Emanuele Caglioti
email: caglioti@mat.uniroma1.it

¸ois Golse
email: francois.golse@math.polytechnique.fr

ON THE BOLTZMANN-GRAD LIMIT FOR THE TWO DIMENSIONAL PERIODIC LORENTZ GAS

Keywords: cit.]. 2000 Mathematics Subject Classification. 82C70, 35B27 (82C40, 60K05) Periodic Lorentz gas, Boltzmann-Grad limit, Kinetic models, Extended phase space, Convergence to equilibrium, Continued fractions, Farey fractions, Threelength theorem

The two-dimensional, periodic Lorentz gas, is the dynamical system corresponding with the free motion of a point particle in a planar system of fixed circular obstacles centered at the vertices of a square lattice in the Euclidian plane. Assuming elastic collisions between the particle and the obstacles, this dynamical system is studied in the Boltzmann-Grad limit, assuming that the obstacle radius r and the reciprocal mean free path are asymptotically equivalent small quantities, and that the particle's distribution function is slowly varying in the space variable. In this limit, the periodic Lorentz gas cannot be described by a linear Boltzmann equation (see [F. Golse, Ann.

The Lorentz gas

The Lorentz gas is the dynamical system corresponding with the free motion of a single point particle in a system of fixed spherical obstacles, assuming that collisions between the particle and any of the obstacles are elastic. This simple mechanical model was proposed in 1905 by H.A. Lorentz [START_REF] Lorentz | Le mouvement des électrons dans les métaux[END_REF] to describe the motion of electrons in a metal -see also the work of P. Drude [START_REF] Drude | Zur Elektronentheorie der metalle[END_REF] Henceforth, we assume that the space dimension is 2 and restrict our attention to the case of a periodic system of obstacles. Specifically, the obstacles are disks of radius r centered at each point of Z 2 . Hence the domain left free for particle motion is

(1.1) Z r = {x ∈ R 2 | dist(x, Z 2 ) > r} ,
where 0 < r < 1 2 . Throughout this paper, we assume that the particle moves at speed 1. Its trajectory starting from x ∈ Z r with velocity ω ∈ S 1 at time t = 0 is denoted by t → (X r , Ω r )(t; x, ω) ∈ R 2 × S 1 . One has (1.2) Ẋr (t) = Ω r (t) , Ωr (t) = 0 , whenever X r (t) ∈ Z r , while (1.3) X r (t + 0) = X r (t -0) , Ω r (t + 0) = R[X r (t)]Ω r (t -0) , whenever X r (t) ∈ ∂Z r .

In the system above, we denote ˙= d dt , and R[X r (t)] is the specular reflection on ∂Z r at the point X r (t) = X r (t ± 0).

Next we introduce the Boltzmann-Grad limit. This limit assumes that r ≪ 1 and that the initial position x and direction ω of the particle are jointly distributed in Z r × S 1 under some density of the form f in (rx, ω) -i.e. slowly varying in x. Given this initial data, we define (1.4) f r (t, x, ω) := f in (rX r (-t/r; x, ω), Ω r (-t/r; x, ω)) whenever x ∈ Z r .

In this paper, we are concerned with the limit of f r as r → 0 + in some sense to be explained below. In the 2-dimensional setting considered here, this is precisely the Boltzmann-Grad limit.

In the case of a random (Poisson), instead of periodic, configuration of obstacles, Gallavotti [START_REF] Gallavotti | Rigorous theory of the Boltzmann equation in the Lorentz gas[END_REF] proved that the expectation of f r converges to the solution of the Lorentz kinetic equation

(1.5)      (∂ t +ω •∇ x )f (t, x, ω) = S 1 (f (t, x, ω -2(ω • n)n)-f (t, x, ω))(ω • n) + dn , f t=0 = f in ,
for all t > 0 and (x, ω) ∈ R 2 × S 1 . Gallavotti's remarkable result was later generalized and improved in [START_REF] Spohn | The Lorentz process converges to a random flight process[END_REF][START_REF] Boldrighini | On the Boltzmann equation for the Lorentz gas[END_REF].

In the case of a periodic distribution of obstacles, the Boltzmann-Grad limit of the Lorentz gas cannot be described by the Lorentz kinetic equation (1.5). Nor can it be described by any linear Boltzmann equation with regular scattering kernel: see [START_REF] Golse | On the statistics of free-path lengths for the periodic Lorentz gas[END_REF][START_REF] Golse | On the periodic Lorentz gas in the Boltzmann-Grad scaling[END_REF] for a proof of this fact, based on estimates on the distribution of free path lengths to be found in [START_REF] Bourgain | On the distribution of free path lengths for the periodic Lorentz gas[END_REF] and [START_REF] Golse | On the distribution of free path lengths for the periodic Lorentz gas II. M2AN Modél[END_REF].

In a recent note [START_REF] Caglioti | The Boltzmann-Grad limit of the periodic Lorentz gas in two space dimensions[END_REF], we have proposed a kinetic equation for the limit of f r as r → 0 + . The striking new feature in our theory for the Boltzmann-Grad limit of the periodic Lorentz gas is an extended single-particle phase-space (see also [START_REF] Golse | The periodic Lorentz Gas in the Boltzmann-Grad Limit[END_REF]) where the limiting equation is posed.

Shortly after our announcements [START_REF] Golse | The periodic Lorentz Gas in the Boltzmann-Grad Limit[END_REF][START_REF] Caglioti | The Boltzmann-Grad limit of the periodic Lorentz gas in two space dimensions[END_REF], J. Marklof and A. Strömbergsson independently arrived at the same limiting equation for the Boltzmann-Grad limit of the periodic Lorentz gas as in [START_REF] Caglioti | The Boltzmann-Grad limit of the periodic Lorentz gas in two space dimensions[END_REF]. Their contribution [START_REF] Marklof | The Boltzmann-Grad limit of the periodic Lorentz gas[END_REF] provides a complete rigorous derivation of that equation (thereby confirming an hypothesis left unverified in [START_REF] Caglioti | The Boltzmann-Grad limit of the periodic Lorentz gas in two space dimensions[END_REF]), as well as an extension of that result to the case of any space dimension higher than 2.

The present paper provides first a complete proof of the main result in our note [START_REF] Caglioti | The Boltzmann-Grad limit of the periodic Lorentz gas in two space dimensions[END_REF]. In fact the method sketched in our announcement [START_REF] Caglioti | The Boltzmann-Grad limit of the periodic Lorentz gas in two space dimensions[END_REF] is different from the one used in [START_REF] Marklof | Kinetic transport in the two-dimensional periodic Lorentz gas[END_REF], and could perhaps be useful for future investigations on the periodic Lorentz gas in 2 space dimensions.

Moreover, we establish some fundamental qualitative features of the equation governing the Boltzmann-Grad limit of the periodic Lorentz gas in 2 space dimensions -including an analogue of the classical Boltzmann H Theorem, a description of the equilibrium states, and of the long time limit for that limit equation.

We have split the presentation of our main results in the two following sections. Section 2 introduces our kinetic theory in an extended phase space for the Boltzmann-Grad limit of the periodic Lorentz gas in space dimension 2. Section 3 is devoted to the fundamental dynamical properties of the integro-differential equation describing this Boltzmann-Grad limit -specifically, we present an analogue of Boltzmann's H Theorem, describe the class of equilibrium distribution functions, and investigate the long time limit of the distribution functions in extended phase space that are solutions of that integro-differential equation.

Main Results I: The Boltzmann-Grad Limit

Let (x, ω) ∈ Z r × S 1 , and define 0 < t 0 < t 1 < . . . to be the sequence of collision times on the billiard trajectory in Z r starting from x with velocity ω. In other words, (2.1) {t j | j ∈ N} = {t ∈ R * + | X r (t; x, ω) ∈ ∂Z r } . Define further (2.2) (x j , ω j ) := (X r (t j + 0; x, ω), Ω r (t j + 0; x, ω)) , j ≥ 0 .

Denote by n x the inward unit normal to Z r at the point x ∈ ∂Z r , and consider

(2.3) Γ ± r = {(x, ω) ∈ ∂Z r × S 1 | ± ω • n x > 0} , Γ 0 r = {(x, ω) ∈ ∂Z r × S 1 | ω • n x = 0}
. Obviously, (x j , ω j ) ∈ Γ + r ∪ Γ 0 r for each j ≥ 0.

2.1. The Transfer Map. As a first step in finding the Boltzmann-Grad limit of the periodic Lorentz gas, we seek a mapping from Γ + r ∪ Γ 0 r to itself whose iterates transform (x 0 , ω 0 ) into the sequence (x j , ω j ) defined in (2.2).

For (x, ω) ∈ (Z r × S 1 ) ∪ Γ + r ∪ Γ 0 r , let τ r (x, ω) be the exit time defined as (2.4)

τ r (x, ω) = inf{t > 0 | x + tω ∈ ∂Z r } .
Also, for (x, ω) ∈ Γ + r ∪ Γ 0 r , define the impact parameter h r (x, ω) as on Figure 2.1 by (2.5) h r (x, ω ′ ) = sin( ω ′ , n x ) .

Denote by (Γ + r ∪ Γ 0 r )/Z 2 the quotient of Γ + r ∪ Γ 0 r under the action of Z 2 by translations on the x variable. Obviously, the map (2.6) (Γ + r ∪ Γ 0 r )/Z 2 ∋ (x, ω) → (h r (x, ω), ω) ∈ [-1, 1] × S 1 coordinatizes (Γ + r ∪ Γ 0 r )/Z 2 , and we henceforth denote by Y r its inverse. For r ∈ ]0, 1 2 [, we define the transfer map (see Figure 2.1)

T r : [-1, 1] × S 1 → R * + × [-1, 1] by (2.7)
T r (h ′ , ω) = (2rτ r (Y r (h ′ , ω)), h r ((X r , Ω r )(τ r (Y r (h ′ , ω)) ± 0; Y r (h ′ , ω)))) . ω 2r

x'

x n x'

x n S/2r T r (h r (x j , ω j ), ω j ) = (2rτ r (x j , ω j ), h r (x j+1 , ω j )) , for each j ≥ 0 , and (2.9)

ω j+1 = R[π -2 arcsin(h r (x j+1 , ω j ))]ω j , for each j ≥ 0 .
The notation

(2.10) R[θ]
designates the rotation of an angle θ.

Notice that, by definition,

h r (x j+1 , ω j ) = h r (x j+1 , ω j+1 ) .
The theorem below giving the limiting behavior of the map T r as r → 0 + was announced in [START_REF] Caglioti | The Boltzmann-Grad limit of the periodic Lorentz gas in two space dimensions[END_REF].

Theorem 2.1. For each Φ ∈ C c (R * + × [-1, 1]) and each h ′ ∈ [-1, 1] (2.11) 1 | ln ǫ| 1/4 ǫ Φ(T r (h ′ , ω)) dr r → ∞ 0 1 -1 Φ(S, h)P (S, h|h ′ )dSdh
a.e. in ω ∈ S 1 as ǫ → 0 + , where the transition probability P (S, h|h ′ )dSdh is given by the formula (2.12)

P (S, h|h ′ ) = 3 π 2 Sη (Sη) ∧ (1-S) + +(ηS -|1-S|) + + (S -1 2 Sη) ∧ (1 + 1 2 Sη) -( 1 2 S + 1 2 Sζ) ∨ 1 + + (S -1 2 Sη) ∧ 1 -( 1 2 S + 1 2 Sζ) ∨ (1 -1 2 Sη) + , with the notation ζ = 1 2 |h + h ′ | , η = 1 2 |h -h ′ | , and a ∧ b = inf(a, b) , a ∨ b = sup(a, b) . Equivalently, for each (S, h, h ′ ) ∈ R + ×] -1, 1[×] -1, 1[ such that |h ′ | ≤ h, (2.13) P (S, h|h ′ ) = 3 π 2 1 ∧ 1 h -h ′ 2 S -(1 + h ′ ) + .
The formula (2.12) implies the following properties of the function P .

Corollary 2.2 (Properties of the transition probability P (S, h|h ′ ).). The function

(S, h, h ′ ) → P (S, h|h ′ ) is piecewise continuous on R + × [-1, 1] × [-1, 1].
1) It satisfies the symmetries (2.14) P (S, h|h ′ ) = P (S, h ′ |h) = P (S, -h| -h ′ ) for a.e. h, h ′ ∈ [-1, 1] and S ≥ 0 , as well as the identities

(2.15)          ∞ 0 1 -1 P (S, h|h ′ )dSdh = 1 , for each h ′ ∈ [1, 1] , ∞ 0 1 -1 P (S, h|h ′ )dSdh ′ = 1 , for each h ∈ [1, 1] .
2) The transition probability P (S, h|h ′ ) satisfies the bounds

(2.16) 0 ≤ P (S, h|h ′ ) ≤ 6 π 2 S 1 1+h ′ < 2 S for a.e. h, h ′ ∈] -1, 1[ such that |h ′ | ≤ h and all S ≥ 4.
Moreover, one has (2.17)

1 -1 1 -1 P (S, h|h ′ )dhdh ′ ≤ 48 π 2 S 3 , S ≥ 4 .
As we shall see below, the family T r (h ′ , ω) is wildly oscillating in both h ′ and ω as r → 0 + , so that it is somewhat natural to expect that T r converges only in the weakest imaginable sense.

The above result with the explicit formula (2.12) was announced in [START_REF] Caglioti | The Boltzmann-Grad limit of the periodic Lorentz gas in two space dimensions[END_REF]. At the same time, V.A. Bykovskii and A.V. Ustinov 1 arrived independently at formula (2.13) in [START_REF] Bykovskii | The statistics of particle trajectories in the inhomogeneous Sinai problem for a two-dimensional lattice[END_REF]. That formulas (2.12) and (2.13) are equivalent is proved in section 6.2 below.

The existence of the limit (2.11) for the periodic Lorentz gas in any space dimension has been obtained by J. Marklof and A. Strömbergsson in [START_REF] Marklof | The distribution of free path lengths in the periodic Lorentz gas and related lattice point problems[END_REF], by a method completely different from the one used in the work of V.A. Bykovskii and A.V. Ustinov or ours. However, at the time of this writing, their analysis does not seem to lead to an explicit formula for P (s, h|h ′ ) such as (2.12)-(2.13) in space dimension higher than 2.

Notice that J. Marklof and A. Strömbergsson as well as V.A. Bykovskii and A.V. Ustinov obtain the limit (2.11) in the weak-* L ∞ topology as regards the variable ω, without the Cesàro average over r, whereas our result, being based on Birkhoff's ergodic theorem, involves the Cesàro average in the obstacle radius, but leads to a pointwise limit a.e. in ω.

In space dimension 2, [START_REF] Marklof | Kinetic transport in the two-dimensional periodic Lorentz gas[END_REF] extends the explicit formula (2.12)-(2.13) to the case of interactions more general than hard-sphere collisions given in terms of their scattering map. The explicit formula proposed by Marklof-Strömbergsson in [START_REF] Marklof | Kinetic transport in the two-dimensional periodic Lorentz gas[END_REF] for the transition probability follows from their formula (4.14) in [START_REF] Marklof | The distribution of free path lengths in the periodic Lorentz gas and related lattice point problems[END_REF], and was obtained independently from our result in [START_REF] Caglioti | The Boltzmann-Grad limit of the periodic Lorentz gas in two space dimensions[END_REF].

Another object of potential interest when considering the Boltzmann-Grad limit for the 2-dimensional periodic Lorentz gas is the probability of transition on impact parameters corresponding with successive collisions, which is essentially the Boltzmann-Grad limit of the billiard map in the sense of Young measures. Obviously, the probability of observing an impact parameter in some infinitesimal interval dh around h for a particle whose previous collision occured with an impact parameter h ′ is Π(h|h ′ ) = ∞ 0 P (S, h|h ′ )dS . 1 We are grateful to J. Marklof for informing us of their work in August 2009.

Explicit formula for Π(h|h

′ ) For |h ′ | < h < 1, one has (2.18) Π(h|h ′ ) = 6 π 2 1 h -h ′ ln 1 + h 1 + h ′
Besides, the transition probability Π(h|h ′ ) satisfies the symmetries inherited from P (S, h|h ′ ):

(2.19) Π(h|h ′ ) = Π(h ′ |h) = Π(-h| -h ′ ) for a.e. h, h ′ ∈ [-1, 1] .

3-obstacle configurations.

Before analyzing the dynamics of the Lorentz gas in the Boltzmann-Grad limit, let us describe the key ideas used in our proof of Theorem 2.1. We begin with an observation which greatly reduces the complexity of billiard dynamics for a periodic system of obstacles centered at the vertices of the lattice Z 2 , in the small obstacle radius limit. This observation is another form of a famous statement about rotations of an irrational angle on the unit circle, known as "the three-length (or three-gap) theorem", conjectured by Steinhaus and proved by V.T. Sós [START_REF] Sós | On the distribution mod. 1 of the sequence nα[END_REF] -see also [START_REF] Suranyi | Über die Anordnung der Vielfahren einer reelen Zahl mod. 1[END_REF].

Assume ω ∈ S 1 has components ω 1 , ω 2 independent over Q. Particle trajectories leaving, say, the surface of the obstacle centered at the origin in the direction ω will next collide with one of at most three, and generically three other obstacles.

Lemma 2.3 (Blank-Krikorian [START_REF] Blank | Thom's problem on irrational flows[END_REF], Caglioti-Golse [START_REF] Caglioti | On the distribution of free path lengths for the periodic Lorentz gas III[END_REF]). Let 0 < r < 1 2 , and ω ∈ S 1 be such that 0 < ω 2 < ω 1 and ω 2 /ω 1 / ∈ Q. Then, there exists (q, p) and (q, p) in Z 2 such that 0 < q < q , q pqp = σ ∈ {±1}

satisfying the following property:

{x + τ r (x, ω)ω | |x| = r and x • ω ≥ 0}
⊂ ∂D((q, p), r) ∪ ∂D((q, p), r) ∪ ∂D((q + q, p + p), r) ,

where D(x 0 , r) designates the disk of radius r centered at x 0 .

The lemma above is one of the key argument in our analysis.

To go further, we need a convenient set of parameters in order to handle all these 3-obstacle configurations as the direction ω runs through S 1 .

For ω as in Lemma 2.3, the sets

{x + tω | |x| = r , x • ω ≥ 0 , x + τ r (x, ω)ω ∈ ∂D((q, p), r) , t ∈ R} and {x + tω | |x| = r , x • ω ≥ 0 , x + τ r (x, ω)ω ∈ ∂D((q, p), r) , t ∈ R}
are closed strips, whose widths are denoted respectively by a and b. The following quantities are somewhat easier to handle:

(2.20)

Q = 2rq ω 1 , Q = 2rq ω 1 , A = a 2r , B = b 2r
(see Figure 2.2 for the geometric interpretation of A, B, Q and Q), and we shall henceforth denote them by (2.21) Q(ω, r) , Q(ω, r) , A(ω, r) , B(ω, r) , together with σ(ω, r)

whenever we need to keep track of the dependence of these quantities upon the direction ω and obstacle radius r -we recall that σ(ω, r) = q p -pq ∈ {±1} .

ε 2r 2rB 2Ar Q/ ε v Q/ ε Q/ Figure 3.
Example of a 3-obstacle configurations, and the parameters A, B, Q and Q. Here ǫ = 2r/ω 1 .

Lemma 2.4. Let 0 < r < 1 2 , and ω ∈ S 1 be such that 0 < ω 2 < ω 1 and ω 2 /ω 1 / ∈ Q. Then, one has

(2.22) 0 < A(ω, r) , B(ω, r) , A(ω, r) + B(ω, r) ≤ 1 , 0 < Q(ω, r) < Q(Ω, r) , and
Q(ω, r)(1 -A(ω, r)) + Q(ω, r)(1 -B(ω, r)) = 1 .
This last equality entails the bound

(2.23) 0 < Q(ω, r) < 1 2 -A(ω, r) -B(ω, r) ≤ 1 .
Therefore, each possible 3-obstacle configuration corresponding with the direction ω and the obstacle radius r is completely determined by the parameters

(A, B, Q, σ)(ω, r) ∈ [0, 1] 3 × {±1}.
The proof of Theorem 2.1 is based on the two following ingredients. The first is an asymptotic, explicit formula for the transfer map T r in terms of the parameters A, B, Q, σ defined above. Proposition 2.5. Let 0 < r < 1 2 , and ω ∈ S 1 be such that 0 < ω 2 < ω 1 and

ω 2 /ω 1 / ∈ Q. Then, for each h ′ ∈ [-1, 1] and each r ∈]0, 1 2 [, one has T r (h ′ , ω) = T A(ω,r),B(ω,r),Q(ω,r),σ(ω,r) (h ′ ) + (O(r 2 ), 0) ,
in the limit as r → 0 + . In the formula above, the map T A,B,Q,σ is defined for each (A, B, Q, σ) ∈ [0, 1] 3 × {±1} in the following manner:

(2.24) T A,B,Q,σ (h ′ ) = (Q, h ′ -2σ(1 -A)) if σh ′ ∈ [1 -2A, 1] , T A,B,Q,σ (h ′ ) = Q, h ′ + 2σ(1 -B) if σh ′ ∈ [-1, -1 + 2B] , T A,B,Q,σ (h ′ ) = Q + Q, h ′ + 2σ(A -B)
otherwise.

For ω = (cos θ, sin θ) with arbitrary θ ∈ R, the map h ′ → T r (h ′ , ω) is computed using Proposition 2.5 by using the symmetries in the periodic configuration of obstacles as follows. Set θ = θ -m π 2 with m = [ 2 π (θ + π 4 )] (where [z] is the integer part of z) and let ω = (cos θ, sin θ). Then (2.25) T r (h ′ , ω) = (s, h) , where (s, sign(tan θ)h) = T r (sign(tan θ)h ′ , ω) .

The second ingredient in our proof of Theorem 2.1 is an explicit formula for the limit of the distribution of ω → (A, B, Q, σ)(ω, r) as r → 0 + in the sense of Cesàro on the first octant S 1 + of S 1 .

Proposition 2.6. Let F be any bounded and piecewise continuous function defined on the compact K = [0, 1] 3 × {±1}. Then

(2.26)

1 | ln ǫ| 1/4 ǫ F (A(ω, r), B(ω, r), Q(ω, r), σ(ω, r)) dr r → K F (A, B, Q, σ)dµ(A, B, Q, σ)
a.e. in ω ∈ S 1 + as ǫ → 0 + , where µ is the probability measure on K given by (2.27)

dµ(A, B, Q, σ) = 6 π 2 1 0<A<1 1 0<B<1-A 1 0<Q< 1 2-A-B dAdBdQ 1 -A (δ σ=1 + δ σ=-1 ) .
This result is perhaps more transparent when stated in terms of the new parameters A, B ′ = B 1-A , Q, σ instead of the original A, B, Q, σ: an elementary change of variables in the integral on the right hand side of (2.26) shows that

K F (A, B, Q, σ)dµ(A, B, Q, σ) = K F (A, (1 -A)B ′ , Q, σ)dµ ′ (A, B ′ , Q, σ) , where dµ ′ (A, B ′ , Q, σ) = 6 π 2 1 0<A<1 1 0<B ′ <1 1 0<Q< 1 1+(1-A)(1-B ′ ) dAdB ′ dQ(δ σ=1 + δ σ=-1 ) .
In other words, the new parameters A, B ′ , Q and σ are uniformly distributed over the maximal domain compatible with the bounds (2.22) and (2.23).

The first part of Theorem 2.1 follows from combining the two propositions above; in particular, for each h ′ ∈ [-1, 1], the transition probability P (S, h|h ′ )dSdh is obtained as the image of the probability measure µ in (2.27) under the transformation (A, B, Q, σ) → T A,B,Q,σ (h ′ ).

The Limiting Dynamics.

With the parametrization of all 3-obstacle configurations given above, we return to the problem of describing the Boltzmann-Grad limit of the Lorentz gas dynamics.

Let (x, ω) ∈ Z r × S 1 , and let the sequence of collision times (t j ) j≥0 , collision points (x j ) j≥0 and post-collision velocities (ω j ) j≥0 be defined as in (2.1) and (2.2). The particle trajectory starting from x in the direction ω at time t = 0 is obviously completely defined by these sequences.

As suggested above, the sequences (t j ) j≥0 , (x j ) j≥0 and (ω j ) j≥0 can be reconstructed with the transfer map, as follows. Set

(2.28)

t 0 = τ r (x, ω) , x 0 = x + τ r (x, ω)ω , h 0 = h r (x 0 , ω) , ω 0 = R[π -2 arcsin(h 0 )]ω .
We then define the sequences (t j ) j≥0 , (x j ) j≥0 inductively, in the following manner:

(2.29)

(2s j+1 , h j+1 ) = T r (h j , ω j ) , t j+1 = t j + 1 r s j+1 , x j+1 = x j + 1 r s j+1 ω j , ω j+1 = R[π -2 arcsin(h j+1 )]ω j ,
for each j ≥ 0.

If the sequence of 3-obstacle configuration parameters b r j = ((A, B, Q, σ)(ω j , r)) j≥0 converges (in some sense to be explained below) as r → 0 + to a sequence of independent random variables (b j ) j≥0 with values in K, then the dynamics of the periodic Lorentz gas in the Boltzmann-Grad limit can be described in terms of the discrete time Markov process defined as

(S j+1 , H j+1 ) = T bj (H j ) , j ≥ 0 . Denote Z j+1 = R 2 × S 1 × R + × [-1, 1] × K n+1
for each j ≥ 0. The asymptotic independence above can be formulated as follows: there exists a probability measure

P 0 on R + × [-1, 1] such that, for each j ≥ 0 and Ψ ∈ C(Z n+1 ), (H) lim r→0 + rZr×S 1 Ψ(x, ω, rτ r ( x r , ω), h r ( x0 r , ω 0 ), b r 0 , . . . , b r n )dxdω = Zn+1 Ψ(x, ω, τ, h, b 0 , . . . , b n )dxdωdP 0 (τ, h)dµ(b 0 ) . . . dµ(b n ) ,
where µ is the measure defined in (2.27). This scenario for the limiting dynamics is confirmed by the following Theorem 2.7. Let f in be any continuous, compactly supported probability density on R 2 × S 1 . Denoting by R[θ] the rotation of an angle θ, let F ≡ F (t, x, ω, s, h) be the solution of (2.30)

         (∂ t + ω • ∇ x -∂ s )F (t, x, ω, s, h) = 1 -1 2P (2s, h|h ′ )F (t, x, R[θ(h ′ )]ω, 0, h ′ )dh ′ , F (0, x, ω, s, h) = f in (x, ω) ∞ 2s 1 -1 P (τ, h|h ′ )dh ′ dτ , where (x, ω, s, h) runs through R 2 × S 1 × R * + ×] -1, 1[, and θ(h) = π -2 arcsin(h). Then the family (f r ) 0<r< 1 2 defined in (1.4) satisfies (2.31) f r → ∞ 0 1 -1 F (•, •, •, s, h)dsdh in L ∞ (R + × R 2 × S 1 ) weak- * as r → 0 + .
Let us conclude this presentation of our main results with a few remarks. Equation (2.30)-(2.31) was proposed first in [START_REF] Caglioti | The Boltzmann-Grad limit of the periodic Lorentz gas in two space dimensions[END_REF], under some additional decorrelation assumption left unverified -specifically, assuming (H). Then, Marklof-Strömbergsson provided a complete, rigorous derivation of that same equation in [START_REF] Marklof | The Boltzmann-Grad limit of the periodic Lorentz gas[END_REF], without any additional assumption, thereby establishing the theorem above.

The main novelty in this description of the Boltzmann-Grad limit of the periodic Lorentz gas is the fact that it involves a Markov process in the extended phase space [START_REF] Blank | Thom's problem on irrational flows[END_REF]. In addition to the space and velocity variables x and ω that are usual in the classical kinetic theory of gases, this extended phase space involves two extra variables: i.e. s, the (scaled) time to the next collision and h, the impact parameter at that next collision, as additional coordinates describing the state of the moving point particle. To the best of our knowledge, the idea of using this extended phase space (and particularly the additional variables s and h) appeared for the first time in our announcements [START_REF] Golse | The periodic Lorentz Gas in the Boltzmann-Grad Limit[END_REF][START_REF] Caglioti | The Boltzmann-Grad limit of the periodic Lorentz gas in two space dimensions[END_REF].

R 2 × S 1 × R + × [-1,

Main Results II: Dynamical Properties of the Limiting Equation

The present section establishes some fundamental mathematical properties of equation (2.30). For simplicity, we henceforth restrict our attention to the case where the space variable x varies in the flat 2-torus

T 2 = R 2 /Z 2 .
3.1. Equilibrium states. As is well-known, in the kinetic theory of gases, the equilibrium states are the uniform Maxwellian distributions. They are characterized as the only distribution functions that are independent of the space variable and for which the collision integral vanishes identically.

In equation (2.30), the analogue of the Boltzmann collision integral is the quantity

1 -1 2P (2s, h|h ′ )F (t, x, R[θ(h ′ )]ω, 0, h ′ )dh ′ + ∂ s F (t, x, ω, s, h) .
On the other hand, the variables (s, h) play in equation (2.30) the same role as the velocity variable in classical kinetic theory.

Therefore, the equilibrium distributions analogous to Maxwellians in the kinetic theory of gases are the nonnegative measurable functions

F ≡ F (s, h) such that -∂ s F (s, h) = 1 -1 2P (2s, h|h ′ )F (0, h ′ )dh ′ , s > 0 , -1 < h < 1 . Theorem 3.1. Define E(s, h) := +∞ 2s 1 -1 P (τ, h|h ′ )dh ′ dτ . 1) Then E(0, h) = 1 , -1 < h < 1 and +∞ 0 1 -1 E(s, h)dhds = +∞ 0 1 -1 1 -1 1 2 SP (S, h|h ′ )dhdh ′ dS = 1 . 2) Let F ≡ F (s, h) be a bounded, nonnegative measurable function on R + × [-1, 1] such that s → F (s, h) is continuous on R + for a.e. h ∈ [-1, 1] and      -∂ s F (s, h) = 1 -1 2P (2s, h|h ′ )F (0, h ′ )dh ′ , s > 0 , -1 < h < 1 , lim s→+∞ F (s, h) = 0 .
Then there exists C ≥ 0 such that

F (s, h) = CE(s, h) , s > 0 , -1 < h < 1 . 3) Define 2 p(t) = lim r→0 + |{(x, ω) ∈ (Z r ∩ [0, 1] 2 ) × S 1 | 2rτ r (x, ω) > t}| |(Z r ∩ [0, 1] 2 ) × S 1 | . Then 1 -1 E(s, h)dh = -2p ′ (2s) , s > 0 , and
1 -1 E(s, h)dh ∼ 1 π 2 s 2 as s → +∞ .
Notice that the class of physically admissible initial data for our limiting equation (2.30) consists of densities of the form

F in (x, ω, s, h) = f in (x, ω)E(s, h)
-see Theorem 2.7. In other words, physically admissible initial data are "local equilibrium densities", i.e. equilibrium densities in (s, h) modulated in the variables (x, ω).

Before going further, we need some basic facts about the evolution semigroup defined by the Cauchy problem (2.30). The existence and uniqueness of a solution of the Cauchy problem (2.30) presents little difficulty. It is written in the form

F (t, •, •, •) = K t F in , t ≥ 0 ,
where (K t ) t≥0 is a strongly continuous linear contraction semigroup on the Banach space

L 1 (T 2 × S 1 × R + × [-1, 1]). It satisfies in particular the following properties: 1) if F in ≥ 0 a.e. on T 2 ×S 1 ×R + ×[-1, 1], then, for each t ≥ 0, one has K t F in ≥ 0 a.e. on T 2 × S 1 × R + × [-1, 1]; 2) for each t ≥ 0, one has K t E = E; 3) if F in ≤ CE (resp. F in ≥ CE) a.e. on T 2 × S 1 × R + × [-1, 1] for some constant C, then, for each t ≥ 0, one has K t F in ≤ CE (resp. K t F in ≥ CE) a.e. on T 2 × S 1 × R + × [-1, 1]; 4) for each F in ∈ L 1 (T 2 × S 1 × R + × [-1, 1]
) and each t ≥ 0, one has

T 2 ×S 1 ×R+×[-1,1] K t F in dxdωdsdh = T 2 ×S 1 ×R+×[-1,1] F in dxdωdsdh ; 5) if F in ∈ C(T 2 × S 1 × R + × [-1, 1]) is continuously differentiable with respect to x and s, i.e. ∇ x F in and ∂ s F in ∈ C(T 2 × S 1 × R + × [-1, 1]), then, for each t ≥ 0, one has K t F in , ∂ t K t F in , ∇ x K t F in and ∂ s K t F in ∈ C(T 2 × S 1 × R + × [-1, 1]), and the function (t, x, ω, s, h) → K t F in (x, ω, s, h) is a classical solution of the equation (2.30) on R * + × T 2 × S 1 × R * + × [-1, 1]
. All these properties follow from straightforward semigroup arguments once (2.30) is established. Otherwise, the semigroup (K t ) t≥0 is constructed together with the underlying Markov process in section 6 of [START_REF] Marklof | The Boltzmann-Grad limit of the periodic Lorentz gas[END_REF] -see in particular Propositions 6.2 and 6.3, formula (6.16) and Theorem 6.4 there.

Instability of modulated equilibrium states.

A well-known feature of the kinetic theory for monatomic gases is that generically, local equilibrium distribution functions -i.e. distribution functions that are Maxwellian in the velocity variable and whose pressure, bulk velocity and temperature may depend on the time and space variables -are solutions of the Boltzmann equation if and only if they are uniform equilibrium distribution functions -i.e. independent of the time and space variables. In other words, the class of local Maxwellian states is generically unstable under the dynamics of the Boltzmann equation. An obvious consequence of this observation is that rarefied gas flows are generically too complex to be described by only the macroscopic fields used in classical gas dynamics -i.e. by local Maxwellian distribution functions parametrized by a pressure, temperature and velocity field. Equation (2.30) governing the Boltzmann-Grad limit of the periodic Lorentz gas satisfies the following, analogous property. Theorem 3.2. Let F be a solution of (2.30) of the form

F (t, x, ω, s, h) = f (t, x, ω)E(s, h) , with f ∈ C 1 (I × T 2 × S 1 ) ,
where I is any interval of R + with nonempty interior. Then f is a constant.

Thus, the complexity of the equation (2.30) posed in the extended phase space

T 2 × S 1 × R + × [-1, 1
] cannot be reduced by postulating that the solution is a local equilibrium, whose additional variables s and h can be averaged out.

As in the case of the classical kinetic theory of gases, this observation is important in the discussion of the long time limit of solutions of (2.30).

H Theorem and a priori estimates.

In this section, we propose a formal derivation of a class of a priori estimates that includes an analogue of Boltzmann's H Theorem in the kinetic theory of gases.

Let h be a convex C 1 function defined on R + ; consider the relative entropy

H h (f E|E) := T 2 ×S 1 ×R+×[-1,1] (h(f ) -h(1) -h ′ (1)(f -1)) (t, x, ω, s, h)E(s, h)dxdωdsdh .
The most classical instance of such a relative entropy corresponds with the choice h(z) = z ln z: in that case h(1) = 0 while h ′ (1) = 1, so that

H z ln z (f E|E) = T 2 ×S 1 ×R+×[-1,1] (f ln f -f +1) (t, x, ω, s, h)E(s, h)dxdωdsdh . Theorem 3.3. Let F ≡ F (t, x, ω, s, h) in C 1 (R + × T 2 × S 1 × R + × [-1, 1]) be such that 0 ≤ F (t, x, ω, s, h) ≤ CE(s, h) , (t, x, ω, s, h) ∈ R + × T 2 × S 1 × R + × [-1, 1] ,
and

(∂ t + ω • ∇ x -∂ s )F (t, x, ω, s, h) = 1 -1 2P (2s, h|h ′ )F (t, x, R[θ(h ′ )]ω, 0, h ′ )dh ′ , with the notations of Theorem 2.7. Then H h (F |E) ∈ C 1 (R + ) and d dt H h (F |E) + T 2 D h (F/E)(t, x)dx = 0 ,
where the entropy dissipation rate D h is given by the formula

D h (f )(t, x) = S 1 ×R+×[-1,1]×[-1,1] 2P (2s, h|h ′ ) h(f (t, x, R[θ(h ′ )]ω, 0, h ′ ))-h(f (t, x, ω, s, h)) -h ′ (f (t, x, ω, s, h) (f (t, x, R[θ(h ′ )]ω, 0, h ′ ) -f (t, x, ω, s, h)) dωdsdhdh ′ .
Integrating the equality above over [0, t], one has

H h (F |E)(t) + t 0 T 2 D h (F/E)(τ, x)dxdτ = H h (F |E)(0)
for each t ≥ 0. Since h is convex, one has

H h (F |E) ≥ 0 and D h (F/E) ≥ 0 ,
and the equality above entails the a priori estimates

         0 ≤ H h (F |E)(t) ≤ H h (F |E)(0) , +∞ 0 T 2 D h (F/E)(t, x)dxdt ≤ H h (F |E)(0) .
That H h (F |E) is a nonincreasing function of time is a general property of Markov processes; see for instance Yosida [START_REF] Yosida | Functional Analysis[END_REF] on p. 392. 3.4. Long time limit. As an application of the analogue of Boltzmann's H Theorem presented in the previous section, we investigate the asymptotic behavior of solutions of (2.30) in the limit as t → +∞.

Theorem 3.4. Let f in ≡ f in (x, ω) ∈ L ∞ (T 2 × S 1 ) satisfy f in (x, ω) ≥ 0 a.e. in (x, ω) ∈ T 2 × S 1 .
Let F be the solution of the Cauchy problem (2.30). Then

F (t, •, •, •, •)⇀CE in L ∞ (T 2 × S 1 × R + × [-1, 1]) weak-*, with C = 1 2π T 2 ×S 1 f in (x, ω)dxdω .
3.5. Speed of approach to equilibrium. The convergence to equilibrium in the long time limit established in the previous section may seem rather unsatisfying. Indeed, in most cases, solutions of linear kinetic models converge to equilibrium in a strong L 2 topology, and often satisfy some exponential decay estimate. While the convergence result in Theorem 3.4 might conceivably be improved, the following result rules out the possibility of a return to equilibrium at exponential speed in the strong L 2 sense. Theorem 3.5. There does not exist any function Φ ≡ Φ(t) satisfying

Φ(t) = o(t -3/2 ) as t → +∞ such that, for each f in ∈ L 2 (T 2 × S 1
), the solution F of the Cauchy problem (2.30) satisfies the bound

(3.1) F (t, •, •, •, •) -f in E L 2 (T 2 ×S 1 ×R+×[-1,1]) ≤ Φ(t) F (0, •, •, •, •) L 2 (T 2 ×S 1 ×R+×[-1,1])
for each t ≥ 0, with the notation

φ = 1 2π T 2 ×S 1 φ(x, ω)dxdω for each φ ∈ L 1 (T 2 × S 1 ).
By the same argument as in the proof of Theorem 3.5, one can establish a similar result for initial data in L p (T 2 × S 1 ), with the L 2 norm replaced with the L p norm in (3.1), for all p ∈]1, ∞[; in that case Φ(t) = o(t -(2p-1)/p ) is excluded.

The case p = 2 discussed in the theorem excludes the possibility of a spectral gap for the generator of the semigroup K t associated with equation (2.30) -that is to say, for the unbounded operator

A on L 2 (T 2 × S 1 × R + × [-1, 1]) defined by Af (x, ω, s, h) = (ω • ∇ x -∂ s )f (x, ω, s, h) - 1 -1 2P (2s, h|h ′ )f (x, R[θ(h ′ )]ω, 0, h ′ )dh ′ , with domain D(A) = {f ∈ L 2 (T 2 ×S 1 ×R + ×[-1, 1]) | (ω•∇ x -∂ s )f ∈ L 2 (T 2 ×S 1 ×R + ×[-1, 1])} .
4. An ergodic theorem with continued fractions 4.1. Continued fractions. Let α ∈ (0, 1) \ Q; its continued fraction expansion is denoted

(4.1) α = [0; a 1 , a 2 , a 3 , . . .] = 1 a 1 + 1 a 2 + 1 a 3 + . . . . Consider the Gauss map (4.2) T : (0, 1) \ Q ∋ x → 1 x - 1 x ∈ (0, 1) ∈ (0, 1) \ Q .
The positive integers a 1 , a 2 , a 3 , . . . are expressed in terms of α as

(4.3) a 1 = 1 α , and
a n = 1 T n-1 α , n ≥ 1 .
The action of T on α is most easily read on its continued fraction expansion:

(4.4) T [0; a 1 , a 2 , a 3 , . . .] = [0; a 2 , a 3 , a 4 , . . .] .
We further define two sequences of integers (p n ) n≥0 and (q n ) n≥0 by the following induction procedure:

(4.5) p n+1 = a n p n + p n-1 , p 0 = 1 , p 1 = 0 , q n+1 = a n q n + q n-1 , q 0 = 0 , q 1 = 1 .
The sequence of rationals ( pn qn ) n≥1 converges to α as n → ∞. Rather than the usual distance | pn qn -α|, it is more convenient to consider (4.6)

d n := |q n α -p n | = (-1) n-1 (q n α -p n ) , n ≥ 0 . Obviously (4.7) d n+1 = -a n d n + d n-1 , d 0 = 1 , d 1 = α .
We shall use the notation

a n (α) , p n (α) , q n (α) , d n (α) ,
whenever we need to keep track of the dependence of those quantities upon α. For each α ∈ (0, 1) \ Q, one has the relation a n (T α) = a n+1 (α), which follows from (4.4) and implies in turn that αd n (T α) = d n+1 (α), for each integer n ≥ 0, by (4.7). Therefore,

(4.8) d n (α) = n-1 k=0 T k α , n ≥ 0 .
While a n (α) and d n (α) are easily expressed in terms of the sequence (T k α) k≥0 , the analogous expression for q n (α) is somewhat more involved. With (4.5) and (4.7), one proves by induction that (4.9)

q n (α)d n+1 (α) + q n+1 (α)d n (α) = 1 , n ≥ 0 . Hence q n+1 (α)d n (α) = 1 - d n+1 (α)d n (α) d n (α)d n-1 (α) q n (α)d n-1 (α) ,
so that, by a straightforward induction (4.10)

q n+1 (α)d n (α) = n j=0 (-1) n-j d n+1 (α)d n (α) d j+1 (α)d j (α) , n ≥ 0 .
Replacing (d k (α)) 0≤k≤n+1 by its expression (4.8) leads to an expression of q n+1 (α) in terms of T k α for 0 ≤ k ≤ n.

4.2. The ergodic theorem. We recall that the Borel probability measure dG(x) = 1 ln 2 dx 1+x on (0, 1) is invariant under the Gauss map T , and that T is ergodic for the measure dG(x) (see for instance [START_REF] Ya | Continued Fractions[END_REF]), and even strongly mixing (see [START_REF] Philipp | Some metrical theorems in number theory[END_REF].)

For each α ∈ (0, 1) \ Q and ǫ ∈ (0, 1], define

(4.11) N (α, ǫ) = inf{n ≥ 0 | d n (α) ≤ ǫ} .
Lemma 4.1. For a.e. α ∈ (0, 1), one has

N (α, ǫ) ∼ 12 ln 2 π 2 ln 1 ǫ , ǫ → 0 + .
See [START_REF] Caglioti | On the distribution of free path lengths for the periodic Lorentz gas III[END_REF] (where it is stated as Lemma 3.1) for a proof. We further define (4.12)

δ n (α, ǫ) = d n (α) ǫ , n ≥ 0 , for each α ∈ (0, 1) \ Q and ǫ ∈ (0, 1].
Theorem 4.2. For m ≥ 0, let f be a bounded measurable function on (R + ) m+1 ; then there exists L m (f ) ∈ R independent of α such that

1 ln(1/η) 1 η f (δ N (α,ǫ) (α), δ N (α,ǫ)-1 (α), . . . , δ N (α,ǫ)-m (α)) dǫ ǫ → L m (f ) and 1 ln(1/η) 1 η (-1) N (α,ǫ) f (δ N (α,ǫ) (α), . . . , δ N (α,ǫ)-m (α)) dǫ ǫ → 0
for a.e. α ∈ (0, 1) as η → 0 + .

Proof. The proof of the first limit is as in [START_REF] Caglioti | On the distribution of free path lengths for the periodic Lorentz gas III[END_REF], and we just sketch it. Write

1 η f (δ N (α,ǫ) (α), δ N (α,ǫ)-1 (α), . . . , δ N (α,ǫ)-m (α)) dǫ ǫ = N (α,ǫ)-1 n=1 dn-1(α) dn(α) f (δ N (α,ǫ) (α), δ N (α,ǫ)-1 (α), . . . , δ N (α,ǫ)-m (α)) dǫ ǫ + d N (α,ǫ)-1 (α) η f (δ N (α,ǫ) (α), δ N (α,ǫ)-1 (α), . . . , δ N (α,ǫ)-m (α)) dǫ ǫ Whenever d n (α) ≤ ǫ < d n-1 (α)
, one has N (α, ǫ) = n so that, for a.e. α ∈ (0, 1),

1 η f (δ N (α,ǫ) (α), δ N (α,ǫ)-1 (α), . . . , δ N (α,ǫ)-m (α)) dǫ ǫ = N (α,η)-1 n=1 dn-1(α) dn(α) f (δ n (α), δ n-1 (α), . . . , δ n-m (α)) dǫ ǫ + O(1) .
Substituting ρ = dn(α) ǫ in each integral on the right hand side of the identity above, one has, for n ≥ m > 1

dn-1(α) dn(α) f (δ n (α), δ n-1 (α), . . . , δ n-m (α)) dǫ ǫ = 1 dn(α)/dn-1(α) f ρ, ρ T n-1 α , . . . , ρ m k=1 T n-m+k-1 α dρ ρ = F m-1 (T n-m α) ,
with the notation

F m-1 (α) = 1 T m-1 α f ρ, ρ T m-1 α , . . . , ρ m k=1 T k α dρ ρ .
Thus, for a.e. α ∈ (0, 1),

1 ln(1/η) 1 η f (δ N (α,ǫ) (α), δ N (α,ǫ)-1 (α), . . . , δ N (α,ǫ)-m (α)) dǫ ǫ = 1 ln(1/η) N (α,η)-1 n=m F m-1 (T n-m α) + O 1 ln(1/η) .
We deduce from Birkhoff's ergodic theorem and Lemma 4.1 that

(4.13) 1 ln(1/η) N (α,η)-1 n=m F m-1 (T n-m α) → L m (f ) := 12 ln 2 π 2 1 0 F m-1 (x)dG(x)
for a.e. α ∈ (0, 1) as η → 0 + , which establishes the first statement in the Theorem.

The proof of the second statement is fairly similar. We start from the identity

1 η (-1) N (α,ǫ) f (δ N (α,ǫ) (α), δ N (α,ǫ)-1 (α), . . . , δ N (α,ǫ)-m (α)) dǫ ǫ = N (α,η)-1 n=1 (-1) n dn-1(α) dn(α) f (δ n (α), δ n-1 (α), . . . , δ n-m (α)) dǫ ǫ + O(1) = N (α,η)-1 n=m (-1) n F m-1 (T n-m α) + O(1) .
Writing

N (α,η)-1 n=m (-1) n F m-1 (T n-m α) = m/2≤k≤(N (α,η)-1)/2 F m-1 (T 2k-m α) -F m-1 (T 2k+1-m α) + O(1)
we deduce from Birkhoff's ergodic theorem (applied to T 2 , which is ergodic since T is mixing, instead of T ) and Lemma 4.1 that, for a.e. α ∈ (0, 1) and in the limit as

η → 0 + , 1 ln(1/η) m/2≤k≤(N (α,η)-1)/2 F m-1 (T 2k-m α) -F m-1 (T 2k+1-m α) → 12 ln 2 π 2 1 0 (F m-1 (x) -F m-1 (T x)) dG(x) = 0 since the measure dG is invariant under T .
This entails the second statement in the theorem.

4.3. Application to 3-obstacle configurations. Consider ω ∈ S 1 such that 0 < ω 2 < ω 1 and ω 2 /ω 1 / ∈ Q, and let r ∈ (0, 1 2 ). The parameters (A(ω, r), B(ω, r), Q(ω, r), σ(ω, r)) defining the 3-obstacle configuration associated with the direction ω and the obstacle radius r are expressed in terms of the continued fraction expansion of ω 2 /ω 1 in the following manner. Proposition 4.3. For each ω ∈ S 1 such that 0 < ω 2 < ω 1 and ω 2 /ω 1 / ∈ Q, and each r ∈ (0, 1 2 ) one has

A(ω, r) = 1 - d N (α,ǫ) (α) ǫ B(ω, r) = 1 - d N (α,ǫ)-1 (α) ǫ - ǫ -d N (α,ǫ)-1 (α) d N (α,ǫ) (α) d N (α,ǫ) (α) ǫ Q(ω, r) = ǫq N (α,ǫ) σ(ω, r) = (-1) N (α,ǫ) where α = ω 2 ω 1 , and ǫ = 2r ω 1 .
By the definition of N (α, ǫ), one has

0 ≤ A(ω, r) , B(ω, r) , Q(ω, r) ≤ 1 .
This is Proposition 2.2 on p. 205 in [START_REF] Caglioti | On the distribution of free path lengths for the periodic Lorentz gas III[END_REF]; see also Blank-Krikorian [START_REF] Blank | Thom's problem on irrational flows[END_REF] on p. 726.

Our main result in this section is

Theorem 4.4. Let K = [0, 1] 3 × {±1}. For each F ∈ C(K), there exists L(F ) ∈ R such that 1 ln(1/η) 1/2 η F (A(ω, r), B(ω, r), Q(ω, r), σ(ω, r)) dr r → L(F )
for a.e. ω ∈ S 1 such that 0 < ω 2 < ω 1 , in the limit as η → 0 + .

Proof. First, observe that

F (A, B, Q, σ) = F + (A, B, Q) + σF -(A, B, Q) with F ± (A, B, Q) = 1 2 (F (A, B, Q, +1) ± F (A, B, Q, - 1

))

. By Proposition 4.3, one has F ± (A(ω, r), B(ω, r), Q(ω, r))

= F ± 1 -δ N (α,ǫ) (α), 1 -δ N (α,ǫ)-1 (α) - 1 -δ N (α,ǫ)-1 (α) δ N (α,ǫ) (α) δ N (α,ǫ) (α), 1 δ N (α,ǫ)-1 (α) d N (α,ǫ)-1 (α)q N (α,ǫ) (α) .
For each m ≥ 0, we define f m,± (δ N , . . . , δ N -m-1 ) :=

F ±   1-δ N , 1-δ N -1 - 1-δ N -1 δ N δ N , 1 δ N -1 N j=(N -m) + (-1) N -j δ N δ N -1 δ j δ j-1   .
Observe that αT α ≤ 1 2 for each α ∈ (0, 1) \ Q , so that, whenever n > m + 1,

q n (α)d n-1 (α) - n j=n-m (-1) n-j d n (α)d n-1 (α) d j (α)d j-1 (α) ≤ d n (α)d n-1 (α) d n-m-1 (α)d n-m-2 (α) ≤ n-1 k=n-m-1 T k α • T k-1 α ≤ 2 -m-1 .
Likewise, whenever n > m + 1 and n > l + 1, (4.14)

n j=n-l (-1) n-j d n (α)d n-1 (α) d j (α)d j-1 (α) - n j=n-m (-1) n-j d n (α)d n-1 (α) d j (α)d j-1 (α) ≤ 2 -min(l,m) . Since δ N (α,ǫ)-1 (α) > 1 by definition of N (α, ǫ), one has (4.15) |f m,± (δ N (α,ǫ) (α), . . . , δ N (α,ǫ)-m-1 (α)) -f l,± (δ N (α,ǫ) (α), . . . , δ N (α,ǫ)-l-1 (α))| ≤ ρ ± (2 -min(m,l) ) ,
where ρ ± is a modulus of continuity for F ± on the compact [0, 1] 3 . The inequality (4.14) implies that (4.16) 1 ln(1/4η)

1/4 η F ± (A(ω, r), B(ω, r), Q(ω, r)) dr r - 1 ln(1/4η) 1/2ω1 2η/ω1 f m,± (δ N (α,ǫ) (α), . . . , δ N (α,ǫ)-m-1 (α)) dǫ ǫ ≤ ρ ± (2 -m-1 )
upon setting α = ω 2 /ω 1 . Moreover, by Theorem 4.2, for each η * > 0 (4.17)

1 ln(η * /η) η * η f m,+ (δ N (α,ǫ) (α), . . . , δ N (α,ǫ)-m-1 (α)) dǫ ǫ → L m+1 (f m,+ ) 1 ln(η * /η) η * η (-1) N (α,ǫ) f m,-(δ N (α,ǫ) (α), . . . , δ N (α,ǫ)-m-1 (α)) dǫ ǫ → 0
a.e. in α ∈ (0, 1) as η → 0 + , and the inequality (4.15) implies that

|L m+1 (f m,+ ) -L l+1 (f l,+ )| ≤ ρ ± (2 -min(m,l) ) , l, m ≥ 0 .
In other words, (L m+1 (f m,+ )) m≥0 is a Cauchy sequence. Therefore, there exists

L ∈ R such that (4.18) L m+1 (f m,+ ) → L as n → ∞ .
Putting together (4.16), (4.17) and (4.18), we first obtain

L m+1 (f m,+ ) -2ρ(2 -m-1 ) ≤ lim η→0 + 1 ln(1/4η) 1/4 η F + (A(ω, r), B(ω, r), Q(ω, r)) dr r ≤ lim η→0 + 1 ln(1/4η) 1/4 η F + (A(ω, r), B(ω, r), Q(ω, r)) dr r ≤ L m+1 (f m,+ ) + 2ρ(2 -m-1 ) , and -2ρ(2 -m-1 ) ≤ lim η→0 + 1 ln(1/4η) 1/4 η σ(ω, r)F -(A(ω, r), B(ω, r), Q(ω, r)) dr r ≤ lim η→0 + 1 ln(1/4η) 1/4 η σ(ω, r)F -(A(ω, r), B(ω, r), Q(ω, r)) dr r ≤ 2ρ(2 -m-1 ) .
Letting m → ∞ in the above inequalities, we finally obtain that 1 ln(1/4η)

1/4 η F + (A(ω, r), B(ω, r), Q(ω, r)) dr r → L 1 ln(1/4η) 1/4 η σ(ω, r)F -(A(ω, r), B(ω, r), Q(ω, r)) dr r → 0
a.e. in ω ∈ S 1 with 0 < ω 2 < ω 1 as η → 0 + . Amplification of Theorem 4.4. The proof given above shows that

L(F ) = L(F + )
for each F ∈ C(K).

Computation of the asymptotic distribution of 3-obstacle configurations: a proof of Proposition 2.6

Having established the existence of the limit L(F ) in Theorem 4.4, we seek an explicit formula for it.

It would be most impractical to first compute L m+1 (f m,+ ) -with the notation of the proof of Theorem 4.4 -by its definition in formula (4.13), and then to pass to the limit as m → ∞.

We shall instead use a different method based on Farey fractions and the asymptotic theory of Kloosterman's sums as in [START_REF] Boca | The distribution of the free path lengths in the periodic twodimensional Lorentz gas in the small-scatterer limit[END_REF].

3-obstacle configurations and Farey fractions.

For each integer Q ≥ 1, consider the set of Farey fractions of order ≤ Q:

F Q := { p q | 1 ≤ p ≤ q ≤ Q , g.c.d.(p, q) = 1} .
If γ = p q < γ ′ = p ′ q ′ are two consecutive elements of F Q , then q + q ′ > Q , and p ′ q -pq ′ = 1 .

For each interval I ⊂ [0, 1], we denote

F Q (I) = I ∩ F Q .
The following lemma provides a (partial) dictionary between Farey and continued fractions.

Lemma 5.1. For each 0 < ǫ < 1, set Q = [1/ǫ]. Let 0 < α < 1 be irrational, and let γ = p q and γ ′ = p ′ q ′ be the two consecutive Farey fractions in

F Q such that γ < α < γ ′ . Then (i) if p q < α < p ′ -ǫ q ′ , then q N (α,ǫ) (α) = q and d N (α,ǫ) (α) = qα -p ; (ii) if p ′ -ǫ q ′ ≤ α ≤ p+ǫ q , then q N (α,ǫ) (α) = min(q, q ′ ) while d N (α,ǫ) (α) = qα -p if q < q ′ , and d N (α,ǫ) (α) = p ′ -q ′ α if q ′ < q ; (iii) if p+ǫ q < α < p ′ q ′ , then q N (α,ǫ) (α) = q ′ and d N (α,ǫ) (α) = p ′ -q ′ α .
Sketch of the proof. According to Dirichlet's lemma, for each integer Q ≥ 1, there exists an integer q such that 1 ≤ q ≤ Q and dist(qα, Z) < 1 Q+1 . If p ′′ q ′′ ∈ F Q is different from p q and p ′ q ′ , then q cannot be equal to q ′′ . For if α < p ′ q ′ < p ′′ q ′′ , then p ′′ -q ′′ α > 1 q ′′ (p ′′ q ′ -p ′ q ′′ ) ≥ 1 q ′′ ≥ 1 Q+1 . Thus q is one of the two integers q and q ′ . In the case (i) p ′ -q ′ α > ǫ > 1 Q+1 so that q = q ′ , hence q = q. Likewise, in the case (iii) qα -p > ǫ > 1 Q+1 so that q = q ′ . In the case (ii), one has

0 ≤ 1 q ′ (1 -qǫ) = 1 q ′ (qp ′ -pq ′ -qǫ) ≤ qα -p ≤ ǫ since q ≤ Q ≤ 1 ǫ . Likewise 0 ≤ p ′ -q ′ α
≤ ǫ, so that q is the smaller of q and q ′ .

In fact, the parameters (A(ω, r), B(ω, r), Q(ω, r)) can be computed in terms of Farey fractions, by a slight amplification of the proposition above. We recall that, for each ω ∈ S 1 such that 0 < ω 2 < ω 1 and ω 2 /ω 1 is irrational, one has

Q(ω, r) = ǫq N (α,ǫ) (α) , with α = ω 2 ω 1 and ǫ = 2r ω 1 .
Under the same conditions on ω, we define

D(ω, r) := d N (α,ǫ) ǫ , again with α = ω 2 ω 1 and ǫ = 2r ω 1 ,
and Q ′ (ω, r) in the following manner. Let Q = [1/ǫ] with ǫ = 2r ω1 , and let γ = p q and γ ′ = p ′ q ′ be the two consecutive Farey fractions in

F Q such that γ < α < γ ′ . Then (i) if p q < α < p ′ -ǫ q ′ , we set Q ′ (ω, r) := ǫq ′ ;
(ii) if p ′ -ǫ q ′ ≤ α ≤ p+ǫ q , we set Q ′ (ω, r) := ǫ max(q, q ′ ) ;

(iii) if p+ǫ q < α < p ′ q ′ , we set Q ′ (ω, r) := ǫq .

With these definitions, the 3-obstacle configuration parameters are easily expressed in terms of Farey fractions, as follows.

Proposition 5.2. Let 0 < r < 1 4 and ω ∈ S 1 be such that 0 < ω 2 < ω 1 and

ω 2 /ω 1 is irrational. Then A(ω, r) = 1 -D(ω, r) , while B(ω, r) = b(ω, r) - b(ω, r) D(ω, r) D(ω, r) , with b(ω, r) = Q(ω, r) -1 + Q ′ (ω, r)D(ω, r) Q(ω,

r) .

Sketch of the proof. We follow the discussion in Propositions 1 and 2 of [START_REF] Boca | The distribution of the free path lengths in the periodic twodimensional Lorentz gas in the small-scatterer limit[END_REF]. Consider the case p q < α < p ′ -ǫ q ′ , and set d = qα -p > 0 and d ′ = p ′ -q ′ α > 0. According to Proposition 1 of [START_REF] Boca | The distribution of the free path lengths in the periodic twodimensional Lorentz gas in the small-scatterer limit[END_REF], ǫB(ω, r) = ǫ -(p k -q k α), with the notation p k = p ′ + kp et q k = q ′ + kq for k ∈ N * chosen so that

p ′ + kp -ǫ q ′ + kq < α < p ′ + (k -1)p -ǫ q ′ + (k -1)q .
In other words, k is chosen so that d ′ -kd ≤ ǫ < d ′ -(k -1)d, where d = qα -p and d ′ = p ′ -q ′ α. That is to say,

-k = ǫ -d ′ d = 1 -d ′ /ǫ D , and
B = 1 - d ′ -kd ǫ = 1 - d ′ ǫ + k d ǫ = b - b D D with b = 1 -d ′ ǫ . Since qp ′ -pq ′ = 1, one has d ′ = (1 -q ′ d)/q, so that b = Q-1-Q ′ D Q .
The other cases are treated similarly. 5.2. Asymptotic distribution of (Q, Q ′ , D). As a first step in computing L(F ), we establish the following

Lemma 5.3. Let f ∈ C([0, 1] 3 ) and J = [α -, α + ] ⊂ (0, 1). Then S 1 + f (Q(ω, 1 2 ǫω 1 ), Q ′ (ω, 1 2 ǫω 1 ), D(ω, 1 2 ǫω 1 ))1 ω2/ω1∈J dω ω 2 1 → |J| [0,1] 3 f (Q, Q ′ , D)dλ(Q, Q ′ , D)
as ǫ → 0 + , where λ is the probability measure on [0, 1] 3 given by

dλ(Q, Q ′ , D) = 12 π 2 1 0<Q,Q ′ <1 1 Q+Q ′ >1 1 0<D< 1-Q Q ′ + 1 Q<Q ′ 1 1-Q Q ′ <D<1 dQdQ ′ dD Q
The proof of this lemma is based on the arguments involving Kloosterman's sums to be found in [START_REF] Boca | The distribution of the free path lengths in the periodic twodimensional Lorentz gas in the small-scatterer limit[END_REF].

Proof. For 0 < ǫ < 1, set Q = [1/ǫ]. Observe that Q(ω, 1 2 ǫω 1 ) - 1 ǫQ Q(ω, 1 2 ǫω 1 ) ≤ 1 Q + 1 Q(ω, 1 2 ǫω 1 ) ǫQ ≤ 1 Q + 1 , Q ′ (ω, 1 2 ǫω 1 ) - 1 ǫQ Q ′ (ω, 1 2 ǫω 1 ) ≤ 1 Q + 1 Q ′ (ω, 1 2 ǫω 1 ) ǫQ ≤ 1 Q + 1 , since Q(ω, 1 2 ǫω 1 ) ≤ ǫQ and Q ′ (ω, 1 2 ǫω 1 ) ≤ ǫQ while |D(ω, 1 2 ǫω 1 ) -ǫQD(ω, 1 2 ǫω 1 )| ≤ ǫ 1 ǫ -Q ≤ ǫ ≤ 1 Q since D(ω, 1 2 ǫω 1 ) ≤ 1. Since f is continuous on the compact set [0, 1] 3 ,
it is uniformly continuous; let ρ be a modulus of continuity for f . Then (5.1)

S 1 + f (Q(ω, 1 2 ǫω 1 ), Q ′ (ω, 1 2 ǫω 1 ), D(ω, 1 2 ǫω 1 )) dω ω 2 1 - S 1 + f 1 ǫQ Q(ω, 1 2 ǫω 1 ), 1 ǫQ Q ′ (ω, 1 2 ǫω 1 ), ǫQD(ω, 1 2 ǫω 1 ) dω ω 2 1 ≤ 3ρ(1/Q) → 0 as ǫ → 0 + .
Henceforth, we seek to prove that (5.2)

S 1 + f 1 ǫQ Q(ω, 1 2 ǫω 1 ), 1 ǫQ Q ′ (ω, 1 2 ǫω 1 ), ǫQD(ω, 1 2 ǫω 1 ) 1 ω2/ω1∈J dω ω 2 1 → |J| [0,1] 3 f (Q, Q ′ , D)dλ(Q, Q ′ , D)
as ǫ → 0 + . Without loss of generality, by an obvious density argument we restrict our attention to the case where the test function f is of the form

f (Q, Q ′ , D) = g(Q, Q ′ )h(D) , with g ∈ C([0, 1] 2 ) and h ∈ C([0, 1]). Then (5.3)
S 1 + g 1 ǫQ Q(ω, 1 2 ǫω 1 ), 1 ǫQ Q ′ (ω, 1 2 ǫω 1 ) h ǫQD(ω, 1 2 ǫω 1 ) 1 ω2/ω1∈J dω ω 2 1 = p/q∈FQ(J) p ′ -ǫ q ′ p q g q Q , q ′ Q h(Q(qα -p))dα + p/q∈FQ(J) p ′ q ′ p+ǫ q g q ′ Q , q Q h(Q(p ′ -q ′ α))dα + p/q∈FQ(J) p+ǫ q p ′ -ǫ q ′ 1 q<q ′ g q Q , q ′ Q h(Q(qα -p))dα + p/q∈FQ(J) p+ǫ q p ′ -ǫ q ′ 1 q ′ <q g q ′ Q , q Q h(Q(p ′ -q ′ α))dα + g L ∞ h L ∞ O sup p/q∈FQ(J) p ′ q ′ - p q + sup p ′ /q ′ ∈FQ(J) p ′ q ′ - p q = I + II + III + IV + O(1/Q)
where p q < p ′ q ′ are consecutive elements of F Q . Notice that the O term above is the contribution of the endpoints of J not being elements of F Q . Define

H(z) = z 0 h(y)dy , for each z ∈ R . Thus I = p/q∈FQ(J) g q Q , q ′ Q 1 Qq H Q 1 -ǫq q ′ II = p/q∈FQ(J) g q ′ Q , q Q 1 Qq ′ H Q 1 -ǫq ′ q III = p/q∈FQ(J) 1 q<q ′ g q Q , q ′ Q 1 Qq H(ǫQ) -H Q 1 -ǫq q ′ IV = p/q∈FQ(J) 1 q ′ <q g q ′ Q , q Q 1 Qq ′ H(ǫQ) -H Q 1 -ǫq ′ q One has H Q 1 -ǫq q ′ -H Q 1 -q/Q q ′ ≤ h L ∞ Q q q ′ 1 Q -ǫ ≤ h L ∞ Q q q ′ 1 Q(Q + 1) ; likewise H Q 1 -ǫq ′ q -H Q 1 -q ′ /Q q ≤ h L ∞ Q q ′ q 1 Q(Q + 1) , while |H(1) -H(ǫQ)| ≤ h L ∞ |1 -ǫQ| = h L ∞ ǫ 1 ǫ -Q ≤ h L ∞ ǫ .
Hence

I = p/q∈FQ(J) g q Q , q ′ Q 1 Qq H 1 -q/Q q ′ /Q + R I with |R I | ≤ g L ∞ h L ∞ p/q∈FQ(J) 1 q ′ Q(Q + 1)
.

Applying Lemmas 1 and 2 in [START_REF] Boca | The distribution of the free path lengths in the periodic twodimensional Lorentz gas in the small-scatterer limit[END_REF] shows that p/q∈FQ(J)

g q Q , q ′ Q 1 Qq H 1 -q/Q q ′ /Q → |J| • 6 π 2 (0,1) 2 g(Q, Q ′ )H 1 -Q Q ′ 1 Q+Q ′ >1 dQdQ ′ Q as Q → ∞ , while |R I | g L ∞ h L ∞ 1 Q • |J| • 6 π 2 (0,1) 2 1 Q+Q ′ >1 dQdQ ′ Q ′ = O(1/Q) , so that (5.4) I → |J| • 6 π 2 (0,1) 2 g(Q, Q ′ )H 1-Q Q ′ 1 Q+Q ′ >1 dQdQ ′ Q as Q → ∞ .
By a similar argument, one shows that (5.5)

II → |J| • 6 π 2 (0,1) 2 g(Q ′ , Q)H 1-Q ′ Q 1 Q+Q ′ >1 dQdQ ′ Q ′ , III → |J| • 6 π 2 (0,1) 2 g(Q, Q ′ ) H(1) -H 1-Q Q ′ 1 Q<Q ′ 1 Q+Q ′ >1 dQdQ ′ Q , IV → |J| • 6 π 2 (0,1) 2 g(Q ′ , Q) H(1) -H 1-Q ′ Q 1 Q ′ <Q 1 Q+Q ′ >1 dQdQ ′ Q ′ , as Q → ∞.
Substituting the limits (5.4)-(5.5) in (5.3) shows that

S 1 + g 1 ǫQ Q(ω, 1 2 ǫω 1 ), 1 ǫQ Q ′ (ω, 1 2 ǫω 1 ) h ǫQD(ω, 1 2 ǫω 1 ) 1 ω2/ω1∈J dω ω 2 1 → |J| • 12 π 2 (0,1) 3 g(Q, Q ′ )h(D)1 Q+Q ′ >1 1 0<D< 1-Q Q ′ dQdQ ′ dD Q +|J| • 12 π 2 (0,1) 3 g(Q, Q ′ )h(D)1 Q<Q ′ 1 Q+Q ′ >1 1 1-Q Q ′ <D<1 dQdQ ′ dD Q
as ǫ → 0, which establishes (5.2).

On account of (5.1), this proves the convergence announced in Lemma 5.3.

Asymptotic distribution of (A, B, Q).

Next we compute the image the probability measure λ under the map (Q, Q ′ , D) → (A, B, Q) defined in Proposition 5.2. In other words:

Lemma 5.4. Let g ∈ C([0, 1] 3 ). Then S 1 + g(A(ω, 1 2 ǫω 1 ), B(ω, 1 2 ǫω 1 ), Q(ω, 1 2 ǫω 1 )) dω ω 2 1 → [0,1] 3 g(A, B, Q)dν(A, B, Q)
as ǫ → 0 + , where ν is the probability measure on [0, 1] 3 given by

dν(A, B, Q) = 12 π 2 1 0<A<1 1 0<B<1-A 1 0<Q< 1 2-A-B dAdBdQ 1 -A .
Proof. We first compute the image Φ * λ of the probability measure λ in Lemma 5.3 under the map

Φ : [0, 1] 3 → [0, 1] × R × [0, 1] (Q, Q ′ , D) → (A, b, Q) = (1 -D, Q-1+Q ′ D Q , Q) The Jacobian of Φ is det∇Φ(Q, Q ′ , D) = 0 1-Q ′ D Q 2 1 0 D Q 0 -1 Q ′ Q 0 = D Q = 1 -A Q ,
A straightforward computation shows that

dΦ * λ(A, b, Q) = 12 π 2 1 0<A,Q<1 1 1-1 Q <b<1-A Q 1 A-A Q <b<0 + 1 b>0 1 b>2-A-1 Q dAdbdQ 1 -A .
This expression can be put in the form

(5.6) dΦ * λ(A, b, Q) = 12 π 2 1 0<A,Q<1 (1 b∈Λ1(A,Q) + 1 b∈Λ2(A,Q) ) dAdbdQ 1 -A where Λ 1 (A, Q) = (A -A Q , (1 -A Q ) ∧ 0) Λ 2 (A, Q) = ((2 -A -1 Q ) ∨ 0, (1 -A Q ))
The probability measure ν is the image of Φ * λ under the map

Ψ : [0, 1] × R × [0, 1] → [0, 1] 3 (A, b, Q) → (A, B, Q) = (A, b -[ b 1-A ](1 -A), Q) In other words, (5.7) dν(A, B, Q) = 12 π 2 1 0<A,Q<1 1 0<B<1-A dAdbdQ 1 -A • n∈Z 1 Λ1(A,Q) (B + n(1 -A)) + 1 Λ2(A,Q) (B + n(1 -A)) = 12 π 2 M (A, B, Q)1 0<A,Q<1 1 0<B<1-A dAdbdQ 1 -A Therefore #{n ∈ Z | B + n(1 -A) ∈ Λ 1 (A, Q)} +#{n ∈ Z | B + n(1 -A) ∈ Λ 2 (A, Q)} = 1 Q≤A + 1 Q>A (1 -A/Q) ∧ 0 -B 1 -A - A -A/Q -B 1 -A + 1 -A/Q -B 1 -A - (2 -A -1/Q) ∨ 0 -B 1 -A = 1 Q≤A + 1 Q>A (1 -A/Q) ∧ 0 -B 1 -A + 1 - (2 -A -1/Q) ∨ 0 -B 1 -A = 1 + 1 Q>A -B 1 -A - (2 -A -1/Q) ∨ 0 -B 1 -A . Now, if Q ≤ 1 2-A , (2 -A -1/Q) ∨ 0 -B 1 -A = -B 1 -A while Q > A if Q > 1 2-A since A + 1/A > 2, so that M (A, B, Q) = #{n ∈ Z | B + n(1 -A) ∈ Λ 1 (A, Q)} + #{n ∈ Z | B + n(1 -A) ∈ Λ 2 (A, Q)} = 1 + 1 Q> 1 2-A -B 1 -A - 2 -A -1/Q -B 1 -A . If 1 2-A < Q ≤ 1 and 0 < B < 1 -A, one has -(1 -A) < 2 -A -1/Q -B < (1 -A) so that -B 1 -A = -1 , 2 -A -1/Q -B 1 -A = 0 , if 2 -A -B ≥ 1/Q , 2 -A -1/Q -B 1 -A = -1 , if 2 -A -B < 1/Q . Therefore (5.9) M (A, B, Q) = 1 + 1 Q> 1 2-A -B 1 -A - 2 -A -1/Q -B 1 -A = 1 -1 Q> 1 2-A 1 2-A-B≥1/Q = 1 Q< 1 2-A-B
whenever B = 0, and this establishes the formula for ν in the lemma.

Computation of L(F ).

With the help of Lemma 5.4, we finally compute L(F ).

Proof of Proposition 2.6. Let F ∈ C(K), and set

F + (A, B, Q) = 1 2 (F (A, B, Q, +1) + F (A, B, Q, -1)) ; obviously F + ∈ C([0, 1] 3 ). By Lemma 5.4 1 ln(η * /η) η * η S 1 + F + (A(ω, 1 2 ǫω 1 ), B(ω, 1 2 ǫω 1 ), Q(ω, 1 2 ǫω 1 )) dω ω 2 1 dǫ ǫ → π 4 [0,1] 3 F + (A, B, Q)dν(A, B, Q) as η → 0 + , for each η * > 0.
On the other hand, by Fubini's theorem

1 ln(η * /η) η * η S 1 + F + (A(ω, 1 2 ǫω 1 ), B(ω, 1 2 ǫω 1 ), Q(ω, 1 2 ǫω 1 )) dω ω 2 1 dǫ ǫ = S 1 + 1 ln(η * /η) η * η F + (A(ω, 1 2 ǫω 1 ), B(ω, 1 2 ǫω 1 ), Q(ω, 1 2 ǫω 1 )) dǫ ǫ dω ω 2 1 = S 1 + 1 ln(η * /η) ω1η * /2 ω1η/2 F + (A(ω, r), B(ω, r), Q(ω, r)) dr r dω ω 2 1 .
By Theorem 4.4, the inner integral on the right hand side of the equality above converges a.e. in ω ∈ S 1 + to L(F + ). By dominated convergence, we therefore conclude that

L(F + ) = [0,1] 3 F + (A, B, Q)dν(A, B, Q) .
Finally, according to the amplification of Theorem 4.4

L(F ) = [0,1] 3 F + (A, B, Q)dν(A, B, Q)
or, in other words,

L(F ) = [0,1] 3 ×{±1} F (A, B, Q, σ)dµ(A, B, Q, σ) where dµ(A, B, Q, σ) = 1 2 dν(A, B, Q) ⊗ (δ σ=+1 + δ σ=-1 ) .
6. The transition probability: a proof of Theorem 2.1 6.1. Computation of P (S, h|h ′ ). We first establish that the image of the probability ν defined in Lemma 5.4 under the map

[0, 1] 3 ∋ (A, B, Q) → T A,B,Q,+1 (h ′ ) ∈ R + × [-1, 1]
is of the form P + (S, h|h ′ )dSdh, with a probability density P + which we compute in the present section. Let

f ∈ C c (R + × [-1, 1]); the identity R+×[-1,1] f (S, h)P + (S, h|h ′ )dSdh = [0,1] 3 f (T A,B,Q,+1 (h ′ ))dν(A, B, Q) defining P + (S, h|h ′ ) is recast as R+×[-1,1] f (S, h)P + (S, h|h ′ )dSdh = [0,1] 3 f (Q, h ′ -2(1 -A))1 1-2A<h ′ <1 dν(A, B, Q) + [0,1] 3 f 1-Q(1-B) 1-A , h ′ + 2(1 -B) 1 -1<h ′ <-1+2B dν(A, B, Q) + [0,1] 3 f 1-Q(B-A) 1-A , h ′ + 2(A -B) 1 -1+2B<h ′ <1-2A dν(A, B, Q) = I + II + III .
In I, we first integrate in B, since the argument of

f does not involve B. Ob- serving that 0 ∨ (2 -A -1 Q ) < B < 1 -A,
we have to distinguish two cases, namely 0 < Q < 1 2-A and 1 < Q < 1, which leads to

I = 12 π 2 f (Q, h ′ -2(1-A))1 (1-h ′ )/2<A<1 1 0<Q< 1 2-A + 1-Q Q(1-A) 1 1 2-A <Q<1 dQdA .
In the right-hand side of the equality above, we set S = Q and h = h ′ -2(1 -A), to find

I = 6 π 2 f (S, h)1 -1<h<h ′ <1 1 0<S< 1 1+(h ′ -h)/2 + 2(1-S) S(h ′ -h) 1 1 1+η <S<1 dSdh .
With the notation η = 1 2 |h -h ′ |, we see that I can be put in the form

I = 6 π 2 f (S, h)1 -1<h<h ′ <1 Sη1 0<S< 1 1+η + (1 -S)1 1 1+(h ′ -h)/2 <S<1 dSdh Sη = 6 π 2 f (S, h)1 -1<h<h ′ <1 1 0<S<1 (Sη) ∧ (1 -S) dSdh Sη .
As for II, given A, we first make the substitution

(S, h) = 1-Q(1-B) 1-A , h ′ + 2(1 -B)
whose Jacobian is

∂(S, h) ∂(Q, B) = -1-B 1-A 0 Q 1-A -2 = 2(1 -B) 1 -A = h -h ′ 1 -A .
Hence

II = 12 π 2 1 0 f (S, h)1 1 1-A+(h-h ′ )/2 <S< 1 1-A 1 A< h-h ′ 2 <1 dSdh |h-h ′ | dA = 12 π 2 f (S, h) 1 0 1 1 1-A+(h-h ′ )/2 <S< 1 1-A 1 A< h-h ′ 2 <1 dA dSdh |h -h ′ | .
The inner integral is recast as

1 0 1 1 1-A+(h-h ′ )/2 <S< 1 1-A 1 A< h-h ′ 2 <1 dA = 1 S>0 1 -1<h ′ <h<1 1 0 1 1-1/S<A<1-1/S+(h-h ′ )/2 1 A< h-h ′ 2 dA = 1 S>0 1 -1<h ′ <h<1 1 S<1 (1 + h-h ′ 2 -1 S ) + +1 S>0 1 -1<h ′ <h<1 1 S>1 ( h-h ′ 2 + 1 S -1) + = 1 S>0 1 -1<h ′ <h<1 ( h-h ′ 2 -| 1 S -1|) + ,
so that, eventually, we find

II = 12 π 2 f (S, h)1 S>0 1 -1<h ′ <h<1 ( h-h ′ 2 -| 1 S -1|) + dSdh h -h ′ = 6 π 2 f (S, h)1 S>0 1 -1<h ′ <h<1 (Sη -|1 -S|) + dSdh Sη
Now for III. Here we make the substitution

(S, h) = ( 1-Q(A-B) 1-A , h ′ + 2(A -B))
whose Jacobian is

∂(S, h) ∂(Q, B) = -A-B 1-A 0 Q 1-A -2 = 2(A -B) 1 -A = h -h ′ 1 -A .
This Jacobian vanishes for A = B; therefore, we further decompose

III = III 1 + III 2
with the notations

III 1 = [0,1] 3 f 1-Q(B-A) 1-A , h ′ + 2(A -B) 1 -1+2B<h ′ <1-2A 1 A<B dν(A, B, Q) , III 2 = [0,1] 3 f 1-Q(B-A) 1-A , h ′ + 2(A -B) 1 -1+2B<h ′ <1-2A 1 A>B dν(A, B, Q) .
We begin with III 1 . With the substitution above, one has

III 1 = 12 π 2 1 0 f (S, h)1 2A-1<(h-h ′ )/2<A 1 h ′ <1-2A 1 -1+2A<h 1 1/(1-A)<S<1/(1-A+(h-h ′ )/4) 1 -1<h<h ′ <1 dSdh |h-h ′ | dA = 12 π 2 f (S, h) 1 0 1 2A-1<(h-h ′ )/2<A 1 h ′ <1-2A 1 -1+2A<h 1 1/(1-A)<S<1/(1-A+(h-h ′ )/4) dA 1 -1<h<h ′ <1 dSdh |h-h ′ | .
The inner integral is

1 0 1 2A-1<(h-h ′ )/2<A 1 h ′ <1-2A 1 -1+2A<h 1 1/(1-A)<S<1/(1-A+(h-h ′ )/4) dA = 1 0 1 A< 1 2 + 1 4 (h-h ′ ) 1 A< 1 2 (1-h ′ ) 1 A< 1 2 (1+h) 1 1-1 S + 1 4 (h-h ′ )<A<1- 1 S dA = (( 1 2 + 1 4 (h -h ′ )) ∧ 1 2 (1 -h ′ ) ∧ 1 2 (1 + h) ∧ (1 -1 S ) -(1 -1 S + 1 4 (h -h ′ )) ∨ 0) + = (( 1 2 -1 4 |h + h ′ | -1 4 (h ′ -h)) ∧ (1 -1 S ) -(1 -1 S -1 4 (h ′ -h)|) ∨ 0) + . Alternatively, using the relations (a+c)∨(b+c) = a∨b+c, (a+c)∧(b+c) = a∧b+c, (-a) ∧ (-b) = -(a ∨ b) and a ∨ b + a ∧ b = a + b, we find that (( 1 2 -1 4 |h + h ′ | -1 4 (h ′ -h)) ∧ (1 -1 S ) -(1 -1 S -1 4 (h ′ -h)) ∨ 0) + = ((-1 2 -1 4 |h + h ′ |) ∧ ( 1 4 (h ′ -h) -1 S )+1-1 4 (h ′ -h) -(1 -1 4 (h ′ -h)) ∨ 1 S + 1 S ) + = ((1 -1 4 (h ′ -h)) ∧ 1 S -( 1 2 + 1 4 |h + h ′ |) ∨ ( 1 S -1 4 (h ′ -h))) + so that III 1 = 12 π 2 f (S, h) 1 0 1 2A-1<(h-h ′ )/2<A 1 h ′ <1-2A 1 -1+2A<h × 1 1/(1-A)<S<1/(1-A+(h-h ′ )/4) dA 1 -1<h<h ′ <1 dSdh |h-h ′ | = 12 π 2 f (S, h)1 -1<h<h ′ <1 (S -1 4 S(h ′ -h)) ∧ 1 -(1 -1 4 S(h ′ -h)) ∨ ( S 2 + 1 4 S|h + h ′ |) + dSdh S|h -h ′ | .
With the notation ζ = 1 2 |h + h ′ |, we recast this last expression as

III 1 = 6 π 2 f (S, h)1 -1<h<h ′ <1 (S -1 2 Sη)∧1-(1-1 2 Sη)∨( 1 2 S + 1 2 Sζ) + dSdh Sη .
The computation of III 2 is fairly similar. Starting with the same substitution as for III 1 , we obtain

III 2 = 12 π 2 1 0 f (S, h)1 2A-1<(h-h ′ )/2<A 1 h ′ <1-2A 1 -1+2A<h 1 1/(1-A+(h-h ′ )/4)<s<1/(1-A) 1 -1<h ′ <h<1 dSdh |h-h ′ | dA = 12 π 2 f (S, h) 1 0 1 2A-1<(h-h ′ )/2<A 1 h ′ <1-2A 1 -1+2A<h 1 1/(1-A+(h-h ′ )/4)<S<1/(1-A) dA 1 -1<h ′ <h<1 dSdh |h -h ′ | .
The inner integral is recast as

1 0 1 2A-1<(h-h ′ )/2<A 1 h ′ <1-2A 1 -1+2A<h 1 1/(1-A+(h-h ′ )/4)<S<1/(1-A) dA = 1 0 1 1 2 (h-h ′ )<A 1 A< 1 2 + 1 4 (h-h ′ ) 1 A< 1 2 (1-h ′ ) 1 A< 1 2 (1+h) 1 1- 1 S <A<1- 1 S + 1 4 (h-h ′ ) dA = 1 0 1 1 2 (h-h ′ )<A 1 A< 1 2 - 1 4 |h+h ′ |+ 1 4 (h-h ′ ) 1 1-1 S <A<1- 1 S + 1 4 (h-h ′ ) dA = (( 1 2 -1 4 |h + h ′ | + 1 4 (h -h ′ )) ∧ (1 -1 S + 1 4 (h -h ′ )) -(1 -1 S ) ∨ 1 2 (h -h ′ )) + .
As in the case of III 1 , we have

(( 1 2 -1 4 |h + h ′ | + 1 4 (h -h ′ )) ∧ (1 -1 S + 1 4 (h -h ′ )) -(1 -1 S ) ∨ 1 2 (h -h ′ )) + = (1 + 1 4 (h -h ′ ) + (-1 2 -1 4 |h + h ′ |) ∧ (-1 S ) -(1 -1 S ) ∨ 1 2 (h -h ′ )) + = (1 + 1 4 (h -h ′ ) -( 1 2 + 1 4 |h + h ′ |) ∨ 1 S -(1 -1 S ) -1 2 (h -h ′ ) + (1 -1 S ) ∧ 1 2 (h -h ′ )) + = ( 1 S -1 4 (h -h ′ ) -( 1 2 + 1 4 |h + h ′ |) ∨ ( 1 S ) + (1 -1 S ) ∧ 1 2 (h -h ′ )) + = ((1 -1 4 (h -h ′ )) ∧ ( 1 S + 1 4 (h -h ′ )) -( 1 2 + 1 4 |h + h ′ |) ∨ 1 S ) + so that III 2 = 12 π 2 1 0 f (S, h)1 2A-1<(h-h ′ )/2<A 1 h ′ <1-2A 1 -1+2A<h 1 1/(1-A+(h-h ′ )/4)<S<1/(1-A) 1 -1<h ′ <h<1 dSdh |h-h ′ | dA = 12 π 2 f (S, h)1 -1<h ′ <h<1 (S -1 4 S(h -h ′ )) ∧ (1 + 1 4 S(h -h ′ )) -( 1 2 S + 1 4 S|h + h ′ |) ∨ 1 + dSdh S|h -h ′ |
In other words

III 2 = 6 π 2 f (S, h)1 -1<h<h ′ <1 (S -1 2 Sη)∧(1+ 1 2 Sη)-( 1 2 S + 1 2 Sζ)∨1 + dSdh Sη .
Summing up the contributions I, II, III 1 and III 2 , we find that

P + (S, h|h ′ ) = 6 π 2 Sη 1 -1<h<h ′ <1 1 0<S<1 (Sη) ∧ (1 -S) + 6 π 2 Sη 1 -1<h ′ <h<1 1 S>0 (Sη -|1 -S|) + + 6 π 2 Sη 1 -1<h<h ′ <1 1 S>0 (S -1 2 Sη)∧1-(1-1 2 Sη)∨( 1 2 S + 1 2 Sζ) + + 6 π 2 Sη 1 -1<h ′ <h<1 1 S>0 (S -1 2 Sη)∧(1+ 1 2 Sη)-( 1 2 S + 1 2 Sζ)∨1 + .
Observe that

1 S>0 (Sη -|1 -S|) + = 1 1 1+η <S< 1 1-η (Sη -|1 -S|) . On the other hand, if (S -1 2 Sη)∧1-(1-1 2 Sη)∨( 1 2 S + 1 2 Sζ) ≥ 0 then (S -1 2 Sη) -(1-1 2 Sη) ≥ 0 so that S ≥ 1; likewise 1-( 1 2 S + 1 2 Sζ) ≥ 0 so that S ≤ 2. Hence 1 S>0 (S -1 2 Sη)∧1-(1-1 2 Sη)∨( 1 2 S + 1 2 Sζ) + = 1 1<S<2 (S -1 2 Sη)∧1-(1-1 2 Sη)∨( 1 2 S + 1 2 Sζ) + . By the same token (S -1 2 Sη)∧(1+ 1 2 Sη)-( 1 2 S + 1 2 Sζ)∨1 ≥ 0 implies that (S -1
2 Sη) -1 ≥ 0, so that S ≥ 1, and hence

1 S>0 (S -1 2 Sη)∧(1+ 1 2 Sη)-( 1 2 S + 1 2 Sζ)∨1 + = 1 S>1 (S -1 2 Sη)∧(1+ 1 2 Sη)-( 1 2 S + 1 2 Sζ)∨1 + . Finally P + (S, h|h ′ ) = 6 π 2 Sη 1 -1<h<h ′ <1 1 0<S<1 (Sη) ∧ (1 -S) + 6 π 2 Sη 1 -1<h ′ <h<1 1 1 1+η <S< 1 1-η (Sη -|1 -S|) + 6 π 2 Sη 1 -1<h<h ′ <1 1 1<S<2 (S -1 2 Sη)∧1-(1-1 2 Sη)∨( 1 2 S + 1 2 Sζ) + + 6 π 2 Sη 1 -1<h ′ <h<1 1 S>1 (S -1 2 Sη)∧(1+
P (S, h|h ′ ) ≤ 6 π 2 S(h -h ′ ) 1 1+h ′ < 2 S .
On the other hand, 1 + h ′ < 2 S and S ≥ 4 imply that h ′ < -1 2 , so that h ≥ |h ′ | > 1 2 ; therefore h -h ′ > 1 and the inequality above entails (2.16).

Starting from (2.16), observe that, because of the symmetries (2.14),

1 -1 1 -1 P (s, h|h ′ )dhdh ′ = 4 0≤|h ′ |<h≤1 P (s, h|h ′ )dhdh ′ ≤ 24 π 2 S 0≤|h ′ |<h≤1 1 1+h ′ < 2 S dhdh ′ = 48 π 2 S 3 ,
which is (2.17).

Proof of Theorem 3.1

The existence of the integral defining E follows from the positivity of P (s, h|h ′ ) and the second identity in (2.15).

That E(0, h) = 1 for |h| ≤ 1 follows from the formula defining E and the second identity in (2.15): this establishes the first part of statement 1).

The definition of E and the first formula in statement 1) show that

-∂ s E(s, h) = 1 -1 2P (2s, h|h ′ )dh ′ = 1 -1 2P (2s, h|h ′ )E(0, h ′ )dh ′ , s > 0 , |h| ≤ 1 .
Using again the positivity of P (s, h|h ′ ) and the second identity in (2.15) shows that, for each h ∈ [-1, 1], the function (s, h ′ ) → P (s, h|h ′ ) belongs to L 1 (R + × [-1, 1]). Therefore E(s, h) → 0 for each h ∈ [-1, 1] as s → +∞. Thus each function F ≡ F (s, h) of the form F (s, h) = CE(s, h) satisfies both conditions in statement 2). Conversely, let F ≡ F (s, h) satisfy the conditions in statement 2), and let Φ(h) := F (0, h) for a.e. h ∈ [-1, 1]. Integrating both sides of the differential equation satisfied by

F in s ∈ R + yields (7.1) Φ(h) = 1 -1 Π(h|h ′ )Φ(h ′ )dh ′ since F (s, h) vanishes as s → +∞, with Π(h|h ′ ) = ∞ 0 P (S, h|h ′ )dS .
Multiplying each side of the identity (7.1) by Φ(h), and integrating in h ∈ [-1, 1], we see that

(7.2) 1 -1 Φ(h) 2 dh = 1 -1 1 -1 Π(h|h ′ )Φ(h)Φ(h ′ )dhdh ′ .
Observe that, by (2.14)

Π(h ′ |h) = Π(h|h ′ ) , for a.e. h, h ′ ∈ [-1, 1] ,
and

1 -1 Π(h|h ′ )dh ′ = 1 -1 Π(h ′ |h)dh ′ = 1 , for a.e. h ∈ [-1, 1] ,
in view of (2.15). Therefore

1 -1 Φ(h) 2 dh = 1 -1 1 -1 Π(h|h ′ )dh ′ Φ(h) 2 dh = 1 -1 1 -1 Π(h|h ′ )dh Φ(h ′ ) 2 dh , so that (7.2) becomes 0 = 1 -1 Φ(h) 2 dh - 1 -1 1 -1 Π(h|h ′ )Φ(h)Φ(h ′ )dhdh ′ = 1 -1 1 -1 Π(h|h ′ ) 1 2 (Φ(h) 2 + Φ(h ′ ) 2 )dhdh ′ - 1 -1 1 -1 Π(h|h ′ )Φ(h)Φ(h ′ )dhdh ′ = 1 -1 1 -1 Π(h|h ′ ) 1 2 (Φ(h) -Φ(h ′ )) 2 dh .
Since Π(h|h ′ ) > 0 a.e. in h, h ′ ∈ [-1, 1] as can be seen from the explicit formula (2.18), this last equality implies that

Φ(h) = C a.e. in h ∈ [-1, 1] ,
for some nonnegative constant C. Therefore

-∂ s F (s, h) = C 1 -1 2P (2s, h|h ′ )dh ′ , and lim s→+∞ F (s, h) = 0 a.e. in h ∈ [-1, 1] ,
so that

F (s, h) = C ∞ 2s 1 -1 P (τ, h|h ′ )dh ′ dτ = CE(s, h) ,
which proves the uniqueness part of statement 2). Now for statement 3); by definition of P (s, h|h ′ ), for each t > 0

1 -1 E(s, h)dh = ∞ 2s 1 -1 1 -1 P (t, h|h ′ )dtdhdh ′ = lim ǫ→0 + 1 | ln ǫ| 1/4 ǫ 1 -1 1 [2s,+∞)×[-1,1] (T r (h ′ , ω))dh ′ dr r = lim ǫ→0 + 1 | ln ǫ| 1/4 ǫ 1 -1 1 [2s,+∞) (2rτ r (Y r (h ′ , ω))dh ′ dr r = 2 lim ǫ→0 + 1 | ln ǫ| 1/4 ǫ ν r ({(x, ω) ∈ Γ + r /Z 2 | 2rτ r (x, ω) ≥ 2s} dr r
where ν r is the probability measure on Γ + r /Z 2 that is proportional to ω • n x dxdω. Using formula (1.3) in [START_REF] Boca | The distribution of the free path lengths in the periodic twodimensional Lorentz gas in the small-scatterer limit[END_REF], which is a straightforward consequence of variant of Santaló's formula established in Lemma 3 of [START_REF] Dumas | Remarks on the notion of mean free path for a periodic array of spherical obstacles[END_REF], we conclude that

1 -1 E(s, h)dh = -2p ′ (2s) , s ≥ 0 ,
which is the first formula in statement 3). The second formula there is a consequence of the expression of p ′′ (s) as a power series in 1/s given in formula (1.5) of [START_REF] Boca | The distribution of the free path lengths in the periodic twodimensional Lorentz gas in the small-scatterer limit[END_REF].

Finally, we establish the second formula in statement 1). Indeed

∞ 0 1 -1 E(s, h)dhds = ∞ 0 -2p ′ (2s)ds = p(0) = 1 ,
while the first equality in that formula follows from the identity defining E and Fubini's theorem.

8. Proof of Theorem 3.3

If F is a solution of (2.30), we set f = F/E, so that, observing that E(0, h

′ ) = 1 for h ′ ∈ [-1, 1], E(s, h)(∂ t + ω • ∇ x -∂ s )f (t, x, ω, s, h) -f (t, x, ω, s, h)∂ s E(s, h) = 1 -1 2P (2s, h|h ′ )f (t, x, R[θ(h ′ )], 0, h ′ )dh ′ , Since ∂ s E(s, h) = - 1 -1 2P (2s, h|h ′ )dh ′ ,
the equation above can be put in the form

E(s, h)(∂ t + ω • ∇ x -∂ s )f (t, x, ω, s, h) = 1 -1 2P (2s, h|h ′ ) (f (t, x, R[θ(h ′ )], 0, h ′ ) -f (t, x, ω, s, h)) dh ′ .
Next we multiply both sides of the equation above by h ′ (f )h ′ (1): the left hand side becomes

(h ′ (f (t, x, ω, s, h)) -h ′ (1))E(s, h)(∂ t + ω • ∇ x -∂ s )f (t, x, ω, s, h) = (∂ t +ω • ∇ x -∂ s ) (E(s, h)(h(f (t, x, ω, s, h)) -h(1) -h ′ (1)(f (t, x, ω, s, h) -1))) +∂ s E(s, h)(h(f (t, x, ω, s, h)) -h(1) -h ′ (1)(f (t, x, ω, s, h) -1)) = (∂ t +ω • ∇ x -∂ s ) (E(s, h)(h(f (t, x, ω, s, h)) -h(1) -h ′ (1)(f (t, x, ω, s, h) -1))) - 1 -1 2P (2s, h|h ′ )(h(f (t, x, ω, s, h)) -h(1) -h ′ (1)(f (t, x, ω, s, h) -1))dh ′ .
Integrating in (ω, s, h) transforms this expression into

∂ t S 1 ∞ 0 1 -1 (h(f ) -h(1) -h ′ (1)(f -1))(t, x, ω, s, h)E(s, h)dhdsdω + div x S 1 ∞ 0 1 -1 ω(h(f ) -h(1) -h ′ (1)(f -1))(t, x, ω, s, h)E(s, h)dhdsdω + S 1 1 -1 (h(f ) -h(1) -h ′ (1)(f -1))) (t, x, ω, 0, h)dhdω - S 1 ∞ 0 1 -1 1 -1 2P (2s, h|h ′ )(h(f )-h(1)-h ′ (1)(f -1))(t, x, ω, s, h)dh ′ dhdsdω .
Using the relation (2.15) simplifies this term into

∂ t S 1 ∞ 0 1 -1 (h(f ) -h(1) -h ′ (1)(f -1))(t, x, ω, s, h)E(s, h)dhdsdω + div x S 1 ∞ 0 1 -1 ω(h(f ) -h(1) -h ′ (1)(f -1))(t, x, ω, s, h)E(s, h)dhdsdω + S 1 1 -1 (h(f ) -h ′ (1)f )(t, x, ω, 0, h ′ )dh ′ dω - S 1 ∞ 0 1 -1 1 -1 2P (2s, h|h ′ )(h(f ) -h ′ (1)f )(t, x, ω, s, h)dh ′ dhdsdω .
On the other hand, multiplying the right hand side by h ′ (f (t, x, ω, s, h))h ′ (1) and integrating in (ω, s, h) leads to

S 1 ∞ 0 1 -1 1 -1 2P (2s, h|h ′ )(f (t, x, R[θ(h ′ )]ω, 0, h ′ ) -f (t, x, ω, s, h)) × (h ′ (f (t, x, ω, s, h)) -h ′ (1))dh ′ dhdsdω = S 1 ∞ 0 1 -1 1 -1 2P (2s, h|h ′ )(f (t, x, R[θ(h ′ )]ω, 0, h ′ ) -f (t, x, ω, s, h)) × h ′ (f (t, x, ω, s, h))dh ′ dhdsdω + h ′ (1) S 1 ∞ 0 1 -1 1 -1 2P (2s, h|h ′ )f (t, x, ω, s, h)dh ′ dhdsdω -h ′ (1) S 1 1 -1 f (t, x, ω, 0, h ′ )dh ′ dω .
after substituting ω for R[θ(h ′ )]ω in the last integral above.

Putting together the left-and right-hand sides, we arrive at the equality

∂ t S 1 ∞ 0 1 -1 (h(f ) -h(1) -h ′ (1)(f -1))(t, x, ω, s, h)E(s, h)dhdsdω + div x S 1 ∞ 0 1 -1 ω(h(f ) -h(1) -h ′ (1)(f -1))(t, x, ω, s, h)E(s, h)dhdsdω + S 1 1 -1 (h(f ) -h ′ (1)f )(t, x, ω, 0, h ′ )dh ′ dω - S 1 ∞ 0 1 -1 1 -1 2P (2s, h|h ′ )(h(f ) -h ′ (1)f )(t, x, ω, s, h)dh ′ dhdsdω = S 1 ∞ 0 1 -1 1 -1 2P (2s, h|h ′ )(f (t, x, R[θ(h ′ )]ω, 0, h ′ ) -f (t, x, ω, s, h)) × h ′ (f (t, x, ω, s, h))dh ′ dhdsdω + h ′ (1) S 1 ∞ 0 1 -1 1 -1 2P (2s, h|h ′ )f (t, x, ω, s, h)dh ′ dhdsdω -h ′ (1) S 1 1 -1 f (t, x, ω, 0, h ′ )dh ′ dω .
All the terms with a factor h ′ (1) in the integral part of the equality above compensate, so that the equality above reduces to

∂ t S 1 ∞ 0 1 -1 (h(f ) -h(1) -h ′ (1)(f -1))(t, x, ω, s, h)E(s, h)dhdsdω + div x S 1 ∞ 0 1 -1 ω(h(f ) -h(1) -h ′ (1)(f -1))(t, x, ω, s, h)E(s, h)dhdsdω + S 1 1 -1 h(f )(t, x, ω, 0, h ′ )dh ′ dω - S 1 ∞ 0 1 -1 1 -1 2P (2s, h|h ′ )h(f )(t, x, ω, s, h)dh ′ dhdsdω - S 1 ∞ 0 1 -1 1 -1 2P (2s, h|h ′ )(f (t, x, R[θ(h ′ )]ω, 0, h ′ ) -f (t, x, ω, s, h) × h ′ (f (t, x, ω, s, h))dh ′ dhdsdω = 0 .
Using again the relation (2.15) and the substitution ω → R[θ(h ′ )]ω, one has

S 1 1 -1 h(f )(t, x, ω, 0, h ′ )dh ′ dω = S 1 ∞ 0 1 -1 1 -1 2P (2s, h|h ′ )h(f )(t, x, ω, 0, h ′ )dh ′ dhdsdω = S 1 ∞ 0 1 -1 1 -1 2P (2s, h|h ′ )h(f )(t, x, R[θ(h ′ )]ω, 0, h ′ )dh ′ dhdsdω
so that the previous identity can be put in the form

∂ t S 1 ∞ 0 1 -1 (h(f ) -h(1) -h ′ (1)(f -1))(t, x, ω, s, h)E(s, h)dhdsdω + div x S 1 ∞ 0 1 -1 ω(h(f ) -h(1) -h ′ (1)(f -1))(t, x, ω, s, h)E(s, h)dhdsdω + S 1 ∞ 0 1 -1 1 -1 2P (2s, h|h ′ ) h(f )(t, x, R[θ(h ′ )]ω, 0, h ′ ) -h(f )(t, x, ω, s, h) -(f (t, x, R[θ(h ′ )]ω, 0, h ′ ) -f (t, x, ω, s, ′ (f (t, x, ω, s, h) dh ′ dhdsdω = 0 .
Integrating both sides of this identity in x ∈ T 2 , we finally obtain

d dt H h (f E|E) + T 2 D h (f )(t, x)dx = 0 ,
with H h (F |E) and D h (f ) defined as in the statement of Theorem 3.3.

Proof of Theorem 3.2

Inserting F (t, x, ω, s, h) = f (t, x, ω)E(s, h) in (2.30), one finds that

E(s, h)(∂ t + ω • ∇ x )f (t, x, ω) -f (t, x, ω)∂ s E(s, h) = 1 -1 2P (2s, h|h ′ )f (t, x, R[θ(h ′ )]ω)dh ′ so that E(s, h)(∂ t + ω • ∇ x )f (t, x, ω) = 1 -1 2P (2s, h|h ′ ) (f (t, x, R[θ(h ′ )]ω) -f (t, x, ω)) dh ′ since -∂ s E(s, h) = 1 -1 2P (2s, h|h ′ )dh ′ , s > 0 , |h| ≤ 1 -see Theorem 3.1.
Integrating in h ∈ [-1, 1] both sides of the penultimate equality, we obtain

1 -1 E(s, h)dh(∂ t + ω • ∇ x )f (t, x, ω) = 1 -1 1 -1 2P (2s, h|h ′ ) (f (t, x, R[θ(h ′ )]ω) -f (t, x, ω)) dh ′ . Since 1 -1 E(s, h)dh ∼ 1 π 2 s 2 as s → +∞ by Theorem 3.1, while 1 -1 1 -1 P (S, h|h ′ )dhdh ′ ≤ C ′ (1 + S) 3 by (2.17), one has 1 -1 E(s, h)dh(∂ t + ω • ∇ x )f (t, x, ω) ∼ 1 π 2 s 2 (∂ t + ω • ∇ x )f (t, x, ω) , and 1 -1 1 -1 2P (2s, h|h ′ ) (f (t, x, R[θ(h ′ )]ω) -f (t, x, ω)) dh ′ dh ≤ 4C ′ (1 + 2s) 3 f (t, •, •) L ∞ (T 2 ×S 1 ) . Hence      (∂ t + ω • ∇ x )f (t, x, ω) = 0 , and
1 -1 ∂ s E(s, h ′ ) (f (t, x, R[θ(h ′ )]ω) -f (t, x, ω)) dh ′ = 0
Integrating the second equality in s > 0, and observing that E s=0 = 1 while

E(s, h) → 0 uniformly in h ∈ [-1, 1] as s → +∞, we see that ∞ 0 1 -1 ∂ s E(s, h ′ ) (f (t, x, R[θ(h ′ )]ω) -f (t, x, ω)) dh ′ ds = 1 -1 ∞ 0 ∂ s E(s, h ′ )ds (f (t, x, R[θ(h ′ )]ω) -f (t, x, ω)) ds = 1 -1 (f (t, x, R[θ(h ′ )]ω) -f (t, x, ω)) dh ′ = 0
or, in other words

f (t, x, ω) = 1 2 1 -1 f (t, x, R[θ(h ′ )]ω)dh ′ =: φ(t, x) .
On the other hand, the first equation implies that

(∂ t + ω • ∇ x )φ(t, x) = 0 so that ∂ t φ = ∂ xj φ = 0 , j = 1, 2 .
Hence φ is a constant, which implies in turn that f (t, x, ω) = F (t,x,ω,s,h)

E(s,h) is a constant. 10. Proof of Theorem 3.4 Let f in ∈ L ∞ (T 2 × S 1
) be such that f in (x, ω) ≥ 0 a.e. in (x, ω) ∈ T 2 × S 1 , and let F be the solution of the Cauchy problem (2.30). Define F 0 (t, x, ω, s, h) = f in (x, ω)E(s, h) and set K j F 0 = F j for j ∈ N, where we recall that (K t ) t≥0 is the evolution semigroup associated to the Cauchy problem (2.30).

Step 1:

Assume first that ∇ m x f in ∈ L ∞ (T 2 × S 1 ) for all m ≥ 0, so that F solves (2.30) in the classical sense. Then, with h(z) = 1 2 z 2 , one has

H(F n+1 |E) + n j=0 (H(F j |E) -H(F j+1 |E)) = H(F 0 |E)
for each n ≥ 0, and since

H(F j |E) -H(F j+1 |E) = H(F j |E) -H(K 1 F j |E) ≥ 0 by Theorem 3.3, one has H(F j |E) -H(F j+1 |E) → 0 as j → +∞ . Now, H(F j |E) -H(F j+1 |E) = 1 0 T 2 D(K t F j /E)dxdt = 1 0 T 2 S 1 ∞ 0 [-1,1] 2 P (2s, h|h ′ )Φ j (t, x, ω, s, h, h ′ ) 2 dhdh ′ dsdωdxdt with the notation Φ j (t, x, ω, s, h, h ′ ) = K t F j (x, ω, s, h) E(s, h) -K t F j (x, R[θ(h ′ )]ω, 0, h ′ ) for a.e. (t, x, ω, s, h, h ′ ) ∈ [0, 1] × T 2 × S 1 × R + × [-1, 1] 2 .
In view of properties 1) and 3) of the evolution semigroup (K t ) t≥0 recalled in section 3, one has

0 ≤ K t F j (x, ω, s, h) E(s, h) ≤ f in L ∞ (T 2 ×S 1 )
for a.e. (x, ω, s, h) ∈ T 2 ×S 1 ×R + ×[-1, 1] and each t ≥ 0, so that, up to extraction of a subsequence,

F j k /E⇀F/E in L ∞ (T 2 × S 1 × R + × [-1, 1
≤ (∂ t + ω • ∇ x -∂ s )K t F j k ≤ 4C ′′ 1 + 2s f in L ∞ (T 2 ×S 1 )
so that, by the usual trace theorem for the advection operator ∂ t + ω • ∇ x -∂ s (see for instance [START_REF] Cessenat | Théorèmes de trace L p pour des espaces de fonctions de la neutronique[END_REF])

K t F j k (x, R[θ(h ′ )]ω, 0, h ′ )⇀K t F (x, R[θ(h ′ )]ω, 0, h ′ ) in L ∞ ([0, 1] × T 2 × S 1 × [-1, 1]
) weak-*. In particular Let us identify the constant C. Property 4) of the semigroup (K t ) t≥0 recalled in section 3 implies that

Φ j k ⇀Φ in L ∞ ([0, 1] × T 2 × S 1 × R + × [-1, 1]
T 2 ×S 1 ∞ 0 1 -1 F j (x, ω, s, h)dhdsdxdω = T 2 ×S 1 f in (x, ω)dxdω ∞ 0 1 -1 E(s, h)dhds = T 2 ×S 1 f in (x, ω)dxdω . Since F j k /E⇀F/E in L ∞ (T 2 × S 1 × R + × [-1, 1]
) weak-* as j k → +∞,

T 2 ×S 1 f in (x, ω)dxdω = T 2 ×S 1 ∞ 0 1 -1 F (x, ω, s, h)dhdsdxdω = T 2 ×S 1 ∞ 0 1 -1 CE(s, h)dhdsdxdω = 2πC . Hence F j k /E⇀C = 1 2π T 2 ×S 1 f in (x, ω)dxdω in L ∞ (T 2 × S 1 × R + × [-1, 1]) weak-*.
Since the sequence (F j /E) j≥0 is relatively compact in L ∞ (T 2 ×S 1 ×R + ×[-1, 1]) weak-* and by the uniqueness of the limit point as j → +∞, we conclude that

F j /E⇀C = 1 2π T 2 ×S 1 f in (x, ω)dxdω in L ∞ (T 2 × S 1 × R + × [-1, 1]
) weak-* as j → +∞. Thus, we have proved that

K j F 0 ⇀CE in L ∞ (T 2 × S 1 × R + × [-1, 1]) weak-*
where

C = 1 2π T 2 ×S 1 f in (x, ω)dxdω = 1 2π T 2 ×S 1 ∞ 0 1 -1
F 0 (x, ω, s, h)dhdsdxdω , whenever ∇ m x f in ∈ L ∞ (T 2 × S 1 ) for each m ≥ 0, with f in ≥ 0 a.e. on T 2 × S 1 . Step2:

The same holds true if 0 ≤ f in ∈ L ∞ (T 2 × S 1 ) without assuming that ∇ m x f in ∈ L ∞ (T 2 × S 1 ) for each m ≥ 1, by regularizing the initial data in the x-variable.

Indeed, if (ζ ǫ ) ǫ>0 is a regularizing sequence in T 2 such that ζ ǫ (-z) = ζ ǫ (z) for each z ∈ T 2 , one has

T 2 ×S 1 ∞ 0 1 -1 (K t (ζ ǫ ⋆ x F 0 ) -K t F 0 )(x, ω, s, h)φ(x, ω, s, h)dhdsdxdω = T 2 ×S 1 ∞ 0 1 -1
K t (F 0 )(x, ω, s, h)(ζ ǫ ⋆ x φ -φ)(x, ω, s, h)dhdsdxdω because K t commutes with translations in the variable x. Since

K t F 0 /E L ∞ (T 2 ×S 1 ×R+×[-1,1]) = F 0 /E L ∞ (T 2 ×S 1 ×R+×[-1,1])
for all t ≥ 0, and

ζ ǫ ⋆ x φ -φ → 0 in L 1 (T 2 × S 1 × R + × [-1, 1]) , we conclude that K t (ζ ǫ ⋆ x F 0 ) -K t F 0 ⇀0 in L ∞ (T 2 × S 1 × R + × [-1, 1]
) weak-* as ǫ → 0 + uniformly in t ≥ 0. Since we have established in Step 1 that

K j (ζ ǫ ⋆ x F 0 )⇀ 1 2π T 2 ×S 1 f in (x, ω)dxdω in L ∞ (T 2 × S 1 × R + × [-1, 1]
) weak-* for each ǫ > 0, we conclude by a classical double limit argument that

K j (F 0 )⇀CE with C = 1 2π T 2 ×S 1 f in (x, ω)dxdω in L ∞ (T 2 × S 1 × R + × [-1, 1]
) weak-* as j → +∞. Summarizing, we have proved that

K j F 0 ⇀ 1 2π T 2 ×S 1 ∞ 0 1 -1 F 0 (x, ω, s, h)dhdsdxdω E in L ∞ (T 2 × S 1 × R + × [-1, 1]
) weak-* as j → +∞. Replacing F 0 with K t F 0 for each t ∈ [0, 1] and noticing that

T 2 ×S 1 ∞ 0 1 -1 K t F 0 (x, ω, s, h)dhdsdxdω = T 2 ×S 1 ∞ 0 1 -1
F 0 (x, ω, s, h)dhdsdxdω we conclude that

K t F 0 ⇀ 1 2π T 2 ×S 1 ∞ 0 1 -1 F 0 (x, ω, s, h)dhdsdxdω E in L ∞ (T 2 × S 1 × R + × [-1, 1]
) weak-* as t → +∞.

11. Proof of Theorem 3.5

The argument used in the proof is reminiscent of the one used in [START_REF] Golse | On the statistics of free-path lengths for the periodic Lorentz gas[END_REF][START_REF] Golse | On the periodic Lorentz gas in the Boltzmann-Grad scaling[END_REF].

Assume the existence of a profile Φ(t) such that the estimate (3.1) holds. Therefore, for each initial data f in ∈ L 2 (T 2 × S 1 ), the solution

F (t, •, •, •, •) = K t (f in (•, •)E(•, •))
of the Cauchy problem satisfies

F (t, •, •, •, •) L 2 (T 2 ×S 1 ×R+×[-1,1]) ≤ f in E L 2 (T 2 ×S 1 ×R+×[-1,1]) + Φ(t) F (0, •, •, •, •) L 2 (T 2 ×S 1 ×R+×[-1,1]) = f in √ 2π E L 2 (R+×[-1,1]) + Φ(t) f in L 2 (T 2 ×S 1 ) E L 2 (R+×[-1,1]
) . Assume f in ≥ 0 a.e. on T 2 × S 1 ; then F ≥ 0 a.e. on R + × T 2 × S 1 × R + × [-1, 1], so that the right-hand side of the equation satisfied by F is a.e. nonnegative on

R + × T 2 × S 1 × R + × [-1, 1]. Thus F ≥ G a.e. on R + × T 2 × S 1 × R + × [-1, 1],
where G is the solution of the Cauchy problem (∂ t + ω • ∇ x -∂ s )G(t, x, ω, s, h) = 0 , G(0, x, ω, s, h) = f in (x, ω)E(s, h) .

The Cauchy problem above can be solved by the method of characteristics, which leads to the explicit formula G(t, x, ω, s, h) = f in (x -tω, ω)E(s + t, h) , for a.e. (t, x, ω, s, h

) ∈ R + × T 2 × S 1 × R + × [-1, 1]. Thus F (t, •, •, •, •) L 2 (T 2 ×S 1 ×R+×[-1,1]) ≥ G(t, •, •, •, •) L 2 (T 2 ×S 1 ×R+×[-1,1]) = f in L 2 (T 2 ×S 1 ) ∞ t 1 -1 E(s, h) 2 dhds 1/2 ≥ f in L 2 (T 2 ×S 1 ) 1 √ 2 ∞ t

Clearly ρǫ

L 1 (T 2 ) = ρ ǫ L 1 (R 2 ) = ǫ 2 ρ L 1 (R 2 ) , ρǫ L 2 (T 2 ) = ρ ǫ L 2 (R 2 ) = ǫ ρ L 2 (R 2 ) .
Choosing f in (x, ω) := ρ ǫ (x) in the inequality above leads to

∞ t 1 -1 E(s, h)dh 2 ds 1/2 ≤ ǫ ρ L 1 (R 2 ) ρ L 2 (R 2 ) + Φ(t) √ 2 E L 2 (R+×[-1,1])
and, letting ǫ → 0 + , we conclude that

∞ t 1 -1 E(s, h)dh 2 ds 1/2 ≤ √ 2Φ(t) E L 2 (R+×[-1,1]) .
That Φ(t) = o(t -3/2 ) as t → +∞ is in contradiction with statement 3) in Theorem 3.1, which implies that

∞ t 1 -1 E(s, h)dh 2 ds 1/2 ∼ 1 √ 3π 2 1 t 3/2
as t → +∞, while statement 1) in the same theorem implies that

Figure 1 .

 1 Figure1. The impact parameter h corresponding with a collision with incoming direction ω or equivalently with outgoing direction ω ′

Figure 2 .

 2 Figure 2. The transfer map

  ρ ǫ (x) := ρ x ǫ for each ǫ ∈ (0, 1). Define ρǫ as the periodicized bump function ρǫ (x) := k∈Z 2 ρ ǫ (x + k) .

E 2 L 2 ( 1 - 1 E 1 - 1 E

 221111 R+×[-1,1]) ≤ ∞ 0 (s, h)E(0, h)dhds = ∞ 0 (s, h)dhds = 1 .

  Proof of Corollary 2.2. As for the statements in Corollary 2.2, observe that the symmetries (2.14) and the fact that P is piecewise continuouson R + × [-1, 1] × [1, 1] follow from(2.12). The first identity in (2.15) follows from the fact that, by definition, (s, h) → P (s, h|h ′ ) is a probability density, while the second identity there follows from the first and the symmetry P (s, h|h ′ ) = P (s, h ′ |h).If S ≥ 4 and h, h ′ ∈] -1, 1[ satisfy |h ′ | ≤ h, the simplified formula (2.13) implies that

	6.3.
	1 2 Sη)-( 1 2 S + 1 2 Sζ)∨1 + .

  P (2s, h|h ′ )Φ(t, x, ω, s, h, h ′ ) 2 dh ′ dhdsdωdxdt -K t F (x, R[θ(h ′ )]ω, 0, h ′ ) = 0 a.e. on [0, 1] × T 2 × S 1 × R + × [-1, 1] 2 . Averaging both sides of this identity in h ′ ∈ [-1, 1] shows that K t F (x, R[θ(h ′ )]ω, 0, h ′ )dh ′ =: f (t, x, ω) By Theorem 3.2, one has f (t, x, ω) = C a.e. in (t, x, ω) ∈ [0, 1] × T 2 × S 1 for some constant C ≥ 0, so that F (x, ω, s, h) = CE(s, h) a.e. in (x, ω, s, h) ∈ T 2 × S 1 × R + × [-1, 1] .

	1					∞	1	1
	0	T 2 S 1 1	0	-1	-1	1
		=	0	T 2	D(K t F/E)dxdt ≤ lim k→∞	0	T 2	D(K t F j k /E)dxdt = 0 ,
	so that
	Φ(t, x, ω, s, h, h ′ ) =	K t F (x, ω, s, h) E(s, h)
	K t F (x, ω, s, h) E(s, h)	= 1 2	1 -1

2

) weak-* with

Φ(t, x, ω, s, h, h ′ ) = K t F (x, ω, s, h) E(s, h) -K t F (x, R[θ(h ′ )]ω, 0, h ′ ) a.e. in [0, 1] × T 2 × S 1 × R + × [-1, 1] 2 .

By convexity and weak limit for a.e. [0, τ ] × T 2 × S 1 × R + × [-1, 1], i.e. K t F (x, ω, s, h) = f (t, x, ω)E(s, h) .

  Therefore, if there exist a profile Φ(t) satisfying the estimate in the statement of the theorem, one has inf in L 2 (T 2 ×S 1 ) + Φ(t) √ 2 E L 2 (R+×[-1,1])for each t > 0, and for eachf in ∈ L 2 (T 2 × S 1 ) s.t. f in ≥ 0 a.e. on T 2 × S 1 . Let ρ ∈ C(R 2 ) such that ρ ≥ 0 , supp(ρ) ⊂ (-1 4 , 1 4 ) 2 , and set

										1	2	1/2
										E(s, h)dh	ds	.
										-1
	t	∞	1 -1	E(s, h)dh	2	ds	1/2	≤	√	2π f

The existence of this limit, and an explicit formula for p are obtained in[START_REF] Boca | The distribution of the free path lengths in the periodic twodimensional Lorentz gas in the small-scatterer limit[END_REF].

where (5.8)

Whenever u, v / ∈ Z, one has

Hence, for a.e. A, Q ∈ [0, 1]

Likewise, for a.e. A, Q ∈

By formula 2.24,

so that P (S, h|h ′ ) = 1 2 (P + (S, h|h ′ ) + P + (S, -h| -h ′ )) , thereby leading to formula (2.11). 6.2. Proof of the simplified formula (2.13).

and denote

Sη . Whenever S ≥ 1, the reader will easily check that (2.12) can be written as

Since L ≥ M , the expression above can be reduced after checking the three cases

the formulas above can be recast as

On the other hand, if S < 1, the last two terms on the right hand side of (2.12) vanish identically so that 2 S(h -h ′ ). Putting together these last two formulas leads to (2.13).


Muhammad Shoaib Saleem

Renault Éric
email: eric.renault@it-sudparis.eu

Djamal Zeghlache
email: djamal.zeghlache@it-sudparis.eu

NetInf Mobile Node Architecture and Mobility Management based on LISP Mobile Node

In this paper, we propose an architecture for Network of Information mobile node (NetInf MN). It bears characteristics and features of basic NetInf node architecture with features introduced in the LISP MN architecture. We also introduce a virtual node layer for mobility management in the Network of Information. Therefore, by adopting this architecture no major changes in the contemporary network topologies is required. Thus, making our approach more practical.

I. INTRODUCTION

4WARD [START_REF]The FP7 4WARD Project[END_REF] is the proposed architecture and design for the Future Internet. The Network of information (NetInf) is one of the components of the 4WARD project. The cornerstone of this architecture is that the information takes the prime position superseding the node-centric approach of the current internet. The genesis of NetInf was influenced by existing major technologies and beyond. The features which NetInf exhibits are the melange of existing and innovative solutions.

Mobility is one of the scenarios considered in NetInf. Late Locator Construction (LLC) [START_REF] Eriksson | Dynamic Internetworking Based on Late Locator Construction[END_REF] is a proposal to handle mobility and multihoming issues in NetInf. It implements the locator/identifier separation idea. Whenever there is a mobility or rehoming event, there is no interruption in the ongoing TCP session between nodes. The notion behind the LLC is to use a Global Locator (GL) for routing between the Core Network (CN) and the Edge Networks (ENs). IPv4/IPv6 can be considered as the CN and nodes, mobile or stationary, forming a topology at the edge of CN, can be considered as ENs. GL is built inside the Locator Construction System (LCS) which is embedded inside the core network. Each Edge Network (EN) and the host attached to these ENs have Attachment Registers (AR) within the LCS. Each AR has IDs of itself and of its neighbors. The overall goal of LLC is to minimize the update signaling to the Locator Construction System (LCS) to update the GL to deal with the scalability problem within the core network. It has also been proposed that the core network scales well as core and edge networks are responsible for their own routing [START_REF] Eriksson | Dynamic Internetworking Based on Late Locator Construction[END_REF].

Location Identifier Separation Protocol (LISP) [START_REF] Farinacci | Locator/ID Separation Protocol (LISP)[END_REF] is a network based approach for locator/identifier separation. It focuses on limiting the size of routing tables and improving scalability and routing system. LISP Mobile Node (MN) [START_REF] Farinacci | LISP Mobile Node[END_REF] is an extention to the classic LISP for mobile nodes. It has multiple design goals including a wide range of communication possibilities in different mobility cases along with multihoming in MN, as well as allowing MN to act as a server. The LISP MN architecture has some LISP features together with additional characteristics to support mobility and rehoming events. What has been envisaged here for NetInf is to provide seamless connectivity between the mobile nodes even if there is a simultaneous roaming. Embedding LISP MN features in NetInf MN can then improve mobility management in LLC for the NetInf architecture.

II. MOBILITY MANAGEMENT IN NETINF

Although LLC is a good proposal, it has however some discrapencies in terms of properly addressing both scalability and mobility issues. LISP MN inherits classic LISP features but lacks complete compatibility to work with LLC. Our proposal consists in a new approach to deal with these issues. The prime goal is the mobility management in NetInf. A NetInf MN should bear charactersitics and features of classic NetInf node architecture together with features introduced by the LISP MN architecture. The design goal of such a node is to deal with mobility in a highly dynamic environment.

A. NetInf MN Architecture

Fig. 1 presents a high-level overview of the NetInf MN architecture. At the bottom both physical and network layers provide services to the transport layer placed above. Within the transport layer is located the transport control engine (TCE). In NetInf, TCE is responsible for the coordination of protocols used for accessing NetInf objects. In this design, both Inner Locator Construction Tunnel Router (ILCTR) and Outer Locator Construction Tunnel Router (OLCTR) include their functionalities in TCE. The goal of these two routers is to work under the conditions when non-NetInf sites are communicating. The working of these routers will be exten- sively explained in the future work. The virtual node layer within the TCE is another feature introduced for the mobility management of information. At the top, the API provides services to users. This API gets services from the transport layer below. It should be noted that NetInf and the nodes are interdependent on each other. Nodes are dependent on NetInf for the network management, while NetInf depends on nodes in scenarios where it has to manage and propagate the resources inside the network.

B. Mobility Scenarios

In order to understand the basic functionalities of our proposed architecture, consider two mobility scenarios. In Fig. 2, MN1 roams into a new edge network EN3 during the ongoing session with a stationary node MN2 in edge network EN2. MN1 first contacts local server LS in EN3 to register its new ID allocated by the DHCP. It also queries the local cache mechanism to find any trace of MN2 activities in EN3. If it is successful in finding one, it can use that cached locator and resume its session. Otherwise, it contacts the core network.

Fig. 3 presents a simultaneous rehoming event. Two mobile nodes MN1 and MN2 are communicating in their respective edge networks EN1 and EN2. A mobility event occurs when both mobile nodes move to a new edge network EN3 at the same time. In this case, it would be more efficient if mobile nodes MN1 and MN2 could access a local server in EN3 and register their IDs to continue their session, instead of connecting to the core network. As IDs are global and unique, no further name resoluton is then required. However, a lookup request must be sent to the local server when both nodes are registered and each node should set TTL. If nodes relocate each other before their TTL expires, the session is reestablished (in case of interruption) or continued. Otherwise, both should contact the core network.

III. MOBILITY MANAGEMENT OF INFORMATION IN NETINF THROUGH VIRTUAL NODE COORDINATION

The main problems faced in the network because of mobility are: the unpredictable motion of nodes and unpredictable availability of nodes i.e. nodes are continuously joining and leaving the network. Since information takes the centric position within the network, the mobility management of the information becomes a major concern. One solution for the management of information mobility is the use of virtual mobile nodes. These virtual nodes can be thought of as a program running on real nodes. Whenever there is a mobility or rehoming event, the node which departs from the local network handovers its connectivity to one of the virtual nodes so that the connectivity of the ongoing session is carried away smoothly. This is made possible in our proposed NetInf MN architecture by the introduction of a new layer, the Virtual Node layer that provides virtual mobile node services in coordination with the real node. All mobile nodes can act as temporary virtual mobile node and can provide services to the local mobile nodes.

For example, in Fig. 2 EN1 hosts two nodes namely MN1 and MN3. These two nodes have the ability to support virtual mobile node features introduced in their respective layered architecture. Lets assume that MN1 is in session with MN2 in EN2. When MN1 moves from EN1 to EN3, it initiates the algorithm to make use of the embedded virtual layer. It contacts MN3 in EN1 and establishes a local connection by informing about its new destination (EN3). MN3 is now acting as a virtual MN1 and is communicating with MN2. This continues until MN1 reaches EN3. When this happens, it gets all the updates from MN3 and reconnects to MN2 again.

IV. CONCLUSION AND FUTURE WORK

There are still some open issues to be addressed for the proposed architecture for mobile entities in NetInf. So far, the work is on the midway. ILCTR and OLCTR services are yet to be finalised. Besides providing services for non-NetInf sites, they can also work as storage spaces in edge networks in case of disconnection from the core network. Different parameters have to be considered to evaluate the proposal. As ILCTR encapsulates packets before sending them to non-NetInf sites, how could the Maximum Transmission Unit (MTU) be maximized. A new efficient protocol shall be proposed to deal with the mobility cases presented above. At present, we are in the process of evaluating our architecture through extensive simulations and real mobility traces. Finally, we plan to address the multihoming issue along with the security issue during mobility.

Fig. 1 .

 1 Fig. 1. NetInf Mobile Node Architecture.

Fig. 2 .

 2 Fig. 2. Single Mobile Node Roaming.

Fig. 3 .

 3 Fig. 3. Simultaneous Mobile Nodes Roaming towards same Edge Network.