Mathias Rousset

Scalable and Quasi-Contractive Markov Coupling of Maxwell Collision

published or not. The documents may come

This paper considers space homgenous Boltzmann kinetic equations in dimension d, with conservative (two-body) collisions of Maxwell type. Emphasis is made on the associated conservative (or Nanbu) N -particle system; a Markov process denoted (up to particle permutations Sym N ) t → V N t = (V N t,(1) , . . . , V N t,(N )

) ∈ R d N [Sym N ], (1)
and satisfying the following conservation laws (for any t ≥ 0):

V N t N = 0 a.s., V N t 2 N = 1 a.s., (2)
where in the above, the bracket denotes the averaging over particles ( . N ≡ 1 N N n=1 ). Moreover, the latter process is reversible with respect to the invariant uniform probability distribution

Unif N 0,1 def = Unif v ∈ R d N
(2) holds , equivalently defined as the Riemannian volume of the associated sphere, or as the conditional distribution associated with the average momentum and energy observables. The Markov dynamics of ( 1) is specified by two-body random collisions of (Levy) jump type, satisfying two-body conservation of momentum and kinetic energy (particles have identical masses). By definition of Maxwell collisions, the rate of the two-body collisions is constant; and by Galilean invariance, a random two-body collision is necessarily an isotropic random step on the euclidean sphere of possible collisional directions. The collisional direction is the direction of the velocity difference of a particle pair. The angular size (θ ∈ [0, π]) of the step is the so-called scattering (or deviation) angle.The angular collision kernel b(dθ) is a positive Levy measure on [0, π] generating the random steps of the scattering angle, with Levy condition

λ def = [0,π]
sin 2 θ b(dθ) < +∞.

(

) 3
More notation on the collision processes of interest are given in Section 2.3, and standard details, especially intended for the unfamiliar audience, are given in Section 3.

It is thus possible to construct an explicit Markov coupling (i.e. a probabilistic coupling of two copies of a Markov process which is itself again Markov) denoted t → (U N t , V N t ) ≡ (U t,(1) , V t,(1) , . . . , U t,(N ) , V t,(N )

) ∈ R d × R d N [Sym N ], (4)
which is globally invariant by particle permutation, such that U N t ∈ R d N and V N t ∈ R d N indpendently satisfy the conservation laws [START_REF] Wilbur | An introduction to multivariate statistical analysis[END_REF], and such that collisions are coupled using the following set of rules:

(i) Collision times and collisional particles are the same.

(ii) Scattering angles are the same.

(iii) The isotropic random step on the collisional direction is coupled using plain parallel transport (in the geometric sense), with no reflexion.

The sphere being a strictly positively curved manifold, the latter coupling is bound to be almost surely decreasing, in the sense that for any initial condition and 0

≤ t ≤ t + h U N t+h -V N t+h 2 N ≤ U N t -V N t 2 N a.s.. (5)
The goal of this paper is to study the quasi-contractivity of the latter coupling, uniformly in the number of particles N . We first compute the coupling creation by computing the time derivative of the average coupling distance

d dt E U N t -V N t 2 N = -E C U N t , V N t , U N * ,t , V N * ,t N ≤ 0. (6)
In the above, and in the rest of the paper, the following notation is used

o u N , v N , u N * , v N * N def = 1 N 2 N n 1 ,n 2 =1 o(u N (n 1 ) , u N (n 1 ) , u N (n 2 ) , v N (n 2 ) ),
in order to account for averages over particles of a two-body observable o :

R d × R d 2 → R.
It turns out that the two-body "coupling creation" functional in the right hand side of ( 6) is given by the following alignement between the velocity difference vv * ∈ R d , and its coupled counterpart uu * ∈ R d :

C(u, v, u * , v * ) = λ c d-1 c d-3 |u -u * | |v -v * | -(u -u * ) • (v -v * ) ≥ 0. (7)
In the above c d = π/2 0 sin d (ϕ) dϕ denotes the d'th Wallis integral, and λ > 0 is the variance of the scattering angle kernel, as defined in [START_REF] Aleksandr | Moment inequalities for the boltzmann equation and applications to spatially homogeneous problems[END_REF].

In order to relate the coupling and the coupling creation, we will introduce in the present paper an original general sharp inequaIity (see [START_REF] Otto | Generalization of an inequality by talagrand and links with the logarithmic sobolev inequality[END_REF] below) that holds for any couple of centered and normalized random variables in R d . In the present context, it takes the following form

f u N -v N 2 N ≤ min κ u N ⊗u N N , κ v N ⊗v N N × u N -u N * 2 v N -v N * 2 -u N -u N
for any vectors u N ∈ (R d ) N and v N ∈ (R d ) N both satisfying the conservation laws (2). In the above, the condition number

κ S def = (1 -|||S|||) -1 ∈ [ d d -1 , +∞] (9)
is a function of the spectral radius |||S||| ≤ 1 of a positive trace 1 symmetric matrix; and

f : [0, 4] → [0, 1] x → x -x 2 4 , (10)
is a positive concave function (used througout the paper) satisfying f (x) ∼ x→0

x, as well as [START_REF] Ea | Strict entropy production bounds and stability of the rate of convergence to equilibrium for the boltzmann equation[END_REF]. Moreover, the equality case in ( 8) is satisfied under the following sufficient two conditions:

f (4 -x) = f (x) which ensures the symmetry v N → -v N in
1. Co-linearity of

u (n) |u(n)| and v (n) |v(n)| for all 1 ≤ n ≤ N ; 2. Isotropy of co-variances u N ⊗ u N N = u N ⊗ v N N = 1 d Id or v N ⊗ v N N = u N ⊗ v N N = 1 d Id.
It is then of interest to compare that the alignement functional in the right hand side of (8) (which is a sharp upper bound of the square coupling distance), and the coupling creation functional [START_REF] Bolley | Uniform convergence to equilibrium for granular media[END_REF]. They differ by a weight of the form |uu * | |vv * | which forbids any strong "coupling/coupling creation" inequality of the form

f u N -v N 2 N C(u N , v N , u N * , u N * ) N ≤ r < +∞
for some universal constant r > 0 independant of N and of the the pair

(u N , v N ) ∈ (R d ×R d ) N
both satisfying the conservation laws (2). However, a direct Hölder inequality yields some weaker power law versions (see Section 2.5), for any α ∈]0, +∞[:

f u N -v N 2 N C u N , v N , u N * , v N * α 1+α N ≤ r α,u N ,v N < +∞, (11)
The inequality [START_REF] Eric A Carlen | Determination of the spectral gap in the kac's model for physical momentum and energy-conserving collisions[END_REF] can be interpreted as a power law (of order α) estimate of the coupling (quasi-)contractivity (i.e. with speed ∼ t→+∞ t -α ); and as expected, r α,u N ,v N can be controlled by any N -averaged finite moments of order > 2 + α of the velocity distributions. Such results are similar to the classical results ( [START_REF] Cercignani | H-theorem and trend to equilibrium in the kinetic theory of gases[END_REF][START_REF] Ea | Strict entropy production bounds and stability of the rate of convergence to equilibrium for the boltzmann equation[END_REF][START_REF] Bobylev | On the rate of entropy production for the boltzmann equation[END_REF][START_REF] Ea Carlen | Propagation of smoothness and the rate of exponential convergence to equilibrium for a spatially homogeneous maxwellian gas[END_REF][START_REF] Toscani | Sharp entropy dissipation bounds and explicit rate of trend to equilibrium for the spatially homogeneous boltzmann equation[END_REF]) that are obtained with entropy methods using "entropy/entropy creation" inequalities. Yet, the coupling method has some noticeable specificities:

Maxwell restriction The analysis is restricted to Maxwell collisions.

Angular condition

The analysis is independent of the scattering angular distribution of collisions.

Particle system size The analysis is independent of the particle system size N . It works similarly for the kinetic equation.

A priori estimates The analysis depends on higher > 2 moments of velocity distributions, and not on regularity estimates. Such estimates are available for the kinetic equation, but unfortunately, up to our knowledge, not directly for the N -particle system in the spirit of [START_REF] Ikenberry | On the pressures and flux of energy in a gas according to {M} axwell\'s kinetic theory[END_REF] . However, they can be obtained (indirectly and at least in principle) by pull-back, by using uniform in time large N propagation of chaos, as proven in [START_REF] Mischler | About kac's program in kinetic theory[END_REF].

Constants Constants are simple and explicit.

Sharpness

The weak "coupling/coupling creation inequalities" are derived via Hölder inequality from the key sharp inequality [START_REF] Ea | Strict entropy production bounds and stability of the rate of convergence to equilibrium for the boltzmann equation[END_REF].

The latter method then yields some lower bound estimates on the contraction rate (of the particle system distribution, or of the kinetic equation) with respect to the L 2 -Wasserstein metric. In the case of the kinetic equation with Maxwell collisions, the latter contraction has been shown to be strictly positive (thus implying uniqueness) in the classical paper by Tanaka [START_REF] Tanaka | Probabilistic treatment of the boltzmann equation of maxwellian molecules[END_REF]. In a sense, the analysis in the present paper makes Tanaka's argument quantitative. We also suggest some negative results in the form of two counterexamples to stronger versions of "coupling/coupling creation inequalities". These are the counterpart of counterexamples to Cercignani's conjecture ( [START_REF] Bobylev | The theory of the nonlinear boltzmann equation for maxwell molecules[END_REF][START_REF] Bobylev | On the rate of entropy production for the boltzmann equation[END_REF][START_REF] Villani | Cercignani's conjecture is sometimes true and always almost true[END_REF]) in the entropy context.

(i) Velocity distributions with sufficiently heavy tails can make the coupling creation vanish. This first counterexample shows that any "coupling/coupling contraction inequality" must involve some higher order (say, > 2) velocity distribution moments.

(ii) There exists a continuous perturbation of the identity coupling at equilibrium for which however the coupling creation is sub-linearly smaller than the coupling itself. This second type of counterexample shows that even with some reasonable moment or coupling domain restrictions, a sub-exponential trend is unavoidable.

Contents

In Section 2.1, we summarize the literature related to the present work. In Section 2.2, we recall some notation and basic concepts related to probabilistic couplings for Markov particle systems. In Section 2.3, we define the random collisions coupling of interest. In Section 2.4, we detail the associated kinetic equations. In Section 2.5, the main results of the present paper are stated. In Section 2.6, the two counter-examples showing the necessity of a sub-exponential trend and of finite higher moments are presented and proved.

In Section 3 and sub-sections there in, standard facts on conservative random collisions and associated particle systems are recalled.

In Section 4.1, an explicit (i.e. coordinate) formulation of spherical couplings is detailed. In Section 4.2, the contractivity of spherical couplings is computed. In Section 4.3, the coupling creation of the coupled particle system is, in turn, computed. In Section 4.4, the proof of the quasi-contractivity in appropriate Wasserstein distance of the kinetic equation and of its associated conservative particle system is detailed.

In Section 5.1, the special inequality between coupling distance and colinearity of coupled pairs is proven. In Section (5.2), the quasi-contractive coupling / coupling creation estimate is deduced.

Context and results

Context

The mathematical literature on the space homogenous Boltzmann kinetic equations (and related models) is extremely vast, especially in the case of Maxwell collisions, and we refer to the classical reviews [START_REF] Cercignani | Mathematical methods in kinetic theory[END_REF][START_REF] Villani | A review of mathematical topics in collisional kinetic theory[END_REF]. In the same way, the use of explicit coupling methods to study the trend to equilibrium of Markov processes (or Markov chains) is now a classical topic on its own, especially for discrete models (see e.g. [START_REF] Asher | Markov chains and mixing times[END_REF]). It is also a well-established topic for continuous models, as well as for non-linear partial differential equations that have an interpretation in terms of Markovian particles. Let us mention some classical papers more closely related to the present study, with a sample of more recent references.

Well-posedness For Maxwell molecules, well-posedness of the kinetic equation as a probability flow using a probabilistic coupling was initiated in [START_REF] Tanaka | Probabilistic treatment of the boltzmann equation of maxwellian molecules[END_REF], with a sophisticated probabilistic technology. In [START_REF] Pulvirenti | The theory of the nonlinear boltzmann equation for maxwell molecules in fourier representation[END_REF][START_REF] Toscani | Probability metrics and uniqueness of the solution to the boltzmann equation for a maxwell gas[END_REF], an ad hoc spectral theory (Fourier) enabled to simplify the argument, and to push forward the theory. A recent example of the use of the coupling method to non-Maxwell molecules is available in [START_REF] Fournier | On the well-posedness of the spatially homogeneous boltzmann equation with a moderate angular singularity[END_REF].

Moments Moments production and moments propagation for the Boltzmann kinetic equation is a well established phenomenon. For Maxwell molecules, some explicit calculations are possible ( [START_REF] Ikenberry | On the pressures and flux of energy in a gas according to {M} axwell\'s kinetic theory[END_REF]), and finite vs. infinite moments strictly propagates. In general, higher moments estimates typically rely on the so-called Pozner inequality ( [START_REF] Wennberg | Entropy dissipation and moment production for the boltzmann equation[END_REF][START_REF] Aleksandr | Moment inequalities for the boltzmann equation and applications to spatially homogeneous problems[END_REF][START_REF] Ea Carlen | Propagation of smoothness and the rate of exponential convergence to equilibrium for a spatially homogeneous maxwellian gas[END_REF]). It is still a field of study ( [START_REF] Lu | On measure solutions of the boltzmann equation, part i: moment production and stability estimates[END_REF][START_REF] Alonso | A new approach to the creation and propagation of exponential moments in the boltzmann equation[END_REF]), with emphasis on moments production without any regularity theory. However, we are not aware of such results directly carried out on the associated N -particle systems, which would be very complementary to the present results.

Trend to equilibrium (i) As said before, most of the results on trend to equilibrium for Boltzmann kinetic equations, rely on "entropy / entropy creation" analysis ( [START_REF] Cercignani | H-theorem and trend to equilibrium in the kinetic theory of gases[END_REF][START_REF] Ea | Strict entropy production bounds and stability of the rate of convergence to equilibrium for the boltzmann equation[END_REF][START_REF] Bobylev | On the rate of entropy production for the boltzmann equation[END_REF][START_REF] Ea Carlen | Propagation of smoothness and the rate of exponential convergence to equilibrium for a spatially homogeneous maxwellian gas[END_REF][START_REF] Toscani | Sharp entropy dissipation bounds and explicit rate of trend to equilibrium for the spatially homogeneous boltzmann equation[END_REF]), which try to circumvent the breakdown of Cercignani's conjecture (counterexamples: [START_REF] Bobylev | The theory of the nonlinear boltzmann equation for maxwell molecules[END_REF][START_REF] Bobylev | On the rate of entropy production for the boltzmann equation[END_REF][START_REF] Villani | Cercignani's conjecture is sometimes true and always almost true[END_REF]). It is of interest to interpret the latter counterexamples with the Markovian viewpoint: they amount ot the impossibility of obtaining a uniform in N log-Sobolev (in the sense of entropy/entropy creation) inequality of the assoicated Nparticle systems. For Maxwell collisions, or Kac's caricature, special simplifications enables to push forward the trend estimations, for instance using Wild's expansion ( [START_REF] Lu | Fast and slow convergence to equilibrium for maxwellian molecules via wild sums[END_REF]), or central limit theorem to obtain sharp exponential rates ( [START_REF] Dolera | Reaching the best possible rate of convergence to equilibrium for solutions of kac equation via central limit theorem[END_REF][START_REF] Dolera | The role of the central limit theorem in discovering sharp rates of convergence to equilibrium for the solution of the kac equation[END_REF]).

Trend to equilibrium (ii) For other models of non-linear partial differential equations, a Markov coupling can give exponential trend to equilibrium, by using a "strong coupling/coupling creation inequalitiy" (see for instance [START_REF] Malrieu | Logarithmic sobolev inequalities for some nonlinear pde's[END_REF][START_REF] Bolley | Convergence to equilibrium in wasserstein distance for fokker-planck equations[END_REF][START_REF] Bolley | Uniform convergence to equilibrium for granular media[END_REF], for non-linear Fokker-Planck models with convexity assumption on potentials). However, the latter cases differ from the case of Boltzmann collisions by their exponential behavior. Indeed, and informally speaking, with respect to the following classical chain of implications (see [START_REF] Otto | Generalization of an inequality by talagrand and links with the logarithmic sobolev inequality[END_REF], using the Otto calculus viewpoint) for classical reversible diffusions on manifolds Contractive Markov coupling ⇒ Log-Sobolev inequality , the underlying constants in the latter models are in fact uniform in N . From the breakdown of Cercignani's conjecture, and assuming as an informal rationale that the latter ideas still hold for jump processes, strictly contractive Markov coupling cannot be constructed for Boltzmann collision processes.

Trend to equilibrium (iii) Direct probabilistic methods studying the trend of equilibrium of the Boltzmann-Maxwel N -particle system (or Kac's caricature) have been undertaken [START_REF] Diaconis | Bounds for kacś master equation[END_REF][START_REF] Eric A Carlen | Determination of the spectral gap for kacś master equation and related stochastic evolution[END_REF][START_REF] Eric A Carlen | Determination of the spectral gap in the kac's model for physical momentum and energy-conserving collisions[END_REF][START_REF] Imbuzeiro | On the convergence to equilibrium of kacś random walk on matrices[END_REF]. The main striking feature of the latter list is the difficulty to achieve the so-called "Kac's program" ( [START_REF] Mischler | About kac's program in kinetic theory[END_REF]), by obtaining a scalable (in N ) analysis of the trend to equilibrium of the particle system. In [START_REF] Eric A Carlen | Determination of the spectral gap for kacś master equation and related stochastic evolution[END_REF][START_REF] Eric A Carlen | Determination of the spectral gap in the kac's model for physical momentum and energy-conserving collisions[END_REF], the ananalysis computes exact spectral gaps, which are not N -scalable measures of trends to equilibrium. In [START_REF] Imbuzeiro | On the convergence to equilibrium of kacś random walk on matrices[END_REF], an explicit coupling method is used but the estimate focus on the contraction constant with optimal scale (O(ln N )) which still is not N -scalable.

Propagation of chaos

In [START_REF] Mischler | About kac's program in kinetic theory[END_REF] have reversed the latter point of view, and proved the trend to equilibrium of the N -particle system by pulling-back the long time stability of the kinetic (mean-field N = +∞ limit) equation with uniform in time propagation of chaos. Moreover, the use of coupling methods in this context have been undertaken in [START_REF] Fournier | Rate of convergence of the nanbu particle system for hard potentials[END_REF] to obtain sharper chaos propagation rates, at the price of time stability. These are nonetheless fairly indirect and impressively technical viewpoints.

In the present work, the whole analysis itself is independant of N , and the study is fully elementary, the only advanced tool being the final Hölder inequality. This generality and simplicity is one of the main motivation of the present work.

Markov coupling and particle systems

Let us now make a brief summary of some basic concepts related to coupling methods for Markov particle systems. Consider an exchangeable (particle permutation symmetric) and coupled N -particle system of the form (4). Strictly speaking, the state space is obtained by quotienting globally R d × R d N /Sym N with particle permutations (the coupling breaks the permutation symmetry, and the system is not in the product space

R d N /Sym N × R d N /Sym N ).
We also assume that the latter is a two-body interaction system, so that its Markov generator has the usual structure:

L N c def = 1 N N n,m=1 L (n,m) c (= N × L c N ) , (12)
where L c is a Markov generator with state space R d × R d 2 , the superscript (n, m) denotes the action on the corresponding pair of particles, and the particle averaging N have been obviously extended to opertaors.

We say that the latter process is a (time homogenous) Markov coupling if the following two conditions hold:

(i) The marginal distribution of the two processes t → U N t ∈ R d N and t → V N t ∈ R d N
are permutation symmetric and Markov with the same generator of the form

L N def = 1 N N n,m=1 L (n,m) (= N × L N ) . ( 13
) (ii) If U N 0 = V N 0 a.s., then U N t = V N t a.
s. for any t ≥ 0. Formally, the first point is direct consequence of the fact that for any

(u, v) ∈ R d × R d L c (ϕ ⊗ l l)(u, v) = L(ϕ)(u), L c (l l ⊗ ϕ)(u, v) = L(ϕ)(v), (14)
for any test function ϕ ranging in a domain of the generator L.

Assuming that the coupling is almost surely increasing ((5) holds), the two-body interaction structure implies that L 2 coupling creation as defined by ( 6) is given by a two-body positive coupling creation functional

C : R d × R d 2 → R +
given by the general formula

C(u, v, u * , v * ) def = -L c • (u, v, u * , v * ) → |u -v| 2 + |u * -v * | 2 . (15)
Next, for any coupled random variables

(U N , V N ) ∈ (R d × R d ) N
, and any α > 0, we will consider weak "coupling / coupling creation" inequalities of the form [START_REF] Eric A Carlen | Determination of the spectral gap in the kac's model for physical momentum and energy-conserving collisions[END_REF]. If the particle system has an invariant probability distribution, and if the constant in the right hand side of [START_REF] Eric A Carlen | Determination of the spectral gap in the kac's model for physical momentum and energy-conserving collisions[END_REF] is uniformly bounded in time and particle size N , a computable N -uniform trend to equilibrium of power law type ∼ t→+∞ t -α can be obtained. If the latter holds for any large α > 0, the coupling may be called N -uniform quasi-contractive.

Weak forms of contractivity in terms of Wasserstein metric can be obtained as a corollary. We can first introduce the "two-step" Wasserstein distance on exchangeable (permutation symmetric) particle systems.

Definition 2.1. Let U N ∈ (R d ) N and V N ∈ (R d ) N two
exchangeable (with Sym N -invariant distribution) random vectors satisfying the conservation laws (2). Denote by

η U N def = 1 N N n=1 δ U (n) ∈ P(R d ) the associated empirical distributions. Then the two-step L 2 -Wasserstein distance is defined by d W N 2 Law(U N ), Law(V N ) def = d W 2 ,(P(R d ),d W 2 ) (Law(η U N ), Law(η V N ))
between Law(U N ) and Law(V N ) is induced by the L 2 -Wasserstein distance between Law(η U N ) and Law(η V N ) on the space of random probability distributions in P(P(R d )), where P(R d ) is itself endowed with the usual R d -euclidean L 2 -Wasserstein distance. It is equivalent to the L 2 -Wassertsein in the quotient space (R d ) N /Sym N endowed with the quotient (orbifold) distance, defined by infimum for

(u N , v N ) ∈ (R d ) N /Sym N × (R d ) N /Sym N : inf σ∈Sym N u N -v N σ( . ) 2 N .
This yields the following general upper bound for any t 0 ≥ 0

d + dt d 2 W N 2 Law(U N t ), Law(V N t ) t=t 0 ≤ -E C U N t 0 , V N t 0 , U N t 0 , * , V N t 0 , * N . ( 16
)
In the above,

( U N , V N ) ∈ (R d × R d )
N is a random variable representation of any optimal coupling and d + dt is the right derivative which exists in [0, +∞] by monotony.

Coupled collisions

As usual, the post-collisional velocities of a particle pair are given by the collision mapping

(n ′ v → (v ′ , v ′ * ) = coll v,v * (n ′ v )) v ′ = 1 2 (v + v * ) + 1 2 |v -v * | n ′ v , v ′ * = 1 2 (v + v * ) -1 2 |v -v * | n ′ v , (17)
where

(n v , n ′ v ) = v-v * |v-v * | , v ′ -v ′ * |v ′ -v ′ * | ∈ S d-1 × S d-1 denote the collisional/post-collisional direc- tions. A Maxwell collision process for two particles in R d ×R d is the Markov process contructed from the following Levy generator on R d × R d : L(ϕ)(v, v * ) def = S d-1 ×[0,π] ϕ(v ′ , v ′ * ) -ϕ(v, v * ) c θ (n v , dn ′ v ) b(dθ), (18)
where in the above ϕ is a test function, b is an angular collisional kernel satisfying the Levy condition (3), and c θ is the isotropic probability transition on the collisional direction (the sphere S d-1 ) with prescribed scattering (or deviation) angle θ.

The 2-body coupled process is then obtained by (i) coupling the scattering angles θ; (ii) coupling the post-collisional directions n ′ u and n ′ v , using parallel transport between n u and n v on the sphere S d-1 (no reflexion). We state without proof (the reader may resort to a drawing here) two equivalent elementary descriptions of the parallel transport coupling on the sphere. We will call the latter spherical coupling.

Definition 2.2. Let (n u , n v ) ∈ S d-1 2 be given with condition n u = -n v . There is a unique rotation of S d-1 denoted n ′ u → n ′ v = Coupl nu,nv (n ′ u ) ∈ S d-1 , called spherical coupling, satisfying n ′ u = n ′ v if n u = n v ,

and equivalently defined as follows for n

u = n v . (i) n ′ v is obtained from n ′ u by performing the elementary rotation in Span(n u , n v ) bringing n u to n v .
(ii) Denote by t u a tangent vector of S d-1 at base point n u of a geodesic of length θ bringing n u to n ′ u . Generate t v from t u by using parallel transport in Span(n u , n v ) from base point n u to base point n v . Generate n ′ v as the endpoint of the geodesic of length θ and tangent to t v at base point n v .

Moreover, it satisfies by construction the symmetry condition

Coupl nv,nu = Coupl -1 nu,nv . (19)
It is necessary to keep in mind that the full mapping (n u , n v ) → Coupl nu,nv is smooth, but has a singularity on the extremity set n u , n v ∈ S d-1 |n u = -n v . This fact has already been pointed out ([31, 20, 19]) in slightly different contexts, and causes difficulty in order to define uniquely regular Levy generators and kinetic equations with such couplings. However, we will avoid such technical issues by using Grad's cut-off (see [START_REF] Malrieu | Logarithmic sobolev inequalities for some nonlinear pde's[END_REF]), and we will consider coupled Levy generators only on the formal level.

Anyway, the resulting probability transition is then a symmetric Markov coupling (by the symmetry condition [START_REF] Fournier | Rate of convergence of the nanbu particle system for hard potentials[END_REF]).

Definition 2.3. A spherically coupled random collision with deviation angle θ is defined by the following probability transition on coupled collisional directions (in S d-1 × S d-1 ):

c c,θ (n u , n v , dn ′ u dn ′ v ) def = l l nu =-nv δ Coupl nu,nv (n ′ u ) (dn ′ v ) + l l nu=-nv δ Coupl nu ,σ (n ′ u ) (dn ′ v )Unif S d-1 (dσ) c θ n u , dn ′ u , (20)
Lemma 2.4. The spherically coupled probability transition (20) verifies the symmetry condition

c c,θ (n u , n v , dn ′ u dn ′ v ) = c c,θ (n v , n u , dn ′ v dn ′ u ). (21)
It is thus a symmetric Markov coupling of c θ .

Proof. By construction, Coupl nu,nv and Coupl nu,σ are isometries. On the other hand, by isotropy, for any isometry R and vector

n v ∈ S d-1 we have R -1 c θ (Rn v , .) = c θ (n v , .
). Finally, the symmetry condition ( 19) yields [START_REF] Ikenberry | On the pressures and flux of energy in a gas according to {M} axwell\'s kinetic theory[END_REF].

In Section 4.2, the following key basic formula will be proved.

Lemma 2.5. The Markov coupling defined by c c,θ is almost surely contractive. Moreover, we have the average quadratic contraction estimate

S d-1 ×S d-1 n ′ u -n ′ v 2 c c,θ (n u , n v , dn ′ u dn ′ v ) -|n u -n v | 2 = -sin 2 θ c d-1 c d-3 |n u -n v | 2 , (22)
where in the above c d = π/2 0 sin d (ϕ) dϕ denotes the d'th Wallis integral.

We can now consider a coupled Nanbu (two-body) N -particle system (see [START_REF] Nanbu | Direct simulation scheme derived from the boltzmann equation. i. monocomponent gases[END_REF]), each particle being endowed with two velocities. It is a Markov process in state space (R d × R d ) N , with generator of the form [START_REF] Lu | Fast and slow convergence to equilibrium for maxwellian molecules via wild sums[END_REF]. The two-body Levy generator L c is a coupled collision operator defined when acting on two particles by:

L c (ψ)(u, v) def = S d-1 ×[0,π] ψ(u ′ , v ′ ) -ψ(u, v) c c,θ (n u , n v , dn ′ u dn ′ v ) b(dθ), (23)
for any test function ψ on (R d × R d ) 2 . The resulting particle system is by construction a formal Markov coupling of the usual Nanbu N -particle system process with two-body Levy generator L given in [START_REF] Ethier | Markov Processes. Characterization and Convergence[END_REF]. We recall that such particle systems conserve total momentum and energy, and have the uniform probability on the sphere defined by these conservation laws as an invariant probability distribution.

In order to be rigorous (at least for uniqueness), let us recall that the latter process can be constructed using Grad's cut-off:

b ε (dθ) def = l l θ≥ε b(dθ) bε def = [0,π] b ε (dθ) < +∞. (24)
(i) Each particle perform a collision with a fixed rate bε , and with a uniformly randomly chosen other particle.

(ii) The scattering angle of each collision is independently sampled according to the probability b ε (dθ) bε .

(iii) The coupled random post-collisional directions (n ′ u , n ′ v ) (with scattering angle prescribed by (ii)) are sampled using the coupled isotropic probability transition on sphere c c,θ .

The general case of Levy grazing collisions can then be considered as a the formal ε → 0 limit of the latter.

Kinetic equations

Consider a N -particle system. We say that propagation of chaos holds if the marginal distribution of k-particles (k being fixed) is converging (in law) to a product measure when N → +∞. Under this assumption, the limit of the one body distribution µ t ∈ P(R d ) of the particle system satisfies formally an evolution equation in closed form with a quadratic non-linearity given by:

d dt R d ϕ dµ t = R d ×R d L (ϕ ⊗ l l) dµ t ⊗ µ t , (25)
where in the above ϕ is a test function of R d . When L is the collision operator [START_REF] Ethier | Markov Processes. Characterization and Convergence[END_REF] with given kernel b, then the non-linear equation ( 25) is the Boltzmann equation in R d with Maxwell collision kernel b. The usual expression on the particle velocity density, denoted f t (v)dv, is then:

d dt f t (v) = R d ×S d-1 ×[0,π] f t (v ′ )f t (v ′ * ) -f t (v)f t (v * ) dv * c θ (n v , dn ′ v * ) b(dθ). (26)
In the above, the collision mapping ( 17) is used implicitly, and detailed balance has been used to remove test functions.

In the same way for the case coupled case, one obtains an evolution equation on the one body coupled particle distribution in the form

d dt R d ×R d ψµ c,t (dudv) = (R d ×R d ) 2 L c (ψ ⊗ l l) µ c,t (dudv)µ c,t (du * dv * ).
When the underlying generator is L c , the coupled collision operator [START_REF] Lu | On measure solutions of the boltzmann equation, part i: moment production and stability estimates[END_REF], the associated nonlinear kinetic equation in measure form is then

d dt R 2d ψ dµ c,t = (R d ×R d ) 2 [0,π]×S d-1 ×S d-1 ψ(u ′ , v ′ ) -ψ(u, v) c c,θ (n u , n v , dn ′ u dn ′ v ) b(dθ)µ c,t (dudv)µ c,t (du * dv * ). (27)
wher µ c,t ∈ P R d × R d is the one body distribution of the coupled velocities. In the case of Boltzmann collisions, there is no hope to obtain on the coupled density (the density of µ c,t ) an explicit simple expression similar to [START_REF] Nanbu | Direct simulation scheme derived from the boltzmann equation. i. monocomponent gases[END_REF]. Indeed, this would require some kind of detailed balance in product space R d × R d , which is broken for contractive coupling.

Here again, the lack of smoothness of the mapping (n u , n v ) → c c,θ (n u , n v ; dn ′ u , dn ′ v ) at the set n u , n v ∈ S d-1 |n u = -n v is causing difficulties to obtain an appropriate Cauchy theory for [START_REF] Imbuzeiro | On the convergence to equilibrium of kacś random walk on matrices[END_REF]. This difficulty is removed under Grad's cut-off (24) (using, say, total variation distance).

Results

We can now detail the results of the present paper.

We first compute the coupling creation functional (7) of the coupled Nanbu particle system as defined by ( 15)-( 23). The fact that the average of this functional can be bounded from above in some way by the coupling distance itself is not obvious. Fortunately, we have found a remarkable general inequality which enables to do so.

Proposition 2.6. Let (U, V ) ∈ R d ×R d a couple of centered and normalized (E |U | 2 = E |V | 2 = 1) random variables in euclidean space. Let (U * , V * ) ∈ R d × R d be an i.i.d. copy. We have: f E |U -V | 2 ≤ min κ E(U ⊗U ) , κ E(V ⊗V ) × E |U -U * | 2 |V -V * | 2 -((U -U * ) • (V -V * )) 2 , (28)
where in the above we have used the notation defined in (10)-(9

) (i.e. f (x) = f (4 -x) = x -x 2 /4, and κ S = (1 -|||S|||) -1 ∈ [d/(d -1)
, +∞]0 is a condition number of a symmetric positive matrix S of trace 1 and maximal eigenvalue |||S|||). Moreover, a sufficient condition for the equality case in (28) is given by the following isotropy and co-linear coupling conditions

(i) U |U | = V |V | a.s.. (ii) Either E (U ⊗ V ) = E (U ⊗ U ) = 1 d Id or E (U ⊗ V ) = E (V ⊗ V ) = 1 d Id.
It is now possible to apply the inequality (28) to coupled particle velocities

(u, v) ∈ (R d × R d ) N with null momentum u N = v N = 0 and normalized energy |u| 2 N = |v| 2 N = 1
on the probability space ([1, N ], Unif ) generated by averaging over particles. We first define the following p-moments:

• For any x N ∈ R d N , m x N ,p def = x N -x N * p 1/p N . • For any random X ∈ R d , with X * ∈ R d an i.i.d. copy m Law(X),p def = E (|X -X * | p ) 1/p . • For any random X N ∈ R d N , m Law(X N ),p def = E X N -X N * p N 1/p .
• For any p 0 ≥ 1 and any random

X N ∈ R d N , m Law(X N ),p 0 ,p def = d -1 d p 0 /p E κ p 0 X N ⊗X N N X N -X N * p N 1/p .
Using Hölder inequality, we obtain the following quasi-contractive estimate:

Proposition 2.7. Let (u N , v N ) ∈ (R d × R d ) N
satisfy the centering and normalization condition (conservation laws) (2). Let α > 0 and p 1 , p 2 > 1 with 1/p 1 + 1/p 2 = 1 be given. Remark that

α 1 + α + 2 + α 1 + α 1 p 1 (2 + α) + 2 + α 1 + α 1 p 2 (2 + α) = 1. ( 29
)
Then, we have the inequality:

f u N -v N 2 N u N -v N u N * -v N * -(u N -v N ) • (u N * -v N * ) α 1+α N ≤ k α min κ u N ⊗u N N , κ v N ⊗v N N m 2+α 1+α u N ,p 1 (2+α) m 2+α 1+α v N ,p 2 (2+α) ,
where in the above we have used the bounded above and below constant

k α def = (1 + α) 2 2 + α 2+α 1+α (-------→ α→0 or +∞ 2 or 1).
Proposition (2.7) is the main result of this paper, and immediately shows that the trend to equilibrium of a Nanbu N -particle system is controlled by the velocity moments of order 2+α > 2. It is useful to state the counterpart of Proposition 2.7 for the formal limit N = +∞. Proposition 2.8. Let (U, V ) ∈ R d × R d be two centered and normalized random vectors, and assume the isotropy condition:

E(U ⊗ U ) = 1 d Id.
Let (U * , V * ) be an i.i.d. copy. Denote f (x) = xx 2 /4. Then we have the inequality:

f E |U -V | 2 (E |U -U * | |V -V * | -(U -U * ) • (V -V * ) N ) α 1+α ≤ k α d d -1 m 2+α 1+α Law(U ),p 1 (2+α) m 2+α 1+α
Law(V ),p 2 (2+α) .

We then obtain consequences on the contraction in Wassertsein distance.

Corollary 2.9. Any Nanbu particle system with Maxwell collisions t → V N t ∈ R d N with centering and normalization condition (2) satisfies the following weak coupling -coupling creation inequality (for any α > 0, 1/p 1 + 1/p 2 = 1)

1 2 d 2 W N 2 (Law(V N t ), Unif N 0,1 ) - 1 λ c d-3 c d-1 d + dt d 2 W N 2 (Law(V N t ), Unif N 0,1 ) α 1+α ≤ d d -1 k α m 2+α 1+α Unif N 0,1 ,p 1 (1+α),p 1 (2+α) m 2+α 1+α Law(V N t ),p 2 (2+α) ,
The modified moment satisfies for any p 0 ≥ 1, p ≥ 1, m Unif N 0,1 ,p 0 ,p < +∞ as soon as p 0 < (d -1)((N -1)d + 1)/2, as well as

lim N →+∞ m Unif N 0,1 ,p 0 ,p = E (|N -N * | p ) 1/p < +∞.
where

(N, N * ) ∈ R d × R d ∼ N 0,1 × N 0,
f d 2 W 2 (µ t , N 0,1 ) - 1 λ c d-3 c d-1 d + dt d 2 W 2 (µ t , N 0,1 ) α 1+α ≤ d d -1 k α m 2+α 1+α N (0,1),p 1 (2+α) m 2+α 1+α
µt,p 2 (2+α) .

Two counter-examples

Let us finally present two negative results, that demonstrates, in coupling-coupling creation inequalities, the necessity of sub-exponential power law estimates on the one hand, and the necessity to resort on higher moments of velocity distribution on the other hand. We give the counter-examples in the form lemmas, with proofs. In both cases, we consider a coupled distribution in the form of random variables

(U, V ) ∈ R d × R d , U ∼ N (0, 1 d Id). ( 30
)
which are centered and normalized, with normal distribution in the first variable. (U * , V * ) is an i.i.d. copy.

Lemma 2.11 (The necessity of > 2-moments). Let (30) holds. Denote the order < 2 moment

m q def = E (|V | q ) 1/q , ( 31
)
for some 0 < q < 1. Then we have the following degeneracy of the coupling creation

lim mq→0 E (|U -U * | |V -V * | -(U -U * ) • (V -V * )) = 0.
Proof. Condition (31) obviously implies via Hölder inequality that lim mq→0 E |V -U | 2 = 2 = 0. Moreover, we have

E (|U -U * | |V -V * | -(U -U * ) • (V -V * )) ≤ 2E (|U -U * | |V -V * |) ≤ E 1/p (|U -U * | p ) E 1/q (|V -V * | q ) ------→ mq→+∞ 0.
Lemma 2.12 (The necessity of sub-exponential rates). Let (30) holds. Consider the co-linear coupling

V |V | def = U |U | a.s.,
with moreover the following radial coupling perturbation on some interval 0 < r -< r + < +∞:

|V | def = |U | l l |U | / ∈[r -,r + ] + E 1/2 |U | 2 | |U | ∈ [r -, r + ] l l |U |∈[r -,r + ] .
Then we have the following degeneracy of the coupling -coupling creation estimate lim

r -→+∞ lim r + →r - f E |V -U | 2 E (|U -U * | |V -V * | -(U -U * ) • (V -V * )) = +∞.
Proof. First, for such isotropic (U is normally distributed) and co-linear couplings, the key inequality ( 28) is in fact an equality. Denoting:

A def = (U -U * ) • (V -V * ) |U -U * | |V -V * | , we obtain R(r -, r + ) def = f E |V -U | 2 E (|U -U * | |V -V * | -(U -U * ) • (V -V * )) = d d -1 E |U -U * | 2 |V -V * | 2 (1 -A 2 ) E (|U -U * | |V -V * | (1 -A))
.

Since A = 1 when both |U | / ∈ [r -, r + ] and |U * | / ∈ [r -, r + ], we have (1-A) ≤ (1-A 2 ) l l |U |∈[r -,r + ] + l l |U * |∈[r -,r + ] a.s., 2(1-A 2 ) ≥ (1-A 2 ) l l |U |∈[r -,r + ] + l l |U * |∈[r -,r + ] a.s.,
and the smoothness of Gaussian density yields

lim r + →r - R(r -, r + ) ≥ d 2(d -1) E |R U -U * | 2 |R U -U * | 2 E (|R U -U * | |R U -U * |) ,
where R U is distributed uniformly on the sphere with radius r -. Taking the limit r -→ +∞ yields the result.

3 Standard notation and facts

Scattering

As usual, the velocities of a pair of collisional particles are denoted

(v, v * ) ∈ R d × R d ,
and the post-collisional quantities are denoted by adding the superscrpit ′ . All particles are assumed to have the same mass so that the conservation of momentum imposes

v ′ + v ′ * = v + v * ,
and conservation of energy

v ′ 2 + v ′ * 2 = |v| 2 + |v * | 2 .
As a consequence, the relative speed is also conserved

v ′ -v ′ * = |v -v * | ,
and a collision can be fully described by only using the normalized velocity difference, called the collisional direction. It is denoted

n v def = v -v * |v -v * | ∈ S d-1 ,
and in the same way n ′ v is called the post-collisional direction. The collisional and postcollisional directions being given, velocities are then determined by the standard involutive collision mapping [START_REF] Dolera | The role of the central limit theorem in discovering sharp rates of convergence to equilibrium for the solution of the kac equation[END_REF]

with inverse v = 1 2 (v ′ + v ′ * ) + 1 2 |v ′ -v ′ * | n v , v * = 1 2 (v ′ + v ′ * ) -1 2 |v ′ -v ′ * | n v .
The scattering or deviation angle θ ∈ [0, π] of the collision is then uniquely defined as the half-line angle between the collisional and the post-collisional direction:

cos θ def = v ′ -v ′ * v ′ -v ′ * • v -v * |v -v * | = n ′ v • n v .

Isotropic random walk on sphere

Isotropy (or invariance by rotation, which is equivalent to the physical Galilean invariance in the case of velocity differences), does not impose any condition on the scattering angle θ; however, it requires other degrees of freedom to be uniformly distributed. This is made precise in the following definition.

Definition 3.1. Let θ ∈ [0, π] be given. The isotropic probability transition on the sphere S d-1 with scattering angle θ ∈ [0, π] is the unique probability transition c θ : S d-1 → P(S d-1 ) invariant under isometries, and generating states at a prescribed angular distance θ. Formally

c θ (n v , dn ′ v ) def = Unif {n ′ v ∈S d-1 | nv•n ′ v =cos θ} dn ′ v , (32)
or equivalently

c θ (n v , .) Law = rot (θ, n v , Σ 1 , Σ 2 ) n v Σ 1 , Σ 2 ∼ Unif S d-1 ⊗ Unif S d-1 , (33)
where in [START_REF] Toscani | Sharp entropy dissipation bounds and explicit rate of trend to equilibrium for the spatially homogeneous boltzmann equation[END_REF], rot (θ, n v , σ 1 , σ 2 ) is the elementary 1 rotation of R d with rotation angle θ, rotation plane Π nv,σ 1 def = Span(n v , σ 1 ), and orientation prescribed by P Πn v ,σ 1 (σ 1 , σ 2 ) where P stands for orthogonal projection.

Clearly, invariance under isometries implies that the uniform distribution is an invariant distribution of c θ . The expression [START_REF] Toscani | Sharp entropy dissipation bounds and explicit rate of trend to equilibrium for the spatially homogeneous boltzmann equation[END_REF] is in fact useful to see why detailed balance (reversibility) hold; using the geometric inversion formula

n ′ v = rot (θ, n v , σ 1 , σ 2 ) n v ⇔ n v = rot (θ, n v , -σ 1 , -σ 2 ) n ′ v ,
and the invariance of the uniform distribution under the parity transformation σ → -σ. This yields:

Lemma 3.2. c θ is reversible with invariant probability the uniform distribution. Formally,

dn v c θ (n v , dn ′ v ) = dn ′ v c θ (n ′ v , dn v ).

Two-body random collisions

The following definition of a random collision will be useful (its properties are directly inherited from those of the isotropic probability transition on the sphere):

Definition 3.3. Consider two collisional particles with velocity pair in euclidean R d × R d . We call a (two body) random collision with scattering angle θ, the probability transition in R d ×R d induced by the collision mapping [START_REF] Dolera | The role of the central limit theorem in discovering sharp rates of convergence to equilibrium for the solution of the kac equation[END_REF] and the isotropic probability transition on sphere c θ (n v , dn ′ v ). It satisfies:

(i) Invariance (in law) under isometries of R d (applied simultaneously to each particle velocity).

(ii) Almost surely conservation of the total energy and momentum (conservation laws).

(iii) Reversibility with respect to uniform distributions. Formally,

dvdv * c θ (n v , dn ′ v ) = dv ′ dv ′ * c θ (n ′ v , dn v ),
where in the above the collision mappings (17)-(3.1) are implicitly used.

In the same way, we can consider continuous time collision (Levy) processes on R d × R d , generated by a ( non-negative ) Levy measure:

b : R + * → M + ([0, π]),
usually called an angular kernel. Recall that Levy measures are measures with finite diffusive intensity λ as defined in [START_REF] Aleksandr | Moment inequalities for the boltzmann equation and applications to spatially homogeneous problems[END_REF]. The latter generates a Markov (isotropic Levy) process on the sphere S d-1 through the generator:

[0,π] ϕ(n ′ v ) -ϕ(n v ) c θ (n v , dn ′ v ) b(dθ).
where in the above ϕ is a test function on S d-1 . By extension using the collision mapping, we can define a two-body collision process on velocity pairs (R d × R d ) with generator [START_REF] Ethier | Markov Processes. Characterization and Convergence[END_REF].

Lemma 3.4. The two-body collision Levy process with generator (18) satisfies the following properties:

Remark 4.8. In the special case N = 2, and if the coupled velocities have the same total momentum and kinetic energy, the system is exponentially contractive (from (38)) with explicit rate:

d dt ln E U N t -V N t 2 + U N * ,t -V N * ,t 2 = - 1 4 λ c d-1 c d-3 < 0.
It is of interest to remark that the exponential rate is in fact uniform in the dimension, since:

lim d→+∞ c d-1 c d-3 = 1.
We can finally compute the coupling creation functional for the coupled Nanbu particle system. Proposition 4.9. Consider the coupled Nanbu particle system in state space R 2d N . Then the coupling creation functional is given by [START_REF] Bolley | Uniform convergence to equilibrium for granular media[END_REF].

Proof. The calculation uses the definition of the Nanbu particle generator, the definition of the coupling creation functional C, and the two-body spherical coupling contractivity in Lemma (4.4). We have

C(u, v, u * , v * ) = L c (u, v, u * , v * ) → |u -v| 2 + |u * -v * | 2 = - [0,π] 2 sin 2 θ sin 2 ϕ b(dθ) sin d-3 ϕdϕ c d-3 (|u -u * | |v -v * | -(u -u * ) • (v -v * )) .
4.4 Proof of Corollary 2.9 and 2.10

The two corollaries 2.9 and 2.10 can now be deduced from Proposition 2.7 and 2.8 respectively.

Proof of Corollary 2.9. First, we need to extedn the moment inequality in Proposition 2.7.

Taking the expectation, and using three-terms Hölder inequality on the expectation (as opposed to the particle averaging) with the power law decomposition [START_REF] Pulvirenti | The theory of the nonlinear boltzmann equation for maxwell molecules in fourier representation[END_REF] yields

Ef U N -V N 2 N ≤ k α E 1/p 1 (1+α) κ p 1 (1+α) U N ⊗U N * N U N -U N * p 1 (2+α) N × E 1/p 2 (1+α) V N -V N * p 2 (2+α) N × E α/(α+1) C U N , U N * , V N , V N * N . (40)
We can the remark that for any pair (µ, ν) ∈ P(R d ) × P(R d ) satisfying the conservations laws, d 2 W 2 (µ, ν) ≤ 2 (try the trivial product coupling). As a consequence, we obtain the general upper bound for random exchangeable particle systems satisfying the conservation laws:

d 2 W 2 (η V N , η U N ) ≤ 2f V N -U N 2 N a.s.,
so that by definition of Wasserstein optimal coupling

d 2 W N 2 Law(V N ), Law(U N ) ≤ 2Ef V N -U N 2 N .
Let now t 0 ≥ 0 be given, and consider as an initial condition ( U N t 0 , V N t 0 ) ∈ (R d × R d ) N , a random variable representation of an d W 2 -optimal coupling between Unif N 0,1 and Law(V N t ). Consider next the solution h → (U N ε,t+h , V N ε,t+h ) of the coupled Nanbu collision process with the latter initial condition, and subject to Grad's cut-off [START_REF] Malrieu | Logarithmic sobolev inequalities for some nonlinear pde's[END_REF] with ε > 0. By construction, the latter satisfies

d 2 W N 2 Law(V N ε,t+h ), Unif N 0,1 -d 2 W N 2 Law(V N ε,t ), Unif N 0,1 ≤ - t+h t E C U N ε,t+h ′ , V N ε,t+h ′ , U N * ,ε,t+h ′ , V N * ,ε,t+h ′ N dh ′ .
We can then combine the inequality (40), with the continuity with respect to weak convergence of ε → Law(V N ε,t+h ) (this is a standard result of approximation of Levy processes by jump processes with bounded intensity, see [START_REF] Ethier | Markov Processes. Characterization and Convergence[END_REF]). Since the Wassertsein distance metrizes weak convergence and moments are continuous bounded observables, the result follows.

Finally, the integrability condition and the limit in (2.9) can be justified using the explicit expression for the probability distribution of eigenvalues of Wishart matrices (see e.g [START_REF] Wilbur | An introduction to multivariate statistical analysis[END_REF][START_REF] Fj Schuurmann | On the distributions of the ratios of the extreme roots to the trace of the wishart matrix[END_REF]). To prove the limit, consider some M > 0 and higher finite l-moments of the following random variable

κ U N ⊗U N N l l κ U N ⊗U N N ≤M .
My Markov inequality, the result will follow from dominated convergence and the fact that κ U N ⊗U N N has finite l-moments uniformly bounded in N for any l ≥ 1. Using the explicit expression of Wishart ensemble distribution (see Section 1 in [START_REF] Fj Schuurmann | On the distributions of the ratios of the extreme roots to the trace of the wishart matrix[END_REF]), such moments can be bounded above by (Γ is the usual so-called function)

Γ(N d/2) Γ((N -l)d/2) d-1 i=0 Γ((N -i -l)/2) d-1 i=0 Γ((N -i)/2) ∼ N →+∞ N ld/2 × (N -l/2 ) d → 1.

The result follows.

Proof of Corollary 2.10. The proof is similar to the one of Corollary 2.9, except that we need to justify the continuity of ε → µ ε,t with respect to weak convergence, as well as uniform control of higher moments, where µ ε,t the solution of the kinetic equation with Grad's cut-off. The continuity can found in Section 5 of [START_REF] Toscani | Probability metrics and uniqueness of the solution to the boltzmann equation for a maxwell gas[END_REF]. The uniform control on higher moments is standard for Maxwell molecules, where explicit computations of the latter can be carried out (see the classical paper [START_REF] Ikenberry | On the pressures and flux of energy in a gas according to {M} axwell\'s kinetic theory[END_REF]).

Coupling / coupling creation inequalities

In this section, the key inequalities between coupling and coupling creation are proven. The key point is to compare the coupling L 2 -distance, with the average parallelogram square area spanned by the difference of two independent copies. This is the content of the first section.

The special inequality

Let (U, V ) be two random vectors in a Euclidean space, and (U * , V * ) an i.i.d. copy. We assume the latter are centered

E(U ) = E(V ) = 0. (41)
If (Z 1 , Z 2 ) are two centered random vectors, we will use the notation

C Z 1 ,Z 2 def = E (Z 1 ⊗ Z 2 ) .
The goal is to bound from above the coupling distance E |U -V | 2 with the following alignement average

E |U -U * | 2 |V -V * | 2 -((U -U * ) • (V -V * )) 2 ,
which is the average parallelogram area spanned by the two vector differences U -U * and V -V * .

The main computation is based on an expansion of the paralleogram area formula and a general trace inequality for positive definite symmetric matrices in even dimension. In the present context, the latter matrix is given by the full co-variance operator

C (U,V ),(U,V ) = C U,U C U,V C V,U C V,V ,
and the trace inequality is detailed in the following lemma.

Lemma 5.1. Let U and V be two centered random vectors in R d . Then we have

Tr (C U,U C V,V )-Tr (C U,V C V,U ) ≤ min |||C U,U ||| Tr (C U,U ) , |||C V,V ||| Tr (C V,V ) Tr (C U,U ) Tr (C V,V ) -Tr (C U,V ) 2 ,
where |||.||| denotes the spectral radius (maximal eigenvalue) of a symmetric non-negative operator. Moreover, the equality case holds if (sufficient condition) either C U,U and C U,V or C V,V and C U,V are co-linear to the identity matrix.

Proof. First assume that C U,U has only strictly positive eigenvalues. In an orthonormal basis where C U,U is diagonal, we have the expression, for i, j, k ∈ [ [1, d]]:

Tr (C U,U C V,V -C U,V C V,U ) = i C i,i U,U C i,i V,V - j,k C j,k U,V C k,j V,U , = i C i,i U,U C i,i V,V -C i,i U,V 2 - j =k C j,k U,V 2
≤0
.

Then, by definition of the maximal eigenvalue of C U,U :

i C i,i U,U C i,i V,V -C i,i U,V 2 ≤ |||C U,U |||   Tr (C V,V ) - i C i,i U,V 2 C i,i U,U    ,
so that by Cauchy-Schwarz inequality

i C i,i U,V 2 ≤    i C i,i U,V 2 C i,i U,U    × i C i,i U,U ,
and we eventually get

Tr (C U,U C V,V -C U,V C V,U ) ≤ |||C U,U ||| Tr (C U,U ) Tr (C V,V ) Tr (C U,U ) -Tr (C U,V ) 2 .
The general case of degenerate eigenvalues is obtained by density.

The expansion of the average square paralellogram area is detailed in the next lemma.

Lemma 5.2. Let U and V be two centered random vector in R d . Let (U * , V * ) be a i.i.d. copy. Then we have the following decomposition: Let us expand the alignement functional (the left hand side of (42)). We have first,

E |U -U * | 2 |V -V * | 2 -((U -U * ) • (V -V * )) 2 = E |U | 2 |V | 2 -(U • V ) 2 ≥0 + Tr ((C U,V -C V,U ) (C V,U -C U,V ))
E |U -U * | 2 |V -V * | 2 = 2E |U | 2 |V | 2 + 2E |U | 2 E |V | 2 + 4E (U • U * V • V * ) + 8 × 0 = 2E |U | 2 |V | 2 + 2Tr (C U,U ) Tr (C V,V ) + 4Tr (C U,V C V,U ) ,
and second,

E ((U -U * ) • (V -V * )) 2 =2E U • V 2 + 2E (U • V ) 2 + 2E U • V 2 * + 2E (U * • V U • V * ) + 8 × 0 =2E U • V 2 + 2Tr (C U,V ) 2 + 2Tr (C U,U C V,V ) + 2Tr C 2 U,V .
On the other hand,

Tr ((C U,V -C V,U ) (C V,U -C U,V )) = -2Tr C 2 U,V + 2Tr (C U,V C V,U ) ,
and the result then follows.

Two remarks.

Remark 5.3. In the papers, the random vectors will be normalized x defined in [START_REF] Eric A Carlen | Determination of the spectral gap for kacś master equation and related stochastic evolution[END_REF]. Then Lemma 5.1 and 5.2 immediately yield Proposition 2.6.

E(|U | 2 ) = E(|V | 2 ) = 1, (43)
Remark 5.4. Assuming that the normalization (43) above holds, and that C U,U = 1 d Id is isotropic, then (42) becomes

f E |U -V | 2 ≤ d d -1 E |U -U * | 2 |V -V * | 2 -((U -U * ) • (V -V * )) 2 (44)
Moreover, a sufficient condition for equality in (44) is given by strongly istropic distributions with co-linear coupling, defined by the fact that the lengths (|U | , |V |) ∈ R 2 + are independant of the identically coupled and uniformly distributed direction U |U | (= V |V | a.s.).

Proof of coupling creation estimates

It is now possible to use Hölder inequality to relate the coupling distance with the coupling contraction functional [START_REF] Bolley | Uniform convergence to equilibrium for granular media[END_REF], by using the special inequality [START_REF] Otto | Generalization of an inequality by talagrand and links with the logarithmic sobolev inequality[END_REF]. This yields Proposition 2.7 and Proposition 2.8. Here are the proofs.

Proof of Proposition 2.7 and 2.8. It is a consequence of Hölder inequality applied to the right hand side of [START_REF] Otto | Generalization of an inequality by talagrand and links with the logarithmic sobolev inequality[END_REF]. We first apply (28) on the pair (u N , v N ) with respect to the probability space generated by the particle averaging operator N . We obtain using the conservation laws (2):

I def = u N -u N * 2 v N -v N * 2 -u N -u N * • v N -v N * 2 N ≥ 1 -u N ⊗ u N * N f ( u N -v N 2 N
).

Next, let us denote

A +/-= u N -u N * v N -v N * + / -u N -u N * • v N -v N * ,
and introduce q = 1 + α, p = 1 + 1/α so that 1/p + 1/q = 1. using Hölder inequality two times yields

I = A + A -N = A + A 1/q -A 1/p - N ≤ q 2 q + 1 1/q+1 u N -u N * 1+1/q v N -v N * 1+1/q A 1/p - N ≤ q 2 q + 1 1/q+1 u N -u N * 1+q v N -v N * 1+q 1/q N A - 1/p N a.s.,

3 Standard notation and facts 15 3 . 1 22 5 26 1

 312226 Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.2 Isotropic random walk on sphere . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.3 Two-body random collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.4 Two-body interacting particle systems . . . . . . . . . . . . . . . . . . . . . . . 18 4 Coupled collisions 18 4.1 Spherical coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.2 Contractivity of spherical couplings . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.3 Coupled Nanbu particle systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.4 Proof of Corollary 2.9 and 2.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . Coupling / coupling creation inequalities 23 5.1 The special inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 5.2 Proof of coupling creation estimates . . . . . . . . . . . . . . . . . . . . . . . . Summary and contents 1.1 Summary

(

  Otto differential calculus) ⇒ Bakry-Emery type convexity criterion (two times time derivation)

≥0 + 2 2 .

 22 Tr (C U,U ) Tr (C V,V ) -Tr (C U,V ) 2 -Tr (C U,U C V,V ) + Tr (C U,V C V,U ) (Lemma 5.1) ≥ 1min |||C U,U ||| Tr (C U,U ) , |||C V,V ||| Tr (C V,V ) Tr (C U,U ) Tr (C V,V ) -Tr (C U,V )Before computing terms, recall that if M, N are two square matrices, then Tr (M N ) = Tr (N M ) = Tr M T N T = Tr N T M T .

  so that we have Tr (C U,U ) = Tr (C V,V ) = 1, andTr (C U,U ) Tr (C V,V ) -Tr (C U,V ) 2 = 1 2 f E |U -V | 2 , with the function f (x) = x -x 2 4 ∼ x→0

  1 is a couple of i.i.d. centered and normalized (E |N | 2 = 1) normal random variables. A similar result holds for the associated kinetic equation Corollary 2.10. The centered and normalized measure solution (µ t ) t≥0 ∈ P R d of the space homogenous kinetic equation with Maxwell collisions (26) satisfies the following weak coupling -coupling creation inequality. Denote N 0,1 the associate reduced normal (Maxwellian) distribution in R d . Then we have

i.e. fixing the orthogonal of a plane

(i) Invariant (in law) under isometries of R d (applied simultaneously to velocity particles), (ii) Almost surely conservation of the total energy and momentum (conservation laws), (iii) Reversibility with respect to the uniform distribution. Remark 3.5. In all the definitions above, Galilean invariance (rotation and translation for velocities) allow that b may depend on the system state (v, v * ) through the absolute collision speed |vv * | (a conserved quantity). Note that the fact that |vv * | is conserved is necessary to keep the reversibility properties. The case where b is in fact independent of |vv * | is exactly what is called a Maxwell collision.

Two-body interacting particle systems

When L is the collision process [START_REF] Ethier | Markov Processes. Characterization and Convergence[END_REF], the N -particle system generated by ( 13) is called the Nanbu particle system. Definition 3.6. The particle system in R d N generated by the generator (13), with Maxwell collision generator (18) is called a Nanbu particle system ( [START_REF] Nanbu | Direct simulation scheme derived from the boltzmann equation. i. monocomponent gases[END_REF]).

The tensorial structure of (13) implies: Lemma 3.7. If L is stationary (or reversible) with state space E and product invariant prob-

As a consequence, the Nanbu particle system satisfies the following properties, directly inherited from the two body case. Lemma 3.8. A Nanbu particle system satisfies (i) Invariance under isometries of R d (applied to each particle velocity).

(ii) Almost surely conservation of total kinetic energy and momentum (conservation laws).

(iii) Reversibility (detailed balance) with uniform invariant distribution.

Coupled collisions

In this section we will consider a pair of collisional particles with coupled velocities

A coupled collision can then be described by expressing the post-collisional velocities

using coupled collision parameters. According to Section 3, it is sufficient in order to obtain the above coupling to express, using the same collision random parameters, the collision and post-collisional directions

. This is naturally done by what we have called in the present work a spherical coupling.

Spherical coupling

We give a special description of the isotropic probability transition with scattering angle θ.

where ϕ ∈ [0, π] is an azimuthal angle and l ∈ S d-1 is such that (n v , m v , l) is an orthonormal triplet. Then the image by the transformation (34) of the probability distribution

is the isotropic probability transition c θ (n v , dn ′ v ) with initial state n v and scattering angle θ (c d-3 denotes the Wallis integral normalization). In particular, the latter does not depend on the choice of m v .

Proof. c θ (n v , dn ′ v ) is defined as the uniform distribution induced by the euclidean structure on the submanifold of S d-1 defined by n ′ v •n v = cos θ. Moreover the expression of volume elements in (hyper)spherical coordinates implies that for any

Of course in the above, only the scattering angle θ has an intrinsic physical meaning, the azimuthal angle ϕ being dependent of the arbitrary choice of the pair (m v , l). This leads to the core analysis of a spherical coupling.

where in the above (n u , m u , l) and (n v , m v , l) are both orthonormal sets of vectors such that (n u , m u ) and (n v , m v ) belong to the same plane have the same orientation with respect to l. Note that if n u = n v , the pair (m u , m v ) and the angle ϕ are defined uniquely up to a common involution (a change of sign of the vectors and the reflexion ϕ → πϕ).

Proof. Assume n u = n v . Denote by R θ the unique elementary rotation bringing n u to n v . By construction R θ m u = m v , and R θ l = l

This immediately implies that the coupled probability transition c c,θ (n u , n v , dn u dn v ) is the image using the mapping (36) above of the uniform probability described in (ϕ, l)-variables in [START_REF] Villani | Cercignani's conjecture is sometimes true and always almost true[END_REF].

Contractivity of spherical couplings

We can now state the contractivity ("coupling creation") equation satisfied by spherical couplings.

Lemma 4.3. Let (n u , n v ) ∈ S d-1 × S d-1 be given, with a spherically coupled pair (n ′ u , n ′ v ) ∈ S d-1 × S d-1 (Definition 2.2). Then the contractivity formula (22) holds.

Proof. Next, we expand n ′ u .n ′ v using [START_REF] Wennberg | Entropy dissipation and moment production for the boltzmann equation[END_REF] and obtain:

Using 1 = cos 2 θ + sin 2 θ cos 2 ϕ + sin 2 θ sin 2 ϕ we obtain

and the result follows.

We can then compute the consequence on velocities.

Lemma 4.4. Consider coupled collisional and post-collisional velocities (u, u * , v, v * ) ∈ R 2d × R 2d , and a spherical coupling as defined by Definition 4.2. Then we have:

The above quantity vanishes if and only if the coupled collision directions are aligned with the same orientation (n u • n v = 1). If moreover the coupled velocities (u, u * ) and (v, v * ) have the same total momentum and energy, then we have the contractivity equality:

Proof. We use the following change of variable:

First remark that

Developing the left hand side of (37), and using the conservation laws (s ′ = s and |d ′ | = |d|), we obtain

Then the contractivity formula [START_REF] Asher | Markov chains and mixing times[END_REF] yields the first result. If (u, u * ) and (v, v * ) have the same momentum and energy, then (39) implies

Coupled Nanbu particle systems

In this section, we detail properties of the coupled Nanbu particle system.

Definition 4.5. A particle system in R 2d × R 2d N with generator (13) associated to the two-body coupled Levy collision generator L c in (23) is called a (spherically) coupled Nanbu particle system.

Note that the possibility of constructing coupled Nanbu system relies on the use of Maxwell collisions. We indeed need a process where collision parameters are independent on the specific invariants (energy and momentum) of a particle pair. Again, uniqueness of such processes requires some further analysis, unless Grad's cut-off (24) is used. Lemma 4.6. Consider a Nanbu particle system with two-body generator defined in [START_REF] Lu | On measure solutions of the boltzmann equation, part i: moment production and stability estimates[END_REF]. Then, the resulting process is a coupling of the Nanbu particle system of Definition 3.6.

Proof. (i) The fact that the marginal processes have the good distribution is a consequence of ( 21). (ii) The fact that an identity coupling (U N 0 = V N 0 a.s.) remains as such (U N t = V N t a.s. ∀t) is implied by the decrease of spherical couplings.

By construction of the spherical coupling, a generated coupled Nanbu system then satisfies the following expected properties. (ii) The coupling is almost surely decreasing (for all 0 ≤ t ≤ t + h):

where in the line before last line, we have use the sharp inequality (1θ)(1 + θ) 1/q ≤ q 2 q + 1 1/q+1 that holds for any θ ∈ [-1, 1]. Then, remarking that k α = q 2 q + 1 1/q and p 1 (1 + q)/qp 1 = α + 2 α + 1 yields the result. The case of Corollary 2.8 is similar with E formally replacing . N .


Thao Phuong Pham

Mourad Rabah
email: mourad.rabah@univ-lr.fr

Pascal Estraillier
email: pascal.estraillier@univ-lr.fr

Agent-based Architecture and Situation-based Scenario for Consistency Management

Keywords: Interactive system, adaptation, misunderstanding, situation-based scenario, consistency management

During interactions, system actors may face up to misunderstandings when their local visions contain inconsistent data about a same fact. Misunderstandings in interaction are likely to reduce interactivity performances (deviation or deadlock) or even affect overall system behavior. In this paper, we present agent-based architecture and scenario-structuring approach to deal with such misunderstandings and consistency. It is based on the notion of "situation" that is an elementary building block dividing the interactions between actors into contextual scenes. This model not only supports the scenario execution, but the consistency management as well. In order to organize and control the interactions, a "situation" contextualizes system's actors' interaction and activity, and includes prevention and tolerances mechanisms to deal with the misunderstandings and their causes. We also simulation experimentation on an Online Distance Learning case study.

I. INTRODUCTION

I N INTERACTIVE SYSTEMS, as games and simulators, the users and the internal agents can modify systems content and progress in real time through input adjustments. The interactive systems may adapt the system execution not only to user's actions, but also to user's profile and behavior, making these systems adaptive... In order to perform the adaptativity, the system must capture users' behaviors from their interactions. Then, according to system's logic and designer's logic, the system adjusts its execution to what it perceives of user's logic. Du to user's actions unpredictability, the execution process of an interactive system is also not predictable.

One of the important problems in interactive system is the potential misunderstanding between the users and the system and more generally between system's actors, virtual or physical. If the system does not capture correctly or confuses user's actions, or if the users do not understand what the system expects, that may lead to an erroneous interpretation of their behavior and an erroneous adaptation of system execution. This misunderstanding may concern user-system interactions, but it can also appear in any kind of interaction between any system's actors. It can be due to the incomplete actors' data or the non-determinism of actors' behavior and cause the interaction deadlock or application failure.

In our recent works, [START_REF] Pham | Handling the Misunderstanding in Interactions : Definition and Solution[END_REF], [START_REF] Trillaud | Online Distant Learning Using Situation-based Scenario[END_REF], we have defined the misunderstanding in interaction as: when two or more system's actors have incoherent data in their local visions about the same fact f and these data is used during their interaction, that can cause an interaction deviation from the planned scenario. An actor may be human user or virtual system's agent. The local vision is actor's own knowledge about its external world (virtual environment, system's resources...), its relations with the others actors (subset of their states) and its own profile (internal state). So, our work focuses on the management of the consistency between the actor's behaviour logic and the system's logic and the consistency between the actor's local visions in order to handle the potential misunderstandings in interactions.

To handle misunderstandings we propose to contextually structure the application execution into interaction sequences called "situations" and including misunderstanding prevention and tolerance mechanisms. Each situation corresponds to a contextual resource-centered sequence of activities and events and is characterized by preconditions and postconditions. That allows the system to control the execution and to establish the casual links between the situations. This model confines actor's interactions in a given context in order to control them and manage the execution consistency. The consistency handling mechanisms, are inspired by techniques from dependability domain since there is an analogy between the misunderstandings in interactive systems and the errors handled in fault tolerant systems [START_REF] Laprie | Basic Concepts and Taxonomy of Dependable and Secure Computing[END_REF].

II. MISUNDERSTANDING IN INTERACTIONS AND RELATED

WORK

In the recent research, we can find several works dealing with the user-system dialogue where the communication is done through a real human language [3]- [START_REF] Rapaport | What Did You Mean by That? Misunderstanding, Negotiation, and Syntactic Semantics[END_REF]. According to Rapaport [START_REF] Rapaport | What Did You Mean by That? Misunderstanding, Negotiation, and Syntactic Semantics[END_REF], negotiation is the key to understanding. A cognitive agent understands by negotiating with the interlocutor or by hypothesizing the meaning of an unknown word from the context... A cognitive agent can negotiate with itself about something external by comparing it perception and internal knowledge in order to change or correct its own misunderstandings. Other works propose to use confidence scores to measure the reliability of each word in a recognized sentence [START_REF] Jiang | Confidence Measures for Speech Recognition: A survey[END_REF]. Besides, Lopez-Cozar proposed to implement a frame correction module, which is independent of speech recognizer [START_REF] Lopez-Cozar | Using Knowledge about Misunderstandings[END_REF]. This module corrects misunderstandings in a sentence, caused by the errors in speech recognition, by replacing the incorrect frame with an adequate one. Karsenty and Botherel applied the adaptable and adaptive transparency strategies to TRAVELS project with the goal of helping the users to understand and react appropriately to system rejections and misunderstandings [START_REF] Karsenty | Transparency strategies to help users handle system errors[END_REF]. The ability of making system's interpretations explicit and informing the users on how to correct misunderstandings are two ways to help users handle them. This strategy is very effective in misunderstanding detection and raises the rate of appropriate user responses after system rejections. All of these works deal with the problem in speech dialogue where the misunderstandings are the more frequent. But the misunderstanding can be found in other forms of interaction like actions, gesture... Our purpose is to define how we can treat the misunderstandings between the actors themselves, besides the user-system misunderstandings. It is not easy to recognize such class of misunderstandings. In the dependability domain [START_REF] Laprie | Basic Concepts and Taxonomy of Dependable and Secure Computing[END_REF], we find the inconsistency problem between systems and operators. The "automation surprise" is inconsistency error occurring when the system behaves differently than its operators expect [START_REF] Combefis | A Bisimulation-Based Approach to the Analysis of Human-Computer Interaction Categories and Subject Descriptors[END_REF]. It may be due to a mismatch between the actual system behavior and the operator's mental model of that behavior [START_REF] King | General Aviation Training for "Automation Surprise[END_REF], and it can lead to "confusion mode" and sometimes critical failures. In general, misunderstandings come from the gap between user's logic and designer's logic, all along action planning between the actors. Many works, particularly in interactive storytelling, have been done to solve the mismatch between users' behaviors and system logic [START_REF] Magerko | Mediating the Tension between Plot and Interaction[END_REF]- [START_REF] Paul | Adaptive Storytelling and Story Repair in a Dynamic Environment[END_REF] by predicting the user's future actions and detecting the invalid ones that deviate the execution from the planned objectives. In general, prediction approach cost is very expensive, such as a short-term player behavior modeling module implanted in [START_REF] Magerko | Mediating the Tension between Plot and Interaction[END_REF] to simulate how the world would change to player's actions. Moreover, this approach seems not well suited to a real-time interactive systems, nor to systems in which user's behaviors cannot be modeled easily by a set of rules.

Our approach will focus on software and component design model to integrate simple prevention and treatment mechanisms. Our solution relies on three points. First, we build robust agent-based architecture with specific additional components in charge of misunderstandings in interaction management. Second, we organize the actors' interactions as a situation-based scenario to facilate the interaction control. Third, we integrate into situations' dynamic execution the consistency management, including data synchronization, misunderstanding detection and treatment inspired and adapted from fault-tolerance techniques. These mechanisms do not try to predict users' behavior but will take into account users' state to adapt the system execution in order to avoid misunderstandings between actors. The system observes and analyzes users' states, detect the misunderstandings or their consequences and act to keep the consistency between actors' logics at the beginning and at the end of interaction sequences.

III. AGENT-BASED GENERAL ARCHITECTURE FOR INTERACTIVE ADAPTATIVE SYSTEM

Several architecture models for interactive systems have been proposed according to the specific purpose of each work. We chose the approach of multi-agent system in [START_REF] Sahaba | Interactive educational games for autistic children with agent-based system[END_REF] as a starting point to build our model. The advantage of this approach is that each agent can be organized and work autonomously and strategically. We added a special agent Scenario agent: It makes decisions about scenario orientation according to user's state, planned scenario and permanent objective defined by the designer. This agent tries to find the best way to orientate the application execution. Scenario agent takes charge of a library of "situations" planned by the designer. These "situations" (defined in section IV) represent scenario components and are the interaction and the activity sequences that can take place in the application as, for instance, all possible scenes in a theater play.

Director agent: This agent receives the decision taken by the scenario agent. He takes in charge the production of the adaptive scenario and realizes a modification, an answer or an action adapted to the users.

Script agent: Its task is to track inconsistency in 3 steps:

• Detection: Detect, confine or partition the inconsistency between situation's actors in order to identify the causes of the misunderstanding.

• Treatment: Apply the handling mechanism or strategy to remove the inconsistency and to correct the deflected state that causes the incoherence.

• Evaluation: Estimate the efficiency of the treatments in order to improve the applied mechanism for the next time.

IV. SITUATION-BASED SCENARIO

A. Approach of interactive storytelling

Our proposition is inspired from the interactive storytelling domain that focuses on scenario execution management.

Interactive storytelling is the unfolding of a story that the player's decisions impact [START_REF] Champagnat | A Storytelling Model For Educational Games : Heros Interactive Journey[END_REF], [START_REF] Lebowitz | Interactive Storytelling for Video Games: A Player-Centered Approach for Creating Memorable Character and Stories[END_REF]. It also refers to how to generate stories which are both interesting and coherent. We consider that the interactions in an interactive application can be organized, strongly or weakly, as a story scenario. That allows us to adapt ideas from storytelling domain to organizing the interactions.

The scenario in interactive storytelling is represented by a series of actions/events linked together by cause and effect as in [START_REF] Karlsson | Applying a Plan-Recognition / Plan-Generation Paradigm to Interactive Storytelling[END_REF] or by ordered link as in [START_REF] Magerko | Mediating the Tension between Plot and Interaction[END_REF], [START_REF] Silva | Bringing interactivity to a virtual storyteller[END_REF] or by Hierarchical Task Network planning as in [START_REF] Paul | Adaptive Storytelling and Story Repair in a Dynamic Environment[END_REF] where each task is decomposed into subtasks until the primitive actions. But all of these scenario structurings are not suited to built complex interaction sequences where the user's actions are free, non predictable and depending on a great amount of context data. Hence, we propose the notion of "situation" that can be seen as a scene encompassing not only interactions execution but also interactions management and resources use. The situations are the basic narrative elements that facilitate interactions' planning and management by characterizing, contextualizing and confining them.

B. Scenario organizing with situations 1) Situation model:

The interactions are split into a set of situations. Each situation is a sequence of interactions between two or more actors in a precise context to achieve a predictive objective, as shows the figure 2. It is characterized by: the preconditions, the postconditions, a set of participating actors and a set of resources. Due to the fact that actors' behaviors, especially human behaviors, are not always precisely modeled, and due to the influence of external events, the progression of a situation can be considered as an execution and adaptation "black box" where the interactions are executed in a non-predictable way. Furthermore, the situation includes consistency management. Itrepresents a set of mechanisms devoted to the prevention, detection and treatment solutions, in order to redress and adjust situation's progression in spite of misunderstanding and inconsistency problems. Consistency management is carried out all along the situation progression Fig. 2: Elementary Situation Structure from the local context initialization to the post-condition completion.

2) Situation Graph and Application Execution:

The situations are considered as the plot structuring elementary blocks. Each application provide a set of situations defining all the possible interaction sequences that can happen during the application execution. They can be grouped and linked together in order to build the overall application scenario. The scenario is then represented by a directed graph of situations. Each node is a situation and each edge is a transition from one situation to another. The situations graph shows the causal relationships between scenario situations. A scenario may have several beginnings and also some possible endings.

The situation-based scenario approach favorizes the execution control and interaction adaptation. The application progression becomes a scenario unfolding from one starting node to one final node on the predefined situation graph (it is taken in charge by the scenario agent in the global architecture). When there is more than one possible situation, the most pertinent one will be chosen by the scenario agent. To increase the adaptability, we can avoid the definition of a predefined graph. In that case, the situation choice is made according to the pre-conditions that best satisfy the global state and decision criteria. This method is flexible, adaptive, and applicable in "real time" during application execution, but it can lead to uncontrollable situation order or infinite loop, if the post-conditions and pre-conditions do not contain sufficient data.

V. CONSISTENCY MANAGEMENT MODEL WITHIN SITUATIONS

A. Handling Mechanisms

The consistency management that we propose consists of a set of specific methods, techniques and mechanisms that aim to handle the misunderstanding problem and to obtain data consistency all along the interactions. They are similar to the dependability techniques [START_REF] Laprie | Basic Concepts and Taxonomy of Dependable and Secure Computing[END_REF].

1) Prevention mechanisms: try to suppress misunderstandings occurrence conditions in order to avoid misunderstandings. To avoid data inconsistency, the proposed technique is the explicit declaration of all shared data before situation's interaction sequence start. It aims to identify and share actors' local visions in order to decrease the possibility of interaction deviation. Once the actors have collected the necessary data, they can start the interactions. The data synchronization is another method intending to compare the actors' local visions at a given moment during the interaction sequence in order to avoid the inconsistency of new perceived data. The synchronization can delay the interactions, so it should be done fast and not too frequently to disturb them as less as possible.

2) Tolerance mechanisms: aim to assure interaction continuation despite misunderstanding occurrence via misunderstanding detection and interaction recovery. Detection: regular check of i) the shared data used during the interactions and ii) the deviation between actors' logics.

Recovery: once a misunderstanding is detected, the system apply one or several of the following techniques : rollback -bringing the system back to a stable state, exempt from misunderstanding, to retry the interactions; rollforwardbringing the system to a new misunderstanding free state from which the interactions can go on; reinforcement -requiring from one or from all participant actors to do some additional interactions.

3) Removal mechanisms: involve misunderstanding detection and correction, followed by reinitialisation of the concerned interaction sequence, or of the whole execution process. The detected misunderstandings will be diagnosed to determine their causes: which data are inconsistent? Which ambiguities exist in the interaction context? Are there protocol faults? After that, an appropriate correction method will be applied to eliminate the related misunderstanding. Finally, the interactions have to be restarted from the last stable point or from the beginning.

B. Inclusion into the Situation Structure

Our situation-based architecture allows the integration of misunderstanding management mechanisms inside the situation in order to control the misunderstandings and their consequences all along situation execution. We define three phases (figure 3). a) Prologue phase: The explicit declarations of interacting content and data are performed, to synchronize actors' local visions before they start to interact. If the initial data of all actors are identical from the beginning, the possibility of misunderstanding is reduced. If the inconsistency exists, a negotiation step will be performed between the inconsistent actors. Then, one or several of them will modify its/their data, or the divergent data will be isolated/removed and not considered during the interactions. b) Interaction or Dialogue phase: when the interactions are carried out, the actors will update their local data step by step, as they continuously observe and perceive each other. Despite the initial local vision agreement, misunderstanding may nevertheless occur during the interactions. This is why their local knowledge is synchronized all along the interaction sequence in order to avoid that local data about same facts diverge in actors' local visions. One or several techniques of reinforcement, rollback, rollforward can be alternatively used. c) Epilogue phase: All the interactions are done in the previous phase. If the post-conditions are fulfilled, we can exit the situation with the expected results. But if, for some reason, we do not reach the expected post-condition, the script Fig. 3: Consistency management mechanisms agent has to detect and settle the existing incoherency in order to avoid the propagation of the misunderstandings to other situations. The system may also require actors to do some reinforcing interactions, or if necessary, make a rollback to a last stable state (in this case there must be systematic state saving mechanism), or, even a restart of the whole situation. The main goal of this phase is to quit the situation with the appropriate post-conditions and without latent or active misunderstanding. But, the rollback or reinforcing interactions may not lead the actors towards the planned post-conditions. Therefore, we add in the situation model a special exit point called "exception" that allows the current situation to be stopped at anytime without expected post-conditions and that leads to exception handling situations.

VI. EXPERIMENT ON ONLINE DISTANCE LEARNING CASE

STUDY

To validate our approach we applied our situation-based methodology in our current online distance learning (ODL) project [START_REF] Trillaud | Online Distant Learning Using Situation-based Scenario[END_REF]. The project is devoted to the development of an online distributed platform that simulates a real classroom: teachers and learners carry out learning sessions as in a real life but by interacting through a virtual class environment. The platform integrates an interactive numeric board, camera, microphone and pedagogic tools (as file sharing system or virtual notebook) to support the courses...The figure 4 shows an example of courses scenario based on 6 situations. However, the users may face many difficulties: class supervision, course quality assessment, misunderstandings due to the weak system's interfaces and mechanisms to catch and manage user behaviors. The interactions between the actors in ODL contains numerous factors that may lead to misunderstandings as: multimeaning or implicit behaviors; supervision tools' observation and interpretation imperfection; system component failures; incomplete, missing, implicit or wrong consigns...

A. "Individual Work" Situation Description

To deal with these various misunderstandings, we applied our situation-based solution including consistency management to a particular situation: "Individual Work" (SU -IW in figure 4). Each learner will work individually and has to do the exercises distributed by the system. The system provides additional exercises each time the learners send the previous exercises report. The expected post-condition is that all the learners reach a required knowledge level "MaxKnowledge".

Because of the long test duration and development for the real platform prototype, we chose to experiment our misunderstanding management mechanisms and agent-based architecture through a multi-agent simulation with the GAMA platform 1 . We have 4 types of agents : "Teacher", "Learner", "Observer" and "ODL System". The Observer's role is to observe the state of sent exercises in order to evaluate learners' accumulated knowledge level. The distribution mechanism based on these observations and learners' skill level evaluations is taken in charge by "ODL System" agent that is a combinaison of 3 other agents in our model: scenario, script and director agent (figure 1). Potential misunderstandings in this situation occur when the system distributes the exercises that are incoherent given the learners' skill and expectation. They can result from wrong learners' exercise state observation or from inappropriate distributed exercise level. The misunderstanding handling is done inside the situation during its 3-phase progression (figure 5).

Prologue phase: The system checks each learner's connection status to begin the exercise series distribution.

Dialogue phase: In this situation, the interactions content refers to the exercises distribution and reporting. During learners' work, each observer agent supervises his associated learner's working state and his exercise report to collect data: partial or total termination, work duration, correctness rate. To avoid the wrong estimation of learner's skill and knowledge level, these data have to be synchronized between the observer and his learner after the exercise report is sent and before a exercise is distributed.

Epilogue phase: To finish the situation the lerners must reach a given skill level after a given number of exercises. If a learner reaches this number without reaching the required skill level, the series will be stopped after a session deadline to avoid an abnormal long series. The system sends a StopSignal message to all learners to confirm the end of the exercise series after a predefined timeout. It refers to the exception treatment.

B. Experimentation Results

We run the simulation of "Individual Work" situation with the following parameters: 50 learners, 1 teacher, max knowledge level = 25, max difficulty level = 20, session deadline = 250 steps of simulation. We will measure a set of 1 https://code.google.com/p/gama-platform/ Fig. 5: Agents main interactions in the simulation important factors influenced by potential misunderstandings: N e : total number of distributed exercises; N notend : total number of real non-finished exercises; N bad : number of bad observation by all observers; N cor : number of system observation corrections while detecting the wrong observed states (it refers to the synchroniaation times where consistency management is performed to remove inconherent data); LI: learners' interest level that increases when the learners succeed and that decreases when they fail their exercises; T total : total session times (in steps) until the last learner has finished his series.

The data are recorded and calculated for the average values from 10 simulations lauching times in each measure. We compare these data between two cases: "with" and "without" the consistency management. The results are summarized in the table I. The total distributed exercises number N e is twice more in "without" case compared to the "with" case. The average number of not finished exercises in "without" series is higher than in "with" series: 747.4 vs 363.4 also depicted in the figure 6(a). It is obvious that the session duration in "without" case is almost 2 times longer than in "with" case. The figure 6(b) shows the number of learners that have finished their whole series during the situation execution in the "with" and "without" consistency management cases. The lines shows that the learners work with more exercises and with longer duration T total in the "without" case. We can make the same observation with the average measure values in table I.

Why do we have this difference result? When the consistency management is integrated in the situation execution to handle the potential misunderstandings, the observers have to adjust their observed data according to learners' "disagree" acknowledgments. Hence, the learner's skill level estimation will converge faster to the real value, and the difficulty level of the distributed exercises is more appropriate to his skill. The result is that learners can finish all the exercises and with higher correctness rate. In contrast, if no mechanism is added to control the inconsistency between learners and observers, a non-finished exercise can be perceived as finished, and vice versa. The skill estimation is less correct: higher or lower than the real one. There is a higher probability that the ODL system gives to the learners too difficult or too easy exercises. That delays the skill level progression and explains why the learners take more time to terminate the series.

VII. CONCLUSION

In this paper, we have presented the situation-based design methodology and consistency management mechanisms to handle the misunderstanding in interactions. Our approach is to contextualize the interactions between actors into "situations" and add to these basic narrative blocks consistency management mechanisms split into 3 steps: the prologue, data declaration and consistency verification, the dialogue, the interaction unfolding, local visions synchronization and misunderstanding treatment, and the epilogue, data update and agreement attainment. We do not seek to find a universal algorithm or a solution to deal with all types of misunderstandings in interaction. Our aim is to provide a management pattern that could be systematically used by the application designers or developers and that allow them to incorporate their own verification, synchronization, prevention and tolerance mechanisms adapted to the specific misunderstandings of their applications.

We have applied our methodology to a case study from an Online Distant Learning project. We have built a simulation of the "Individual Work" situation and integrated into it the proposed solutions to show how the consistency management operates on a simulation example. From the experimentation results, we have found out that our mechanisms reduce the incoherent data between learners and observers and improve the performance of exercise distribution: shorter session duration, lower exercise number, faster required level attainment... Even if the simulation is simple and does not cover exhaustively all the possible interactions that can occur in such situation, it illustrates the benefits of misunderstanding management during interaction progression. The next step of our work is to perform the same experimentation and measures on the prototype under development with live case study and to check the relevance of our approach in other e-Learning "situations".

Fig. 1 :

 1 Fig. 1: General agent-based architecture for interactive system

Fig. 4 :

 4 Fig. 4: Situation-based scenario example

  (a) Number of distributed exercices. (b) Number of learners finishing exercise series.

Fig. 6 :

 6 Fig. 6: Comparason between 2 cases "With" et "Without".

TABLE I :

 I Statistical data comparason between 2 cases: With (Wi) et Without (Wo) the consistency management

		Ne		N notend		N bad	N obsnon		Ncor	LI			T total
		Wi	Wo	Wi	Wo	Wi	Wo	Wi	Wo	Wi	Wo	Wi	Wo	Wi	Wo
	1	330	735	23	64	83	103	93	114	83	0	78.98	66.1	988	2692
	2	363	692	45	28	87	76	110	88	87	0	77.73	76.41	1104	2640
	3	361	744	44	55	94	97	114	114	94	0	76.35	69.06	1076	2700
	4	383	768	47	73	110	99	129	118	110	0	77.31	65.18	1160	2724
	5	347	744	32	60	87	99	109	115	87	0	77.94	67.31	1024	2688
	6	360	806	37	65	111	117	122	135	111	0	77.55	64.68	1108	2760
	7	392	737	66	59	93	92	118	108	93	0	73.06	66.88	1188	2692
	8	379	752	42	66	117	100	131	112	117	0	77.65	65.88	1140	2712
	9	353	722	37	69	96	100	111	111	96	0	77.49	67.55	1048	2672
	10	361	774	40	62	93	115	117	134	93	0	74.18	65.92	1084	2728
	Ave.	363.4	747.4	42.4	60.1	97.1	99.8	115.9	114.9	17.8	0	76.82	67.71	1092	2641