L Halaga Čka

M Vanwolleghem

F Vaurette

J Ben Youssef

K Postava

J Pištora

And B Dagens

Magnetoplasmonic nanograting enables optical nonreciprocity sign reversal

We experimentally demonstrate a disruptive approach to control magnetooptical nonreciprocal effects. It has been known that the combination of a magneto-optically (MO) active substrate and extraordinary transmission (EOT) effects through deep-subwavelength nanoslits of a noble metal grating, leads to giant enhancements of the magnitude of the MO effects that would normally be obtained on just the bare substrate. We show here that even more than just an enhancement, the MO effects can also undergo a sign reversal by achieving a hybridization of the different types of resonances at play in these EOT nanogratings. By tuning the geometrical profile of the grating's slits, one can engineer -for a fixed wavelength and fixed magnetization -the transverse MO Kerr effect (TMOKE) reflectivity of such a magnetoplasmonic system to be enhanced, extinguished or inversely enhanced. We have fabricated gold gratings with varying nanoslit widths on a Bi-substituted gadolinium iron garnet and experimentally confirmed such a behavior using a customized magneto-optic Mueller matrix ellipsometer. This demonstration allows new design paradigms for integrated nonreciprocal circuits and biochemical sensors with increased sensitivity and reduced footprint.

Introduction

Magnetoplasmonics [START_REF] Armelles | Magnetoplasmonics: Combining Magnetic and Plasmonic Functionalities[END_REF] -a recently coined term referring to any physical system exploiting the combined effect of a magnetic and a plasmonic functionality -is attracting a surge of interest in recent years. This research domain knows two main driving forces. On the one hand, the use of a magnetic field to add an active control, such as switching and modulation, to plasmonic-based nanophotonic circuitry [START_REF] Stockman | Nanoplasmonics: past, present, and glimpse into future[END_REF] has been proven a compact, reliable and fast alternative [START_REF] Temnov | Active magneto-plasmonics in hybrid metal-ferromagnet structures[END_REF][START_REF] Belotelov | Plasmon-mediated magneto-optical transparency[END_REF][START_REF] Temnov | Ultrafast acousto-magneto-plasmonics[END_REF][START_REF] Wurtz | Controlling optical transmission through magneto-plasmonic crystals with an external magnetic field[END_REF] to more complex control agents such as nonlinearities [START_REF] Macdonald | Ultrafast active plasmonics[END_REF][START_REF] Macdonald | Active plasmonics: Current status[END_REF]. Secondly and more importantly, plasmonic field concentration has been exploited to enhance the unique time reversal breaking properties of magneto-optic (MO) materials [START_REF] Jalas | What is -and what is not -an optical isolator[END_REF]. In recent years numerous experimental reports have demonstrated the strong enhancement of all traditional nonreciprocal MO phenomena -complex Kerr and Faraday rotation and complex transverse Kerr phase shift -due to coupling with propagating surface plasmon polaritons or excitation of localised plasmon resonances. A wide range of magnetoplasmonic configurations has been investigated: semi-transparent dielectric magnetic iron garnet films coupled to noble metal nanogratings [START_REF] Wurtz | Controlling optical transmission through magneto-plasmonic crystals with an external magnetic field[END_REF][START_REF] Grunin | Surface-plasmon-induced enhancement of magneto-optical kerr effect in all-nickel subwavelength nanogratings[END_REF][START_REF] Belotelov | Enhanced magneto-optical effects in magnetoplasmonic crystals[END_REF][START_REF] Chin | Nonreciprocal plasmonics enables giant enhancement of thin-film Faraday rotation[END_REF][START_REF] Halagacka | Experimental demonstration of anomalous nonreciprocal optical response of 1D periodic magnetoplasmonic nanostructures[END_REF][START_REF] Floess | Plasmonic analog of electromagnetically induced absorption leads to giant thin film faraday rotation of 14[END_REF], nanogratings of pure ferromagnetic (FM) metal thin films [START_REF] Ctistis | Optical and magnetic properties of hexagonal arrays of subwavelength holes in optically thin cobalt films[END_REF][START_REF] Papaioannou | Magneto-optic enhancement and magnetic properties in Fe antidot films with hexagonal symmetry[END_REF], FM nanoparticles [START_REF] Bonanni | Designer magnetoplasmonics with nickel nanoferromagnets[END_REF][START_REF] Maccaferri | Tuning the Magneto-Optical Response of Nanosize Ferromagnetic Ni Disks Using the Phase of Localized Plasmons[END_REF] and nanowires [START_REF] González-Díaz | Enhanced Magneto-Optics and Size Effects in Ferromagnetic Nanowire Arrays[END_REF] on dielectric substrates, hybrid ferromagnetic/noble metal sandwiches both in nanoparticle or nanowires [START_REF] González-Díaz | Plasmonic Au/Co/Au nanosandwiches with enhanced magneto-optical activity[END_REF][START_REF] Armelles | Magnetoplasmonic nanostructures: systems supporting both plasmonic and magnetic properties[END_REF][START_REF] Regatos | Suitable combination of noble/ferromagnetic metal multilayers for enhanced magneto-plasmonic biosensing[END_REF][START_REF] Armelles | Magneto-optical properties of core-shell magneto-plasmonic au-co x fe3-x o4 nanowires[END_REF], systems of coupled plasmonic and ferromagnetic nanoparticles [START_REF] Zubritskaya | Magnetic control of the chiroptical plasmonic surfaces[END_REF], inverted Babinet nanohole layouts [START_REF] Armelles | Magnetic Field Modification of Optical Magnetic Dipoles[END_REF][START_REF] Barsukova | Magneto-optical response enhanced by mie resonances in nanoantennas[END_REF] and propagating SPP layout [START_REF] Safarov | Magneto-optical Effects Enhanced by Surface Plasmons in Metallic Multilayer Films[END_REF][START_REF] Kaihara | Enhancement of magneto-optical Kerr effect by surface plasmons in trilayer structure consisting of double-layer dielectrics and ferromagnetic metal[END_REF], and magnetic/plasmonic core-shell nanoparticles [START_REF] Levin | Magnetic-plasmonic core-shell nanoparticles[END_REF][START_REF] Wang | Plasmonics and enhanced magneto-optics in core-shell co-ag nanoparticles[END_REF][START_REF] Meneses-Rodríguez | Probing the electromagnetic field distribution within a metallic nanodisk[END_REF].

All these demonstrations share the same conceptual approach to boost the magneto-optic properties of the system: by operating closely to a high quality factor plasmonic resonance the impact of an otherwise weak MO phenomenon is sharply enhanced. A particularly convincing example of plasmonic enhanced magneto-optics is the demonstration of giant transverse MO Kerr reflectivity (TMOKE) when applying a one-dimensional plasmonic slit grating on a transparent magneto-optic iron garnet substrate (which can be possibly used as an optical waveguide) that is magnetized in-plane and parallel to the grating's slits [START_REF] Belotelov | Extraordinary transmission and giant magneto-optical transverse Kerr effect in plasmonic nanostructured films[END_REF][START_REF] Kreilkamp | Waveguide-plasmon polaritons enhance transverse magneto-optical kerr effect[END_REF]. By themselves, lossless MO materials (such as iron garnets in the near infrared region) can fundamentally not exhibit nonreciprocal TMOKE intensity effects [START_REF] Višňovský | Optics in magnetic multilayers and nanostructures[END_REF]Eq. (1.138)]. However by coupling a transparent MO garnet to the Wood anomalies of a gold extraordinary transmission grating, nonreciprocal intensity reflectivities are obtained that even exceed those of strongest lossy FM metals.

In many optimized layouts for nonreciprocal circuits (such as isolators and circulators) it is inevitable to have antisymmetric magnetized regions in order to break also the spatial inversion symmetries [START_REF] Takeda | Compact optical one-way waveguide isolators for photonic-band-gap microchips[END_REF][START_REF] Khanikaev | One-Way Extraordinary Optical Transmission and Nonreciprocal Spoof Plasmons[END_REF], to optimize the MO interaction with the symmetries of the modes at play [START_REF] Wang | Optical circulators in two-dimensional magneto-optical photonic crystals[END_REF][START_REF] Kono | Nonreciprocal microresonators for the miniaturization of optical waveguide isolators[END_REF] or to achieve a more powerful push-pull configuration for the nonreciprocal effect [START_REF] Shoji | Magneto-optical isolator with silicon waveguides fabricated by direct bonding[END_REF][START_REF] Dulal | Optimized magneto-optical isolator designs inspired by seedlayer-free terbium iron garnets with opposite chirality[END_REF]. Gaining control over the sign of the nonreciprocity by plasmonic engineering would allow realizing such layouts without the need for complicated inverted magnetic domains and just having a uniform magnetization in the whole structure.

Differently and adding to these demonstrations of magnetoplasmonic nonreciprocity enhancement, we report here on the anomalous control of the sign of the nonreciprocal phenomena when exploiting the different resonances in a magnetoplasmonic system [START_REF] Halagačka | Coupled mode enhanced giant magnetoplasmonics transverse Kerr effect[END_REF]. Apart from the obvious interest of investigating controlled tuning of resonances in magnetoplasmonic systems and their impact on the MO enhancement, there is another important motivation for controlling the sign of the nonreciprocity at play. Time reversal breaking (or thus nonreciprocity) is governed by the sense of the magnetization in the system. Up to first order, inverting the magnetization will invert the sign of the nonreciprocal phenomenon at play. Naturally, in design of the magnetoplasmonic gratings and arrays, the optical reciprocity can be tuned by controlling of the wavevector/angle of incidence [START_REF] Belotelov | Enhanced magneto-optical effects in magnetoplasmonic crystals[END_REF]. We demonstrate additional degree of freedom in controlling of the magnetooptical nonreciprocity based on tuning of the coupling between resonances without changes of the electromagnetic wave properties.

Experimental demonstration of sign control and enhancement of magnetoplasmonic effect

Magnetoplasmonic grating design, fabrication, and characterization

To investigate magnetoplasmonic sign control and enhancement, we have designed and fabricated one-dimensional (1D) lamellar gold nanogratings on the 4 µm thick, compositionnally optimized, Bi-substituted gadolinium iron garnet (Bi:GIG), Gd The first set of samples was fabricated by varying the e-beam exposure dose per written patch: the width r of the nanoslit is varied from patch to patch (see Fig. 1b). In this way we expect all developed patches have the same period and thickness of deposited gold. Therefore, the effect of the nanoslit width on the magnetoplasmonic properties of the grating is directly comparable. To complete our study in investigation of the sign control, two additional sets of patches with gold thickness of 118 nm and 134 nm were fabricated. The new fabricated sets of patches were analyzed and two patches with similar nanoslit widths were selected. Therefore, MO response of the patches which are different only in the thickness can be directly compared.

The grating's optical response have been probed by using a spectroscopic Mueller matrix ellipsometer Wooollam RC2-DI (J.A. Woollam Co.) operating in spectral range from 0.74 eV to 6.42 eV. To measure magnetooptical response of the samples, the ellipsometer was extended with in-house design computer controlled in-plane permanent magnet circuit. See Ref. [START_REF] Halagačka | Mueller matrix optical and magneto-optical characterization of bi-substituted gadolinium iron garnet for application in magnetoplasmonic structures[END_REF] for more details about parametrized model of the studied plasmonic gratings and used optical optical and magnetooptical functions of materials. Clearly at a given photon energy, for example 1.68 eV, the TMOKE response can be positive (nanoslit width r=120 nm) or negative (r=70 nm). In other words, for a given optical signal and a given magnetization the reciprocity sign is inverted by changing only the slit width.

M = ± |M | x), δR = R (+M) -R (-M).

Measured magnetooptical response of plasmonic nanogratings with different nanoslit width

Measured magnetooptical response of plasmonic nanogratings with different thickness of the gold gratings and comparable slit width

For complete analysis of the FP and SPP modes coupling here we also demonstrate tuning of TMOKE via plasmonic grating thickness. According to Eq. ( 2) the dispersion of the FP modes is strongly affected by the length of the resonant cavity, which is in our case presented by thickness of the gold grating h. For direct demonstration of the induced coupling, two sets of the grating samples with different thicknesses were fabricated by the same procedure of electron beam dose variation as was described above. Samples were measured and characterized with the Mueller matrix ellipsometry. By the detailed analysis of all fabricated patches we have determined thickness and nanoslits width of all samples. The gold grating thickness of new set of samples was determined to be h = 118 nm and h = 142 nm, respectively. For direct demonstration of impact of SPP and FP modes coupling on MO response, the two patches with comparable nanoslit widths r = 148 nm and r = 137 nm were selected. Figure 3 shows detail on MO response δR observed for both mentioned samples at position of the 2 nd SPP peak. In our case it is clearly visible that measured MO response (blue and red circles) at 1.7 eV (black line in Fig. 3) is positive for the patch with thickness h = 118 nm and negative for the patch with grating thickness h = 142 nm. The MO sign inverting phenomena by the grating thickness is observed for the 2 nd SPP peak, the same SPP peak as used for the switching demonstration by the grating nanoslit width. In addition in Fig. 3 calculated MO response is shown with red and blue solid lines. The MO response was calculated from model used for determination of the geometry of individual patches. Very good agreement between measured and calculated data proves validity of the used model. In order to understand the origin of this property and to be able to control it, we have analyzed more deeply the reflectivity spectrum of such samples, firstly without magnetization. In the next step, the influence of the magnetooptical activity is introduced in model of the SPPs EOT resonances.

Operational principle of resonant modes in magnetoplasmonic grating

Optical and magnetooptical activity of samples were studied in planar diffraction configuration.

With the plane of incidence perpendicular to the grating direction, the ellipsometric specular intensity ratio R p R s obtained in such measurements will contain the signatures of the surface plasmon polariton (SPP) anomalies of the EOT nanoslit grating. In particular, due to the strong subwavelength nature of the slits the structure has a smooth and overall strong spectral s-reflectivity R s , while the p-reflectivity R p exhibits sharp peaks at the spectral position of the diffractively folded surface plasmon polaritons,

k SPP (E ph ) = E ph hc Au Bi:GIG Au + Bi:GIG + m 2π
Λ (m = ±1, ±2, . . . ). The SPP EOT transmission maxima occur slightly blueshifted with respect to these peaks. [START_REF] Aguanno | Transmission resonances in plasmonic metallic gratings[END_REF][START_REF] García De Abajo | Colloquium : Light scattering by particle and hole arrays[END_REF]. Due to the overall smooth R s , the sharp Fano-like minima in R p R s therefore correspond to the detection of the Wood plasmon transmission anomalies of the gold grating.

The measured ellipsometric spectrum for an incidence angle of ϕ = 20°on a 100 nm thick gold grating formed by a periodic arrangement of strong sub-λ slits (r = 63 nm and Λ = 500 nm) show systematic shift of the reflectivity R dip from 1.68 eV to 1.7 eV for the nanoslit width change from 63 nm to 130 nm (Fig. 4b). As the inset of the Fig. 4a shows, the observed Fano resonances indeed appear close to the crossings of the 20°-light line (solid black) with the theoretically calculated dispersion of the folded SPPs on a flat Au/garnet interface (solid blue). In addition, experimental data were measure for configuration in witch SPPs modes are separates in spectrum. Therefore, the interaction between different SPPs in minimal and their interaction with FP resonant modes can be studied directly.

Interestingly, when zooming in on the measured reflectivity spectra close the +2 nd order garnet/Au SPP, a pronounced redshift of the extraordinary reflection resonance is observed for gratings with decreasing width of nanoslit (Fig. 4b). The Rigorous Coupled Wave Algorithm (RCWA) was used for optical modeling of the plasmonic grating. The widths of nanoslits have been obtained by a single global least-squares fit of model to measured ellipsometric data. The measured spectra for 15 patches with different nanoslit have been processed in a single step on the same garnet substrate. The fitting procedure accounted for the presence of a top roughness layer on the fabricated gold gratings and for depolarization effects due to an angular spread of the incident Gaussian beam [START_REF] Halagačka | Analysis and modeling of depolarization effects in mueller matrix spectroscopic ellipsometry data[END_REF]. It must be noted that the fitted nanoslit widths r follow the trend of the values estimated by SEM observation r SEM (as shown on Fig. 1b).

Optical observation of the effect of coupling between SPP and FP resonances

In the following we will explain the phenomena of tuning the plasmon resonances and MO sign conversion by a simple model of coupling SPP and Fabry-Pérot (FP) modes. In a first approximation, this resonance shift as a function of nanoslit width r cannot be explained by the diffractive folding of the Au/garnet SPP dispersion, as this latter is only parametrically dependent on the grating's period Λ, the incidence angle ϕ, and the Au and Bi:GIG permittivity, ε Au and ε Bi:GIG . In previous theoretical work we demonstrated how variations of the grating's geometry (in particular its thickness h) causes an anticrossing interaction of the diffractively coupled SPP resonances and the FP slit resonances undergone by the guided TM mode in the subwavelength metal/insulator/metal slit. [START_REF] Dionne | Plasmon slot waveguides: Towards chip-scale propagation with subwavelength-scale localization[END_REF] Increasing thickness leads to an increasing geometric phase of the fundamental TM slit mode, ω c n eff,TM 0 h, thereby redshifting the FP EOT resonances. Due to anticrossing the SPP reflection anomalies will therefore eventually also redshift with the grating thickness h.

In a first approximation the spectral position of the grating's resonances as a function of its geometrical parameters (h, r, andΛ), can be obtained by solving the following dispersion equations for the photon energy E:

SPP : ±k SPP (E) ∓ m 1 2π Λ = E c ε i (E) sin(ϕ inc ), m 1 ∈ Nandi ∈ {air, Bi : GIG}, k SP (E)= E c ε Au (E )ε i (E ) ε Au (E )+ε i (E ) (1)
FP :

E c 2n eff,FP (E)h + φ r air (E) + φ r Bi:GIG (E) = 2m 2 π, m 2 ∈ N 0 . (2)
Here n eff,FP is the effective index of the fundamental TM mode of the Au/air/Au plasmonic waveguide formed by the nanoslit and obtained by solving [START_REF] Dionne | Plasmon slot waveguides: Towards chip-scale propagation with subwavelength-scale localization[END_REF] tanh

r E 2 c n 2 eff,FP -1 = - n 2 eff,FP -ε Au (E) ε Au (E) n 2 eff,FP -1 . (3)
The reflection phase shifts, φ r i , at both ends of the slit cavity can in first approximation be those of normal plane wave incidence,

r i = |r i | exp(iφ r i ) = n eff,FP - √ ε i n eff,FP + √ ε i
. Beside the SPP resonances depend only on Λ and ϕ inc , and are independent of the grating's geometrical parameters governing the FP spectral location, h and r, as seen from Eqs. 1-3. Varying therefore only, for instance, the nanoslit width r at fixed incidence angle and grating periodicity may lead to SPP-FP resonance coupling. Fig. 5b illustrates this for the case of ϕ inc = 20°and h = 100 nm, corresponding to the experimental conditions of Fig. 2a. From Eq. (3) one can deduce how the decreasing slit width r will cause an increase of n eff,FP , which in turn by Eq. (2) will redshift the FP resonance. The FP slit resonance (blue) is seen to cross the 2 nd -order Au/garnet SPPs (red) for nanoslits widths between 20 and 40 nm. Due to anticrossing mode coupling the otherwise fixed energy of the 2 nd -order SPPs will be perturbed even for slit widths values beyond this range. This is demonstrated in Fig. 5c where we have numerically calculated the specular reflectivity spectrum R p for a range of slit widths (at ϕ inc = 20°) using an extended coupled-mode formalism [(ECMF), see Ref. [START_REF] De León-Pérez | Theory on the scattering of light and surface Plasmon polaritons by arrays of holes and dimples in a metal film[END_REF]]. The reflection minima corresponding to the different mentioned grating resonances can be distinguished, but more importantly their coupling and anticrossing in the region predicted in Fig. 5b is convincingly observed. Fig. 5a zooms in on the redshift of the 2 nd -order Au/garnet SPP. The experimentally observed redshift of the plasmonic grating reflection anomaly can therefore be correctly attributed to a coupling between FP and SPP resonances. This experimental observation confirms for the first time the previously suggested theoretical possibility of tuning the strength of the EOT effects only by tuning of one geometrical parameter [START_REF] Halagačka | Coupled mode enhanced giant magnetoplasmonics transverse Kerr effect[END_REF][START_REF] Anishur Rahman | Extraordinary optical transmission: coupling of the Wood-Rayleigh anomaly and the Fabry-Perot resonance[END_REF].

Study of magnetooptical perturbation of the SPPs EOT modes

Turning our attention to the magnetic character of the garnet substrate of the studied plasmonic nanograting, we explore the impact of the observed coupling between the grating's resonances on the magneto-optic activity of the system. In MO materials the presence of a magnetization creates antisymmetric off-diagonal components in the permittivity tensor, ← → ε =

ε +ig z -ig y -ig z ε +ig x +ig y -ig x ε
, where in first order the gyration vector g = g1 M is parallel to the magnetization and its magnitude is linearly proportional to that of M, g ∼ |M|. For lossless MO media ← → ε must be hermitian, so that both ε and g must be real. In the most general case this gives rise to a nonreciprocal polarization change of a polarized beam [START_REF] Zvezdin | Modern Magnetooptics and Magnetooptical Materials[END_REF] when reflected from a surface of a MO layer. The fundamental process induced by the magnetization is the Lorentz force on carriers. Therefore, in a unique configuration of a magnetization perfectly perpendicular to the incidence plane, the s-wave having its E-field perfectly parallel to the magnetization, will not undergo any MO effect. In this so-called transverse configuration only the p-polarization will undergo a nonreciprocal correction. The ensuing transverse MO Kerr effect leads to a nonreciprocal first-order correction on the Fresnel reflection coefficient, r p = r 0 p (1 + igδr p ), where r 0 p is the isotropic conventional reciprocal Fresnel coefficient (for M = 0) and the small effect correction δr p = r 0 p gk y 2ε MO k z,MO . On a lossless MO material and below its critical incidence angle, TMOKE is then solely a reflection phase shift since Im(gδr p ) = 0, and the power reflectivity R p remains perfectly reciprocal, ∆R p = |r p (M)| 2 -|r p (-M)| 2 = 0. Nonreciprocal power reflectivity can only be realized on absorbing media. However on bare lossy ferromagnetic metals such as Co or Fe, ∆R p doesn't exceed 10 -3 . [START_REF] Zvezdin | Modern Magnetooptics and Magnetooptical Materials[END_REF] On the other hand, TMOKE power nonreciprocity on a lossless MO garnet can be obtained by cladding it with a plasmonic grating. [START_REF] Belotelov | Extraordinary transmission and giant magneto-optical transverse Kerr effect in plasmonic nanostructured films[END_REF]. The gold/garnet SPP dispersion gets a small nonreciprocal correction due to the nonreciprocal phase shift by the transverse MO garnet substrate. In combination with the sharp reflection resonances of the grating's FP, this leads to a huge enhancement. [START_REF] Halagačka | Coupled mode enhanced giant magnetoplasmonics transverse Kerr effect[END_REF] Including the effect of the gyrotropy in the boundary conditions at the interface between gold and a transversely magnetized garnet, leads to the following nonreciprocal dispersion equation for the surface magnetoplasmon polaritons:

ε Au (ε Bi:GIG -n 2 eff,SPP ) + ε Bi:GIG ε Au -n 2 eff,SPP ε Bi:GIG -n 2 eff,SPP -g 2 ε Bi:GIG = ign eff,SPP ε Au -n 2 eff,SPP wheren eff,SPP = k y k 0 (4)
The gyrotropy clearly breaks the inversion symmetry n eff,SPP → -n eff,SPP . Linearizing this equation with respect to g -for the considered garnet [START_REF] Halagačka | Mueller matrix optical and magneto-optical characterization of bi-substituted gadolinium iron garnet for application in magnetoplasmonic structures[END_REF] in the near infrared g ≈ 0.003 (g = 0.0096 @ 1.67 eV) -and solving to first order of g:

n eff,SPP = ± ε Au ε Bi:GIG ε Au + ε Bi:GIG + igε 2 Au ε 2 Au -ε 2 Bi:GIG √ ε Au + ε Bi:GIG . (5)
Similarly with non-MO configuration described by Eqs. (1-3) FP-SPP coupling occurs. The dependence of the imaginary part of n eff,SPP with g induce non-reciprocal response of coupled FP-SPP modes. The model of MO SPP describes spectral shift of the SPP mode by MO effect by a perturbation linearly dependent on gyrotropy g. It could thus be used to predict the effect and thus to design devices based on non-reciprocal reflection or transmission with locally inverted sign. In Fig. 2b this n eff,SPP dependence introduces spectral shift of typically ≈ 30 meV whatever the slit.

Conclusion

In this paper we have demonstrated unique approach for fine local tuning and switching of the magnetooptical effect in 1D periodic plasmonic gratings. By tuning geometrical parameters of the grating, the non-reciprocal reflectivity not only can be enhanced but also tuned in terms of sign and magnitude. The MO tuning was demonstrated experimentally and numerically for two cases: variation of the grating nanoslit width r and variation of the grating thickness h. We have proved correctness of our models used for determination of individual gratings geometry by direct comparison of measured and calculated MO response. We have also demonstrated how these significant effects can be correctly modeled with a linear analytic approach, giving easy design tool. This offers possibility of additional enhancement thanks to push-pull configuration by playing on technologically realistic combination of grating of the same thickness and different slits for advanced circuits including non-reciprocal functions.

Fig. 1 .

 1 Fig. 1. (a) Definition of parameters in the model describing the magnetoplasmonic structure in planar diffraction configuration, (b) fabricated set of diffraction gratings with SEM images of developed plasmonic gratings showing good quality of the grating lamelas and observed widths of nanoslits r.

Fig. 2a and

  Fig. 2a and 2b directly compare measured MO data on 15 samples having different width of nanoslit with MO numerical simulation data. Experimental data were measured at the nominal angle of incidence of 20 • . Numerical simulations were calculated for the incident angle of 20 • . The systematic shift of the peak position in model proves observed trend in the experiment.Clearly at a given photon energy, for example 1.68 eV, the TMOKE response can be positive (nanoslit width r=120 nm) or negative (r=70 nm). In other words, for a given optical signal and a given magnetization the reciprocity sign is inverted by changing only the slit width.

Fig. 2 .Fig. 3 .

 23 Fig. 2. Experimentally observed MO response (a) (δR) measured at incident angle of 20 • on 15 different patches of samples with various opening r is compared with numerical model calculated for various nanoslit width r. (b) Black dashed line at photon energy 1.67 eV highlighting the change of the MO peak sign with the grating nanoslit width r .

Fig. 4 .

 4 Fig. 4. Spectra of relative reflectance R = R p /R s measured for the angle of incidence ϕ = 20 • , grating nanoslit width r = 63 nm, and the grating thickness h = 100 nm. In subplot (a) the full spectra is presented with marked position of the 1 st and 2 nd garnet SPP peaks. The inserted detail of light-cone includes dispersion curves of the predicted SPP (solid blue) crossing the 20 • -light line (solid black). The area for air SPP is marked as red line. Subplot (b) detail red/shift of the +2 nd garnet SPP peak measured for increasing width of the nanoslit r.

Fig. 5 .

 5 Fig. 5. Analysis of coupling between FP and SPP modes base on variation of the grating air-gap width r. Subplot (a) shows red-shift of coupled SPP/FP mode at cross-cuts for air-gap width r = 40, 60, 80, and 100 nm. Subplot (b) shows positions of the SPP modes without geometrical dispersion and dispersive FP mode (calculated from Eq. (1, 2)). Subplot (c) shows dispersion map of SPPs and FP modes calculated using extended coupled-mode formalism.

Acknowledgment

This work has been partially supported by the french RENATECH network. The authors gratefully acknowledge the support by the Grant Agency of the Czech Republic (Project 18-22102S). CZ.02.1.01/0.0/0.0./16_013/0001791, and IT4Innovations National Supercomputing Center -LM2015070.


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

Florent Bouly
email: florent.bouly@univ-lille.fr

Moving average Multifractional Processes with Random Exponent: lower bounds for local oscillations

Keywords: Fractional Brownian Motion, varying Hurst parameter, pointwise Hölder regularity, Itô integral AMS Subject Classification (2010). 60G17, 60G22, 60G18

In the last few years Ayache, Esser and Hamonier introduced a new Multifractional Process with Random Exponent (MPRE) obtained by replacing the Hurst parameter in a moving average representation of Fractional Brownian Motion through Wiener integral by an adapted Hölder continuous stochastic process indexed by the integration variable. Thus, this MPRE can be expressed as a moving average Itô integral which is a considerable advantage with respect to another MPRE introduced a long time ago by Ayache and Taqqu. Thanks to this advantage, very recently, Loboda, Mies and Steland have derived interesting results on local Hölder regularity, self-similarity and other properties of the recently introduced moving average MPRE and generalizations of it. Yet, the problem of obtaining, on an universal event of probability 1 not depending on the location, relevant lower bounds for local oscillations of such processes has remained open. We solve it in the present article under some conditions.

Introduction and statement of the main result

Roughly speaking multifractional stochastic processes (see e.g. [START_REF] Ayache | Multifractional stochastic fields: wavelet strategies in multifractional frameworks[END_REF]) are continuous real-valued stochastic processes with non-stationary increments which extend the wellknown Fractional Brownian Motion (see e.g. [START_REF] Embrechts | Selfsimilar processes[END_REF][START_REF] Samorodnitsky | Stable non-Gaussian random processes: stochastic models with infinitive variance[END_REF]); yet, in contrast with it, their local path roughness can be prescribed via a functional Hurst parameter and thus can change from point to point. For a generic multifractional process Y = {Y (t)} t∈R + , path roughness in a neighborhood of any arbitrary fixed point τ ∈ R * + := R + \{0} = (0, +∞) is usually measured through α Y (τ ), the pointwise Hölder exponent at τ , defined as:

α Y (τ ) := sup α ∈ [0, 1] : lim sup r→0 + r -α Osc Y (τ, r) < +∞ , (1.1)
where, for all real number r > 0 small enough,

Osc Y (τ, r) := sup |Y (t ) -Y (t )| : (t , t ) ∈ [τ -r, τ + r] 2 (1.2)
is the oscillation of the process Y on the interval [τ -r, τ + r] ⊂ R * + . When a stochastic process is the functional Hurst parameter of Y , then Y is said to be a Multifractional Process with Random Exponent (MPRE). Such kind of process turned out to be useful in stock prices modeling (see for instance [START_REF] Bianchi | Modeling and simulation of currency exchange rates using MPRE[END_REF][START_REF] Bianchi | Modeling stock prices by multifractional Brownian motion: an improved estimation of the pointwise regularity[END_REF][START_REF] Bianchi | Multifractional processes in finance[END_REF]). A long time ago, [START_REF] Ayache | Multifractional Processes with Random Exponent[END_REF] introduced a first type of MPRE which is given by a random wavelet series, but unfortunately cannot be represented through the usual Itô integral. In order to avoid the latter drawback, another type of MPRE was introduced in the last few years in [START_REF] Ayache | A new multifractional process with random exponent[END_REF], and was generalized very recently in [START_REF] Loboda | Regularity of multifractional moving average processes with random Hurst exponent[END_REF]. Let (Ω, (F s ) s∈R , F, P) be a filtered probability space, and let B = {B(s)} s∈R be a standard Brownian motion with respect to the filtration (F s ) s∈R , the MPRE X = {X(t)} s∈R + studied in [START_REF] Loboda | Regularity of multifractional moving average processes with random Hurst exponent[END_REF] is defined, for each fixed t ∈ R + , as the Itô integral:

X(t) := t -∞ g(t, s) dB(s) , (1.3)
where, for each fixed t ∈ R + , the stochastic process {g(t, s)} s∈(-∞,t) is adapted to the filtration (F s ) s∈(-∞,t) and satisfies almost surely t -∞ |g(t, s)| 2 ds < +∞, which guarantees the existence of Itô integral in (1.3). In order to show that X has a modification with continuous paths and to conveniently bound from below its pointwise Hölder exponents, a well adapted extension of the Kolmogorov-Chentsov Hölder continuity Theorem has been derived in [START_REF] Loboda | Regularity of multifractional moving average processes with random Hurst exponent[END_REF] (see Theorem 2.1 and Corollary 2.2 in [START_REF] Loboda | Regularity of multifractional moving average processes with random Hurst exponent[END_REF]). Thus, the latter article has established that such a nice modification of X exists as soon as the associated integrand g satisfies the 3 conditions (A), (B) and (C) that we are now going to give.

Condition (A). For each fixed s ∈ R, the random function t → g(t, s) is differentiable on the open interval (s∨0, +∞). Moreover, there exist (F s ) s∈R -adapted processes {H(s)} s∈R , {L(s)} s∈R and {R(s)} s∈R , such that H(s) ∈ (0, 1), L(s) > 0, R(s) > 1/2, and it holds, for all t ∈ R + , |g(t, s)| ≤ L(s)|t -s| H(s)-1/2 , for every s ∈ (t -1, t), |∂ t g(t, s)| ≤ L(s)|t -s| H(s)-3/2 , for every s ∈ (t -1, t), |∂ t g(t, s)| ≤ L(s)|t -s| -R(s) , for every s ∈ (-∞, t -1].

Condition (B).

There are deterministic real numbers H, H, L and R, such that, for all s ∈ R, one has 0 < H ≤ H(s) ≤ H < 1 , 0 < L(s) ≤ L and R(s) ≥ R > 1/2 .

(1.4)

Condition (C). One has almost surely, for all (s , s

) ∈ R 2 , H(s ) -H(s ) ≤ µ |s -s | , (1.5)
where µ is some deterministic function from R + to R + which is continuous and increasing and satisfies µ(0) = 0.

Among many other things, the following theorem has been obtained in [START_REF] Loboda | Regularity of multifractional moving average processes with random Hurst exponent[END_REF].

Theorem 1.1 [START_REF] Loboda | Regularity of multifractional moving average processes with random Hurst exponent[END_REF] Assume that the conditions (A), (B) and (C) hold, then the process X has a modification with continuous paths which is identified with X from now on. Moreover, its pointwise Hölder exponents satisfy

P ∀ τ ∈ R * + , α X (τ ) ≥ H(τ ) = 1 . (1.6)
In other words, there exists Ω * 1 an universal event of probability 1 not depending on τ such that one has

α X (τ, ω) ≥ H(τ, ω) , for all (τ, ω) ∈ R * + × Ω * 1 . (1.7)
In order to obtain the further result on pointwise Hölder exponents of X stated below, the article [START_REF] Loboda | Regularity of multifractional moving average processes with random Hurst exponent[END_REF] has assumed that the integrand g satisfies the following additional condition (A * ) in which {L(s)} s∈R denotes the same process as in Condition (A).

Condition (A * ). One has, for all t ∈ R + , g(t, s) -σ(s)(t -s) H(s)-1/2 ≤ L(s)|t -s| H(s)-1/2+ρ , for every s ∈ (t -1, t), ∂ t g(t, s) -∂ t σ(s)(t -s) H(s)-1/2 ≤ L(s)|t -s| H(s)-3/2+ρ , for every s ∈ (t -1, t),
where ρ is some positive deterministic real number which satisfies H + ρ < 1 and does not depend on t and s, and where {σ(s)} s∈R is a continuous (F s ) s∈R -adapted process not depending on t and satisfying 0 < |σ(s)| ≤ L(s) , for all s ∈ R.

(1.8) Theorem 1.2 [START_REF] Loboda | Regularity of multifractional moving average processes with random Hurst exponent[END_REF] Under the conditions (A), (A * ), (B), (C) and the additional condition lim

ε→0 + µ(ε) log(ε) = 0 , (1.9)
one has

P α X (τ ) = H(τ ) = 1 , for all τ ∈ R * + . (1.10)
Let us point out that the keystone of the proofs of Theorems 1.1 and 1.2 given in [START_REF] Loboda | Regularity of multifractional moving average processes with random Hurst exponent[END_REF] is the important Burkholder-Davis-Gundy inequality (see for instance [START_REF] Mao | Stochastic differential equations and applications[END_REF][START_REF] Protter | Stochatic integration and differential equations[END_REF]) as formulated in the following proposition:

Proposition 1.3 Let p ∈ [1,
+∞[ be arbitrary and fixed. There is an universal deterministic finite constant a(p) for which the following result holds: for any (F s ) s∈Radapted stochastic process f = {f (s)} s∈R satisfying almost surely

+∞ -∞ |f (s)| 2 ds < +∞, one has E +∞ -∞ f (s) dB(s) p ≤ a(p)E +∞ -∞ |f (s)| 2 ds p/2 , (1.11)
where +∞ -∞ f (s) dB(s) denotes the Itô integral of f on R.

Remark 1.4 In fact the additional information brought by Theorem 1.2 with respect to Theorem 1.1 is that, for each fixed τ ∈ R * + , there exists Ω(τ ) an event of probability 1 which a priori depends on τ , such that α X (τ, ω) ≤ H(τ, ω) , for all ω ∈ Ω(τ ).

(1.12)

Notice that, in view of (1.1) the inequality (1.12) can equivalently be expressed as follows in terms of the local oscillations of X in the vicinity of τ :

lim sup r→0 + r -H(τ,ω)-θ Osc X (τ, r, ω) = +∞ , for all (θ, ω) ∈ R * + × Ω(τ ). (1.13)
The main goal of our article is to show that when the condition (C) is strengthened to the condition (C * ) given below then a significantly more strong result than (1.13) holds, namely: Theorem 1.5 Suppose that the conditions (A), (A * ), (B) and (C * ) are satisfied. Then, there exists Ω * 2 an universal event of probability 1 not depending on τ such that one has

lim inf r→0 + r -H(τ,ω)-θ Osc X (τ, r, ω) = +∞ , for all (θ, τ, ω) ∈ R * + × R * + × Ω * 2 . (1.14)
Condition (C * ). There are two deterministic constants κ ∈ (0, +∞) and γ ∈ (0, 1) such that one has almost surely, for all (s , s

) ∈ R 2 , H(s ) -H(s ) + σ(s ) -σ(s ) ≤ κ |s -s | γ . (1.15)
It is worth mentioning that a straightforward consequence of Theorems 1.1 and 1.5 and (1.1) is that:

Corollary 1.6 Suppose that the conditions (A), (A * ), (B) and (C * ) hold. Let Ω * 1 and Ω * 2 be the two universal events of probability 1 which were introduced in Theorems 1.1 and 1.5. Then Ω * := Ω * 1 ∩ Ω * 2 is an universal event of probability 1 such that α X (τ, ω) = H(τ, ω) , for all (τ, ω) ∈ R * + × Ω * .

(1.16) Remarks 1.7

(i) The conditions (B), (C) and (C * ) might seem restrictive, yet they can be relaxed by a localization procedure via stopping times (see for instance Section 4.4.1 in [START_REF] Jacod | Discretization of processes[END_REF]) which is explained in the setting of the MPRE X in the introduction of [START_REF] Loboda | Regularity of multifractional moving average processes with random Hurst exponent[END_REF].

In particular, it is important to mention that Theorem 1.5 remains true when the condition (1.15) only holds for all s and s belonging to every fixed compact interval I ⊂ R, and the constant κ in it is no longer a deterministic real number but an almost surely finite random variable which may depend on I.

(ii) Usually, in the literature relevant lower bounds for local oscillations of a stochastic process Y valid on an universal event of probability 1 not depending on the location (that is the point τ in our setting) are obtained via the classical strategy (see e.g. [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][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]) which consists in showing that the local time associated to Y is regular in the set variable uniformly in the space variable.

For deriving such a regularity result for the local time explicit and exploitable formulas for the characteristic functions of the finite-dimensional distributions of Y need to be available. Unfortunately, such formulas are not available for the MPRE X. Thus, for proving Theorem 1.5 we make use of another strategy which is to some extent reminiscent of that introduced in the last few years in [START_REF] Ayache | Lower bound for local oscillations of Hermite processes[END_REF] in the framework of Hermite processes. However several significant modifications of the strategy of [START_REF] Ayache | Lower bound for local oscillations of Hermite processes[END_REF] are needed since there is a wide difference between the latter selfsimilar chaotic processes with stationary increments and the MPRE X. Among other things the Burkholder-Davis-Gundy inequality provided by Proposition 1.3 plays a crucial role in our proof of Theorem 1.5.

(iii) In the framework of multifractal analysis (see e.g. [START_REF] Jaffard | The multifractal nature of Lévy processes[END_REF][START_REF] Balança | Fine regularity of Lévy processes and linear (multi)fractional stable motion[END_REF][START_REF] Ayache | Wavelet construction of Generalized Multifractional Processes[END_REF]) one is very often interested in determining singularity spectra of sample paths of stochastic processes. For a generic process Y = {Y (t)} t∈R + with nowhere differentiable paths the singularity spectrum ρ Y = {ρ Y (α)} α∈[0,1] is defined, for each ω ∈ Ω (the underlying probability space) and α ∈ [0, 1], as the Hausdorff dimension (see e.g. [START_REF] Falconer | Fractal geometry -mathematical foundations and applications[END_REF]) of the level set τ ∈ R * + : α Y (τ, ω) = α , where α Y (τ ) is the pointwise Hölder exponent of Y . Thus, in contrast with (1.10) the more strong result (1.16) draws close connections between the singularity spectrum of the MPRE X and its functional random Hurst parameter H.

Proof of the main result

Let σ and H be the same processes as in the conditions (A) and (A * ). One assumes that the condition (B) is satisfied and one denotes by Z = {Z(t)} t∈R + and R = {R(t)} t∈R + the two processes defined, for all t ∈ R + , as:

Z(t) := t -1 σ(s)(t -s) H(s)-1/2 dB(s) = t -∞ σ(s)(t -s) H(s)-1/2 1l [-1,t) (s) dB(s) (2.1) and R(t) := X(t) -Z(t) . (2.2)
One can easily derive from Theorem 1.1 applied to Z that:

Lemma 2.1 Suppose that the conditions (B) and (C) hold, then the process Z has a modification with continuous paths which is identified with Z from now on. Moreover, its pointwise Hölder exponents satisfy

P ∀ τ ∈ R * + , α Z (τ ) ≥ H(τ ) = 1 . (2.3)
On the other hand, combining (1.3), (2.2) and (2.1), for all t ∈ R + , one has

R(t) = t -∞ g(t, s) dB(s), where g(t, s) := g(t, s) -σ(s)(t -s) H(s)-1/2 1l [-1,t) (s), for every s ∈ (-∞, t
). Moreover, one can derive from the conditions (A) and (A * ), imposed to g, that g satisfies, for all t ∈ R + , a slightly modified version of the condition (A) in which H(s) is replaced by H(s) + ρ, for each s ∈ (t -1, t). Thus, similarly to Theorem 1.1 it can be shown that:

Lemma 2.2 Assume that the conditions (A), (A * ), (B) and (C) hold, then the process R has a modification with continuous paths which is identified with R from now on. Moreover, its pointwise Hölder exponents satisfy

P ∀ τ ∈ R * + , α R (τ ) ≥ H(τ ) + ρ = 1 . (2.4) Remark 2.3
In view of Lemma 2.2, (1.1) and (1.2), in order to show that Theorem 1.5 holds, it is enough to prove it in the particular case where X = Z. On the other hand, in view of the fact that the interval R * + := (0, +∞) can be expressed as the countable union of the open, bounded and overlapping intervals 2 -1 q, 1 + 2 -1 q , q ∈ Z + , 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, we will only prove it when q = 0, that is τ ∈ (0, 1); its proof can be done in a similar way for any other q.

For any integers j ≥ 2 and k ∈ {0, . . . , 2 j -1}, one denotes by ∆(j, k) the increment of the process Z such that

∆(j, k) := Z(d j,k+1 ) -Z(d j,k ) , (2.5)
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 . (2.6)
Observe that, in view of (2.1), the increment ∆(j, k) can be expressed as:

∆(j, k) = d j,k+1 -1 σ(s) (d j,k+1 -s) H(s)-1/2 -(d j,k -s) H(s)-1/2 + dB(s) , (2.7)
with the convention that, for each (y, α) ∈ R 2 , one has

y α + := y α , if y > 0, 0, else. (2.8)
In all the sequel the parameter b ∈ (0, 1/2) is arbitrary and fixed. For each integer j ≥ 2, one sets

e j := 2 jb , (2.9)
where • is the integer part function, 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 . (2.10)
Observe that the cardinality of L j satisfies, for some positive finite constant c not depending on j, Card(L j ) ≤ c 2 j(1-b) .

(2.11) Also, observe that, for any l ∈ L j , the random variable ∆(j, le j ), defined through (2.7) with k = le j , can be expressed as ∆(j, le j ) = ∆(j, le j ) + ∆(j, le j ) ,

where ∆(j, le j ) :=

d j,le j +1 d j,(l-1)e j +1 σ(s) (d j,le j +1 -s) H(s)-1/2 -(d j,le j -s) H(s)-1/2 + dB(s) (2.13)
and ∆(j, le j ) :=

d j,(l-1)e j +1 -1 σ(s) (d j,le j +1 -s) H(s)-1/2 -(d j,le j -s) H(s)-1/2 dB(s) . (2.14)
Let us now focus on the study of asymptotic behavior of the random variables ∆(j, le j ) when j goes to +∞. 2 j(H(d j,le j )+b(1-H)/2) ∆(j, le j ) = 0 .

(2.15)

For showing that Lemma 2.4 holds one needs two preliminary results. In order to state them one first has to introduce some additional notations. Let η ∈ (0, 1) be arbitrary and fixed and such that 5κη

γ < b(1 -H) , (2.16)
where κ, γ, b and H are as in (1.15), (2.9) and (1.4). One assumes that the integer J 0 ≥ 6 is chosen so that one has e j /2 j ≤ 2 -j(1-b) < η/2, for all integer j ≥ J 0 . Then, one can derive from (2.14) that ∆(j, le j ) = ∆ (j, le j ) + ∆ (j, le j ) ,

where ∆ (j, le j ) :=

d j,le j -η -1 σ(s) (d j,le j +1 -s) H(s)-1/2 -(d j,le j -s) H(s)-1/2 dB(s) (2.18)
and ∆ (j, le j ) :=

d j,(l-1)e j +1 d j,le j -η σ(s) (d j,le j +1 -s) H(s)-1/2 -(d j,le j -s) H(s)-1/2 dB(s) . (2.19)
Lemma 2.5 There is a finite deterministic constant c > 0 such that, for all integer j ≥ J 0 , one has

sup l∈L j d j,le j -η -1 |σ(s)| 2 (d j,le j +1 -s) H(s)-1/2 -(d j,le j -s) H(s)-1/2 2 ds ≤ c 2 -2j . (2.20)
Proof of Lemma 2.5 Using the mean value Theorem, (2.6), (1.8), and (1.4), one gets, for all j ≥ J 0 , l ∈ L j and s

∈ [-1, d j,le j -η], that |σ(s)| 2 (d j,le j +1 -s) H(s)-1/2 -(d j,le j -s) H(s)-1/2 2 ≤ L 2 η 2(H-H) (d j,le j -s) 2H-3 2 -2j .
Thus setting c := L 2 η 2(H-H) 2 η x 2H-3 dx < +∞, it follows that (2.20) holds.

Lemma 2.6 There is a finite deterministic constant c > 0 such that, for all integer j ≥ J 0 + 2b -1 , one has sup l∈L j d j,(l-1)e j +1 d j,le j -η

2 2jH(d j,le j -η) |σ(s)| 2 (d j,le j +1 -s) H(s)-1/2 -(d j,le j -s) H(s)-1/2 2 ds ≤ c 2 -2j(4b(1-H)/5) . (2.21)
Proof of Lemma 2.6 One can derive from the mean value Theorem, (2.6), (1.8), (1.4), (2.9) and (1.15) that, for all j ≥ J 0 + 2b -1 , l ∈ L j and s ∈ [d j,le j -η, d j,(l-1)e j +1 ],

|σ(s)| 2 (d j,le j +1 -s) H(s)-1/2 -(d j,le j -s) H(s)-1/2 2 ≤ L 2 (d j,le j -s) 2H(d j,le j -η)-2κη γ -3 2 -2j ,
and consequently that

d j,(l-1)e j +1 d j,le j -η |σ(s)| 2 (d j,le j +1 -s) H(s)-1/2 -(d j,le j -s) H(s)-1/2 2 ds ≤ L 2 2 -2j d j,(l-1)e j +1 d j,le j -η (d j,le j -s) 2H(d j,le j -η)-2κη γ -3 ds ≤ L 2 2 -2j 2(1 + κη γ -H(d j,le j -η)) 2 -j ( 2 jb -1)
2(H(d j,le j -η)-κη γ -1)

≤ 4 κη γ +1 L 2 2 -2j 2(1 -H) × 2 -2j(1-b)(H(d j,le j -η)-κη γ -1) ≤ 4 κη γ +1 L 2 2(1 -H) × 2 -2j(H(d j,le j -η)-(1-b)κη γ +b(1-H)) ≤ 16L 2 2(1 -H) × 2 -2j(H(d j,le j -η)+4b(1-H)/5) , (2.22)
where the last inequality follows from (2.16). Finally, (2.21) results from (2.22).

We are now ready to prove Lemma 2.4.

Proof of Lemma 2.4 It easily follows (2.17) and the triangle inequality that, for all integer j ≥ J 0 + 2b -1 , one has

P sup l∈L j
2 jH(d j,le j ) ∆(j, le j ) > 2 -j(8b(1-H)/15)+1 ≤ P sup l∈L j 2 jH(d j,le j ) ∆ (j, le j ) > 2 -j(8b(1-H)/15) (2.23) +P sup l∈L j 2 jH(d j,le j ) ∆ (j, le j ) > 2 -j(8b(1-H)/15) .

In order to conveniently bound each one of the two probabilities in the right-hand side of (2.23), one denotes by p a fixed real number such that 2 jH(d j,le j ) ∆ (j, le j ) > 2 -j(8b(1-H)/15) ≤ P sup 2 jH(d j,le j ) ∆ (j, le j ) > 2 -j(8b(1-H)/15))

p > 15 b(1 -H) > 15 . ( 2
l∈L j ∆ (j, le j ) > 2 -j(H+8b(1-H)/15) ≤ l∈L j P ∆ (j, le j ) > 2 -j(H+8b(1-H)/15) ≤ 2 jp(H+8b(1-H)/15) l∈L j E ∆ (j, le j ) p ≤ c 1 2 j((1-b)-p(1-H-8b(1-H)/15)) ≤ c 1 2 j(1-p(7(1-H)/15)) , ( 2
≤ P sup l∈L j 2 jH(d j,le j -η)+jκη γ ∆ (j, le j ) > 2 -j(8b(1-H)/15)

≤ P sup 2 jH(d j,le j ) ∆(j, le j ) > 2 -j(8b(1-H)/15)+1 < +∞.

l∈L j 2 jH(d j,le j -η) ∆ (j, le j ) > 2 -j(11b(1-H)/15) ≤ l∈L j P 2 jH(d j,le j -η) ∆ (j, le j ) > 2 -j(11b(1-H)/15) ≤ 2 jp(11b(1-H)/15) l∈L j E 2 jH(d j,le j -η) ∆ (j, le j ) p ≤ c 2 2 j((1-b)-pb(1-H)/
Thus, one can derive from the Borel-Cantelli Lemma that there is Ω * 3 (b) an event of probability 1 depending b on which one has sup j≥2 sup l∈L j 2 j(H(d j,le j )+8b(1-H)/15) ∆(j, le j ) < +∞ .

(2.27)

Finally, it is clear that (2.27) implies that (2.15) is satisfied on Ω * 3 (b).

Let us now focus on the study of asymptotic behavior of the random variables ∆(j, le j ) (see (2.13)) when j goes to +∞. First, one needs to introduce some additional notations. For each fixed integer j ≥ 6, the integer M j ≥ 1 denotes the integer part of j -1 (2 j /e j ) -2 , that is

M j := j -1 (2 j /e j ) -2 .
(2.28)

Moreover, (U j m ) m∈{0,1,...,M j } denotes the subdivision of the interval 1, (2 j /e j ) -1 by the M j + 1 integer points such that: U j M j := (2 j /e j ) -1 and U j m := 1 + mj , for all m ∈ {0, 1, . . . , M j -1};

(2.29) notice that

j ≤ U j M j -U j M j -1 < 2j . (2.30)
For all integers j ≥ 6 and m ∈ {1, . . . , M j }, let L j m be the non-empty finite set of positive integers defined as:

L j m := N ∩ U j m-1 , U j m ; (2.31)
observe that, (2.10), (2.29), (2.30) and (2.31) entail that

L j = M j m=1 L j m
(2.32) and j < Card(L j m ) ≤ 2j , for all m ∈ {1, . . . , M j }.

(2.33)

For every integers j ≥ 6, m ∈ {1, . . . , M j } and l ∈ L j m , the 3 random variables ∆ m (j, le j ), ∆ m (j, le j ) and ∆ m (j, le j ) are defined as:

∆ m (j, le j ) := d j,le j +1 d j,(l-1)e j +1 σ(ζ j,m ) (d j,le j +1 -s) H(ζ j,m )-1/2 -(d j,le j -s) H(ζ j,m )-1/2 + dB(s) , (2.34) ∆ m (j, le j ) := d j,le j +1 d j,(l-1)e j +1 σ(s)-σ(ζ j,m ) (d j,le j +1 -s) H(ζ j,m )-1/2 -(d j,le j -s) H(ζ j,m )-1/2 + dB(s) (2.35) and ∆ m (j, le j ) := d j,le j +1 d j,(l-1)e j +1 σ(s) (d j,le j +1 -s) H(s)-1/2 -(d j,le j -s) H(s)-1/2 + -(d j,le j +1 -s) H(ζ j,m )-1/2 -(d j,le j -s) H(ζ j,m )-1/2 + dB(s) , (2.36)
where the dyadic number ζ j,m is defined as:

ζ j,m := d j,(U j m-1 -1)e j +1 .
(2.37)

Notice that, in view of (2.13), (2.34), (2.35) and (2.36), one has ∆(j, le j ) = ∆ m (j, le j ) + ∆ m (j, le j ) + ∆ m (j, le j ) , for all l ∈ L j m .

(2.38)

The following lemma allows to understand the reason for which one has introduced the random variables ∆ m (j, le j ), l ∈ L j m .

Lemma 2.7 For all integers j ≥ 6 and m ∈ {1, . . . , M j }, and for each finite sequence (z l ) l∈L j m of real numbers, one has, almost surely,

E exp i l∈L j m z l ∆ m (j, le j ) F ζ j,m (2.39) = exp -2 -1 l∈L j m z 2 l d j,le j +1 d j,(l-1)e j +1 σ(ζ j,m ) (d j,le j +1 -s) H(ζ j,m )-1/2 -(d j,le j -s) H(ζ j,m )-1/2 + 2 ds .
Notice that (2.39) means that conditionally to the sigma-algebra F ζ j,m , the random variables ∆ m (j, le j ), l ∈ L j m , have independent centred Gaussian distributions with variances

E | ∆ m (j, le j )| 2 F ζ j,m , l ∈ L j m , satisfying E | ∆ m (j, le j )| 2 F ζ j,m = d j,le j +1 d j,(l-1)e j +1 σ(ζ j,m ) (d j,le j +1 -s) H(ζ j,m )-1/2 -(d j,le j -s) H(ζ j,m )-1/2 + 2 ds ≥ 2 -1 |σ(ζ j,m )| 2 2 -2jH(ζ j,m ) . (2.40)
Proof of Lemma 2.7 The main idea of the proof consists in the observation that, for all integers j ≥ 6, m ∈ {1, . . . , M j } and l ∈ L j m , the Brownian motion B in (2.34) can be replaced by the Brownian motion W j,m = {W j,m (x)} x∈R + := {B(x+ζ j,m )-B(ζ j,m )} x∈R + which is independent on the sigma-algebra F ζ j,m . Therefore W j,m is independent on the integrand in (2.34), denoted by K j,m,l , which is F ζ j,m -measurable. Having made this observation the proof becomes classical: it can be done in a standard way by approximating, for each fixed l ∈ L j m , the integrand K j,m,l = {K j,m,l (s)} s∈[d j,(l-1)e j +1 ,d j,le j +1 ] by a sequence K n j,m,l n∈N = {K n j,m,l (s)} s∈[d j,(l-1)e j +1 ,d j,le j +1 ] n∈N of elementary processes of the form:

K n j,m,l (s) = q-1 p=0 A p 1l [tp,t p+1 ) (s) ,
where the random variables A p , 0 ≤ p < q, are F ζ j,m -measurable, and the finite sequence (t p ) 0≤p≤q is a subdivision of the interval [d j,(l-1)e j +1 , d j,le j +1 ].

For every fixed integer j ≥ 6, let (Λ j,m ) m∈{1,...,M j } , ( Λj,m ) m∈{1,...,M j } , ( Λ j,m ) m∈{1,...,M j } , ( Λ j,m ) m∈{1,...,M j } , ( Λ j,m ) m∈{1,...,M j } and ( Λ j,m ) m∈{1,...,M j } be the 6 finite sequences of 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) Λj,m := sup ∆(j, le j ) : l ∈ L j m , (2.42) Λ j,m := sup ∆(j, le j ) : l ∈ L j m , (2.43)

Λ j,m := sup ∆ m (j, le j ) : l ∈ L j m , (2.44)
Λ j,m = sup ∆ m (j, le j ) : l ∈ L j m , (2.45)
and Λ j,m := sup ∆ m (j, le j ) : l ∈ L j m .

(2.46)

The main ingredient of the proof of Theorem 1.5 is the following proposition.

Proposition 2.8 There exists Ω * 2 (b) an event of probability 1 depending on b on which one has lim inf 2 j(H(d j,le j )+b(1-H)/2) ∆(j, le j ) = sup

j→+∞ inf 1≤m≤M j 2 jH(ζ j,m ) |σ(ζ j,m )| -1 Λ j,m ≥ 1/4 . ( 2
1≤m≤M j sup l∈L j m 2 j(H(d j,le j )+b(1-H)/2) ∆(j, le j ) ≥ 2 -2κ j γ+1 2 -j(1-b)γ sup 1≤m≤M j 2 j(H(ζ j,m )+b(1-H)/2) Λj,m ≥ c 0 sup 1≤m≤M j 2 j(H(ζ j,m )+b(1-H)/2) Λj,m , (2.49)
where the strictly positive constant c 0 := inf j∈N 2 -2κ j γ+1 2 -j(1-b)γ > 0. Then combining (2.15) and (2.49) one obtains (2.48).

Lemma 2.10 There exists Ω * 4 (b) an event of probability 1 depending on b on which one has lim inf

j→+∞ inf 1≤m≤M j 2 jH(ζ j,m ) |σ(ζ j,m )| -1 Λ j,m ≥ 1/4 . (2.50)
Proof of Lemma 2.10 Let Ω * 4 (b) be the event defined as:

Ω * 4 (b) := +∞ J=6 +∞ j=J M j m=1 ω ∈ Ω : 2 jH(ζ j,m ,ω) |σ(ζ j,m , ω)| -1 Λ j m (ω) ≥ 1/4 . (2.51)
In order to show that the lemma holds, it is enough to prove that P(Ω * 4 (b)) = 1 which is equivalent to prove that

P(Ω \ Ω * 4 (b)) = 0 . (2.52)
Notice that, in view of (2.51) the event Ω \ Ω * 4 (b) can be expressed as:

Ω \ Ω * 4 (b) := +∞ J=6 +∞ j=J M j m=1 ω ∈ Ω : Λ j m (ω) < 4 -1 2 -jH(ζ j,m ,ω) |σ(ζ j,m , ω)| .
Thus, one knows from the Borel-Cantelli Lemma that in order to derive (2.52) it is enough to prove that

+∞ j=6 P M j m=1 ω ∈ Ω : Λ j m (ω) < 4 -1 2 -jH(ζ j,m ,ω) |σ(ζ j,m , ω)| < +∞ . ( 2

.53)

One clearly has, for every j ≥ 6,

P M j m=1 ω ∈ Ω : Λ j m (ω) < 4 -1 2 -jH(ζ j,m ,ω) |σ(ζ j,m , ω)| ≤ M j m=1 P Λ j m < 4 -1 2 -jH(ζ j,m ,ω) |σ(ζ j,m , ω)| (2.54) = M j m=1 P l∈L j m ω ∈ Ω : ∆(j, le j , ω) < 4 -1 2 -jH(ζ j,m ,ω) |σ(ζ j,m , ω)| ,
where the last equality follows from (2.44). Moreover, using the fact that, for each m ∈ {1, . . . , M j },

P l∈L j m ω ∈ Ω : ∆(j, le j , ω) < 4 -1 2 -jH(ζ j,m ,ω) |σ(ζ j,m , ω)| = E l∈L j m 1l | ∆(j,le j )|<4 -1 2 -jH(ζ j,m ) |σ(ζ j,m )| = E E l∈L j m 1l | ∆(j,le j )|<4 -1 2 -jH(ζ j,m ) |σ(ζ j,m )| F ζ j,m
and Lemma 2.7, one gets that

P l∈L j m ω ∈ Ω : ∆(j, le j , ω) < 4 -1 2 -jH(ζ j,m ,ω) |σ(ζ j,m , ω)| = E l∈L j m 2π E | ∆ m (j, le j )| 2 F ζ j,m -1/2 × R exp -2 -1 E | ∆ m (j, le j )| 2 F ζ j,m -1 x 2 1l |x|<4 -1 2 -jH(ζ j,m ) |σ(ζ j,m )| dx ≤ E l∈L j m π -1/2 |σ(ζ j,m )| -1 2 jH(ζ j,m ) R 1l |x|<4 -1 2 -jH(ζ j,m ) |σ(ζ j,m )| dx ≤ 2 -1 π -1/2 Card(L j m ) < 2 -1 π -1/2 j , (2.55)
where the first inequality and the last inequality respectively follow from (2.40) and (2.33). Finally, putting together (2.54), (2.55) and (2.28), one obtains for every j ≥ 6, that

P M j m=1 ω ∈ Ω : Λ j m (ω) < 4 -1 2 -jH(ζ j,m ,ω) |σ(ζ j,m , ω)| ≤ M j 2 -1 π -1/2 j ≤ π -j/2 ,
which shows that (2.53) holds.

In view of Lemma 2.10 and of the fact that one has, almost surely

|σ| inf := inf s∈[0,1] |σ(s)| > 0 , (2.56)
the following two lemmas basically show that sup 1≤m≤M j 2 jH(ζ j,m ) Λ j,m and sup 1≤m≤M j 2 jH(ζ j,m ) Λ j,m are almost surely asymptotically negligible with respect to inf 1≤m≤M j 2 jH(ζ j,m ) |σ(ζ j,m )| -1 Λ j,m when j goes to +∞.

Lemma 2.11 There is Ω * 5 (b) an event of probability 1 depending on b on which one has lim sup

j→+∞ sup 1≤m≤M j 2 j(H(ζ j,m )+(1-b)γ/2) Λ j,m = 0 . (2.57) Lemma 2.
12 There is Ω * 6 (b) an event of probability 1 depending on b on which one has lim sup

j→+∞ sup 1≤m≤M j 2 j(H(ζ j,m )+(1-b)γ/2) Λ j,m = 0 . (2.58)
Before proving Lemmas 2.11 and 2.12 let us give the proof of Proposition 2.8.

Proof of Proposition 2.8 First notice that the event of probability 1

Ω * 2 (b) is defined as: Ω * 2 (b) := 6 q=3 Ω * q (b)
, where the Ω * q (b), 3 ≤ q ≤ 6, are the same events of probability 1 as in Lemmas 2.9, 2.10, 2.11 and 2.12. Using (2.12), (2.41), (2.42), (2.43) and the triangle inequality, one obtains, for all integers j ≥ 6 and m ∈ {1, . . . , M j }, that Λ j,m ≥ Λ j,m -Λj,m . Thus, combining the latter inequality with (2.56) one gets that inf

1≤m≤M j 2 jH(ζ j,m ) |σ(ζ j,m )| -1 Λ j,m ≥ inf 1≤m≤M j 2 jH(ζ j,m ) |σ(ζ j,m )| -1 Λ j,m -|σ| -1 inf sup 1≤m≤M j 2 jH(ζ j,m ) Λ j,m .
Then, one can derive from Lemma 2.9 that the following inequality holds on Ω * 2 (b):

lim inf j→+∞ inf 1≤m≤M j 2 jH(ζ j,m ) |σ(ζ j,m )| -1 Λ j,m ≥ lim inf j→+∞ inf 1≤m≤M j 2 jH(ζ j,m ) |σ(ζ j,m )| -1 Λ j,m .
(2.59) Moreover, (2.38), (2.43), (2.44), (2.45), (2.46) and the triangle inequality entail, for all integer j ≥ 6 and m ∈ {1, . . . , M j }, that Λ j,m ≥ Λ j,m -Λ j,m -Λ j,m . Thus, combining the latter inequality with (2.56) one obtains that inf

1≤m≤M j 2 jH(ζ j,m ) |σ(ζ j,m )| -1 Λ j,m ≥ inf 1≤m≤M j 2 jH(ζ j,m ) |σ(ζ j,m )| -1 Λ j,m -|σ| -1 inf sup 1≤m≤M j 2 jH(ζ j,m ) Λ j,m + sup 1≤m≤M j 2 jH(ζ j,m ) Λ j,m .
Then, one can derive from Lemmas 2.10, 2.11 and 2.12 that the following inequality holds on Ω * 2 (b): lim inf

j→+∞ inf 1≤m≤M j 2 jH(ζ j,m ) |σ(ζ j,m )| -1 Λ j,m ≥ 1/4 . (2.60)
Finally, (2.47) results from (2.59) and (2.60).

For proving Lemmas 2.11 and 2.12 one needs some preliminary results.

Lemma 2.13 Let J 1 ≥ 6 be a fixed integer such that

κ j γ 2 -j(1-b)γ ≤ 4 -1 H for all j ≥ J 1 .
(

2.61)

There is a finite deterministic constant c > 0 such that, for all integers j ≥ J 1 and m ∈ {1, . . . , M j }, one has

2 2jH(ζ j,m ) sup l∈L j m d j,le j +1 d j,(l-1)e j +1 σ(s) 2 (d j,le j +1 -s) H(s)-1/2 -(d j,le j -s) H(s)-1/2 + -(d j,le j +1 -s) H(ζ j,m )-1/2 -(d j,le j -s) H(ζ j,m )-1/2 + 2 ds ≤ c j 2(1+γ) 2 -2j(1-b)γ . (2.62)
Proof of Lemma 2.13 Throughout the proof the integers j ≥ J 1 , m ∈ {1, . . . , M j } and l ∈ L j m are arbitrary and fixed. Let s ∈ [d j,(l-1)e j +1 , d j,le j ) be arbitrary and fixed; using the mean value Theorem, (1.15), the inequalities 0 < (d j,le j -s) < (d j,le j +1 -s) < 1, (2.37), (2.9), (2.6), (2.31), (2.29), (2.30) and the triangle inequality one obtains, for some real number

β belonging to interval [H(s) ∧ H(ζ j,m ), H(s) ∨ H(ζ j,m )], that (d j,le j +1 -s) H(s)-1/2 -(d j,le j -s) H(s)-1/2 -(d j,le j +1 -s) H(ζ j,m )-1/2 -(d j,le j -s) H(ζ j,m )-1/2 ≤ (d j,le j +1 -s) β-1/2 log(d j,le j +1 -s) -(d j,le j -s) β-1/2 log(d j,le j -s) |H(s) -H(ζ j,m )| ≤ 2κ j γ 2 -j(1-b)γ (d j,le j +1 -s) β-1/2 log(d j,le j +1 -s) -(d j,le j -s) β-1/2 log(d j,le j -s) ≤ 2κ j γ 2 -j(1-b)γ (d j,le j +1 -s) H(ζ j,m )-2κ j γ 2 -j(1-b)γ -1/2 | log(d j,le j +1 -s)| +(d j,le j -s) H(ζ j,m )-2κ j γ 2 -j(1-b)γ -1/2 | log(d j,le j -s)| 1l [d j,le j -1 ,d j,le j ) (s) +2κ j γ 2 -j(1+(1-b)γ) (d j,le j -s) H(ζ j,m )-2κ j γ 2 -j(1-b)γ -3/2 1 + | log(d j,le j -s)| 1l [d j,(l-1)e j +1 ,d j,le j -1 ) (s) . (2.63) Moreover, notice that c 1 := sup j∈N 2 2κj γ+1 2 -j(1-b)γ < +∞ . (2.64)
One can derive from (2.63), (1.8), (1.4), the change of variables u = 2 j (s -d j,le j ) and v = -u, (2.64), and (2.61) that d j,le j d j,(l-1)e j +1 σ(s)

2 (d j,le j +1 -s) H(s)-1/2 -(d j,le j -s) H(s)-1/2 -(d j,le j +1 -s) H(ζ j,m )-1/2 -(d j,le j -s) H(ζ j,m )-1/2 2 ds ≤ 4κ 2 L 2 j 2γ 2 -j(1+2(1-b)γ) 0 -1 (2 -j -2 -j u) H(ζ j,m )-2κ j γ 2 -j(1-b)γ -1/2 | log(2 -j -2 -j u)| +(-2 -j u) H(ζ j,m )-2κ j γ 2 -j(1-b)γ -1/2 | log(-2 -j u)| 2 du +4κ 2 L 2 j 2γ 2 -j(3+2(1-b)γ) -1 -∞ (-2 -j u) 2H(ζ j,m )-4κ j γ 2 -j(1-b)γ -3 1 + | log(-2 -j u)| 2 du ≤ c 2 j 2(1+γ) 2 -2j(H(ζ j,m )+(1-b)γ) 1 0 (1 + v) H(ζ j,m )-2κ j γ 2 -j(1-b)γ -1/2 1 + | log(1 + v)| +v H(ζ j,m )-2κ j γ 2 -j(1-b)γ -1/2 1 + | log(v)| 2 dv +c 2 j 2(1+γ) 2 -2j(H(ζ j,m )+(1-b)γ) +∞ 1 v 2H(ζ j,m )-4κ j γ 2 -j(1-b)γ -3 2 + | log(v)| 2 dv ≤ c 3 j 2(1+γ) 2 -2j(H(ζ j,m )+(1-b)γ) , (2.65)
where the constant finite constant c 2 := 4κ 2 L 2 c 2 1 and the finite constant

c 3 := c 2 1 0 (1 + v) H-1/2 1 + | log(1 + v)| + v H /2-1/2 1 + | log(v)| 2 dv +c 2 +∞ 1 v 2H-3 2 + | log(v)| 2 dv .
Similarly to (2.65) it can be shown that

d j,le j +1 d j,le j σ(s) 2 (d j,le j +1 -s) H(s)-1/2 -(d j,le j +1 -s) H(ζ j,m )-1/2 2 ds ≤ c 4 j 2(1+γ) 2 -2j(H(ζ j,m )+(1-b)γ) , (2.66)
where the finite constant

c 4 := c 2 1 0 (1 -u) H-1 2 + | log(1 -u)| 2 du .
Finally, (2.62) results from (2.8), (2.65) and (2.66).

Lemma 2.14 There are two finite deterministic constants 0 < c < c such that, for all integers j ≥ 6, m ∈ {1, . . . , M j } and l ∈ L j m , one has c 2 -2jH(ζ j,m ) ≤ d j,le j +1 d j,(l-1)e j +1 (d j,le j +1 -s) H(ζ j,m )-1/2 -(d j,le j -s)

H(ζ j,m )-1/2 + 2 ds ≤ c 2 -2jH(ζ j,m ) .
(2.67)

Proof of Lemma 2.14 In view of (2.6), it is clear that (d j,le j +1 -s)

d j,le j +1 d j,le j (d j,le j +1 -s) 2H(ζ j,m )-1 ds = 2 -2jH(ζ j,m ) 2H(ζ j,m ) . ( 2
H(ζ j,m )-1/2 -(d j,le j -s) H(ζ j,m )-1/2 2 ds + d j,le j +1 d j,le j (d j,le j +1 -s) 2H(ζ j,m )-1 ds , (2.69)
it clearly follows from (2.68) and (1.4) that the first inequality in (2.67) is satisfied. Thus, from now on, our goal is to show that the second inequality in (2.67) holds. Using (2.6), the change of variable x = 2 j (d j,le j -s), and standard computations, one gets that d j,le j d j,(l-1)e j +1 (d j,le j +1 -s)

H(ζ j,m )-1/2 -(d j,le j -s) H(ζ j,m )-1/2 2 ds ≤ 2 -2jH(ζ j,m ) +∞ 0 (x + 1) H(ζ j,m )-1/2 -x H(ζ j,m )-1/2 2 dx .
(2.70)

One can easily derive from (1.4) that

1 0 (x+1) H(ζ j,m )-1/2 -x H(ζ j,m )-1/2 2 dx ≤ 2 1 0 (x+1) 2H-1 dx+2 1 0 x 2H-1 dx . (2.71)
Moreover, it follows from the mean value Theorem and (1.4) that

+∞ 1 (x + 1) H(ζ j,m )-1/2 -x H(ζ j,m )-1/2 2 dx ≤ +∞ 1 x 2H-3 dx . (2.72)
Finally, letting c be the finite constant defined as

c := (2H ) -1 + 2 1 0 (x + 1) 2H-1 dx + 2 1 0 x 2H-1 dx + +∞ 1
x 2H-3 dx , one can derive from (1.4) and (2.68) to (2.72) that the second inequality in (2.67) holds.

Lemma 2.15 There is a finite deterministic constant c > 0 such that, for all integers j ≥ 6 and m ∈ {1, . . . , M j }, one has

2 2jH(ζ j,m ) sup l∈L j m d j,le j +1 d j,(l-1)e j +1 σ(s) -σ(ζ j,m ) 2 × (d j,le j +1 -s) H(ζ j,m )-1/2 -(d j,le j -s) H(ζ j,m )-1/2 + 2 ds ≤ c j 2γ 2 -2j(1-b)γ . ( 2
P sup 1≤m≤M j 2 jH(ζ j,m ) Λ j,m > 2 -2j(1-b)γ/3 ≤ M j m=1 P 2 jH(ζ j,m ) Λ j,m > 2 -2j(1-b)γ/3 ≤ M j m=1 l∈L j m P 2 jH(ζ j,m ) ∆ m (j, le j ) > 2 -2j(1-b)γ/3 ≤ 2 2jp(1-b)γ/3 M j m=1 l∈L j m E 2 jH(ζ j,m ) ∆ m (j, le j ) p ≤ c 1 M j Card(L j m ) j pγ 2 -jp(1-b)γ/3 ≤ c 2 j pγ 2 -j(1-b)(pγ/3-1) , (2.76)
where c 1 and c 2 are two finite constants not depending on j. Next, combining (2.76) and (2.75), one obtains that

+∞ j=6 P sup 1≤m≤M j 2 jH(ζ j,m ) Λ j,m > 2 -2j(1-b)γ/3 < +∞.
Thus, one can derive from the Borel-Cantelli Lemma that there is Ω * 5 (b) an event of probability 1 depending b on which one has sup j≥6 sup 1≤m≤M j 2 j(H(ζ j,m )+2(1-b)γ/3) Λ j,m < +∞ .

(

P sup 1≤m≤M j 2 jH(ζ j,m ) Λ j,m > 2 -2j(1-b)γ/3 ≤ M j m=1 P 2 jH(ζ j,m ) Λ j,m > 2 -2j(1-b)γ/3 ≤ M j m=1 l∈L j m P 2 jH(ζ j,m ) ∆ m (j, le j ) > 2 -2j(1-b)γ/3 ≤ 2 2jp(1-b)γ/3 M j m=1 l∈L j m E 2 jH(ζ j,m ) ∆ m (j, le j ) p ≤ c 1 M j Card(L j m ) j p(1+γ) 2 -jp(1-b)γ/3 ≤ c 2 j p(1+γ) 2 -j(1-b)(pγ/3-1) , (2.78)
where c 1 and c 2 are two finite constants not depending on j. Next, combining (2.78) and (2.75), one obtains that

+∞ j=J 1 P sup 1≤m≤M j 2 jH(ζ j,m ) Λ j,m > 2 -2j(1-b)γ/3 < +∞.
Thus, one can derive from the Borel-Cantelli Lemma that there is Ω * 6 (b) an event of probability 1 depending b on which one has sup We are now in position to complete the proof of Theorem 1.5. End of the proof of Theorem 1.5 In view of Remark 2.3 we will only prove that (1.14) holds in the case where X is the process Z defined through (2.1) and τ ∈ (0, 1). For any integer j ≥ 6, one sets l j (τ ) := 2 j τ /e j , (

j≥J 1 sup 1≤m≤M j 2 j(H(ζ j,m )+2(1-b)γ/3) Λ j,m < +∞ . ( 2
where • denotes as usual the integer part function and e j is as in (2.9). It is clear that l j (τ ) satisfies 0 ≤ τ -l j (τ )e j 2 j < e j 2 j ≤ 2 -j(1-b) .

(2.81) Moreover, it easily follows from (2.80), the inequalities 0 < b < 1/2, 0 < τ < 1, (2.10) and (2.9) that there exists a positive integer J 2 (τ ) ≥ 6, such that, for all integer j ≥ J 2 (τ ), one has l j (τ ) ∈ L j (2.82)

and [τ -12j2 -j(1-b) , τ + 12j2 -j(1-b) ] ⊂ [τ -2 -j(1-2b)+1 , τ + 2 -j(1-2b)+1 ] ⊂ (0, 1) . Next observe that in view of (2.82) and (2.32), there is m j (τ ) ∈ {1, . . . , M j } such that l j (τ ) ∈ L j m j (τ ) .

(2.83) Thus, one can derive from (2.83), (2.31), (2.29) and (2.30) that l j (τ ) -l < 2j , for all l ∈ L j m j (τ ) .

(2.84)

Next, combining (2.84), (2.81), (2.9) and the triangle inequality, one gets that τ -le j 2 j < (2j + 1)e j 2 j ≤ 3je j 2 j ≤ 3j2 -j (1-b) , for all l ∈ L j m j (τ ) , (2.85) and τ -le j + 1 2 j < (2j + 1)e j + 1 2 j ≤ 3je j 2 j ≤ 3j2 -j (1-b) , for all l ∈ L j m j (τ ) . ( 2 Hence, for all ω ∈ Ω * 2 (b), there exists an integer j 3 = j 3 (τ, ω) ≥ J 2 (τ ) such that, for all integer j ≥ j 3 , one has 2 jH(τ,ω) Osc Z τ , 2 -j (1-2b) , ω ≥ 5 -1 c 1 |σ| inf (ω) .

(2.88)

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

(2.89)

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

(2.90)

Then, (2.88), (2.89), (2.90) and (1.2) imply that, for all ρ ∈ [j 3 , +∞), one has ρ -H(τ,ω) Osc Z τ , (2ρ) 1-2b , ω ≥ 2 j * (ρ)H(τ,ω) Osc Z τ , 2 -j * (ρ) (1-2b) , ω ≥ 5 -1 c 1 |σ| inf (ω) .

Thus, for all r ∈ [2j 3 , +∞), one has r -H(τ,ω) Osc Z τ , r 1-2b , ω ≥ 10 -1 c 1 |σ| inf (ω) . (2.92)

Finally, denoting by Ω * 2 the event of probability 1 defined as Ω * 2 := b∈(0,1/2)∩Q Ω * 2 (b), where Q is the countable set of the rational numbers, it results from (2.92), the inequality c 1 |σ| inf > 0 and the equality lim b→0 + H(τ )/(1 -2b) = H(τ ) that (1.14) holds when X = Z and τ ∈ (0, 1).

Lemma 2 . 4

 24 There is Ω * 3 (b) an event of probability 1 depending on b on which one has lim sup j→+∞ sup l∈L j

  .68) Thus, setting c := (2H ) -1 > 2 -1 and noticing that (see (2.8))d j,le j +1 d j,(l-1)e j +1 (d j,le j +1 -s) H(ζ j,m )-1/2 -(d j,le j -s) H(ζ j,m )-1/2 + 2 ds = d j,le j d j,(l-1)e j +1

15 ) that 2

 152 jH(τ ) Osc Z τ , 3j2 -j(1-b) ≥ 2 jH(τ ) Λ j m j (τ ) ≥ 2 -j|H(τ )-H(ζ j,m j (τ ) )| 2 jH(ζ j,m j (τ ) ) Λ j m j (τ ) ≥ 2 -jκ(2j+1) γ 2 -jγ(1-b) inf 1≤m≤M j 2 jH(ζ j,m ) Λ j m ≥ c 1 inf 1≤m≤M j 2 jH(ζ j,m ) Λ j m ,(2.87)where the deterministic constant c 1 := inf j∈N 2 -jκ(2j+1) γ 2 -jγ/2 > 0 (recall that b ∈ (0, 1/2)). Thus, one can derive from (2.87), Proposition 2.8, (2.56) and (1.2) that the following inequalities hold on Ω * 2 (b) (the event of probability 1 introduced in Proposition 2.8):lim inf j→+∞ 2 jH(τ ) Osc Z τ , 2 -j(1-2b) ≥ lim inf j→+∞ 2 jH(τ ) Osc Z τ , 3j2 -j(1-b) ≥ 4 -1 c 1 |σ| inf .

  (τ ) Osc Z τ , r 1-2b ≥ 10 -1 c 1 |σ| inf , on Ω * 2 (b),which amounts to saying that lim infr→0 + r -H(τ )/(1-2b) Osc Z (τ , r) ≥ 10 -1 c 1 |σ| inf , on Ω * 2 (b).

  .25) where c 1 is a constant not depending on j. Moreover, it results from (1.15), (2.16), (2.11), the Markov inequality, (1.11), (2.19) and (2.21) that

	P sup
	l∈L j

  .86) Next it follows from (2.41), (2.5), (2.6), (1.2), (2.85), (2.86), (2.81), (2.83), (2.31), (2.37), (2.9), (2.29), (2.30) and (1.

Acknowledgements

This work has been partially supported by the Labex CEMPI (ANR-11-LABX-0007-01) and the GDR 3475 (Analyse Multifractale et Autosimilarité).


Paul Valensi

Patrick Henry

Franck Boccara

Emmanuel Cosson

Gaetan Prevost

Joseph Emmerich

Laura Ernande

Dany Marcadet

Elie Mousseaux

François Rouzet

Risk stratification and screening for coronary artery disease in asymptomatic patients with diabetes mellitus: Position paper of the French Society of Cardiology and the French-speaking Society of Diabetology

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.


Paul Valensi

Patrick Henry

Franck Boccara

Emmanuel Cosson

Gaëtan Prévost

Joseph Emmerich

Laura Ernande

Dany Marcadet

Elie Mousseau

François Rouzet

Risk stratification and screening for coronary artery disease in asymptomatic patients with diabetes mellitus

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.


Timothy Daly
email: timothy.daly@paris-sorbonne.fr.

Marion Houot

Anouk Barberousse

Amélie Petit

Stéphane Epelbaum

Lathe R Itzhaki Rf

Ball MJ Balin Bj

Bullido Mj

Cosby Sl

Del Tredici

Griffin Ws

Haas J Hudson

Kamer Ap

Kell Ar

Licastro F, Letenneur L, Lövheim Db

Palamara At

A Proposal to Make Biomedical Research into Alzheimer's Disease More Democratic Following an International Survey with Researchers

Keywords: Alzheimer's disease, amyloid-␤, dementia prevention, diversity in science, gender, lifestyle factors, lifestyle interventions, pharmaceutical industry, tau protein, women in science

Background: Therapeutic research into Alzheimer's disease (AD) has been dominated by the amyloid cascade hypothesis (ACH) since the 1990s. However, targeting amyloid in AD patients has not yet resulted in highly significant disease-modifying effects. Furthermore, other promising theories of AD etiology exist.

Objective: We sought to directly investigate whether the ACH still dominates the opinions of researchers working on AD and explore the implications of this question for future directions of research.

Methods: During 2019, we undertook an international survey promoted with the help of the Alzheimer's Association with questions on theories and treatments of AD. Further efforts to promote a similar study in 2021 did not recruit a significant number of participants.

Results: 173 researchers took part in the 2019 survey, 22% of which held "pro-ACH" opinions, tended to have more publications, were more likely to be male, and over 60. Thus, pro-ACH may now be a minority opinion in the field but is nevertheless the hypothesis on which the most clinical trials are based, suggestive of a representation bias. Popular vote of all 173 participants suggested that lifestyle treatments and anti-tau drugs were a source of more therapeutic optimism than anti-amyloid treatments.

Conclusion:

We propose a more democratic research structure which increases the likelihood that promising theories are published and funded fairly, promotes a broader scientific view of AD, and reduces the larger community's dependence on a fragile economic model.

INTRODUCTION 33

Disagreement is an obvious fact of science and 34 medicine, but how much is a good thing, and for how 35 long, is worth asking. The community of clinicians 36 and researchers working on Alzheimer's disease 37 (AD) is an amalgam of distinct communities with other drug targets, given the uncertainty around the clinical utility of anti-A␤ strategies [1]. Indeed, this shift bears witness to the existence of a variety of promising theories for AD with compelling evidence in favor of them (two examples being microbes [3] and tau protein initiation [START_REF] Arnsten | Hypoth-580 esis: Tau pathology is an initiating factor in sporadic Alzhei-581 mer's disease[END_REF]).

Scientists are guided in their decision-making by scientific data, but also by opinion. Zollman [START_REF] Zollman | The epistemic benefit of transient diver-583 sity[END_REF] studied how extreme beliefs and the unequal distribution of information within the research community can lead to "harmful homogeneity in science" (p. [START_REF] Reiss | Biomedical research, neglected diseases, and well-ordered science[END_REF]). The religious language used to describe debates around the suitability of therapeutic targets (defenders of A␤ and tau proteins as targets being termed BAptists and TAUists, for example [START_REF] Mudher | Alzheimer's disease -do 585 tauists and baptists finally shake hands?[END_REF]), while perhaps used jokingly, is nevertheless suggestive of the possibility of extreme opinions in the AD community.

An empirical study of productivity in AD research suggests that "a small percentage of researchers" [START_REF] Sorensen | Alzheimer's disease research: Scien-588 tific productivity and impact of the top 100 investigators in 589 the field[END_REF] has access to a large portion of the research apparatus, and while this does not entail that information is distributed unequally, it does suggest the strong Fig. 1. A decision tree revealing pro-ACH/non-ACH differences according to the participant's view on whether or not there is problematic adherence to the ACH. We cut the depth of the tree to 5. Leave nodes (i.e., the final node, colored in the figure) present the number of pro-ACH participants on the right and the number of non-ACH participants on the left. They are light blue to dark blue as a function of the proportion of non-ACH in the leave node, or they are light green to dark green as a function of the proportion of pro-ACH in the leave node. In the non-ACH group, 63 participants (47.37%) not identifying as male argue that there is problematic adherence to the ACH, compared to only 7 (18.42%) of the pro-ACH group with these characteristics. Conversely, on the other end of the scale, 6 males (or preferred not to say) of the pro-ACH group (15.79%) argued that there was no problematic adherence to the ACH and had more than 113 median publications. None of the non-ACH had this profile.

U n c o r r e c t e d A u t h o r P r o o f

T. Daly

METHODS

Ethical approval

The project received ethical approval from the 1).

Table 2 The popular vote of all researchers (pro-ACH and non-ACH taken together) toward therapeutic priorities in AD research, tabulated according to participants' gender. Concerning pharmacological treatments, anti-tau drugs offered more optimism than drug classes inspired by the ACH (anti-A␤ antibodies and/or BACE inhibitors). The top three therapeutic targets at preclinical, prodromal, and established AD were also investigated. Lifestyle interventions were a top-3 therapeutic priority at all stages of AD. Taken as a whole, the data suggest a favorable opinion regarding lifestyle factors and tau protein intervention. Gender differences in therapeutic priority were only significant for preclinical AD, with significantly more males arguing in favor of anti-A␤ strategies at this stage. Herrup [14] that "clinging to an inaccurate disease model" was the worst option for the future facing the community.

Finally, in order to better discern pro-ACH versus 224 non-ACH opinions, we used a decision tree (Fig. 1).

225

We also studied the popular vote of all participants

Fricker

  as "wrong done to someone specifically in their capacity as a knower"[START_REF] Fricker | Epistemic Injustice: Power and the Ethics of Knowing[END_REF].Fricker draws on examples from literature and history where factors such as race and gender have led to points of view being ignored and condemned. Within AD research, there are surely examples of intellectually honest researchers defending controversial hypotheses of AD who have struggled to get data published, receive funding, and retain their place within academia. In other words, certain hypotheses might be rejected not because of scientific argument but rather the social structure of the field of biomedical research. We finish by noting that the results from our small sample tentatively suggest that the majority of women do not support the ACH and may therefore be particularly vulnerable to the negative consequences of a community gatekeeping bias. Taking the example of hypotheses concerning the role of microbes in AD [3], Fig. 1 from Liu et al. (2019) showed that " . . . up to 2019 . . . 0.5% of trials tested the virus hypothesis" [1]. Concerning this "fringe theory . . . now, researchers are taking it seriously" [27], but the fact that up to 2019 only 1 in 200 clinical trials were dedicated to testing a direct viral contribution to AD, does beg the question: are theories of AD 505 being funded according to fair-share principle? Ruth 506 Itzhaki, first author on the previously cited [3] edi-507 torial on microbes in AD, has described "a series 508 of battles . . . awful problems getting [research] 509 published" [28]. This example does point to the pos-510 sibility of epistemic injustice in AD research and 511 suggests the existence of perspectives whose contri-512 bution to improving the lives of AD patients has not 513 yet been fully taken into account. This seems like 514 community-wide oversight, since the perspectives of 515 marginalized individuals in institutionalized social 516 structures may offer particularly insightful contribu-517 tions to research, since they may recognize patterns 518 in the world that those in more dominant groups may 519 be blinded to [29]. 520 CONCLUSION 521 The recent, controversial accelerated FDA 522 approval of Biogen/Eisai's Aducanumab for use 523 in mild AD is a testament to the influence of the 524 ACH and its defenders on the scientific and wider 525 community. The tentative results found in our survey 526 suggest that there is a complex scientific landscape 527 behind the scenes which risks becoming even more 528 polarized following such divisive decision-making 529 [30]. Given the hardships of the research community 530 in finding a disease-modifying treatment for AD, 531 we argue that further efforts should be made to 532 explore democratic solutions to overcome research 533 monopolies so that their potential consequences 534 for patients and scientists can be reduced, and 535 clinically useful treatments for AD be found as soon 536 as possible. It appears that the optimism toward the 537 ACH which has motivated industry and the recent 538 FDA decision may well not be shared by the majority 539 of researchers working on AD. This study offers one 540 tool to study this otherwise silent majority, whose 541 collective wisdom, we argue, could and should be 542 taken into further consideration for the future of vital 543 research into this devastating, complex disease. 544 ACKNOWLEDGMENTS 545 The authors thank the Alzheimer's Association for 546 their invaluable help promoting the survey, Yves Agid 547 and Marc Dhenain for help with formulating some 548 survey questions, and three anonymous reviewers at 549 JAD Reports who all provided helpful feedback. TD 550 thanks the Fondation Médéric Alzheimer for financial

  et al. / A Proposal to Make Biomedical Research into Alzheimer's Disease 3 influence of an unrepresentative minority holding a hierarchical sway over the broader direction of the field, at least at the level of publications.

	Furthermore, scientific "gatekeeping" in the form
	of peer review and broader editorial policy offers the
	advantage of improving the mean quality of published
	science, yet also increases the risk that more uncon-
	ventional work is rejected [8]. Indeed, critiques of
	current funding and publishing models for biomedi-
	cal research exist beyond the AD field, arguing that
	conformity to dominant models tends to lead to more
	funding [9].
	These conditions suggest there might well be
	"harmful homogeneity" in AD research. However,
	the presence of an influential minority suggests that
	getting access to most researchers' opinions about

AD should not be done via published literature, which cannot adequately represent most researchers working on this disease. We therefore decided to opt for direct access to researchers' opinions about theories and treatments of AD, creating the first anonymous survey into researcher opinions towards theories and treatments of AD. Firstly, we wanted to test Hunter et al.

[START_REF] Hunter | Time for a change 596 in the research paradigm for Alzheimer's disease: The value 597 of a chaotic matrix modeling approach[END_REF]

's hypothesis of "two broad groups; those that support the amyloid cascade hypothesis and those that do not" (p. 254). Secondly, we tested the constituent characteristics of the "pro-ACH" group, before thirdly, looking at possible gender differences in the popular vote toward treatments for AD at different disease stages.

Table 1

 1 Differences in the constitutive characteristics and opinions towards the ACH of pro-ACH and non-ACH groups identified in the 173 survey participants. Gender differences were significant between the pro-ACH and non-ACH groups, with significantly more men being pro-ACH. Taken together, these results suggest an association between having pro-ACH opinions and more publications, industry money, and self-identifying as a key opinion leader. ‡ Fisher's exact test was used to compare groups for categorical variables

	147

Research Ethics Community of Université Paris

Descartes and the data analysis complied with the French Commission nationale de l'informatique et des libertés (CNIL) guidelines. All the data were anonymous and were analyzed in aggregate form. All of the raw survey data are available as Supplementary Material.

Survey design and promotion

Questions and responses comprised two categories: research (on theories and treatments of AD) versus personal (age, profession, country of primary affiliation, clinical versus academic researcher). All questions were optional and multiple choice, based on extensive literature review and consultation with colleagues. The participant filled out the form by folwere due to the optional nature of every question. 146 Hypothesis 1: Pro-ACH responders will account * Questions regarding the ACH's validity ACH drugs are NOT a source of optimism for treating human AD. 86 (54.09%) 86 (71.67%) 0 (0.00%) < 0.001 * Beta-amyloid is NOT the #1 therapeutic priority either at preclinical, early, or late-stage AD. * Moving forwards (2019-), the ACH is a useful tool to guide research. * Agree with Tanzi (2015): "The clinical trials are failing the hypothesis, the hypothesis is not failing the trial." * Agree with Tanzi (2017): "we need to find people with amyloid buildup on their brain early" and target it. * Agree with Davies (2016): "we're flogging a dead horse" (A-beta) * gender; country: USA; number of publications; clinical researcher (versus academic); key opinion leader (KOL); has received money from the pharmaceutical industry; whether or not the researcher thinks that "there is problematic adherence to the ACH from either industry, academia, associations or funding bodies". We used the term adherence so as to insist upon the ACH's ability to guide research. Hypothesis 3: There will be gender differences in the popular vote towards treatments of AD We investigated the top three therapeutic targets at pre-clinical, early-stage, and established AD according to popular vote of all the survey participants, pro-ACH and non-ACH taken together. Furthermore, if there are gender differences to be found in the 183 pro-ACH/non-ACH groups, we might expect to find 184 gender differences in the popular vote. Only par-185 ticipants identifying as M/F were included in the 186 gender differences so as to use comparable group 187 sizes for significance testing (n = 7 of "trans/prefer 188 not to say/other"). 189 RESULTS 190 One hundred and seventy-three participants from 191 across the world filled out the questionnaire, with 192 a median age of 35, 83 (49.7%) being women. We 193 identified 38 (22.0%) "pro-ACH" participants, the 194 majority (65.8%) of whom were men (Table

wisdom," the empirical finding that informed col-