Jean-Baptiste Gouéré
email: jbgouere@univ-orleans.fr

Percolation in a multiscale Boolean model

We consider percolation in a multiscale Boolean model. This model is defined as the union of scaled independent copies of a given Boolean model. The scale factor of the n th copy is ρ -n . We prove, under optimal integrability assumptions, that no percolation occurs in the multiscale Boolean model for large enough ρ if the rate of the Boolean model is below some critical value.

The following description may be more intuitive. Let χ denote the projection of ξ on R d . With probability one this projection is one-to-one. We can therefore write: ξ = {(c, r(c)), c ∈ χ}.

Write µ = mν where ν is a probability measure. Then, χ is a Poisson point process on R d with density m. Moreover, given χ, the sequence (r(c)) c∈χ is a sequence of independent random variable with common distribution ν. We shall not use this point of view.

Percolation in the Boolean model

Let C denote the connected component of Σ that contains the origin. We say that Σ percolates if C is unbounded with positive probability. We refer to the book by Meester and Roy [START_REF] Meester | Continuum percolation[END_REF] for background on continuum percolation. Set: λ c (µ) = inf{λ > 0 : Σ(λµ) percolates}.

One easily check that λ c (µ) is finite as soon as µ has a positive mass. In [START_REF] Gouéré | Subcritical regimes in the Poisson Boolean model of continuum percolation[END_REF] we proved that λ c (µ) is positive if and only if:

r d µ(dr) < ∞.
The only if part had been proved earlier by Hall [START_REF] Hall | On continuum percolation[END_REF]. For all A, B ⊂ R d , we write A ↔ Σ B if there exists a path in Σ from A to B. We denote by S(c, r) the Euclidean sphere or radius r centered at c :

S(c, r) = {x ∈ R d : x -c 2 = r}.
We write S(r) when c = 0.

The critical parameter λ c (µ) can also be defined as follows:

λ c (µ) = sup λ > 0 : P {0} ↔ Σ(λµ) S(r) → 0 as r → ∞ ,

We shall need two other critical parameters:

λ c (µ) = sup λ > 0 : P S(r/2) ↔ Σ(λµ) S(r) → 0 as r → ∞ , λ c (µ) = sup λ > 0 : r d P {0} ↔ Σ(λµ) S(r) → 0 as r → ∞ .

We have (see Lemma 14) :

λ c (µ) ≤ λ c (µ) ≤ λ c (µ). (1)
When the support of µ is bounded,

P {0} ↔ Σ(λµ) S(r)
decays exponentially fast to 0 as soon as λ < λ c (µ) (see for example [START_REF] Meester | Continuum percolation[END_REF], Section 12.10 in [START_REF] Grimmett | of Grundlehren der Mathematischen Wissenschaften[END_REF] in the case of constant radii or the papers [START_REF] Meester | Nonuniversality and continuity of the critical covered volume fraction in continuum percolation[END_REF], [START_REF] Men | Coincidence of critical points in Poisson percolation models[END_REF], [START_REF] Zuev | Continuous models of percolation theory[END_REF] and [START_REF] Zuev | Continuous models of percolation theory[END_REF]). Therefore:

λ c (µ) = λ c (µ) = λ c (µ)
as soon as the support of µ is bounded.

(

) 2
Remarks.

-The treshold parameter λ c (µ) is positive if and only if r d µ(dr) is finite (i.e., if and only if λ c (µ) is positive). See Lemma 15. -Using ideas of [START_REF] Gouéré | Subcritical regimes in the Poisson Boolean model of continuum percolation[END_REF], we can check that λ c (µ) is positive if and only if

x d ∞ x r d µ(dr) → 0 as x → ∞.
If we only use results stated in [START_REF] Gouéré | Subcritical regimes in the Poisson Boolean model of continuum percolation[END_REF], we can easily get the following weaker statements. Let D(λµ) denote the Euclidean diameter of the connected component of Σ(λµ) that contains the origin. Note that λ c (µ) is positive if and only if there exists λ such that:

r d P (D(λµ) ≥ r) → 0, as r → ∞. ( 3
) If E(D(λµ) d ) is finite then (3) holds. If (3) holds then E(D(λµ) d-ε
) is finite for any small enough ε > 0. By Theorem 2.2 of [START_REF] Gouéré | Subcritical regimes in the Poisson Boolean model of continuum percolation[END_REF] we thus get the following implications:

+∞ 0 r 2d µ(dr) < ∞ implies λ c (µ) > 0 implies ∀ε > 0 : +∞ 0 r 2d-ε µ(dr) < ∞.

A multiscale Boolean model

Let ρ > 1 be a scale factor. Let (Σ n ) n≥0 be a sequence of independent copies of Σ(µ). In this paper, we are interested in percolation properties of the following multiscale Boolean model:

Σ ρ (µ) = n≥0 ρ -n Σ n . (4)
We shall sometimes write Σ ρ to simplify the notations. As before, we say that Σ ρ percolates if the connected component of Σ ρ that contains the origin is unbounded with positive probability. This model seems to have been first introduced as a model of failure in geophysical medias in the 80 ′ . We refer to the paper by Molchanov, Pisarenko and Reznikova [START_REF] Molchanov | Multiscale models of failure and percolation[END_REF] for an account of those studies. For more recent results we refer to [START_REF] Broman | Universal behavior of connectivity properties in fractal percolation models[END_REF], [START_REF] Meester | Continuum percolation[END_REF], [START_REF] Meester | Nonuniversality and continuity of the critical covered volume fraction in continuum percolation[END_REF], [START_REF] Menshikov | On the connectivity properties of the complementary set in fractal percolation models[END_REF], [START_REF] Menshikov | On a multiscale continuous percolation model with unbounded defects[END_REF] and [START_REF] Yu | A note on percolation of Poisson sticks[END_REF].

This model is related to a discrete model introduced by Mandelbrot [START_REF] Mandelbrot | Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier[END_REF]. We refer to the survey by L. Chayes [START_REF] Chayes | Aspects of the fractal percolation process[END_REF] and, for more recent results, to [START_REF] Broman | Large-N limit of crossing probabilities, discontinuity, and asymptotic behavior of threshold values in Mandelbrot's fractal percolation process[END_REF], [START_REF] Orzechowski | On the phase transition to sheet percolation in random Cantor sets[END_REF] and [START_REF] White | On the value of the critical point in fractal percolation[END_REF].

In [START_REF] Menshikov | On the connectivity properties of the complementary set in fractal percolation models[END_REF], Menshikov, Popov and Vachkovskaia considered the case where the radii of the unscaled process Σ 0 equal 1. They proved the following result.

Theorem 1 ( [START_REF] Menshikov | On the connectivity properties of the complementary set in fractal percolation models[END_REF]) If λ < λ c (δ 1 ) then, for all large enough ρ, Σ ρ (λµ) does not percolate.

In [START_REF] Menshikov | On a multiscale continuous percolation model with unbounded defects[END_REF] the same authors considered the case where the radii are random and can be unbounded. They considered the following sub-autosimilarity assumption on the measure µ:

lim a→∞ sup r≥1/2 a d µ([ar, +∞[) µ([r, +∞[) = 0 (5)
with the convention 0/0 = 0. They proved the following result.

Theorem 2 ( [START_REF] Menshikov | On a multiscale continuous percolation model with unbounded defects[END_REF]) Assume that the measure µ satisfies [START_REF] Gouéré | Subcritical regimes in some models of continuum percolation[END_REF]. Assume that λ c (µ) is positive. If λ < λ c (µ) then, for all large enough ρ, Σ ρ (λµ) does not percolate.

Note that (5) is fulfilled for any measure with bounded support. Because of (2), Theorem 2 is then a generalization of Theorem 1.

In [START_REF] Gouéré | Subcritical regimes in some models of continuum percolation[END_REF] we proved the following related result in which ρ is fixed.

Theorem 3 ([5]) Let ρ > 1.
There exists λ > 0 such that Σ ρ (λµ) does not percolate if and only if:

[1,+∞[ β d ln(β)µ(dβ) < ∞. (6)
The main result of this paper is the first item of the following theorem. The second item is easy and already contained in Theorem 3. Recall that, by Lemma 15, λ c (µ) is positive as soon as r d µ(dr) is finite and therefore as soon as (6) holds.

Theorem 4

1. Assume [START_REF] Grimmett | of Grundlehren der Mathematischen Wissenschaften[END_REF]. Then, for all λ < λ c (µ), there exists ρ(λ) > 1 such that, for all ρ ≥ ρ(λ): P S(r/2) ↔ Σ ρ (λµ) S(r) → 0 as r → ∞ [START_REF] Hall | On continuum percolation[END_REF] and therefore Σ ρ (λµ) does not percolate.

2. Assume that (6) does not hold. Then, for all λ > 0 and for all ρ > 1, Σ ρ (λµ) percolates.

The proof is given in Section 2. The ideas of its proof and the ideas of the proofs of Theorems 1 and 2 are given in Subsection 2.2.

The first item of Theorem 4 is a generalization of Theorem 2 and thus of Theorem 1. Indeed, by [START_REF] Broman | Large-N limit of crossing probabilities, discontinuity, and asymptotic behavior of threshold values in Mandelbrot's fractal percolation process[END_REF], one has λ < λ c as soon as λ < λ c . Moreover, by the second item of Theorem 4,[START_REF] Grimmett | of Grundlehren der Mathematischen Wissenschaften[END_REF] has to be a consequence of the assumptions of Theorem 2. For example, one can check that ( 6) is a consequence of (5)1 . Alternatively, one can check that ( 6) is a consequence of λ c (µ) > 0 (see the remarks at the end of Section 1.2).

Let us denote by λ c (m ρ ∞ ) and λ c (m ρ ∞ ) the λ c and λ c critical tresholds for the multiscale model with scale parameter ρ. Theorems 3 and 4 yield the following result:

1. If (6) holds then λ c (m ρ ∞ ) > 0 (and actually the proof of Theorem 3 yields λ c (m ρ ∞ ) > 0) for all ρ > 1 and λ c (m ρ ∞ ) → λ c (µ) > 0 as ρ → ∞. 2. Otherwise, λ c (m ρ ∞ ) = λ c (m ρ ∞ ) = 0 for all ρ > 1.
Let us denote by D ρ (λµ) the diameter of the connected component of Σ ρ (λµ) that contains the origin. The following result is an easy consequence of Theorem 4 above and Theorems 2.9 and 1.2 in [START_REF] Gouéré | Subcritical regimes in some models of continuum percolation[END_REF].

Theorem 5 Let s > 0, λ > 0 and ρ > 1. 1. If [1,+∞[ β d+s µ(dβ) < ∞ and (7) holds, then E (D ρ (λµ)) s < ∞. 2. If [1,+∞[ β d+s µ(dβ) = ∞ then E (D ρ (λµ)) s = ∞.
The proof is given is Section 3.

Superposition of Boolean models with different laws

Using the same arguments as in the proof of Theorem 4, we could prove similar results for infinite superpositions n≥0 ρ -n Σ n where the Boolean models Σ n are independent but not identically distributed. We will not give such a result here. However, we wish to give a weaker result for the superposition of two independent Boolean models at different scales. As we consider only two scales the proof is easier than the proof of Theorem 4. The proof uses Lemmas 7, 8 and 9 and is given in Section 4. The result gives some insight on the critical treshold in the case of balls of random radii. This result, in the case where the supports of ν 1 and ν 2 are bounded, is already implicit in [START_REF] Meester | Nonuniversality and continuity of the critical covered volume fraction in continuum percolation[END_REF] in their proof of non universality of critical covered volume (see (8) below). See also [START_REF] Molchanov | Multiscale models of failure and percolation[END_REF].

Proposition 6 Let ν 1 and ν 2 be two finite measures on ]0, +∞[. We assume that the masses of ν 1 and ν 2 are positive. Let 0 < α < 1. Then, for all ρ > 1,

λ c (αν 1 + (1 -α)H ρ ν 2 ) ≤ min λ c (αν 1 ), λ c ((1 -α)H ρ ν 2 ) = min λ c (ν 1 ) α , λ c (ν 2 ) 1 -α .
Moreover,

λ c (αν 1 + (1 -α)H ρ ν 2 ) → min λ c (ν 1 ) α , λ c (ν 2 ) 1 -α as ρ → ∞.
The above convergence is uniform in α.

We now make some remarks about this result and about some related numerical results. For a finite measure µ on ]0, +∞[, we denote by φ c (µ) the critical covered volume:

φ c (µ) = P 0 ∈ Σ(λ c (µ)µ) = 1 -exp -λ c (µ) v d r d µ(dr) (8)
where v d is the volume of the unit Euclidean ball in R d . This is the mean volume occupied by the critical Boolean model and this is scale invariant. Let us assume that

ν 1 = ν 2 = δ 1 .
By [START_REF] Broman | Universal behavior of connectivity properties in fractal percolation models[END_REF], by Proposition 6 and with the above notation we have:

φ c (αδ 1 + (1 -α)H ρ δ 1 ) → 1 -exp -v d λ c (δ 1 ) min 1 α , 1 1 -α . (9)
There are several numerical studies of the above critical covered volume when d = 2 and d = 3. To the best of our knowledge, the most acccurate values when d = 2 are given in [START_REF] Quintanilla | Asymmetry in the percolation thresholds of fully penetrable disks with two different radii[END_REF]. Let us assume henceforth that d = 2. In [START_REF] Quintanilla | Asymmetry in the percolation thresholds of fully penetrable disks with two different radii[END_REF], the authors give:

φ c (δ 1 ) = 1 -exp(-v 2 λ c (δ 1 )) ≈ 0.6763475(6). ( 10
)
In Figure 1 we reproduce the graph of critical covered volume φ(α, ρ) as a function of α when ρ = 2, ρ = 5 and ρ = 10 (see [START_REF] Quintanilla | Asymmetry in the percolation thresholds of fully penetrable disks with two different radii[END_REF] for more results). We also represent the graph of the right-hand side of ( 9), that we denote by φ(α, ∞), as a function of α. We use [START_REF] Meester | Nonuniversality and continuity of the critical covered volume fraction in continuum percolation[END_REF] to get an approximate value of v 2 λ c (δ 1 ).

Remarks

-When ρ → ∞, the critical covered volume φ(•, ρ) converges to φ(•, ∞) which is symmetric: φ(α, ∞) = φ(1 -α, ∞).
When ρ is finite, the critical covered volume may also look symmetric but Quintanilla and Ziff showed, based on their numerical simulations and statistical analysis, that this was not the case. -When ρ is finite, the critical covered volume looks concave as a function of α.

However φ(•, ∞) is not concave as soon as φ c (δ 1 ) < 1exp(-2). Based on [START_REF] Meester | Nonuniversality and continuity of the critical covered volume fraction in continuum percolation[END_REF], φ(•, ∞) is therefore not concave. As a consequence, at least for large enough ρ, φ(•, ρ) is not concave. -The numerical results suggests that the minimum of the critical covered fraction is reached when all the disks have the same radius. (Equivalently, for all ρ and all α, φ(α, ρ) ≥ φ(0, ρ) = φ(1, ρ) = φ c (δ 1 ).) However there is neither a proof nor a disproof of such a result. -The numerical results also suggest some monotonicity in ρ. This has not been proven nor disproven. 2 Proof of Theorem 4

Some notations

In the whole of Section 2, we make the following assumptions: -µ satisfies (6). -1 < λ c (µ). For all η > 0, we denote by T η µ the measure defined by T η µ(A) = µ(Aη). In other words, we can built Σ(T η µ) from Σ(µ) by adding η to each radius.

For all ρ > 1, we denote by H ρ µ the measure defined by H ρ µ(A) = ρ d µ(ρA). With this definition, ρ -1 Σ(µ) is a Boolean model driven by the measure H ρ µ. For all n ≥ 0, we let:

m ρ n = n k=0 H ρ k µ.
With this definition and the notations of (4),

n k=0 ρ -k Σ k
is a Boolean model driven by m ρ n . We also let:

m ρ ∞ = k≥0 H ρ k µ.
So, Σ ρ (µ) is a Boolean model driven by the locally finite measure m ρ ∞ . Let p(a, µ) denote the probability of existence of a path from S(a/2) to S(a) in Σ(µ): p(a, µ) = P (S(a/2) ↔ Σ(µ) S(a)).

We aim at proving that, for large enough ρ, p(a, m ρ ∞ ) → 0 as a tends to infinity and Σ ρ (µ) does not percolate. The first item of Theorem 4 follows by applying this result to the measure λµ. Recall that the second item of Theorem 4 is contained in Theorem 3.

Ideas

In this subsection we first sketch the proof of the existence of ρ and a such that p(a, m ρ ∞ ) is small. This gives the main ingredients of the proof of the first item of Theorem 4. A full proof is given in Subsection 2.3. We then give the ideas of the proof of Theorems 1 and 2 by Menshikov, Popov and Vachkovskaia. Their basic strategy is similar but the implementation of the proofs are different.

Sketch of the proof of the first item of Theorem 4

Consider a small ε 1 > 0. Fix a small η > 0 and a large a such that (see Lemma 7):

p(a, T η µ) ≤ ε 1 /2. ( 11
)
For all n ≥ 1, write:

m ρ n = H ρ m ρ n-1 + µ. If the event {S(a/2) ↔ Σ(m ρ n ) S(a)
} occurs, then either the event {S(a/2) ↔ Tηµ S(a)} occurs (with a natural coupling between the Boolean models) either in Σ(H ρ m ρ n-1 )∩B(a) one can find a component of diameter at least η. We use this observation through its following crude consequence (see Lemma 8):

p(a, m ρ n ) ≤ p(a, T η µ) + Ca d η -d p(η/2, H ρ m ρ n-1 ).
By scaling and by [START_REF] Menshikov | On the connectivity properties of the complementary set in fractal percolation models[END_REF], this yields:

p(a, m ρ n ) ≤ ε 1 /2 + Ca d η -d p(ρη/2, m ρ n-1 ). ( 12
)
But for any ε 2 , any small enough ε 1 and any large enough a we can find τ such that (see Lemmas 9 and 10):

p(τ a, m ρ n-1 ) ≤ ε 2 as soon as p(a, m ρ n-1 ) ≤ ε 1 . (13)
An important fact is that τ does not depends on n nor on ρ, provided ρ ≥ ρ 0 where ρ 0 is an arbitrary constant strictly larger than 1. Here we use assumption [START_REF] Grimmett | of Grundlehren der Mathematischen Wissenschaften[END_REF] to bound error terms due to the existence of large balls.

We choose ε 2 such that:

Ca d η -d ε 2 = ε 1 /2.
We set ρ = 2τ a/η. Then, ( 12) and ( 13) can be rewritten as follows:

p(a, m ρ n ) ≤ ε 1 /2 + Ca d η -d p(τ a, m ρ n-1 ) (14) Ca d η -d p(τ a, m ρ n-1 ) ≤ ε 1 /2 as soon as p(a, m ρ n-1 ) ≤ ε 1 . (15)
As moreover [START_REF] Menshikov | On the connectivity properties of the complementary set in fractal percolation models[END_REF] implies p(a, m ρ 0 ) ≤ ε 1 we get, by induction and then sending n to infinity (see Lemma 11):

p(a, m ρ ∞ ) ≤ ε 1 . The convergence of p(a, m ρ ∞ ) to 0 is then extracted from the above result for a small enough ε 2 and from arguments behind [START_REF] Men | Coincidence of critical points in Poisson percolation models[END_REF] applied to m ρ ∞ and other ε.

Sketch of the proofs of Theorems 1 and 2 by Menshikov, Popov and Vachkovskaia

Let us quickly describe the ideas of the proofs of Menshikov, Popov and Vachkovskaia. Those ideas are used in their papers [START_REF] Menshikov | On the connectivity properties of the complementary set in fractal percolation models[END_REF] and [START_REF] Menshikov | On a multiscale continuous percolation model with unbounded defects[END_REF] through a discretization of space ; we describe them in a slightly more geometric way. For simplicity we only consider two scales: ρ -1 Σ 1 and Σ 0 . For simplicity, we also assume that the radius is one in the unscaled model (µ = λδ 1 ). We assume that the scale factor ρ is large enough. Assume that C is a connected component of ρ -1 Σ 1 ∪ Σ 0 whose diameter is a least α (it can be much larger) for a small enough constant α > 0. Then, C is included in the union of the following kind of sets:

1. connected components of ρ -1 Σ 1 whose diameter is at least α ; 2. balls of Σ 0 enlarged by α (same centers but the radii are 1 + α instead of 1).

Then, they show that the union of all those sets is stochastically dominated by a Boolean model similar to Σ 0 but with radii enlarged by a factor α and with density of centers 1 + α ′ times the corresponding density for Σ 0 for a suitable α ′ > 0. This part uses λ < λ c . In some sense, one can therefore control percolation in the union of two models by percolation in one model. Iterating the argument with some care in the constants α and α ′ , one sees that -for large enough ρ -one can control percolation in the multiscale model by percolation in a subcritical model. This yields the result.

Proof of Theorem 4

As 1 < λ c (µ), we know that p(a, µ) tends to 0 as a tends to infinity. We need the following slightly stronger consequence.

Lemma 7 There exists η > 0 such that p(a, T η µ) tends to 0.

Proof. Let ε > 0 and x > 0. We have:

H 1+ε T ε 2 µ([x, +∞[) = (1 + ε) d T ε 2 µ([x(1 + ε), +∞[) = (1 + ε) d µ([x(1 + ε) -ε 2 , +∞[) ≤ κ(ε)(1 + ε) d µ([x, +∞[) (16)
where

κ(ε) = µ(]0, +∞[) µ([ε, +∞[) .
The inequality is proven as follows

. If x ≥ ε, then [x(1 + ε) -ε 2 , +∞[⊂ [
x, +∞[ and the result follows from κ(ε) ≥ 1. If, on the contrary, x < ε, then the left hand side is bounded above by (1 + ε) d µ(]0, +∞[) which is itself bounded above by the right hand side. Note that κ(ε)(1 + ε) d tends to 1 as ε tends to 0. Let us say that a measure ν is subcritical if λ c (ν) > 1. As µ is subcritical, we get that κ(ε)(1 + ε) d µ is subcritical for small enough ε. We fix such an ε. By [START_REF] Yu | A note on percolation of Poisson sticks[END_REF] we can couple a Boolean model driven by H 1+ε T ε 2 µ and a Boolean model driven by κ(ε)(1 + ε) d µ in such a way that the first one is contained in the second one. Therefore the first one is subcritical. By scaling, a Boolean model driven by T ε 2 µ is then subcritical. We take η = ε 2 .

Lemma 8 Let ν 1 and ν 2 be two finite measures on ]0, +∞[. One has, for all η > 0 and a ≥ 4η:

p(a, ν 1 + ν 2 ) ≤ p(a, T η ν 1 ) + C 1 a d η -d p(η/2, ν 2 )
where C 1 = C 1 (d) > 0 depends only on the dimension d.

Proof. Let (x i ) i∈I be a family of points such that :

-The balls B(x i , η/4), i ≤ I, cover B(a).

-There are at most C 1 a d η -d points in the family where C 1 = C 1 (d) depends only on the dimension d. We couple the different Boolean model as follows. Let Σ(ν 1 ) be a Boolean model driven by ν 1 . Let Σ(ν 2 ) be a Boolean model driven by ν 2 . Assume that Σ(ν 1 ) and Σ(ν 2 ) are independent. Then Σ(ν 1 ) ∪ Σ(ν 2 ) is a Boolean model driven by ν 1 + ν 2 . We set Σ(ν 1 + ν 2 ) = Σ(ν 1 ) ∪ Σ(ν 2 ). We also consider Σ(T η ν 1 ), the Boolean model obtained by adding η to the radius of each ball of Σ(ν 1 ). Thus Σ(T η ν 1 ) is driven by T η ν 1 .

Let us prove the following property:

{S(a/2) ↔ Σ(ν 1 +ν 2 ) S(a)} ⊂ {S(a/2) ↔ Σ(Tην 1 ) S(a)}∪ i∈I {S(x i , η/4) ↔ Σ(ν 2 ) S(x i , η/2)}. (17) Assume that Σ(ν 1 + ν 2 ) = Σ(ν 1 ) ∪ Σ(ν 2 ) connects S(a/2) with S(a). Recall a ≥ 4η.
If the diameter of all connected components of Σ(ν 2 ) ∩ B(a) are less or equal to η, then Σ(T η ν 1 ) connects S(a/2) with S(a). Otherwise, let C be a connected component of Σ(ν 2 ) ∩ B(a) with diameter at least η. Let x, y be two points of C such that xy > η. The point x belongs to a ball B(x i , η/4). As y does not belong to B(x i , η/2), the component C connects S(x i , η/4) to S(x i , η/2). Therefore, Σ(ν 2 ) connects S(x i , η/4) to S(x i , η/2). We have proven [START_REF] Quintanilla | Asymmetry in the percolation thresholds of fully penetrable disks with two different radii[END_REF]. The lemma follows.

The following lemma is essentially the first item of Proposition 3.1 in [START_REF] Gouéré | Subcritical regimes in the Poisson Boolean model of continuum percolation[END_REF]. For the sake of completeness we nevertheless provide a proof.

Lemma 9 Let ν be a finite measure on ]0, +∞[. There exists a constant C 2 = C 2 (d) > 0 such that, for all a > 0:

p(10a, ν) ≤ C 2 p(a, ν) 2 + C 2 [a,+∞[ r d ν(dr).
Proof. Let K be a finite subset of S(5) such that K + B(1/2) covers S(5). Let L be a finite subset of S(10) such that L + B(1/2) covers S [START_REF] Meester | Nonuniversality and continuity of the critical covered volume fraction in continuum percolation[END_REF]. Let A be the following event: there exists a random ball B(c, r) of Σ(ν) such that r ≥ a and B(c, r) ∩ B(10a) is non empty. We have:

{S(5a) ↔ Σ(ν) S(10a)} \ A ⊂ {S(5a) ↔ ≤a Σ(ν) S(10a)}
where, in the last event, we ask for a path using only balls of Σ(ν) of radius at most a. Let us prove the following:

{S(5a) ↔ Σ(ν) S(10a)} \ A ⊂ k∈K,l∈L {S(ak, a/2) ↔ ≤a Σ(ν) S(ak, a)} ∩ {S(al, a/2) ↔ ≤a Σ(ν) S(al, a)}. ( 18
)
Assume that the event on the left hand side occurs. Then, by the previous remark, there exists a path from a point x ∈ S(5a) to a point y ∈ S(10a) that is contained in balls of Σ(ν) of radius at most a. As Ka + B(a/2) covers S(5a), there exists k ∈ K such that x belongs to B(ka, a/2). Using the previous path, one gets that the event {S(ak, a/2) ↔ ≤a Σ(ν) S(ak, a)} occurs. By a similar arguments involving y we get [START_REF] White | On the value of the critical point in fractal percolation[END_REF].

Observe that, for all k ∈ K and l ∈ L, the events {S(ak, a/2) ↔ ≤a Σ(ν) S(ak, a)} and {S(al, a/2) ↔ ≤a Σ(ν) S(al, a)} are independent. Indeed, the first one depends only on balls with centers in B(ak, 2a), the second one depends only on balls with centers in B(al, 2a), and ak -al ≥ 5a. Using this independence, stationarity and (18), we then get: P ({S(5a) ↔ Σ(ν) S(10a)}) ≤ CP (S(a/2) ↔ ≤a Σ(ν) S(a)}) 2 + P (A) where C is the product of the cardinality of K by the cardinality of L. The probability P (A) is bounded above by standard computations.

From the previous lemma, we deduce the following result.

Lemma 10 Let ε > 0. There exists C 3 = C 3 (d) > 0, a 0 = a 0 (d, µ) and k 0 = k 0 (d, µ, ε) such that, for all N, all ρ ≥ 2 and all a ≥ a 0 : if p(a, m ρ N ) ≤ C 3 then for all k ≥ k 0 , p(a10 k , m ρ N ) ≤ ε.

Proof. For all ρ ≥ 2 and all a ≥ 1 we have: Therefore, for all ρ ≥ 2, all N, all a ∈ [a 1 , 10a 1 ] and all η ∈ [η 1 , 10η 1 ]:

[a,+∞[ r d m ρ ∞ (dr) = k≥0 ρ kd ]0,+∞[ 1 [a,+∞[ (rρ -k )(rρ -k ) d µ(dr) = ]0,+∞[ k≥0
C 1 a d η -d p(a10 k , m ρ N ) ≤ C 3 2 for all k ≥ k 0 as soon as p(a, m ρ N ) ≤ C 3 . Fix k ≥ k 0 , a ∈ [a 1 , 10a 1 ] and η ∈ [η 1 , 10η 1 ]. Set: ρ = 2a10 k η -1 .
Note ρ ≥ 8 ≥ 2 as a ≥ a 1 ≥ 40η 1 ≥ 4η. By Lemma 8 we have, for all N:

p(a, m ρ N +1 ) ≤ p(a, T η µ) + C 1 a d η -d p(η/2, m ρ N +1 -µ) = p(a, T η µ) + C 1 a d η -d p(η/2, H ρ m ρ N )
. By definition of a 1 , by a 1 ≤ a, by η ≤ 10η 1 , by scaling and by definition of ρ we get, for all N:

p(a, m ρ N +1 ) ≤ C 3 2 + C 1 a d η -d p(ρη/2, m ρ N ) = C 3 2 + C 1 a d η -d p(a10 k , m ρ N ).
Combining this inequality with the property defining k 0 , we get that p(a, m ρ

N ) ≤ C 3 implies p(a, m ρ N +1 ) ≤ C 3 . As p(a, m ρ 0 ) = p(a, µ) ≤ p(a, T 10η 1 µ) ≤ C 3 /2 we get p(a, m ρ N ) ≤ C 3 for all integer N.
Let ε > 0. Using again Lemma 10 we get the existence of an integer k ′ 0 such that p(a10 k ′ , m ρ N ) ≤ ε for all k ′ ≥ k ′ 0 as soon as p(a, m ρ N ) ≤ C 3 . But we have proven the latter property. Therefore p(a10 k ′ , m ρ N ) ≤ ε for all N and all k ′ ≥ k ′ 0 . By Lemma 11, we get p(a10 k ′ , m ρ ∞ ) ≤ ε for all k ′ ≥ k ′ 0 . Using the freedom on the choice of k ≥ k 0 and η ∈ [η 1 , 10η 1 ], we get that the previous result holds for all ρ ≥ 2a10 k 0 -1 η -1 1 and then for all ρ ≥ 2a 1 10 k 0 η -1 1 . Moreover, using the freedom on the choice of a ∈ [a 1 , 10a 1 ] and k ′ ≥ k ′ 0 , we get: p(r, m ρ ∞ ) ≤ ε for all r ≥ a 1 10 k ′ 0 and all ρ ≥ 2a 1 10 k 0 η -1 1 . Therefore, p(r, m ρ ∞ ) tends to 0 as r tends to infinity. As a consequence, Σ ρ (µ) does not percolate for any ρ ≥ 2a 1 10 k 0 η -1 1 .

3 Proof of Theorem 5 Lemma 12 Let s > 0 and ρ > 1. The following assumptions are equivalent:

1. ]0,+∞[ r d+s µ(dr) < ∞.

[1,+∞

[ r d+s m ρ ∞ (dr) < ∞.
Proof. We have:

[1,+∞[ r d+s m ρ ∞ (dr) = k≥0 ρ kd ]0,+∞[ 1 [1,+∞[ (rρ -k )(rρ -k ) d+s µ(dr) = [1,+∞[ k≥0 1 [1,+∞[ (rρ -k )ρ -ks r d+s µ(dr). Now, let 0 < α < 1 and let λ = min λ c (ν 1 )(1 -ε) α , λ c (ν 2 )(1 -ε) 1 -α .
By (26), ( 29), ( 27) and (28) we get that Assumptions 1 , 2 and 3 of Lemma 13 are fulfilled for the measures αλν 1 and (1α)λν 2 and for ρ ≥ ρ 0 . Therefore, we get

λ c (αλν 1 + (1 -α)λH ρ ν 2 ) ≥ 1
and thus:

λ c (αν 1 + (1 -α)H ρ ν 2 ) ≥ λ = (1 -ε) min λ c (ν 1 ) α , λ c (ν 2 ) 1 -α .
Therefore, as soon as ρ ≥ ρ 0 , we have:

0 ≤ min λ c (ν 1 ) α , λ c (ν 2 ) 1 -α -λ c (αν 1 + (1 -α)H ρ ν 2 ) ≤ ε min λ c (ν 1 ) α , λ c (ν 2 ) 1 -α ≤ ε max(2 λ c (ν 1 ), 2 λ c (ν 2 )).
This yields the proposition.

A Critical parameters

Lemma 14 λ c (µ) ≤ λ c (µ) ≤ λ c (µ).

Proof. The second inequality is a consequence of the following inclusion:

{{0} ↔ Σ S(r)} ⊂ {S(r/2) ↔ Σ S(r)}.

The first inequality can be proven as follows. Let r ≥ 1. By the FKG inequality, we get: The first inequality stated in the lemma follows.

P ({0} ↔ Σ S(

Lemma 15

The treshold parameter λ c (µ) is positive if and only if r d µ(dr) is finite.

Proof. If λ c (µ) is positive, then there exists λ > 0 such that Σ(λµ) does not percolate. By Theorem 2.1 of [START_REF] Gouéré | Subcritical regimes in the Poisson Boolean model of continuum percolation[END_REF] this implies that r d µ(dr) is finite.

Let us assume now that r d µ(dr) is finite. We need to prove the existence of λ > 0 such that p(a, λµ) tends to 0. This is proven, as an intermediate result, in the proof of Theorem 1.1 in [START_REF] Gouéré | Subcritical regimes in some models of continuum percolation[END_REF]. As the result is an easy consequence of Lemma 9, we find it more convenient to provide a proof here. Let C 2 be the constant given by Lemma 9. For all a > 0 and λ > 0 we have:

1

  Introduction and statement of the main result 1.1 The Boolean model Let d ≥ 2. Let µ be a finite measure on ]0, +∞[. We assume that the mass of µ is positive. Let ξ be a Poisson point process on R d ×]0, +∞[ whose intensity is the product of the Lebesgue measure on R d by µ. With ξ we associate a random set Σ(µ) defined as follows: Σ(µ) = (c,r)∈ξ B(c, r) where B(c, r) is the open Euclidean ball of radius r centered at c. The random set Σ(µ) is the Boolean model with parameter µ. When shall sometimes write Σ to simplify the notations.

Figure 1 :

 1 Figure 1: Critical covered volume as a function of α for different values of ρ. From bottom to top: ρ = 2, ρ = 5, ρ = 10 and the limit as ρ → ∞.

1

  [a,+∞[ (rρ -k )r d µ(dr) = [a,+∞[ ln(r/a) ln(ρ) -1 + 1 r d µ(dr) ≤ [a,+∞[ (ln(r) ln(2) -1 + 1)r d µ(dr).

  r)) ≥ P (B(0, 1) ⊂ Σ and S(1) ↔ Σ S(r)) ≥ CP (S(1) ↔ Σ S(r))where C = P (B(0, 1) ⊂ Σ) > 0 does not depend on r. For all large enough r, we can cover S(2r) by at most C ′ r d balls B(x i , 1) where C ′ only depends on the dimension d. If there is a path in Σ from S(2r) to S(4r), then there exists i and a path in Σ from S(x i , 1) to S(x i , r). (Consider the ball B(x i , 1) that contains the initial point of the path.) By stationarity and by the previous inequality we thus get:P (S(2r) ↔ Σ S(4r)) ≤ C ′ r d P (S(1) ↔ Σ S(r))≤ C ′ C -1 r d P ({0} ↔ Σ S(r)).

C 2 pr

 2 (10a, λµ) ≤ (C 2 p(a, λµ)) 2 + λC 2 2 [a,+∞[ r d µ(dr).(30)For all 0 < a ≤ 1 we have, by standard computations:C 2 p(a, λµ) ≤ C 2 P (a ball of Σ(λµ) touches B(a)) ≤ C 2 v d λ ]0,+∞[ (1 + r) d µ(dr)where v d is the volume of the unit Euclidean ball. As r d µ(dr) is finite, we can therefore fix λ > 0 such that: d µ(dr) ≤ 1/4 for all a > 0 and C 2 p(a, λµ) ≤ 1/2 for all 0 < a ≤ 1. (31) By (30), (31) and by induction we get C 2 p(a, λµ) ≤ 1/2 for all a > 0. Therefore, we have 0 ≤ lim sup C 2 p(a, λµ) ≤ 1/2. But (30) also yields the inequality lim sup C 2 p(a, λµ) ≤ (lim sup C 2 p(a, λµ))2 . As a consequence we must have lim sup C 2 p(a, λµ) = 0 and then p(a, λµ) → 0.

From[START_REF] Gouéré | Subcritical regimes in some models of continuum percolation[END_REF] one gets the existence of a > 1 such that, for all r ≥ a, one has µ([r, +∞[) ≤ 2 -1 a -d µ([r/a, +∞[). By induction and standard computations this yields, for all r ≥ a, µ([r, +∞[) ≤ Ar -ln(2)/ ln(a)-d . Therefore, for a small enough η > 0, one has r d+η µ(dr) < ∞.

Let C 2 be the constant given by Lemma 9. By [START_REF] Grimmett | of Grundlehren der Mathematischen Wissenschaften[END_REF] we can chose a 0 = a 0 (d, µ) ≥ 1 such that C 2 2 [a 0 ,+∞[ (ln(r) ln(2) -1 + 1)r d µ(dr) ≤ 1 4 .

Let C 3 = (2C 2 ) -1 . Let N, ρ and a be as in the statement of the lemma. From Lemma 9 we get:

Let (u k ) be a sequence defined by u 0 = 1/2 and, for all k ≥ 0:

Note that the sequence (u k ) only depends on d and µ.

Assume that p(a, m ρ N ) ≤ C 3 . We then have C 2 p(a, m ρ N ) ≤ u 0 . Using a ≥ a 0 and (21), we then get C 2 p(a10 k , m ρ N ) ≤ u k for all k. Therefore, it sufficies to show that the sequence (u k ) tends to 0.

Using ( 22), [START_REF] Zuev | Continuous models of percolation theory[END_REF] and u 0 = 1/2 we get 0 ≤ u k ≤ 1/2 for all k. Therefore, 0 ≤ lim sup u k ≤ 1/2. By (22) and by the convergence of the integrale we also get lim sup u k ≤ (lim sup u k ) 2 . As a consequence, lim sup u k = 0 and the lemma is proven.

Lemma 11 For all a > 0 and ρ > 1 the following convergence holds:

Proof. The sequence of events

} is increasing (we use the natural coupling between our Boolean models). Therefore, it suffices to show that the union of the previous events is

. By a compactness argument, this path is included in a finite union of ball of Σ(m ρ ∞ ). Therefore, there exists N such that the path is included in Σ(m ρ N ) and A N occurs. This proves A ⊂ ∪A N . The other inclusion is straightforward.

Proof of the second item of Theorem 4. By Lemma 7, we can fix η 1 > 0 such that p(a, T 10η 1 µ) tends to 0 as a tends to ∞. Let C 1 be given by Lemma 8. Let a 0 and C 3 be as given by Lemma 10. Fix a 1 ≥ max(40η 1 , a 0 , 1) such that p(a, T 10η 1 µ) ≤ C 3 /2 for all a ≥ a 1 . Let k 0 be given by Lemma 10 with the choice:

Therefore:

This yields the result.

Proof of the first item of Theorem 5. By the discussion at the beginning of Section 1.5 in [START_REF] Gouéré | Subcritical regimes in some models of continuum percolation[END_REF], Σ ρ (λµ) is driven by a a Poisson point process whose intensity is the product of the Lebesgue measure by the locally finite measure λm ρ ∞ . Let us check the three items of Theorem 2.9 in [START_REF] Gouéré | Subcritical regimes in some models of continuum percolation[END_REF] with ρ = 10 (ρ is not use in the same way in [START_REF] Gouéré | Subcritical regimes in some models of continuum percolation[END_REF]). We refer to Section 2.1 of [START_REF] Gouéré | Subcritical regimes in some models of continuum percolation[END_REF] for definitions.

1. The first item is fulfilled thanks to [START_REF] Hall | On continuum percolation[END_REF] 2. For all β > 0 and all x ∈ R d , the event G(x, 0, β) only depends on balls B(c, r) ∈ Σ ρ (λµ) such that c belongs to B(x, 3β). By the independance property of Poisson point processes, we then get that G(0, 0, β) and G(x, 0, β) are independent whenever x ≥ 10β. Therefore I(10, 0, β) = 0 and the second item of Theorem 2.9 is fulfilled.

3. The third item (note that µ in [START_REF] Gouéré | Subcritical regimes in some models of continuum percolation[END_REF] is m ρ ∞ in this paper) is fulfilled thanks to Lemma 12 Theorem 2.9 in [START_REF] Gouéré | Subcritical regimes in some models of continuum percolation[END_REF] yields the result.

Proof of the second item of Theorem 5.

If r d µ(dr) is infinite then, Σ(λµ) percolates for all λ > 0 (see the dicussion of Section 1.2). Therefore Σ ρ (λµ) percolates for all ρ > 1 and λ > 0. Therefore D ρ (λµ) = ∞ with positive probability for all ρ > 1 and λ > 0. Now, assume that r d µ(dr) is finite. Then, by the discussion at the beginning of Section 1.5 in [START_REF] Gouéré | Subcritical regimes in some models of continuum percolation[END_REF], Σ ρ (λµ) is driven by a a Poisson point process whose intensity is the product of the Lebesgue measure by the locally finite measure λm ρ ∞ . We can therefore apply Theorem 1.2 in [START_REF] Gouéré | Subcritical regimes in some models of continuum percolation[END_REF]. By Lemma 12, assumption (A3) of Theorem 1.2 in [START_REF] Gouéré | Subcritical regimes in some models of continuum percolation[END_REF] is not fulfilled (note that µ in [START_REF] Gouéré | Subcritical regimes in some models of continuum percolation[END_REF] is m ρ ∞ in this paper). Theorem 1.2 in [START_REF] Gouéré | Subcritical regimes in some models of continuum percolation[END_REF] then yields the result.

Proof of Proposition 6

We first need a lemma, which is a consequence of Lemmas 8 and 9.

Lemma 13 Let ν 1 and ν 2 be two finite measures on ]0, +∞[. Let η > 0 and a 0 ≥ 4η. Let ρ > 1. There exists

) ≥ 1 as soon as the following conditions hold:

For all a ∈ [a 0 , 10a 0 ] we have, by Lemma 8 applied to ν 1 and H ρ (ν 2 ), by scaling and by the assumptions of the lemma:

But for all a ≥ a 0 we have, by Lemma 9 and by the assumptions of the lemma:

By ( 23) and (25) we get C 2 p(a, ν) ≤ 1/2 for all a ≥ a 0 and therefore 0 ≤ lim sup C 2 p(a, ν) ≤ 1/2. By (24) and the third assumption of the lemma, we get lim sup C 2 p(a, ν) ≤ (lim sup C 2 p(a, ν)) 2 . Therefore, we must have lim sup C 2 p(a, ν) = 0 and the lemma is proven.

Proof of Proposition 6. The inequality is straightforward. To prove the inequality, we note that, by scaling, λ c (H ρ ν 2 ) = λ c (ν 2 ). Let us prove the convergence. We can assume λ c (ν 1 ) > 0 and λ c (ν 2 ) > 0, otherwise the convergence is obvious. Therefore, by Lemma 15, the integrals r d ν 1 (dr) and r d ν 2 (dr) are finite.

Let C 4 be the constant given by Lemma 13. Let 0 < ε < 1. Note:

Therefore, by Lemma 7 (in which ( 6) is not used), we can fix η > 0 such that p(a, T η (1ε) λ c (ν 1 )ν 1 ) → 0.

We can then fix a 0 ≥ 4η such that:


R Alexandre

LANDAU LINEARIZED OPERATOR AND HYPOELLIPTICITY

Keywords: Mathematics Subject Classification. 35H10, 76P05, 84C40 Landau equation, hypoellipticity. 1

We consider the linearized Landau operator for which we provide simple proofs of hypoellipticity, and in particular we recover the recent results of Hérau and . Our arguments are elementary and in particular avoids any use of pseudo-differential calculus.

Introduction

We consider hypoelliptic effects associated with the time version of a kinetic equation related to the linearized Landau equation and studied recently by Herau and Pravda-Starov [START_REF] Herau | Anisotropic hypoelliptic estimates for Landau-type operators[END_REF] (1.1)

∂ t f + v.∇ x f -∇ v .λ(v)∇ v f -(v ∧ ∇ v ).µ(v)(v ∧ ∇ v f ) + F(v) f (v) = h
where t ∈ R, x ∈ R 3 and v ∈ R 3 . f and h will be supposed to be in L 2 , where here and below L 2 denotes the usual space w.r.t. full variables (t, x, v). In fact, as it will be clear from the proofs, it is also possible to work in weighted L 2 spaces and even in weighted Sobolev spaces. The norm in L 2 will be denoted by . and its associated scalar product by (. , .). We shall work with real functions f and h though there is absolutely no difficulties in considering complex cases, up to the addition of the real parts when necessary.

As regards the coefficients appearing in (1.1), we assume that the positive functions λ, µ and F satisfy the following coercive type lower bounds

(1.2) λ(v) < v > γ , µ(v) < v > γ and F(v) < v > γ+2
where -3 ≤ γ ≤ 1, and

(1.3) |D m v λ| < v > γ-m , |D m v µ| < v > γ-m and |D m v F(v)| < v > γ+1
-m for any m of order at most two.

We set (1.4)

L( f ) = -∇ v .λ(v)∇ v f -(v ∧ ∇ v ).µ(v)(v ∧ ∇ v f ) + F(v) f (v) = L 1 ( f ) + L 2 ( f ) + L 3 ( f )
so that (1.1) writes also (1.5)

∂ t f + v.∇ x f + L( f ) = h.
As explained in [START_REF] Herau | Anisotropic hypoelliptic estimates for Landau-type operators[END_REF], (1.1) or its version (1.5) is related to the linearized Landau operator, which plays a crucial role in Plasma Physics, see for example [START_REF] Alexandre | A review on Boltzmann equation with singular kernels[END_REF][START_REF] Villani | A review of mathematical topics in collisional kinetic theory[END_REF] and the references therein. Moreover the above assumptions (1.2) and (1.3) are natural in view of Guo's work [START_REF] Guo | The Landau equation in a periodic box[END_REF]. However, note that the model (1.1) does not take into account the kernel which is naturally associated with the true Landau linearized equation. In particular, for applications to the true nonlinear Landau model near equilibrium, the present results need to be adapted, though the main issue is related to the macroscopic part.

Our main concern is to give a shorter proof of the following result about optimal hypoelliptic results which was first proven in [START_REF] Herau | Anisotropic hypoelliptic estimates for Landau-type operators[END_REF] (in the time independent version) Theorem 1.1 ). Assume f and h belong to L 2 . Then, under the hypothesis (1.2) and (1.3), we have

< v > γ+2 f 2 + < v > γ |D v | 2 f 2 + < v > γ |v ∧ ∇ v | 2 f 2 + < v > γ 3 |D x | 2 3 f 2 + < v > γ 3 |v ∧ ∇ x | 2 3 f 2 f 2 + h 2 .
The proof in [START_REF] Herau | Anisotropic hypoelliptic estimates for Landau-type operators[END_REF] uses pseudo-differential calculus, together with Wick calculus. In the continuation of a previous work of Morimoto and Xu [START_REF] Morimoto | Hypoelliticity for a class of kinetic equations[END_REF], a similar study was also performed in [START_REF] Lerner | Hypoelliptic Estimates for a Linear Model of the Boltzmann Equation without Angular Cutoff[END_REF] in order to deduce hypoelliptic results for a fractional order kinetic equation, and again the proof therein was using such tools.

Recently, we have provided in [START_REF]Fractional order kinetic equations and hypoellipticity[END_REF] a very simple proof of the results of [START_REF] Lerner | Hypoelliptic Estimates for a Linear Model of the Boltzmann Equation without Angular Cutoff[END_REF] by using arguments originally introduced by Bouchut [START_REF] Bouchut | Hypoelliptic regularity in kinetic equations[END_REF] and Perthame [START_REF] Perthame | Higher moments for kinetic equations: the Vlasov-Poisson and Fokker-Planck cases[END_REF].

Again herein, we shall give a different proof of this main Theorem by using simple and standard arguments. Moreover, we keep the regularity of the coefficients as low as possible, this point being connected with the second order commutators estimations which are needed in the proofs. As a byproduct, other estimations will emerge from our computations. Note also that, as usual, time derivatives estimates are also available, but we do not detail this point. All in all, together with our previous work [START_REF]Fractional order kinetic equations and hypoellipticity[END_REF], we provide extremely simple arguments to deduce hypoelliptic results for kinetic equations with a diffusive part. It is to be expected that the underlying arguments are sufficiently simple to enable the study of different questions related to diffusive models arising from scaled kinetic equations. We hope to get back on this issue soon. Furthermore as it will be clear from the proofs, the Cauchy problem can be also analyzed with the same methods up to some minor changes. It is expected that such simple methods will provide other methods for the analysis of the Cauchy problem associated with fully nonlinear kinetic equations such as Boltzmann or Landau equations, see the quoted works in the bibliography.

We shall always assume that all functions f and h are smooth.The paper is organized as follows. We first deduce in Section 2 some estimates from a transport equation. These are used in Section 3, in particular to control cross products terms.

Preliminary results

First of all, by multiplying the equation by f , integration over all variables and using the assumptions (1.2), we get Lemma 2.1. One has

< v > γ 2 ∇ v f 2 + < v > γ 2 (v ∧ ∇ v f ) 2 + < v > γ 2 +1 f 2 f h .
The next two Lemmas are the adaptions of some of the steps which appear in Bouchut's paper [START_REF] Bouchut | Hypoelliptic regularity in kinetic equations[END_REF]. They are related to transport type equations, with a given right hand side, and assuming that we know already some kind of regularity w.r.t. velocity variable, they give some informations about regularity of some spatial derivatives.

Lemma 2.2. Assume f ∈ L 2 , g ∈ L 2 , < v > γ |D v | 2 f ∈ L 2 and ∂ t f + v.∇ x f = g. Then < v > γ 3 |D x | 2 3 f < v > γ-1 f 1 3 + < v > γ |D v | 2 f 1 3 g 2 3 .
Observe that the first term in the r.h.s. of this estimate is controlled by the L 2 norm of f , since γ ≤ 1.

Proof. Following [START_REF] Bouchut | Hypoelliptic regularity in kinetic equations[END_REF], we start from the formula

∂ x j f = ∂ v j g -(∂ t + v.∇ x )∂ v j f. Then we write < v > γ 3 |D x | 2 3 f 2 = < v > γ 3 |D x | -1 3 ∂ x j f 2 = (< v > 2γ 3 |D x | -2 3 ∂ x j f , ∂ x j f ) = (< v > 2γ 3 |D x | -2 3 ∂ x j f , ∂ v j g -(∂ t + v.∇ x )∂ v j f ) = -4 γ 3 (< v > 2γ 3-1 v.∇ x (|D x | -2 3 f, g) -(< v > 2γ 3 |D x | -2 3 ∂ x j ∂ v j f , g) + (< v > 2γ 3 |D x | -2 3 ∂ x j ḡ, ∂ v j f ). Thus (2.6) < v > γ 3 |D x | 2 3 f 2 = -4 γ 3 (< v > 2 γ 3 -1 v.∇ x (|D x | -2 3 f, g)-2Re(< v > 2γ 3 |D x | -2 3 ∂ x j ∂ v j f , g) = I+II.
Starting from

II 2 < v > 2γ 3 |D x | -2 3 ∂ x j ∂ v j f 2 g 2 , it follows that < v > 2γ 3 |D x | -2 3 ∂ x j ∂ v j f 2 (< v > 4γ 3 D x | 2 3 ∂ v j f, ∂ v j f ) = +(< v > 4γ 3 D x | 2 3 f, |D v | 2 f ) + 8γ/3(< v > 4γ 3 -1 v j |D x | 2 3 f, ∂ v j f ) = A + B. Now, one has A < v > γ 3 |D x | 2 3 f < v > γ |D v | 2 f while for B, we have B = 8 γ 3 (< v > 4γ 3 -1 |D x | 2 3 f, v.∇ v f ) = 8 γ 3 (< v > 4γ 3 -1 [|D x | 1 3 f ], v.∇ v [|D x | 1 3 f ]) which is of the form (g, β(v).∇ v g) ≃ (g, g div v [β])
where

β(v) =< v > 4γ 3 -1 v. Therefore div β ≃< v > 4γ 3 -1 + < v > 4γ 3 -3 < v > 2 . Thus B (< v > 4γ 3 -1 g, g) = (< v > 4γ 3 -1 [|D x | 1 3 f ], [|D x | 1 3 f ]) = (< v > γ 3 |D x | 2 3 f, < v > γ-1 f ) < v > γ 3 |D x | 2 3 f < v > γ-1 f and A + B < v > γ 3 |D x | 2 3 f < v > γ |D v | 2 f + < v > γ-1 f . Therefore II < v > γ 3 |D x | 2 3 f 1 2 < v > γ-1 f 1 2 + < v > γ |D v | 2 f 1 2
g .

For I, (using Fourier transform w.r.t. variable x)

I < v > 2 γ 3 -1 2 |D x | 1 3 f g
and by the same computations, we get

I < v > γ 3 |D x | 2 3 f 1 2 < v > γ-1 f 1 2 g . Thus I + II < v > γ 3 |D x | 2 3 f 1 2 < v > γ-1 f 1 2 + < v > γ |D v | 2 f 1 2
g and finally in view of (2.6)

< v > γ 3 |D x | 2 3 f < v > γ-1 f 1 3 + < v > γ |D v | 2 f 1 3 g 2 3 .
This ends the proof of the Lemma.

Lemma 2.3. Assume f ∈ L 2 , < v > γ+2 f ∈ L 2 , g ∈ L 2 , < v > γ |v ∧ D v | 2 f ∈ L 2 and ∂ t f + v.∇ x f = g. Then < v > γ 3 < v ∧ D x > 2 3 f < v > γ 3 f + + g 2 3 < v > γ+2 f 1 3 + < v > γ |D v | 2 f 1 3 + < v > γ |v ∧ D v | 2 f 1 3 .
Proof. We want to estimate

< v > γ 3 < v ∧ D x > 2 3 f 2 = < v > γ 3 < v ∧ k > 2 3 f 2 and, in view of Lemma 2.2, it is enough to work for < v ∧ k >≥< k >, ie < v ∧ k >< k > -1 ≥ 1 (using Parseval relation w.r.t. variable x).
Let φ be a positive function which is 0 for small values and 1 for large values. To simplify notations, write φ

(v, k) = φ(< v ∧ k >< k > -1
). Then we need to estimate

< v > γ 3 φ(v, D x ) < v ∧ D x > 2 3 f 2 . We write < v > γ 3 φ(v, D x ) < v∧D x > 2 3 f 2 = (< v > γ 3 φ(v, D x ) < v∧D x > 2 3 f , < v > γ 3 φ(v, D x ) < v∧D x > 2 3 f ) (< v > 2γ 3 φ2 (v, D x ) < v ∧ D x > 2 3 f , f ) = (< v > 2γ 3 φ2 (v, D x ) < v ∧ D x > -2 3 < v ∧ D x > 2 f , f ) = (< v > 2γ 3 φ2 (v, D x ) < v ∧ D x > -2 3 f , f ) + (< v > 2γ 3 φ2 (v, D x ) < v ∧ D x > -2 3 |v ∧ D x | 2 f , f ). Note that the first term is (for example) bounded by < v > γ 3 f 2 . So we concentrate on the second one Imp = (< v > 2γ 3 φ2 (v, D x ) < v ∧ D x > -2 3 |v ∧ D x | 2 f , f ) = -((< v > 2γ 3 φ2 (v, D x ) < v ∧ D x > -2 3 v ∧ D x f , v ∧ D x f ). Now we use the fact that ∇ x f = ∇ v g -v.∇ x ∇ v f , and therefore v ∧ ∇ x f = (v ∧ ∇ v g) -v.∇ x (v ∧ ∇ v f ). Thus Imp = -(< v > 2γ 3 φ2 (v, D x )|v∧D x | -2 3 v∧∇ v ḡ, v∧∇ x f )+(< v > 2γ 3 φ2 (v, D x )|v∧D x | -2 3 v.∇ x (v∧∇ v f ), v∧∇ x f ) = -(< v > 2γ 3 φ2 (v, D x )|v ∧ D x | -2 3 v ∧ ∇ v ḡ, v ∧ ∇ x f ) -(< v > 2γ 3 φ2 (v, D x )|v ∧ D x | -2 3 v ∧ ∇ v f , v ∧ ∇ x g). Set S =< v > 2γ 3 φ2 (v, D x )|v ∧ D x | -2 3 .Then Imp = -(S v ∧ ∇ v ḡ, v ∧ ∇ x f ) -(S v ∧ ∇ v f , v ∧ ∇ x g).
We introduce some notations (though there is also another line of proof which avoids such notations. However, the arguments are simple enough). Let e j be the canonical basis of R 3 . Then we can write

v ∧ ∇ x F = (v ∧ ∇ x F).e j e j = -[(v ∧ e j ).∇ x F]e j . Setting X j = X j (v, ∂ x )F = (v ∧ e j ).∇ x F, we get v ∧ ∇ x F = -X j (F)e j . Similarly, we can write v ∧ ∇ v G = (v ∧ ∇ v G).e j e j = -(v ∧ e j ).∇ v Ge j . Letting V j = V j (v, ∂ v )G = (v ∧ e j ).∇ v G, then v ∧ ∇ v G = -V j (G)e j . Note that X * j = -X j and V * j = -V j . Now we can write (with summation of indices) (2.7)          Imp = -(S V j (ḡ).X j ( f )) -(S V j ( f ).X j (g)) = -(S V j (ḡ).X j ( f )) + (X j S V j ( f ).g) = (ḡ.V j S X j ( f )) + (S X j V j ( f ).g) = 2Re(ḡ, S X j V j ( f )) + (ḡ, [V j , S X j ]( f )) = Imp 1 + Imp 2 .
Let us look to Imp 2 . We use Fourier transform w.r.t. x variables. Then (summation over j)

[V j , S X j ]( f ) = iV j [S (k, v)(v ∧ e j ).k f (k)] -iS (k, v)(v ∧ e j ).k(V j ( f )) = iV j [S (k, v)(v ∧ e j ).k] f (k) where S (k, v) =< v > 2γ 3 φ 2 (< v ∧ k >< k > -1 )|v ∧ k| -2 3 . Set below ψ = φ 2 also. Note that (v ∧ e j ).k = (e j ∧ k).v. Then ∇ v [(v ∧ e j ).k] = ∇ v [(e j ∧ k).v] = e j ∧ k and V j [v ∧ e j ).k] = (v ∧ e j ).(e j ∧ k).. Note that V j ((e j ∧ k).v) = (v ∧ e j ).(e j ∧ k) = -(v ∧ (e j ∧ k)).e j and v ∧ (e j ∧ k) = (v.k).e j -(v.e j )k. Thus -V j ((e j ∧ k).v) = (v.k) -(v.e j )(k.e j ) = 0. Next ∇ v < v > 2γ 3 ∼< v > 2γ 3 -2 v, and therefore V j ∇ v < v > 2γ 3 = 0. On the whole V j [S (k, v)(v ∧ e j ).k] =< v > 2γ 3 (v ∧ e j ).kV j [ψ(< v ∧ k >< k > -1 ))|v ∧ k| -2 3 ]. Now note that |v ∧ k| -2 3 = (|v| 2 |k| 2 -(v.k) 2 ) - 1
3 , and thus

∇ v [|v ∧ k| -2 3 ] = - 1 3 (|v| 2 |k| 2 -(v.k) 2 ) -4 3 [|k| 2 2v -2(v.k)k] and V j [|v ∧ k| -2 3 ] = 2 3 |v ∧ k| -8 3 (v.k)(v ∧ e j ).k. Finally ∇ v [ψ(< v ∧ k >< k > -1 ))] = ψ ′ (< v ∧ k >< k > -1 )) < k > -1 ∇ v [1 + |v ∧ k| 2 ] 1 2 = ψ ′ (< v ∧ k >< k > -1 )) < k > -1 [1 + |v ∧ k| 2 ] -1 2 [|k| 2 2v -2(v.k)k]. Thus V j [ψ(< v ∧ k >< k > -1 ))] = -2ψ ′ (< v ∧ k >< k > -1 )) < k > -1 < v ∧ k > -1 (v.k)(v ∧ e j ).k and at last (2.8)        V j [S (k, v)(v ∧ e j ).k] =< v > 2γ 3 ψ(< v ∧ k >< k > -1 )) 2 3 |v ∧ k| -8 3 (v.k)|v ∧ k| 2 -2 < v > 2γ 3 ψ ′ (< v ∧ k >< k > -1 )) < k > -1 < v ∧ k > -1 (v.k)|v ∧ k| 2 |v ∧ k| -2 3 . By definition of ψ ′ , the second term in (2.8) is bounded by (because |v ∧ k| ∼ |k|) < v > 2 γ 3 +1 < k > 1 3
, and therefore going back we have a contribution to Imp 2 as

< v > 2 γ 3 +1 < k > 1 3 f g . Since < v > 2 γ 3 +1 < k > 1 3 f 2 = (< v > 4 γ 3 +2 < k > 2 3 f , f ) = (< v > γ 3 < k > 2 3 f , < v > γ+2 f ) < v > γ 3 |D x | 2 3 f < v > γ+2 f the second term in (2.8) gives a contribution to Imp 2 as < v > γ 3 |D x | 2 3 f 1 2 < v > γ+2 f 1 2 g .
For the first term in (2.8), it is also bounded by the same form. All in all, we have shown that (2.9)

Imp 2 < v > γ 3 |D x | 2 3 f 1 2 < v > γ+2 f 1 2
g . Now we can turn to Imp 1 from (2.7) to get first of all

Imp 1 g S X j V j ( f ) .
Then, it follows that

(2.10) S X j V j ( f ) 2 = (S X j V j ( f ), S X j V j ( f )) = -(S 2 X 2 j V j ( f ), V j ( f )) = -([S 2 X 2 j , V j ]( f ), V j ( f )) + (S 2 X 2 j ( f ), V 2 j ( f ) = A 1 + B 1 . B 1 is estimated as follows B 1 < v > -γ S 2 X 2 j ( f ) < v > γ V 2 j ( f ) < v > γ 3 |v ∧ D x | 2 3 f < v > γ |v ∧ D v | 2 f .
For A 1 , this is again a commutator estimation: by Fourier transform w.r.t. x, we have, using the previous computations

[S 2 X 2 j , V j ]( f ) = -iS (k, v)(v ∧ e j ).k)V j [S (k, v)(v ∧ e j ).k] f (k) = -i < v > 2γ 3 φ 2 (< v ∧ k >< k > -1 )|v ∧ k| with V j [S (k, v)(v ∧ e j ).k] =< v > 2γ 3 (v ∧ e j ).kV j [ψ(< v ∧ k >< k > -1 ))|v ∧ k| -2 3 ] =< v > 2γ 3 ψ(< v ∧ k >< k > -1 )) 2 3 |v ∧ k| -8 3 (v.k)|v ∧ k| 2 -2 < v > 2γ 3 ψ ′ (< v ∧ k >< k > -1 )) < k > -1 < v ∧ k > -1 (v.k)|v ∧ k| 2 |v ∧ k| - 2
3 . Thus we have two contributions, the first one being given by

A 11 = -i < v > 2γ 3 φ 2 (< v∧k >< k > -1 )|v∧k| -2 3 (v∧k).e j < v > 2γ 3 ψ(< v∧k >< k > -1 )) 2 3 |v∧k| - 8
3 (v.k)|v∧k| 2 ≃ i < v > 4γ 3 φ 4 (< v ∧ k >< k > -1 )|v ∧ k| - 4
3 (v ∧ k).e j (v.k)
which is bounded from above by < v > 4 γ 3 +1 |k| 2 3 . Thus we may write

A 11 ≃ i < v > γ 3 -1 φ 4 (< v ∧ k >< k > -1 )|v ∧ k| - 4
3 (v ∧ k).e j (v.k)× < v > γ+1
and the contribution given by A 11 is estimated by

< v > γ 3 |D x | 2 3 f < v > γ+1 (v ∧ D v ) f . Then < v > γ+1 (v ∧ D v ) f 2 = (< v > γ |v ∧ D v | 2 f, < v > γ+2 f ) < v > γ |v ∧ D v | 2 f < v > γ+2 f
and therefore the contribution by A 11 is estimated by

< v > γ 3 |D x | 2 3 f < v > γ |v ∧ D v | 2 f 1 2 < v > γ+2 f 1 2 .
Now we turn to the other contribution in A 1 . We have

A 12 ≃ 2i < v > 2γ 3 φ 2 (< v ∧ k >< k > -1 )|v ∧ k| -2 3 (v ∧ k).e j < v > 2γ 3 ψ ′ (< v ∧ k >< k > -1 )) × < k > -1 < v ∧ k > -1 (v.k)|v ∧ k| 2 |v ∧ k| -2 3 ≃ 2i < v > 4γ 3 φ 2 (< v ∧ k >< k > -1 )ψ ′ (< v ∧ k >< k > -1 )) < v ∧ k > -1 < k > -1 (v.k)(v ∧ k).e j |v ∧ k| 2 3 ≃ 2i < v > γ 3 -1 φ 2 (< v∧k >< k > -1 )ψ ′ (< v∧k >< k > -1 )) < v∧k > -1 < k > -1 (v.k)(v∧k).e j |v∧k| 2
3 . < v > γ+1 Then, the contribution by this term is estimated by (or by

|D x | instead of v ∧ D x ) < v > γ 3 |v ∧ D x | 2 3 f < v > γ+1 v ∧ D v f
and therefore the total contribution by A 11 + A 12 gives

A 1 < v > γ 3 |D x | 2 3 f < v > γ |v ∧ D v | 2 f 1 2 < v > γ+2 f 1 2 .
Thus

A 1 + B 1 < v > γ 3 |D x | 2 3 f < v > γ |v ∧ D v | 2 f 1 2 < v > γ+2 f 1 2 + < v > γ |v ∧ D v | 2 f which gives Imp 1 g < v > γ 3 |D x | 2 3 f < v > γ |v ∧ D v | 2 f 1 2 < v > γ+2 f 1 2 + < v > γ |v ∧ D v | 2 f 1 2 that is Imp 1 g < v > γ 3 |D x | 2 3 f 1 2 < v > γ |v ∧ D v | 2 f 1 4 < v > γ+2 f 1 4 + < v > γ |v ∧ D v | 2 f 1 2
and therefore

Imp g . < v > γ 3 |D x | 2 3 f 1 2 < v > γ |v∧D v | 2 f 1 4 < v > γ+2 f 1 4 + < v > γ |v∧D v | 2 f 1 2 + < v > γ+2 f 1 2 . Thus < v > γ 3 φ(v, D v ) < v ∧ D x > 2 3 f < v > γ 3 f + + g 1 2 < v > γ 3 |D x | 2 3 f 1 4 < v > γ |v∧D v | 2 f 1 8 < v > γ+2 f 1 8 + < v > γ |v∧D v | 2 f 1 4 + < v > γ+2 f 1 4
and all in all

< v > γ 3 < v ∧ D x > 2 3 f < v > γ 3 f + < v > γ 3 |D x | 2 3 f + g 1 2 < v > γ 3 |D x | 2 3 f 1 4 < v > γ |v∧D v | 2 f 1 8 < v > γ+2 f 1 8 + < v > γ |v∧D v | 2 f 1 4 + < v > γ+2 f 1 4
that is also

< v > γ 3 < v ∧ D x > 2 3 f < v > γ 3 f + < v > γ 3 |D x | 2 3 f + g 1 2 < v > γ 3 |D x | 2 3 f 1 4 < v > γ |v ∧ D v | 2 f 1 4 + < v > γ+2 f 1 4 .
Using Lemma 2.2, it follows that

< v > γ 3 < v ∧ D x > 2 3 f < v > γ 3 f + < v > γ-1 f 1 3 + < v > γ |D v | 2 f 1 3 g 2 3 + g 2 3 < v > γ-1 f 1 12 + < v > γ |D v | 2 f 1 12 < v > γ |v ∧ D v | 2 f 1 4 + < v > γ+2 f 1 4
and thus

< v > γ 3 < v ∧ D x > 2 3 f < v > γ 3 f + + g 2 3 < v > γ+2 f 1 3 + < v > γ |D v | 2 f 1 3 + < v > γ |v ∧ D v | 2 f 1 3 ,
which concludes the proof.

Scalar Products between elements of L and the transport part

The main idea is to get an estimate on the square of the norms of each L i , and then conclude with the Lemma from the previous sections.

First Step

We shall first of all start first by getting an estimate on L 3 f , that is on < v > γ+2 f , which is the easiest to obtain. It will be also helpful in order to control other scalar products. Lemma 3.1. We have

< v > γ+2 f h + f .
Proof. We take the equation, multiply by < v > γ+2 f (or by L 3 f ) and integrate to get

(3.11) (λ(v)∇ v f, ∇ v (< v > γ+2 f ))+(µ(v)(v∧∇ v f ), (v∧∇ v (< v > γ+2 f )))+ < v > γ+2 f 2 h < v > γ+2 f .
and we consider the first and second terms on the l.h.s. of this inequality, denoted by J and K respectively, that we need to bound from above (and removing any positive contribution).

The first term on the l.h.s. of (3.11) is

J = (λ(v)∇ v f, ∇ v (< v > γ+2 f )) = -(< v > γ+2 ∇ v .(λ(v)∇ v f ), f ) = -(∇ v .(< v > γ+2 λ(v)∇ v f ), f ) + (λ(v)∇ v < v > γ+2 ∇ v f, f
). We can forget the first term because it is positive, i.e. let

J ≃ (λ(v)∇ v < v > γ+2 , ∇ v f ) ≃ ( λv.∇ v f, f ) with λ = λ < v > γ . Now ( λv.∇ v f, f ) = -( f, ∇ v .(v λ f )) = -( f, ∇ v .(v λ) f ) -( λv.∇ v f, f ). and thus J ≃ ( f, ∇ v .(v λ) f ) (< v > 2γ f, f ) = (< v > γ+2 f, < v > γ-2 f ) ε < v > γ+2 f 2 + C ε f 2 .
Similarly, the second term on the l.h.s. of (3.11) is

K = (µ(v)(v ∧ ∇ v f ), (v ∧ ∇ v (< v > γ+2 f )))
that we can write as

K = (µ(v)V j f, V j (< v > γ+2 f )) = (µ(v)V j f, V j [< v > γ+2 ] f ) + (µ(v)V j f, < v > γ+2 V j [ f ])
and again we can forget the second term since it is positive, to write

K ≃ (µ(v)V j f, V j [< v > γ+2 ] f ). Since β j = µV j [< v > γ+2 ] = 0, it follows that K ≃ (β j V j f, f ) = -( f, V j [β j f )) = -( f, β j V j f )) -( f, V j [β j ] f ) and therefore K ≃ ( f, V j [β j ] f ) = 0.
In view of the estimates on J and K just obtained, we can go back to (3.11), ending the proof.

As a corollary of the proof, note that we have also (though we do not use it)

(3.12) < v > γ+1 (v ∧ ∇ v f ) + < v > γ+1 ∇ v f h + f .
Step 2: A preliminary inequality Below, we set g = h -L( f ). We start from

L( f ) = -∂ t f -v.∇ x f + h. Then L( f ) 2 = -(v.∇ x f, L( f )) + (h, L( f )) (v.∇ x f, L( f )) + h L( f )
By expanding the square, and using Holder inequality with a parameter ε, we obtain (recall that we have already obtained a control for L 3 f 2 from Lemma 3.1)

(3.13) 2(L 1 ( f ), L 2 ( f ) + L 1 f 2 + L 2 f 2 (v.∇ x f, L 1 ( f ) + L 2 ( f )) + h 2 + f 2 .
Step 3: Scalar product with the transport operator Now we compute the scalar product of v.∇ x f with L 1 ( f ) + L 2 ( f ) which appears in (3.13), for which we have Lemma 3.2. With the above notations, we have

(3.14)          (v.∇ x f, L 1 ( f ) + L 2 ( f )) f 2 + h 2 + g f 1 2 + < v > γ+2 f 1 2 + < v > γ |D v | 2 f 1 2 + < v > γ |v ∧ D v | 2 f 1 2
where again recall that g = h -L( f ).

Proof. We have

(v.∇ x f, L 1 ( f ) + L 2 ( f )) = -(v.∇ x f, ∇ v .λ(v)∇ v f (v)) -(v.∇ x f, (v ∧ ∇ v ).µ(v)(v ∧ ∇ v )( f )) = (∂ v j (v.∇ x f ), λ(v)∂ v j f ) + ((v ∧ ∇ v )(v.∇ x f ), µ(v)(v ∧ ∇ v f )) = (∇ x f, λ(v)∇ v f )) + (V j (v.∇ x f ), µ(v)V j ( f )). Now note that V j (v i ) = (v ∧ e j ).∇ v (v i ) = (v ∧ e j ).e i . Thus (v.∇ x f, Q( f )) = (∇ x f, λ(v)∇ v f )) + ((v ∧ e j ).∇ x f, µ(v)V j ( f )) = (∇ x f, λ(v)∇ v f )) -(v ∧ ∇ x f, µ(v)v ∧ ∇ v f ) = J 1 + J 2 .
The term J 1 is estimated as

J 1 = (< v > γ 3 ∇ x |D x | -1 3 f, λ(v) < v > -γ 3 ∇ v |D x | 1 3 f ). Noticing that λ(v) = λ(v). < v > -γ 3 < v > 2γ 3 , we get J 1 < v > γ 3 |D x | 2 3 f λ(v) < v > -γ 3 ∇ v |D x | 1 3
f , and we are reduced to study

λ(v) < v > -γ 3 ∇ v |D x | 1 3 f . We take |D x | 1 3 of the equation, multiply by λ < v > -2γ 3 = λ 1 , multiply by |D x | 1 3 f and integrate to get -(λ 1 ∇ v .λ∇ v |D x | 1 3 f, |D x | 1 3 f )-(λ 1 (v∧∇ v )µ(v)(v∧∇ v )|D x | 1 3 f, |D x | 1 3 f )+(λ 1 F|D x | 2 3 f, f ) = (λ 1 |D x | 1 3 h, |D x | 1 3 f ).
We can forget the third term on the left hand side also. The first term on the right hand side is

-(λ 1 ∇ v .λ∇ v |D x | 1 3 f, |D x | 1 3 f ) = +(λ∇ v |D x | 1 3 f, ∇ v λ 1 |D x | 1 3 f ) + (λ 1 λ∇ v |D x | 1 3 f, ∇ v |D x | 1 3 f ).
The second term is what we want to estimate. So we need to upper bound the first one. Set β = λ∇ v λ 1 as a vector field. Then this is of the form (and by symmetry)

(β.∇ v g, g) ≃ (g, g div v β). Now div β ∼ λ∆ v λ 1 + ∇ v λ.∇ v λ 1 Since in fact λ ∼< v > γ and λ 1 ∼< v > γ 3 , then β ∼< v > 4(γ/3
)-1 and thus div β ∼< v > 4(γ/3)-2 , we have something similar to

(< v > 4(γ/3)-2 |D x | 1 3 f, |D x | 1 3 f ) < v > γ 3 |D x | 2 3 f < v > γ-2 f
. This computation also adapts to the other term, and we get

λ(v) < v > -γ 3 ∇ v |D x | 1 3 f 2 < v > γ 3 |D x | 2 3 f < v > γ-2 f + h that is λ(v) < v > -γ 3 ∇ v |D x | 1 3 f < v > γ 3 |D x | 2 3 f 1 2 < v > γ-2 f 1 2 + h 1 2
and thus

J 1 < v > γ 3 |D x | 2 3 f 3 2 < v > γ-2 f 1 2 + h 1 2 .
Using Lemma 2.2 (with g = h -L( f )), one has

< v > γ 3 |D x | 2 3 f 3 2 < v > γ-1 f 1 2 + < v > γ |D v | 2 f 1 2
g and we get

J 1 < v > γ-1 f 1 2 + < v > γ |D v | 2 f 1 2 g < v > γ-2 f 1 2 + h 1 2 .
We now turn to J 2 , recalling that

J 2 = -(v ∧ ∇ x f, µ(v)v ∧ ∇ v f
), and we proceed as for J 1 . We write

J 2 = -(< v > γ 3 v ∧ ∇ x |v ∧ ∇ x | -1 3 f, µ(v) < v > -γ 3 v ∧ ∇ v |v ∧ ∇ x | 1 3 f ) < v > γ 3 |v ∧ D x | 2 3 f µ < v > -γ 3 v ∧ ∇ v |v ∧ ∇ x | 1
3 f and therefore we need to study

µ < v > -γ 3 v ∧ ∇ v |v ∧ ∇ x | 1 3 f . We take |v ∧ ∇ x |
-(µ 1 ∇ v .λ∇ v |v∧D x | 1 3 f, |v∧D x | 1 3 f )-(µ 1 (v∧∇ v )µ(v)(v∧∇ v )|v∧D x | 1 3 f, |v∧D x | 1 3 f ) = (µ 1 |v∧D x | 1 3 h, |v∧D x | 1 3 f ).
where again recall that g = h -L( f ).

Note the exponents on the r.h.s. of this inequality which are less than 2.

Step 4: The remaining scalar product We consider the last scalar product which appears in (3.13) (recall that it is on the l.h.s of the inequality we want to control and therefore we can forget any positive term)

(L 1 ( f ), L 2 ( f ) = (∇ v .λ(v)∇ v f, (v ∧ ∇ v )µ(v)(v ∧ ∇ v ) f ).
If we introduce B k = √ λ∂ v k , and W j = √ µV j , then this term is also

(B * k B k f, W * j W j f ) = (B k f, B k W * j W j f ) = (B k f, [B k , W * j ]W j f ) + (B k f, W * j B k W j f ) = (B k f, [B k , W * j ]W j f ) + (W j B k f, B k W j f ) = (B k f, [B k , W * j ]W j f ) + (W j B k f, [B k , W j ] f ) + (W j B k f, W j B k f
). If we exchange the role of B k and W j we find

= (W j f, [W j , B * k ]B k f ) + (B k W j f, [W j , B k ] f ) + (B k W j f, B k W j f ) = ([W j , B * k ] * W j f, B k f ) + (B k W j f, [W j , B k ] f ) + (B k W j f, B k W j f ) = ([B k , W * j ]W j f, B k f ) + (B k W j f, [W j , B k ] f ) + (B k W j f, B k W j f ). Thus adding two lines we find 2(B * k B k f, W * j W j f ) = 2([B k , W * j ]W j f, B k f ) + (W j B k f, [B k , W j ] f ) -(B k W j f, [B k , W j ] f ) +(W j B k f, W j B k f ) + (B k W j f, B k W j f ) so we have (3.16) 2(L 1 ( f ), L 2 ( f )) = 2(B * k B k f, W * j W j f ) = 2([B k , W * j ]W j f, B k f ) -([B k , W j ] f, [B k , W j ] f ) + (W j B k f, W j B k f ) + (B k W j f, B k W j f
). Note that the last two terms are positive (so we can forget them). We compute

[B k , W j ]( f ) = [ √ λ∂ v k , √ µV j ]( f ) = √ λ∂ v k [ √ µV j ( f )] - √ µV j [ √ λ∂ v k ( f )] = - √ λ∂ v k [ √ µ(∇ v f ∧ e j ).v] - √ µ(v ∧ e j ).∇ v [ √ λ∂ v k ( f )] √ λ∂ v k [ √ µ]V j ( f ) + √ λ √ µV j (∂ v k f ) + √ λ √ µ(e k ∧ e j ).∇ v f - √ µ(v ∧ e j ).∇ v [ √ λ]∂ v k ( f ) - √ λ √ µ(v ∧ e j ).∇ v ∂ v k f = √ λ∂ v k [ √ µ]V j ( f ) + √ λ √ µ(e k ∧ e j ).∇ v f - √ µ(v ∧ e j ).∇ v [ √ λ]∂ v k ( f ) = √ λ∂ v k [ √ µ]V j ( f ) + √ λ √ µ(e k ∧ e j ).∇ v f - √ µV j [ √ λ]∂ v k ( f ) Therefore [B k , W j ]( f ) 2 √ λ∂ v k [ √ µ]V j ( f ) 2 + √ λ √ µ(e k ∧ e j ).∇ v f 2 + √ µV j [ √ λ]∂ v k ( f ) 2 .
We note that the weight for the first term is similar to < v > γ-1 , for the second one to < v > γ and for the last to < v > γ also. Thus

[B k , W j ]( f ) 2 < v > γ ∇ v f 2 = (< v > 2γ ∇ v f, ∇ v f ) = -(< v > 2γ ∆ v f, f ) -(∇ v < v > 2γ .∇ v f, f ) and by symmetry (< v > 2γ ∆ v f, f ) + (∆ v < v > 2γ f, f ). Thus [B k , W j ]( f ) 2 ε < v > γ |D v | 2 f 2 + C ε < v > γ 2 +1 f 2 .
For the term in (3.16

) involving 2([B k , W * j ]W j f, B k f ), we can substract W * j to write 2([B k , W * j ]W j f, B k f ) = 2([B k , W * j -W j ]W j f, B k f ) + 2([B k , W j ]W j f, B k f ) = 2([B k , V j ( √ µ)]W j f, B k f ) + 2([B k , W j ]W j f, B k f ) Now [B k , V j ( √ µ)]( f ) = √ λ∂ v k [V j ( √ µ
)] f and we note that the weight is < v > γ-1 . Thus the first term is estimated by something similar to

(< v > γ-1 W j f, B k f ) < v > γ 2 -1/2 W j f 2 + < v > γ 2 -1/2 B k f 2 f h
For the other remaining term 2([B k , W j ]W j f, B k f ), we have in view of the commutators that it looks as something similar to

(< v > γ-1 V j √ µV j f, √ λ∂ v k f ) + (< v > γ (e k ∧ e j ).∇ v √ µV j f, √ λ∂ v k f ) + (< v > γ ∂ v k √ µV j f, √ λ∂ v k f )
and thus it looks as

< v > 2γ-1 V 2 j f, ∂ v k f ) + (< v > 2γ V j , ∆ v f ) < v > γ V 2 j f . < v > γ-1 ∇ v f + < v > γ |D v | 2 f < v > γ V j f < v > γ V 2 j f . < v > γ 2 ∇ v f + < v > γ |D v | 2 f < v > γ V j f ε < v > γ V 2 j f 2 + C ε < v > γ 2 ∇ v f 2 + ε < v > γ |D v | 2 f 2 + C ε < v > γ V j f 2 . But we can also write that < v > γ V j f 2 = (< v > 2γ V j f, V j f ) = (< v > γ V 2 j f, < v > γ f ) < v > γ |v ∧ D v | 2 f < v > γ f thus < v > γ |D v | 2 f < v > γ V j f < v > γ |D v | 2 f 3 2 < v > γ f 1 2
and by using Holder with exponent and a small exponent ε again, we find that it is less than

ε < v > γ |D v | 2 f 2 + C ε < v > γ f 2 .
All in all, we have shown that all terms from the scalar product (but for the positive) in (3.16) can be absorbed with previous ones. That is the scalar product on the l.h.s. of (3.13) can be absorbed with all the other terms, taking into account also (3.15). This is enough to conclude the proof of the main Theorem, in view of the exponents appearing on the r.h.s. Remark 3.3. We make some comments about the case of the Cauchy problem, when working say on a time interval (0, T ) (possibly with T = +∞), with an initial value at time t = 0 given by f 0 .

Let φ 1 = φ 1 (t) be a smooth and compactly supported positive function being one for small positive values of time. Let φ 2 = φ 2 (t) be again a smooth and positive function, but being one for larger values of t and such that φ 1 + φ 2 ∼ 1. Let f 1 = φ 1 f and f 2 = φ 2 f . We shall also assume that eventually φ 2 has a compact support on the right of the real axis.

We first consider f 2 . Then (3.17)

∂ t f 2 + v.∇ x f 2 -∇ v .λ(v)∇ v f 2 -(v ∧ ∇ v ).µ(v)(v ∧ ∇ v f 2 ) + F(v) f 2 (v) = φ 2 h + φ ′ 2 f ≡ h 2 .
Now considering the fact that this equation for f 2 holds true all over R, the previous results apply. Of course, we need f to be in L 2 in all variables. This statement might be assumed, but note that it also follows from the equation if we assume that γ ≥ -2 by using the coercivity from the third term of the diffusive part.

We can now turn to f 1 which also satisfies:

(3.18) ∂ t f 1 +v.∇ x f 1 -∇ v .λ(v)∇ v f 1 -(v∧∇ v ).µ(v)(v∧∇ v f 2 )+ F(v) f 1 (v) = φ 1 h+φ ′ 1 f ≡ h 1 , f 1 | t=0 = f 0 .
A careful look at the previous proof shows that if we assume that L i f 0 belongs to L 2 , then we have again the same conclusion as in the main theorem.

Finally, a last (formal) comment can be made if we assume h = 0 from the beginning. In that case, as it is standard in semi group theory, we can get formally more regularity. As explained above, we assume γ ≥ -2 to ensure that we have automatically an L 2 bound. Then following formally the arguments from [START_REF] Cazenave | An Introduction to semilinear evolution equations[END_REF], we see that, using only norms w.r.t. (x, v) variables

(-v.∇ x f -L( f )) f (t) ≤ 1 t √ 2 f 0
By using the main Theorem (without time dependence), we see that we have hypoelliptic results on f (t) in terms of the initial data, and with a singular behavior as times tends to 0.

1 3

 1 of the equation, multiply by µ < v > -2γ 3 = µ 1 , multiply by f and integrate to get

(v ∧ k).e j V j [S (k, v)(v ∧ e j ).k] f (k)

Above the last term involving F was omitted. We look for the second term:

The first term of this equality is the one we are looking for. For the second one, set β j = V j [µ 1 ]µ, then this is of the form (and by anti symmetry):

and thus

From Lemma 2.3, it follows that

and thus

.

Using Lemma 3.1, we get (though of course it is not optimal), and taking into account that γ ≤ 1 and Lemma 2.1, using the fact that γ 3 ≤ γ 2 + 1

which simplifies to yield (3.15)


Ondine Filippi-Codaccioni

Jean-Pierre Moussus

Jean-Paul Urcun

Fre ´de ´ric Jiguet

F O Bairlein

J.-P Filippi-Codaccioni

Á F Moussus

Jiguet

J.-P Urcun

Advanced departure dates in long-distance migratory raptors

Keywords: Climate change, Migrating birds, Trans-Saharan, Mean passage dates, Phenology

Evidences for phenological changes in response to climate change are now numerous. One of the most documented changes has been the advance of spring arrival dates in migratory birds. However, the effects of climate change on subsequent events of the annual cycle remain poorly studied and understood. Moreover, the rare studies on autumn migration have mainly concerned passerines. Here, we investigated whether raptor species have changed their autumn migratory phenology during the past 30 years at one of the most important convergent points of western European migration routes in France, the Organbidexka pass, in the Western Pyrenees. Eight out of the 14 studied raptor species showed significant phenological shifts during 1981-2008. Long-distance migrants displayed stronger phenological responses than short-distance migrants, and advanced their mean passage dates significantly. As only some short-distance migrants were found to delay their autumn migration and as their trends in breeding and migrating numbers were not significantly negative, we were not able to show any possible settling process of raptor populations. Negative trends in numbers of migrating raptors were found to be related to weaker phenological responses. Further studies using data from other migration sites are necessary to investigate eventual changes in migration routes and possible settling process.

Introduction

Living organisms displayed various responses to 30 years of climate warming at the end of the twentieth century [START_REF] Walther | Ecological responses to recent climate change[END_REF]. Among the most documented responses are temporal shifts in species phenology, witnessed thanks to long-term monitoring datasets. Many studies reported advances in the seasonal start of breeding or growth in numerous animal and plant species. Among them, the ones on migratory birds' spring arrival dates in Europe have been particularly noteworthy [START_REF] Sokolov | Long-term trends in the timing of spring migration of passerines on the Courish Spit of the Baltic Sea[END_REF][START_REF] Lehikoinen | Arrival and departure dates[END_REF][START_REF] Sparks | Examining the total arrival distribution of migratory birds[END_REF][START_REF] Jonze ´n | Rapid advance of spring arrival dates in long-distance migratory birds[END_REF][START_REF] Zalakevicius | Spring arrival response to climate change in birds: a case study from eastern Europe[END_REF]). However, the effects of climate change on subsequent events of the annual cycle, like autumn migration, remain poorly studied and understood (but see [START_REF] Jenni | Timing of autumn bird migration under climate change: advances in long-distance migrants, delays in short-distance migrants[END_REF][START_REF] Tøttrup | Changes in timing of autumn migration in North European songbird populations[END_REF][START_REF] Thorup | Patterns of phenological changes in migratory birds[END_REF]. Evidence for a delayed or advanced autumn migration was equivocal in numerous studies with some species departing earlier and others later [START_REF] Gatter | Zugzeiten und Zugmuster im Herbst: Einfluss des Treibhauseffektes auf den Vogelzug?[END_REF][START_REF] Bairlein | Changes and risks[END_REF][START_REF] Gilyazov | Change in the timing of migration of common birds at the Lapland Nature Reserve (Kola Peninsula, Russia) during 1931-1999[END_REF][START_REF] Lehikoinen | Arrival and departure dates[END_REF]). In fact, the phenological response often depends on specific life history traits [START_REF] Lehikoinen | Arrival and departure dates[END_REF]). Indeed, [START_REF] Jenni | Timing of autumn bird migration under climate change: advances in long-distance migrants, delays in short-distance migrants[END_REF] showed for 65 passerine species that the temporal trend in peak passage date depended on several specific life history traits like the migration strategy, the number of broods or the main food type.

As most studies have been undertaken on passerines, taking advantage of the numerous migration observatories, raptor migration phenology-as well as its phenological changes according to specific life-history traits-remains poorly investigated.

Here, we chose to study these possible phenological shifts using migration counts of visually-detected migrants at one of the most important convergent points of major western European migration routes in France, the Organbidexka pass, Western Pyrenees. To achieve this, we computed mean passage dates weighted by species abundance for 14 raptor species over a 28-year period , and looked for long-term trends in these mean dates. We further tested whether among-species variations in these long-term trends depended on several traits. Firstly, migration strategy: we predicted long-distance migrants to advance their migration dates in autumn. Indeed, there are several advantages to departing earlier. Their survival could increase if they are able to track ecological changes due to climate change en route as well as on the wintering grounds (i.e. the onset of the dry season in the Sahel from September onwards) [START_REF] Jenni | Timing of autumn bird migration under climate change: advances in long-distance migrants, delays in short-distance migrants[END_REF]. They could also find better winter territories if arriving earlier than other non-nomadic wintering species thus resulting in better body condition in preparation for spring migration [START_REF] Alerstam | Bird migration[END_REF][START_REF] Newton | The migration ecology of birds. Academic, London R Development Core Team[END_REF]. Some evidence also shows that raptors are able to advance their breeding phenology due to increased spring temperature which enables them to depart earlier [START_REF] Lehikoinen | Climate forcing on avian life history[END_REF][START_REF] Lehikoinen | Reproduction of the common buzzard at its northern range margin under climatic change[END_REF]. In contrast, we predicted short-distance migrants to delay their departure dates. Indeed, as they winter in areas offering better conditions resulting from global warming, they may delay autumn migration or even winter in the breeding grounds [START_REF] Berthold | Climate change and unequal phenological changes across four trophic levels: constraints or adaptations?[END_REF]). They could thus achieve higher survival rates and/or obtain higher quality breeding territories by arriving earlier in spring [START_REF] Berthold | Climate change and unequal phenological changes across four trophic levels: constraints or adaptations?[END_REF][START_REF] Jenni | Timing of autumn bird migration under climate change: advances in long-distance migrants, delays in short-distance migrants[END_REF][START_REF] Newton | The migration ecology of birds. Academic, London R Development Core Team[END_REF]. Secondly, moult strategy: arrival and departure dates could be linked to moult timing. Moulting some flight feathers is necessary each year in raptors but cannot occur simultaneously with migration because it reduces their gliding ability. Species replacing large flight feathers (wings and tail) before migration could be more constrained and depart later than the others. And thirdly, generation time: if selection for a changing phenology in response to a changing climate is occurring, it could occur more rapidly in species with a shorter generation time.

Next, we looked at the relationship between population trends (as available for European breeding populations) and our estimated temporal trends in species phenology of migration in the 14 raptor species considered. In European birds [START_REF] Møller | Populations of migratory bird species that did not show a phenological response to climate change are declining[END_REF], most declining species could be those displaying the least phenological responses to climate change. We made this prediction with trends related to breeding populations. As trends of migrating species could both signify a settling process (reduction of the migratory behaviour within a constant population size) or a decline in population size, we also found it of interest to test this relationship with trends of migrating raptors at Organbidexka.

Methods

Study site

The Western Pyrenees and their passes constitute one of a few convergent points along the most important western European migration flyways. The main pass there for raptor migration is the Organbidexka pass (43°02 0 19.06 00 N, 1°00 0 21.68 00 W). It is located in the Basque Country, in the department of Pyre ´ne ´es-Atlantiques (south-west France), lying 1,283 m above sea level. It is one of the most important migration sites in Western Europe together with Falsterbo (Sweden) and the Strait of Gibraltar (Spain), particularly for the observation of raptors, storks, cranes and pigeons.

Data on bird migration

At the pass, annual autumn surveys are conducted from 15 July to 15 November from dawn to dusk by varying numbers of observers among which permanent skilled ones are present throughout the period. Annual counts of raptors average *40,000. We used a dataset comprising 28 years of migration counts, from 1981 to 2008, with the same data recording methodology used during the whole period [START_REF] Devisse | Mise en place du suivi de populations europe ´ennes d'oiseaux migrateurs transpyre ´ne ´ens, Fasc. I. Rapport SRETIE/MERE/8815[END_REF]. Daily numbers of birds flying south were used for analyses.

Tested traits were taken from different sources: generation time and migration strategy from (Birdlife International 2004), moult strategies from Cramp et al. (1977Cramp et al. ( -1984)). Trends for European breeding populations were taken from Birdlife International (2004) and transformed using a 6-level scale (as in [START_REF] Møller | Populations of migratory bird species that did not show a phenological response to climate change are declining[END_REF], ranging from large decline to moderate increase (mentioned as ''TrendEU'' hereafter), while local trends in numbers migrating at Organbidexka were calculated as slopes of total annual counts by species as a function of years (hereafter ''TrendOrg'') (Table 1). Two categories designed migration strategy: short-distance migrants which winter in Europe and North Africa, and long-distance migrants which winter south of the Sahara. The Marsh Harrier (Circus aeruginosus) was classified as a long-distance migrant because birds passing the Pyre ´ne ´es originate from northern European breeding populations, which winter mainly in sub-Saharan Africa [START_REF] Strandberg | Complex timing of Marsh Harrier Circus aeruginosus migration due to pre-and post-migratory movements[END_REF].

Statistical analyses

Various phenological modelling methods designed to compute phenological shifts exist in the literature. These include first appearance dates [START_REF] Sparks | Examining the total arrival distribution of migratory birds[END_REF]; reviewed in [START_REF] Lehikoinen | Arrival and departure dates[END_REF]), mean or median dates [START_REF] Sokolov | Long-term trends in the timing of spring migration of passerines on the Courish Spit of the Baltic Sea[END_REF][START_REF] Va ¨ha ¨talo | Spring arrival of birds depends on North Atlantic oscillation[END_REF][START_REF] Sparks | Examining the total arrival distribution of migratory birds[END_REF], different percentiles [START_REF] Jonze ´n | Rapid advance of spring arrival dates in long-distance migratory birds[END_REF] or smoothing techniques [START_REF] Knudsen | Characterizing bird migration phenology using data from standardized monitoring at bird observatories[END_REF]. Although the accuracies of these methods have still not been extensively compared, first appearance dates and other methods relying on a small portion of the total phenological distribution have aroused major criticism [START_REF] Tryjanowski | Is the detection of the first arrival date of migrating birds influenced by population size? A case study of the red-backed shrike Lanius collurio[END_REF][START_REF] Miller-Rushing | Bird migration times, climate change, and changing population sizes[END_REF]). The issue is even more crucial with raptor migration phenology as some species display a bimodal phenological distribution due to different passage dates between males and females or between adults and juveniles. Moreover, there was an important variation in sample size between years in our data, with up to 50% variations between the annual counts. To deal with these issues, the two most robust methods are the mean passage dates and the smoothing techniques which both take the entire phenological distribution into account and compute sensible estimates for bimodal phenological distributions (Moussus et al., unpublished results). The results which are presented here relate to the estimates computed using the mean passage dates, while the results we obtained with phenological shift estimates from the smoothing method are very similar. Indeed, there is a strong correlation between estimations of the phenological shifts between the two methods which range between r = 0.55, n = 28, P = 0.002 for the Merlin (Falco columbarius), and r = 0 0.98, n = 28, P \ 0.001 for the European Sparrowhawk (Accipiter nisus) with a mean Pearson's coefficient r = 0.90 ± 0.11 for the 14 species. Methodological details of the smoothing method are available in [START_REF] Moussus | A method to estimate phenological variation using data from large-scale abundance monitoring programmes[END_REF].

The temporal trend of mean passage dates was modelled using linear models and tested using analyses of variance (ANOVAs). Adjusted effects of moult date, migration strategy and generation time on the temporal trend in mean passage dates were tested using linear models and ANOVAs. Relationships between the temporal trend of mean passage dates and species population trends were modelled using linear models and statistical significance of trends tested with ANOVAs. Population trends effect was first adjusted to migration and moult strategies as well as to generation time. We computed the contribution of each explanatory term to the variation in the responses after fitting all other terms in the models.

To avoid considering too imprecise data, Va ¨ha ¨talo et al. ( 2004) omitted from their analysis the species data for years when less than 20 individuals were counted. In our case, this restriction would apply only for the Merlin. If we restrict the species-specific dataset to years with at least 20 individuals detected, 8 years have to be deleted. Considering only the remaining years with more than 20 birds changed the temporal trend of mean migration dates from -0.01 ± 0.08 to -0.09 ± 0.09, but did not affect the outputs of the other analyses.

All analyses were conducted using the R statistical software (R Development Core Team 2006).

Results

Phenological shifts during the last 28 years were significant for 8 out of the 14 species. They ranged from an advance of 10 days for the Marsh Harrier to a delay of 9 days for the Common Kestrel (Falco tinnunculus) (Table 1). In addition, two species, Booted Eagle (Aquila pennata) and Montagu's Harrier (Circus pygargus), had P values lower than 0.10 (Table 1).

Relationship to life history traits

Migration strategy was found to explain significantly the changes in phenology (F 1,10 = 5.83, P = 0.03) when adjusted for moult strategy (F 1,10 = 0.84, P = 0.37) and generation time (F 1,10 = 1.72, P = 0.21). The model with migration strategy as a single predictor explained 33% of the total variance (F 1,12 = 5.92, P = 0.03). Other traits did not relate to the temporal trend in mean passage dates when tested separately. Long-distance migrants showed a significant negative slope in their temporal trend in mean passage dates (mean ± SE: -0.14 ± 0.06, P = 0.04) whereas short-distance migrants showed a non-significant positive slope (mean ± SE: 0.09 ± 0.07, P = 0.23; Fig. 1).

None of the studied long-distance migrants (8 species) delayed its autumn migration during the last 30 years, while 3 out of the 6 short-distance migrants delayed their passage dates significantly (e.g. Fig. 2; Table 1).

Within the same genus, comparisons between species are noted. For example, Montagu's and Hen Harriers (Circus cyaneus) display very different responses to climate change. The first is a long-distance migrant and has advanced its migration date, while the second winters in Europe and North Africa and has delayed its migration. Among the genus Falco, responses are also contrasted. Only the Common Kestrel which can be classified as a short-distance migrant, displayed a strong response (as a delay) in mean passage dates temporal trends whereas the Merlin did not respond at all. As a long-distance migrant, the Eurasian Hobby (Falco subbuteo) did not show significant earlier migration dates.

Relationship to population trends

We failed to find a significant negative relationship between the trends in European breeding populations (TrendEU) and the temporal trends in mean passage dates (F 1,10 = 1.17, R 2 = 10.5%, P = 0.30), whereas the trends in migrant numbers at Organbidexka (TrendOrg) and the temporal trends of mean passage dates were found to be negatively and significantly correlated (F 1,12 = 8.73, R 2 = 42%, P = 0.01) (Fig. 3). When adjusted for the effect of migration and moult strategies and of generation time, the relationship between breeding populations trends and trends in phenology remained non-significant, while the one between migrating numbers trends and temporal trends in mean passage dates become non-significant (TrendEU: F 1,7 = 0.48, P = 0.50; TrendOrg: F 1,9 = 2.98, P = 0.11).

We found a significant difference in trends of migrating raptor numbers according to the migration strategy (F 1,376 = 41.15, P \ 0.001).

Migrating and breeding trends were not correlated when considering all species (F 1,10 = 0.32, R 2 = 0.03, P = 0.57). There was no difference in this relationship between migration strategies (TrendEU 9 migration strategy: F 1,8 = 0.58, P = 0.46).

Discussion

Phenological trends

The main result is that long-distance migrants advanced their autumn migration during this 28-year period, while some, though not all, short-distance migrants delayed their passage. Apparently, raptors display the same dichotomic response as passerines according to their migration strategy [START_REF] Jenni | Timing of autumn bird migration under climate change: advances in long-distance migrants, delays in short-distance migrants[END_REF] which is in favour of a common pattern of the response among birds to global climate change, the same constraints driving the same adaptations even for bird species with very different demographic traits.

However, some exceptions remain within short-distance migrants suggesting that other traits could be responsible for the change in migratory phenology. For example, the Eurasian Sparrowhawk showed a great advance in its mean passage dates whereas it is classified as short-distance migrant. It could be possible that an important part of individuals passing at Organbidexka are long-distance Fig. 1 The temporal trends in mean passage dates as a function of the migration strategy (short-or long-distance) for 14 raptor species migrating at the Organbidexka pass, Western Pyrenees in southern France Fig. 2 Examples of changes in the migratory phenology during migrants. Another explanation could be that the Sparrowhawk is a predator specialized on passerines, in contrast to the other short-distance delaying species (Hen Harrier, Common Kestrel and Common Buzzard) which feed on small mammals. Small mammals do not migrate but occur all over the Europe year round while passerines are mainly migratory. Sparrowhawks are possibly trying to migrate with the passerines whereas mammal eaters do not need to do so [START_REF] Lehikoinen | Climate forcing on avian life history[END_REF]. [START_REF] Lehikoinen | Climate forcing on avian life history[END_REF] found the same advance of autumn migration in Sparrowhawks. The advance was in accordance with an earlier breeding schedule. The author found differences between early migrating young that have advanced their autumn migration and late-phase migrants, mainly representing adults, which have not advanced their autumn schedule. As we here referred to mean migration dates, it is possible that our pattern of advanced migration is explained mainly by juvenile Sparrowhawks which would be more numerous than adults [START_REF] Thorup | Patterns of phenological changes in migratory birds[END_REF]). Two hypotheses have been proposed regarding the effects of climate change on the autumn phenology of migratory birds. [START_REF] Jenni | Timing of autumn bird migration under climate change: advances in long-distance migrants, delays in short-distance migrants[END_REF] suggested that single-brooded short-distance migrants, such as Sparrowhawks, should show delayed rather than advanced autumn phenology despite advancing spring arrival [START_REF] Lehikoinen | Climate forcing on avian life history[END_REF]. Because of the warming climate, it would be beneficial for those species to winter as close to breeding areas as possible with respect to earlier arrival and territory establishment [START_REF] Jenni | Timing of autumn bird migration under climate change: advances in long-distance migrants, delays in short-distance migrants[END_REF][START_REF] Newton | The migration ecology of birds. Academic, London R Development Core Team[END_REF]. On the other hand, [START_REF] Tøttrup | Changes in timing of autumn migration in North European songbird populations[END_REF] have documented advancing autumn migration times in many northern European shortand long-distance migrating passerines. These authors hypothesized that the time spent by the species on their breeding areas is constant, and thus earlier spring arrival and breeding would lead to an earlier autumn migration of both adult and young birds [START_REF] Thorup | Patterns of phenological changes in migratory birds[END_REF]. Taken as a whole, our migration data on Sparrowhawks would provide support for the hypotheses of [START_REF] Thorup | Patterns of phenological changes in migratory birds[END_REF] as we found an advance of autumn migration for this species despite it being a short-distance migrant (Cramp et al. 1977(Cramp et al. -1994;;Birdlife International 2004). However, a closer look at the differential migration between ages or sexes could highlight that different groups of the same species may respond differently to climate change, making it possible to test both hypotheses [START_REF] Lehikoinen | Climate forcing on avian life history[END_REF].

We did not find any significant response in temporal trends of mean passage dates of one short-distance species, the Merlin, and of one long-distance migrant, the Eurasian Hobby which did not show significant earlier migration dates. Organbidexka is possibly not the best migration site to study phenological trends of these two falcons, as these species present the smallest mean annual counts out of our 14 species (Table 1) and therefore significant phenological shifts could be harder to detect. This could prevent us from finding any temporal trend in mean passage dates.

Relationship between phenological shifts and population trends

No relationship was found between breeding population trends and temporal trends in mean passage dates. We could have expected species whose breeding populations are increasing to display the strongest advances in autumn migration, as found by [START_REF] Møller | Populations of migratory bird species that did not show a phenological response to climate change are declining[END_REF] for 100 European species using breeding population trends and spring arrival timing. As autumn migration timing is often related to the timing of spring events [START_REF] Sokolov | Effect of global warming on the timing of migration and breeding of passerine birds in the 20th century[END_REF]), we suggest that species which advanced their spring migration would have also advanced their autumn migration. Consequently, It is possible either that our sample size is not large enough to give evidence of such effects or that there is no such relationship in raptors.

Conversely, species-specific decreases in migrating numbers tend to be associated with the latest mean passage dates. However, if a decrease in counted numbers on the breeding grounds can be interpreted as a decline of breeding population numbers, this does not apply to migration counts. A decrease in migrating numbers can also ensue from a disappearance of the migratory behaviour even if the global population size is stable, a case of some putative ongoing settling process. We here have no proof of such possible ongoing settling process for ''European'' migrant raptors as only some short-distance migrants delayed their migration and as this species group did not show significant negative trends in migration counts.

Further investigations on the link between recapture distances taken from ringing database for those species suspected to have begun a settling process (Common Buzzard, Common Kestrel) would be necessary to highlight this possible phenomenon. We can add that evidence of a shortening of the distance between breeding and wintering sites has been recently shown for 24 bird species and that it has been proposed as a possible explanation for the stronger advancement of arrival date found in several short-distance species relative to long-distance migrants [START_REF] Visser | Climate change leads to decreasing bird migration distances[END_REF]). Among them, Hen Harrier, Common Kestrel and Common Buzzard showed significant decreases in their winter recapture distances-and delayed their mean passage date at Organbidexka-while Sparrowhawk and Marsh Harrier did not show such significant decreases-both migrate earlier at Organbidexka. Comparisons between trends detected at other migration sites like Gibraltar and Falsterbo would also be worthwhile.

Zusammenfassung

Verfru ¨hte Herbstzugtermine bei langstreckenziehenden Greifvo ¨geln

Hinweise fu ¨r pha ¨nologische Vera ¨nderungen als Antwort auf den Klimawandel sind inzwischen zahlreich. Eine der am ha ¨ufigsten dokumentierten Vera ¨nderungen sind die fru ¨heren Ankunftsdaten der Zugvo ¨gel im Fru ¨hjahr. Jedoch bleibt bis jetzt der Einfluss von Klimavera ¨nderungen auf aufeinanderfolgende Ereignisse des Jahreslaufes schlecht untersucht und verstanden. Daru ¨ber hinaus bescha ¨ftigten sich die wenigen Studien zum Herbstzug hauptsa ¨chlich mit Singvo ¨geln. In dieser Studie untersuchten wir, ob es in den letzten 30 Jahren im Herbstzug von Greifvo ¨geln eine pha ¨nologische Vera ¨nderung an einem der wichtigsten Durchzugspunkte der westeuropa ¨ischen Zugrouten in Frankreich, dem Organbidexka Pass der westlichen Pyrena ¨en, gab. Acht der vierzehn untersuchten Greifvogelarten zeigten pha ¨nologische Verschiebungen wa ¨hrend der Jahre 1981-2008. Langstreckenzieher zeigten deutlichere pha ¨nologische Reaktionen als Kurzstreckenzieher und verfru ¨hten ihren mittleren Durchzugstermin signifikant. Da nur einige Kurzstreckenzieher ihren Herbstzug verspa ¨teten und da ihr Trend in Brut-und Zugzahlen nicht signifikant negativ war, konnten wir keine mo ¨glichen Vera ¨nderungen ihrer Zugbereitschaft nachweisen. Ru ¨ckga ¨nge in den Zahlen durchziehender Greifvo ¨gel korrelierten mit schwa ¨cheren pha ¨nologischen Reaktionen. Weitere Studien mit Daten anderer Durchzugsgebiete sind notwendig, um mo ¨gliche A ¨nderungen in Zugrouten und Zugbereitschaft zu untersuchen.

Fig. 3

 3 Fig. 3 Relationship between the temporal trends in mean passage dates and a trends of European breeding populations and b trends of migrants counted at Organbidexka

Table 1

 1 Species traits tested on the temporal trend in the mean passage date during 28 years at Organbidexka pass for 14 raptor species

	P value		\0.001	0.00	0.00	0.02	0.08	0.00	0.06	0.69	0.82	0.91	0.94	0.05	0.02	0.03
	Difference	(days/28 years)	-9.73	-8.80	-5.55	-4.67	-3.78	-3.39	-3.30	-0.79	-0.50	-0.25	-0.15	8.15	8.55	8.63
	Slope	(days/year) ± SE	-0.36 ± 0.08	-0.33 ± 0.10	-0.21 ± 0.06	-0.17 ± 0.07	-0.14 ± 0.08	-0.13 ± 0.04	-0.12 ± 0.06	-0.03 ± 0.07	-0.02 ± 0.08	-0.01 ± 0.08	-0.01 ± 0.08	0.30 ± 0.14	0.32 ± 0.13	0.32 ± 0.14
	TrendOrg		0.02	0.04	0.08	0.04	0.01	-0.01	0.00	0.04	0.00	-0.02	0.01	0.00	-0.06	-0.01
	TrendEU		2	1	-3	2	0	0	2	-1	0	-2	NA	NA	1	-2
	Generation time		6	4	6	9	11	9	6	13	5	6	3	6	8	3
	Moult		BM	BM	BAM	BAM	BM	BAM	BAM	BM	BAM	BM	BM	BM	BM	BM
	Migration		Long	Short	Long	Long	Long	Long	Long	Long	Long	Short	Short	Short	Short	Short
	Temporal	trend	-0.36	-0.33	-0.21	-0.17	-0.14	-0.13	-0.12	-0.03	-0.02	-0.01	-0.01	0.30	0.32	0.32
	Mean-counts	(Min-max)	217.82 (74-408)	274.46 (129-599)	12,478.28 (1,406-33,304)	104.96 (45-198)	66.78 (37-108)	10,811.82 (6,481-21,504)	87.85 (47-145)	107.00 (41-176)	49.39 (22-86)	3,080.50 (1,521-4,529)	25.25 (11-41)	85.53 (40-158)	191.64 (34-378)	120.89 (29-252)
	Species		Circus aeruginosus	Accipiter nisus	Milvus migrans	Pandion haliaetus	Aquila pennata	Pernis apivorus	Circus pygargus	Circaetus gallicus	Falco subbuteo	Milvus milvus	Falco columbarius	Circus cyaneus	Buteo buteo	Falco tinnunculus

Mean counts

Mean annual numbers of each raptor species with minimum and maximum numbers, Temporal trend the slope between the mean passage date and years (n = 14), Migration migration strategy (short or long-distance migrant), Generation time mean age of breeding females, Moult moult strategy (BM before migration, BAM before and after migration), TrendEU population trends on a 6-point scale from large decline to moderate increase, TrendOrg slopes of raptor numbers at Organidexka in function of years, Slope phenological shift (slope of the mean passage date in function of years at Organbidexka pass ± SE) and P values for each of the 14 raptor species

J Ornithol (2010) 151:687-694

years by a a short-distance migrant, the Common Buzzard (Buteo buteo) and b a long-distance migrant, the Booted Eagle (Aquila pennata). Mean dates are given in number of days, day 1 being 12 July

Acknowledgments We would like to thank the many volunteers counting birds at Organbidexka for many years, since the beginning of the migration observatory. We are grateful to Marc Ke ´ry and Aleksi Lehikoinen for their helpful comments. We also thank the Ligue de Protection des Oiseaux and Organbidexka Col Libre, joined into the ORMO Program (Regional Observatory of Bird Migration in Aquitaine), for funding this research and providing the data.