L Khemakhem
email: khemakhemh@yahoo.fr

A Maalej

A Kabadou

A Ben Salah

A Simon

And M Maglione

Modulation of Relaxor Behaviour by Chemical Substitution in the System Ba 1-x Ca x Ti 1-y (Zn 1/3 Nb 2/3 ) y O 3

Keywords: Dielectric, ferroelectric, relaxors and modulation

This paper reports the structural and dielectric properties of Ba 1-x Ca x Ti 1-y (Zn 1 / 3 Nb 2 / 3 ) y O 3 ceramics with y is fixed to 0.2 and x = 0.2 and 0.3. Single-phase solid solutions of the simples were determined by X-ray diffraction. For the solid solution, BaTi 1-y (Zn 1 / 3 Nb 2 / 3 ) y O 3 , the evolution of dielectric behaviour from a sharp ferroelectric peak (for y < 0.075) and to ferroelectric relaxor (for 0.075 < y < 0.2) was observed with increasing of Zn and Nb concentration but in this case the temperature of the maximum of dielectric permittivity T m and T m are lowers . To modulate this relaxor character we have substitute the Ba by Ca to increase T m and T m .

Introduction

The recent discovery on the very high piezoelectric and electro-strictive strain properties in lead-based relaxor ferroelectric single crystals [START_REF] Kuwata | Dielectric and Piezoelectric Properties of 0.91Pb(Zn 1/3 Nb 2/3 )O 3 -0.09PbTiO 3 Single Crystals[END_REF][START_REF] Park | Ultrahigh strain and piezoelectric behavior in relaxor based ferroelectric single crystals[END_REF][START_REF] Paik | E-field induced phase transition in 001-oriented rhombohedral 0.92Pb(Zn 1/3 Nb 2/3 )O 3 -0.08PbTiO 3 crystals[END_REF][START_REF] Liu | Electric field dependence of piezoelectric properties for rhombohedral 0.955Pb(Zn 1/3 Nb 2/3 )O 3 -0.045PbTiO 3 single crystals[END_REF] has stimulated a renewed interest in searching for high strain behaviour in environmentally friendly lead-free materials. BaTiO 3 (BT), is one end member of the industrially interesting solid-solution series. The materials with a perovskite structure of general formula ABO 3 where A is a mono or divalent ion, B is a tri, tetra or pentavalent ion have been found to be very useful and interesting for different solid-state devices [START_REF] Aydi | New ferroelectric and relaxor ceramics in the mixed oxide system NaNbO 3 -BaSnO 3[END_REF][START_REF] Khemakhem | Relaxor or classical ferroelectric behavior in ceramics with composition Ba 1-x Na x Ti 1-x Nb x O 3[END_REF]. Much attention has been given to the interests of perovskite compounds with disordered cations (complex perovskite-type) [START_REF] Uchino | Relaxor ferroelectric devices[END_REF].

Lead-free compositions could be of great interest for environmentally friendly applications. BaTiO 3 , one of the displacive-type ferroelectric materials, is a typical ABO 3 type simple perovskite oxide. Substitution of other ions for host cations at A or B site in BaTiO 3 leads to remarkable changes in various characteristics [START_REF] Bahri | Dielectric and Raman studies on the solid solution (1-x)BaTiO 3 /xNaNbO 3 ceramics[END_REF][START_REF] Abdelmoula | Structural and dielectric studies of relaxor ferroelectric Ba 1-x La x(1-y)/2 Eu xy/2 Na x/2 TiO 3 ceramics[END_REF][START_REF] Burfoot | Polar dielectrics and their applications[END_REF][START_REF] Lines | Principles and applications of ferroelectrics and related Materials[END_REF].

However, lead is a toxic element and with increasing demands of the environment protection, lead-free materials are highly desirable. Their dielectric behaviour is dependent on the chemical composition and statistical distribution of B-site cations in the perovskite lattice [START_REF] Randall | Nanostructural-properties relation in complex lead perovskites[END_REF]. In recent years there has been considerable interest in the development of a Ba(Zn 1/3 Nb 2/3 )O 3 (BZN) based system [START_REF] Varma | Effect of dopants on microwave dielectric properties of Ba(Zn 1/3 Nb 2/3 )O 3 ceramics[END_REF]. Onada et al. [START_REF] Onada | Ba(Zn 1/3 Nb 2/3 )O 3 -Sr(Zn 1/3 Nb 2/3 )O 3 Solid Solution Ceramics with Temperature-Stable High Dielectric Constant and Low Microwave Loss[END_REF] and Endo et al. [START_REF] Endo | Dielectric Properties of Ceramics in Ba(Co 1/3 Nb 2/3 )O 3 -Ba(Zn 1/3 Nb 2/3 )O 3 Solid Solution[END_REF] respectively investigated the effects of mixed Ba/Sr and Zn/Co sites in BZN and high lighted the versatility and tuning capabilities of the BZN system.

In our last works, we have studied the system BaTi 1-y (Zn 1/3 Nb 2/3 ) y O 3 (BTZN) to characterize the relaxor behaviour. In the present work, and to modulate the relaxor character in the region (0.075 < y < 0.2), we have study the Ca ion modified BTZN and we have choose the composition y = 0.2 because it have the lowest T m .

Experimental

Ceramic sample with the chemical formula of Ba 1-x Ca x Ti 1-y (Zn 1/3 Nb 2/3 ) y O 3 (BTZN) was prepared using a conventional solid state reaction method by reacting the stoechiometric proportions of high purity (up 99%) BaCO 3 , CaCO 3 , ZnO, Nb 2 O 5 and TiO 2 powders. The mixture was ball-milled, dried and then calcined at 1100 • c for 15h under oxygen atmosphere and cooled to room temperature. The calcined pellets were reground, pressed and sintered at 1300 • C for 2 h.

X-ray diffraction analysis at room temperature was carried out on a Phillips diffractometer using MoK α radiation (λ = 0.710690 Å) in the angle range 5 ≤ 2θ ≤ 60 • with 10s counting time for each step of 0.02 in order to determine the structure for this ceramic composition. The dielectric measurements were performed under dry helium on ceramic disc, after deposition of gold electrodes on the circular faces by cathodic sputtering, using a Wayne Kerr 6425 component analyzer.

X-ray Diffraction Analysis

Figs. 1 and2, show the X-ray diffraction pattern of some compositions of the system Ba 1-x Ca x Ti 0.8 (Zn 1/3 Nb 2/3 ) 0.2 O 3 with x = 0.2 and 0.3 taken at room temperature which reveal that ceramics are single phased. All the reflection peaks of the X-ray profile were indexed, and lattice parameters were determined using a global profilemaching method with the fullprof software [START_REF] Onada | Ba(Zn 1/3 Nb 2/3 )O 3 -Sr(Zn 1/3 Nb 2/3 )O 3 Solid Solution Ceramics with Temperature-Stable High Dielectric Constant and Low Microwave Loss[END_REF]. In table 1, we present all crystallographic parameters.

Dielectric Studies

Before measurements, the samples were first heated at 473K for 4h under vacuum (This heat treatment was used to eliminate, as much as possible, the water content in the ceramic pores). The evolution of the real (ε r ) part of the complex permittivity for BaTi 1-y (Zn 1/3 Nb 2/3 ) y O 3 ceramics sample as a function of both temperature and frequency shows three different dielectric behaviours.

For 0 ≤ y ≤ 0.05, the sample is of normal ferroelectric behaviour. We note that there is no significant frequency dispersion for T < T c .

For 0.075 ≤ y ≤ 0.2 and 0.75 ≤ y ≤ 0.975, there is only one broad peak in the thermal evolution of ε r at T m . A frequency dispersion occurs for T < T m with a shift of T m with frequency [START_REF] Khemakhem | New relaxor ceramic with composition BaTi 1-x (Zn 1/3 Nb 2/3 ) x O 3[END_REF]. This frequency dispersion is the highest at y = 0.95. In the region 0.075 ≤ y ≤ 0.2, the relaxor character is not pronounced T m and T m are lowers and the lowest Tm is for the composition y = 0.2 (Fig. 3). So that, we have consider this composition as reference in the present study. To modulate this behaviour we have substitute Ba 2+ by Ca 2+ in the system Ba 1-x Ca x Ti 0.8 (Zn 1/3 Nb 2/3 ) 0.2 O 3 (BTZN). Figs. 4 and5 show the temperature dependencies of the real part of permittivity ε r at different frequencies of samples with compositions x = 0.2 and x = 0.3 respectively. Diffuse transition with frequency dispersion above and around the ε r peak were observed. The temperature and frequency variations of ε r were also specific: the temperature T m of ε r maximum was shifted to higher values on increasing frequency. The evolution of the frequency dispersion was here characterized by an increase in ε r when frequency increased. This shows that these compositions are of a relaxor type. We remark that T m presents an evolution to higher temperatures. This behaviour can be considered as important for applications.

The relaxor characteristics for these compositions are reported in Table 2 as the T m shift, T m = T m (10 5 Hz) -T m (10 2 Hz) and the frequency relative dispersion ε r /ε r = ε r (10 2 Hz) -ε r (10 5 Hz)/ε r (10 2 Hz).

It can be noted that the partial substitution of Ba 2+ by Ca 2+ in Ba 1-x Ca x Ti 1-y (Zn 1/3 Nb 2/3 ) y O 3 has considerable effects: apparition of the relaxor behaviour and increase of T m . The degree of disorder of the Ba 1-x Ca x Ti 1-y (Zn 1/3 Nb 2/3 ) y O 3 was evaluated using an empirical formula developed by Uchino and Nomura [START_REF] Uchino | Critical exponents of the dielectric constants in diffused-phase-transition crystals[END_REF]:

1 ε r - 1 ε r max = (T -T ε r max ) γ c
where C is a constant and γ is a measure of diffuseness; γ value is between 1 and 2. The parameters γ give information on the character of the phase transition and only depending on the composition of the specimens. The limiting value γ = 1 and γ = 2 reduce the equation to Curie-weiss law for the case of normal ferroelectric and for ideal relaxor ferroelectric respectively [START_REF] Viehland | The glassy behavior of relaxor ferroelectrics[END_REF].

The logarithmic plots related to this equation for all doped compositions (x = 0.2 and x = 0.3) are shown in Fig. 6.

The slope of the fitting is used to determine the value of γ at 10 kHz. These parameters are γ = 1.909 and γ = 1.9175 respectively for x = 0.2 and x = 0.3.

Conclusion

Ceramic samples of calcium Ca doped barium Ba with general composition Ba 1-x Ca x Ti 1-y (Zn 1/3 Nb 2/3 ) y O 3 have been elaborated by solid state reaction method. It was shown by X-ray diffraction at room temperature that the structure of the compounds is of perovskite type. Dielectric measurements on the two compositions show that the phase

Table 2

Values of T m , T m and ε r /ε r characteristics of the relaxor behavior of some Ba 1-x Ca x Ti 1-y (Zn

Figure 1 .

 1 Figure 1. X-ray spectrum for the composition x = 0.2 of Ba 1-x Ca x (Zn 1/3 Nb 2/3 ) 0.2 Ti 0.8 O 3 solid solution at room temperature.

Figure 2 .

 2 Figure 2. X-ray spectrum for the composition x = 0.3 of Ba 1-x Ca x (Zn 1/3 Nb 2/3 ) 0.2 Ti 0.8 O 3 solid solution at room temperature

Figure 3 .

 3 Figure 3. The real part of permittivity evolution as a function of temperature and frequency for Ba 1-x Ca x (Zn 1/3 Nb 2/3 ) 0.2 Ti 0.8 O 3 (x = 0).

Figure 4 .

 4 Figure 4. The real part of permittivity evolution as a function of temperature and frequency for Ba 1-x Ca x (Zn 1/3 Nb 2/3 ) 0.2 Ti 0.8 O 3 (x = 0.2).

Figure 5 .

 5 Figure 5. The real part of permittivity evolution as a function of temperature and frequency for Ba 1-x Ca x (Zn 1/3 Nb 2/3 ) 0.2 Ti 0.8 O 3 (x = 0.3).

2 Figure 6 .

 26 Figure 6. Variation of Ln(1/ε -1/ε max ) as a function of Ln(T-T m ) for x = 0.2 and x = 0.3.

Table 1

 1 Crystallographic parameters of the compositions x = 0.2 and x = 0.3 of the solid solution Ba 1-x Ca x (Zn 1/3 Nb 2/3 ) 0.2 Ti 0.8 O 3

	x	structure	a = b = c( Å)
	0,2	cubic	4.0720(2)
	0,3	cubic	4,0232(2)

  1/3 Nb 2/3 ) y O 3 ceramics and characterizes a relaxor behaviour. The amelioration in the relaxor character introduced by the substitution of barium by calcium represents the originality of this study. This result can be considered as important for applications.

	transition becomes diffuse
		with various compositions
	x	T m (K) at 1kHz	T m (K)	ε r /ε r
	0	112	12	0.05700
	0,2	118	40	0.05833
	0,3	134	42	0.01837

Acknowledgment

The authors would kindly like to thank Dr. Regnault von der Muhll for his technical helps.


I Negreanu

G Gary

D Mohr

Temperature and emissivity measurement in dynamic experiments using an advanced infrared detector array technique

A new thermography method is presented to measure the absolute temperature history in dynamic experiments of very short duration. By superposing a known infrared radiation to the naturally emitted specimen radiation, both the specimen's emissivity and temperature are de termined from the measurements of a custom-made detector array. As a result, this method can be used without prior knowledge of the specimen's optical properties. At the same time, the tempera ture measurement accuracy is substantially improved in comparison with other established methods. Since the specimen's emissivity is measured in addition to the temperature, this method may also be used to study the coupling between the evolution of the optical and mechanical properties under dynamic loading.

INTRODUCTION

Reliable measurements of the temperature history during high strain rate experiments provide an important basis for the development of thermo-mechanical constitutive models of engineering materials. Due to the short duration of high strain rate experiments, infrared techniques have been proposed in the literature to measure the temperature rise during Split Hopkinson Pressure Bar (SHPB) experiments [l-4]. Using a special optical system, the radiation coming from the specimen is recor ded by an infrared detector. The relationship between the recorded signal and the specimen te mperature is typically established through a calibration procedure on undeformed specimens.

As an alternative, the specimen's surface temperature may be calculated based on the detector signal using basic physical relationships. For this calculation, the specimen's surface emissivity need s to be known. The relationship between the surface emissivity and the temperature may be inferre d from the above calibration procedure. However, the emissivity is also affected by surface roug hness changes throughout a thermo-mechanical experiment. To take this effect into account, so me authors performed the calibration procedure using both un � eformed and deformed sp eci mens [START_REF] Kapoor | Determination of temperature rise during high strain rate deformation[END_REF][START_REF] Rittel | Thermomechanical characteri zation of pure polycrystalline tantalum[END_REF]. The results of Kapper and Nemat-Nasser [START_REF] Kapoor | Determination of temperature rise during high strain rate deformation[END_REF] show that deformation-induced sur face alterations may change the temperature estimates by as much as l 5 °C for a total tem peratur e rise of about 90°c. In the present paper, we propose a new experimental method that pro vides both real-time temperature and surface emissivity measurements throughout high strain rate experim ents.

TE MPE RATURE MEASUREMEN T TECHNIQUE

2-I Theoreti cal backgro und

We briefl y outline the theoretical basis of the proposed infrared temperature me � sure � ent tec hni que. The reader is referred to [START_REF] Herve | New infrared technique for both temperature and emissivity measurement[END_REF] for further details. The main feature of the technique 1s the sim ulta neous determination of the surface temperature and surface emissivity. This is achieved through two simultaneous infrared radiation measurements: we measure the radiation coming from the specimen surface with and without the addition of the reflection of the radiation emitted from a well-known source. In the following, we outline the mathematical description of this experiment which yields a system of equations that can be used to identify the temperature and surface emissivity of the specimen based on the radiation measurements. The optical system is built such that the output signal S of an infrared detector is proportional to the energy (rate) of the electromagnetic waves radiated by the specimen. This proportionality factor a depends on the wave length A of the radiation, a= a(l); it is a function of the optical system, notably the sensitivity of the detector, the transmission coefficients of the mirrors and the solid angles of the system. Its dependency can be determined prior to a dynamic experiment as it is does not depend on the specimen properties. For a specific wave length, the radiation coming from a specimen of temperature 0 s corresponds to the sum of the emitted and reflected radiation, S/a = eL 0 (0 s ) + rL 0 (0A)-

(1)
e = e(A, 0 s , <:) and r = r(A, 0 s , <:) denote the wave length and temperature dependent surface emissivity and surface reflection factors, respectively. Furthermore, e and r depend on the deformation E of the specimen. According to the definition of the emissivity of a material, the wave length dependent function L 0 ( 0) is given by Plank's radiation law for a black body

L o (A 0 ) = 2hc 2 i-s (2)
' he ekJ...0 -I with Planck's constant h, Boltzmann's constant k and the speed of light c. The second term of the sum in (1) accounts for the reflection of the radiation coming from the environment of known temperature 0 A• If the specimen is opaque for radiation of wave length A, we have the relationship r = I -e and Eq. I may be rewritten as S/a = eL 0 (0 s ) + (I -e)L 0 (0A)-

As mentioned above, measurements are also performed when the radiation from an additio nal source of temperature 0 E is reflected at the specimen surface. In this case, the signal !lS is superposed to the measurement S,

l:iS / a = A ( 1 -e) L O ( 0 E). (4)
Here, A represents the fraction of the reflection lobe corresponding to the solid angle used for the measurement. A se � ies of experiments has shown that A is independent of the wavelength, but dependent on the specimen surface roughness. It is reemphasized that all other variables used in Eqs. 3 and 4 depend on the wave length.

If the experiments are performed for two distinct wave lengths ..t 1 and ..t 2 , we have 4 distinct ?1 easurements: S, = S(l, 0 5 ) , S 2 = S(2 2 , 0 5 ) , �S, = �S(l 1 , 0 s ) and �S 2 = �S(l 2 , 0 s ). Denot ing the known calibration factors by a 1 = a(2 1 ) and a 2 = a(l 2 ) , we may determine the unknown specimen temperature 0 s and the three unknown coefficients A , e1 = e1 (l 1 , 0 s , e-) and e 2 = e x (l2, 0s, c) for a given strain€ from the following non-linear system of equations:

Experimental set-up

S,/a, = e,L 0 (J,,0s)

+ (I -e i )L 0 (l 1 ,0A) Sif a2 = e2L 0 (A2, 0s) + (I -e2)L 0 (..t 2 , 0 A ) llS,/a, = A(l -e 1 )L 0 (..t 1 , 0 £ ) �Szla2 = A(l -e2)L 0 (l 2 , 0 £ ) ( 5 )
Figures l and 2 show a schematic and photograph of the experimental set-up, respectively. Two 2-color infrared detector units are used to perform four simultaneous measurements. The detectors are cooled with liquid nitrogen to maintain an operating temperature of 77 K. Each detector unit contains an InSb and an HgCdTe detector. The response time of the detectors is of the order of a few nanoseconds and thus much smaller than the characteristic time step of our measurements. DC amplifiers are used with a passing band of up to l MHz. The view angles of the two 2-color detectors are symmetric with respect to the normal n at the surface of the sample. Thus, the radiation detected by each detector unit is modulated by the same spectral emissivity. At the same time, it is assumed that there is no retro-reflection [START_REF] Herve | New infrared technique for both temperature and emissivity measurement[END_REF]. The recording of purely emitted radiation (without any reflected radiation added) requires a view angle that is exactly opposite to the lighting angle of the reflected radiation. Here, we employed a special optical system that is composed of a set of flat and parabolic off-axis gold treated mirrors. Note that the gold surface treatment of the mirror is necessary to ensure a very good reflection over the spectral ranges used for the detection (about 98% in our case).

RESULTS AND DISCUSSION

Two SHPB tests on two polycarbonate samples are performed. The_ SHPB system comprises 12 mm diameter steel bars and a corresponding 0.4 m long striker bar. Using a striker speed of abou t 26 m/s, a strain rate of about 2400 s-1 is achieved. The two samples have the same di mensio ns and mechanical properties, but we intentionally applied a different surface finish duri ng machining. After verifying the specimen equilibrium, we calculated the stress-strain curve from the direct estimates of the output force based stress and the average strain (Figure 3).

The temperature rise during these tests is determined using the proposed infrared method. Figu re 4a shows the corresponding temperature evolution for both experiments. According to our measu rements, the temperature rises by about l 0 °C as the total strain increases from 0 to 0.37. The measured curves feature a high frequency noise of a maximum amplitude of± 3 °C. Note that the curves for both tests lie on top of each other (Figure 4). This indicates that the temperature measu reme nt technique is not affected by the surface roughness of the specimens. For reference, we also calculated the mechanical work done throughout the experiment (smooth black curves). Subs eque ntly, the temperature evolution is estimated based on the assumption that this work is converted into heat. The comparison with the infrared measurements clearly shows that this assumption is not valid for the present material. Moreover, we also plotted the evolution of the emissivity for the specimen of low surface roughness in Figure 4b. The InSb detectors show an increase of the emissivity from about 0.9 to 0.98. In the range of wavelengths of the HgCdTe detectors, the computed change in emissivity is much smaller; it increases from 0.75 to about 0.8. We compared these results with the emissivity measurements for the undeformed specimens. The comparison of these curves suggests that deformation-induced emissivity changes need to be taken into account.

CONCLUSION

A new infrared technique for measuring the temperature evolution during SHPB tests has been presented. The unique feature of this technique is its ability to measure both the temperature and surface emissivity of a specimen throughout a dynamic experiment. This is achieved by using two pairs of infrared detectors of different wavelength along with a well-known source of radiation that is reflected on the specimen surface. The results for a SHPB experiment at a strain rate of 2400/s are shown to illustrate the measurement capabilities of the proposed method.

Figure 1 .

 1 Figure 1. Schematic of experimental set-up.

Figure 2 .

 2 Figure 2. Photograph of optical detector system.

Figure 3 .

 3 Figure 3. (a) Recorded waves, (b) stress-strain curve for both tests at 2400/s.

  ,-----�---,----.-----,---...

Figure 4 .

 4 Figure 4. Thermal measurements: (a) evolution of the temperature as a function of the applied strain, (b) evolution of the emissivity as a function of the strain energy.

Acknowledgements

The authors are grateful to Professor Philipe Herve (Laboratoire d'Energetique et d'Economie d'Energie, Universite Paris X) who outlined the physics of this new method. Thanks are also due to his junior colleague Professor J. Cedelle. Their guidance and advice throughout this study are gratefully acknowledged.


Loïc Chaumont

Thomas Pellas

Creeping of Lévy processes through curves

Keywords: Mathematics Subject Classification. 60G50 Lévy process, bivariate subordinator, creeping, Ornstein-Uhlenbeck process

come

Introduction

A real Lévy process X starting from 0 under P is said to creep through the level x > 0 if (1.1) P(X τ + x = x) > 0 , where τ + x = inf{t : X t > x}. The first study of this property is due to Millar [START_REF] Millar | Exit properties of stochastic processes with stationary independent increments[END_REF] who, according to his own words, gives a 'reasonably complete solution' of the creeping problem, that is conditions bearing on the characteristics of X for (1.1) to hold. It is proved in [START_REF] Millar | Exit properties of stochastic processes with stationary independent increments[END_REF] that if (1.1) holds for some x > 0, then it holds for all x > 0. We will then simply say that X creeps upward. According to the same paper, when X has bounded variation, it creeps upward if and only if it has positive drift. When X has unbounded variation, it creeps upward if it has a Brownian part or if its Lévy measure π satisfies 1 0 xdπ(x) < ∞. It does not creep upward if it has no Brownian part and 0 -1 |x|dπ(x) < ∞. When X has no Brownian part and 1 0 xdπ(x) = 0 -1 |x|dπ(x) = ∞, both possibilities can occur and an analytic criterion in terms of the characteristic exponent of X is given. As pointed out by Vigon [START_REF] Vigon | Votre Lévy rampe-t-il ?[END_REF], this criterion has the disadvantage of involving both the characteristic exponent of X and its Lévy measure and we rarely know both at the same time. This gap is filled in [START_REF] Vigon | Votre Lévy rampe-t-il ?[END_REF] where an integral test only bearing on the Lévy measure is established for a Lévy process with unbounded variation and no Brownian part to creep upward. Let us also mention the work of Rogers [START_REF] Rogers | A new identity for real Lévy processes[END_REF] who deduced some results of Millar. It seems that it was in the latter article that the term 'creeping' first appeared.

According to a remark first made in [START_REF] Millar | Exit properties of stochastic processes with stationary independent increments[END_REF] and then used in [START_REF] Vigon | Votre Lévy rampe-t-il ?[END_REF], the upward ladder height process H of X satisfies H S + x = X τ + x , x > 0, where S + x = inf{t : H t > x}, so that creeping of general Lévy processes can in fact be reduced to that of subordinators. More specifically, X creeps upward if and only if H creeps upward, which holds if and only if the drift d H of H is positive according to Neveu [START_REF] Neveu | Une généralisation des processus à accroissements positifs indépendants[END_REF] and Kesten [START_REF] Kesten | Hitting probabilities of single points for processes with stationary independent increments[END_REF]. These authors also proved that in this case, the renewal measure of the ladder height process admits a continuous and bounded density u(x) and that the creeping probability is then given by (1.2) P(X τ + x = x) = d H u(x) . So the whole creeping problem for a Lévy process X boils down to giving conditions on its characteristics for d H to be positive. Let us also mention a more recent result in this direction by Griffin and Maller [START_REF] Griffin | The time at which a Lévy process creeps[END_REF] who studied the law of the creeping time, that is the law of the first passage time conditionally on the creeping event.

In the present work, we are interested in the probability for Lévy processes to creep through the graph of continuous non increasing functions. We first consider the case of a bivariate subordinator (Y, Z), that is a two dimensional Lévy process issued from 0, whose both coordinates are non decreasing. Let f : (0, ∞) → (0, ∞) be a continuous non increasing function. Our first main result gives an expression of the probability that the path {(Y t , Z t ) : t > 0} crosses the graph {(t, f (t)) : t > 0} in a continuous way, that is

P(Z S = f (Y S ), u 0 < Y S < u 1 )
, where S = inf{t : Z t > f (Y t )}, for all 0 ≤ u 0 < u 1 ≤ ∞. This expression involves the drifts of Y and Z and the renewal function of the process (Y, Z), see Theorem 2.1. In particular, for (Y, Z) to creep through the graph of f with positive probability, it is necessary that at least one of the drifts of Y and Z is positive. This result will then be applied in Theorem 2.2 to the probability for a real Lévy process X to creep at its supremum through the graph of f . More formally, define the first passage time above the curve of f by

T f = inf{t > 0 : X t > f (t)} .
Then we give an explicit expression of the probability P(X T f = X T f = f (T f ), t 0 < T f < t 1 ), for all 0 ≤ t 0 < t 1 ≤ ∞, where X T f = sup t≤T f X t . Our result shows that this probability is composed of two terms. The first one represents the contribution of the accumulation of jumps of the ladder height process. This shows that the function acts as a fixed (horizontal) barrier through which the process creeps upward. Then the second term is due to the accumulation of jumps of the ladder time process. When its drift is positive, that is when 0 is not regular for (-∞, 0), this term represents a kind of creeping forward, as if the function acted as a vertical barrier. Note that our result actually provides the law of T f conditionally on the event {X T f = X T f = f (T f )}. Our only assumption is that the renewal measure of the bivariate upward ladder process has a continuous density. As proved in the present work, it is ensured whenever X t has a bounded density for each t > 0. The latter result can then be applied to the creeping of the Lévy process X conditioned to stay positive. Thanks to a time reversal argument, we obtain in Theorem 2.3 the probability for this process to creep at its last passage time below the function f . The problem of creeping through fixed levels by Ornstein-Uhlenbeck processes is at the origin of our work. Indeed, the process Z t = z + X t -γ t 0 Z s ds, t ≥ 0, γ > 0, z < 0 creeps through y ∈ (z, 0) if and only if X creeps through the adapted, continuous decreasing functional t → y -z +γ t 0 Z s ds. This problem is still an open question. However, when X is stable, it can be reduced to the creeping of X through a deterministic function. Then we prove, as an application of our main result, that Ornstein-Uhlenbeck processes driven by stable processes with index less than 1 always creep through some fixed levels.

We present our main results in the next section. Theorems 2.1, 2.2 and 2.3 will be proved in Sections 5, 6 and 7, respectively. We give some examples for which the creeping probabilities can be computed explicitly in Section 3, and in Section 4 we present the above-mentioned application of our results to the creeping property of stable Ornstein-Uhlenbeck processes.

Main results

2.1. Creeping of bivariate subordinators through curves. A bivariate subordinator (Y, Z) is a possibly killed two dimensional Lévy process starting from 0, whose coordinates Y and Z are non decreasing. We set

Y t = Z t = ∞, for t ∈ [ζ, ∞],
where ζ is the lifetime of (Y, Z). In what follows, the drift of a (univariate) subordinator Y will be denoted by d Y . Moreover, a measure on some subset of R d is said to be absolutely continuous if it is absolutely continuous with respect to the Lebesgue measure on this subset.

Our first theorem concerns the creeping of a bivariate subordinator (Y, Z) through the curve defined by the graph of a continuous non increasing function f . More specifically, we are interested in the probability that the path {(Y t , Z t ) : t ≥ 0} crosses continuously the graph {(u, f (u)) : u ≥ 0} of f . This event can be written as {Z S = f (Y S )}, where S = inf{t : Z t > f (Y t )}, see Figure 1. When it holds, we will say that (Y, Z) creeps through the graph of f . Theorem 2.1. Let (Y, Z) be a bivariate subordinator with semigroup P(Y t ∈ dy, Z t ∈ dz) = p t (dy, dz), t > 0, y, z ∈ [0, ∞). Assume that the renewal measure v(dy, dz) := +∞ 0 p t (dy, dz) dt is absolutely continuous on (0, ∞) 2 and that its density (y, z) → v(y, z) is continuous on (0, ∞) 2 . Let f : (0, ∞) → (0, ∞) be a continuous, non increasing function. Set f (0) = lim t→0+ f (t), f (∞) = lim t→∞ f (t) and define,

S = inf{t : Z t > f (Y t )} .
1. Then almost surely, 0 < S < ∞, Y S-< ∞ and Z S-< ∞. Moreover, the process (Y, Z) can neither jump on the graph {(u, f (u)) : u > 0} nor jump from this graph, that is

P(Z S-= f (Y S-), ∆(Y, Z) S = 0) = P(Z S = f (Y S ), ∆(Y, Z) S = 0) = 0 , where ∆(Y, Z) S = (Y, Z) S -(Y, Z) S-. 2. Let u 0 , u 1 be such that 0 ≤ u 0 < u 1 ≤ ∞. Then the creeping probability of (Y, Z)
through the graph of f is given by,

(2.3) P(Z S = f (Y S ), u 0 < Y S < u 1 ) = d Z u 1 u 0 v(u, f (u)) du -d Y u 1 u 0 v(u, f (u)) df (u) .
3. Assume that the measure p t (dy, dz) is absolutely continuous on (0, ∞) 2 for all t > 0 and that the densities (t, y, z) → p t (y, z) are continuous on (0, ∞) 3 . Then for all t 0 , t 1 , u 0 , u 1 such that 0 ≤ t 0 < t 1 ≤ ∞ and 0 ≤ u 0 < u 1 ≤ ∞,

P(Z S = f (Y S ), u 0 < Y S < u 1 , t 0 < S < t 1 ) = d Z u 1 u 0 t 1 t 0 p t (u, f (u)) dt du -d Y u 1 u 0 t 1 t 0 p t (u, f (u)) dt df (u) . (2.4)
Let us mention that an intermediary result between those of part 2. and part 3. of Theorem 2.1 can be stated as follows:

2 . Assume that for some 0 ≤ t 0 < t 1 ≤ ∞, the measure t 1 t 0 p t (dy, dz) dt is absolutely continuous on (0, ∞) 2 , with density v(y, z) and that v is continuous on (0, ∞) 2 . Then for all u 0 , u 1 such that 0 ≤ u 0 < u 1 ≤ ∞,

P(Z S = f (Y S ), u 0 < Y S < u 1 , t 0 < S < t 1 ) = d Z u 1 u 0 v(u, f (u)) du -d Y u 1 u 0 v(u, f (u)) df (u) .
The proof of 2 . is quite similar to this of part 3. of Theorem 2.1, see subsections 5.3 and 5.4, so it is omitted. (Y, Z) t > a}. This is simply obtained from Theorem 2.1 by choosing f (y) = a 2 -y 2 , so that S = U a .

0 Y S Y t Z t Z S = f (Y S )
Corollary 2.1. Let (Y, Z) be a bivariate subordinator. Then with the same notation and assumptions than in Theorem 2.1,

(2.5) P( (Y, Z) Ua = a) = d Z a 0 v(u, √ a 2 -u 2 ) du + d Y a 0 uv(u, √ a 2 -u 2 ) √ a 2 -u 2 du .
The second direct application concerns the probability for a (one dimensional) subordinator to creep through the graph of a continuous non increasing function. Let X be a subordinator whose semigroup has continuous densities (t, x) → p t (x) on (0, ∞) 2 and let us apply identity (2.3) to the bivariate subordinator (Y t , Z t ) = (t, X t ). The renewal measure of (Y, Z) is then v(t, x) dt dx = p t (x) dt dx. Hence it satisfies the assumption of part 2. of Theorem 2.1. Moreover, since Y t = t, one has Y S = inf{t : X t > f (t)}. This leads to the following corollary.

Corollary 2.2. Assume that X is a subordinator whose semigroup P(X t ∈ dx) is absolutely continuous on (0, ∞) for each t > 0, with continuous densities (t, x) → p t (x) on (0, ∞) 2 . Set T f = inf{t > 0 : X t > f (t)}, where f is as in the statement of Theorem 2.1.

Then for all t 0 and t 1 such that 0 ≤ t 0 < t 1 ≤ ∞,

P(X T f = f (T f ), t 0 < T f < t 1 ) = d X t 1 t 0 p u (f (u)) du - t 1 t 0 p u (f (u)) df (u) . (2.6)
Let us mention that the creeping probability for subordinators with no drift has been determined in [START_REF] Chi | On exact sampling of the first passage event of a Lévy process with infinite Lévy measure and bounded variation[END_REF] when f is differentiable. We will actually extend (2.6) to the creeping probability of any Lévy process at its supremum in the next section.

Note that if f is decreasing, then (2.3) can also be written as

P(Z S = f (Y S ), u 0 < Y S < u 1 ) = d Y u 1 u 0 v(f -1 (u), u) du -d Z u 1 u 0 v(f -1 (u), u) df -1 (u) ,
and this applies to (2.5) and (2.6).

The question of finding necessary and sufficient conditions for a bivariate subordinator (Y, Z) to creep through a given continuous, non increasing function is still open. In particular, nothing is known when the renewal measure v(dy, dz) is not absolutely continuous. Looking at (2.3) in Theorem 2.1, one is tempted to think that when d Y > 0 and d Z > 0, the probability P(Z S = f (Y S )) is always positive and when d Y = 0 and d Z = 0, this probability is always equal to 0. Let us also mention that the absolute continuity condition of Theorem 2.1 is discussed in Subsection 5.6.

It seems hardly possible to obtain an expression of the creeping probability when f is a general continuous function. When f is continuous and non decreasing we can still obtain an upper bound for the creeping probability as the following proposition shows. Proposition 2.1. Let (Y, Z) be a bivariate subordinator which satisfies the same assumptions as in part 2. of Theorem 2.1. Keep the same notation as in this theorem. Let f : (0, ∞) → (0, ∞) be a continuous non decreasing function such that lim t→0+ f (t) > 0. Set f (0) = lim t→0+ f (t), f (∞) = lim t→∞ f (t) and define S = inf{t : Z t > f (Y t )}.

1. Then P(S > 0) = 1 and the process (Y, Z) can neither jump on the graph {(u, f (u)) :

u > 0} nor jump from this graph, that is

P(Z S-= f (Y S-), ∆(Y, Z) S = 0, Y S < ∞) = P(Z S = f (Y S ), ∆(Y, Z) S = 0, Y S < ∞) = 0 . 2. Let u 0 , u 1 be such that 0 ≤ u 0 < u 1 ≤ ∞.
Then the creeping probability of (Y, Z) through the graph of f is bounded from above as follows,

(2.7)

P(Z S = f (Y S ), u 0 < Y S < u 1 ) ≤ d Z u 1 u 0 v(u, f (u)) du .
Proposition 2.1 implies that when d Z = 0, the process (Y, Z) never creeps through the graph of any continuous non decreasing function. This result and its consequence in Proposition 2.2 will be useful for our application to Ornstein-Uhlenbeck processes in Section 4.

2.2.

Creeping of real Lévy processes at their supremum. Let us now consider a non killed real Lévy process X. We will always assume in the sequel that X 0 = 0, a.s. Since our interest lies in creeping upward of X, it is natural to assume throughout this paper that -X is not a subordinator. Moreover the case of subordinators has already been dealt with in Corollary 2.2. Therefore in this subsection, we will assume that |X| is not a subordinator. We write X for the supremum process, that is,

X t = sup s≤t X s , t ≥ 0 .
Then let us first recall a few basic definitions. It is well known that the reflected process X -X is strongly Markovian under P. Let L be the local time at 0 of X -X (a proper definition is given in Subsection 6.1) and denote by (τ, H) the upward ladder process of X, that is the bivariate Lévy process whose coordinates are the following (possibly killed) subordinators: τ t = inf{s : L s > t} and H t = X τt , t ≥ 0 , where τ t = H t = ∞, for t ≥ L ∞ . The process τ (resp. H) is called the upward ladder time (resp. height) process of X. We will denote by d τ and d H their respective drift coefficients. Let us introduce the renewal measure U on [0, ∞) 2 of the ladder process (τ, H), that is,

U (dt, dh) = ∞ 0 P(τ u ∈ dt, H u ∈ dh) du .
We specify that when the measure U (dt, dh) is absolutely continuous on (0, ∞) 2 , its density will be denoted by q * t (h). This notation may seem unnatural but it comes from the fact that this density corresponds to the entrance law of the reflected excursions, see Subsection 6.1, which is thus noted in older references, see [START_REF] Chaumont | On the law of the supremum of Lévy processes[END_REF] and [START_REF] Chaumont | On the asymptotic behavior of the density of the supremum of Lévy processes[END_REF] for instance.

Throughout this paper, for any function f : (0, ∞) → R, we set

T f := inf{t > 0 : X t > f (t)} .
When a Lévy process X satisfies P(X T f = X T f = f (T f ), t 0 < T f < t 1 ) > 0, we will say that X creeps at its supremum through the function f over the interval (t 0 , t 1 ). Omission of (t 0 , t 1 ) will simply mean that X creeps over the whole half line (0, ∞). The following result gives an expression of the probability of this event when f is a continuous, non increasing function.

Theorem 2.2. Let X be a real Lévy process and f : (0, ∞) → (0, ∞) be a continuous, non increasing function.

1. Then P(T f > 0) = 1 and the process X can neither jump on the graph {(u, f (u)) : u > 0} nor jump from this graph, that is

P(X T f -= f (T f ) < X T f , T f < ∞) = P(X T f -< X T f = f (T f ), T f < ∞) = 0 . 2.
Assume that the renewal measure U (dt, dx) of the ladder process (τ, H) has a continuous density (t, x) → q * t (x) on (0, ∞) 2 . Then X creeps at its supremum through f if and only if its ladder process (τ, H) creeps through the graph of f . Moreover, for all t 0 and t 1 such that 0 ≤ t 0 < t 1 ≤ ∞, (2.8)

P(X T f = X T f = f (T f ), t 0 < T f < t 1 ) = d H t 1 t 0 q * u (f (u)) du -d τ t 1 t 0 q * u (f (u)) df (u) .
The creeping probability of part 2. of Theorem 2.2 reveals two types of creeping at the supremum, each of which corresponds to one of the two terms involved. When d H > 0, the process creeps upward either continuously (in absence of positive jumps) or through an accumulation of jumps of H as in the case of a fixed barrier. In the case where d τ > 0 (which is equivalent to the fact that 0 is not regular for (-∞, 0)), another type of creeping occurs. Then the process creeps forward through an accumulation of jumps of τ provided f decreases at the creeping time. Recall that the later jumps correspond to the lengths of the excursions of X -X. This new kind of creeping happens as if f were acting as a vertical barrier.

0 T f X Tf = X Tf = f (T f ) Figure 2.
A sample path of X creeping at its supremum through f

We point out that the assumption of absolute continuity in part 2. of the above result is not as strong as it may appear. Indeed, we will see in Proposition 6.1, that it is satisfied whenever the transition semigroup of X admits densities, x → p t (x), x ∈ R, which are bounded for all t > 0, and for all c ≥ 0, the process (|X t -ct|, t ≥ 0) is not a subordinator. Moreover, from the same proposition, q * t is positive on (0, ∞), for all t > 0, which ensures the positivity of the creeping probability (2.8) whenever d H > 0 or d τ > 0 and f is decreasing. On the other hand, when the assumption of Theorem 2.2 is not satisfied, we can still expect that the process creeps at its supremum through f with positive probability, but finding necessary and sufficient conditions for this to hold is an open question which is closely related to the question raised by Theorem 2.1, see the discussion after Corollary 2.2.

Recall from

(3.3) in [4] that the function x → u(x) = ∞ 0 q * t (x)
dt is the potential density of H (this is true by definition when X is a subordinator since q * t (x) = p t (x)). Hence by taking f ≡ x in the statement of the above theorem, as well as in Corollary 2.2, we recover the classical creeping result for Lévy processes recalled in the introduction.

When P(X T f = f (T f ), t 0 < T f < t 1 )
> 0 we will say that X creeps through the function f (over the interval (t 0 , t 1 )). Finding an expression for the probability P(X

T f = f (T f ), t 0 < T f < t 1 ) seems more complicated than for P(X T f = X T f = f (T f ), t 0 < T f < t 1 )
. Indeed, in the second case, we only need to ensure that X stays below its past supremum before time T f , whereas in the first case the condition is that the whole path of X stays below the curve of f before time T f . Therefore, the expression of this probability must strongly depend on the behaviour of whole paths of X with respect to the curve of f before time T f and it seems hardly possible to perform this computation for a general function. However the example below and Corollary 2.3 give conditions for X to creep through a function f in some particular cases.

Let X be the Lévy process X t = S t -at, t ≥ 0, where S is a subordinator without drift and a > 0. Then 0 is not regular for (0, ∞) and it is regular for (-∞, 0), so that d H = d τ = 0 (see Subsection 6.1) and from Theorem 2.2, the process does not creep at its supremum through any continuous non increasing function. However, it does creep through the function f (t) = 1 -t, since for any a < 1, the subordinator with positive drift S t + (1 -a)t creeps through the level 1. This simply means that the creeping time of X through f is a.s. not a time at which X reaches its past supremum. From this example, we can build a more general result regarding the probability to creep through a function (not necessarily at the supremum) of Lévy processes with bounded variation.

Corollary 2.3. Let X be a Lévy process with bounded variation and nonnegative drift. Assume that for all c ≥ 0, the process (|X t -ct|, t ≥ 0) is not a subordinator and that for all t > 0, the distribution P(X t ∈ dx) is absolutely continuous on R with a bounded density. Let f : (0, ∞) → R be a function such that for some 0 ≤ t 0 < t 1 ≤ ∞ and a > 0, f (t) + at is positive, continuous and non increasing over the interval (t 0 , t 1 ). Then X creeps through f over the interval (t 0 , t 1 ), that is

P(X T f = f (T f ), t 0 < T f < t 1 ) > 0 .
The above corollary will be useful for our application to the creeping of Ornstein-Uhlenbeck processes in Section 4.

From Theorem 2.2, for X to creep at its supremum through f , it is necessary that d H > 0 or d τ > 0. In particular, if 0 is not regular for [0, ∞), then d τ = d H = 0 (see Subsection 6.1) and X never creeps at its supremum through a continuous non increasing function. On the other hand, when X has unbounded variation, d τ = 0 and hence the process can creep at its supremum through f only if d H > 0. We conjecture that when X has unbounded variation and d H = 0, then X never creeps through any continuous, non increasing function. Indeed, from the results of this section it is reasonable to think that the only chance for a Lévy process X to creep through such a function is that either d H > 0 or X has bounded variation.

We end this subsection with the case of a continuous non decreasing function.

Proposition 2.2. Let X be a real Lévy process and f : (0, ∞) → (0, ∞) be a continuous, non decreasing function such that lim t→0+ f (t) > 0.

1. Then P(T f > 0) = 1 and the process X can neither jump on the graph {(u, f (u)) :

u > 0} nor jump from this graph, that is

P(X T f -= f (T f ) < X T f , T f < ∞) = P(X T f -< X T f = f (T f ), T f < ∞) = 0 .
2. Assume that the renewal measure U (dt, dx) of the ladder process (τ, H) has a continuous density (t, x) → q * t (x) on (0, ∞) 2 . Then for all t 0 and t 1 such that

0 ≤ t 0 < t 1 ≤ ∞, P(X T f = f (T f ), t 0 < T f < t 1 ) ≤ d H t 1 t 0 q * u (f (u)) du .
Proposition 2.2 implies that a Lévy process such that d H = 0 never creeps through a continuous non decreasing function. Note that since f is non decreasing, when the process X creeps through f , it necessarily creeps at its supremum, that is

{X T f = f (T f )} = {X T f = X T f = f (T f )}.

2.3.

Creeping of Lévy processes conditioned to stay positive. Recall that X is a non killed real Lévy process such that X 0 = 0, a.s. Moreover, we assume again that |X| is not a Lévy process. The process X conditioned to stay positive is a Doob h-transform of the process killed at its first hitting time of the negative half-line. Let us briefly recall the definition of this process of which one will find a more complete description in [START_REF] Chaumont | On Lévy processes conditioned to stay positive[END_REF]. Let us set,

τ - -x = inf{t > 0 : X t < -x} , for x ≥ 0 and let h(x) = ∞ 0 P(H * t ∈ [0, x]
) dt be the renewal function of the downward ladder height process H * of X (see Subsection 6.1 for a full definition). Then the process X conditioned to stay positive is a (possibly non conservative) strong Markov process with state space (0, ∞), which we denote by X x,↑ when it is issued from x and whose semigroup is given by

P(X x,↑ t ∈ dy) = 1 h(x) P(x + X t ∈ dy, t < τ - -x ) , x, y > 0 .
It is proved in [START_REF] Chaumont | On Lévy processes conditioned to stay positive[END_REF] that when 0 is regular for (0, ∞) and X, the process X x,↑ converges weakly in the Skohorod's space, as x tends to 0 toward a non degenerate process which will be denoted here by X ↑ . This process satisfies X ↑ 0 = 0, a.s. and if moreover lim sup t→∞ X t = ∞, a.s., then (X x,↑ , x > 0) is conservative and lim t→∞ X ↑ t = ∞, a.s. Let us now define the future infimum process of X ↑ and its last passage time below the graph of a function f respectively by,

X ↑ t = inf s≥t X ↑ s , t ≥ 0 and σ f = sup{t : X ↑ t ≤ f (t)} .
Theorem 2.3. Let X be a real Lévy process such that lim sup t→∞ X t = ∞, a.s. and assume that 0 is regular for (0, ∞). Let X ↑ be the process X conditioned to stay positive and let f : (0, ∞) → (0, ∞) be a continuous, non increasing function.

1. Then 0 < σ f < ∞, a.s. and at time σ f , the process X ↑ can neither jump on the graph {(u, f (u)) : u > 0} nor jump from this graph, that is

P(X ↑ σ f -= f (σ f ) < X ↑ σ f ) = P(X ↑ σ f -< X ↑ σ f = f (σ f )) = 0 .
2. Assume that the renewal measure U (dt, dx) of the ladder process (τ, H) has a continuous density (t, x) → q * t (x) on (0, ∞) 2 . Then for all t 0 and t 1 such that

0 ≤ t 0 < t 1 ≤ ∞, P(X ↑ σ f = X ↑ σ f = f (σ f ), t 0 < σ f < t 1 ) = d H t 1 t 0 q * u (f (u)) du -d τ t 1 t 0 q * u (f (u)) df (u) .
It appears that the creeping probability of X ↑ at its future infimum through the graph of f has the same expression as the creeping probability of X at its past supremum stated in Theorem 2.2. This can be explained by the following time reversal property

[(X T f -X (T f -t)-, 0 ≤ t ≤ T f ) | X T f = X T f ] (d) = [(X ↑ t , 0 ≤ t ≤ σ f ) | X ↑ σ f = X ↑ σ f ] ,
where X 0-= X 0 = 0. The later identity in law is itself a consequence of the proof of Theorem 2.3, see Section 7.

Examples

Illustrating Theorem 2.1 with a bivariate subordinator (Y, Z) such that none of the coordinates Y and Z is a pure drift and whose renewal density measure is explicit seems to be quite challenging. Our first example regards Corollary 2.2, that is the case when one of the subordinators is a pure drift. a) Let X be a stable subordinator with index 1/2. Then for t > 0, X t has density

p t (x) = t √ 2πx 3 e -t 2 2x 1 (0,∞) (x) .
From Corollary 2.2, the distribution of the creeping time is then

P(X T f = f (T f ), T f ∈ dt) = - t 2πf (t) 3 e -t 2 /(2f (t)) df (t) , t > 0 .
Taking f (t) = 1/t 2 gives the creeping probability through f over the half-line (0, ∞) and the distribution of the creeping time conditionally on this event:

P(X T f = f (T f )) = 2 π ∞ 0 te -t 4 /2 dt = 1 2 P(T f ∈ dt | X T f = f (T f )) = 2 2 π te -t 4 /2 dt , t > 0 .
Our second example illustrates Theorem 2.2.

b) Standard Brownian motion creeping at its supremum. When X is standard Brownian motion, 0 is regular for both half-lines (-∞, 0) and (0, ∞) and hence d τ = 0. On the other hand, our normalisation of the local time L (see Subsection 6.1) gives d H = 1/ √ 2 and for t > 0,

q * t (x) = x √ πt 3 e -x 2 /2t 1 (0,∞) (x) .
From part 2. of Theorem 2.2, the distribution of the creeping time at the supremum with respect to any continuous non increasing function f : (0, ∞) → (0, ∞) is given by

P(X T f = X T f = f (T f ), T f ∈ dt) = f (t) √ 2πt 3 e -f (t) 2 /2t dt , t > 0 .
For instance, the probability for the standard Brownian motion to creep at its supremum through the function f (t) = 1/t on (0, ∞) and the distribution of the creeping time T f conditionally on this event are respectively,

P(X T f = X T f = f (T f )) = ∞ 0 1 √ 2πt 5 e -1/2t 3 dt = 1 3
P(T f ∈ dt | X T f = X T f = f (T f )) = 3 √ 2πt 5/2 e -1/2t 3 dt , t > 0 .
A slight generalisation of the above computation can be obtained by considering the case of Brownian motion with drift µ ∈ R. The expression of q * t (x) is then x √ πt 3 e -(x+µ) 2 /2t . The density (t, x) → q * t (x) of the renewal measure can actually be made explicit in very few cases and the few known expressions are too complicated to allow the calculation of the creeping probability.

Actually both examples a) and b) are closely related to each other. Indeed, for standard Brownian motion, the ladder process (τ, H) is such that τ is a stable subordinator with index 1/2 and H is a pure drift process, that is

H t = d H • t, t ≥ 0. So, example b) is also a consequence of Corollary 2.2.
4. Application : creeping of α-stable Ornstein-Uhlenbeck processes Let X be a non killed real Lévy process such that |X| is not a subordinator and X 0 = 0, a.s. The Ornstein-Uhlenbeck process starting from z ∈ R and driven by X is the unique strong solution of the following stochastic differential equation:

(4.9) Z t = z + X t -γ t 0 Z s ds , t ≥ 0 , γ > 0 .
It is explicitly given by Z t = e -γt (z + t 0 e γs dX s ). For x ∈ R, we set T + x = inf{t : Z t > x} and T - x = inf{t : Z t < x} and we say that Z creeps through the state x = z if either x > z and P(Z

T + x = x, T + x < ∞) > 0 or x < z and P(Z T - x = x, T - x < ∞) > 0.
We are interested here in the creeping property of Z, when X is a stable Lévy process with index α ∈ (0, 1). Actually stable Lévy processes never creep across any level. Creeping of an Ornstein-Uhlenbeck process Z driven by a stable Lévy process with index α ∈ (0, 1) is actually due to the smoothing effect of the functional t → t 0 Z s ds, as the proof of the next proposition shows. Recall that when X is stable with any index α ∈ (0, 2], Z fulfils the following representation:

(4.10) (Z t , t ≥ 0) (d) = e -γt (z + X e αγt -1 αγ
), t ≥ 0 , see for instance E 18.17, p.116 in [START_REF] Sato | Lévy processes and infinitely divisible distribution[END_REF]. When moreover X is symmetric and α ∈ (0, 1), it is proved in [START_REF] Jakubowski | On Harnack inequality for α-stable Ornstein-Uhlenbeck processes[END_REF] (see Proposition 12 on p.618) that Z creeps through x ∈ (z, 0) when z < 0 and through x ∈ (0, z), when z > 0. Proposition 4.1 below provides an extension of this result.

Proposition 4.1. Let Z be the Ornstein-Uhlenbeck process solution of (4.9) with γ > 0, starting point z ∈ R and driven by a stable Lévy process X with index α ∈ (0, 1) such that |X| is not a subordinator.

1. If z ∈ (-∞, 0), then Z creeps through x if and only if x ∈ (z, 0). 2. If z ∈ (0, ∞), then Z creeps through x if and only if x ∈ (0, z).

3. If z = 0, then Z does not creep through any level.

Proof. From the representation (4.10), we can write by setting s = (e αγt -1)/(αγ),

T + x = 1 αγ log(αγ inf{s : X s > x(αγs + 1) 1/α -z} + 1) T - x = 1 αγ log(αγ inf{s : X s < x(αγs + 1) 1/α -z} + 1)
, so that Z creeps through x if and only if X creeps through f (s) = x(αγs + 1) 1/α -z.

Let z ∈ (-∞, 0) and x ∈ (z, 0). Then recall that X has bounded variation and zero drift and that for all t > 0, P(X t ∈ dx) is absolutely continuous on R with a bounded density, see Theorem 53.1, p.404 in [START_REF] Sato | Lévy processes and infinitely divisible distribution[END_REF]. Therefore, X satisfies the conditions of Corollary 2.3. On the other hand, note that f (0) = x -z > 0 and that f is continuous. Moreover, f (s) + as is non increasing on (0, ∞) whenever 0 < a < -xγ. Let us take such a value a and note that there is t 1 > 0 such that f (s) + as is positive on (0, t 1 ). Then the result follows from Corollary 2.3.

If z < 0 ≤ x, then f (0) > 0 and f is non decreasing. Since X satisfies d H = 0 (H is a stable subordinator), the result follows from Proposition 2.2.

If x < z < 0 then f (0) < 0 and f is non increasing and the result follows again from Proposition 2.2 by considering -X and -f . The proof of 2. is deduced from this of 1. by considering the Ornstein-Uhlenbeck process -Z which is solution of (4.9) with γ > 0, starting point -z ∈ R \ {0} and driven by the stable process -X.

If z = 0 and x > 0, then f is non decreasing and the result follows again from Proposition 2.2. For x < 0, we derive the result by considering -Z from the same argument as above.

Looking at the previous proof and in accordance with the remarks made in Subsection 2.2, we conjecture that when α > 1, since X has unbounded variation, the process Z never creeps across any level. However, as proved in [START_REF] Chaumont | Markovian Bridges: Weak continuity and pathwise constructions[END_REF], points are not polar for Z when the leading Lévy process is stable with any index α ∈ (0, 2].

Finally let us mention that dealing with the creeping through x ∈ (z, 0) of an Ornstein-Uhlenbeck process Z starting from z < 0 and driven by any Lévy process X would require to study the creeping property of X through the continuous non increasing adapted functional t → x -z + γ t 0 Z s ds, as equation (4.9) shows. This raises the more general question of the creeping of a Lévy process through any adapted, continuous and non increasing functional.

Proof of Theorem 2.1 and Proposition 2.1

Let us note that from monotone convergence, we do not lose any generality by proving Theorem 2.1 in the case where t 0 , u 0 > 0 and t 1 , u 1 < ∞. Therefore in all this section, t 0 , u 0 , t 1 and u 1 will be chosen so that 0 < u 0 < u 1 < ∞ and 0 < t 0 < t 1 < ∞ .

5.1.

Proof of the first assertion in Theorem 2.1. The following lemma actually shows a much stronger result than the first assertion of Theorem 2.1. It will also be used in further results. Lemma 5.1. Let L be any R d -valued Lévy process with infinite lifetime and set ∆L t = L t -L t-, t > 0. Let G : R d → R be any Borel function such that for all x ∈ R d , P(G(L t + x) = 0) = 0, for λ-a.e. t ≥ 0 (λ being the Lebesgue measure on [0, ∞)). Then (5.11)

P (∃ t ≥ 0, G(L t-) = 0, ∆L t = 0) = P(∃ t ≥ 0, G(L t ) = 0, ∆L t = 0) = 0 .
Proof. Recall that t → ∆L t is a Poisson point process with characteristic measure ν, where ν is the Lévy measure of L. Then from the compensation formula for Poisson point processes, for every ε > 0,

E t≥0 1I {G(L t-)=0} 1I { ∆Lt >ε} = ν({x : x > ε})E ∞ 0 1I {G(Lt)=0} dt .
But from our assumption, the last term is 0. Then the equality P (∃ t ≥ 0, G(L t-) = 0, ∆L t = 0) = 0 in (5.11) is obtained by taking ε to 0 and using monotone convergence.

We prove the second equality through the same argument by writing

E t≥0 1I {G(L t-+∆Lt)=0} 1I { ∆Lt >ε} = ∞ 0 dtE { x >ε} ν(dx)1I {G(L t-+x)=0} = { x >ε} ν(dx) ∞ 0 P(G(L t + x) = 0) dt = 0
and the conclusion follows as for the first equality.

Let us now prove the first assertion of Theorem 2.1. First of all, since f is positive and non increasing, it is clear that 0 < S < ∞, a.s., so that Y S-, Z S-are well defined and Y S-< ∞ and Z S-< ∞, a.s. Then let us apply Lemma 5.1 to the Lévy process L = (Y, Z) and the Borel function G(y, z) = f (y)-z under the assumption that L has infinite lifetime. The condition: for all x = (y, z) ∈ R 2 , P(G(L t + x) = 0) = P(f (Y t + y) = Z t + z) = 0, for λ-a.e. t ≥ 0 is satisfied since, by assumption, the renewal measure v(dy, dz) of (Y, Z) is absolutely continuous. Then part 1. of Theorem 2.1 follows from Lemma 5.1 and the equalities The case of P(Z S = f (Y S ), ∆(Y, Z) S = 0) = 0 is handled in the same way.

P(Z S-= f (Y S-), ∆(Y, Z) S = 0) = P(∃t ≥ 0, Z t-= f (Y t-), ∆(Y, Z) t = 0) P(Z S = f (Y S ), ∆(Y, Z) S = 0) = P(∃t ≥ 0, Z t = f (Y t ), ∆(Y, Z) t = 0) .

Preliminary lemmas.

Recall that (Y, Z) is a bivariate subordinator as defined in Subsection 2.1. Its semigroup and its renewal measure are respectively denoted by p t (dy, dz) and v(dy, dz). In all the remainder of this paper we set for all y, z ≥ 0,

T Y y = inf{t : Y t > y} and T Z z = inf{t : Z t > z} .
Lemma 5.2. Assume that the renewal measure v(dy, dz) is absolutely continuous on (0, ∞) 2 and let us denote by v(y, z) its density, then for y, z ∈ (0, ∞),

P(Y T Y y = y, Z T Y y ∈ dz) dy = d Y v(y, z) dy dz (5.12) P(Z T Z z = z, Y T Z z ∈ dy) dz = d Z v(y, z) dy dz . (5.13)
Assume that for all t > 0, the semigroup p t (dy, dz) is absolutely continuous on (0, ∞) 2 and let us denote by p t (y, z) its densities, then for t, y, z ∈ (0, ∞), =s} ds, for all y ≥ 0. Hence for α, β, γ > 0,

P(Y T Y y = y, Z T Y y ∈ dz, T Y y ∈ dt) dy = d Y p t (
∞ y=0 t,z∈[0,∞) 2 e -αy-βz-γt P(Y T Y y = y, Z T Y y ∈ dz, T Y y ∈ dt) dy = E ∞ 0 e -αY T Y y -βZ T Y y -γT Y y d Y dT Y y = E ∞ 0 e -αYt-βZt-γt d Y dt = ∞ y=0 t,z∈[0,∞) 2
e -αy-βz-γt d Y p t (dy, dz) dt , which proves both identities. The proof of (5.13) and (5.15) is the same.

Let us mention that expressions similar to (5.12) and (5.13) have been obtained in Section 4 of [START_REF] Griffin | The time at which a Lévy process creeps[END_REF], see Theorem 4.2 therein. We also use the latter theorem for the proof of the next Lemma.

Lemma 5.3. Assume that the renewal measure v(dy, dz) is absolutely continuous on (0, ∞) 2 and that its density (y, z) → v(y, z) is locally bounded. Then there are functions ε 1 and ε 2 satisfying ε 1 (h), ε 2 (h) → 0, as h → 0 and such that for all h ∈ (0, u 1 -u 0 ) and u ∈ [u 0 , u 1 -h],

P(Y T Y u+h -∈ (u, u + h), Z T Y u+h -∈ (f (u + h), f (u))) ≤ hε 1 (h) , P(Z T Z f (u) -∈ (f (u + h), f (u)), Y T Z f (u) -∈ (u, u + h)) ≤ hε 2 (h) . Proof.
Let us denote by π Y and π Z the Lévy measures of Y and Z and set πY (y) = π Y (y, ∞) and πZ (z) = π(z, ∞). Recall that ζ is the lifetime of (Y, Z) and let us denote by q ∈ [0, ∞) its rate. Applying Theorem 4.2 in [START_REF] Griffin | The time at which a Lévy process creeps[END_REF] we obtain,

P(Y T Y u+h -∈ (u, u + h), Z T Y u+h -∈ (f (u + h), f (u)), T Y u+h < ζ) = f (u) z=f (u+h) u+h y=u v(y, z)π Y (y) dydz ≤ πY (u 0 )Ch sup u∈[u 0 ,u 1 -h] (f (u) -f (u + h)) , (5.16)
where C is a bound for v on [u 0 , u 1 ] × [f (u 1 ), f (u 0 )] (recall that u 0 > 0). On the other hand,

P(Y T Y u+h -∈ (u, u + h), Z T Y u+h -∈ (f (u + h), f (u)), T Y u+h = ζ) ≤ P(Y ζ-∈ (u, u + h), Z ζ-∈ (f (u + h), f (u))) = q f (u) z=f (u+h) u+h y=u v(y, z) dydz ≤ qCh sup u∈[u 0 ,u 1 -h] (f (u) -f (u + h)) . (5.17)
Then the first inequality of the statement follows from (5.16), (5.17) and the uniform continuity of f on [u 0 , u 1 ].

Similarly, we have from Theorem 4.2 in [START_REF] Griffin | The time at which a Lévy process creeps[END_REF],

P(Z T Z f (u) -∈ (f (u + h), f (u)), Y T Z f (u) -∈ (u, u + h), T Z f (u) < ζ) = f (u) z=f (u+h) u+h y=u v(y, z)π Z (z) dydz ≤ πZ (f (u 1 ))Ch sup u∈[u 0 ,u 1 -h] (f (u) -f (u + h)) ,
and the same conclusion follows from the same argument.

Proof of the lower bound in identities (2.3) and (2.4).

Let us first prove the lower bound in equation (2.4), that is the inequality,

P(Z S = f (Y S ), u 0 < Y S < u 1 , t 0 < S < t 1 ) d Z u 1 u 0 t 1 t 0 p t (u, f (u)) dt du -d Y u 1 u 0 t 1 t 0 p t (u, f (u)) dt df (u) . (5.18)
We define the sequence (σ n ) n≥1 of subdivisions of [u 0 , u 1 ],

(5. [START_REF] Vigon | Votre Lévy rampe-t-il ?[END_REF])

σ n = u n k = u 0 + k n (u 1 -u 0 ), 0 k n , n ≥ 1 ,
and we set, for all n ≥ 1,

B n = n-1 k=0 B (1) n,k ∪ B (2)
n,k , where

B (1) n,k = Z T Z f (u n k ) = f (u n k ), u n k Y T Z f (u n k ) < u n k+1 , t 0 < T Z f (u n k ) < t 1 (5.20) B (2) n,k = Y T Y u n k+1 = u n k+1 , f (u n k+1 ) Z T Y u n k+1 < f (u n k ), t 0 < T Y u n k+1 < t 1 . (5.21) Then let us check that (5.22) lim sup n B n ⊂ {Z S = f (Y S ), u 0 ≤ Y S ≤ u 1 , t 0 ≤ S ≤ t 1 } , almost surely .
Let us first note that

ω ∈ B (1) n,k ⇒ (Y T Z f (u n k ) , Z T Z f (u n k ) ) -(u n k , f (u n k )) ≤ u n k+1 -u n k = u 1 -u 0 n and ω ∈ B (2) n,k ⇒ (Y T Y u n k+1 , Z T Y u n k+1 ) -(u n k+1 , f (u n k+1 )) ≤ f (u n k ) -f (u n k+1 ) ≤ ε(n) ,
where in the last inequality, the mapping n → ε(n) does not depend on k and satisfies lim n→∞ ε(n) = 0. This follows from the uniform continuity of f on [u 0 , u 1 ]. Therefore if ω ∈ lim sup n B n , then there is a subsequence u in jn (ω) := u n of (u n k ) n≥1, 0≤k≤n which converges to u ∈ [u 0 , u 1 ] and such that (Y

T Z f (un) , Z T Z f (un) ) converges to (u, f (u)) or (Y T Y un , Z T Y un ) converges to (u, f (u)). Assume for instance that (Y T Z f (un) , Z T Z f (un) ) converges to (u, f (u)
). Since Y and Z are non decreasing, this means that T Z f (un) tends to some value S such that (Y S-, Z S-) = (u, f (u)). Moreover t 0 ≤ S ≤ t 1 and u 0 ≤ Y S-≤ u 1 . In particular,

Z S-= f (Y S-), so that S = inf{t : Z t ≥ f (Y t )}, since Y and Z are non decreasing. If (Y T Y un , Z T Y un ) converges to (u, f (u))
, then the same argument leads to the same conclusion. Then (5.22) follows from part 1. of Theorem 2.1. Now, let us prove that (5.23)

lim n→+∞ P(B n ) = d Z u 1 u 0 t 1 t 0 p t (u, f (u)) dt du -d Y u 1 u 0 t 1 t 0 p t (u, f (u)) dt df (u) .
We first notice that, for all n ∈ N and all integer 0 k n -1, B

n,k ∩ B (2) n,k = ∅. Moreover, σ n being a subdivision of [u 0 , u 1 ], for all integers n ≥ 2, k , k such that k = k and 0 k , k n -1, (B (1)
n,k ∪ B (2) n,k ) ∩ (B (1) n,k ∪ B (2) (1)
n,k ) = ∅. On the other hand, from (5.14) and (5.15) in Lemma 5.2, we obtain

P B (1) n,k = d Z u n k+1 u n k t 1 t 0 p t (u, f (u n k )) dt du and P B (2) n,k = d Y f (u n k ) f (u n k+1 ) t 1 t 0 p t (u n k+1 , s) dt ds , so that P(B n ) = n-1 k=0 P B (1) n,k + P B (2) n,k = d Z n-1 k=0 u n k+1 u n k t 1 t 0 p t (u, f (u n k )) dt du + d Y n-1 k=0 f (u n k ) f (u n k+1 ) t 1 t 0 p t (u n k+1 , s) dt ds .
Recall that 0 < t 0 < t 1 < ∞ and 0 < u 0 < u 1 < ∞ and assume first that f (u 1 ) < f (u 0 ), in particular 0 < f (u 1 ) < f (u 0 ) < ∞. Then we derive from the continuity of (t, y, z)

→ p t (y, z) on [t 0 , t 1 ] × [u 0 , u 1 ] × [f (u 1 ), f (u 0 )] that the mapping (y, z) → t 1 t 0 p t (y, z) dt is uniformly continuous on [u 0 , u 1 ] × [f (u 1 ), f (u 0 )]. Set v(y, z) := t 1 t 0 p t (y, z) dt. We derive from the uniform continuity of (y, z) → v(y, z) on [u 0 , u 1 ] × [f (u 1 ), f (u 0 )] and this of the mapping u → f (u) on [u 0 , u 1 ] that there are functions α(n), β(n) → 0, as n → ∞, such that for all 0 ≤ k ≤ n -1, |v(u, f (u n k )) -v(u n k , f (u n k ))| ≤ α(n), for all u ∈ [u n k , u n k+1 ] and |v(u n k+1 , s) -v(u n k+1 , f (u n k+1 ))| ≤ β(n), for all s ∈ [f (u n k+1 ), f (u n k )].
These arguments and the Riemann-Stieltjes integrability of the function u → v(u, f (u)) with respect to the measures du and df (u) imply that

lim n→∞ n-1 k=0 u n k+1 u n k v(u, f (u n k )) du = lim n→∞ n-1 k=0 v(u n k , f (u n k ))(u n k+1 -u n k ) = u 1 u 0 v(u, f (u)) du ,
and that

lim n→∞ n-1 k=0 f (u n k ) f (u n k+1 ) v(u n k+1 , s) ds = lim n→∞ n-1 k=0 v(u n k+1 , f (u n k+1 ))(f (u n k ) -f (u n k+1 )) = - u 1 u 0 v(u, f (u)) df (u) , (5.24)
which proves (5.23) in this case. If f (u 0 ) = f (u 1 ) := w, then the result follows from the same arguments. We only need to replace the mapping (y, z) → t 1 t 0 p t (y, z) dt by the mapping y → t 1 t 0 p t (y, w) dt and the function u → v(u, f (u)) by the function u → v(u, w). (The term (5.24) is then clearly equal to 0.) Finally, recall that lim sup

n-→∞ B n ⊂ {Z S = f (Y S ), u 0 ≤ Y S ≤ u 1 , t 0 ≤ S ≤ t 1 }, a.s.
Then by Fatou's lemma,

P(Z S = f (Y S ), u 0 ≤ Y S ≤ u 1 , t 0 ≤ S ≤ t 1 ) P(lim sup n-→∞ B n ) lim n-→∞ P(B n ) ,
and (5.23) implies that

P(Z S = f (Y S ), u 0 ≤ Y S ≤ u 1 , t 0 ≤ S ≤ t 1 ) d Z u 1 u 0 t 1 t 0 p t (u, f (u)) dt du -d Y u 1 u 0 t 1 t 0 p t (u, f (u)) dt df (u) .
Since the latter inequality is valid for all u 0 , u 1 , t 0 and t 1 , large inequalities in the left member can be replaced by strict inequalities, so that (5.18) holds. The proof of the corresponding inequality in (2.3) uses (5.12) and (5.13) instead of (5.14) and (5.15) in the above arguments. It is actually simpler so it is omitted. 5.4. Proof of the upper bound in identities (2.3) and (2.4). Let us first prove the upper bound in equation (2.4), that is the inequality

P(Z S = f (Y S ), u 0 < Y S < u 1 , t 0 < S < t 1 ) d Z u 1 u 0 t 1 t 0 p t (u, f (u)) dt du -d Y u 1 u 0 t 1 t 0 p t (u, f (u)) dt df (u) . (5.25)
Recall the definition (5.19) of the sequence (σ n ) n≥1 of subdivisions of [u 0 , u 1 ]. We define the two following sequences of events which are slightly different from those introduced in (5.20) and (5.21), B

n,k = Z T Z f (u n k ) = f (u n k ), u n k Y T Z f (u n k ) u n k+1 , t 0 T Z f (u n k ) t 1 + o 1 (n) B (1)
n,k = Y T Y u n k+1 = u n k+1 , f (u n k+1 ) Z T Y u n k+1 f (u n k ), t 0 T Y u n k+1 t 1 + o 2 (n) , (2)
where

o 1 (n) = d -1 Z max 0≤k≤n-1 (f (u n k )-f (u n k+1 )), if d Z > 0 and o 2 (n) = (nd Y ) -1 , if d Y > 0. If d Y = 0 (resp. d Z = 0), we set o 2 (n) = ∞ (resp. o 1 (n) = ∞), for all n ≥ 1. Note that from uniform continuity of f on [u 0 , u 1 ], if d Z > 0, then o 1 (n) → 0, as n → ∞ (which is also clearly verified for o 2 , when d Y > 0)
. Then let us also define the sequences, B

n,k = f (u n k+1 ) Z T Z f (u n k ) - f (u n k ), u n k Y T Z f (u n k ) - u n k+1 B (3)
n,k = u n k Y T Y u n k+1 - u n k+1 , f (u n k+1 ) Z T Y u n k+1 - f (u n k ) , (4)
for n ≥ 1 and k = 0, . . . , n -1 and let us check the inclusion, (5.26)

A := {Z S = f (Y S ), u n k Y S < u n k+1 , t 0 < S < t 1 } ⊂ B (1)
n,k ∪ B (2) n,k ∪ B (3) n,k ∪ B (4)
n,k .

Assume that A holds and note that

T Z f (u n k ) ≥ S and T Y u n k+1 ≥ S.
Then, only the two events described below in (a) and (b) can occur:

(a) In the first case, either Z creeps through the level

f (u n k ) before time T Y u n k+1 or Y creeps through the level u n k+1 before time T Z f (u n k ) . Let us consider the subcase where Z creeps through the level f (u n k ) before time T Y u n k+1 . Then we have Z T Z f (u n k ) = f (u n k ) and since u n k Y S < u n k+1 and S ≤ T Z f (u n k ) ≤ T Y u n k+1 , we also have u n k Y T Z f (u n k ) u n k+1 . Moreover, recall that S ≥ t 0 , therefore T Z f (u n k ) ≥ t 0 . On the other hand since Z S = f (Y S ), then from time S, the process Z will reach the level f (u n k ) in a time which is at most equal to d -1 Z (f (u n k ) -f (Y S )). Hence T Z f (u n k ) ≤ S + d -1 Z (f (u n k ) -f (Y S )) ≤ t 1 + o 1 (n). We conclude that B (1)
n,k holds in this subcase. In the subcase where Y creeps through the level u n k+1 before time T Z f (u n k ) , we prove in the same way that B

n,k holds or Y jumps through the level u n k+1 and the event B

n,k holds. Note the both events can occur at the same time.

Then the inclusion (5.26) is proved.

u n k u n k+1 f u n k+1 f (u n k ) Y T Z f ( u n k ) - Z T Z f (u n k ) - u n k u n k+1 f u n k+1 f (u n k ) Y T Y u n k+1 - Z T Y u n k+1 - Figure 3. An occurrence of the event B (3)
n,k on the left and B

n,k on the right

Let us set v (n) 1 (y, z) := t 1 +o 1 (n) t 0 p t (y, z) dt and v (n) 2 (y, z) := t 1 +o 2 (n) t 0
p t (y, z) dt. Then the inclusion (5.26) implies that

P(Z S = f (Y S ), u 0 < Y S < u 1 , t 0 < S < t 1 ) n-1 k=0 P(Z S = f (Y S ), u n k ≤ Y S ≤ u n k+1 , t 0 < S < t 1 ) n-1 k=0 P(B (1) n,k ) + n-1 k=0 P(B (2) n,k ) + n-1 k=0 P(B (3) n,k ) + n-1 k=0 P(B (4) n,k )
n,k ) + n-1 k=0 P(B (1)
n,k ) + n-1 k=0 P(B (2)
n,k ) + n-1 k=0 P(B (3)
Z u n k+1 u n k v (n) 1 (u, f (u n k )) du + n-1 k=0 d Y f (u n k ) f (u n k+1 ) v (n) 2 (u n k+1 , s) ds + n-1 k=0 1 n (ε 1 (n) + ε 2 (n)) . (4) n,k ) n-1 k=0 d
We prove in a similar way as in the previous subsection that the sum of the two first sums in the last expression tends to

d Z u 1 u 0 v(u, f (u)) du -d Y u 1 u 0 v(u, f (u)) df (u)
, when n tend to ∞. The third sum tends clearly to 0, as n tends to ∞, so that inequality (5.25) is proved.

The upper bound in equality (2.3) is proved in the same way. This achieves the proof of Theorem 2.1. 5.5. Proof of Proposition 2.1. Part 1. is proved as part 1. of Theorem 2.1, see Subsection 5.1. (Note that P(S > 0) = 1 since lim t→0+ f (t) > 0. In particular, Y S-and Z S- are well defined.)

The proof of part 2. follows very similar arguments as those used for the proof of the upper bound in identities (2.3) and (2.4) which is given in Subsection 5.4. We only describe it here in broad outline. Recall the sequence (σ n ) n≥1 of subdivisions of [u 0 , u 1 ] defined in (5.19). Then define the two following sequences of events

C (1) n,k = Z T Z f (u n k+1 ) = f (u n k+1 ), u n k Y T Z f (u n k+1 ) u n k+1 C (2) n,k = f (u n k ) Z T Z f (u n k+1 ) -< f (u n k+1 ), u n k Y T Z f (u n k+1 ) -< u n k+1 .
Since f is non decreasing, the inclusion, (5.27)

C := {Z S = f (Y S ), u n k Y S < u n k+1 } ⊂ C (1) n,k ∪ C (2) n,k
holds for all n sufficiently large and k = 0, . . . , n -1. Indeed, by definition of S, there is ε > 0 such that the path of (Y, Z) stays above the graph of f on the interval [S, S + ε]. This implies that for all n sufficiently large, if

t ≥ 0 is such that Y S+t ∈ [u n k , u n k+1 ]
for some k = 0, 1, . . . , n -1, then Z S+t ≥ f (S + t) so that either the process Z creeps above the level f (u n k+1 ), that is C

n,k holds, or Z jumps above this level, that is C

n,k holds. There is no other possibility.

The remainder of the proof is then very similar to the arguments of the end of Subsections 5.1 and 5.4, the main difference being that the term due to the creeping of Y is now absent. 5.6. On assumptions of Theorem 2.1. The aim of this subsection is to show that Theorem 2.1 holds whenever the transition semigroup of (Y, Z) is absolutely continuous with bounded densities. Let d ≥ 1 and let L be any possibly killed R d valued Lévy process.

Proposition 5.1. Let Ψ be the characteristic exponent of L, that is E(e i ξ,L 1 , 1 < ζ) = e -Ψ(ξ) , ξ ∈ R d , where ζ is the lifetime of L. Then the following assertions are equivalent:

(i) The transition semigroup of L is absolutely continuous and there is a version of its densities, denoted by x -→ l t (x), x ∈ R d , which are bounded for all t > 0. (ii) For all t > 0, e -tΨ(ξ) ∈ L 1 (R d ) .

If these conditions are satisfied, then the function (t, x) → l t (x) is continuous on (0, ∞) × R d . In particular, if all coordinates of L are subordinators, then l t (0) = 0, for all t > 0.

Proof. The proof follows similar arguments as those used in the beginning of Section 2 in [START_REF] Chaumont | On the asymptotic behavior of the density of the supremum of Lévy processes[END_REF]. Boundness of l t implies that l t ∈ L 2 (R d ) and consequently e -tΨ(ξ) ∈ L 2 (R d ), for all t > 0, which implies that e -tΨ(ξ) ∈ L 1 (R d ), for all t > 0. Conversely, if e -tΨ(ξ) ∈ L 1 (R d ), for all t > 0, then Fourier inversion theorem implies (i). Indeed, by the Riemann-Lebesgue lemma, l t ∈ C 0 (R d ). Furthermore, from Fourier inversion theorem we have, for all (t, x) ∈ (0, ∞) × R d , (5.28)

l t (x) = 1 2π R d e -i
x,ξ e -tΨ(ξ) dξ .

Let (t 0 , x 0 ) ∈ (0, ∞) × R d and K be a compact neighborhood of (t 0 , x 0 ) such that K ⊂ (0, ∞)×R d . Then, there exists t 1 > 0 such that for all (t, x, ξ) ∈ K ×R d , |e -i x,ξ e -tΨ(ξ) | = e -tRe(Ψ(ξ)) |e -t 1 Ψ(ξ) | = e -t 1 Re(Ψ(ξ)) , (recall that Re(Ψ(ξ)) ≥ 0). Since e -t 1 Ψ(ξ) ∈ L 1 (R d ), we obtain from (5.28) and Lebesgue's dominated convergence theorem, that (t, x) → l t (x) is continuous in (t 0 , x 0 ) and finally on (0, ∞) × R d .

Recall from Subsection 5.1 that the absolute continuity of v(dy, dz) is actually enough for part 1. of Theorem 2.1 to hold. Therefore, condition (i) (or (ii)) of Proposition 5.1 implies part 1. of Theorem 2.1. Moreover, condition (i) (or (ii)) of Proposition 5.1 applied to the bivariate subordinator (Y, Z) immediately implies the assumption of part 3. of Theorem 2.1. The mapping (t, y, z) → p t (y, z) is then locally bounded on (0, ∞) 3 . Hence the assumption of assertion 2'. (stated after Theorem 2.1) is satisfied for all t 0 , t 1 such that 0 < t 0 < t 1 < ∞ and this implies part 2. of Theorem 2.1 by monotone convergence.

Let us finally notice that assumptions of Theorem 2.1 could be weakened. Indeed, it appears from the limits in (5.24) that the continuity assumption of v could be replaced by an assumption closed to the Riemann integrability of this function. (Recall that continuity of v is not required in lemmas 5.3 and 5.2.) 6. Proof of Theorem 2.2, Corollary 2.3 and Proposition 2.2

Recall that X is a real non killed Lévy process and that we have assumed that |X| is not a subordinator. 6.1. Some reminders in fluctuation theory. Recall that X t = sup{X s : 0 ≤ s ≤ t} and let X t = inf{X s : 0 ≤ s ≤ t}. It is well known that both processes X -X and X -X are strongly Markovian. The state 0 is regular for itself, for the process X -X (resp. X -X) if and only if 0 is regular for [0, ∞) (resp. (-∞, 0]) for the process X. When it is the case, we will simply write that [0, ∞) (resp. (-∞, 0]) is regular. Similarly, we will write that (0, ∞) (resp. (-∞, 0)) is regular when 0 is regular for (0, ∞) (resp. (-∞, 0)) for the process X. Recall also that when X is not a compound Poisson process, at least one of the half lines (-∞, 0) or (0, ∞) is regular. Moreover compound Poisson processes are the only Lévy processes for which [0, ∞) and (-∞, 0] are regular but not (0, ∞) and (-∞, 0).

Let us briefly recall the definition of the local time at 0 of the process X -X which we denote by L. If [0, ∞) is regular, L is a continuous, increasing, additive functional such that L 0 = 0, a.s., and the support of the measure dL t is the set {t : X t = X t }. Moreover L is the unique process, up to a multiplicative constant, satisfying these properties. We can normalize it for instance in the following way,

E ∞ 0 e -t dL t = 1 .
When [0, ∞) is not regular, the set {t : (X -X) t = 0} is discrete and the local time at 0 of the process X -X is defined as follows

L t = Rt k=0 e (k) ,
where R 0 = 0, for t > 0, R t = Card{s ∈ (0, t] : X s = X s } and e (k) , k = 0, 1, . . . is a sequence of independent and exponentially distributed random variables with parameter

γ = 1 -E(e -τ + 0 ) -1
, and τ + 0 = inf{t > 0 : X t > 0}. The choice of the parameter γ is consistent with the normalisation E ∞ 0 e -t dL t = 1 of the regular case. The process L is called the local time at the supremum of X. The local time at the infimum is defined in the same way with respect to the process X -X and is denoted by L * . It corresponds to the local time at the supremum of the dual process X * := -X.

Let us now recall the definition of the ladder processes. The ladder time processes τ and τ * , and the ladder height processes H and H * are the following (possibly killed) subordinators: , where σ represents the Brownian part of X in its Lévy-Khintchine decomposition (this follows from the Wiener-Hopf factorisation of the characteristic exponent of X, see p.166 in [START_REF] Bertoin | Lévy processes[END_REF]). When X has bounded variation, we define its drift as the almost sure limit d := lim t→0 X t /t. It was proved in Section 2. of [START_REF] Millar | Exit properties of stochastic processes with stationary independent increments[END_REF] that in this case, d > 0 if and only if d H > 0. When X has unbounded variation and no Brownian part, Theorem Kaa of [START_REF] Vigon | Votre Lévy rampe-t-il ?[END_REF] asserts that d H > 0 if and only if

τ t = inf{s : L s > t} , τ * t = inf{s : L * s > t} , H t = X τt , H * t = -X τ * t ,
1 0 x 0 -x π1 (u)du π(x) dx < ∞ , where π is the Lévy measure of X, for x ∈ R, π(x) = R 1I {0<x≤u} + 1I {u≤x<0} dπ(u) and for x ∈ [-1, 0), π1 (x) = x -1 π(u)du. Note also that if d τ > 0 and d H > 0, then d H = dd τ , since lim t→0 Ht t = d H = lim t→0 Xτ t τt τt t = dd τ , a.s.
Let us assume for the remainder of this section that X is not a compound Poisson process. The Itô measure n * of the excursions away from 0 of the process X -X is the characteristic measure of the Poisson point process

t →    {(X -X) τ * t-+s , 0 ≤ s < τ * t -τ * t-} if τ * t-< τ * t δ if τ * t-= τ * t ,
where δ is some isolated point added to the space of excursions. We refer to [START_REF] Bertoin | Lévy processes[END_REF], Chap. IV, [START_REF] Kyprianou | Fluctations of Lévy processes with applications[END_REF], Chap. 6 and [START_REF] Doney | Fluctuation theory for Lévy processes[END_REF] for a more detailed definition n * . The measure n * is a Markovian measure whose semigroup is this of the killed Lévy process when it enters in the negative half line. More specifically, for x > 0, let us denote by Q * x the law of the process

(X t 1I {t<τ - 0 } + ∞ • 1I {t≥τ - 0 } , t ≥ 0) under P x , where τ - 0 = inf{t > 0 : X t < 0}. That is for Λ ∈ F t , (6.30) Q * x (Λ, t < ζ) = P x (Λ, t < τ - 0 )
. Then for all Borel positive functions f and g and for all s, t > 0,

(6.31) n * (f (X t )g(X s+t ), s + t < ζ) = n * (f (X t )E Q * Xt (g(X s )), s < ζ) , where E Q *
x means the expectation under Q * x and ζ is the lifetime of the excursions. Moreover, we define the entrance law (q * t (dx), t > 0) of the excursion measure n * as

(6.32) n * (f (X t ), t < ζ) = ∞ 0 f (x)q * t (dx) .
The Itô measure of the excursions away from 0 of the reflected process at its supremum X -X = X * -X * is defined in the same way as for X -X. It will be denoted by n and its entrance law by q t (dx). We define the probability measures Q x in the same way as in (6.30) with respect to the dual process X * = -X. Let us finally recall that the entrance laws q t (dh) and q * t (dh) are related to the renewal measures U (dt, dh) and U * (dt, dh) of (τ, H) and (τ * , H * ) as follows,

d τ * δ {(0,0)} (dt, dh) + q * t (dh) dt = U (dt, dh) (6.33) d τ δ {(0,0)} (dt, dh) + q t (dh) dt = U * (dt, dh). (6.34)
These identities between measures on [0, ∞) 2 can be found in Lemma 1 of [START_REF] Chaumont | On the law of the supremum of Lévy processes[END_REF]. In particular, under the assumption of Theorem 2.2, that is when U (dt, dh) has a density on (0, ∞) 2 , this density is the same as this of q * t (dh) dt, which explains the notation q * t (h), t, h ∈ (0, ∞) in this theorem. 6.2. Proof of Theorem 2.2. Part 1. of Theorem 2.2 is a direct application of Lemma 5.1. It suffices to consider L t = (X t , t) and G(u, v) = u -f (v) in this lemma. (Note that P(T f > 0) = 1 follows from the fact that f is positive. In particular, X T f -is well defined on {T f < ∞}.)

Then part 2. is a consequence of Lemma 5.1 and Theorem 2.1. Recall first from Subsection 6.1 that (τ, H) is a bivariate subordinator. If the time T f = inf{t : X t > f (t)} satisfies X T f = X T f = f (T f ), then it is a zero of X -X and hence there is S such that T f = τ S or T f = τ S -. Therefore H S = f (τ S ) or X τ S -= X τ S -= f (τ S -) but we know, applying again Lemma 5.1 to the function G as above, that in the latter case, X is necessarily a.s. continuous at time τ S -. In particular, H S -= X τ S --= f (τ S -). On the other hand, applying again Lemma 5.1, it follows that if H S -= f (τ S -), then (τ, H) is necessarily a.s. continuous at time S , so that τ S = τ S -. This proves that H S = f (τ S ), a.s. Moreover, since the subordinator H is not lattice, the time S is necessarily unique, so that S = S = inf{t ≥ 0 : H t > f (τ t )}. This proves in particular that τ S = T f .

Conversely assume that S satisfies H S = X τ S = X τ S = f (τ S ). Since X is not lattice, and f is non increasing, a time T satisfying X T = X T = f (T ) is necessarily unique. Moreover, when X is not a compound Poisson process, τ S = inf{t : X t > X τ S }. Hence, since f is non increasing, τ S = inf{t : X t > f (t)} = T f , so that X T f = X T f = f (T f ) and we have proved the identity

{H S = f (τ S ), u 0 < τ S < u 1 } = {X T f = X T f = f (T f ), u 0 < T f < u 1 }
which allows us to conclude thanks to part 2. of Theorem 2.1. 6.3. On assumptions of Theorem 2.2. It is assumed in part 2. of Theorem 2.2 that the renewal measure of the ladder process (τ, H) has continuous densities (t, x) → q * t (x) on (0, ∞) 2 . The next proposition provides conditions for this to hold. Moreover, when the later are satisfied, q * t is positive on (0, ∞), for all t > 0. This ensures the positivity of the creeping probability (2.8) whenever d H > 0 or d τ > 0 and f (t 1 ) < f (t 0 ). Proposition 6.1. Assume that X is a non killed real Lévy process which satisfies the equivalent conditions (i) and (ii) of Proposition 5.1, then 1. the semigroup q * t (x, dy) of the killed process (X, Q * x ) defined in (6.30) is absolutely continuous and its densities (t, x, y) → q * t (x, y) are continuous in (0, ∞) 2 × [0, ∞). 2. Assume moreover that for all c ≥ 0, the process (|X t -ct|, t ≥ 0) is not a subordinator. Then the entrance law, q * t (dx) is absolutely continuous on [0, ∞) for all t > 0 and there is a version q * t (x), x ≥ 0, t > 0 of its densities such that the function (t, x) → q * t (x) is continuous on (0, ∞) × [0, ∞). Moreover, for all t > 0, q * t is positive on (0, ∞).

Denote by h the renewal function of H, that is h(x) = ∞ 0 P(H t ≤ x) dt, then lim y→0+ q * t (x, y) = h(0)q t (x). Moreover for all t > 0, lim x→0+ q * t (x) = h(0)p t (0)/t. We then set q * t (x, 0) = h(0)q t (x) and q * t (0) = h(0)p t (0)/t, for all t, x > 0. Proof. The first assertion is a slight reinforcement of Lemma 2 in [START_REF] Uribe | Bridges of Lévy processes conditioned to stay positive[END_REF] and our proof follows the same arguments. Recall first the definition (6.30) of the killed process (X, Q *

x ) and let us denote by q * t (x, dy) its semigroup. Then from Lemma 2 in [START_REF] Uribe | Bridges of Lévy processes conditioned to stay positive[END_REF], under our assumption and thanks to Proposition 5.1, this semigroup admits densities q * t (x, y) which satisfy for all t > 0, x > 0, y ≥ 0, q * t (x, y) = P t x,y (Λ t )p t (y -x) , where P t x,y is the law of the bridge of (X, P) starting from x and ending in y at time t and Λ t = {X t > 0} ∪ {X s > 0, 0 ≤ s < t, X t = 0}. Here, in order to define q * t (x, 0) for all t > 0 and x > 0, we have included y = 0 since the right member is then well defined. From Corollary 1 of [START_REF] Chaumont | Markovian Bridges: Weak continuity and pathwise constructions[END_REF], the measures P t

x,y are weakly continuous in (t, x, y) ∈ (0, ∞)×R 2 . Then from the same arguments as in Lemma 2 of [START_REF] Uribe | Bridges of Lévy processes conditioned to stay positive[END_REF], these measures satisfy P t

x,y (∂Λ t ) = 0 for all t, x > 0 and y ≥ 0. Hence from portemanteau theorem and Proposition 5.1, the mapping (t, x, y) → P t

x,y (Λ t )p t (y -x) is continuous in (0, ∞) 2 × [0, ∞). In order to prove 2., recall from part 3. of Lemma 1 in [START_REF] Chaumont | On the law of the supremum of Lévy processes[END_REF] that q * t (dx) is absolutely continuous under our assumptions. Let us denote by q * t (x) the corresponding densities and recall the Chapman-Kolmogorov equation, (6.35) q * t+s (y) = ∞ 0 q * s (x)q * t (x, y) dx , s, t, y > 0 , which can be derived for instance from (6.31). From Proposition 1 in [START_REF] Chaumont | On the asymptotic behavior of the density of the supremum of Lévy processes[END_REF], for all x > 0 and t > 0, lim y→0+ q * t (x, y) = h(0)q t (x). Let us set q * t (x, 0) = h(0)q t (x). Let Ψ be the characteristic exponent of X, that is E(e iξX 1 ) = e -Ψ(ξ) . From Lemma 2 in [START_REF] Uribe | Bridges of Lévy processes conditioned to stay positive[END_REF], q * t (x, y) ≤ p t (y -x), for all t > 0 and x, y ∈ (0, ∞) 2 , so that, (6.36) q * t (x, y) ≤ p t (y -x) = 1 2π R e -i(y-x)ξ e -tΨ(ξ) dξ ≤ 1 2π R |e -tΨ(ξ) |dξ , and this bound can be extended to y = 0. From the same arguments as in the proof of Proposition 5.1, the mapping t → R |e -tΨ(ξ) |dξ is continuous on (0, ∞). Let (t 0 , y 0 ) ∈ (0, ∞) × [0, ∞), then from (6.36) and the continuity of t → R |e -tΨ(ξ) |dξ, there is ε > 0 and C ε > 0 such that q * t (x, y) ≤ C ε , for all x > 0, y ≥ 0 and t ∈ (t 0 -ε, t 0 + ε). Moreover, from (6.31) ∞ 0 q * s (x) dx = n * (s < ζ) < ∞, hence, (6.35) can be extended to y = 0 and

Figure 1 .

 1 Figure 1. A sample path of (Y, Z) creeping through the graph of f

  Now let L = ( Ỹ , Z) be a Lévy process with infinite lifetime such that L is obtained by killing L at an independent exponential time e. Denote by ζ the lifetime of L. Then,P(Z S-= f (Y S-), ∆(Y, Z) S = 0) = P(∃t ≤ ζ, Z t-= f (Y t-), ∆L t = 0)= P(∃t ≤ e, Zt-= f ( Ỹt-), ∆ Lt = 0)≤ P( ZS-= f ( ỸS-), ∆( Ỹ , Z) S = 0) = 0 .

  holds. (b) In the second case, either Z jumps through the level f (u n k ) and the event B

  Proof. Let us prove (5.12) and(5.14). Note that the time T Y y satisfies d Y T Y y =

	y 0 1I {Y T Y s

y, z) dt dy dz (5.14)

P(Z T Z z = z, Y T Z z ∈ dy, T Z z ∈ dt) dz = d Z p t (

y, z) dt dy dz . (5.15)

  t ≥ 0 , where τ t = H t = +∞, for t ≥ L ∞ and τ * t = H * t = +∞, for t ≥ L * ∞ . Recall that the processes (τ, H) and (τ * , H * ) are bivariate subordinators. The renewal measures of the ladder processes (τ, H) and (τ * , H * ) are the measures on [0, ∞) 2 defined by, ∈ dt, H u ∈ dh) du and U * (dt, dh) = The drifts d τ and d τ * of the subordinators τ and τ * satisfy Recall that d τ > 0 if and only if (-∞, 0) is not regular. Similarly, d τ * > 0 if and only if 0 is not regular for (0, ∞). Hence if X is not a compound Poisson process, then d τ d τ * = 0 holds since 0 is necessarily regular for at least one of the half lines. If X is a compound Poisson process, then d τ > 0 and d τ * > 0. Given our normalisation of the local time L, the drifts d H and d H * of H and H * satisfy 2d H d H * = σ 2

	U (dt, dh) =	∞	P(τ u ∞	P(τ * u ∈ dt, H * u ∈ dh) du .
	0		0
	(6.29)

t 0 1I {Xs=Xs} ds = d τ L t , t 0 1I {Xs=X s } ds = d τ * L * t .

we can set q * t (0) = lim y→0+ q * t (y), t > 0. Moreover, continuity of (t, y) → q * t (y) at (t 0 , y 0 ) follows from (6.35), continuity of (t, x, y) → q * t (x, y) in (0, ∞) 2 ×[0, ∞) proved in assertion 1. and Lebesgue theorem of dominated convergence.

The positivity of q * t for all t > 0 follows from Lemma 2 in [START_REF] Chaumont | On the asymptotic behavior of the density of the supremum of Lévy processes[END_REF] and the two last assertions in 2. follow from Proposition 1 in [START_REF] Chaumont | On the asymptotic behavior of the density of the supremum of Lévy processes[END_REF]. 6.4. Proof of Corollary 2.3. As recalled in Subsection 6.1, since the process Y t := X t + at, t ≥ 0 has bounded variation and positive drift, its upward ladder height process also has positive drift. Moreover, by assumption, for all t > 0 the distribution P(X t ∈ dx) is absolutely continuous with a bounded density, and for all c ≥ 0, the process (|X t -ct|, t ≥ 0) is not a subordinator. Then these conditions are clearly also satisfied by the process Y . Let us denote by q * ,a t (x) the density of the renewal measure of the ladder process associated to Y . Then from part 2. of Proposition 6.1, (t, x) → q * ,a t (x) is continuous on (0, ∞) 2 . Moreover, q * ,a t is positive on (0, ∞) for all t > 0. These properties imply, from part 2. of Theorem 2.2, that the process Y creeps at its supremum through the function f (t) + at over the interval (t 0 , t 1 ), so that X creeps through the function f (t) over the interval (t 0 , t 1 ). 6.5. Proof of Proposition 2.2. Proposition 2.2 is a consequence of Proposition 2.1 and its proof follows the same arguments as those used for the proof of Theorem 2.2 in Subsection 6.2. The details are omitted.

Proof of Theorem 2.3

The proof is a direct application of the extension of Tanaka's construction of random walks conditioned to stay positive obtained in [START_REF] Doney | s construction for random walks and Lévy processes[END_REF]. Let us first state this transformation. Recall that X t = sup s≤t X s and let us set for t > 0,

Then the process defined by

has the law of X conditioned to stay positive, X ↑ . According to this construction, the process W is obtained by time reversing of each excursion of the reflected process X -X and then by gluing them back together.

The assertion 0 < σ f < ∞, a.s. is straightforward since lim t→0+ f (t) > 0 and lim t→∞ X ↑ t = ∞, almost surely. Then we derive from the above construction that almost surely, X t = W t , for all t ≥ 0.

Moreover, the sets {t ≥ 0 : X t = X t } and {t ≥ 0 :

, almost surely and that X T f = X T f if and only if W T f = W T f . Then it remains to observe that on the set {X T f = X T f } = {W T f = W T f }, T f = sup{t : W t ≤ f (t)} = σ f and this proves parts 1. and 2. of Theorem 2.3.