapid assay for spore detection which is able to compete in sensitivity with conventional costly and time-consuming laboratory analyses We acknowledge financial support from the French National Agency for Research (ANR-21-CE21-0009 SIENA project) and the University Paris-Saclay (Poc in labs N° 2022-2626 "SportAlert"). We thank Jennifer Upfold (INRAE, France) and Microbadapt team (INRAE, France) for discussion and critical reading