Thomas Duquesne
email: thomas.duquesne@upmc.fr

Guanying Wang

EXCEPTIONALLY SMALL BALLS IN STABLE TREES *

Keywords: AMS 2000 subject classifications: Primary 60G57, 60J80. Secondary 28A78 Continuum Random Tree, Lévy trees, stable trees, mass measure, small balls

.

Introduction

Stable trees are particular instances of Lévy trees that form a class of random compact metric spaces introduced by Le Gall and Le Jan in [START_REF] Gall | Branching processes in Lévy processes: the exploration process[END_REF] as the genealogy of Continuous State Branching Processes (CSBP for short). The class of stable trees contains Aldous's continuum random tree that corresponds to the Brownian case (see Aldous [START_REF] Aldous | The continuum random tree I[END_REF][START_REF] Aldous | The continuum random tree III[END_REF]). Stable trees (and more generally Lévy trees) are the scaling limit of Galton-Watson trees (see [START_REF] Duquesne | Random Trees, Lévy Processes and Spatial Branching Processes[END_REF] Chapter 2 and [START_REF] Duquesne | A limit theorem for the contour process of conditioned Galton-Watson trees[END_REF]). Various geometric and distributional properties of Lévy trees (and of stable trees, consequently) have been studied in [START_REF] Duquesne | Probabilistic and fractal aspects of Lévy trees[END_REF] and in Weill [START_REF] Weill | Regenerative real trees[END_REF]. Stable trees have been also studied in connection with fragmentation processes: see Miermont [START_REF] Miermont | Self-similar fragmentations derived from the stable tree I: splitting at heights[END_REF][START_REF] Miermont | Self-similar fragmentations derived from the stable tree II: splitting at nodes[END_REF], Haas and Miermont [START_REF] Haas | The genealogy of self-similar fragmentations with negative index as a continuum random tree[END_REF], Goldschmidt and Haas [START_REF] Goldschmidt | Behavior near the extinction time in self-similar fragmentation I: the stable case[END_REF] for the stable cases and see Abraham and Delmas [START_REF] Abraham | Fragmentation associated to Lévy processes using snake[END_REF] for related models concerning more general Lévy trees. To study Brownian motion on stable trees, D. Croydon in [START_REF] Croydon | Scaling limits for simple random walks on random ordered graph trees[END_REF] got partial results on balls with exceptionnally small mass measure.

Before stating the results, let us briefly explain the definition of stable trees before stating the main results of the paper. Let us fix the stable index γ ∈ (1, 2] and let X = (X t ) t≥0 be a spectrally positive γ-stable Lévy process that is defined on the probability space (Ω, F , P). More precisely, we suppose that E[exp(-λX t )] = exp(tλ γ ), t, λ ∈ [0, ∞). Note that X is a Brownian motion when γ = 2 and we shall refer to this case as to the Brownian case. As shown by Le Gall and Le Jan [START_REF] Gall | Branching processes in Lévy processes: the exploration process[END_REF] (see also [START_REF] Duquesne | Random Trees, Lévy Processes and Spatial Branching Processes[END_REF] Chapter 1), there exists a continuous process H = (H t ) t≥0 such that for any t ∈ [0, ∞), the following limit holds true in probability

H t := lim ε→0 1 ε t 0 1 {I s t <Xs<I s t +ε} ds. (1)
Here I s t stands for inf s≤r≤t X r . The process H is the γ-stable height process. Note that in the Brownian case, H is simply a reflected Brownian motion. Theorems 2.3.2 and 2.4.1 in [START_REF] Duquesne | Random Trees, Lévy Processes and Spatial Branching Processes[END_REF] show that H is the scaling limit of the contour function (or the depthfirst exploration process) of a i.i.d. sequence of Galton-Watson trees whose offspring distribution is in the domain of attraction of a γ-stable law.

As in the discrete setting, the process H encodes a family of continuous trees: each excursion of H above 0 corresponds to the exploration process of a single continuous tree of the family. Let us make this statement more precise thanks to excursion theory. Recall that X has unbounded variation sample paths. We set I t = inf s∈[0,t] X s , that is the infinimum process of X. Basic results on fluctuation theory (see Bertoin [4] VII.1) entail that X -I is a strong Markov process in [0, ∞) and that 0 is regular for (0, ∞) and recurrent with respect to this Markov process. Moreover, -I is a local time at 0 for X -I (see Bertoin [START_REF] Bertoin | Lévy Processes[END_REF] Theorem VII.1). We denote by N the corresponding excursion measure of X -I above 0. We denote by (a j , b j ), j ∈ I, the excursion intervals of X -I above 0, and by X j = X (aj +•)∧bj -I aj , j ∈ I, the corresponding excursions. Next, observe that if t ∈ (a j , b j ), the value of H t only depends on X j . Moreover, one can show that j∈I (a j , b j ) = {t ≥ 0 : H t > 0}. This allows to define the height process under N as a certain measurable function H(X) of X, that we simply denote by (H t ) t≥0 . For any j ∈ I, we then set H j = H (aj +•)∧bj and the point measure j∈I δ (-Ia j ,H j ) [START_REF] Aldous | The continuum random tree I[END_REF] is distributed as a Poisson point measure on [0, ∞) × C([0, ∞), R) with intensity ℓ ⊗ N, where ℓ stands for the Lebesgue measure on [0, ∞). Note that X and H under N are paths with the same lifetime given by

ζ := inf{t ∈ [0, ∞) : ∀s ∈ [t, ∞) , H s = H t } .
Standard results in fluctuation theory imply that 0 < ζ < ∞, N-a.e. and that

N 1 -e -λζ = λ 1/γ , λ ∈ [0, ∞) .
Thus, N(ζ ∈ dr) = C r -1 γ -1 ℓ(dr), where 1/C = γΓ(1 -1 γ ) (here, Γ stands for Euler's Gamma function). Note that N-a.e. H t > 0 iff t ∈ (0, ζ) and H 0 = H t = 0, for any t ∈ [ζ, ∞). We refer to Chapter 1 in [START_REF] Duquesne | Random Trees, Lévy Processes and Spatial Branching Processes[END_REF] for more details.

The excursion (H t ) 0≤t≤ζ under N is the depth-first exploration process of a continuous tree that is defined as the following metric space: for any s, t

∈ [0, ζ], we set b(s, t) = min s∧t≤r≤s∨t H r and d(s, t) = H t + H s -2b(s, t) . (3)
The quantity b(s, t) is the height of the branching point between the vertices visited at times s and t and d(s, t) is therefore the distance in the tree of these vertices. We easily show that d is a pseudo-metric and we introduce the equivalence relation ∼ on [0, ζ] by setting s ∼ t iff d(s, t) = 0. We then define the γ-stable tree as the quotient metric space

T , d = [0, ζ]/∼ , d .
We denote by p : [0, ζ] → T the canonical projection. It is easy to see that p is continuous. Thus the γ-stable tree (T , d) is a (random) connected compact metric space. More precisely, Theorem 2.1 in [START_REF] Duquesne | Probabilistic and fractal aspects of Lévy trees[END_REF] asserts that (T , d) is a R-tree, namely a metric space such that the following holds true for any σ, σ ′ ∈ T .

(a) There is a unique isometry

f σ,σ ′ from [0, d(σ, σ ′ )] into T such that f σ,σ ′ (0) = σ and f σ,σ ′ (d(σ, σ ′ )) = σ ′ . We set [[σ, σ ′ ]] = f σ,σ ′ ([0, d(σ, σ ′ )]) that is the geodesic joining σ to σ ′ . (b) If g : [0, 1] → T is continuous injective, then g([0, 1]) = [[g(0), g(1)]].
We refer to Evans [START_REF] Evans | Probability and real trees[END_REF] or to Dress, Moulton and Terhalle [START_REF] Dress | T-theory: an overview[END_REF] for a detailed account on R-trees. An intrinsic approach of continuous trees has been developped by Evans Pitman and Winter in [START_REF] Evans | Rayleigh processes, real trees, and root growth with re-grafting[END_REF] (see also [START_REF] Duquesne | Probabilistic and fractal aspects of Lévy trees[END_REF]): Theorem 1 in [START_REF] Evans | Rayleigh processes, real trees, and root growth with re-grafting[END_REF] asserts that the set of isometry classes of compact R-trees equipped with the Gromov-Hausdorff distance (see Gromov [START_REF] Gromov | Metric Structures for Riemannian and non-Riemannian Spaces[END_REF]) is a Polish space. This geometric point of view has been used in [START_REF] Duquesne | Probabilistic and fractal aspects of Lévy trees[END_REF] and [START_REF] Duquesne | Growth of Lévy trees[END_REF] to study Lévy trees. We need to introduce two additional features of γ-stable trees. First, we distinguish a special point ρ := p(0) in T , that is called the root. We also equip T with the measure m that is induced by the Lebesgue measure ℓ on [0, ζ] via the canonical projection p. Namely, for any Borel subset A of (T , d),

m(A) = ℓ p -1 (A) .
The measure m is called the mass measure of the γ-stable tree T . Note that m(T ) = ζ. One can prove that m is diffuse and that its topological support is T . Moreover, m is carried by the set of leaves of T that is the set of the points σ such that T \{σ} remains connected (see [START_REF] Duquesne | Probabilistic and fractal aspects of Lévy trees[END_REF] for more details). The measured tree (T , d, m) is thus a continuum tree, as defined by Aldous in [START_REF] Aldous | The continuum random tree III[END_REF].

Let us discuss briefly the scaling property of T . From the scaling property of X and from (1), we see that for any r ∈ (0, ∞), under P, (r γ-1 γ H t/r ) t≥0 has the same distribution as H. Then, by ( 2), (r γ-1 γ H t/r ) 0≤t≤rζ under r -1 γ N has the same "distribution" as (H t ) 0≤t≤ζ under N. Thus, the rescaled measured tree (T , r γ-1 γ d , rm) under r -1 γ N has the same "distribution" as (T , d, m) under N. This allows to define for any r ∈ (0, ∞), a probability distribution N(

• | ζ = r) on C([0, ∞), R) such that r → N( • | ζ = r) is weakly continuous and such that N = ∞ 0 N( • | ζ = r) N(ζ ∈ dr) . Moreover, (r γ-1 γ H t/r ) 0≤t≤r under N( • | ζ = 1) has the same distribution as (H t ) 0≤t≤r under N( • | ζ = r). Since m(T ) = ζ, the tree T under N( • | ζ = 1
) is interpreted as the γ-stable tree conditioned to have total mass equal to 1 and it is simply called the normalised γ-stable tree. When γ = 2, it corresponds (up to a scaling constant) to Aldous Continuum Random Tree as defined in [START_REF] Aldous | The continuum random tree I[END_REF] (see also Le Gall [START_REF] Gall | The uniform random tree in a Brownian excursion[END_REF] for a definition via the normalised Brownian excursion). The normalised γ-stable tree is the weak limit when n goes to infinity of a rescaled Galton-Watson trees conditionned to have total size n and whose offspring distribution belongs to the domain of attraction of a γ-stable law: see Aldous [START_REF] Aldous | The continuum random tree III[END_REF] for the Brownian case and see [START_REF] Duquesne | A limit theorem for the contour process of conditioned Galton-Watson trees[END_REF] for the general case.

The mass measure m is in some sense the most spread out measure on T and it plays a crucial role in the study of stable trees. For instance Theorem 1.1 in [START_REF] Duquesne | The exact packing measure of Lévy trees[END_REF] asserts that for any γ ∈ (1, 2], N-a.e. the mass measure m is equal to a deterministic constant times the g γ -packing measure where the gauge function is given by

g γ (r) = r γ γ-1 (log log 1/r) 1 γ-1 , r ∈ (0, e -1 ) . (4)
Actually, this result holds true for general Lévy trees (with a more involved gauge function). Here, the power exponent γ/(γ -1) reflects the scaling property. This value is also equal to the packing dimension of T , and to its Hausdorff and its box counting dimensions (see [START_REF] Duquesne | Probabilistic and fractal aspects of Lévy trees[END_REF]). The function g γ is also the lower density of m at typical points. More precisely, denote by B(σ, r) the open ball in T with center σ ∈ T and radius r ∈ (0, ∞). Then, Theorem 1.2 in [START_REF] Duquesne | The exact packing measure of Lévy trees[END_REF] asserts that there exists a constant

C γ ∈ (0, ∞) such that N-a.e. for m-almost all σ, lim inf r→0 m B(σ, r) g γ (r) = C γ . ( 5
)
Theorem 1.2 in [START_REF] Duquesne | The exact packing measure of Lévy trees[END_REF] also holds true for general Lévy trees and the constant is unknown. However, in the stable cases, we are able to compute explicitely C γ , as shown by the following proposition.

Proposition 1.1 For any γ ∈ (1, 2], C γ = γ -1.
We also discuss the balls with exceptionally small mass measure. More precisely, we investigate the behaviour of inf σ∈T m B(σ, r) when r goes to 0. Our main result is the following.

Theorem 1.2 For any γ ∈ (1, 2], we set

f γ (r) = r γ γ-1 (log 1/r) 1 γ-1 , r ∈ (0, 1) . ( 6
)
Then, there exist k γ , K γ ∈ (0, ∞) such that N-a.e.

k γ ≤ lim inf r→0 1 f γ (r) inf σ∈T m B(σ, r) ≤ lim sup r→0 1 f γ (r) inf σ∈T m B(σ, r) ≤ K γ . (7)
To study Brownian motion on stable trees, D. Croydon in Proposition 5.1 [START_REF] Croydon | Scaling limits for simple random walks on random ordered graph trees[END_REF] states a partial lower bound for inf σ∈T m B(σ, r) that is sufficient to his purpose (but that does not provide the right scale function). When 1 < γ < 2, there is no exact upper density of m at typical points (see Proposition 1.9 in [START_REF] Duquesne | Hausdorff and packing measure of stable trees[END_REF]). Moreover, Theorem 1.10 in [START_REF] Duquesne | Hausdorff and packing measure of stable trees[END_REF] shows that T has no exact Hausdorff measure whose gauge function is regularly varying. We also strongly believe that there is no exact asymptotic function for r → sup σ∈T m B(σ, r) , when r goes to 0, but we will not consider this problem in this paper.

In the Brownian case, Theorem 1.1 in [START_REF] Duquesne | The Hausdorff measure of stable trees[END_REF] asserts that N-a.e. the mass measure m is equal to a deterministic constant times the g-Hausdorff measure where the gauge function g is given by g(r) = r 2 log log 1/r , r ∈ (0, e -1 ) .

In the proof of Theorem 1.1 [START_REF] Duquesne | The Hausdorff measure of stable trees[END_REF], it is proved that g is the upper density of m at typical points. In this paper we obtain the following specific constant. Then, there exist k, K ∈ (0, ∞) such that N-a.e.

k ≤ lim inf r→0 1 f (r) sup σ∈T m B(σ, r) ≤ lim sup r→0 1 f (r) sup σ∈T m B(σ, r) ≤ K . ( 9
)
Observe that, in the Brownian case, Theorem 1.2 and Theorem 1.4 immediately imply the following result. The paper is organised as follows. Section 2 recalls useful technical results on the height process and basic geometric properties of stable trees. Section 3 is devoted to the tail estimates of the mass measure of specific subsets of stable trees. Section 4 is devoted to the proof of the results.

2 Preliminaries and basic results.

Results on the stable height process.

Local times of the height process. Let H be the γ-stable height process under its excursion measure N as defined in the introduction. It is possible to define the local times of H under the excursion measure N as follows. For any

b ∈ (0, ∞), let us set v(b) = N(supt∈[0,ζ] H t > b)
. The continuity of H under P and the Poisson decomposition (2) obviously imply that v(b) < ∞, for any b > 0. It is moreover clear that v is non-increasing and lim b→∞ v(b) = 0. For every a ∈ (0, ∞), we then define a continuous increasing process (L a t ) 0≤t≤ζ , such that for every b ∈ (0, ∞) and for any t ∈ [0, ∞), one has

lim ε→0 N 1 {sup H>b} sup 0≤s≤t∧ζ 1 ε s 0 dr1 {a-ε<Hr ≤a} -L a s = 0. (10)
See [START_REF] Duquesne | Random Trees, Lévy Processes and Spatial Branching Processes[END_REF] Section 1.3 for more details. The process (L a t ) 0≤t≤ζ is the a-local time of the height process. For any a, λ, µ ∈ [0, ∞), we set

κ a (λ, µ) := N 1 -e -µL a ζ -λ a 0 1 {H t <a} dt . (11)
The function κ is the Laplace exponent of a specific additive functional of a γ-stable Continuous States Branching Process (γ-stable CSBP, for short). An elementary result on CSBPs, whose proof can be found in Le Gall [START_REF] Gall | Spatial Branching Processes, Random Snakes and Partial Differential Equations[END_REF] Section II.3, entails that a → κ a (λ, µ) is the unique solution of the following ordinary differential equation:

∂κ a ∂a (λ, µ) = λ -κ a (λ, µ) γ and κ 0 (λ, µ) = µ .
For more details, see [START_REF] Duquesne | The Hausdorff measure of stable trees[END_REF] Section 4 page 405 or [9] Section 2.3 page 106. We note the following:

if µ = λ 1/γ , then κ a (λ, µ) = λ 1/γ . If µ < λ 1/γ , (resp. if µ > λ 1/γ ), then a → κ a (λ, µ) is increasing (resp. decreasing).
A simple change of variable implies that κ satisfies the following integral equation

κa(λ,µ) µ du λ -u γ = a, for any a, λ, µ ∈ [0, ∞) such that µ = λ 1 γ . ( 12
)
As a consequence, we get

κ a+b (λ, µ) = κ a λ, κ b (λ, µ) , a, b, λ, µ ∈ [0, ∞) . ( 13
)
We also derive from ( 12) the following scaling property:

c 1 γ-1 κ a c -γ γ-1 λ, c -1 γ-1 µ = κ a/c (λ, µ) , a, c, λ, µ ∈ [0, ∞) . (14)
When γ = 2 and λ > 0, it seems difficult to compute κ explicitly. However, when λ = 0, [START_REF] Duquesne | Probabilistic and fractal aspects of Lévy trees[END_REF] implies that for any a, µ ∈ [0, ∞),

κ a (0, µ) = N 1 -e -µL a ζ = (γ -1)a + 1 µ γ-1 -1 γ-1 . ( 15
)
It is convenient to interpret these quantities in terms of the γ-tree T . For any a ∈ (0, ∞), first define the a-local time measure ℓ a as the measure induced by dL a

• via the canonical projection p : [0, ζ] → T . Namely,

ℓ a , f = ζ 0 f (p(s)) dL a s ,
for any positive measurable application f on T . Here dL a • stands for the Stieltjes measure associated with the non-decreasing function s → L a s . Note that the topological support of ℓ a is included in the a-level set

T (a) = {σ ∈ T : d(ρ, σ) = a} . (16)
Next, we set

Γ(T ) = sup σ∈T d(ρ, σ) (17)
that is the total height of T . Then, observe that

ℓ a = L a ζ , m B(ρ, a) = ζ 0 1 {Ht<a} dt and Γ(T ) = sup t∈[0,ζ] H t , (18)
where ℓ a stands for the total mass of ℓ a . This implies

κ a (λ, µ) = N 1 -e -µ ℓ a -λ m(B(ρ,a)) . (19)
We recall from [START_REF] Duquesne | Random Trees, Lévy Processes and Spatial Branching Processes[END_REF] Chapter 1 (proof of Corollary 1.4.2 page 41) that

N-a.e. 1 {sup H>a} = 1 {L a ζ =0} . (20)
By letting µ go to ∞ in (15), we get

v(a) = N(Γ(T ) > a) = N( ℓ a = 0) = (γ -1)a -1 γ-1 . (21)
For any a ∈ (0, ∞), we next define the probability measure N a by setting

N a = N • Γ(T ) > a = N • ℓ a = 0 . ( 22
)
This combined with [START_REF] Duquesne | Growth of Lévy trees[END_REF] implies that

N a e -µ ℓ a = 1 -1 + 1 (γ -1)aµ γ-1 -1 γ-1 . ( 23
)
By differentiating this equality at µ = 0, one gets

N a ℓ a = (γ -1)a 1 γ-1 = 1 v(a) . (24)
The branching property. We now describe the distribution of excursions of the height process above a given level (or equivalently of the corresponding stable subtrees above this level). We fix a ∈ (0, ∞) and we denote by (l a j , r a j ), j ∈ J a , the connected components of the open set {t ∈ (0, ζ) : H t > a}. For any j ∈ J a , denote by H a,j . the corresponding excursion of H that is defined by H a,j s = H (l a j +s)∧r a ja, s ∈ [0, ∞). This decomposition is interpreted in terms of the tree as follows. Denote the closed ball in T with center ρ and radius a by B(ρ, a). Observe that the connected components of the open set T \ B(ρ, a) are the subtrees T a,o j

:= p((l a j , r a j )), j ∈ J a . The closure T a j of T a,o j is simply {σ a j } ∪ T a,o j
, where σ a j = p(l a j ) = p(r a j ), that is the point in the a-level set T (a) at which T a,o j is grafted. Observe that the rooted measured tree T a j , d, σ a j , m( • ∩ T a j ) is isometric to the tree coded by H a,j .

. We define the following point measure on [0, ∞) × C([0, ∞), R):

M a (dxdH) = j∈Ja δ (L a l a j ,H a,j ) (25)
For any s ∈ [0, ∞), we also set Ha s = H τ a s , where the time-change τ a s is given by

τ a s = inf t ≥ 0 : t 0 dr 1 {Hr≤a} > s , s ∈ (0, ∞) .
The process Ha = ( Ha s ) s≥0 is the height process below a and the rooted compact Rtree ( B(ρ, a), d, ρ) is isometric to the tree coded by Ha . Let G a be the sigma-field generated by Ha augmented by the N-negligible sets. From the approximation [START_REF] Duquesne | The exact packing measure of Lévy trees[END_REF], it follows that L a ζ is measurable with respect to G a . Recall notation N a from [START_REF] Gall | The uniform random tree in a Brownian excursion[END_REF]. The branching property at level a then asserts that under N a , conditionally given G a , M a is distributed as a Poisson point measure with intensity 1 [0,L a ζ ] (x)ℓ(dx) ⊗ N(dH). For more details, we refer to Proposition 1.3.1 in [START_REF] Duquesne | Random Trees, Lévy Processes and Spatial Branching Processes[END_REF] or to the proof of Proposition 4.2.3 [START_REF] Duquesne | Random Trees, Lévy Processes and Spatial Branching Processes[END_REF] (see also Theorem 4.2 [START_REF] Duquesne | Probabilistic and fractal aspects of Lévy trees[END_REF]). We apply the branching property to prove the following lemma. Lemma 2.1 For any a, b, λ ∈ (0, ∞), we set

Φ a,b (λ) = N a e -λm(B(ρ,a+b)) . Then, Φ a,b (λ) = v(a) -1 κ a (λ, ∞) -κ a+b (λ, 0) ,
where κ a (λ, ∞) stands for lim µ→∞ κ a (λ, µ), which is well-defined and finite.

Proof: first note that

m(B(ρ, a + b)) = ζ 0 1 {Hr <a+b} dr = ζ 0 1 {Hr<a} dr + ζ 0 1 {a≤Hr <a+b} dr = ζ 0 1 {Hr <a} dr + j∈Ja ζ a j 0 1 {H a,j r <b} dr ,
where ζ a j = r a j -l a j stands for the lifetime of H a,j . The branching property then implies that

N a e -λm(B(ρ,a+b)) G a = e -λ ζ 0 1 {Hr <a} dr exp -L a ζ N 1 -e -λ ζ 0 1 {Hr <b} dr = e -κ b (λ,0)L a ζ -λ ζ 0 1 {Hr <a} dr .
Monotone convergence implies

N a e -κ b (λ,0)L a ζ -λ ζ 0 1 {Hr<a} dr = lim µ→∞ v(a) -1 N (1 -e -µL a ζ )e -κ b (λ,0)L a ζ -λ ζ 0 1 {Hr <a} dr = lim µ→∞ v(a) -1 κ a λ, µ+κ b (λ, 0) -κ a (λ, κ b (λ, 0) ,
which easily implies the desired result thanks to [START_REF] Duquesne | The Hausdorff measure of stable trees[END_REF].

From Lemma 2.1 and the scaling property [START_REF] Duquesne | On the re-rooting invariance property of Lévy trees[END_REF], we get

Φ a,b (λ) = Φ 1,b/a a γ γ-1 λ , which implies that for any a, b ∈ (0, ∞), m B(ρ, a + b) under N a (law) = a γ γ-1 m B(ρ, 1 + a b ) under N 1 . (26)
Spinal decomposition. We recall another decomposition of the height process that is proved in [START_REF] Duquesne | Random Trees, Lévy Processes and Spatial Branching Processes[END_REF] Chapter 2 (see [START_REF] Duquesne | Probabilistic and fractal aspects of Lévy trees[END_REF] for a more specific statement and see [START_REF] Duquesne | On the re-rooting invariance property of Lévy trees[END_REF] for further applications). This decomposition is used in the proof of Proposition 1.1 and Proposition 1.3. Let h : [0, ∞) → [0, ∞) be a continuous function with compact support. Let us assume for clarity that h(0) > 0. We view h as the depth-first exploration process of a tree. Thus, the exploration starts at a vertex with height h(0) > 0 that we call the initial vertex. We obtain the subtrees grafted along the ancestral line of the initial vertex as follows: set h(s) = inf [0,s] h and denote by (l i , r i ), i ∈ I(h), the excursion intervals of hh away from 0 that are the connected components of the open set {s ∈ [0, ∞) : h(s)h(s) > 0}. For any i ∈ I(h), set

h i (s) = (h -h)((l i + s) ∧ r i ) s≥0 .
Then, the subtrees along the ancestral line of the initial vertex are coded by the functions (h i ; i ∈ I(h)), and the tree coded by h i is grafted at distance h(0)-h(l i ) from the initial vertex. We next define N (h) as the point measure on

[0, ∞) × C([0, ∞), R) given by N (h) = i∈I(h) δ (h(0)-h(li) , h i ) .
Recall that H stands for the γ-height process under its excursion measure N. For any t ∈ (0, ζ), we set Ĥt := (H (t-s)+ ) s≥0 ; here, ( •) + stands for the positive part function. We also set Ȟt := (H (t+s)∧ζ ) s≥0 , and we define the random point measure

N t on [0, ∞) × C([0, ∞), R) by N t = N ( Ĥt ) + N ( Ȟt ) := j∈Jt δ (r t j , H (t),j ) . ( 27
)
This point measure records the subtrees grafted along the ancestral line of the vertex visited at time t in the coding of T by H. Namely, set σ = p(t) ∈ T . Then, the geodesic [[ρ, σ]] is the ancestral line of σ. Denote by T o j , j ∈ J , the connected components of the open set T \[[ρ, σ]] and denote by T j the closure of T o j . Then, there exists a point

σ j ∈ [[ρ, σ]] such that T j = {σ j } ∪ T o j .
The specific coding of T by H entails that for any j ∈ J there exists a unique j ′ ∈ J t such that d(σ, σ j ) = r t j ′ and such that the rooted compact R-tree (T j , d, σ j ) is isometric to the tree coded by H (t),j ′ .

. The law of N t when t is chosen "at random" according to the Lebesgue measure is given as follows. To simplify the argument, we assume that the random variables we mention are defined on the same probability space (Ω, F , P). Let (U t ) t≥0 be a (γ -1)-stable subordinator with initial value U 0 = 0, and whose Laplace exponent is given bylog E[exp(-λU 1 )] = γλ γ-1 . Let

N * = j∈I * δ (r * j , H * j ) (28)
be a random point measure on [0, ∞) × C([0, ∞), R) such that a regular version of the law of N * conditionally given U is that of a Poisson point measure with intensity dU r ⊗ N(dH). Here, dU r stands for the (random) Stieltjes measure associated with the non-decreasing path r → U r . For any a ∈ (0, ∞), we also set

N * a = j∈I * 1 [0,a] (r * j ) δ (r * j , H * j ) . (29)
By Lemma 3.4 in [START_REF] Duquesne | Probabilistic and fractal aspects of Lévy trees[END_REF], for any nonnegative measurable functional F ,

N ζ 0 F H t , N t dt = ∞ 0 E [F (a, N * a )] da . (30)
We shall refer to this identity as to the spinal decomposition of H at a random time.

We use the spinal decomposition to compute the law of the mass measure of random balls in T . To that end, we first fix t ∈ (0, ζ) and we express m B(p(t), r) in terms of N t as follows. The definition of (N ( Ĥt ), N ( Ȟt )) then entails

m B(p(t), r) = j∈Jt 1 [0,r] (r t j ) • ζ t j 0 1 {H (t),j s <r-r t j } ds, (31)
where ζ t j stands for the lifetime of the path H * t,j . For any a ∈ (0, ∞) and for any r ∈ [0, a], we next set

M * r (a) = j∈I * 1 [0,r∧a] (r * j ) • ζ * j 0 1 {H * j s ≤r-r * j } ds , (32)
where ζ * j stands for the lifetime of the path H * j . Then, (M * r (a)) 0≤r≤a is a function of N * a . It is a cadlag non-decreasing process and the spinal decomposition (30) implies for any bounded measurable functional

F : D([0, ∞), R) → R, we have N ζ 0 F m(B(p(t), r)) r≥0 dt = ∞ 0 E F (M * r (a)) r≥0 da ,
which can be rephrased as follows in terms of the tree

N T F m(B(σ, r)) r≥0 m(dσ) = ∞ 0 E F (M * r (a)) r≥0 da . ( 33
)
For any 0 ≤ r ′ ≤ r ≤ a, we next set

M * r ′ ,r (a) = j∈I * 1 (r ′ ,r] (r * j ) ζ * j 0 1 {H * j s <r-r * j } (34)
Note that M * r (a) = M * 0,r (a). The random variables M * r ′ ,r (a) play an important role in the proof of Proposition 1.1 and Proposition 1.3. We gather in the following lemma their basic properties that are easy consequences of the definition (we refer to Lemma 2.11 and to Remark 2.12 in [START_REF] Duquesne | Hausdorff and packing measure of stable trees[END_REF] page 115 for more details). Lemma 2.2 Let us fix a ∈ (0, ∞). The following holds true.

(i) Let (r n ) n≥0 be a sequence such that 0 < r n+1 ≤ r n ≤ a and lim n→∞ r n = 0.

Then, the random variables (M * rn+1,rn (a)) n≥0 are independent. (ii) The increments of r ∈ [0, a] → M * r (a) are not independent. However, for any 0 ≤ r ′ ≤ r ≤ a, we have

M * r (a) -M * r ′ (a) = M * r ′ ,r (a) + j∈I * 1 [0,r ′ ] (r * j ) ζ * j 0 1 {r ′ -r * j ≤H * j s <r-r * j } . It implies that M * r (a) ≥ M * r (a) -M * r ′ (a) ≥ M * r ′ ,r (a). Note that M * r ′ ,r (a) is independent of M * r ′ (a). The law of M * r ′ ,r (
a) is characterised by its Laplace transform:

E exp(-λM * r ′ ,r (a)) = 1 - κ r-r ′ (λ, 0) γ λ , λ ∈ [0, ∞) . ( 35
)
Proof of (35): recall that conditionally given U , N * is a Poisson point process with intensity dU r ⊗ N. Thus,

E exp(-λM * r ′ ,r (a)) U = exp - (r ′ ,r]
dU s κ r-s (λ, 0) .

We get E[exp(-λM * r ′ ,r (a))] = exp(-γ r-r ′ 0 κ s (λ, 0) γ-1 ds). By a change of variable based on [START_REF] Duquesne | Probabilistic and fractal aspects of Lévy trees[END_REF], we obtain γ r-r ′ 0 κ s (λ, 0) γ-1 ds = log λlog λκ r-r ′ (λ, 0) , which entails (35).

We also introduce the following notation:

M * := M * 0,1 (1) = M * 1 (1) . (36)
Then, (35) and the scaling property [START_REF] Duquesne | On the re-rooting invariance property of Lévy trees[END_REF] imply that for any 0 ≤ r ′ ≤ r ≤ a,

(r -r ′ ) -γ γ-1 M * r ′ ,r (a)
(law)

= M * . (37)
We see in particular that r -γ γ-1 M * r (a) has the same law as M * . The tail at 0+ of the distribution of M * is studied in Section 3.

Balls and truncated subtrees.

Recall that (H t ) 0≤t≤ζ stands for the excursion of the γ-stable process under N and that (T , d) is the γ-stable tree coded by H. Recall that for any σ, σ ′ ∈ T , [[σ, σ ′ ]] stands for the (unique) geodesic joining σ to σ ′ . For any σ ∈ T , we set

T σ = σ ′ ∈ T : σ ∈ [[ρ, σ ′ ]] ,
that is the subtree stemming from σ. We then set Γ(T σ ) = sup σ ′ ∈Tσ d(σ, σ ′ ) that is the total height of T σ . Next, for any a, ε ∈ (0, ∞), we set

T (a) = σ ∈ T : d(ρ, σ) = a and T (a, ε) = σ ∈ T (a) : Γ(T σ ) > ε .
Since T is a compact metric space T (a, ε) is a finite subset and we set Z a (ε) = #T (a, ε). Then,

T (a, ε) = σ 1 , . . . , σ Za(ε) ,
where, the σ i is the i-th point to be visited by H. For any η ∈ (0, ∞), we set

D a,ε,η = T i ; 1 ≤ i ≤ Z a (ε)
where

T i = T σi ∩ B(σ i , η).
The T i s are the subtrees above level a that are higher than ε and that are truncated at height η. We simply call them the (a, ε)-subtrees truncated at height η.

Recall from (21) that v(ε) = N(Γ(T ) > ε) = ((γ -1)ε) -1 γ-1 . Recall from the subsection stating the branching property, that G a stands for the sigma-field generated by the height process below a. Recall from [START_REF] Gall | The uniform random tree in a Brownian excursion[END_REF] that N a = N( • | sup H > a). By the branching property at level a, (i) under N a and conditionally given G a , Z a (ε) is a Poisson random variable with parameter v(ε) ℓ a .

Moreover, each T σi is coded by an excursion above level a that is higher than ε. Therefore, (ii) conditionally given Z a (ε), the truncated subtrees T i are independent and distributed as B(ρ, η) under N ε .

Consequently, for any integer k ≥ 1, for any measurable functions

F 1 , . . . , F k : [0, ∞) → [0, ∞), we have N a 1 {Za(ε)=k} 1≤i≤k F i m(T i ) G a = (v(ε) ℓ a ) k k! e -v(ε) ℓ a 1≤i≤k N ε F i m(B(ρ, η) . ( 38
)
The distribution of m(B(ρ, η)) under N ε plays an important role in the proofs and it is studied in Section 3.

The two following lemmas are used in the proofs of Theorem 1.2 and Theorem 1.4. We first show that any ball contains a reasonably large truncated subtree. Lemma 2.3 Let r ∈ (0, 1/2). Let n r be the positive integer given by 2 -nr < r ≤ 2 -nr+1 . Let σ ∈ T be such that d(ρ, σ) ≥ 2.2 -nr-2 and let k ≥ 1 be the integer such that (k + 1)2 -nr-2 ≤ d(ρ, σ) < (k + 2)2 -nr-2 .

Then, there exists a unique truncated subtree T ∈ D k2 -nr -2 ,2 -nr-2 ,2 -nr -2 such that T ⊂ B(σ, r).

Proof: to simplify notation, we set a = k2 -nr-2 and ε = η = 2 -nr-2 . There is a unique i ∈ {1, . . . , Z a (ε)} such that σ i ∈ [[ρ, σ]] (and note that d(ρ, σ i ) = a = k2 -nr-2 ). Then,

T i = T σi ∩ B(σ i , η) ⊂ B(σ, r). Indeed, for any σ ′ ∈ T i , d(σ, σ ′ ) ≤ d(σ, σ i ) + d(σ ′ , σ i ). Since σ ′ ∈ B(σ i , η), d(σ ′ , σ i ) ≤ η = 2 -nr-2 , and since σ i ∈ [[ρ, σ]], we get d(σ, σ i ) = d(ρ, σ) -d(ρ, σ i ) ≤ (k + 2)2 -nr-2 -a = 2.2 -nr-2 .
Thus, d(σ, σ ′ ) ≤ 3.2 -nr-2 < r, which completes the proof.

Conversely, one proves that any ball is contained in a reasonably small truncated subtree.

Lemma 2.4 Let r ∈ (0, 1/2). Let n r be the positive integer given by 2 -nr < r ≤ 2 -nr+1 . Let σ ∈ T be such that d(ρ, σ) ≥ 2.2 -nr+1 and let l ≥ 1, be the integer such that (l + 1)2 -nr+1 ≤ d(ρ, σ) < (l + 2)2 -nr+1 .

Then, there exists a unique truncated subtree T ∈ D l2 -nr +1 ,2 -nr+1 ,3.2 -nr +1 such that B(σ, r) ⊂ T .

Proof: to simplify notation, we set a = l2 -nr+1 , ε = 2 -nr+1 and η = 3ε. There is a unique

i ∈ {1, . . . , Z a (ε)} such that σ i ∈ [[ρ, σ]] (and note that d(ρ, σ i ) = a = l2 -nr+1 ). We check that B(σ, r) ⊂ T i = T σi ∩ B(σ i , η). Indeed, let σ ′ ∈ B(σ, r) and let σ ∧ σ ′ be the branching point of σ and σ ′ . Namely, [[ρ, σ ∧ σ ′ ]] = [[ρ, σ]] ∩ [[ρ, σ ′ ]]. We get d(ρ, σ ∧ σ ′ ) = min d(ρ, σ), d(ρ, σ ′ ) > d(ρ, σ) -r ≥ (l + 1)2 -nr+1 -2 -nr+1 = l2 -nr+1 . Thus, d(ρ, σ i ) ≤ d(ρ, σ ∧ σ ′ ) and since σ i and σ ∧ σ ′ belongs to [[ρ, σ]], it implies that σ ′ ∈ T σi . Moreover, d(σ ′ , σ i ) ≤ d(σ ′ , σ) + d(σ, σ i ) < r + 2.2 -nr+1 ≤ 3.2 -nr+1 = η, which completes the proof.
3 Tail estimates. Recall that v(1) = N(sup H > 1) and that N 1 = N( • | sup H > 1). Recall from [START_REF] Duquesne | Random Trees, Lévy Processes and Spatial Branching Processes[END_REF] the definition of κ a (λ, µ) and Lemma 2.1 that asserts that for any c ∈ [0, ∞),

N 1 e -λm(B(ρ,1+c)) = Φ 1,c (λ) = v(1) -1 κ 1 (λ, ∞) -κ 1+c (λ, 0) . ( 39
)
Lemma 3.1 For any γ ∈ (1, 2] and any c ∈ [0, ∞), we get

-log N 1 m(B(ρ, 1 + c) ) ≤ y ∼ y→0+ γ -1 y γ-1
.

Proof: by De Bruijn's Tauberian theorem, we only need to get an equivalent to log Φ 1,c (λ) when λ goes to infinity. To that end, we use (39) and we first get an estimate of κ 1 (λ, ∞): we take a = 1 in ( 12) and we let µ go to infinity to obtain

∞ κ1(λ,∞) du u γ -λ = 1 .
We set a(λ) = log κ1(λ,∞) γ λ -1 and we use the change of variable y = log(λ -1 u γ -1) in the previous integral equation to get

γλ 1-1 γ = ∞ a(λ)
dy

(1 + e y ) 1-1 γ .
Note that a(λ) decreases to -∞ when λ goes to ∞. Thus, there exists λ 0 ∈ (0, ∞) such that a(λ) < 0, for any λ > λ 0 . Next, observe that

Q 0 (λ):= ∞ 0 dy (1 + e y ) 1-1 γ - 0 a(λ) 1 - 1 (1 + e y ) 1-1 γ dy ---→ λ→∞ Q 0 (∞) ∈ R , (40)
and that γλ 1-1 γ = Q 0 (λ)a(λ), for any λ > λ 0 . Namely,

κ 1 (λ, ∞) = λ 1 γ 1 + exp Q 0 (λ) -γλ γ-1 γ 1 γ . ( 41
)
Similarly, we get an estimate for κ 1+c (λ, 0): we take a = 1 + c and µ = 0 in (12):

κ1+c(λ,0) 0 du λ -u γ = 1 + c .
We set b(λ) =log 1 -κ1+c(λ,0) γ λ and we take y =log(1λ -1 u γ ) to get

(1 + c)γλ 1-1 γ = b(λ) 0 dy (1 -e -y ) 1-1 γ = b(λ) + Q 1 (λ) ,
where

Q 1 (λ) := b(λ) 0 1 (1 -e -y ) 1-1 γ -1 dy ---→ λ→∞ Q 1 (∞) ∈ [0, ∞) . ( 42
)
Thus,

κ 1+c (λ, 0) = λ 1 γ 1 -exp Q 1 (λ) -(1 + c)γλ γ-1 γ 1 γ . ( 43
)
By ( 39), ( 40), ( 41), ( 42) and ( 43), we get

-log N 1 e -λm(B(ρ,1+c)) ∼ λ→∞ γλ γ-1 γ
, and De Bruijn's Tauberian theorem entails the desired result (see Theorem 4.12.9 page 254 in [START_REF] Bingham | Encyclopaedia of mathematics and its applications[END_REF]).

In the Brownian case, computations are explicit: we easily derive from ( 12) that for any a, λ, µ

∈ [0, ∞) such that √ λ = µ, κ a (λ, µ) = √ λ • e a √ λ ( √ λ + µ) -e -a √ λ ( √ λ -µ) e a √ λ ( √ λ + µ) + e -a √ λ ( √ λ -µ) . ( 44
)
Recall that coth(x) = (e x + e -x )/(e xe -x ) = 1/ tanh(x), and note that v(1) = 1. Thus, (39) implies that

N 1 e -λm(B(ρ,1+c)) = √ λ coth( √ λ) -tanh((1 + c) √ λ) .
We next use the well-known formulas

x tanh(x) = n≥0 2x 2 x 2 + π 2 4 (2n + 1) 2 and x coth(x) = 1 + n≥1 2x 2 x 2 + π 2 n 2 . (45) Therefore, ∞ 0 N 1 m(B(ρ, 1 + c)) ≥ y e -λy dy = λ -1 1 -N 1 e -λm(B(ρ,1+c)) = 2 1+c n≥0 1 λ + π(2n+1) 2(1+c) 2 -2 n≥1 1 λ + π 2 n 2 .
This easily implies the following. Lemma 3.2 Consider the Brownian case: γ = 2. Then, for any y ∈ [0, ∞),

N 1 m(B(ρ, 1 + c)) ≥ y = 2 1+c n≥0 exp - π 2 (2n+1) 2 4(1+c) 2 y -2 n≥1 exp(-π 2 n 2 y) ∼ y→∞ 2 1+c exp -π 2 4(1+c) 2 y .
This result shall be used in the proof of Therem 1.4.

Tail of the distribution of M * .

Recall from (36) that M * = M * 1 (1). We set L(λ) = E exp(-λM * ) and from (35), we get

L(λ) = E exp(-λM * ) = 1 - κ 1 (λ, 0) γ λ , λ ∈ [0, ∞) . (46)
The following lemma provides an equivalent of the tail at 0+ of the distribution of M * that is used in the proof of Proposition 1.1.

Lemma 3.3 For any γ ∈ (1, 2] we have the following estimate.

lim y→0+ y -γ-1 2 exp 1 y γ-1 P M * ≤ (γ -1) y = e Cγ γ(γ -1) 2π .
where C γ is a constant given by

C γ = 1 0 u -1 (1 -u) -γ-1 γ -1 du = n≥1 1 n 1-γ γ n , (47)
Proof: first observe that for any u ∈ [0, 1), we have

(1 -u) -γ-1 γ = n≥0 (-1) n 1-γ γ n u n = 1 + n≥1 1-γ γ n u n . (48)
This easily entails the second equality in (47). For any y ∈ [0, 1), we set

F (y) = 1 y du u(1 -u) γ-1 γ
By [START_REF] Duquesne | Probabilistic and fractal aspects of Lévy trees[END_REF] and by a simple change of variable, F (L(λ)) = γλ γ-1 γ . Note that

F (y) = 1 y u -1 du + 1 0 u -1 (1 -u) -γ-1 γ -1 du - y 0 u -1 (1 -u) -γ-1 γ -1 du = -log y + C γ -h(y) .
Here

h : [0, 1] → [0, ∞) is increasing, h(0) = 0, h (1)
= C γ , and h(y) = n≥1 a n y n , where for any n ≥ 1,

h (n) (0) n! = 1 n 1-γ γ n = 1 n n k=1 1 - 1 γ k ∈ (0, 1) . (49)
Thus,

L(λ) = e Cγ exp -γλ γ-1 γ exp (-h(L(λ))) . (50)
We next use Fubini for sums of nonnegative real numbers to prove that for any y ∈ [0, 1], and any integer m ≥ 1,

h(y) m = n≥m y n • q1,...,qm≥1 q1+...+qm=n a q1 . . . a qm = n≥m y n • p1+...+pn=m p1+2p2+...+npn=n m! p 1 ! . . . p n ! a p1 1 . . . a pn n . (51)
Thus, for any y

∈ [0, 1], exp(h(y)) = 1 + n≥1 d n y n , d n = a p1 1 . . . a pn n /(p 1 ! . . . p n !)
, where the sum is over all the p 1 , . . . , p n ≥ 0 such that p 1 + 2p 2 + . . . + np n = n. Standard arguments on analytic functions imply that there exists r 1 > 0 such that

exp(-h(y)) = 1 + n≥1 c n y n , y ∈ [0, r 1 )
and (51) easily entails that for any n ≥ 1,

c n = p1,...,pn≥0 p1+2p2+...+npn=n (-1) p1+...+pn p 1 ! . . . p n ! a p1 1 . . . a pn n .
Consequently, |c n | ≤ d n , and r 1 ≥ 1, which implies that

n≥0 |c n | ≤ exp(h(1)) = e Cγ . (52)
The previous arguments and (50) imply

L(λ) = e Cγ exp -γλ γ-1 γ + n≥1 e Cγ c n exp -γλ γ-1 γ L(λ) n , λ ∈ [0, ∞) . (53)
We next introduce a non-negative random variable S defined on (Ω, F , P), that has a stable distribution whose Laplace transform is given by

E exp -λS = exp -γλ γ-1 γ
.

We use the following standard tail estimate due to Skorohod [START_REF] Skorokhod | Asymptotic formulas for stable distribution laws[END_REF] (see also Example 4.1 in [START_REF] Jain | Lower tail probability estimates for subordinators and nondecreasing random walks[END_REF]).

lim y→0+ y -γ-1 2 exp 1 y γ-1 P S ≤ (γ -1) y = γ(γ -1) 2π . (54)
We denote by q the density of S and by µ the distribution of M * . The bound (52) implies that R(dx) := e Cγ q(x)dx

+ n≥1 e Cγ c n (q * µ * n )(dx)
is a Borel signed measure on [0, ∞) whose total variation is bounded by 2 exp(2C γ ). Moreover (53), implies that [0,∞) e -λx R(dx) = L(λ), λ ∈ [0, ∞). Standard arguments on Laplace transform imply that R = µ. Denote by (Y n ) n≥1 , a sequence of i.i.d. copies of M * that are also independent of S. Since R = µ, for any y ∈ [0, ∞),

P(M * ≤ (γ -1)y) = e Cγ P(S ≤ (γ -1)y) + n≥1 e Cγ c n P(S + Y 1 + . . . + Y n ≤ (γ -1)y).
The obvious bound

P S + Y 1 + . . . + Y n ≤ (γ -1)y ≤ P S ≤ (γ -1)y P M * ≤ (γ -1)y n entails P(M * ≤ (γ -1)y) e Cγ P(S ≤ (γ -1)y) -1 ≤ n≥1 |c n | P M * ≤ (γ -1)y n -----→ y→0+ 0 ,
which entails the desired result thanks to (54).

In the Brownian case, the computations are explicit. By (44), we get

∞ 0 e -λy P(M * ≥ y) = λ -1 1 -L(λ) = λ -1 tanh 2 ( √ λ) .
Observe that tanh 2 (x) = tanh ′ (0)tanh ′ (x). If we set a n = π 2 (2n + 1) 2 /4, then (45) implies

tanh 2 (x) = n≥0 2x 2 a n (x 2 + a n ) + 4x 2 (x 2 + a n ) 2 . Thus ∞ 0 e -λy P(M * ≥ y) = n≥0 2 a n (λ + a n ) + 4 (λ + a n ) 2 ,
which easily implies the following lemma. Lemma 3.4 Consider the Brownian case: γ = 2. Then, for any y ∈ [0, ∞),

P M * ≥ y = n≥0 4 2 π 2 (2n+1) 2 + y exp -π 2 4 (2n + 1) 2 y ∼ y→∞ 4y exp -π 2 4 y .
This lemma shall be used in the proof of Proposition 1.3.

4 Proofs.

4.1 Proof of Theorem 1.2.

We fix γ ∈ (1, 2] and we consider the γ-stable tree (T , d) with root ρ coded by the γheight process H under its excursion measure N, as defined in the introduction. Recall from Section 2.2 the definition of the (a, ε)-subtrees truncated at height η, whose set is denoted by

D a,ε,η = {T i ; 1 ≤ i ≤ Z a (ε)}. Recall that f γ (r) = r γ γ-1 (log 1/r) 1 γ-1
, r ∈ (0, 1) . By Lemma 3.1, there is a constant q ∈ (0, ∞) that only depends on γ, such that for any n ≥ 4,

N 1 m(B(ρ, 1)) ≤ α(log 2 n ) -1 γ-1 ≤ exp -α -(γ-1) q log 2 n .
This inequality combined with (57) entails that for any n ≥ 4,

N V (n)1 {Γ(T )>2 -n } ≤ R 0 (γ -1) -1 γ-1 exp -α -(γ-1) q log 2 - γ γ-1 log 2 n , which implies (56) if α < (γ-1)q γ 1 γ-1 .
Since V (n) ∈ N, (56) implies that N-a.e. for any sufficiently large n, we have V (n)1 {Γ(T )>2 -n } = 0. Since N(Γ(T ) = 0) = 0, it implies that N-a.e. for any sufficiently large n, V (n) = 0. We next use (55), to get the following: there exists α γ ∈ (0, ∞), that only depends on γ, such that for any positive integer R 0 , N-a.e. ∃n 0 ∈ N s.t. ∀n ≥ n 0 , inf

σ∈B(ρ,R0) m B(σ, 2 -n+3 ) ≥ α γ f γ (2 -n ) .
Note that N-a.e. there exists R 0 such that B(ρ, R 0 ) = T . Since α γ does not depend on R 0 , it entails

N-a.e. lim inf n→∞ 1 f γ (2 -n ) inf σ∈T m B(σ, 2 -n+3 ) ≥ α γ . (58)
Upper Bound. Let R 0 be a positive integer and let h 0 ∈ (0, ∞). We also fix β ∈ (0, ∞), that is specified further. We introduce the following event

A n = Γ(T ) > h 0 ∩ inf σ∈B(ρ,R 0 ) m(B(ρ, 2 -n )) > βf γ (2 -n ) .
We assume that n ≥ 4, and that h 0 > 2 -n+1 . Let l ≥ 1 be such that (l + 1)2 -n ≤ h 0 < (l + 2)2 -n . We argue on the event A n : let σ ∈ T be such that d(ρ, σ) = h 0 ; we apply Lemma 2.4 with r = 2 -n ; thus n r = n + 1 (namely, 2 -n-1 = 2 -nr < r = 2 -nr+1 = 2 -n ); it implies that there exists a truncated subtree T ∈ D l2 -n ,2 -n ,3.2 -n such that B(σ, 2 -n ) ⊂ T . This proves

A n ⊂ B n := Γ(T ) > l2 -n ∩ ∀T i ∈ D l2 -n ,2 -n ,3.2 -n : m(T i ) > βf γ (2 -n ) . ( 59
)
To simplify notation we set Z(l, n) = Z l2 -n (2 -n ). We then use (38) to get

N(B n ) = N(Γ(T ) > l2 -n ) N l2 -n 1≤i≤Z(l,n) 1 {m(Ti)>βfγ (2 -n )} = v(l2 -n ) N l2 -n N 2 -n m(B(ρ, 3.2 -n )) > βf γ (2 -n ) Z(l,n) .
Recall that under N l2 -n , conditionally given G l2 -n , the random variable Z(l, n) has a Poisson distribution with mean v(2 -n ) ℓ l2 -n . We then get

N(B n ) = v(l2 -n ) N l2 -n exp -z n ℓ l2 -n , (60)
where we have set

z n = v(2 -n ) N 2 -n m(B(ρ, 3.2 -n )) ≤ βf γ (2 -n ) .
By [START_REF] Jain | Lower tail probability estimates for subordinators and nondecreasing random walks[END_REF] and the scaling property ( 14), we get

z n = (γ -1) -1 γ-1 2 n γ-1 N 1 m(B(ρ, 3) ≤ β(log 2 n ) -1 γ-1
.

By Lemma 3.1, there exists q, q ′ ∈ (0, ∞) that only depend on γ such that for any n ≥ 4, z n ≥ w n := q ′ exp 1 γ-1 log 2β -(γ-1) q log 2 n .

We fix β > (γ -1)q 1 γ-1 so that θ γ := 1 γ-1 log 2β -(γ-1) q log 2 > 0. Thus, w n = q ′ exp(θ γ n) → ∞, when n → ∞. By (60), we get

N(B n ) ≤ v(l2 -n ) N l2 -n exp -w n ℓ l2 -n .
Recall that h 0 > 2 -n+1 , which implies that l ≥ 1. Thus l2 -n ≥ h 0 /3. Since v decreases, v(l2 -n ) ≤ v(h 0 /3). Next, recall [START_REF] Gall | Spatial Branching Processes, Random Snakes and Partial Differential Equations[END_REF] and observe that a → N a (exp(-µ ℓ a )) is decreasing. Thus,

N(B n ) ≤ v(h 0 /3) N h0/3 exp -w n ℓ h0/3 . ( 61
)
Since lim n→∞ w n = ∞, we easily derive from ( 23) with a = h 0 /3 that

N h0/3 exp -w n ℓ h0/3 ∼ n→∞ 3 (γ-1) 2 (q ′ ) γ-1 h0 exp -(γ -1)θ γ n .
Thus, and ( 59) and (61) immediately entail n≥4 N(A n ) < ∞. By Borel-Cantelli, there exists β γ ∈ (0, ∞), that only depends on γ, such that for any h 0 , R 0 ∈ (0, ∞), N-a.e. for any sufficiently large n, 1 An = 0. Since N-a.e. 0 < Γ(T ) < ∞, one gets N-a.e. lim sup

n→∞ 1 f γ (2 -n ) inf σ∈T m B(σ, 2 -n ) ≤ β γ . ( 62
)
Since f γ is regularly varying at 0, (58) and (62) entail Theorem 1.2.

Proof of Theorem 1.4.

The proof is close to that of Theorem 1.2. Here, we fix γ = 2 and we recall that f (r) = r 2 log 1/r, r ∈ (0, 1).

Upper Bound. We fix a positive integer R 0 and a real number α ∈ (0, ∞) that is specified further. For any integer n ≥ 4, we set

W (n) = 1 {m(B(ρ,3.2 -n )≥αf (2 -n )} + 1≤l<R02 n # T i ∈ D k2 -n ,2 -n ,3.2 -n : m(T i ) ≥ αf (2 -n ) .
Arguing as previously, we apply Lemma 2.4 with r = 2 -n to prove that

W (n) = 0 =⇒ sup σ∈B(ρ,R0) m B(σ, 2 -n ) ≤ α f (2 -n ) . (63)
We next claim that it is possible to find α such that

n≥4 N W (n)1 {Γ(T )>2 -n } < ∞ . (64)
To simplify notation we first set

x n = N W (n)1 {Γ(T )>2 -n } and y n = N m(B(ρ, 3.2 -n )) ≥ αf (2 -n ) and Γ(T ) > 2 -n = v(2 -n )N 2 -n m(B(ρ, 3.2 -n )) ≥ αf (2 -n ) .
We also set Z(l, n) = Z l2 -n (2 -n ). Then, we get the following.

x n ≤ y n + 1≤l<R02 n N 1≤i≤Z(l,n) 1 {m(Ti)≥αf (2 -n )} ≤ y n + 1≤l<R02 n v(l2 -n )N l2 -n 1≤i≤Z(l,n) 1 {m(Ti)≥αf (2 -n )} ≤ y n + 1≤l<R02 n v(l2 -n ) N l2 -n Z(l, n) N 2 -n m(B(ρ, 3.2 -n )) ≥ αf (2 -n ) ≤ y n + 1≤l<R02 n v(l2 -n )N l2 -n ℓ l2 -n y n .
Here, we used (38) in the third line. Recall from (24) that v(l2 -n )N l2 -n ℓ l2 -n = 1. Thus, x n ≤ R 0 2 n y n . We next get an equivalent of y n : by ( 21) with γ = 2, we have v(2 -n ) = 2 n ; the scaling property (26) and Lemma 3.2 with c = 2 imply

y n = 2 n N 2 -n m(B(ρ, 3.2 -n )) ≥ αf (2 -n ) = 2 n N 1 m(B(ρ, 3)) ≥ α log 2 n ∼ n→∞ 2 3 exp - π 2 log 2 36 α -log 2 n . Thus, x n ≤ R 0 2 n y n ∼ n→∞ 2 3 R 0 exp - π 2 log 2 36 α -2 log 2 n
which implies (64) if α > 72/π 2 . We argue as in the proof of Theorem 1.2 to prove that (64) implies N-a.e. lim sup

n→∞ 1 f (2 -n ) sup σ∈T m B(σ, 2 -n ) ≤ 73 π 2 . ( 65
)
Lower Bound. Let R 0 be a positive integer and let h 0 ∈ (0, ∞). We also fix β ∈ (0, ∞), that is specified further. We introduce the following event

C n = Γ(T ) > h 0 ∩ sup σ∈B(ρ,R 0 ) m(B(ρ, 2 -n+3 )) < βf (2 -n )
We assume that n ≥ 4, and that h 0 ≥ 2 -n+1 . Let k ≥ 1 be such that (k + 1)2 -n ≤ h 0 < (k + 2)2 -n . We argue on the event C n : let σ ∈ T be such that d(ρ, σ) = h 0 ; we apply Lemma 2.3 with r = 2 -n+3 (and thus n r = n -2) to prove that there exists

T ∈ D k2 -n ,2 -n ,2 -n such that T ⊂ B(σ, 2 -n+3 ). Thus, C n ⊂ D n := Γ(T ) > k2 -n ∩ ∀T i ∈ D k2 -n ,2 -n ,2 -n : m(T i ) < βf (2 -n ) . ( 66
)
To simplify notation we set Z(k, n) = Z k2 -n (2 -n ). We then use (38) to get

N(D n ) = N(Γ(T ) > k2 -n ) N k2 -n 1≤i≤Z(k,n) 1 {m(Ti)<βf (2 -n )} = v(k2 -n ) N k2 -n N 2 -n m(B(ρ, 2 -n )) < βf (2 -n ) Z(k,n)
.

Recall that under N k2 -n , conditionally given G k2 -n , the random variable Z(k, n) has a Poisson distribution with mean v(2 -n ) ℓ k2 -n . We then get 1) .

N(D n ) = v(k2 -n ) N k2 -n exp -z n ℓ k2 -n , ∼ n→∞ Qc γ-1 2 (log 1/̺) -c -(γ-1) (log n) -1/2 n -c -(γ-
By Borel-Cantelli, for any c < 1, P-a.s. lim inf n→∞ M * ̺ n (a)/g γ (̺ n ) ≥ (γ -1)c. An easy argument, entails that P-a.s.

lim inf n→∞ M * ̺ n (a) g γ (̺ n ) ≥ γ -1 .
For any r ∈ (0, 1/̺), let n(r) ∈ N be such that ̺ n(r) < r ≤ ̺ n(r)-1 . Thus,

M * r (a) g γ (r) ≥ ̺ γ γ-1 log(log(̺ 1-n(r) )) log(log(̺ -n(r) )) 1 γ-1 M * ̺ n(r) (a) g γ (̺ n(r) )
.

Thus, for any ̺ ∈ (0, 1), P-a.s. lim inf r→0 M * r (a)/g γ (r) ≥ ̺ γ γ-1 (γ -1), and by letting ̺ go to 1, we get

P-a.s. lim inf r→0 M * r (a) g γ (r) ≥ γ -1 . (71)
Upper bound in (69). For any n ≥ 2, we set r n = (log n) -n and ε n = 1 {M * rn (a)≤(γ-1)gγ (rn)} , and S n = ε 2 + . . . + ε n . Then, (70) and Lemma 3.3 imply that

E [ε n ] ∼ n→∞ Q (log log n) -1 (log n) -1/2 n -1 . (72) Therefore, lim n→∞ E[S n ] = ∞. Next observe that E[S 2 n ] = E[S n ] + 2 2≤k<l≤n E[ε k ε l ] . (73)
We then use the following lemma.

Lemma 4.1 There exists a constant q ∈ (0, ∞) that only depend on γ such that for any

2 ≤ k < l, E[ε k ε l ] ≤ qE[ε k ]E[ε l ].
Proof: first recall that (U t , t ≥ 0) is a subordinator defined on (Ω, F , P) with Laplace exponent λ → γλ γ-1 . Then, recall that N * = j∈I * δ (r * j , H * j ) is a random point measure on [0, ∞) × C([0, ∞), R) defined on (Ω, F , P) such that conditionally given U , N * is distributed as a Poisson point measure with intensity dU r ⊗ N(dH). Next recall for any 0 ≤ r ′ ≤ r ≤ a, the notation The previous lemma and (73) imply that lim sup n→∞ E[S 2 n ]/(E[S n ]) 2 ≤ q. By the Kochen-Stone Lemma, we get P( n≥2 ε n = ∞) ≥ 1/q > 0. Namely, with the lower bound (71), it entails that P lim inf r→0 M * r (a)/g γ (r) = γ -1 ≥ 1/q > 0 .

M * r ′ ,r (a) = j∈I * 1 (r ′ ,r] (r * j ) ζ * j 0 1 {H * j
Standard arguments on Poisson point processes imply that lim inf r→0 M * r (a)/g γ (r) is measurable with respect to the tail sigma-field of U at 0+. By Blumenthal zero-one law, (74) entails (69), which completes the proof of Proposition 1.1.

Proof of Proposition 1.3.

Let us fix γ = 2 and let us recall that g(r) = r 2 log log 1/r, (0, e -1 ). Recall from (32) the definition of M * r (a). We only need to prove the following: for any a ∈ (0, ∞), P-a.s. lim sup Upper bound in (75). Recall from (37) that r -2 M * r (a) has the law as M * . We fix ̺ ∈ (0, 1) and c ∈ (0, ∞). By Lemma 3.4, we get P(M * ̺ n (a) ≥ cg(̺ n )) = P(M * ≥ c log log ̺ -n ) ∼ n→∞ 4c (log 1/̺) -π 2 4 c n -π 2 4 c log n .

Borel-Cantelli and an easy argument imply that P-a.s.

lim sup n→∞ M * ̺ n (a) g(̺ n ) ≤ 4 π 2 .
Let r ∈ (0, 1). There exists n(r) ∈ N such that ̺ n(r)+1 < r ≤ ̺ n(r) . Thus,

M * r (a) g(r) ≤ 1 ̺ 2 M * ̺ n(r) (a) g(̺ n(r) )
.

Consequently, for any ̺ ∈ (0, 1), we have P-a.s. lim sup r→0 M * r (a)/g(r) ≤ 4(π̺) -2 . By letting ̺ go to 1, we get for any a ∈ (0, ∞), P-a.s. lim sup Lower bound in (75). For any 0 ≤ r ′ ≤ r ≤ a, recall from (34) the definition of M * r ′ ,r (a) and recall from Lemma 2.2 that for any sequence (r n ) n≥0 such that 0 ≤ r n+1 ≤ r n ≤ a, and lim n→∞ r n = 0, the random variables M * rn+1,rn (a) , n ≥ 0, are independent. Here, we take r n = ̺ n , with ̺ ∈ (0, 1). We fix c ∈ (0, ∞) and for any n ≥ 1, we set

Proposition 1 . 3 Theorem 1 . 4

 1314 Consider the Brownian case: γ = 2. Then N-a.e. for m-almost all σ, lim sup r→0 the Brownian case, the balls with exceptionally large mass have also an exact asymptotic function as shown by the following theorem. Consider the Brownian case: γ = 2. Let us set f (r) = r 2 log 1/r , r ∈ (0, e -1 ) .

Corollary 1 . 5

 15 Consider the Brownian case: γ = 2. Then, there are two constants c, C ∈ (0, ∞) such thatN-a.e. ∃r 0 ∈ (0, ∞) : ∀r ∈ (0, r 0 ), ∀σ ∈ T , c log 1/r ≤ r -2 m B(σ, r) ≤ C log 1/r.Note that Proposition 1.1, Theorem 1.2, Proposition 1.3, Theorem 1.4 and Corollary 1.5 hold true under the normalised law N( • | ζ = 1).

0 1 0 1

 00 First, recall from (3) the definition of b(s, t) and d(s, t). Note that if H s = b(s, t) with s = t, then p(s) cannot be a leaf of T . Let us fix a radius r in [0, H t ]. Since the leaves of T have zero m-measure, we get m B(p(t), r) = ζ {d(s,t)<r} ds = ζ {0<Hs-b(s,t)<r-Ht+b(s,t)} ds.

3. 1

 1 Tail of the distribution of m(B(ρ, 1 + c)) under N 1 .

s 1 γ- 1 E[ε l ] ≤ P ( 1 - 1 γ- 1 Eγ 1 γ- 1 ∼

 1111111 ≤r-r * j } ds , where ζ * j stands for the lifetime of H * j , for any j ∈ I * . Recall from Lemma 2.2 that M * r ′ ,r (a) ≤ M * r (a) and that M * r ′ ,r (a) is independent from M * r ′ (a). Thus, observe that for any 2 ≤ k < l,M * r k (a) ≤ (γ -1)g γ (r k ) ∩ M * r l (a) ≤ (γ -1)g γ (r l ) ⊂ M * r l ,r k (a) ≤ (γ -1)g γ (r k ) ∩ M * r l (a) ≤ (γ -1)g γ (r l ) .Thus,E[ε k ε l ] ≤ P M * r l ,r k (a) ≤ (γ -1)g γ (r k ) E[ε l ] .Next recall from (37) that (r -r ′ ) -γ γ-1 M * r ′ ,r (a) has the same law as M * . Consequently,E[ε k ε l ] ≤ P (1 -(r l /r k )) γ γ-1 M * ≤ (γ -1)(log log 1/r k ) -(r k+1 /r k )) γ γ-1 M * ≤ (γ -1)(log log 1/r k ) -[ε l ] . Now observe that r k+1 /r k = (log k) -1 + O((log k) -2) and log log 1/r k = log k + log log log k .By Lemma 3.3, we getP M * ≤ (γ -1)(1 -(r k+1 /r k )) -γ γ-1 (log log 1/r k ) log log 1/r k = log k + log log log kγ + O log log log k log k , we get, P M * ≤ (γ -1)(1 -(r k+1 /r k )) -γ γ-1 (log log 1/r k ) -k→∞ e γ Q(log log k) -1 (log k) -1/2 k -1 ,which easily completes the proof of the lemma by (72).

1

 1 {lim sup r→0 m(B(σ,r))/g(r) = 4/π 2 } m(dσ) = 0 , that immediately entails Proposition 1.3.

ε n = 1

 1 M * ̺ n+1 ,̺ n (a) ≥ cg(̺ n ).The scaling property (37) and Lemma 3.4 entailE[ε n ] = P ̺ 2n (1 -̺) 2 M * ≥ cg(̺ n ) ∼ n→∞ 4c (1 -̺) 2 (log 1/̺) ̺) 2 c log n. If c ≤ 4(1 -̺) 2 π -2 , then n≥1 E[ε n ] = ∞.Since the ε n 's are independent, the usual converse of Borel-Cantelli entails that P-a.s. n≥1 ε n = ∞. Now observe that for any n ≥ 1,ε n ≤ 1 M * ̺ n (a)≥cg(̺ n ).Therefore, for any ̺ ∈ (0, 1) and any c ∈ (0, ∞) such that c ≤ 4(1 -̺) 2 π -2 we P-a.s.It easily entails the desired lower bound. It completes the proof of (75) and that of Proposition 1.3.

* This research was supported by the grant ANR A3, Projet BLAN-****

Lower Bound. Let fix a positive integer R 0 and a real number α ∈ (0, ∞), that is specified further. For any integer n ≥ 4, we set

We first prove that

Indeed, we apply Lemma 2.3 with r = 2 -n+3 . Thus, n r = n -2 and we have 2 -nr < r = 2 -nr+1 . Let σ ∈ T be such that d(ρ, σ) ≤ R 0 . We first consider the case where

We next consider the case where d(ρ, σ) < 2 -n+1 . Then it is easy to see that

We next claim that it is possible to find α such that

We first set

To simplify notation, we also set

We get the following.

Recall from the definition of the branching property that G k2 -n stands for the sigmafield generated by the tree below k2 -n . Recall from Section 2.2 that conditionnally given G k2 -n , Z(k, n) is a Poisson random variable with parameter v(2 -n ) ℓ k2 -n and by (38), we get

We next use [START_REF] Gall | Branching processes in Lévy processes: the exploration process[END_REF] that implies v(k2 -n )N k2 -n ℓ k2 -n = 1. Thus, we get x n ≤ R 0 2 n y n . Namely,

By [START_REF] Jain | Lower tail probability estimates for subordinators and nondecreasing random walks[END_REF],

γn γ-1 and the scaling property [START_REF] Miermont | Self-similar fragmentations derived from the stable tree II: splitting at nodes[END_REF] implies

where we have set

We next apply ( 21) and ( 23) with γ = 2 to get v(a) = 1/a and N a (exp(-µ ℓ a )) = (1 + aµ) -1 . Since h 0 ≥ 2 -n+1 , we have k2 -n ≥ h 0 /3 and we get

We next apply ( 14) and Lemma 3.2 with c = 0, to obtain

Then, for any β < 4/π 2 , (66) and (67) entail that n≥4 N C n < ∞. Thus, for any h 0 , R 0 ∈ (0, ∞), N-a.e. for any sufficiently large n, 1 Cn = 0. Since N-a.e. 0 < Γ(T ) < ∞, one easily gets

Since f is regularly varying at 0, (65) and (68) entail Theorem 1.4.

Proof of Proposition 1.1.

Let us fix γ ∈ [START_REF] Abraham | Fragmentation associated to Lévy processes using snake[END_REF][START_REF] Aldous | The continuum random tree I[END_REF]. Recall that

, r ∈ (0, e -1 ) .

Recall from (32) the definition of M * r (a). We only need to prove the following: for any a ∈ (0, ∞), P-a.s. lim inf Lower bound in (69). We fix a ∈ (0, ∞). Recall from (36) and (37) that for any r ∈ (0, a], r -


Dynamic Visual Servoing with Image Moments for a Quadrotor Using a Virtual Spring Approach

Ryuta Ozawa and Franc ¸ois Chaumette,

Abstract-This paper presents an image-based visual servoing for controlling the position and orientation of a quadrotor using a fixed downward camera observing landmarks on the level ground. In the proposed method, the negative feedback to the image moments is used to control the vertical motion and rotation around the roll axis. On the other hand, the negative feedback cannot be used to control the horizontal motion due to under-actuation of a quadrotor. Thus, a novel control method is introduced to control the horizontal motion. Simulations are presented to validate the proposed method.

I. INTRODUCTION

Control of unmanned aerial vehicles (UAV) is an active research topic with civilian applications including search and rescue, wild fire monitoring, traffic monitoring, pipeline patrol and so on [START_REF] Degarmo | Issues concerning integration of unmanned aerial vehicles in civil airspace[END_REF]. Attitude control [START_REF] Tayebi | Attitude stabilization of a VTOL quadrotor aircraft[END_REF] and pose control [START_REF] Hamel | Dynamic modelling and configuration stabilization for an X4-flyer[END_REF] of UAVs have become increasingly important to be applied to those applications, gyroscopes are used for attitude control and inertial sensors or GPSs are often used to sense the current pose. However, it is difficult to obtain accurate current position using those sensors [START_REF] Thrun | Probabilistic robotics[END_REF]. A camera is a candidate to know the current state, and visual servoing is a powerful tool to control the pose using a camera. Visual servoing can be divided into two main classes [START_REF] Chaumette | Visual servo control part I: Basic approaches[END_REF]; position-based visual servoing (PBVS) and image-based visual servoing (IBVS). PBVS requires an accurate kinematic model to estimate the pose and is sensitive to image measurement errors and kinematic model errors. On the other hand, IBVS uses image features directly, and is less sensitive to those errors than PBVS, though it has its own problems. It is generally more difficult to build controllers with IBVS than those with PBVS due to the complexity of kinematics between control variables and image features.

Much of the existing researches for visual servoing of UAVs have used PBVS [START_REF] Altug | Control of a quadrotor helicopter using visual feedback[END_REF], [START_REF] Mejias | Visual servoing of an autonomous helicopter in urban areas using feature tracking[END_REF], [START_REF] Azrad | Visual servoing of quadrotor micro-air vehicle using color-based tracking algorithm[END_REF]. An image-based PID controller using a stationary camera has been proposed to control the pose of a helicopter under the assumption the roll and pitch motion can be neglected [START_REF] Watanabe | Imagebased visual PID control of a micro helicopter using a stationary camera[END_REF]. The desired pose is calculated using the interaction matrix, and a PID controller is used to converge to this pose. An IBVS of an under-actuated system with an eye-in-hand camera has been designed with passivity-like properties in spherical image space and the measurement of translational velocity [START_REF] Hamel | Visual servoing of an under-actuated dynamic rigid-body system: An image-based approach[END_REF]. Based on the passivity-like properties, a new visual error term has been considered to design a dynamic controller [START_REF] Guenard | A practical visual servo control for an unmanned aerial vehicle[END_REF]. Both the controllers stabilize the position with backstepping method, but an extra controller is needed to control the yaw angle. Bourquardez et al. [START_REF] Bourquardez | Image-based visual servo control of the translation kinematics of a quadrotor aerial vehicle[END_REF] have compared several kinematic IBVS algorithms experimentally using a quadrotor. They found that perspective image moments are useful to design a IBVS translational control law, but the stability problem is still open. On the other hand, in the case of the spherical image moment, the stability problem has been solved but the behavior on the vertical axis is not good.

This paper proposes an image based visual servoing for controlling the pose of a quadrotor, which is a UAV with four fixed propellers and is often used for the UAV research [START_REF] Altug | Control of a quadrotor helicopter using visual feedback[END_REF], [START_REF] Azrad | Visual servoing of quadrotor micro-air vehicle using color-based tracking algorithm[END_REF], [START_REF] Bouabdallah | Design and control of an indoor micro quadrotor[END_REF], [START_REF] Guenard | A practical visual servo control for an unmanned aerial vehicle[END_REF]. We assume that a camera is fixed to the quadrotor and is directed downward to observe landmarks on the level ground [START_REF] Tahri | Point-based and region-based image moments for visual servoing of planar objects[END_REF]. Perspective image moments are then computed from these landmarks. In the proposed method, the transposed Jacobian control [START_REF] Takegaki | A new feedback method for dynamic control of manipulators[END_REF] is used to control the pose of the quadrotor. It can be applied to control the translation along and the rotation around the vertical axis. However, it cannot be applied to control the translations in the horizontal plane due to the under-actuation of a quadrotor. To control these translations, positive image feature feedback method with virtual spring is proposed based on the kinematic properties of a quadrotor. Some benefits of the proposed approach are that it is compact, requires small amounts of computation and sensor information, does not need a supplementary method for controlling the yaw rotation, and does not neglect the rotational dynamics of the quadrotor. Previous approaches use the knowledge of the translational velocity [START_REF] Altug | Control of a quadrotor helicopter using visual feedback[END_REF]- [START_REF] Guenard | A practical visual servo control for an unmanned aerial vehicle[END_REF] and the rotational matrix between the base and UAV frames, which includes rotation around yaw axis [START_REF] Altug | Control of a quadrotor helicopter using visual feedback[END_REF], [START_REF] Mejias | Visual servoing of an autonomous helicopter in urban areas using feature tracking[END_REF], [START_REF] Hamel | Visual servoing of an under-actuated dynamic rigid-body system: An image-based approach[END_REF], [START_REF] Guenard | A practical visual servo control for an unmanned aerial vehicle[END_REF]. On the other hand, the proposed approach only needs the knowledge of the angular velocity and the unit gravity vector, which does not contain the yaw rotation, in the body frame, and does not require any translational velocity.

Simulation results are finally presented to validate the effectiveness and robustness of the controller.

II. MODELING OF A QUADROTOR

First, we consider the kinematics of a quadrotor as shown in Fig. 1. Let x = (x, y, z) and θ = (θ 1 , θ 2 , θ 3 ) be the position and the orientation of the quadrotor in a static reference frame Σ w , and w R r (θ) be the rotational matrix from Σ w to Σ r , referred to as roll, pitch, and yaw. Then, the velocity of the quadrotor r v can be described as r v = r R w (θ) ẋ, and the angular velocity of the quadrotor is given

as r Ω Ω Ω = [ r ω] × = r R w (θ) w Ṙr (θ, θ) = -ṙR w (θ, θ) r R w T (θ). (1)
The vector expression of the angular velocity (1) is given as

r ω = G(θ) θ with G(θ) = ⎡ ⎣ 1 0 sinθ 2 0 cosθ 1 -sin θ 1 cos θ 2 0 sinθ 1 cos θ 1 cos θ 2 ⎤ ⎦ . (2)
Thus,

r ż = r v r ω = V ż, where V = r R w 0 0 0 0 0 0 G(θ) , (3)
and ż = ( ẋ, θ). The Lagrangian of the quadrotor is given as follows:

L = 1 2 m r v 2 + 1 2 r ω T Î r ω + mg T x. ( 4
)
where m is the mass of the quadrotor, Î is the inertia moment, g = (0, 0, -g r ) is the gravitational vector. Then, by solving the variational problem using Eq. ( 4) [START_REF] Goldstein | Classical Mechanics[END_REF] and by adding damping terms, the dynamical equation of the quadrotor is given as follows:

mI 3 0 0 0 0 0 0 H(θ) ẍ θ + 0 0 0 1 2 d dt H(θ) θ + S(θ, θ) θ - mg 0 0 0 + b 1 b 2 = V T A T u, (5)
where H(θ) = G T (θ) ÎG(θ), u is the force generated by the propellers, S(θ, θ) is the skew-symmetric matrix, b 1 and b 2 are the translational and rotational friction components, which are modeled as

b 1 = d 1 ⎡ ⎣ ẋ ẋ ẏ ẏ ż ż ⎤ ⎦ , and b 2 = d 2 ⎡ ⎣ θ1 θ1 θ2 θ2 θ3 θ3 ⎤ ⎦ , (6)
A = ⎡ ⎢ ⎢ ⎣ 0 0 1 0 - κ 0 0 1 0 -κ 0 0 1 0 κ 0 0 1 - 0 -κ ⎤ ⎥ ⎥ ⎦ , ( 7
)
is the length of the moment arm, and κ is a ratio between the translational force to the rotational torque. The rank of the matrix A is four so that the system is under-actuated.

It is important to understand the feature of the driving force for designing a controller for the quadrotor. Fig. 2 shows the explanation of the net force/torque acted on the quadrotor. The translational movement in the z axis is generated as shown in Fig. 2 (a). The summation of all the propellers is equivalent to the translational force. The rotational motions around the x, y and z axes are shown in Fig. 2 (b),(c) and (d) respectively. The rotation around the x and y axes are generated by the difference between two propellers. Each propeller generates the reaction torque to the quadrotor, and the summation of those reactive torque generates the rotation around the z axis.

Thus, the dynamics of quadrotor given by ( 5) can be rewritten as follows (see, e.g., [START_REF] Guenard | A practical visual servo control for an unmanned aerial vehicle[END_REF]),

m ẍ + b 1 -mg = w R r r ê τ 1 (8) H(θ) θ + 1 2 Ḣ(θ) + S(θ, θ) θ + b 2 = G T (θ)τ 2 , ( 9
)
where r ê = (0, 0, 1) is the unit vector of the direction of the translational driving force, τ 1 is the translational force for quadrotor and τ 2 is the rotational torque w.r.t. Σ r . From Eq. ( 2), note that the dynamical equations ( 8) and ( 9) are effective in the case that θ 2 does not cross ±π/2 (rad). The quaternion representation [START_REF] Tayebi | Attitude stabilization of a VTOL quadrotor aircraft[END_REF] is useful to avoid this singularity. However, in the case that θ 2 = ±π/2, the quadrotor lies in the vertical plane and cannot be controlled anymore. Thus, in this paper, we treat the behavior of the quadrotor only in the region where -π/2 < θ 2 < π/2.

III. IMAGE SPACE

First we consider the general framework of image errors, and interaction matrices [START_REF] Chaumette | Visual servo control part I: Basic approaches[END_REF]. Let s i and s * i be the current and the desired image feature of the i th landmark, Δs i = s i -s * i be the image error, c ż = ( c v , c ω) be the camera velocity in the camera coordinates Σ c . Then,

Δ ṡi = L si c ż = L si1 L si2 c ż , ( 10
)
where L si is the interaction matrix. The camera velocity is given as

c ż = c S r V ż, (11)
where

c S r = c R r -c R r [ r t c ] × 0 0 0 c R r , ( 12
) c R r = diag.(1, -1, -1)
and r t c is the position vector of the camera from the origin of Σ r w.r.t. Σ r . Thus, from Eqs. ( 10), ( 11) and ( 12),

Δ ṡi = P i V ż = P i1 P i2 V ż (13)
where

P i1 = L si1 c R r , and P i2 = -L si1 c R r [ r t c ] × + L si2 c R r .
We now give the interaction matrices of perspective image moments as shown in [START_REF] Tahri | Point-based and region-based image moments for visual servoing of planar objects[END_REF]. As will be described later, the translational parts of these matrices are independent each others, and this property is useful to design our new controller. Let n points form a landmark on the level ground. Then, the moments are defined by:

m ij = n k=1 x i k y j k , ( 14
)
where (x i , y i ) is the position of the i th landmark in image space. Let (x g , y g ) be the center of gravity of the landmarks in the image. The centered moments are defined by:

μ ij = n k=1 (x k -x g ) i (y k -y g ) j , ( 15
)
where x g = m 10 /n and y g = m 01 /n, m 00 = n. Let a be defined by

a = μ 20 + μ 02 . ( 16
)
As the visual features to control the translation, we choose s n = (x n , y n , a n ) which are given by

x n = a n x g , y n = a n y g , a n = Z * a * a , ( 17
)
where a * and Z * are the value of a and the depth when the quadrotor is in the desired configuration. Then,

Δ ṡn = P sn1 P sn2 V ż. ( 18
)
where, when the points are coplanar and parallel to the image plane [START_REF] Tahri | Point-based and region-based image moments for visual servoing of planar objects[END_REF], -1, -1), thus we can see that these translational motions are independent each other. It will be used to control the translations in the horizontal plane.

P sn1 = -c R r and P sn2 = c R r [ r t c ] × + L sn2 c R r , L sn2 = ⎡ ⎣ L xn2 L yn2 L an2 ⎤ ⎦ = ⎡ ⎣ a n 11 -a n (1 + 12 ) y n a n (1 + 21 ) -a n 22 -x n -a n 31 a n 32 0 ⎤ ⎦ , with ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 11 = n 11 + x g (
n ij = μ ij /m 00 . c R r = diag.(1,
The third element of Δs n , Δa n , is important to control the translation on the vertical axis. For Δa n , the interaction matrix is given as follows,

P an = P an1 P an2 = r êT r êT [ r t c ] × + L an2 c R r . ( 19
)
Let α be the object orientation angle defined as

α = 1 2 tan -1 2μ 11 μ 20 -μ 02 . ( 20
)
The time derivative of Δα = αα * is

Δ α = P α V ż, (21)
when the points are coplanar and parallel to the image plane, where,

P α = P α1 P α2 = 0 0 0 L α2 c R r , ( 22
)
with

L α2 = α wx α wy -1 , ( 23
)
and 2 11 , with β = 4 and γ = 2. P α1 is the zero matrix, and we can see that α is independent from the translational motion. In the next section, we design a controller based on the visual features selected, that is s n and α.

α wx = (β[μ 12 (μ 20 -μ 02 ) + μ 11 (μ 03 -μ 21 )] + γx g [μ 02 (μ 20 -μ 02 ) -2μ 2 11 ] + γy g μ 11 [μ 20 -μ 02 ])/d, α wy = (β[μ 21 (μ 02 -μ 20 ) + μ 11 (μ 30 -μ 12 )] + γx g μ 11 [μ 20 + μ 02 ] + γ[μ 20 (μ 02 -μ 20 ) -2μ 2 11 ])/d, d = (μ 20 -μ 02 ) 2 + 4μ

IV. CONTROLLER DESIGN FOR A QUADROTOR

The control objective is to make a quadrotor stabilize the point such that (s n , α, θ 1 , θ 2 )=(s * n , α * , 0, 0), where θ 1 and θ 2 are roll and pitch angles of the quadrotor. The desired visual features s * n and α * can be chosen to reach any particular configuration of the quadrotor such that it is parallel to the level ground.

To control the quadrotor, we need to sense the unit vector of the gravity, which contains the roll and pitch angles of the quadrotor, while previous approaches need to sense w R r [START_REF] Altug | Control of a quadrotor helicopter using visual feedback[END_REF]- [START_REF] Guenard | A practical visual servo control for an unmanned aerial vehicle[END_REF] , which also requires the yaw angles. It is defined as

r y sen = r R w e g , (24)
where e g = (0, 0, -1) is the gravitational unit vector w.r.t. Σ w . The time derivative is calculated using Eqs. ( 1) and ( 2) as

r ẏsen = r y sen × G(θ) θ. (25)
To control the translational and rotational motions in and around the vertical axis in Fig. 3, we consider the case where the quadrotor is parallel to the level ground above the target. We will see in the simulation results that the controller is also effective when it is not the case.

Here, we design a controller based on a task-space PD feedback with gravity compensation [START_REF] Takegaki | A new feedback method for dynamic control of manipulators[END_REF] to control the translational and rotational motion on the z axis using the interaction matrices (19) and ( 22) as follows,

τ 1 = - mg r r êT r y sen -γ 1 Δa n , ( 26
)
τ 2 = v -γ 1 P T α2 Δa n -γ 2 P T α2 Δα -γ 3 G(θ) θ. ( 27
)
The first term in τ 1 is for the gravitational compensation. The second terms of τ 1 and τ 2 are the visual feedback terms to eliminate the vertical error Δa n . As shown in Eq. (18), Δa n affects only in the z direction of the translational motion. The third term in τ 2 is the image feedback of Δα. It does not affect τ 1 because the first three columns of (22) are zeros. The last term in τ 2 is the rotational damping. v in τ 2 is an additional control input, and we will discuss it later.

Next, we consider the translational motion in the horizontal plane. The sideway motion cannot be generated using the Image error and the orientation of the quadrotor. The error can be eliminated by moving on the left, and the quadrotor must be inclined to move on the left. (b) Positive visual feedback effect. It provides translational force, but the quadrotor over-rotates and the system become unstable. A virtual spring approach prevents from turning the UAV over, and is effective to eliminate the position errors. image feedback directly due to under-actuation of a quadrotor. As shown in Fig. 4 (a), when the quadrotor inclines around the y axis and generates the gravity compensation force, the vertical force is canceled and the translational force in the x direction is generated implicitly. Then, the quadrotor can move in the x direction. (For y translational force, see Fig. 4 (b).) We now consider the relationship between the image error and the translational motion. As shown in Fig. 5(a), the quadrotor would like to move to the left for eliminating the image error (Phase 1). The quadrotor inclines to generate the positive force in x direction, and the image error becomes larger (Phase 2). Then, the quadrotor moves till the image error is eliminated (Phase 3). Thus, to generate the translational force in the x direction, the image error must be fed back positively as shown in Fig. 5 (b) left. However, the positive visual feedback makes the system unstable, and we must prevent the quadrotor from giving the overthrow so that a virtual spring is added as shown in Fig. 5 (b) right. Thus,

v = v 1 + v 2 (28)
is composed of the virtual spring force v 1 and the positive feedback v 2 .

To realize the virtual spring effect, we consider the following artificial potential.

U k = 1 2 Δ r y sen T KΔ r y sen , ( 29
)
where Δ r y sen = r y sen -e g . (30)

Then, from Eq. ( 25),

v 1 = - ∂U k ∂θ T = G(θ) T [ r y sen ] × KΔ r y sen . ( 31
)
To design the positive feedback of the image error, we consider the static equilibrium in Eq. ( 8) as follows,

mg -w R r r ê mg r r êT r y sen -γ 1 w R r r ê Δa n (32) = f g -γ 1 Δa n w R r P T an1 = 0 0 0, ( 33
)
where f g is the effective force in Fig. 4 and is defined as

f g = mg r w R r r y sen -r ê 1 r êT r y sen . ( 34
)
If f g is generated by a virtual force h v = γ 0 Δs n in the image space as a result of the image error Δs n , then it could be expressed using the transpose of the interaction matrix as

f g G T (θ)v 21 = V T P T sn γ 0 h v = w R r P T sn1 G(θ) T P T sn2 h v , ( 35
)
where v 21 is a component of v 2 . However, f g cannot be generated as a result of h v due to under-actuation of the quadrotor.

Next we consider the relationship between the rotation and the image error as shown in Fig. 5. The quadrotor inclines by the image error, and we can find the torque v 22 which is equivalent to h v . If the translational motion of the quadrotor can be neglected, then from the principle of the virtual work,

δθ T G T v 22 -δ(Δs n ) T h v = 0. (36)
From the first raw of Eq. ( 35), h v can be calculated as

h v = ( w R r P T sn1 ) -1 f g = -c R r r R w f g = -mg r c R r r y sen -r ê 1 r êT r y sen . ( 37
)
From h v = γ 0 Δs n , the virtual displacement of Δs n can be calculated as

δ(Δs n ) = δ 1 γ 0 h v = 1 γ 0 ∂h v ∂θ δθ = 1 γ 0 H v Gδθ ( 38
)
where

H v = -mg r c R r I 3 + r ê r êT ( r êT r y sen ) 2 [ r y sen ] × . (39)
Then, Eq. (36) can be calculated using Eqs. (37)-(39) as

δθ T G T v 22 -δ(Δs n ) T h v = δθ T v 22 -δ( 1 γ 0 h v ) T γ 0 Δs n = δθ T G T v 22 -H T v Δs n = 0 0 0.
The virtual displacement δθ is an arbitrary vector, and we can see that

v 22 = H T v s n . ( 40
)
Substituting Eqs. ( 35) and (40) into Eq. ( 28), we can get

v 2 = P T sn2 h v + H T v Δs n . (41)
By substituting Eq. (41) into Eq. ( 27), the control input for the rotational part can be calculated as follows,

τ 2 = [ r y sen ] × KΔ r y sen + H T v ( r y sen )Δs n + P T sn2 h v -γ 1 P T an2 Δa n -γ 2 P T α2 Δα -γ 3 G θ. (42)
The first term of τ 2 is the virtual spring term, the second and the third terms are for the positive feedback, the fourth and the fifth terms are negative feedback of Δa n and Δα, and the last term is the damping of the angular velocity, respectively. The damping term can be omitted when the quadrotor is enough damped by the natural damping b i (i = 1, 2). Note that the controllers ( 26) and (42) use only the sensing of r y sen , the landmarks and the angular velocity θ. G is function of θ, but uses only the roll and the pitch angles (see Eq. ( 2)), which can be calculated from r y sen . We omit the stability analysis due to space limitation, but we can prove that the equilibrium point (Δs n , Δα, θ 1 , θ 2 , ż) = 0 0 0 of the system ( 8) and ( 9) controlled by ( 26) and ( 27) is locally asymptotically stable by the Lyapunov's direct method in the vicinity of the equilibrium point where the UAV is parallel to the image plane.

Note that the interaction matrices (18), (21), and ( 22) are obtained under the assumption that the visual feature points lie in the plane parallel to the image plane. The robustness of these approximations will be confirmed in the simulations.

V. SIMULATION

In this section, five selected simulations are presented to validate the effectiveness of the proposed controller using the conditions given in TABLE I. The following parameters are used for the simulations; m = 0.90 (kg), = 0.30 (m), Î = (1.20, 1.20, 2.00) × 10 -3 (kgm 2 ), = 0.30 (m), and κ = 0.0158 (m). The visual data is updated every 33 (ms) except the case 4, while r y sen and θ are measured and the control input is updated every 1 (ms). We use four feature points, which are distributed around the origin of the base frame. As shown in TABLE II, in Cases 1 to 4, the initial configuration is assigned so that the quadrotor goes from the vicinity to the above of the base frame. In Case 5, the initial configuration is assigned so that the quadrotor goes away from the above of the base frame. The differences in each configuration are shown in the image sampling time, the feedback gain γ 3 and the coefficient of the damping terms [START_REF] Altug | Control of a quadrotor helicopter using visual feedback[END_REF]. Figs. 6-10 show the translation and orientation errors of each case as well as the image features error and the image points trajectory. In Case 1, a good damping is used, and the errors converge to zero around 15(s). In Case 2, less damping in the translational directions is used. Then, the system is under-damped and the convergence is retarded compared to Case 1. In Case 3, less damping in the rotational directions is used. Then, residual errors remain and the motion becomes vibrationally as in a limit cycle. This is due to the sampling time of the visual data. This motion is improved using 1(ms) image sampling time as shown in Case 4, as well as using A B (100, 0.5) 1.5 33 † Point A is that x =(0, 0, 0,5)(m) and θ3 =0(rad), and Point B is that x =(0.5, 0.5, 5.5)(m) and θ3 = 0.2π(rad). larger rotational damping as shown in Case 1. In Case 5, the convergence is a little bit slower than in Case 1 because of the difference of the desired configuration. We can get the better performance when the landmarks are just below the quadrotor at a desired configuration. Indeed, the interaction matrix plays role of the moment arm of the image error, and the desired configuration in Case 5 becomes more sensitive to the image error than in Case 1-4.

VI. CONCLUSION

This paper proposed an image-based visual servoing for controlling the position and the orientation of a quadrotor with a fixed camera, which points the downward direction. The transpose Jacobian control from image moments is used to control the translational and the rotational motions in the vertical axis. On the other hand, the sideway motion cannot be generated with this method due to the under-actuation of quadrotors. Positive feedback structure of the image errors to the sideway motion was revealed, and a virtual spring was introduced to stabilize the pose of the quadrotor. Simulations showed that this method is effective to control the pose of a quadrotor, even if the low sampling rate of the visual data may cause some residual errors. We show also that it is better to put the landmarks just below the desired pose of the quadrotor. Future works will be devoted to realize experiments on a real platform. The current controller depends on the perspective image moment, and the controller is also undergoing to be generalized to algorithms, which can use any other image features, and camera configuration.

9 Fig. 1 .

 91 Fig. 1. Coordinates of a quadrotor. R. Ozawa is with the Department of Robotics, Ritsumeikan University, Shiga, 525-8577 JAPAN e-mail: ryuta@se.ritsumei.ac.jp. F. Chaumette is with INRIA Rennes-Bretagne Atlantique and IRISA, 35 042 Rennes, France e-mail: Francois.Chaumette@irisa.fr.

Fig. 2 .

 2 Fig. 2. The effects of the propellers on the motion of the quadrotor. The red arrows indicate increment of force/torque generated by the propellers and acted on the quadrotor from the equilibrium. The black thick arrows indicate the net force/torque acted on the quadrotor. The internal forces are eliminated in this figure. (a) Translational motion in z direction. (b) Rotational motion around x axis. (c) Rotational motion around y axis. (d) Rotational motion around z axis. As shown in the circle, the action to rotate the propeller generates the reaction torque, which is used to rotate the quadrotor around z axis.

Fig. 3 .

 3 Fig. 3. Translational and rotational motions of the quadrotor in the vertical plane.(a) translational motion, (b) rotational motion.

Fig. 4 .Fig. 5 .

 45 Fig. 4. The effect of Rotation for translational motion in horizontal plane. (a) x direction, (b) y direction (a)

Fig. 6 .

 6 Fig. 6. Simulation results in Case 1. (a) Position error, (b) orientation error, (c) image error, and (d)image trajectories.

Fig. 7 .

 7 Fig. 7. Simulation results in Case 2, configured as Fig.6.

Fig. 8 .

 8 Fig. 8. Simulation results in Case 3, configured as Fig.6.

Fig. 9 .

 9 Fig. 9. Simulation results in Case 4, configured as Fig.6.

Fig. 10 .

 10 Fig. 10. Simulation results in Case 5, configured as Fig.6.

  y g -31 ),[START_REF] Bourquardez | Image-based visual servo control of the translation kinematics of a quadrotor aerial vehicle[END_REF] = n 20 + x g (x g -32 ), 21 = n 02 + y g (y g -31 ), 22 = n 11 + y g (x g -32 ), 31 = y g + (y g μ 02 + x g μ 11 + μ 21 + μ 03 )/a, 32 = x g + (x g μ 20 + y g μ 11 + μ 12 + μ 30 )/a, and

We would like to thank Dr. Nicolas Guenard from CEA/LIST for his valuable information of the parameters used in the simulation.