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H Boutayeb
email: h.boutayeb@polymtl.ca

K Wu

K Mahdjoubi

On the impedance matching of left-handed materials to free-space

Using an original approach, this work shows that the characteristic impedance, as it is usually defined, is negative for a left-handed medium (LHM) and a method to match lefthanded media to free space is proposed. A full-wave technique is used to validate our analysis and proposed scheme. Our work is not in contradiction with previous studies on left-handed media and does not mean that the LHM is active, but we explain why one can encounter difficulty in simulating the excitation by an electromagnetic wave of an interface air/LHM if the LHM is considered homogeneous and we show that it is possible to excite the forward wave of a left-handed medium instead of the backward wave as it is usually done.

INTRODUCTION

Veselago 1 explored the properties of isotropic media where both the permittivity and the permeability are negative. He observed that in such media, the wave vector k, the electric field vector E, and the magnetic field vector H form a left-handed system. For this reason, these materials are called left-handed materials (LHMs). He deduced that these media possess negative refractive indexes and proposed to use them for the design of flat lenses, by taking benefits of the negative index of refraction. This idea has been recently reintroduced and further analyzed by Pendry [START_REF] Pendry | Negative refraction makes a perfect lens[END_REF] . Different methods have been proposed to design LHMs by using artificial periodic structures and experimental prototypes have been fabricated [START_REF] Shelby | Experimental verification of a negative index of refraction[END_REF][START_REF] Chen | Left-handed materials composed of S-shaped resonators[END_REF][START_REF] Parazzoli | Experimental verification and simulation of negative index of refraction using Snell's law[END_REF][START_REF] Houck | Experimental observations of left-handed material that obeys Snell's law[END_REF] . These artificial materials are also called metamaterials, because they have characteristics that can not be found in materials of the nature. Although a homogeneous non-dispersive LHM has never been observed in the nature, it could be interesting to facilitate the analysis by considering an artificial LHM as a homogeneous non-dispersive medium, around a frequency of interest. However, several authors have noticed that the theoretical analysis of a homogeneous non-dispersive LHM slab excited by a plane wave or a line source can lead to a divergent solution or to physical contradictions [START_REF] Hooft | Comment on 'Negative refraction makes a perfect lens[END_REF][START_REF] Garcia | Left-handed materials do not make a perfect lens[END_REF][START_REF] Maystre | Perfect lenses made with left-handed materials: Alice's mirror?[END_REF] . In this work, we present a new analysis of media with negative permittivity and permeability. According to our analysis, it is shown that, contrary to the common belief, the characteristic impedance, as it is usually defined, is negative for an LHM. This result is not in contradiction with previous studies on left-handed media and does not mean that the left-handed medium is active. It is simply due to the definition of the characteristic impedance and to the backward wave property of left-handed medium. Our result and interpretation allow us to explain why one can encounter difficulty when one simulates the excitation by an electromagnetic wave of a homogeneous left-handed medium. We also show that frequency-dispersion is not a necessary condition for the existence of a passive LHM, but the inhomogeneousness is necessary. Furthermore, it is shown that by using a sheet of resistors and a sheet of amplifiers, it is possible to match the LHM to air, for forward wave. To validate our approach, we present the analysis of the transmission and reflection coefficients of an LHM slab illuminated by a plane wave at normal incidence, by using a transmission line model, a Fabry-Perot cavity model and a full-wave time-domain technique.

CHARACTERISTIC IMPEDANCE OF A MEDIUM HAVING NEGATIVE PERMITTIVITY AND PERMEABILITY

The objective of this section is to determine the characteristic impedance of a medium having, at a given frequency, the following constitutive parameters : = -0 and µ = -µ 0 . 0 and µ 0 are the permittivity and permeability of air, respectively. Let us assume a medium, called Medium A, that has the permittivity 0 /p and the permeability µ 0 p, where p is a real. For this medium, Maxwell's equations can be written

∇ × E(r, t) = -pµ 0 ∂H(r, t) ∂t , ∇ × H(r, t) = 0 p ∂E(r, t) ∂t (1)
Because p is a scalar, it is obvious that (1) can be written as follows

∇ × E(r, t) = -µ 0 ∂pH(r, t) ∂t , ∇ × pH(r, t) = 0 ∂E(r, t) ∂t (2)
Solutions obtained for air can be used for Medium A by using pH instead of H. The impedance of Medium A is obtained by calculating the ratio between an electric field component E z and a magnetic field component H y . From the previous statement, Ez pH y = Z 0 = 120π, which leads to

Z M ediumA = E z H y = pZ 0 = p120π (3)
This equation can be tested for positive values of p. For example, for p = 0.5 ( = 2 0 and µ = 0.5µ 0 ), we obtain Z M ediumA = p120π = 60π (= µ/ ). If we consider p = -1, Medium A is a LHM ( = -0 and µ = -µ 0 ) and the impedance can be written Z LHM = -Z 0 = -120π (Note that Chew has shown that the signs of current and charge change in a left-handed medium [START_REF] Chew | Some reflections on double negative materials[END_REF] , which seems to be in accordance with our result).

INTERPRETATION OF THE NEGATIVE CHARACTERISTIC IMPEDANCE: FORWARD AND BACKWARD WAVES

The result in the previous section seems to be in contradiction with previous studies on left-handed media and seems to indicate that the LHM is active. However, we have to think further what we understand by the characteristic impedance of a medium, how the LHM is made and what is the property of the LHM.

The characteristic impedance of a transmission line is determined by considering the usual convention that the transmission line is infinite, that the wave is outgoing from the source (forward wave) and that no wave is going back to the source (backward wave). This convention works well for usual transmission line but it can cause misunderstanding for backward wave transmission line.

We have to note also that the characteristic impedance, although real and looking like a resistance, is a lossless and nondissipative impedance. Nothing gets hot as a result of supplying energy to this resistance. All that happens is that energy is transferred from the generator and stored temporarily in the transmission line, which transmits it elsewhere.

Concerning the left-handed medium, we first have to remind that the medium must be designed with a periodic structure. If a plane wave is sent to this medium, there are multiple wave-reflections between each partially reflection surface that composes the medium. Because of the property of the left-handed medium (angular-dispersion diagram), the total backward wave (the wave that goes to the source) is predominant compared to the total forward wave (the outgoing wave from the source). By neglecting the forward wave inside the LHM, the LHM seems to work with a backward wave. Now, by taking the convention for current flow to be from "generator" end to "load", we can make the following important remarks: -In air, the forward wave has positive characteristic impedance Z 0 , and the backward wave has mathematically negative characteristic impedance -Z 0 .

-In LHM, the backward wave has positive characteristic impedance Z 0 and the forward wave has negative characteristic impedance -Z 0 .

Usually the LHM slab is used with a backward wave, because for this case the LHM and air are matched. However, the simulation of this problem is not possible by considering a homogeneous medium (but possible and with a sense if we consider a periodic structure, see Fig. 1): indeed, in the case of a homogeneous LHM, where does the backward wave can come from? This impossibility of simulating an LHM working in backward wave by using a homogeneous medium was also confirmed by tests that we had carried out with our home-made Finite Difference Time Domain (FDTD) code. By sending a plane wave to a air/homogeneous left-handed medium interface, the FDTD solution becomes unstable. This unstableness can also be explained by the mismatch between the forward waves' impedances of air and LHM. Finally, thanks to our statements, we can note that the frequency-dispersion characteristic is not a necessary condition for the existence of a passive LHM, but the inhomogeneousness is necessary. Indeed, the negative total power (P = |E| 2 + µ|H| 2 ) for the LHM means that the wave goes to the source (backward wave) and confirms the negative characteristic impedance of the LHM. Step 2: because of the angular-dispersive nature of the LHM and because the LHM is matched to air for backward waves, the total forward wave is negligible as compared to the total backward wave. Thus, the LHM is equivalent to a medium with an index near -1. However, if one starts his numerical analysis by considering a homogeneous medium with index -1 (step 2), one will encounter a big problem!: where does the backward wave can come from!? In conclusion, an LHM has to be inhomogeneous for testing the backward wave impedance matching to air. In regards to these facts, the frequency-dispersive nature of an LHM becomes not a necessary condition for the existence of a passive LHM but only the inhomogeneousness. Indeed, the negative total power (P = |E| 2 + µ|H| 2 ) inside the LHM only means that the wave goes to the source.

|Transmission coefficient| ≈ 1 Reflection coefficient ≈ 0 n ≈ -1 (a) Step 1 (b) Step 2

METHOD TO MATCH AN LHM SLAB TO FREE SPACE FOR FORWARD WAVES

Let us consider a slab of a homogeneous lossless and dispersionless LHM in free space. Note that we can consider the medium homogeneous because we force the matching between forward waves' impedances of air and LHM. We assume that a plane wave is illuminating this slab at normal incidence. We consider that two sheets of resistors are placed in both sides of the LHM as illustrated in Fig. 2. According to Figs. 2(b) and (c), the impedance matching conditions, for forward waves, at the left and right interfaces, respectively, can be written as

-Z 0 R 1 -Z 0 + R 1 = Z 0 (4)
and

Z 0 R 2 Z 0 + R 2 = -Z 0 (5) Which result in R 1 = Z 0 2 = 60π (6)
and

R 2 = - Z 0 2 = -60π (7)
The matching sheets can be designed by using 2-D periodic structures and are assumed to be characterized in the air. From this, R 1 corresponds to an absorbtion of the wave and R 2 corresponds to an amplification of the wave, which can be obtained in practice by using linear amplifiers.

Air LHM -ε 0 -µ 0 ε 0 µ 0 Air ε 0 µ 0 Sheet of resistors Incident plane wave Sheet of resistors (a) R 2 Z 0 R 1 (b) ( c ) -Z 0 Z 0 -Z 0 ( R 1 ) ( R 2 )

TRANSMISSION AND REFLECTION CHARACTERISTICS OF A MATCHED OR IMPERFECTLY MATCHED LHM SLAB IN AIR

In this section, we use a transmission line model to calculate the transmission and reflection coefficients of the LHM slab with matching sheets. The effect of imperfectly matching is also analyzed.

k = 2πf /c = ω/c
is the propagation constant of air, f is the frequency and c is the speed of light in air. The propagation constant of the left-handed medium can be expressed by

k LHM = n LHM k = -k (8)
In this paper, we use the convention exp(-jωt) to express the propagation of a wave with time.

From this, the term exp(-jkx) expresses the propagation of a plane wave in positive x direction in air. If we consider a plane wave propagating from x = 0 to x = x 1 , the transmission coefficient of this line section of width x 1 can be written exp(-jkx 1 ) and x 1 /c express the time that the wave makes for travelling from x = 0 to x = x 1 .

The ABCD matrix (or chain matrix) of a line section of air of width d can be written by

cos(kd) jsin(kd) jsin(kd) cos(kd) (9)
The ABCD matrix of a line section of LHM of width d can be written as cos(kd) -jsin(kd) -jsin(kd) cos(kd) (10) The ABCD matrix of a sheet of resistors R can be written as

1 0 Z0 R 1 ( 11
)
The chain matrix of the system composed of the LHM and the two sheets of resistors can be expressed as follows :

A B C D = 1 0 Z 0 R 1 1 cos(kd) -jsin(kd) -jsin(kd) cos(kd) 1 0 Z 0 R 2 1 = 1 0 2 1 cos(kd) -jsin(kd) -jsin(kd) cos(kd) 1 0 -2 1 = cos(kd) + 2jsin(kd) -jsin(kd) 3jsin(kd) cos(kd) -2jsin(kd) (12)
The transmission and reflection factors of the matched LHM slab, excited from the left, T M LHM and R M LHM , respectively, are given by

T M LHM = 2 A + B + C + D = exp(-jkd) ( 13
)
R M LHM = A + B -C -D A + B + C + D = O (14)
which are the same results as those obtained for a wave propagating in air. It can be concluded that the time for the wave to pass through the matched LHM slab is the same as that for the wave to propagate if the LHM is replaced by air. The normalized transmission coefficient of the slab is then written as T = T LHM /T AIR = 1.

Let us analyze the case where the matching conditions are not perfectly achieved. We consider that the equivalent impedance of the sheets are not equal to the ideal case due to an additional imaginary admittance, jδ/Z 0 , which leads to the followings values of the equivalent impedances of the sheets, Z 1 and Z 2 :

Z 1 = Z 0 2 + jδ (15)
and

Z 2 = - Z 0 2 + jδ ( 16
)
where δ is a real close to zero. The chain matrix of the system composed of the LHM and the two imperfect matching sheets can be written by

A B C D = 1 0 2 + jδ 1 cos(kd)
-jsin(kd) -jsin(kd) cos(kd)

1 0 -2 -jδ 1 (17)
After some algebraic manipulation, one can obtain the transmission and reflection coefficients of the imperfectly matched LHM slab, T IM LHM and R IM LHM :

T IM LHM = 2 A + B + C + D = 2 2exp(jkd) + (4δj -δ 2 )jsin(kd) (18) R IM LHM = A + B -C -D A + B + C + D = -(2δ + δ 2 )jsin(kd) 2exp(jkd) + (4δj -δ 2 )jsin(kd) (19)
Note that δ = 0 corresponds to the perfectly matched LHM slab.

FABRY-PEROT CAVITY MODEL AND ANALYSIS OF CONVERGENCE CONDITIONS FOR IMPERFECTLY MATCHED LHM

In this section, we use the Fabry-Perot cavity model to analyze the transmission coefficient of the imperfectly matched LHM slab. In other words, we analyze the multiple wave-reflections between the two interfaces. This model shows more clearly the physical insight of the characteristics of the slab and allows to determine the conditions that must be verified to ensure the stability of the solution.

We consider that an imaginary admittance jδ/Z 0 is added to the equivalent admittance of the matching sheets. The reflection coefficients at the left and right sides of the slab, respectively, can be written as r = -δj + 4 δj + 2 (20) and r = -δj δj + 2 (21)

The transmission coefficients at the left and right interfaces are

t = t = 2 δj + 2 (22)
By considering the multiple reflections of the plane waves between the two interfaces, and by adding the successive transmitted rays, the transmission coefficient of the imperfectly matched LHM slab can be written by

T IM LHM = tt T M LHM ∞ i=0 (rr T 2 M LHM ) i = 4 (δj + 2) 2 exp(-jkd) ∞ i=0 δj(δj + 4) (δj + 2) 2 exp(-2jkd) i (23)
To ensure the convergence of the sum of geometrical series in (23), i.e. to ensure that the sum of the successive transmitted rays is finite, we need to verify the following relation

δj(δj + 4) (δj + 2) 2 < 1 (24)
When this relation is satisfied, (23) can be written

T IM LHM = 4 (δj+2) 2 exp(-jkd) 1 -δj(δj+4) (δj+2) 2 exp(-2jkd) = 2 2exp(jkd) + (4δj -δ 2 )jsin(kd) (25)
This is the same result as (18). However, by using the Fabry-Perot cavity model we have now also the information on the convergence condition. If one want to simulate this problem with a numerical method, it is clear that the convergence condition (24) must be satisfied, independently of the numerical technique that is used. Note that jδ = -2 corresponds to the LHM without matching sheets. If jδ tends to -2, it can be seen from ( 24) that the solution diverges.

NUMERICAL RESULTS USING THE FDTD TECHNIQUE

In this section, the full-wave simulation results of a homogeneous non-dispersive LHM slab with the proposed matching sheets and excited by a plane wave in air, are presented. We use a Finite Difference Time Domain (FDTD) code, which has been well validated for modeling and simulating a wide range of electromagnetic problems. This method is based on a direct discretization in space and time domains of Maxwell's equations.

In the FDTD method, we use a T M z polarized plane wave and a gaussian pulse in the frequency band 0 -5 GHz. Figure 3(a) shows the computational domain. Perfect Electric Conductors, Perfect Magnetic Conductors, and Perfect Matched Layers 11 are used at the boundaries to limit the computational domain. Two observation points, P 1 and P 2 , before and after the slab, respectively, are used for the calculation of the reflection and transmission coefficients. At the interfaces, matching resistors are located in each cells in y-z plane, as illustrated in Fig. 3(b). Simulations without the LHM slab were also carried out for normalization. The space mesh in all the directions is ∆ = 5mm. The LHM is made by using a medium with the following parameters : = -0 and µ = -µ 0 . Note that no arithmetic mean was used to force the permittivity at the interface. This introduces a space shift of a half-space mesh between the sheets of resistors and the actual interfaces between the two media 12 . A plane wave with a gaussian pulse comes from the left side. The position of the LHM is indicated in the figure. Note that the wave appears to propagate in the LHM as in the air (except for a small reflected energy part that is due to imperfection of the matching).

Figure 4 shows the amplitude of normalized field E z versus x-direction at different time instants. This figure demonstrates that the results converge with a wide frequency band excitation. These curves show also that the direction of the wave propagation does not change inside the LHM slab. Inside the LHM slab, the wave propagates like in free space, except for a small reflected energy part that is due to the imperfections in impedances values at the interfaces, as it will be developed later. Figure 5(a) presents simulated amplitudes of the transmission and reflection coefficients, T and R, for different slab widths. One can note that the transmission coefficient is not equal to one for all frequencies. This is due to the fact that the sheets are not at the same positions as the interfaces, which introduces additional terms in the equivalent admittance values of the interfaces. However, the transmission coefficient is equal to one at resonant frequencies at which the slab widths are equal to a multiple of the half-wavelength. In Fig. 5(b), is plotted the phase of the transmission coefficient for different slab widths. From these curves, it can be seen that the phase is never positive, which indicates that the slab does not accelerate the propagation of wave, but can only introduce a time-delay. The fact that the speed of light can not be surpassed is in accordance with the theory of relativity [START_REF] Einstein | Zur Elektrodynamik bewegter Körper[END_REF] . When the LHM is matched (T = 1 and R = 0), the time-delay is null. On the other hand, if we look into the distribution of the phase of the normalized field, at a resonant frequency, as shown in Fig. 6(a), the phase can be positive inside the slab. This is due to the stationary waves regime that occurs inside the slab, as it can be seen from the distribution of amplitude of the normalized field in Fig.

6(b).

The previous numerical results show that the matching sheets are not perfect. This is due to the fact that the distance between the sheets and the boundary between the two media are spaced by half space mesh, i.e. ∆/2, due to the FDTD technique. Physically, this distance represents the electrical distance between a matching sheet and the LHM. At the interface between the two media, the matching conditions impose that the admittance is null. Therefore, the equivalent admittance of the interface at the sheet position is given by

jδ/Z 0 = jtan(k∆/2)/Z 0 (26)
where ∆ = 5mm. jδ/Z 0 represents the additional term that appears in the equivalent admittances of the sheets. Equation ( 26) is introduced in Eqs. ( 18)-( 19). In Fig. 7(a), are plotted the amplitude of the transmission and reflection coefficients and in Fig. 7(b) is plotted the phase of the transmission coefficient. In the same Figures, the FDTD results are also added. These curves show that the numerical and analytical calculations yield the same results. We have also validated the analytical results for other values of d (not shown here). From this, we can conclude that our theory is validated.

If the mesh ∆ tends to zero, i.e. if the space between the sheet and the interface tends to zero, then δ tends to zero and the LHM slab becomes perfectly matched to air. Analytical results

FDTD results

Frequency (Ghz) In this work, a new theory on the characteristic impedance of media with simultaneously negative permittivity and permeability, called left-handed media (LHMs), has been proposed. This theory shows that the characteristic impedance, as it is usually defined, is negative for these media. An interpretation of this result has been proposed and it has been explained why one can encounter difficulties when one considers a homogeneous LHM. Another result of our analysis is that only the non-homogeneousness is a required condition for the existence of a passive LHM and not the frequency-dispersive property which can be a consequence of the inhomogeneousness. In other words, there is a limit in the homogenization of a periodic structure, and this limit should be clearly understand to avoid misconception and surprising results. A method has been proposed to match, for forward waves, an LHM slab to free space, by using sheets of resistors and amplifiers. We have also considered the effect of an imperfectly matching. The transmission and reflection characteristics of an imperfectly matched LHM slab has been calculated by using analytical methods (transmission line and multiple-reflections methods). The proposed scheme allows to excite the forward waves of the LHM instead of the backward waves.

The numerical results presented in this paper can be tested easily by researchers active in the electromagnetic domain, without developing new numerical methods. Future work will concentrate on the analysis of the excitation by plane wave at oblique incidence of an LHM slab and on the flat lens, by using the proposed scheme. We will also verify our theory with artificial LHMs made with periodic structures.

Figure 1 :

 1 Figure 1: LHM homogenization: principle and limit. (a) Step 1: multiple wave reflections occur inside the periodic structure (b)Step 2: because of the angular-dispersive nature of the LHM and because the LHM is matched to air for backward waves, the total forward wave is negligible as compared to the total backward wave. Thus, the LHM is equivalent to a medium with an index near -1. However, if one starts his numerical analysis by considering a homogeneous medium with index -1 (step 2), one will encounter a big problem!: where does the backward wave can come from!? In conclusion, an LHM has to be inhomogeneous for testing the backward wave impedance matching to air. In regards to these facts, the frequency-dispersive nature of an LHM becomes not a necessary condition for the existence of a passive LHM but only the inhomogeneousness. Indeed, the negative total power (P = |E| 2 + µ|H| 2 ) inside the LHM only means that the wave goes to the source.

Figure 2 :

 2 Figure 2: Principle of the matched left-handed material for forward wave. (a) An incident plane wave illuminates a matched LHM. (b) Matching condition at the left-side interface. (c) Matching condition at the right-side interface.

Figure 3 :

 3 Figure 3: (a) Schematic for the calculation of the transmission and refection coefficients in the FDTD method. (b) Model of the resistors sheet in the FDTD method. Note that for a T M z wave excitation, only the resistors in z-direction are required.

Figure 4 :

 4 Figure 4: Amplitude of the field versus x-direction at different time instants. The slab width is d = 40 mm.A plane wave with a gaussian pulse comes from the left side. The position of the LHM is indicated in the figure. Note that the wave appears to propagate in the LHM as in the air (except for a small reflected energy part that is due to imperfection of the matching).

Figure 5 :

 5 Figure 5: (a) Computed amplitudes of the transmission and reflection coefficients, T and R, of the matched LHM slab vs. frequency, for different values of the slab width (FDTD). (b) Phase of the transmission coefficient vs. frequency, for different values of the slab width.

Figure 6 :

 6 Figure 6: Distributions of the phase and the amplitude of the normalized field in x-direction for d = 40 mm at 3.7 GHz (d ≈ λ/2). At this frequency the LHM is matched (T = 1, R = 0). The position of the LHM is indicated in the figures. (a) Distribution of the phase. (b) Distribution of the amplitude. Note that the amplitude varies inside the LHM, which indicates that a stationary waves regime appears.

8

 8 Phase of transmission coefficient (rad)

Figure 7 :

 7 Figure 7: Comparison between analytical and FDTD results for the characteristics of the LHM slab with matching sheets vs. frequency, for d = 40 mm and ∆ = 5mm: (a) amplitudes of the transmission and reflection coefficients, T and R (b) phase of the transmission coefficient

ACKNOWLEDGMENT

The authors gratefully acknowledge the support of this research from National Science Engineering Research Council of Canada (NSERC). The authors wish also to thank Dr. Claude Kondo Assi, Prof. Christophe Caloz, and Prof. Giuseppe Conciauro, for helpful discussions or comments concerning the negative characteristic impedance.


Emmanuel Gobet
email: emmanuel.gobet@imag.fr

Céline Labart
email: celine.labart@inria.fr.

Sharp estimates for the convergence of the density of the Euler scheme in small time

Keywords: stochastic differential equation, Euler scheme, rate of convergence, Malliavin calculus. AMS classification: 65C20 60H07 60H10 65G99 65M15 60J60

In this work, we approximate a diffusion process by its Euler scheme and we study the convergence of the density of the marginal laws. We improve previous estimates especially for small time.

Introduction

Let us consider a d-dimensional diffusion process (X s ) 0≤s≤T and a q-dimensional Brownian motion (W s ) 0≤s≤T . X satisfies the following SDE dX i s = b i (s, X s )ds + q j=1 σ ij (s, X s )dW j s , X i 0 = x i , ∀i ∈ {1, • • • , d}.

(1.1)

We approximate X by its Euler scheme with N (N ≥ 1) time steps, say X N , defined as follows. We consider the regular grid

{0 = t 0 < t 1 < • • • < t N = T } of the interval [0, T ], i.e. t k = k T N .
We put X N 0 = x and for all i ∈ {1, • • • , d} we define

X N,i u = X N,i t k + b i (t k , X N t k )(u -t N k ) + q j=1 σ ij (t k , X N t k )(W j u -W j t k ), for u ∈ [t k , t k+1 [. (1.2)
The continuous Euler scheme is an Itô process verifying

X N u = x + u 0 b(ϕ(s), X N ϕ(s) )ds + u 0 σ(ϕ(s), X N ϕ(s) )dW s
where ϕ(u) := sup{t k : t k ≤ u}. If σ is uniformly elliptic, the Markov process X admits a transition probability density p(0, x; s, y). Concerning X N (which is not Markovian except at times (t k ) k ), X N s has a probability density p N (0, x; s, y), for any s > 0. We aim at proving sharp estimates of the difference p(0, x; s, y)p N (0, x; s, y).

It is well known (see [START_REF] Bally | The law of Euler scheme for stochastic differential equations: convergence rate of the density[END_REF], [START_REF] Konakov | Edgeworth type expansions for Euler schemes for stochastic differential equations[END_REF], Guyon (2006)) that this difference is of order 1 N . However, the known upper bounds of this difference are too rough for small values of s. In this work, we provide tight upper bounds of |p(0, x; s, y)p N (0, x; s, y)| in s (see Theorem 3), so that we can estimate quantities like

E[f (X N T )] -E[f (X T )] or E T 0 f (X N ϕ(s) )ds -E T 0 f (X s )ds (1.3)
(without any regularity assumptions on f ) more accurately than before (see Theorem 5).

For other applications, see [START_REF] Labart | [END_REF]. Unlike previous references, we allow b and σ to be time-dependent and assume they are only C 3 in space. Besides, we use Malliavin's calculus tools.

Background results

The difference p(0, x; s, y)p N (0, x; s, y) has been studied a lot. We can found several results in the literature on expansions w.r.t. N . First, we mention a result from [START_REF] Bally | The law of Euler scheme for stochastic differential equations: convergence rate of the density[END_REF] (Corollary 2.7). The authors assume Hypothesis 1 σ is elliptic (with σ only depending on x) and b, σ are C ∞ (R d ) functions whose derivatives of any order greater or equal to 1 are bounded.

By using Malliavin's calculus, they show that

p(0, x; T, y) -p N (0, x; T, y) = 1 N π T (x, y) + 1 N 2 R N T (x, y), (1.4) with |π T (x, y)| + |R N T (x, y)| ≤ K(T ) T q exp(-c |x-y| 2 T )
, where c > 0, q > 0 and K(•) is a non decreasing function. We point out that q is unknown, which doesn't enable to deduce the behavior of pp N when T → 0.

Besides that, [START_REF] Konakov | Edgeworth type expansions for Euler schemes for stochastic differential equations[END_REF] have proposed an analytical approach based on the so-called parametrix method to bound p(0, x; 1, y)p N (0, x; 1, y) from above. They assume Hypothesis 2 σ is elliptic and b, σ are C ∞ (R d ) functions whose derivatives of any order are bounded.

For each pair (x, y) they get an expansion of arbitrary order j of p N (0, x; 1, y). The coefficients of the expansion depend on N p(0, x; 1, y)p N (0, x;

1, y) = j-1 i=1 1 N i π N,i (0, x; 1, y) + O( 1 N j ).
(1.5)

The coefficients have Gaussian tails : for each i they find constants

c 1 > 0, c 2 > 0 s.t. for all N ≥ 1 and all x, y ∈ R d , |π N,i (0, x; 1, y)| ≤ c 1 exp(-c 2 |x -y| 2 ).
To do so, they use upper bounds for the partial derivatives of p (coming from [START_REF] Friedman | Partial differential equations of parabolic type[END_REF]) and prove analogous results on the derivatives of p N . Strong though this result may be, nothing is said when replacing 1 by t, for t → 0. That's why we present now the work of Guyon (2006).

Guyon ( 2006) improves (1.4) and (1.5) in the following way.

Definition 1. Let G l (R d ), l ∈ Z be the set of all measurable functions π :

R d × (0, 1] × R d → R s.t.
• for all t ∈ (0, 1], π(•; t, •) is infinitely differentiable,

• for all α, β ∈ N d , there exist two constants c 1 ≥ 0 and c 2 > 0 s.t. for all t ∈ (0, 1] and

x, y ∈ R d , |∂ α x ∂ β y π(x; t, y)| ≤ c 1 t -(|α|+|β|+d+l)/2 exp(-c 2 |x -y| 2 /t).
Under Hypothesis 2 and for T = 1, the author has proved the following expansions

p N -p = π N + π N N 2 , (1.6) p N -p = j-1 i=1 π N,i N i + j i=2 t - ⌊N t⌋ N i π ′ N,i + π ′′ N,j N j , (1.7) where π ∈ G 1 (R d ) and (π N , N ≥ 1) is a bounded sequence in G 4 (R d ). For each i ≥ 1, (π N,i , N ≥ 1) is a bounded family in G 2i-2 (R d )
, and (π ′ N,i , N ≥ 1), (π ′′ N,i , N ≥ 1) are two bounded families in G 2i (R d ). These expansions can be seen as improvements of (1.4) and (1.5) : it also allows infinite differentiations w.r.t. x and y and makes precise the way the coefficients explode when t tends to 0.

As a consequence (see Guyon (2006, Corollary 22)), one gets

|p(0, x; s, y) -p N (0, x; s, y)| ≤ c 1 N s d+2 2 e -c 2 |x-y| 2 s , (1.8)
for two positive constants c 1 and c 2 , and for any x, y and s ≤ 1. This result should be compared with the one of Theorem 3 (when T = 1), in which the upper bound is tighter (s has a smaller power).

Main Results

Before stating the main result of the paper, we introduce the following notation Definition 2. C k,l b denotes the set of continuously differentiable bounded functions φ : (t, x) ∈ [0, T ] × R d with uniformly bounded derivatives w.r.t. t (resp. w.r.t. x) up to order k (resp. up to order l).

The main result of the paper, whose proof is postponed to Section 4, is established under the following Hypothesis

Hypothesis 3 σ is uniformly elliptic, b and σ are in C 1,3 b and ∂ t σ is in C 0,1 b .
Theorem 3. Assume Hypothesis 3. Then, there exist a constant c > 0 and a non decreasing function K, depending on the dimension d and on the upper bounds of σ, b and their derivatives s

.t. ∀(s, x, y) ∈]0, T ] × R d × R d , one has |p(0, x; s, y) -p N (0, x; s, y)| ≤ K(T )T N s d+1 2 exp - c|x -y| 2 s .
Corollary 4. Assume Hypothesis 3. From the last inequality and Aronson's inequality (A.1), we deduce

p(0, x; T, x) -p N (0, x; T, x) p(0, x; T, x) ≤ K(T ) N √ T . (2.1)
This inequality yields p(0, x; T, x) ∼ p N (0, x; T, x) when T → 0.

Theorem 3 enables to bound quantities like in (1.3) in the following way Theorem 5. Assume Hypothesis 3. For any function

f such that |f (x)| ≤ c 1 e c 2 |x| , it holds E[f (X N T )] -E[f (X T )] ≤ c 1 e c 2 |x| K(T ) √ T N , E T 0 f (X N ϕ(s) )ds -E T 0 f (X s )ds ≤ c 1 e c 2 |x| K(T ) T N .
Had we used the results stated by Guyon (2006) (and more precisely the one recalled in (1.8)), we would have obtained

E[f (X N T )] -E[f (X T )] = O( 1 N ).
Intuitively, this result is not optimal: the right hand side doesn't tend to 0 when T goes to 0 while it should. Analogously, regarding E

T 0 f (X N ϕ(s) )ds -E T 0 f (X ϕ(s) )ds , we would obtain O( T ln N N ) instead of O( T N ). Proof of Theorem 5. Writing E[f (X N T )]-E[f (X T )] as R d f (y)(p N (0,
x; T, y)-p(0, x; T, y))dy and using Theorem 3 yield the first result.

Concerning the second result, we split

E T 0 (f (X N ϕ(s) ) -f (X s ))ds in two terms E T 0 (f (X N ϕ(s) ) -f (X ϕ(s) ))ds and E T 0 (f (X ϕ(s) ) -f (X s ))ds . First, using Theorem 3 leads to E T 0 (f (X N ϕ(s) ) -f (X ϕ(s) ))ds = R d dy T T N dsf (y)(p N (0, x; ϕ(s), y) -p(0, x; ϕ(s), y)) , ≤ K(T )T N c 1 e c 2 |x| T T N ds ϕ(s) ,
where we use the easy inequality

R d e c 2 |y| e -c|x-y| 2 s s d/2 dy ≤ K(T )e c 2 |x| . Since ϕ(s) ≥ s -T N , we get E T 0 (f (X N ϕ(s) ) -f (X ϕ(s) ))ds ≤ K(T )T 3/2 N c 1 e c 2 |x|
. Second, we write

E T 0 (f (X ϕ(s) ) -f (X s ))ds ≤ c 1 e c 2 |x| T N + R d dy T T N dsc 1 e c 2 |y| s ϕ(s) du|∂ u p(0, x; u, y)|.
Then, Proposition 13 yields

E T 0 (f (X ϕ(s) ) -f (X s ))ds ≤ c 1 e c 2 |x| T N + C T T N ln( s ϕ(s) )ds . Moreover, T T N ln( s ϕ(s) )ds = N -1 k=1 t k+1 t k ln( s t k )ds = T N N -1 k=1 ((k + 1) ln( k+1 k ) -1) ≤ C T N , using a second order Taylor expansion. This gives E T 0 (f (X ϕ(s) ) -f (X s ))ds ≤ c 1 e c 2 |x| K(T ) T N .
In the next section, we give results related to Malliavin's calculus, that will be useful for the proof of Theorem 3.

Basic results on Malliavin's calculus

We refer the reader to [START_REF] Nualart | Malliavin Calculus and Related Topics (2nd Edition)[END_REF], for more details. Fix a filtered probability space (Ω, F, (F t ), P) and let (W t ) t≥0 be a q-dimensional Brownian motion. For h(•)

∈ H = L 2 ([0, T ], R q ), W (h) is the Wiener stochastic integral T 0 h(t)dW t . Let S denote the class of random variables of the form F = f (W (h 1 ), • • • , W (h n )) where f is a C ∞ function with derivatives having a polynomial growth, (h 1 , • • • , h n ) ∈ H n and n ≥ 1. For F ∈ S, we define its derivative DF = (D t F := (D 1 t F, • • • , D q t F )) t∈[0,T ]
as the H valued random variable given by

D t F = n i=1 ∂ x i f (W (h 1 ), • • • , W (h n ))h i (t).
The operator D is closable as an operator from L p (Ω) to L p (Ω; H), for p ≥ 1. Its domain is denoted by D 1,p w.r.t. the norm F 1,p = [E|F | p + E( DF p H )] 1/p . We can define the iteration of the operator D, in such a way that for a smooth random variable F , the derivative D k F is a random variable with values on H ⊗k . As in the case k = 1, the operator

D k is closable from S ⊂ L p (Ω) into L p (Ω; H ⊗k ), p ≥ 1. If we define the norm F k,p = [E|F | p + k j=1 E( D j F p H ⊗j )] 1/p ,
we denote its domain by D k,p . Finally, set D k,∞ = ∩ p≥1 D k,p , and D ∞ = ∩ k,p≥1 D k,p . One has the following chain rule property

Proposition 6. Fix p ≥ 1. For f ∈ C 1 b (R d , R), and F = (F 1 , • • • , F d ) * a random vector whose components belong to D 1,p , f (F ) ∈ D 1,p and for t ≥ 0, one has D t (f (F )) = f ′ (F )D t F, with the notation D t F =    D t F 1 . . . D t F d    ∈ R d ⊗ R q .
We now introduce the Skorohod integral δ, defined as the adjoint operator of D.

Proposition 7. δ is a linear operator on

L 2 ([0, T ] × Ω, R q ) with values in L 2 (Ω) s.t.
• the domain of δ (denoted by

Dom(δ)) is the set of processes u ∈ L 2 ([0, T ] × Ω, R q ) s.t. |E( T 0 D t F • u t dt)| ≤ c(u)|F | L 2 for any F ∈ D 1,2 . • If u belongs to Dom(δ), then δ(u)
is the one element of L 2 (Ω) characterized by the integration by parts formula

∀F ∈ D 1,2 , E(F δ(u)) = E T 0 D t F • u t dt .
Remark 8. If u is an adapted process belonging to L 2 ([0, T ] × Ω, R q ), then the Skorohod integral and the Itô integral coincide : δ(u) = T 0 u t dW t , and the preceding integration by parts formula becomes

∀F ∈ D 1,2 , E F T 0 u t dW t = E T 0 D t F • u t dt . (3.1)
This equality is also called the duality formula.

This duality formula is the corner stone to establish general integration by parts formula of the form

E[∂ α g(F )G] = E[g(F )H α (F, G)]
for any non degenerate random variables F . We only give the formulation in the case of interest F = X N t .

Proposition 9. We assume that σ is uniformly elliptic and b and σ are in C 0,3 b . For all p > 1, for all multi-index α s.t. |α| ≤ 2, for all t ∈]0, T ], all u, r, s ∈ [0, T ] and for any functions f and g in C |α| b , there exist a random variable H α ∈ L p and a function K(T ) (uniform in N, x, s, u, r, t, f and g) s.t.

E[∂ α x f (X N t )g(X N u , X N r , X N s )] = E[f (X N t )H α ], (3.2) with |H α | Lp ≤ K(T ) t |α| 2 g C |α| b . (3.3)
These results are given in the article of [START_REF] Kusuoka | Applications to the Malliavin calculus I. Stochastic Analysis[END_REF] Although this upper bound seems to be quite standard, to our knowledge such a result has not appeared in the literature before, except in the case of time homogeneous coefficients (see [START_REF] Konakov | Edgeworth type expansions for Euler schemes for stochastic differential equations[END_REF], proof of Theorem 1.1).

Proof. The inequality (1.32) of Kusuoka and Stroock (1984, Theorem 1.31) gives p N (0, x; s, y) ≤ K(T ) s d/2 for any x and y. This implies the required upper bound when |x -y| ≤ √ s. Let us now consider the case |x -y| > √ s. Using the same notations as in [START_REF] Kusuoka | Applications to the Malliavin calculus I. Stochastic Analysis[END_REF], we denote ψ(y) = ρ( |y-x| r ) where r > 0 and ρ is a

C ∞ b function such that 1 {[3/4,∞[} ≤ ρ ≤ 1 {[1/2,∞[} .
Then, combining inequality (1.33) of Kusuoka and Stroock (1984, Theorem 1.31)

Proof of Theorem 3

In the following, K(•) denotes a generic non decreasing function (which may depend on b and σ). To prove Theorem 3, we take advantage of Propositions 9 and 10. The scheme of the proof is the following

• Use a PDE and Itô's calculus to write the difference p N (0, x; s, y)p(0, x; s, y)

= s 0 E d i=1 (b i (ϕ(r), X N ϕ(r) ) -b i (r, X N r ))∂ x i p(r, X N r ; s, y) + 1 2 d i,j=1 (a ij (ϕ(r), X N ϕ(r) ) -a ij (r, X N r ))∂ 2 x i x j p(r, X N r ; s, y)   dr := E 1 + E 2 . (4.1) • Prove the intermediate result ∀(r, x, y) ∈ [0, s[×R d × R d and c > 0 E exp -c |y -X N r | 2 s -r ≤ K(T ) s -r s d 2 exp -c ′ |x -y| 2 s , (4.2)
where c ′ > 0.

• Use Malliavin's calculus, Proposition 10 and the intermediate result, to show that each term E 1 and E 2 (see (4.1)) is bounded by

K(T )T N 1 s d+1 2 exp(-c |x-y| 2 s ).
Definition 11. We say that a term E(x, s, y) satisfies property

P if ∀(x, s, y) ∈ R d ×]0, T ]×R d |E(x, s, y)| ≤ K(T )T N 1 s d+1 2 exp -c |x -y| 2 s . (P)
4.1 Proof of equality (4.1)

First, the transition density function (r, x) -→ p(r, x; s, y) satisfies the PDE

(∂ r + L (r,x) )p(r, x; s, y) = 0, ∀r ∈ [0, s[, ∀x ∈ R d ,
where

L (r,x) is defined by L (r,x) = i,j a ij (r, x)∂ 2 x i x j + i b i (r, x)∂ x i , and a ij (r, x) = 1 2 [σσ * ] ij (r, x).
The function, as well as its first derivatives, are uniformly bounded by a constant depending on ǫ for |s -r| ≥ ǫ (see Appendix A). Second, since p N (0, x; s, y) is a continuous function in s and y (convolution of Gaussian densities), we observe that

p N (0, x; s, y) -p(0, x; s, y) = lim ǫ→0 E[p(s -ǫ, X N s-ǫ ; s, y) -p(0, x; s, y)].
Then, for any ǫ > 0, Itô's formula leads to

E[p(s -ǫ, X N s-ǫ ; s, y) -p(0, x; s, y)] = E s-ǫ 0 ∂ r p(r, X N r ; s, y)dr + E   s-ǫ 0 d i=1 b i (ϕ(r), X N ϕ(r) )∂ x i p(r, X N r ; s, y)dr + 1 2 s-ǫ 0 d i,j=1 a ij (ϕ(r), X N ϕ(r) )∂ 2 x i x j p(r, X N r ; s, y)dr   .
From the PDE, the above equality becomes

E[p(s -ǫ, X N s-ǫ ; s, y)-p(0, x; s, y)] = E s-ǫ 0 d i=1 (b i (ϕ(r), X N ϕ(r) ) -b i (r, X N r ))∂ x i p(r, X N r ; s, y)dr + 1 2 E   s-ǫ 0 d i,j=1 (a ij (ϕ(r), X N ϕ(r) ) -a ij (r, X N r ))∂ 2 x i x j p(r, X N r ; s, y)dr   , := s-ǫ 0 E[φ(r)]dr, where φ(r) = d i=1 (b i (ϕ(r), X N ϕ(r) ) -b i (r, X N r ))∂ x i p(r, X N r ; s, y) + 1 2 d i,j=1 (a ij (ϕ(r), X N ϕ(r) ) - a ij (r, X N r ))∂ 2 x i x j p(r, X N r ; s, y).
To get (4.1), it remains to prove that E(φ(r)) is integrable over [0, s]. We check it by looking at the rest of the proof.

Proof of the intermediate result (4.2)

We prove inequality (4.

2). E[exp(-c |y-X N r | 2 s-r )] = R d exp(-c |y-z| 2
s-r )p N (0, x; r, z)dz. Using Proposition 10, we get

E exp -c |y -X N r | 2 s -r ≤ K(T ) r d 2 R d exp -c |y -z| 2 s -r exp -c ′ |x -z| 2 r dz ≤ K(T )Π d i=1 R 1 √ r exp -c |y i -z i | 2 s -r exp -c ′ |x i -z i | 2 r dz i ,
and

R 1 2π (s-r) 2c exp(-c |y i -z i | 2 s -r ) 1 2π r 2c ′ exp(-c ′ |x i -z i | 2 r
)dz i is the convolution product of the density of two independant Gaussian random variables N (-x i , r 2c ′ ) and N (y i , s-r 2c ) computed at 0. Hence, the integral is equal to 1

2π( r 2c ′ + s-r 2c ) exp - |x i -y i | 2 r c ′ + s-r c . Then, R 1 √ r exp -c |y i -z i | 2 s -r exp -c ′ |x i -z i | 2 r dz i ≤ C s -r s 1 2 exp -c ′′ |x i -y i | 2 s
and (4.2) follows.

Upper bound for E 1

We recall that

E 1 = s 0 E d i=1 (b i (ϕ(r), X N ϕ(r) ) -b i (r, X N r ))∂ x i p(r, X N r ; s, y) dr.
For each i, we apply Itô's formula to b i (u, X N u ) between u = ϕ(r) and u = r. We get

b i (ϕ(r), X N ϕ(r) ) -b i (r, X N r ) = r ϕ(r) α i u du + r ϕ(r) q k=1 β i,k u dW k u , (4.3)
where α i u depends on ∂ t b, ∂ x b, ∂ 2 x b, σ, and

β i u = -∇ x b i (u, X N u )σ(ϕ(r), X N ϕ(r) ). Since b, σ belong to C 1,3
b , α i and (β i,k ) 1≤k≤q uniformly bounded. Using (4.3) and the duality formula (3.1) yield

E 1 = d i=1 s 0 {E[ r ϕ(r) ∂ x i p(r, X N r ; s, y)α i u du + E[ r ϕ(r) D u (∂ x i p(r, X N r ; s, y)) • β i u du]}dr := E 11 + E 12 , (4.4)
where β i u is a row vector of q components. We upper bound E 11 and E 12 .

Bound for Besides that, from Proposition 13, |∂ x p(r, X N r ; s, y)| ≤ K(T ) (s-r)

E 11 = d i=1 s 0 E[ r ϕ(r) ∂ x i p(r, X N r ; s, y)α i u du]dr. Since | d i=1 ∂ x i p(r, X N r ; s, y)α i u | ≤ |α u ||∂ x p(
d+1 2 exp -c |y-X N r | 2 s-r . Then, |E 11 | ≤ K(T ) T N s 0 1 (s -r) d+1 2 E exp -c |y -X N r | 2 s -r dr.
Using the intermediate result (4.2) yields

|E 11 | ≤ K(T ) T N s 0 1 √ s -r 1 s d 2 exp -c |x -y| 2 s dr ≤ K(T ) T N 1 s d-1 2 exp -c |x -y| 2 s
and thus, E 11 satisfies property P (see Definition 11).

Bound for

E 12 = d i=1 s 0 E[ r ϕ(r) D u (∂ x i p(r, X N r ; s, y)) • β i u du]dr.
To rewrite E 12 , we use the expression of β i u and Proposition 6, which gives D u (∂ x i p(r, X N r ; s, y)) = ∇ x (∂ x i p(r, X N r ; s, y))σ(ϕ(r), X N ϕ(r) ). Then,

E 12 = - s 0 dr r ϕ(r) d i,k=1 E[∂ 2 x i x k p(r, X N r ; s, y)[(σσ * )(ϕ(r), X N ϕ(r) )(∇ x b i (u, X N u )) * ] k ]du. (4.5)
Using the integration by parts formula (3.2), we get that

E 12 = -
p(r, X N r ; s, y)| d+1 d ] d/(d+1) ≤ K(T ) (s -r) d+1 2 s -r s d 2 2(d+1) exp -c |y -x| 2 s . (4.6)
Hence, E 12 is bounded by

K(T ) s d 2 2(d+1) T N exp -c |y -x| 2 s s 0 1 r 1/2 1 (s -r) d+1 2 -d 2 2(d+1) dr.
The above integral equals s

1 2 -d+1 2 + d 2 2(d+1) B( 1 2 , 1 2(d+1) ) where B is the function Beta. Thus |E 12 | ≤ K(T ) s d/2 T N exp(-c |y-x| 2 s )
, and E 12 satisfies property P.

Upper bound for E 2

We recall

E 2 = 1 2 s 0 E[ d i,j=1 (a ij (ϕ(r), X N ϕ(r) ) -a ij (r, X N r ))∂ 2
x i x j p(r, X N r ; s, y)]dr. As we did for E 1 , we apply Itô's formula to a ij (u, X N u ) between ϕ(r) and r. We get a ij (ϕ(r),

X N ϕ(r) ) - a ij (r, X N r ) = r ϕ(r) γ ij u du + r ϕ(r) δ ij u dW u , where γ ij u depends on σ, ∂ t σ, ∂ x σ, b, ∂ 2 xx σ and δ ij u is a row vector of size q, with l-th component (δ ij u ) l = -d k=1 ∂ x k a ij (u, X N u )σ kl (ϕ(r), X N ϕ(r)
). Then, the duality formula (3.1) leads to

E 2 = d i,j=1 s 0 {E[ r ϕ(r) ∂ 2 x i x j p(r, X N r ; s, y)γ ij u du + E[ r ϕ(r) D u (∂ 2 x i x j p(r, X N r ; s, y)) • δ ij u du]}dr := E 21 + E 22 .
Bound for

E 21 = d ij=1 s 0 E[ r ϕ(r) ∂ 2 x i x j p(r, X N r ; s, y)γ ij u du]dr. As σ, b, ∂ t σ, ∂ x σ, ∂ 2 x σ are C 1 b in space, γ ij u has the same smoothness properties as the term [(σσ * )(ϕ(r), X N ϕ(r) )(∇ x b i (u, X N u )
) * ] k appearing in (4.5). Thus, E 21 can be treated as E 12 and satisfies to the same estimate.

Bound for E

22 = d i,j=1 s 0 E[ r ϕ(r) D u (∂ 2 x i x j p(r, X N r ; s, y)) • δ ij u du]dr.
To rewrite E 22 , we use the expression of δ ij u and Proposition 6, which asserts

D u (∂ 2 x i x j p(r, X N r ; s, y)) = ∇ x (∂ 2 x i x j p(r, X N r ; s, y))σ(ϕ(r), X N ϕ(r)
). Thus,

E 22 = - d i,j,k=1 s 0 dr r ϕ(r) E[∂ 3 x i x j x k p(r, X N r ; s, y)[(σσ * )(ϕ(r), X N ϕ(r) )(∇ x a ij (u, X N u )) * ] k ]du.
To complete this proof, we split E 22 in two terms : E 1 22 (resp E 2 22 ) corresponds to the integral in r from 0 to s 2 (resp. from s 2 to s).

• On [0, s 2 ], E A Bounds for the transition density function and its derivatives

We bring together classical results related to bounds for the transition probability density of X defined by (1.1).

Proposition 12 [START_REF] Aronson | Bounds for the fundamental solution of a parabolic equation[END_REF]). Assume that the coefficients σ and b are bounded measurable functions and that σ is uniformly elliptic. There exist positive constants K, α 0 , α 1 s.t. for any x, y in R d and any 0 ≤ t < s ≤ T , one has

K -1 (2πα 1 (s -t))

  : (3.3) is owed to Theorem 1.20 and Corollary 3.7. Another consequence of the duality formula is the derivation of an upper bound for p N . Proposition 10. Assume σ is uniformly elliptic and b and σ are in C 0,2 b . Then, for any x, y ∈ R d , s ∈]0, T ], one has p N (0, x; s, y) ≤ K(T ) s d/2 e -c |x-y| 2 constant c and a non decreasing function K, both depending on d and on the upper bounds for b, σ and their derivatives.

  This easily completes the proof in the case |x -y| ≥ √ s.

  r, X N r ; s, y)| and α u is uniformly bounded in u, we have |E 11 | ≤ C T N s 0 E|∂ x p(r, X N r ; s, y)|dr.

  x i p(r, X N r ; s, y)H i e k (r, u)]du where e k is a vector whose k-th component is 1 and other components are 0. From (3.3), we deduceE[|H i e k (r, u)| p ] 1/p ≤ C K(T ) r 1/2 ,where C only depends on |σ| ∞ , |∂ x σ| ∞ , |∂ x b| ∞ , |∂ 2 xx b| ∞ . By the Hölder inequality, it follows that |E 12 | ≤ K(T ) leads to |∂ x p(r, X N r ; s, y)| ≤ K(T ) ), and combining this inequality with the intermediate result (4.2) yields x

  3x i x j x k p(r, X N r ; s, y)|]dr. Using Proposition 13 and (4.2), it gives r) 3/2 dr.Hence, E 22 satisfies P.• On [ s 2 , s], we use the integration by parts formula (3.2) of Proposition 9, with |α| = 2. x i p(r, X N r ; s, y)H i e jk ]du, where e jk is a vector full of zeros except the j-th and the k-th components. Using Hölder's inequality and (3.3) (remember that σ ∈ C 1,3 b ), we obtain |E 2 22 | ≤ K(T ) x p(r, X N r ; s, y)|

  s-t) ≤ p(t, x; s, y) ≤ K 1 (2πα 2 (st))Proposition 13[START_REF] Friedman | Partial differential equations of parabolic type[END_REF]). Assume that the coefficients b and σ are Hölder continuous in time, C 2 b in space and that σ is uniformly elliptic. Then, ∂ m+a x ∂ b y p(t, x; s, y) exist and are continuous functions for all 0 ≤ |a| + |b| ≤ 2, |m| = 0, 1. Moreover, there exist two positive constants c and K s.t. for any x, y in R d and any 0 ≤ t < s ≤ T , one has |∂ m+a x ∂ b y p(t, x; s, y)| ≤ K (st) (|m|+|a|+|b|+d)/2 exp -c |y -x| 2 st .


Alexandrina Orzan

Jean-Marc Hasenfratz

Omnidirectional texturing of human actors from multiple view video sequences

In 3D video, recorded object behaviors can be observed from any viewpoint, because the 3D video registers the object's 3D shape and color. However, the real-world views are limited to the views from a number of cameras, so only a coarse model of the object can be recovered in real-time. It becomes then necessary to judiciously texture the object with images recovered from the cameras. One of the problems in multi-texturing is to decide what portion of the 3D model is visible from what camera. We propose a texture-mapping algorithm that tries to bypass the problem of exactly deciding if a point is visible or not from a certain camera. Given more than two color values for each pixel, a statistical test allows to exclude outlying color data before blending.

Introduction

Currently, visual media such as television and motion pictures only present a 2D impression of the real world. In the last few years, increasingly more research activity has been devoted to investigate 3D video from multiple camera views. The goal is to obtain a free-viewpoint video, where the user is able to watch a scene from an arbitrary viewpoint chosen interactively.

The possible applications are manifold. A freeviewpoint system can increase the visual realism of telepresence technology 1 , thus enabling users in different locations to collaborate in a shared, simulated environment as if they were in the same physical room. Also, special effects used by the movie industry, such as freeze-and-rotate camera, would be made accessible to all users.

For free-viewpoint video, a scene is typically captured by N cameras. From the views obtained by the cameras a 3D video object, with its shape and appearance, is created. The shape can be described by polygon meshes, point samples or voxels. In order to make the model more realistic, appearance is typically described by the textures captured from the video streams. Appearance data is mapped onto the 3D shape, thus completing the virtual representation of the real object. The 3D video object can be seamlessly blended into existing content, where it can be interactively viewed from different directions, or under different illumination.

Since people are central to most visual media content, research has been dedicated in particular to the extraction and reconstruction of human actors.However, the system used in this article is not restricted to human actors, as [START_REF] Carranza | Free-viewpoint video of human actors[END_REF]. Moreover, it allows the acquisition of multiple objects present in the scene.

The rest of the paper proceeds with a review of related work in section 2. Section 3 will be dedicated to describing the proposed method of texture-mapping, after which results and future tasks are discussed.

Previous Work

Over the last few years, several systems with different model reconstruction and different ways of texturing the 3D model have been proposed.

3D Model reconstruction

Two different approaches of model reconstruction have been studied in the recent years: model-free and modelbased reconstruction.

Model-free reconstruction makes no a priori assumptions on scene geometry, allowing the reconstruction of complex dynamic scenes. In human modeling it allows the reproduction of detailed dynamics for hair and loose clothing.

Most model-free methods aim to estimate the visual hull, an approximate shell that envelopes the true geometry of the object [START_REF] Laurentini | The visual hull concept for silhouette-based image understanding[END_REF]. To achieve this, object silhouettes are extracted from each camera image by detecting the pixels not belonging to the background.

The visual hull can then be reconstructed either by voxel-based or polyhedron-based approaches. The first approach discretizes a confined 3D space in voxels and carves away those voxels whose projection fall outside the silhouette of any reference view [START_REF] Hasenfratz | A real-time system for full body interaction[END_REF]. Polyhedronbased approaches represent each visual cone as a polyhedral object and computes the intersection of all visual cones [START_REF] Li | Online accelerated rendering of visual hulls in real scenes[END_REF][START_REF] Matusik | Polyhedral visual hulls for real-time rendering[END_REF][START_REF] Matsuyama | Generation, visualization, and editing of 3d video[END_REF].

The visual hull allows real-time reconstruction and rendering, yet it needs a large number of views to accurately represent a scene, otherwise the obtained model is not very exact.

Model-based reconstruction assumes that the real object is a human body and uses a generic humanoid model, which is deformed to fit the observed silhouettes [START_REF] Carranza | Free-viewpoint video of human actors[END_REF][START_REF] Hilton | Model-based multiple view reconstruction of people[END_REF][START_REF] Hilton | Multiple view reconstruction of people[END_REF]. Although it results in a more accurate model and permits motion tracking over time, this approach is restricted to a simple model and does not allow complex clothing movements. Moreover, it places a severe limitation on what can be captured (i.e. a single human body) and it is not real-time.

In this paper, the 3D model used is the one created in the context of CYBER-II project 2 , a polyhedron-based model obtained in real-time.

Multi-view texture mapping

Original images from multiple viewpoint are often mapped onto recovered 3D geometry in order to achieve realistic rendering results [START_REF] Debevec | Modeling and rendering architecture from photographs: A hybrid geometry-and imagebased approach[END_REF]. Proposed methods for 2 http://artis.imag.fr/Projects/Cyber-II/ multi-texture mapping are either view-dependent or view-independent.

View-dependent texture mapping considers only the camera views closest to the current viewpoint. In between camera views, two to four textures are blended together in order to obtain the current view image [START_REF] Debevec | Efficient view-dependent image-based rendering with projective texture-mapping[END_REF][START_REF] Goldlücke | Realtime microfacet billboarding for free-viewpoint video rendering[END_REF]. This method exhibits noticeable blending artifacts in parts where the model geometry does not exactly correspond to the observed shape. What's more, the result is usually blurred and the passing from one camera view to another does not always go unnoticed.

View-independent texture mapping selects the most appropriate camera for each triangle of the 3D model, independently of the viewer's viewpoint [START_REF] Carranza | Free-viewpoint video of human actors[END_REF][START_REF] Hilton | Model-based multiple view reconstruction of people[END_REF][START_REF] Matsuyama | Generation, visualization, and editing of 3d video[END_REF]. The advantage of this method is that it does not change the triangle texture when the user changes the viewpoint. Moreover, the blurred effect is less noticeable. However, the problem is that the best camera is not the same from patch to patch, even if they are neighboring. Here also, blending between visible views is necessary in order to reduce the abrupt change in texture at triangle edges.

Blending is done using various formulas that depend of:

-the angle between the surface normal and the vector towards the considered camera -the angle between the surface normal and the vector towards the viewpoint -the angle the vector towards a camera and the vector towards the viewpoint Blending weights can be computed per vertex or per polygon [START_REF] Carranza | Free-viewpoint video of human actors[END_REF][START_REF] Debevec | Efficient view-dependent image-based rendering with projective texture-mapping[END_REF][START_REF] Goldlücke | Realtime microfacet billboarding for free-viewpoint video rendering[END_REF]. Matsuyama [START_REF] Matsuyama | Generation, visualization, and editing of 3d video[END_REF] proposes using this method for determining each vertex color and then paints the triangles with the colors obtained by linearly interpolating the RGB values of its vertices. However, for large triangles, small details like creases in the clothes are lost.

Li and Magnor [START_REF] Li | A hybrid hardware-accelerated algorithm for high quality rendering of visual hulls[END_REF] compute the blending for each rasterized fragment, which results in a more accurate blending.

Visibility

Visibilities with respect to reference views are very important for multi-view texture mapping. For those parts that are invisible in a reference view, the corresponding color information should be ignored when blending multiple textures.

Debevec et al. [START_REF] Debevec | Modeling and rendering architecture from photographs: A hybrid geometry-and imagebased approach[END_REF] splits the object triangles so that they are either fully visible or fully invisible to any source view. This process takes a long time even for a moderately complex object and is not suitable for real-time applications. Matusik [START_REF] Matusik | Polyhedral visual hulls for real-time rendering[END_REF] proposes computing the vertex visibility at the same time that the visual hull is generated. Magnor et al. [START_REF] Li | A hybrid hardware-accelerated algorithm for high quality rendering of visual hulls[END_REF] solves the visibility problem per fragment, using shadow mapping. However, they require rendering the scene from each input camera viewpoint and is not real-time even with a hardwareaccelerated implementation.

We propose a per pixel method that checks only polygon visibility and eliminates the wrong colors by considering only those colors that are close to a computed average.

Texture mapping algorithm

Model constraints

The polyhedron-based model-free method recreates the geometrical object at each frame. The number of polygons, their form and position in space vary greatly in time, so we cannot track vertices from one frame to another.

This means that it is impossible to decide the color of the polygons only once, at the beginning of the video. Color values have to be computed in real-time, for each frame.

Algorithm description

To achieve realistic rendering results, we use the projective texture mapping, a method introduced by Segal [START_REF] Segal | Fast shadows and lighting effects using texture mapping[END_REF] and included in the OpenGL graphics standard. But the current hardware implementation of projective texture mapping in OpenGL lets the texture pass through the geometry and be mapped onto all back-facing and occluded polygons. Thus it is necessary to perform visibility check so that only polygons visible to a particular camera are texture mapped with the corresponding image.

A point p on the object's surface is visible from a camera c i if (1) the triangle t j to which the point belongs faces the camera and (2) the point is not occluded by any other triangles.

The first condition can be fast determined by checking the equation n t j • v c i →t j < 0, where n t j is the triangle normal vector and v c i →t j is the viewing direction from c i towards the centroid of t j .

Still, in a per-pixel approach, we do not have the geometrical data. We solve this problem by an additional rendering of the object from the current viewpoint, where we use the polygon ID as its color. Thus, we can determine what polygons are visible from the viewpoint and exactly which pixel of the current image view belongs to which triangle.

Determining if a point viewed by the viewer is occluded or not to the cameras is a less obvious problem. Methods to determine what points are occluded were briefly presented in the previous section. We propose to bypass the occlusion checking by doing a basic statistical test. The strong condition that has to be fulfilled is that for each pixel at least three cameras have to pass the first visibility test, and the majority has to see the correct color. Still, this is usually the case with a system having an evenly distributed camera configuration.

As all the cameras are calibrated prior to use and the images are acquired at the same time and in the same lighting conditions, we can compare colors and calculate distances in the RGB space [START_REF] Agathos | Colour texture fusion of multiple range images[END_REF][START_REF] Grammatikopoulos | On automatic orthoprojection and texture-mapping of 3d surface models[END_REF].

If a sufficient number of color values are available for a pixel, we compute the mean (µ) and the standard deviation (τ) for each of the R, G, B channels. Individual colors falling outside the range µ ± β • τ for at least one channel are excluded. The factor β permits us to modify the confidence interval for which the colors are accepted. The classical normal deviation test considers β is 1. We experimentally concluded that it was best to set it at 1.17, to allow for slight errors in manipulating the color-values.

If less than three possible colors are available for a pixel, we do not exclude any of them.

A weighted mean of all contributing images is finally used for texturing each particular pixel. The blending weight is computed using the value of the cos(angle(n t j , v c i →t j )).

If the pixel is invisible for all cameras, we compute its color using the color values of the neighbours whose color was already decided.

The algorithm runs as follows:

1: for all polygons in the 3D model do

Results

We tested this algorithm with the system used by the CYBER-II project. The system has 6 cameras, 4 in the front and 2 in the back, as seen in Figure 1 3 . For the front views, the algorithm succeeded in eliminating the wrong colors and in seamlessly mixing data from various cameras. Moreover, the pixel color doesn't change with the change of viewpoint. Images comparing view-dependent and view-independent algorithms, without occlusion checking, and our method can be seen in Figure 2.

However, for the back views, where the object is seen by at most 2 cameras, the algorithm does only a weighted average, without color elimination. We set the resolution of the rendered novel view to 512x512, and we tested the algorithm for a model of about 5000 polygons. On a Intel 2.40GHz CPU and a GeForce4 Ti 4800 graphic card, the frame rate is of 17 fps.

Conclusions and Future work

A per-pixel algorithm for multi-view texture mapping has been implemented. It succeeds in eliminating wrong colors for pixels viewed by more that 2 cameras, without doing a time-consuming occlusion checking.

Yet, further enhancements are both necessary and feasible. Thus, a hardware-implementation should be considered, since the mane time-consuming task in our algorithm is transferring information from the framebuffer to the CPU. Moreover, we would like to consider a continuity in time of the computed pixel colors and a dynamic deactivation of the unused cameras.

  actually present in a different place or time (S. Fisher & B. Laurel, 1991) or enables objects from a different place to feel as if they are actually present (T. Lacey & W. Chapin, 1994).

2 :for all cameras do 6 :

 26 check if they are at least partially visible from the current view 3: end for 4: for all pixels in the image view do 5:if the polygon that colored the pixel faces the camera then

3

  video sequences were acquired with the Grimage platform of Inria Rhône-Alpes

Figure 1 :

 1 Figure 1: Camera setting

Figure 2 :

 2 Figure 2: a) View dependent, b) View independent, c) Our method


Objectives

• Observe individuals' reactions and strategies when facing inequalities of income.

-When facing an inequality (whether advantageous/disadvantageous), how do individuals react? -Do people care about the difference between their situation and others' situation? -Can the inequality induce the individuals to make additional efforts in order to increase their situation (Negative action afterwards)? -Can the inequality induce individuals to destroy other's situation (Positive action afterwards) ?

The Good, the Bad and the Selfish

• Study the intensity of individuals' reactions in front of inequalities.

-Do people react identically to inequalities? -Do people exhibit higher efforts (whether positive or destructive) when the difference is at their advantage or at their disadvantage ?

• Does the absolute effect exerts an influence on individuals' behaviour?

-Are individuals' actions similar in the case when the absolute difference is high than when the absolute difference is low?

The Good, the Bad and the Selfish

The Good, the Bad and the Selfish Experimental Protocol

• Strategy Method (Selten).

• Subjects randomly and anonymously paired.

• Partner Design.

• Each player receives an initial endowment of 10 ECUs (Experimental Currency Unit). • A subject can only face one kind of inequality (whether advantageous, equal or disadvantageous). • The experiment consists in 3 steps.

Conversion rate : 10 ECUs = 1 €

The Good, the Bad and the Selfish The Good, the Bad and the Selfish

First step :

All subjects receive information about :

-Their own endowment (conditional on the treatment). -The endowment of the other member of the pair (opponent afterwards). -The value of action C.

-The value of the multiplicative coefficient.

The Good, the Bad and the Selfish Second step :

• Each subject has to choose a specific action among 3 distinct actions:

-Action A; consists in investing part of the initial capita in order to reduce the opponent's payoff. -Action B; consists in investing part of the initial capita in order to increase the subject's own payoff. -Action C; consists in receiving the highest amount possible but also increases the opponent's payoff (subject renounces to his initial capita).

Any integer amount invested in actions A or B is multiplied by a coefficient (which vary among treatments), identical for both actions. Any amount kept is not multiplied.

The Good, the Bad and the Selfish Third step : (120 ; 120)

12

(120 ; 240)

-120 12In red, the subject's payoff. In italics, absolute difference between the subject's payoff and the payoff of the opponent.

Treatments

The Good, the Bad and the Selfish

Global results

Action • Most actions undertaken are actions B (positive actions), i.e. individuals are willing to increase their own payoff without increasing/decreasing the difference between their position and others' position.

•Actions A only represent 10% of total decisions. Sample = 30 subjects. Sessions organized at LEEM (Montpellier).

The Good, the Bad and the Selfish •Whereas actions A are little, worst-off players took up to three times more negative actions than well-off players and equal players.

Distribution of actions among treatments

•Worst-off and well-off players invested a significantly more important part of their initial capita in negative actions than equal players.

the Bad and the Selfish

	Advantageous	Equality	Disadvantageous
	Inequality			Inequality
	• If choosing action A or B, subjects have to (60; 30) (30 ; 30) (30 ; 60)
	determine the value of the action. +30 3 3	-30 3
	• Subjects have to indicate the (integer) amount (80; 40) (40 ; 40) (40 ; 80)
	of the initial capita (10 ECU) they want to +40 4 4 -40 4
	invest in their action. (120; 60) (60, 60)		(60 ; 120)
	+60 6 • The value of the action is determined by the 6 -60 6 (160 ; 80) (80 ; 80) (80 ; 160) amount invested (≤10 ECU) multiplied by a +80 8 8 -80 8 coefficient. (240 ; 120)
	+120 12

The Good,

The Good, the Bad and the Selfish

	The Good, the Bad and the Selfish
	Individual(s) results : About disadvantaged players? Individual(s) results : About advantaged Individual(s) results : About equal players? players?
	Use of actions concerning disadvantaged players : Use of actions concerning equal players :
	Opponent's Opponent's Opponent's	Opponent's Opponent's Opponent's	Opponent's Opponent's Opponent's
		action A action A action A	action B action B action B	action C action C action C
	Action A Action A Action A	46,15% 50% 20%	30,77% 40% 30%	23,08% 10% 50%
	Action B Action B Action B	39,70% 29,20% 39,30%	35,30% 31,40% 36,70%	25% 39,40% 24%
	Action C Action C Action C	19,64% 38,88% 27,87%	32,14% 36,12% 29,50%	48,22% 25% 42,70%

Use of actions concerning advantaged players

Graphic evolution of actions A chosen

The Good, the Bad and the Selfish

Annex: Mean effort by action A and by inequality

37,33%

•Worst-off and well-off players invested a significantly more important part of their initial capita in negative actions than equal players.


inria-00281381, version 1 -22 May 2008

Exploring a type-theoretic approach to accessibility constraint modelling Sylvain Pogodalla LORIA/INRIA Accessibility constraints When dealing with anaphora resolution, anaphoric expressions are often considered to have access to a restricted set of discourse referents. The choice of the relevant discourse referent inside this set may depends on various kinds of information (morphosyntactic features, semantic features, salience. . . ). But theories that give a dynamic interpretation of language also provide some ways to restrict the set of discourse referents by stating accessibility constraints. The description of these constraints depends on the chosen formal theory of discourse interpretation. For instance, DRT [START_REF] Kamp | From Discourse to Logic[END_REF] describes the accessibility constraints using DRS subordination. Taking into account the hierarchical structure of discourse, SDRT [START_REF] Asher | Logics of conversation[END_REF] uses DRS subordination together with an outscoping relation between discourse units. In both cases, the constraints are based on the structure of the discourse: for DRT, the structure arises from the negation while for SDRT it also arises from the nature of the relation between discourse units. In the latter case, it gives a formal definition of the Right Frontier Constraint (RFC) [START_REF] Asher | Troubles on the right frontier[END_REF].

[3] proposes a modelling of DRT that fits Montague's semantic framework. In this approach, the logical quantifiers have their standard scoping definitions and free and bound variables also have their standard definitions. One of the advantages is to use standard notions (such as De Bruijn's indices for λ-calculus) for implementing variable renaming in semantic systems based on λ-calculus, instead of implementing specific and complex work-around. [START_REF] De | Towards a montagovian account of dynamics[END_REF] also advocates its independence from any specific theory. While [START_REF] De | Towards a montagovian account of dynamics[END_REF] exemplifies the DRT view on accessibility constraints, the work we propose here consists in exploring this approach in modelling two other kinds of accessibility constraints: the accessibility to proper nouns in DRT and the RFC.

A Montagovian flavour of DRT In [START_REF] De | Towards a montagovian account of dynamics[END_REF], the syntactic type of sentences s is interpreted with the semantic type: s = γ → (γ → t) → t. It means that (the semantic interpretation of) a sentence, instead of being a proposition, requires two arguments to produce a proposition. The first argument is its left context (we can think about it as the current set of discourse referents). The second argument is its right context, or its continuation: something able to return a proposition if fed with a (potentially updated) set of discourse referents.

Two sentences s 1 and s 2 are combined in the following way: s 1 .s 2 = λeφ. s 1 e (λe ′ . s 2 e ′ φ). It means that the result of the combination of s 1 and s 2 takes as input an environment e and a continuation φ. This environment is the same as the one s 1 has access to. So s 1 takes e as first parameter. Then, the continuation of s 1 is made of something taking e ′ as left environment 1 and the final result is the one we get by feeding s 2 with the new left environment e ′ and the same continuation as the s 1 .s 2 combination.

With this interpretation and the right semantic recipes (see [START_REF] De | Towards a montagovian account of dynamics[END_REF] for the detailed lexical semantics) we get the following interpretation for John loves a woman: λeφ.∃y.woman y ∧ love j y ∧ φ (y :: e) where :: (of type e → γ → γ) is a function that adds a discourse referent to a set of discourse referents. Note that the continuation of this sentence, φ, will be provided with the y discourse referent, bound in the standard way by the existential quantifier.

Modelling other accessibility constraints

To state as in DRT that negation blocks the access to discourse referents, [START_REF] De | Yet another dynamic logic[END_REF] interprets: (doesn't VP)S = λeφ.¬((VP S) e (λe ′ .⊤)) ∧ φ e. The continuation of the sentence (φ) has no access to what VP and S could introduce as discourse referents, it only has access to the discourse referents of e2 .

Note that an alternative could have been: (doesn't VP)S = λeφ.¬((VP S) e (λe ′ .φ e ′ )). But in this case, the continuation would also have been in the scope of the negation: the rest of the discourse would also be negated, which is not what we expect.

We propose to introduce the type of logical connectives κ = t → t → t and to interpret the syntactic type of sentences as s = κ → γ → (κ → γ → t) → t. Then negation can be interpreted as: (doesn't VP) S = λceφ.¬((VP S) (¬c) e(λc ′ e ′ .¬(φ c ′ e))). Without all the details, if we think as c being the logical connective ∧, we have the result to be interpreted as ¬((VP S) ∨ ¬(φ e)) ≡ (¬(VP S)) ∧ (φ e) which is now the expected result. Interestingly, we see that it could also give access to the discourse referents introduced by VP S to φ using the continuation λc ′ e ′ .¬(φ c ′ e ′ ). This, of course, is not expected, except for proper nouns (which are "always on top" in DRT). We then see how to add an additional left environment parameter for proper nouns (say e 1 ) so that negation blocks the discourse referents introduced by existentials (in e 2 ) but propagates the ones introduced by proper nouns. With the new value of thanks to the following equivalence (c is either ∧ or ∨):

s = κ → γ → γ → (κ → γ → γ → t) → t
.⊤)x c ′ e ′ 1 e ′ 2 φ ′ )(φ ′ c ′ e ′ 1 e ′ 2
(c(car y)(φ ′ c e 1 e 2 )) ∧ (c(own x y)(φ ′ ce 1 e 2 ))) ≡ c(car y ∧ own x y)(φ ′ ce 1 e 2 )
Finally, a discourse d such as John doesn't own a car. It is red with the empty context nil and the empty continuation φ e = λce 1 e 2 .¬(c ⊤⊥) (which returns ⊤ under the ∧ connective and ⊥ under the ∨ connective) is interpreted as:

d (∧) nil nil φ e = ¬(∃y.(car y ∧ own j y) ∨ (¬(red(sel((j :: nil) ∪ nil))) ∨ ⊥))
= ¬(∃y.(car y ∧ own j y) ∨ (¬(red(sel(j :: nil))))) ≡ (¬∃y.(car y ∧ own j y)) ∧ red(sel(j :: nil))

It shows that whereas the discourse referent introduced by y is not accessible to the sel operator, the discourse referent introduced by j would be accessible, in spite of the negation. We extend this approach to model the RFC, introducing a parameter for the type of discourse relations: subordinating or coordinating, and two ways of combining sentences: s 1 . c s 2 and s 1 . s s 2 instead of just one s 1 .s 2 . The interpretation of these two combinations manage the set of discourse referents so that the environment given to s 2 only includes the set of discourse referents defined by the RFC. As for now, we only deal with discourse structures that can be described by a formula (or syntactic tree) made of the . c and of the . s connectives. Discourse pops or discourse structure that are not trees, for instance, are not yet considered.

We consider a new type κ = γ → γ → γ and

s = κ → γ → γ → (κ → γ → γ → t) → t,

Conclusion

We show how [START_REF] De | Towards a montagovian account of dynamics[END_REF]'s approach is flexible enough to model various accessibility constraints, such as the ones for discourse referents introduced by proper nouns or by the hierarchical structure of the discourse. Moreover, it proves to be able to combine the different constraints. We hope this could be helpful to give an account of the hierarchy of referential expressions and their adequacy to the RFC [START_REF] Asher | Troubles on the right frontier[END_REF] by combining various constraints for pronouns or definite descriptions for instance.

  )) red = λP xce 1 e 2 φ.(P x c e 1 e 2 φ) ∧ (red x) we can interpret: own ( a car ) = λS.S(λxce 1 e 2 φ.∃y.[λφ ′ .(c(car y)(φ ′ c e 1 e 2 )) ∧ (c(own x y)(φ ′ ce 1 e 2 ))] (λc ′ e ′ 1 e ′ 2 .φ c e ′ 1 (y :: e ′ 2 ))) ≡ λS.S(λxce 1 e 2 φ.∃y.[λφ ′ .c((car y) ∧ (own x y))(φ ′ c e 1 e 2 )] (λc ′ e ′ 1 e ′ 2 .φ c e ′ 1 (y :: e ′ 2 ))) = λS.S(λxce 1 e 2 φ.∃y.c((car y) ∧ (own x y))(φ c e 1 (y :: e 2 )))

  and of sentence composition:s 1 .s 2 = λce 1 e 2 φ. s 1 c e 1 e 2 (λc ′ e ′ 1 e ′ 2 . s 2 c ′ e ′ 1 e ′ 2 φ))3 , the negation is now interpreted as: doesn't = λV Sce 1 e 2 φ.¬((V S) (¬c) e 1 e 2 (λc ′ e ′ 1 e ′

[START_REF] Asher | Logics of conversation[END_REF]

.¬(φ c ′ e ′ 1 e 2 ))). Together with the following interpretations:

John = λP ce 1 e 2 φ.P j c (j :: e 1 ) e 2 φ own = λOS.S(λx.O(λyc ′ e ′ 1 e ′ 2 φ ′ .c ′ (own x y)(φ ′ c ′ e ′ 1 e ′ 2 ))) car = λxce 1 e 2 φ.c(car x)(φ c

e 1 e 2 ) a = λP Qce 1 e 2 φ.∃x.[λφ ′ .(P x c e 1 e 2 φ ′ ) ∧ (Q x c e 1 e 2 φ ′ )](λc ′ e ′ 1 e ′ 2 .φce ′ 1 (x :: e ′ 2 )) it = λP.P (sel (e 1 ∪ e 2 )) is = λAS.S(λxc ′ e ′ 1 e ′ 2 φ ′ .c ′ (A(λyc ′′ e ′′ 1 e ′′ 2 φ ′′

  and we define Coord = λe 1 e 2 .e 2 and Sub = λe 1 e 2 .e 1 ∪ e 2 . We give the following definition of sentence composition:s 1 . s s 2 = λce 1 e 2 φ.s 1 c e 1 e 2 (λc ′ e ′ 1 e ′ 2 . s 2 Sub e ′ 1 (c e 1 e 2 ) φ) s 1 . c s 2 = λce 1 e 2 φ. s 1 c e 1 e 2 (λc ′ e ′ 1 e ′ 2 . s 2 Coord e ′ 1 (c e 1 e 2 ) φ) Then s 1 . c (s 2 . c s 3 ) = λce 1 e 2 φ. s 1 c e 1 e 2 (λc ′ e ′ 1 e ′ 2 . s 2 Coord e ′ 1 (c e 1 e 2 ) (λc ′′′ e ′′′ 1 e ′′′ 2 . s 3 Coord e ′′′ 1 (c e 1 e 2 ) φ)) s 1 . c (s 2 . s s 3 ) = λce 1 e 2 φ. s 1 c e 1 e 2 (λc ′ e ′ 1 e ′ 2 . s 2 Coord e ′ 1 (c e 1 e 2 ) (λc ′′′ e ′′′ 1 e ′′′ 2 . s 3 Sub e ′′′ 1 (c e 1 e 2 ) φ)) s 1 . s (s 2 . c s 3 ) = λce 1 e 2 φ. s 1 c e 1 e 2 (λc ′ e ′ 1 e ′ 2 . s 2 Sub e ′ 1 (c e 1 e 2 ) (λc ′′′ e ′′′ 1 e ′′′ 2 . s 3 Coord e ′′′ 1 (e ′ 1 ∪ (c e 1 e 2 )) φ)) s 1 . s (s 2 . s s 3 ) = λce 1 e 2 φ. s 1 c e 1 e 2 (λc ′ e ′ 1 e ′ 2 . s 2 Sub e ′ 1 (c e 1 e 2 ) (λc ′′′ e ′′′ 1 e ′′′ 2 . s 3 Sub e ′′′ 1 (e ′ 1 ∪ (c e 1 e 2 )) φ)) which means that in addition to the environment introduced by their previous discourse unit (e ′ 1 for s 2 and e ′′ 1 for s 3 ), s 2 and s 3 have access to (c e 1 e 2 ) whose value is: either e 2 if the whole part of the discourse is in a coordinating relation with what comes before (and then only s 1 should access e 1 ), or e 1 ∪ e 2 if the whole part is in a subordinating relation with what comes before (and the previous unit dominates s 1 , s 2 and s 3 and each have access to e 1 and e 2 ).

Note that the actual value of e ′ will be provided by the semantic recipe of s1. That is s1 can choose to update or not e before it gives it as parameter to its continuation.

⊤ stands for true. That is, λe.⊤ is the continuation that always returns true.

We use here a currified notation that introduces as many arrows as functions have parameters. We could of course use a product or record type to keep only one parameter.