domain: spi phys

text:
Daniel Prost
email: daniel.prost@onera.fr

François Issac
email: francois.issac@onera.fr

Maxime Romier
email: maxime.romier@cnes.fr

Imaging electric and magnetic near field of radiating structures by infrared thermography

Keywords: Electromagnetism, Infrared thermography, High Impedance Surface, Near-field visualization

The characterization and visualization of the radiated electromagnetic field is a way to diagnose and check microwaves sources, in particular in the space industry. Beyond the traditional 3D scanning of the radiating element, infrared thermography is an alternative way that may give key results in a reduced time, since a few seconds are sufficient to obtain the emitted field map even for a complex multi-cells antenna. This method, called EMIR (ElectroMagnetic InfraRed) has been developed and used for years at ONERA. We have recently extended this technique to the magnetic field, which will lead to a complete characterization of the near field structure. We illustrate this feature in the particular cases of antennas for space application and for a High Impedance Surface (HIS), a metamaterial structure that could be used to reduce the coupling between antennas confined in compact structures (typically size constraints of the spatial components).

I. INTRODUCTION

The electromagnetic infrared (EMIR TM ) method has been developed and used at ONERA [START_REF] Levesque | Capteur vectoriel de champs électromgnétique par thermographie infrarouge[END_REF] in order to visualize and measure microwave electric fields. This method consists in inserting specific conductive thin films in the microwave electric field to be measured. Because of the low electrical resistivity of the film, induced electric currents generate ohmic losses inside and induce a heating which is recorded by the infrared camera. The measured increase of temperature is post-processed to evaluate the incident microwave electric field. It is also possible to measure the magnetic component of the wave when the field structure is not a plane wave (typically in the near-filed zone): in this case, a thin ferromagnetic film is used and ferromagnetic losses also initiate a heating captured by the camera [START_REF] Prost | Infrared Thermography of Microwave Electromagnetic Fields[END_REF][START_REF] Vernieres | Ferromagnetic microstructured thin films with high complex permeability for microwave applications[END_REF].

We present here some recent measurements achieved on components that are likely to be used in space systems. The EMIR TM method may be very useful for rapidly characterizing and diagnosing array antennas or other components (like metasurface) for space industry, which is becoming a hot topic in the context of growing production of much smaller satellites. The EMIR TM method is also well suited to provide experimentally 2D field representations in an original, fast and convenient way.

We present in this paper near-field measurement results of two different antennas: a Ku/Ka-band horn antenna (section III), and a multibeam antenna. Such devices are more and more used on communication satellites (section IV). In the section V we provide images of both electric and magnetic near field of a High Impedance Surface (HIS) developed at ONERA. To our knowledge, these are the first experimental visualizations of both electric and magnetic field of a metamaterial structure.

II. EMIR METHOD OF MEASUREMENT

The EMIR measurement is illustrated by Fig. 1.

-For electric field measurement, the conductive film is a carbon loaded polyamide with a surface impedance Z s of almost 2 kΩ/sq (Z s is the resistance per square, Z s = 1/(σe) where σ is the conductivity and e the thickness of the film). The absorbed power ratio is estimated at almost 15 %, which is sufficient to obtain field patterns, still low enough to avoid modifying the electric field structure. -For the magnetic field measurement, the sensing film is a 25-µm thick commercial FLEX-TOKIN composed of ferromagnetic micro pellets embedded in a polymer matrix. The magnetic losses are proportional to the imaginary part of the relative permeability which is evaluated to 100 in our frequency range (1 to 2 GHz).

The incident EM field is radiated by the component under test structure (antenna or HIS) fed by a RF amplifier properly driven by a signal generator providing a low frequency modulation; camera recordings are then demodulated at the same low frequency which suppresses convective heating noise and other continuous components of the heating. A few Watts injected on the radiated element allows to get a few degrees heating dynamic recorded by the camera. After having registered the demodulated heating, we can deduce the field amplitude. Indeed the thermal frames are related to the amplitude of the electric or magnetic field since we have:

-In the case of the electric field (Kapton film):

(1) -In the case of the magnetic field (Tokin film):

(2)

Where k E and k H are coefficients that depend on the heat transfer coefficient, the heat capacity, the density of the film, and the modulation frequency.

Therefore we can derive from equations ( 1) and ( 2) the electric or magnetic field amplitude as a function of the heating through a factor that depends only on film characteristics and modulation frequency. These factors can be considered as calibration factors depending on the parameters above. However, and more directly (without calibration), raw thermal frames can be interpreted as nonnormalized electromagnetic power density images (normalization not mandatory for getting relative power information through a dB scale).

III. KU-BAND HORN ANTENNA SOURCE

This Ku/Ka-band spline-profiled square horn antenna has been designed for space application [START_REF] Adam | Design, Manufacturing and Test of a Spline-Profile Square Horn for Focal Array Applications[END_REF], is linearly polarized. has been conveniently used to monitor the influence of the thin film on return losses. When the aperture is "sealed" by the film (worst case), mismatches remain below -10dB.

Obtaining the electric field thermal frame takes a few seconds; any major defect can easily be detected without any signal processing. This appears clearly on the heating images, as illustrated by the Fig. 2. Thermal frames can be produced at different distances and/or at different frequencies to obtain general information on the antenna behavior. For instance the Fig. 3 shows how the emission mode evolves as frequency increases.

IV. MULTIPLE FEED PER BEAM FOCAL ARRAY

The Multiple Feed per Beam focal array [START_REF] Leclerc | Ka-Band Multiple Feed per Beam Focal Array Using Interleaved Couplers[END_REF] is circularly polarized and has been designed for space applications in Ka-band. Its geometry is illustrated by the Fig. 4 below: In our measurement setup, the focal array is fed via the central waveguide by a circularly polarized TE11 mode (more details in [START_REF] Leclerc | Ka-Band Multiple Feed per Beam Focal Array Using Interleaved Couplers[END_REF]). Because of slot coupling, RF power is distributed to a sub-array of 7 feed horn. Fig. 5 (left) provides heating measurement at 3 mm above the antenna aperture. To maximize image contrast, it is worth normalizing the heating to its peak value; we obtain the power density P(x,y) in dB, on Fig. We achieved measurements at different distances from 1 mm to a few wavelengths (50 mm corresponding to 3 wavelengths). As a validation process, in Fig. 6 we present measurement and simulation results, for 2 distances (3 and 30 mm). The simulations have been computed with CST MWS using the time domain solver. The depicted distribution shows time-averaged squared amplitude of the tangential electric field. The noticeable agreement between the two pictures, although the presence of the film was not accounted in the simulation, confirms its low intrusive behavior. A slight "twist" of the pattern can be observed, which direction switches when the polarization is changed: indeed we have observed that behavior instantly by switching the beam polarization. Thus the thermal signature can detect relatively slight phenomena -through their impact on field amplitudesuch as field polarization, or unbalanced beamforming.

V. HIGH IMPEDANCE SURFACE (HIS)

HIS is a special passive electromagnetic structure (based on metamaterial technology) designed for reducing mutual coupling between adjacent patch antennas (in particular for applications where size constraints are crucial; like space systems) [START_REF] Capet | Amélioration du découplage inter élément par surface haute impédance pour antennes réseaux GNSS compactes[END_REF]. As illustrated by Fig. 7, the HIS cell consists of a two layers dielectric substrate with a first 4.8-mm thick layer of TLY (ε r = 2.2) with ground plane on its bottom face. The second layer, 0.127-mm thick, is made of the same material. A high capacitance value is obtained with coupled patches, which are printed on both sides of this thin dielectric substrate, to obtain a small unit cell (almost 11x19 mm²). Triangular and hexagonal elements are designed for bottom and top patches respectively. Vertical vias in the triangle centers ensure the patch to ground plane connection and create the inductive effect. This HIS is therefore composed of periodic capacitive and inductive elements, which exhibit resonance frequencies, in particular around 2 GHz.

We fed our HIS with a microwave power (modulated at 0.2 Hz) of the order of 2 to 3 W. The HIS structure is connected to a 50 Ω load.

Compared to previous measurements [START_REF] Prost | Electric field imaging of a high impedance surface for GNSS array decoupling application[END_REF], we are now able to visualize both magnetic and electric field frames.

Indeed, using successively our electric and magnetic losses films, we visualize electric and magnetic field patterns of the HIS over a large frequency range (0.8 to 2.3GHz). The last Fig. 8 and 9 are given in arbitrary units. These illustrate the differences of the near field structures between resonant (2 GHz, Fig. 8) and passing (2.25 GHz, Fig. 9) behaviors:

-At 2 GHz we observe the resonant behavior of the HIS cells. It can be noticed that geometric details appear: the current flowing on the triangular patches (Fig. 8) is exhibited by the magnetic field frame in particular. The thermography method could therefore be used as a diagnose tool.

-At 2.25GHz the HIS exhibits passing behavior (Fig. 9) ; some details of the structure can still be distinguished, but of course much less current is present on the patches.

VI. CONCLUSION

In conclusion, we have shown how the EMIR TM method could provide accurate and replicable radiation signatures, and highlight antenna behaviors as well as metamaterial structures almost instantly. More generally, it is also an experimental tool that provides 2D maps of the electric or magnetic fields that are usually only obtained by simulation. In the EMC context, it allows, in a low intrusive and fast way, to check RF behavior at different levels (antenna, system, integration -including detection electromagnetic leaks through small apertures) and gives 2D fields cartographies in a large frequency range.

Fig. 1 :

 1 Fig.1:The EMIR method: the fed component (antenna or else) radiates and heats the film; the infrared camera records this heating in which convection part is eliminated through low frequency modulation

Fig. 2 :Fig. 3 :

 23 Fig. 2: Ku-band horn antenna near electric field at 12 GHz, without and with an insertion defect

Fig. 4 :

 4 Fig. 4: Multiple Feed per Beam focal array, and computed E field amplitude for a TE11 mode

3 )Fig. 5 :

 35 Fig. 5: Heating image of the MFB source (at 3 mm of the radiating surface), and related power density in dB

Fig. 6 :

 6 Fig. 6: Power density at 3 (up) and 30 mm (bottom) ; measurement on left, simulation on right

Fig. 7 :

 7 Fig. 7: Structure of the HIS embedded in the transmission line (a), elementary cell (b), picture of the realized structure (c), thin film implementation (d)

Fig. 8 :

 8 Fig. 8: Electric and magnetic near field at resonance frequency (2 GHz)


Laurent Serlet
email: laurent.serlet@math.univ-bpclermont.fr

Explicit laws for the records of the perturbed random walk on Z

Keywords: perturbed random walk, once-reinforced random walk, perturbed Brownian motion, records, invariance principle, recurrence

Introduction

We consider a process (X n ) n≥0 with values in Z, started at 0 (X 0 = 0), which is a nearest neighbor random walk on Z that is, for every n ≥ 0, we have X n+1 ∈ {X n -1, X n + 1}. We denote its maximum and minimum up to time n by X n = max{X 0 , X 1 , . . . , X n } and X n = min{X 0 , X 1 , . . . , X n }. We say that (X n ) n≥0 is a perturbed random walk (PRW) with reinforcement parameters β, γ ∈ (0, +∞) if the transition probability P X n+1 = X n + 1 X 0 , X 1 , . . . , X n is equal to

• 1/2 if X n < X n < X n or n = 0 • 1/(1 + β) if X n = X n and n ≥ 1 • γ/(1 + γ) if X n = X n and n ≥ 1.
When β = γ = 1, we obtain a standard random walk (SRW). When β = γ, we obtain a symmetric perturbed random walk (SPRW) with parameter β ∈ (0, +∞). We interpret the case β > 1 as a self-attractive walk whereas for β ∈ (0, 1) the walk is self-repulsive. This process belongs to the broad class of processes with reinforcement which has generated an important amount of literature. Pemantle gives in [START_REF] Pemantle | A survey of random processes with reinforcement[END_REF] a nice survey with lots of references. More precisely, our PRW is sometimes called the once-reinforced random walk. This once-reinforced random walk can also be defined in Z d for d > 1 and some fundamental questions are still open in these dimensions but, in the present paper, we stay in dimension 1 which enables a much easier treatment and in particular, explicit computations of laws.

In [START_REF] Davis | Weak limits of perturbed random walks and the equation Yt = Bt + α sup{Ys : s ≤ t} + β inf{Ys; s ≤ t}[END_REF] and [START_REF] Davis | Brownian motion and random walk perturbed at extrema[END_REF], Davis introduces a diffusive rescaling by setting X n t = 1 √ n X nt (after linear interpolation of X between integer times) and he proves that the process (X n t ) t≥0 converges in law to a process (W t ) t≥0 which is the unique solution of the equation

W t = B t -(β -1) sup s≤t W s + (γ -1) inf s≤t W s (1)
where (B t ) t≥0 is a Brownian motion. The solutions of (1) have been studied by several authors under the name "Brownian motion perturbed at its extrema", see for instance [START_REF] Carmona | Beta variables as times spent in [0, +∞) by certain perturbed Brownian motions[END_REF], [START_REF] Werner | Some remarks on perturbed reflecting Brownian motion[END_REF], [START_REF] Davis | Weak limits of perturbed random walks and the equation Yt = Bt + α sup{Ys : s ≤ t} + β inf{Ys; s ≤ t}[END_REF], [START_REF] Davis | Brownian motion and random walk perturbed at extrema[END_REF], [START_REF] Perman | Perturbed Brownian motions[END_REF], [START_REF] Chaumont | Upper and lower limits of doubly perturbed Brownian motion[END_REF] and the references therein.

The present paper uses a different approach since it is based on explicit computations for the random walk and we get results on the limiting continuous objects as by-products. The methods are similar to those in [START_REF] Serlet | Hitting times for the perturbed reflecting random walk[END_REF] where we treated the case of the reflecting random walk perturbed at the maximum. We will of course refer to this paper for several proofs which are identical to that case. However since our approach is based on explicit computations, the non-reflecting case generates different formulas than the reflecting case and the non-reflecting case also adds new questions related to the signs, that we address in the present paper.

Since the PRW behaves as a SRW when it stays away from the extrema, we concentrate on the study of the process when it reaches an extremum, in particular for the first time and in that case we call it a record time. More precisely, we set

V k = X k -X k + 1
for the number of visited points up to time k i.e. the number of distinct values in the set {X 0 , X 1 , . . . , X k }. Then we define T 0 = 0 and for n ≥ 1, we call

T n = inf{k ≥ 1; V k = n + 1}
the time of the n-th record. Then R n = X Tn is the value of the n-th record and the sign of R n is denoted by χ n ∈ {-1, 1}. As we will see the sequence (χ n ) n≥1 is a time-inhomogeneous Markov chain for which the transition matrix is easily computed. As a consequence we will derive an invariance principle. We will also note that the record values (R n ) n≥0 can be reconstructed from the sequence of signs (χ n ) n≥1 and thus derive an invariance principle for the record values.

Then we introduce the rescaled record time process (τ n t ) t≥0 by

τ n t = 1 n 2 T [nt] (2)
where [•] denotes the integer part. We want an invariance principle for this process. First we work conditionally on the record signs and then without conditioning. Unfortunately in this latter case, we are unable to obtain a result in the general case and we restrict to symmetric perturbation (β = γ).

This invariance principle has consequences on the process of the number of visited points (V k ) k≥1 . As in the standard case, V k is of order √ k and we obtain in particular the asymptotic law of V k / √ k. Finally we will examine the possibility of "positive recurrence" for the PRW.

The paper is organized as follows. The next section is a precise statement of our main results, ending with two open questions. The following sections are devoted to proofs, beginning with a section of technical preliminary lemmas.

Statement of the results

Most of the processes that we consider in this section have their trajectories in the space d([0, +∞), R) of càdlàg functions that we endow with the usual Skorohod topology. Weak convergence of probability laws on this space is simply called in the sequel "convergence in law". However special care will be needed in Proposition 2 where we have to restrict to compact intervals of (0, +∞). In the sequel the notations cosh, sinh and tanh refer to the usual functions of hyperbolic trigonometry.

Let (X n ) n≥0 denote a PRW with parameters β, γ ∈ (0, +∞). First note that the sequence of record values (R n ) n≥0 is easily reconstructed from the sequence of record signs (χ n ) n≥1 because, for any n ≥ 1,

R n = χ n n k=1 1 {χ k =χn} (3)
and it justifies that we first focus on (χ n ) n≥1 . We start with an easy fact.

Proposition 1 The sequence of the signs of records (χ n ) n≥1 of the PRW is a time-inhomogeneous Markov chain with transition matrix

Q n = β+n γ+β+n γ γ+β+n β γ+β+n γ+n γ+β+n . ( 4
)
We notice that the off-diagonal terms of this transition matrix are of order 1/n so we speed up time by factor n to get a limit in law.

Asymptotic results for the signs of records

Proposition 2 Let the rescaled sequence of the record signs of the PRW be defined by ∀t > 0,

χ n t = χ [nt] . (5)
There is a process (x t ) t∈(0,+∞) with values in {-1, +1} uniquely defined in law such that, for any a > 0,

P(x a = -1) = β β + γ ; P(x a = 1) = γ β + γ (
T (s, t) = 1 β + γ   β + γ s t β+γ γ 1 -s t β+γ β 1 -s t β+γ γ + β s t β+γ   . (8)
For all 0 < a < b, the sequence of processes (χ n t ) t∈ Let us remark that setting xt = x e t for t ∈ R gives a new process (x t ) t∈R which is a time-homogeneous Markov jump process on {-1, 1} or, in other words, an alternating renewal process.

Consequences on the sequence of records

We derive a corollary which is the continuous-time counterpart of (3).

Corollary 3 Let (y t ) t>0 be the process defined by ∀t > 0, y t = x t t 0 1 {xu=xt} du where (x t ) t>0 is the process introduced in Proposition 2. Then, the process (R [nt] /n) t>0 converges in law to the process (y t ) t>0 .

In particular R n /n converges in law to y 1 . One must not be deceived by the formula

R n n = -1 n n k=1 1 {χ k =-1} if χ n = -1 1 n n k=1 1 {χ k =1} if χ n = 1 (9)
(which is a reformulation of (3)) and the fact that (χ n ) n≥1 is a Markov chain converging in law to the probability β/(β + γ) δ -1 + γ/(β + γ) δ +1 .

In the time-homogeneous case, (9) would imply by the ergodic Theorem that R n /n accumulates almost surely on the two limit points -β β+γ and γ β+γ . But here the Markov chain (χ n ) n≥1 is time-inhomogeneous and the almost sure behaviour of R n /n is completely different as one can see in the following proposition which holds whatever the values of β, γ ∈ (0, +∞).

Proposition 4 Almost surely, the sequence {R n /n; R n > 0} has limsup 1 and liminf 0 and similarly {R n /n; R n < 0} has limsup 0 and liminf -1 that is, for all ε ∈ (0, 1) and N ≥ 1 there exist n 1 , n 2 , n 3 , n 4 ≥ N such that

R n1 n 1 > 1 -ε, 0 < R n2 n 2 < ε, -ε < R n3 n 3 < 0, R n4 n 4 < -1 + ε.

Invariance principle for the record times, conditionally on the signs

We pass to the study of (τ n t ) t≥0 the rescaled process of the record times of the PRW as defined by (2).

We have seen in the previous subsection the convergence in law of the rescaled record signs (on compact sets away from 0). By the Skorohod representation Theorem, we could suppose -concerning the properties that involve the law-that this convergence holds almost surely. In that case, we want to show the convergence of the conditional law of (rescaled) record times knowing these (rescaled) record signs. Let us introduce some notation. Let (x n (t)) t>0 be a sequence of càdlàg functions taking their values in {-1, +1} which converges to a function (x(t)) t>0 with respect to the Skorohod topology when t varies in any compact sets of (0, +∞). We set, for 0 ≤ s < t, D(x; s, t) = {r ∈ (s, t); x(r-) = x(r)} for the set of discontinuities of x(•) and similarly D(x n ; •, •) for x n (•). We suppose that D(x n ; 0, +∞) ⊂ 1 n N for all n and that D(x; s, t) is finite for all 0 < s < t < +∞.

Proposition 5 As n → +∞, the conditional law of (τ n t ) t≥0 knowing (χ n t ) t>0 = (x n (t)) t>0 converges weakly to the law of a process (τ (x) t ) t≥0 -defined conditionally on (x(t)) t>0 -such that it has independent non-negative increments with distribution given, for 0 < s < t, by the Laplace transform

E e -µ 2 2 (τ (x) t -τ (x) s ) =   r∈D(x;s,t) µ r sinh(µ r)   × exp t s δ(x(u)) 1 u -µ coth µu du (10)
where

δ(y) = β 1 {y=1} + γ 1 {y=-1} .
In the case of the SPRW (i.e. β = γ), this formula simplifies into

E e -µ 2 2 (τ (x) t -τ (x) s ) = sinh(µ s) µ s β   r∈D(x;s,t) µ r sinh(µ r)   µ t sinh(µ t) β . (11)

Invariance principle for the record times of the SPRW

Our goal is to state an invariance principle for (τ n t ) t≥0 without any conditioning. But the same approach as the one leading to Proposition 5 stumbles over a computational difficulty that we will explain later and we are compelled to restrict to the case of symmetric perturbation.

Theorem 6 Let (τ n t ) t≥0 be the rescaled record process of the SPRW (β = γ). Then, as n → +∞, the process (τ n t ) t≥0 converges in law to a process (τ t ) t≥0 with independent non-negative increments whose law is given, for 0 < s < t, by the Laplace transform

E e -µ 2 2 (τt-τs) = cosh( µ 2 s) cosh( µ 2 t) 2 β . ( 12
)
This process has strictly increasing trajectories and is self-similar :

∀a > 0, (τ a t ) t≥0 (d) = (a 2 τ t ) t≥0 . (13)
For any t > 0, the density of τ t on (0, +∞) is a (signed) mixture of 1/2-stable laws :

φ τt : x → 2 2β √ 2π +∞ k=0 -2 β k (β + k) t x 3/2 e -(β+k) 2 t 2 2 x (14)
(using the usual notation ( 19)). Moreover we have the representation

τ t = t 0 R+
x N (ds dx)

where N (ds dx) is a Poisson point measure on R + × R + with intensity f s (x) ds dx where

f s (x) = 2 π 2 s 3 β +∞ n=1 (2n -1) 2 e -(2n-1) 2 π 2 2 x s 2 .

Criterion of positive recurrence for the PRW

From the value of the transition matrix (4), it is clear that the signs of records cannot be asymptotically constant; indeed the infinite product of the diagonal terms of that transition matrix tends to 0. The PRW, whatever its parameters β, γ ∈ (0, +∞), is thus recurrent in the sense that every level is visited infinitely often, as it is the case for the SRW. But compared to the SRW which is null recurrent, the PRW can also become "positive recurrent" if the reinforcement parameters are high enough that is if the process is sufficiently self-attractive. We define the notion of positive recurrence in the non-Markov setting of PRW as follows but note that this property clearly implies that the return time to any level has finite mean. For q ≥ 1, we introduce C q = inf{j > q; χ j = -χ q } as the index of the first record after record q of opposite sign. We say that the PRW is positive recurrent if and only if, for every integer q ≥ 1,

E T Cq -T q < +∞.
Theorem 7 The PRW is positive recurrent if and only if β, γ ∈ (2, +∞).

Number of visited points for the SPRW

The number of visited points of the PRW is the inverse of the record time process :

V k = inf{n ≥ 1; T n > k}.
In the case of the SPRW (β = γ) the invariance principle stated in Theorem 6, with limit process (τ t ) t≥0 , implies an invariance principle for (V k ) k≥1 .

Proposition 8 Let (Y s ) s≥0 be the non-decreasing process defined by Y s = inf{t; τ t > s}. This process has continuous trajectories and is self-similar :

∀a > 0, (Y a s ) s≥0 (d) = ( √ a Y s ) s≥0 . Its marginal laws are ∀s > 0, Y s (d) = 1 τ 1/ √ s (d) = s τ 1 (15)
and, for any s > 0, the variable Y s admits the density on R + given by :

φ Ys (x) = 2 s x 3 φ τ1 s x 2 (16) = 2 2β+1 √ 2π s +∞ k=0 -2β k (β + k) e -(β+k) 2 x 2 2 s . ( 17
)
Moreover, as k → +∞, the process

V [ks] √ k s≥0
converges in law to the process

(Y s ) s≥0 .
As a corollary of the invariance principle for the PRW proved by Davis, we have the representation

Y s = max u∈[0,s] W u -min u∈[0,s] W u (18)
where (W u ) is the perturbed Brownian motion as introduced in (1). Also the process (τ t ) t≥0 obtained in Theorem 6 is the inverse of (Y s ) s≥0 . As a consequence it can be interpreted in terms of the perturbed Brownian motion via the representation (18) of (Y s ) s≥0 given above.

Open questions

Here are two questions we were unable to solve :

• find the law of y 1 ; this will describe how "non-centered" the range can be, asymptotically; • obtain a generalization of Theorem 6 to the case β = γ.

As we will see later, both questions amount to find the value of an infinite matrix product. Alternatively the question about the law of y 1 can also be stated in terms of alternating renewal process.

Preliminary lemmas

In the sequel we will use the classical notation of matrix exponential :

exp[M ] = +∞ k=0 1 k! M k
for any real or complex square matrix M . We also recall the usual notation for generalized binomial coefficients

-β k = k j=1 -β -j + 1 j = (-1) k k! Γ (k + β) Γ (β) (19)
where Γ ( • ) is the classical Gamma function. Let 0 < s < t be fixed. We introduce a notion of approximate equality of two quantities a(k, n) and b(k, n) up to terms of order 1/n 2 by

a(k, n) ≈ b(k, n) ⇔ sup n≥1; [ns]≤k≤[nt] n 2 |a(k, n) -b(k, n)| < +∞. ( 20
)
Denote by M d×d the set of real d×d matrices.

If A(k, n) = (A x,y (k, n)) 1≤x,y≤d and B(k, n) = (B x,y (k, n))
f (k, n) ≈ I + 1 n g k n .
Then

lim n→+∞ [nt] k=[ns] f (k, n) = lim n→+∞ [nt] k=[ns] exp 1 n g k n (21)
provided the limit on the right-hand side exists. Moreover, when g(x) g(y) = g(y) g(x) for all x, y ∈ [s, t], then

lim n→+∞ [nt] k=[ns] f (k, n) = exp t s g(x) dx . ( 22
)
Proof. As all matrix norms induce the same topology we may choose a matrix norm || • || which has the supplementary property that ||A B|| ≤ ||A|| ||B|| for all matrices A, B and, as a consequence, || exp[A]|| ≤ e ||A|| . Also, using this property of the norm, the assumption easily implies that

f (k, n) = exp 1 n g k n + 1 n 2 R(k, n) with sup n≥1 sup [ns]≤k≤[nt] ||R(k, n)|| < +∞.
Then we have

[nt] k=[ns] f (k, n) = [nt] k=[ns] exp 1 n g k n + Rem
where the remainder term is Rem =

[nt]-[ns]+1 j=1 J:#(J)=j [nt] k=[ns] exp 1 n g k n 1 {k ∈J} + 1 n 2 R(k, n) 1 {k∈J} .
But the term in the product on the right-hand side has a norm bounded by e H∞/n if k ∈ J and by

R ∞ /n 2 if k ∈ J where H ∞ and R ∞ are two constants. It follows that ||Rem|| ≤ [nt]-[ns]+1 j=1 [nt] -[ns] + 1 j e H∞ n [nt]-[ns]+1-j R ∞ n 2 j = e H∞ n + R ∞ n 2 [nt]-[ns]+1 -e H∞ n [nt]-[ns]+1 = e H∞ n [nt]-[ns]+1   1 + R ∞ e -H∞ n n 2 [nt]-[ns]+1 -1   .
The first term is bounded and the second one tends to zero ; hence the remainder term converges to zero and we get (21). When the commutation property is satisfied by g, the classical property of the matrix exponential entails Lemma 10 Let (Y n ) n≥1 be a sequence of nonnegative random variables, ν be a probability on R + and denote its Laplace transform by

L ν (µ) = +∞ 0 e -µ 2 2 x ν(dx).
Assume that

lim n→+∞ E 1 cosh(µ/n) Yn = L ν (µ)
uniformly with respect to µ belonging to a compact neighborhood of any positive value.

Then Yn n 2 converges in law to ν. Proof : easy and omitted.

The following elementary lemma is for instance a part of Lemma 12 of [START_REF] Serlet | Hitting times for the perturbed reflecting random walk[END_REF] but we recall it for the convenience of the reader.

Lemma 11 (Time spent in a strip by a SRW) Let (X n ) n≥0 be a SRW started at 1 and ξ be the hitting time of {0, k}. Then

J +1 (k, λ) = E (cosh λ) -(1+ξ) 1 {X ξ =0} = 1 - tanh λ tanh(kλ) (23) and J -1 (k, λ) = E (cosh λ) -(1+ξ) 1 {X ξ =k} = tanh λ sinh(kλ) . ( 24
)
Lemma 12 (Time needed for the PRW to exit the strip of visited points) Let k ∈ {1, 2, 3, . . . } and (X n ) n≥0 be the Markov chain on Z whose transition probabilities (p(x, y); x, y ∈ Z) are given by

• p(x, x + 1) = p(x, x -1) = 1/2 if x ∈ {1, . . . , k -1} • p(0, 1) = γ 1+γ , p(0, -1) = 1 1+γ • p(k, k + 1) = 1 1+β , p(k, k -1) = β 1+β ,
[ • ] = E[ • | X 0 = a] and define, G k (λ, 1, 1) = E k (cosh λ) -ζ 1 {X ζ =k+1} (25)
G k (λ, -1, -1) = E 0 (cosh λ) -ζ 1 {X ζ =-1} (26)
G k (λ, 1, -1) = E k (cosh λ) -ζ 1 {X ζ =-1} (27)
G k (λ, -1, 1) = E 0 (cosh λ) -ζ 1 {X ζ =k+1} . (28)
Then we have

G k (λ, 1, 1) = cosh λ sinh(kλ) + γ sinh λ cosh(kλ) D k (λ) (29)
G k (λ, -1, -1) = cosh λ sinh(kλ) + β sinh λ cosh(kλ)

D k (λ) (30) G k (λ, 1, -1) = β sinh λ D k (λ) (31) G k (λ, -1, 1) = γ sinh λ D k (λ) (32)
where

D k (λ) = sinh(kλ) 1 + (1 + βγ) sinh 2 λ + β + γ 2 sinh(2λ) cosh(kλ). ( 33
)
Proof. Let us start with G k (λ, -1, -1). To simplify notation we set z = 1/(cosh λ). We condition on the value of X 1 to get

G k (λ, -1, -1) = 1 1 + γ z + γ 1 + γ E 1 z 1+ξ+ζ•θ ξ 1 {X ξ+ζ•θ ξ =-1}
where ξ is the duration needed to reach {0, k} and ζ • θ ξ is the duration needed after that time to hit -1 or k + 1. We now use the strong Markov property at the stopping time ξ to get

E 1 z 1+ξ+ζ•θ ξ 1 {X ξ+ζ•θ ξ =-1} = E 1 z 1+ξ 1 {X ξ =0} E 0 z ζ 1 {X ζ =-1} + E 1 z 1+ξ 1 {X ξ =k} E k z ζ 1 {X ζ =-1} = J +1 (k, λ) G k (λ, -1, -1) + J -1 (k, λ) G k (λ, 1, -1)
where J -1 (k, λ) and J 1 (k, λ) are the functions introduced in Lemma 11. So we get a first equation on the G k 's as displayed on the first line below and we add three more equations by similar reasoning :

G k (λ, -1, -1) = 1 1 + γ z + γ 1 + γ J +1 (k, λ) G k (λ, -1, -1) + γ 1 + γ J -1 (k, λ) G k (λ, 1, -1) G k (λ, 1, 1) = 1 1 + β z + β 1 + β J +1 (k, λ) G k (λ, 1, 1) + β 1 + β J -1 (k, λ) G k (λ, -1, 1) G k (λ, -1, 1) = γ 1 + γ J +1 (k, λ) G k (λ, -1, 1) + γ 1 + γ J -1 (k, λ) G k (λ, 1, 1) G k (λ, 1, -1) = β 1 + β J +1 (k, λ) G k (λ, 1, -1) + β 1 + β J -1 (k, λ) G k (λ, -1, -1).
From the second and third equations we derive

G k (λ, -1, 1) = 1 1+β γ 1+γ z J -1 (k, λ) 1 -β 1+β J 1 (k, λ) 1 -γ 1+γ J 1 (k, λ) -β 1+β γ 1+γ J 2 -1 (k, λ)
.

We replace J -1 (k, λ) and J 1 (k, λ) by their explicit values in terms of hyperbolic trigonometric functions. After a few lines of computation, we get (32). Then, by the third equation of the system above we obtain (29). Finally, (31) and ( 30) can be obtained by the substitution γ ↔ β.

Signs of records of the PRW

We start with the proof of Proposition 1. The fact that (χ n ) n≥1 is a (time-inhomogeneous) Markov chain is clear. For n ≥ 1 and x, y ∈ {-1, 1}, the transition probabilities are given by

p n (x, y) = P(χ n+1 = y | χ n = x) = lim λ→0 G n (λ, x, y) (34)
where the G n 's were introduced in (25-28) (for the second equality above apply Lebesgue's dominated convergence Theorem). Using the explicit values given by (29-32), it is straightforward to compute these limits and this completes the proof of (4). We pass to the proof of Proposition 2. The requirements on (x t ) t∈(0,+∞) impose the finite-dimensional marginal laws thus the uniqueness in law of (x t ) t∈(0,+∞) is clear. The existence of this law is a consequence of the standard Kolmogorov extension Theorem, the compatibility condition following from the invariance of the probability defined by (6) for the transition matrices T (s, t).

In order to prove the convergence in law of (χ n t ) t∈[r,A] , for any 0 < r < A, we first show the tightness. Recalling for instance Corollary 7.4 of [START_REF] Ethier | Markov processes; Caracterization and convergence[END_REF], it suffices, in the present context, to show that, for any η > 0, we may find δ > 0 such that for all n large enough, the probability that (χ n t ) t∈[r,A] has 2 jumps separated by less than δ is lower than η. But this probability is lower than

[nA] k=[nr] [nδ] j=1 P (χ k+1 = -χ k , χ k+j+1 = -χ k+j ) ≤ [nA] k=[nr] [nδ] j=1 β ∨ γ β + γ + k β ∨ γ β + γ + k + j . ( 35
)
Using the usual expansion of the partial sums of the harmonic series, we have

[nδ] j=1 β ∨ γ β + γ + k + j ≤ c log 1 + nδ k
so that we can bound the expression (35) above by

c [nA] k=[nr] 1 k log 1 + nδ k ≤ c [nA]/n ([nr]-1)/n 1 x log 1 + δ x dx
and this quantity tends to 0 as δ ↓ 0, uniformly in n large enough. This ends the proof of tightness.

To complete the proof of the Proposition, it suffices to show firstly that, for r > 0, the law of χ n r converges to the law given by (6) which will follow from

lim n→+∞ [nr] k=1 Q k = 1 β + γ β γ β γ (36)
and secondly that the transition kernels also converge that is, for all t > s > 0,

lim n→+∞ [nt] k=[ns] Q k = T (s, t). (37)
We will only prove (37) since (36) is similar. Note that

Q k = I + 1 β+γ+k A where A = -γ γ β -β = Ω 0 0 0 -β -γ Ω -1
with

Ω = 1 γ 1 -β and Ω -1 = 1 β + γ β γ 1 -1 . We deduce that lim n→+∞ [nt] k=[ns] Q k = Ω 1 0 0 L Ω -1 (38)
where

L = lim n→+∞ [nt] k=[ns] 1 - β + γ β + γ + k (39)
but this limit L is easily shown to be (s/t) β+γ , for instance by expressing the product in values of the Gamma function and using the fact that, for a > 0, as x → +∞,

Γ (x + a) Γ (x) ∼ x a . (40)
Finally we check that the matrix product on the right-hand side of (38) is equal to T (s, t) and it concludes the proof.

Record values of the PRW

Let us start with the proof of Corollary 3. We rewrite (3) as

R k = χ k 1 + k 1 1 {χ [u] =χ k } du
Changing the variable in the integral and substituting [nt] for k we get

1 n R [nt] = χ n t 1 n + [nt]/n 1/n 1 {χ n y =χ n t } dy .
For our purpose of convergence in law, by Skorohod representation Theorem and Proposition 2, we can suppose that, almost surely, (χ n t ) converges to (x t ) in the Skorohod topology over every compact of (0, +∞). It follows easily that the right-hand side above converges -again with respect to the Skorohod topology over all compacts of (0, +∞)-to the process (y t ) defined in the statement of the proposition.

We now give a proof of Proposition 4. As was noticed by Formula (3), the sequence (R n ) n≥1 can be reconstructed from the sequence of record signs (χ n ) n≥1 . The idea is to show that there are long sequences of records with the same sign.

Let q be a large integer. For k ≥ 1 we introduce the events

A 1 k = {χ q 4k = 1}, A 2 k = ∀j ∈ [q 4k , q 4k+1 ], χ j = χ q 4k = 1 , A 3 k = ∃!j 0 ∈ (q 4k+1 , q 4k+2 ), χ j0+1 = -χ j0 , A 4 k = ∀j ∈ [q 4k+2 , q 4k+3 ], χ j = χ q 4k+2 = -1 , and A k = A 1 k ∩ A 2 k ∩ A 3 k ∩ A 4 k . The probability of A 1 k converges to γ/(β + γ) as k → +∞. The probability of A 1 k ∩ A 2 k is equal to P(A 1 k ) multiplied by q 4k+1 -1 j=q 4k 1 - β β + γ + j = Γ (β + γ + q 4k ) Γ (γ + q 4k+1 ) Γ (γ + q 4k ) Γ (β + γ + q 4k+1
) .

This term converges to q -β > 0 using (40).

Thus we claim that P(A

1 k ∩ A 2 k ) is bounded from below by a positive constant. Now P(A 1 k ∩ A 2 k ∩ A 3 k ) is equal to P(A 1 k ∩ A 2 k ) multiplied by q 4k+2 -1 j0=q 4k+1 +1 j0-1 j=q 4k+1 1 - β β + γ + j β β + γ + j 0 q 4k+2 -1 j=j0+1 1 - γ β + γ + j
which is bounded from below by a positive constant, by the same arguments as above. Now

P(A k ) = P(A 1 k ∩ A 2 k ∩ A 3 k ∩ A 4 k ) is equal to P(A 1 k ∩ A 2 k ∩ A 3 k ) multiplied by q 4k+3 -1 j=q 4k+2 1 - γ β + γ + j
and repeating once more the same arguments we obtain finally that P(A k ) is bounded from below by a positive constant. We deduce that with positive probability the events A k holds infinitely often. Note that on A k , we have R q 4k+1 q 4k+1 ≥ q 4k+1 -q 4k q 4k+1 = 1 -1 q , R q 4k+3 q 4k+3 ≤ -

q 4k+3 -q 4k+2 q 4k+3 = -1 - 1 q , 0 > R j0+1 j 0 + 1 ≥ -q 4k q 4k+1 = - 1 q , 0 < R j1 j 1 ≤ q 4k+2 q 4k+3 = 1 q ,
where j 1 denotes the time of the first change of sign of χ n after time q 4k+3 . From these remarks we deduce that the statements of Proposition 4 hold with positive probability. To conclude with probability 1 and thus complete the proof, we use the following zero-one law.

Proposition 13 (Zero-one law) Every event in the asymptotic σ-algebra

A(χ) = n σ(χ k ; k ≥ n)
has probability zero or one.

Q k = 1 β + γ β γ β γ
so that both quantities in square brackets in (41) tend to zero as n → +∞. As a consequence P(A|B) = P(A) i.e. A is independent of B. Since this holds for all B of the specified form, in particular for all m, we deduce that A(χ) is independent of the σ-algebra generated by all the variables χ i , i ≥ 1. But this σ-algebra contains A(χ). Hence A(χ) is independent of itself which ends the proof.

6 Record times of the PRW : conditional case Let us prove Proposition 5. We first concentrate on the convergence of finite-dimensional marginal laws. We will give the justification of tightness at the end of section 7.

For x, y ∈ {-1, 1} and k ≥ 1, the quantity

Gk (λ, x, y) = E 1 cosh λ T k+1 -T k χ k = x, χ k+1 = y (42)
is equal to G k (λ, x, y)/p k (x, y) because of (25-28) and (34). For further reference we gather the explicit values in a matrix :

Gk (λ, -1, -1) Gk (λ, -1, 1) Gk (λ, 1, -1) Gk (λ, 1, 1) = k + β + γ D k (λ) • • • • • • × cosh λ sinh kλ+β sinh λ cosh kλ k+β sinh λ sinh λ cosh λ sinh kλ+γ sinh λ cosh kλ k+γ , (43)
recalling the D k (λ) is given by (33). The increments of the record times T k+1 -T k , k ≥ 1 are independent random variables, even conditionally on the record signs (χ k ) k≥1 . It follows that

E 1 cosh λ T [nt] -T [ns] (χ k ) k≥1 = (x n (k/n)) k≥1 = [nt]-1 k=[ns] Gk λ, x n k n , x n k + 1 n = [ns] ≤ k < [nt] x n (k/n) = x n ((k + 1)/n) (sinh λ) k + β + γ D k (λ) × [ns] ≤ k < [nt] x n (k/n) = x n ((k + 1)/n) (k + β + γ) cosh λ sinh(kλ) + δ(x n (k/n)) sinh λ cosh(kλ) D k (λ) k + δ(x n (k/n)) lim n→+∞ k+1 n ∈D(x n ;s,t) ϕ(k/n) = r∈D(x;s,t) ϕ(r).
As a consequence we deduce that, lim n→+∞ k∈D(x n ;s,t)

sinh k µ n k + δ x n k n cosh µ n sinh k µ n + δ x n k n sinh µ n cosh k µ n = r∈D(x;s,t)
µr sinh µr .

Moreover by inspecting all the proof we see that the limits above are uniform for µ varying in any compact neighborhood of a fixed positive value. By Lemma 10, the proof of the convergence of finite-dimensional marginal laws is complete.

7 Record times of the PRW : unconditional case

We start with the general case β, γ ∈ (0, +∞) to see how far we can go before being compelled to restrict to β = γ. The main step is to compute the limit in law of a rescaled increment (T [nt] -T [ns] )/n 2 and, as before, this is done by computing the limit of

E 1 cosh µ n T [nt] -T [ns]
.

We set λ = µ/n and z = 1/ cosh λ, as before. By the repeated use of the Markov property and the definition of G k (•, •, •), we get, for fixed x

[ns] ∈ {-1, 1}, E z T [nt] -T [ns] χ [ns] = x [ns] = x k ∈ {-1, 1} [ns] < k ≤ [nt] E     [nt]-1 k=[ns] z T k+1 -T k   [nt] k=[ns]+1 1 {χ k =x k } χ [ns] = x [ns]   = x k ∈ {-1, 1} [ns] < k ≤ [nt] [nt]-1 k=[ns] G k (λ, x k , x k+1 ) = x [nt] ∈{-1,1}   [nt]-1 k=[ns] G k (λ)   (x [ns] , x [nt] ) (46)
where

G k (λ) = (G k (λ, x, y)) x,y∈{-1,1} is a 2 × 2 matrix. We recall that G k (λ) = cosh λ sinh(kλ) D k (λ) 1 + β tanh λ cotanh(k λ) γ tanh λ sinh(kλ) β tanh λ sinh(kλ) 1 + γ tanh λ cotanh(k λ)
.

Changing λ into µ/n and without conditioning, Formula (46) writes as

E 1 cosh µ/n T [nt] -T [ns] = x∈{-1,1} P(χ [ns] = x) y∈{-1,1}   [nt]-1 k=[ns] G k (µ/n)   (x, y) . (47)
We want to pass to the limit n → +∞. It is easy to see, using again the notation (20), that

G k µ n ≈ I + 1 n H k n where H(x) = -γ µ cotanh(µ x) γ µ sinh -1 (µ x) β µ sinh -1 (µ x) -β µ cotanh(µ x) .
Then we would like to apply Lemma 9 and conclude that lim n→+∞

[nt]-1 k=[ns] G k (µ/n) = lim n→+∞ [nt]-1 k=[ns] exp 1 n H k n
provided the limit on the right-hand side exists. Unfortunately we are not able to prove the existence of the limit in the general case. The problem is that the matrices H(•) do not commute in the general case. We listed this limit as one of the open problems of Section 2.7. But in the particular case of symmetric perturbation β = γ, the matrices H( Moreover we know that the probabilities P(χ [ns] = x), x ∈ {-1, 1} appearing in (47) simply converge in this case to 1/2. Combining all these facts, the passage to the limit in (47) gives = cosh( µ 2 s) cosh( µ 2 t)

2 β
which leads to the desired convergence of finite-dimensional marginal laws, via the usual argument. Now we want to address the problem of tightness of the laws of the processes ((τ n t ) t≥0 , n ≥ 1). We can use for instance the criterion stated in [START_REF] Billingsley | Convergence of Probability Measures[END_REF] Also we omit the proof of Proposition 8 which is identical to the proof of Proposition 2 in [START_REF] Serlet | Hitting times for the perturbed reflecting random walk[END_REF].

Positive recurrence

We now want to prove Theorem 7. By symmetry, it suffices to prove that β > 2 is equivalent to E T inf{j>q; χj =-1} -T q χ q = 1 < +∞.

This (conditional) expectation equals (54)

But, by Proposition 1,

P (χ q+1 = • • • = χ j = 1 = -χ j+1 | χ q = 1) =   j-1 k=q 1 - β β + γ + k   β β + γ + j ( 55
)
and it is easy to see that this quantity is equivalent to C j -β-1 where C is a constant. We discussed at the end of the previous section the procedure to get the conditional means of T k+1 -T k , see (52) and (53). This procedure shows also that, as k, j → +∞, Combining this with the estimate for (55) already obtained, we get that the expression of the (conditional) expectation given by (54) behaves like j j 1-β hence is finite for β > 2 as announced.

  [a,b] converges in law to (x t ) t∈[a,b] , as laws on the space D([a, b], {-1, 1}) of càdlàg functions from [a, b] to {-1, 1} endowed with the Skorohod topology.

  follows immediately by a Riemann sum argument.

Proof.

  Let A belong to A(χ) and take B ∈ σ(χ k ; k ≤ m) of the form B = {χ 1 = x 1 , . . . , χ m = x m } where x 1 , . . . , x m ∈ {-1, 1}. Take any n > m. Since A ∈ σ(χ k ; k ≥ n) we may write P(A | B) -P(A) = P(A|χ m = x m ) -P(A) = P(A|χ n = 1) [P(χ n = 1|χ m = x m ) -P(χ n = 1)] + P(A|χ n = -1) [P(χ n = -1|χ m = x m ) -P(χ n = -1)]. (41) By the same computation as the one leading to (36), we obtain lim n→+∞ P(χ n = -1|χ m = -1) P(χ n = 1|χ m = -1) P(χ n = -1|χ m = 1) P(χ n = 1|χ m = 1)

T

  exp [H(t) -H(s)] 1 1 (48)where, of course, the right-hand side should be read as a product of three matrices. But the matrix H(t) -H(s) has the special formH(t) -H(s) = β [nt] -T [ns]

  k+1 -T k | χ k+1 = χ k = 1)   + E(T j+1 -T j | χ j = 1 = -χ j+1 )    × P (χ q+1 = • • • = χ j = 1 = -χ j+1 | χ q = 1) .

E

  (T k+1 -T k | χ k+1 = χ k = 1) ∼ 2β 3 k and E(T j+1 -T j | χ j = 1 = -χ j+1 ) ∼ j 2 3 .

  1≤x,y≤d belong to M d×d , we extend the previous notion by settingA(k, n) ≈ B(k, n) if and only if A x,y (k, n) ≈ B x,y (k, n) for all x, y.

Lemma 9 Let us suppose that g(•) is a M d×d -valued function which is continuous on [s, t] and f (k, n) is a M d×d -valued function such that

  other transition probabilities being irrelevant for what follows. Let ζ be the hitting time of {-1, k + 1}. We use the notation E a

where δ(x) = γ if x = 1 and δ(x) = β if x = -1. We denote the set of discontinuities of the càdlàg function x n (•) over [s, t] by D(x n ; s, t) = {r ∈ (s, t); x n (r-) = x n (r)}. In order to get the asymptotic behaviour, we regroup the terms and set λ = µ n :

Let us now perform asymptotic expansions up to the order 1/n 2 in the sense of (20). We obtain that

Also we get

recalling that δ(y) = β + γ -δ(y). Then we do similarly for the other terms in (44). Now we use Lemma 9 (1-dimensional case) to deduce that lim n→+∞

To be precise we apply this Lemma a finite number of time, on each interval where δ(x n (•)) is constant and at the limit, reunite all the integrals over these intervals into a single one. The set D(x; s, t) is finite and the convergence of (x n (r)) s≤r≤t toward (x(r)) s≤r≤t with respect to the Skorohod topology implies that, for any continuous function ϕ, Theorem 15.6 which consists, for any T > 0, in finding a nondecreasing continuous function F such that, for all 0 ≤ t 1 ≤ t ≤ t 2 ≤ T and all n large enough,

Note that only the case t 2 -t 1 ≥ 1/n has to be considered, otherwise the left-hand side vanishes. Let us recall that trivially

By the Definition (42) we derive easily

The expressions of Gk (λ, x, y) for x, y ∈ {-1, 1} are explicitly given by (43). So it suffices to differentiate Gk (λ, x, y) and replace every hyperbolic trigonometric function by its Taylor expansion around 0 (up to order 3) to get the value of the limit in (51). The computations are a bit tedious and left to the reader but the important fact is that there exists a constant c such that, for all k,

and

Moreover we have seen that, for a certain (other) constant c,

)/n. Using the independence of the increments, (49) easily follows, with a linear function F (•) and tightness is assured.

Let us come back shortly to the conditional case where we claim that a similar proof of tightness can be constructed on every compact interval of (0, +∞). Indeed the same argument works on a time interval where the signs of the corresponding records are constant. Any time interval [ε, T ] with T > ε > 0 can be decomposed for every n into a (finite) partition such that on each interval of this partition the signs of the corresponding records, given by x n (•) are constant. Because of the convergence in Skorohod topology of x n (•) toward x(•), these partitions converge to the partition ruling the sign of x(•) over [ε, T ]. So the relative compactness of the conditional laws over [ε, T ] can be deduced from the tightness garanteed on each sub-interval of the partition.

The end of the proof of Theorem 6 is similar to the proof of Theorem 1 in [START_REF] Serlet | Hitting times for the perturbed reflecting random walk[END_REF], except the multiplication by a power of 2 from places to places.


Nghi Van

Sébastien Nguyen

Claude-Henri Baguet

Régis Lamarque

Dufour

Lamarque

Van-Nghi Nguyen

Sébastien Baguet

Claude-Henri Lamarque

Régis Dufour

Alternative mass sensing techniques based on nonlinear phenomena in M/NEMS resonators

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

Introduction

Mass sensing is one important application of M/NEMS resonators. The principle of a resonant sensor consists in electrostatically actuating a microbeam on its fundamental mode and in measuring the shift in frequency induced by an external perturbation. For this purpose, the resonator can be driven in the linear or in the nonlinear regime. In the linear regime, the model is simpler, however vibrations are limited to small amplitudes which may not exceed thermo-mechanical noise. One way to improve the sensitivity of the resonator is to improve the signal-to-noise ratio and the quality factor. Another way is to downscale the beam's dimension to the nanoscale [START_REF] Hanay | Single-protein nanomechanical mass spectrometry in real time[END_REF]. A third way consists in working in the nonlinear regime and in using nonlinear phenomena [START_REF] Younis | Exploration of new concepts for mass detection in electrostatically-actuated structures based on nonlinear phenomena[END_REF] such as parametric resonance [START_REF] Zhang | Application of parametric resonance amplification in a single-crystal silicon micro-oscillator based mass sensor[END_REF][START_REF] Thomas | Efficient parametric amplification in microresonators with integrated piezoelectric actuation and sensing capabilities[END_REF], pull-in [START_REF] Khater | Nonlinear phenomena in mems and nems -a micro sensor for measuring minute gas and biological masses[END_REF] or multi-stability and bifurcations [START_REF] Kumar | Modeling, analysis, and experimental validation of a bifurcation-based microsensor[END_REF][START_REF] Nguyen | Bifurcation-based micro/nano-electromechanical mass detection[END_REF]. An alternative detection technique that takes advantage of multi-stability and bifurcations observed in hysteretic cycles is investigated here.

Nonlinear micro/nanobeam-based electromechanical resonator with added mass

The model for the nonlinear electrostatically-actuated clamped-clamped microbeam with added mass of Figure 1 is based on the model developed by Kacem et al. for inertial resonant sensors [START_REF] Kacem | Nonlinear dynamics of M&NEMS resonant sensors: design strategies for performance enhancement[END_REF].The nondimensional equation of motion is given by

∂ 4 w ∂x 4 dx + ∂ 2 w ∂t 2 dx + δ x0 (x)m ∂ 2 w ∂t 2 + c ∂w ∂t dx -N + α 1 1 0 ∂w ∂x 2 dx ∂ 2 w ∂x 2 dx = α 2 V dc + V ac cos(Ωt) 2 (1 -w) 2 dx (1)
where w(x, t) is the nondimensional bending displacement of the beam, m is the ratio between the added mass and the microbeam mass, i.e. m = m added /m beam and δ x0 (x) is the Dirac function used to locate the added mass. The left-hand side of Equation ( 1) represents the inertial, damping, elastic and geometrically nonlinear mechanical forces, whereas the right-hand side stands for the nonlinear electrostatic forces with V dc , V ac , Ω being the DC polarization voltage, the amplitude of the AC voltage, and the excitation frequency respectively. A reduced-order finite-degree-of-freedom model is generated by modal decomposition with the undamped linear mode shapes of the straight microbeam as basis functions for the Galerkin procedure. Response curves of the resonator (see Figure 2) are then computed by the Harmonic Balance combined with the Asymptotic Numerical Method for continuation analysis by means of the ManLab software [START_REF] Kacem | Computational and quasi-analytical models for nonlinear vibrations of resonant mems and nems sensors[END_REF].

g b h x y z V(t) w(x,t) l O l = 40µm g = 0.1µm b = 1.5µm h = 1.5µm Q = 10000 V dc = 100V ac = 1.9V

Mass detection techniques

The softening response curve of the nonlinear resonator with and without added mass is shown in Figure 2. Classical mass sensing relies on the shift in frequency of the response curve in presence of an added mass. However, this shift becomes all the smaller and thus more difficult to detect as the amount of added mass decreases. An alternative bifurcation-based detection technique that exploits nonlinear dynamical phenomena such as the bi-stability of a piezoelectrically actuated microcantilever was recently proposed by Kumar et al. [START_REF] Kumar | Modeling, analysis, and experimental validation of a bifurcation-based microsensor[END_REF]. This technique is illustrated in Figure 2. When operating at a constant excitation frequency Ω op slightly lower than the frequency Ω lim of the fold bifurcation point A lim , the frequency response curve without added mass (red curve) is multi-valued with two stable solutions A 1 and A 2 . When a mass m is added, the response curve is shifted to the left (blue curve) and becomes single-valued at frequency Ω op . As a consequence, a jump from point A 1 to point B is observed, resulting in a sudden increase in amplitude. This detection technique was numerically investigated in [START_REF] Nguyen | Bifurcation-based micro/nano-electromechanical mass detection[END_REF] and it was shown that after the removal of the mass the solution can either return to point A 1 or jump up to point A 2 depending on the amount of added mass. For small mass ratios the jump always occurs towards the upper point A 2 and consequently only a small jump in amplitude from point A 2 to point B happens when another mass arrives on the beam, which is not satisfying. In order to ensure the robustness of this detection technique, the system must always return to its initial stable position A 1 , so that a large jump in amplitude takes place each time a mass hits the resonator. A simple solution for enforcing the reinitialization, based on a sequential decrease and increase of the operating frequency, was proposed in [START_REF] Nguyen | Bifurcation-based micro/nano-electromechanical mass detection[END_REF] but it suffers from many shortcomings. For instance, the instant when this reinitialization has to be operated is unknown. This problem can be overcome by using a slow time-varying frequency sweep such as Ω(t) = Ω op + ∆Ω cos( πt + φ) where << Ω op is the sweep velocity and Ω max = Ω op + ∆Ω and Ω min = Ω op -∆Ω are the frequency sweep boundaries. When t changes, successive sweep-up and sweep-down in frequency are performed. Without added mass, the response goes back-and-forth between 0 and 0 . There is no hysteretic cycle nor associated amplitude jump. When an added mass arrives on the resonator (at point 1) the response goes through the hysteretic cycle shown in Figure 3 according to the path defined by the following points: 1-2/3-4-5-6-7-8/3-4-5-6-7-8/. . . , and there are amplitude jumps from 3 to 4 and from 6 to 7 which can be used for detection and quantification. When the added mass leaves the resonator, the response goes back between 0 and 0 and the jumps disappear. This new mass sensing technique enables automatic reinitialization as well as real-time detection. Moreover, it allows for a fine tuning of the minimal and maximal mass that can be detected, Ω 0 lim -Ω max being the threshold of mass detection for instance. Experimental validation is under progress to obtain a proof of concept and demonstrate the interest of the proposed strategy for ultrasensitive mass sensing.

Conclusions

An alternative mass sensing technique based on the nonlinear phenomena in M/NEMS resonators has been numerically investigated. This detection takes advantage of the bistability and bifurcations of the hysteretic nonlinear responses and permits real-time mass detection and quantification.

Figure 1 :

 1 Figure 1: Geometry and data of the resonant microbeam with added mass.

Figure 2 :Figure 3 :

 23 Figure 2: Shift in frequency of the due to added mass m = 5.10 -5 and jump phenomenon from A1 to B.


M Zaki Dakel

Sébastien Baguet

Régis Dufour

Prediction of the non-linear dynamic behavior of on-board rotors under extreme solicitations

The implicit Newmark time-step integration algorithm combined with the Newton-Raphson iterative technique is used to solve the equations of motion. When sinusoidal motions of the support lead to periodic (harmonic and sub-harmonic) responses of the rotor, the shooting method is optimized and employed to solve the equations of motion.


Mélyssa Cambronel
email: melyssa.cambronel@etu.univ-rouen.fr

Sophie Rodrigues

Florian Fécamp

Olivier Maillot

Marc Feuilloley

Nathalie Connil

 Conclusion

 Introduction

Nowadays, hospital-acquired infections are a huge problem of public health in the world. According to the last InVS report, several micro-organisms can induce these infections as Escherichia coli (for 23.6% of the nosocomial cases), Staphylococcus aureus (13.8%), Enterococcus faecalis (6.5%), Pseudomonas aeruginosa (6.3%), and Klebsiella pneumonia (5.6%). E. faecalis has been incriminated in endocarditis, bacteremia and urinary tract infection. It is also a bacterium which belongs to the commensal human gut microbiota, where a multitude of molecules are secreted and needed for the proper functioning of the body. Bacteria are constantly in contact with these substances, some of them can respond to these signals, by adapting their physiology and pathogenicity. This is the case of the bacteria Escherichia coli O157:H7, which is able to modulate its capacity to form biofilm and its mobilities as well as Vibrio harveyi, under exposure to epinephrine and/or norepinephrine. Bacteria are able to respond to these molecules thanks to two component systems, like QseC/QseB in E. coli O157:H7 or the BasS/BasR system in Salmonella Typhimurium. Here, we study the effect of epinephrine on growth, biofilm formation, adhesion on abiotic and biotic surfaces, hydrophobicity, autoaggregation and cytotoxicity of E. faecalis MMH 594.

Supported by :

 Methods 1 -> The capacity of MMH 594 to adhere on abiotic surface is increased upon exposure to 10 µM and 1 µM of epinephrine. Epinephrine does not modify its capacity to form biofilm or its adhesion towards the intestinal Caco2/TC7 cell line.

Physiological modulations of the pathogen Enterococcus faecalis MMH594 by epinephrine

Epinephrine 100 , 10, 1 and 0,1 µM This study showed that epinephrine can impact some aspects of Enterococcus faecalis MMH 594. One micromolar of epinephrine does not modify its growth curve but can modulate its capacity to adhere on abiotic surface and its cytotoxicity towards the intestinal cell line Caco2/TC7. Epinephrine can also, at a concentration of 10 µM, increase its adhesion to abiotic surface. However, the biofilm is not impacted upon exposure to various concentrations of epinephrine as well as hydrophobicity and autoagreggation. Complementary studies like biofilm including an adhesion step are necessary, along with qRT-PCR of genes involved in adhesion.

The strain MMH 594 contains many putative two component systems (TCS) that may be interesting candidates to identify the sensor able to perceive epinephrine. Molecular docking and mutants construction could help to find the sensor of epinephrine in this particular strain.

  . et al., 2012 "Pathogen espionage: multiple bacterial adrenergic sensors eavesdrop on host communication systems" Molecular Microbiology, vol. 87, no. 3, p455-465 Lesouhaitier O. et al., 2009 "Gram-negative bacterial sensors for eukaryotic signal molecules" Sensors, vol. 9, no. 9, p6967-6990 Bansal T. et al., 2007 "Differential effects of epinephrine , norepinephrine, and indole on Escherichia coli O157:H7 chemotaxis, colonization, and gene expression" Infection and Immunity, vol. 75, no. 9, p4597-4607 Yang. Q et al., 2014 "Norepinephrine and dopamine increase motility, biofilm formation, and virulence of Vibrio harveyi" Frontiers in Microbiology, vol. 5Enquête de prévalence des infections nosocomiales et des traitements anti-infectieux en établissement de santé, France, mai-juin 2012 , Institut de veille sanitaire Enquête de prévalence des infections nosocomiales et des traitements anti-infectieux en établissement de santé, France, mai-juin 2017 , Institut de veille impact of 1 µM of epinephrine on these parameters.Parameters related to the capacity of adhesion of E. faecalis.

two component systems (TCS) homologous to epinephrine's sensor

					-> Exposure to 1
					µM of epinephrine
					does not modify
					the growth curve
					of the strain
					E. faecalis MMH
					594.
		120%
		100%
	control)	80%
	of
	(%	60%
	cells
	Viable	40%
		20%
		0%
		Control	Triton	MMH 594	MMH 594 / 1µM Epi

 Putative