Xan Duhalde
email: xan.duhalde@upmc.fr

Uniform Hausdorff measure of the level sets of the Brownian tree

Keywords: AMS 2000 subject classifications: Primary 60G57, 60J80 Secondary 28A78 Brownian tree, Hausdorff measure, CRT, Brownian excursion, local time

Uniform Hausdorff measure of the level sets of the

Brownian tree.

1 Introduction.

The Continuum Random Tree was introduced by Aldous [START_REF] Aldous | The continuum random tree[END_REF] as a random compact metric space (T 1 , d, m 1 ), endowed with a mass measure m 1 such that almost surely m 1 (T 1 ) = 1. It appears as the scaling limit of a large class of discrete models of random trees, and can be alternatively encoded by a normalised Brownian excursion (see Le Gall [START_REF] Gall | Brownian excursions, trees and measure-valued branching processes[END_REF]). This encoding procedure will be the viewpoint of the present paper, but for the sake of simplicity, we will not ask the total mass to be equal to one. Instead, we work on the tree encoded by a Brownian excursion (e t , t 0), under its excursion measure N. Let us mention that our result remains true for the CRT.

The Brownian tree has a distinguished vertex ρ called the root, so it makes sense to define, for all a ∈ (0, ∞) the a-level set T (a) = {σ ∈ T : d(ρ, σ) = a}. Moreover, one can define the collection of measures (ℓ a (dσ), σ ∈ T , a ∈ (0, ∞)), as the image of the local times on the levels of the excursion. Those measures are called local time measures. Indeed, N-a.e. for all a ∈ (0, ∞), the topological support of ℓ a is included in T (a). Duquesne and Le Gall [START_REF] Duquesne | The Hausdorff measure of stable trees[END_REF] showed that for a fixed level a, one has N-a.e. ℓ a (•) = cH g ( • ∩ T (a)),

where H g stands for the Hausdorff measure associated with the gauge function g(r) = r log log 1/r and c ∈ (0, ∞) is a multiplicative constant. In this paper, we prove that c = 1 2 and that the result holds N-a.e. simultaneously for all levels a. Let us mention that the value 1 2 depends on the normalisation chosen for the excursion measure N. A similar result has been obtained by Perkins [START_REF] Perkins | A space-time property of a class of measure-valued branching diffusions[END_REF][START_REF] Perkins | The Hausdorff measure of the closed support of super-Brownian motion[END_REF] for Super Brownian Motion. Briefly, let (Z a , a 0) a version of this measure-valued process on R d , defined on (Ω, F, P). Perkins proves that if the dimension d of the space is such that d 3 (which corresponds to the supercritical dimension case), there exists two constants c d , C d in (0, ∞), only depending on d such that the following holds

P-a.s. ∀a ∈ (0, ∞) c d H g (• ∩ Supp(Z a )) Z a (•) C d H g (• ∩ Supp(Z a )) , (2)
where Supp(Z a ) is the topological support of the measure Z a and H g is the Hausdorff measure associated to the gauge function g(r) = r 2 log log 1/r. In this paper, we use the ideas and techniques of [START_REF] Perkins | A space-time property of a class of measure-valued branching diffusions[END_REF][START_REF] Perkins | The Hausdorff measure of the closed support of super-Brownian motion[END_REF] to get a result similar to [START_REF] Bertoin | Lévy processes[END_REF], an equality being accessible in the setting of trees.

Before stating formally our result, let us recall precisely basic facts. A metric space (T, d) is a real tree if and only if the following two properties hold for any σ 1 , σ 2 in T :

(i) There is a unique isometric map

f σ 1 ,σ 2 from [0, d(σ 1 , σ 2 )] into T such that f σ 1 ,σ 2 (0) = σ 1 and f σ 1 ,σ 2 (d(σ 1 , σ 2 )) = σ 2 . We set σ 1 , σ 2 = f σ 1 ,σ 2 ([0, d(σ 1 , σ 2 )]
) that is the geodesic path joining σ 1 and σ 2 .

(ii) If q is a continuous injective map from [0, 1] into T , such that q(0) = σ 1 and q(1) = σ 2 , we have q([0, 1]) = f σ 1 ,σ 2 ([0, d(σ 1 , σ 2 )]).

If σ 1 ∈ ρ, σ 2 , we will say that σ 1 is an ancestor of σ 2 (σ 2 is a descendant of σ 1 ).

Real trees can be derived from continuous functions that represent their contour functions. Namely, let us consider a (deterministic) excursion e, that is to say a continuous function for which there exists ζ ∈ (0, ∞) such that : ∀t ζ, e(0) = e(t) = 0, and ∀t ∈ (0, ζ), e(t) > 0. A real tree T can be associated with e in the following way. For s, t ∈ [0, ζ], we set d(s, t) = e(s) + e(t) -2 inf r∈ [s∧t,s∨t] e(r).

It is easy to see that d is a pseudo-distance on [0, ζ]. Defining the equivalence relation s ∼ t iff d(s, t) = 0, one can set

T = [0, ζ]/ ∼ . (3)
The function d induces a distance on the quotient set T . For a fixed excursion e, let

p : [0, ζ] -→ (T, d) (4)
be the canonical projection. Clearly p is continuous, which implies that (T, d) is a compact metric space. Moreover, it can be shown (see [START_REF] Duquesne | Probabilistic and fractal aspects of Lévy trees[END_REF] for a proof) that (T, d) is a real tree We take ρ = p(0) as the root of T . For all a ∈ (0, ∞), the a-level set T (a) = {σ ∈ T : d(ρ, σ) = a} is the image by p of the set {t ∈ [0, ζ] : e(t) = a}. The total height of the tree is defined by h(T ) = sup {d(ρ, σ); σ ∈ T } .

We define the Brownian tree as the metric space (T , d) coded by the Brownian excursion. More precisely, let (Ω, F, P) a probability space, large enough to carry all the random variables we need. We consider on that space a process (X t , t ∈ [0, ∞)) such that ( 1 √ 2 X t , t ∈ [0, ∞)) is a standard real-valued Brownian motion (the choice of the normalizing constant √ 2 is explained below). Let us set X t = inf s∈[0,t] X s . Then, the reflected process X -X is a strong Markov process, and the state 0 is instantaneous in (0, ∞) and recurrent (see [START_REF] Bertoin | Lévy processes[END_REF], chapter VI). We denote by N the excursion measure associated with the local time -X; N is a sigma-finite measure on the space of continuous functions on [0, ∞), denoted C 0 in this work. More precisely, let j∈J (l j , r j ) = {t > 0 : X t -X t > 0} be the excursion intervals of the reflected process, and for all j ∈ J , we set e j (s) = X (l j +s)∧d j -X l j , s ∈ [0, ∞). Then,

M(dt, de) = j∈J δ (-X l j ,e j )
is a Poisson point measure on [0, ∞) × C 0 of intensity dtN(de). Let us recall that the two processes (|X t |, 2L t ) t 0 and (X t -X t , -X t ) t≥0 have the same law under P by a celebrated result of Lévy (see Blumenthal [START_REF] Blumenthal | Excursions of Markov processes[END_REF], Th. II 2.2) where the process (L t , t 0) is defined by the approximation L t = lim ε→0 (2ε) -1 t 0 1 {|Xs| ε} ds that holds uniformly in t on compact subsets of [0, ∞).

We shall denote by (e t , t 0) the canonical process on C 0 . Under N, it is a strong Markov process, with transition kernel of the original process X killed when it hits 0 (see [START_REF] Blumenthal | Excursions of Markov processes[END_REF] III 3(f)). The following properties hold for the process N-a.e. : there exists a unique real ζ ∈ (0, ∞) such that ∀t ∈ (0, ζ), e(t) > 0, and ∀t ∈ [ζ, ∞), e(t) = e(0) = 0. Moreover, with our normalization, one has (see [START_REF] Blumenthal | Excursions of Markov processes[END_REF]

IV 1.1) ∀λ ∈ [0, ∞), N(1 -e -λζ ) = √ λ and N(ζ ∈ dr) = r -3/2 2 √ π dr. (6)
One can show that

N (• | ζ ∈ [1 -ε, 1 + ε]
) converges when ε goes to 0, towards a probability measure that is denoted by N(• | ζ = 1). It can be seen as the law of the excursion of X -X conditioned to have length one. The tree encoded by e under N(• | ζ = 1) is the CRT defined in [START_REF] Aldous | The continuum random tree[END_REF]. The choice of the normalising constant √ 2 is explained by the following. Let τ n be uniformly distributed as the set of rooted planar trees with n vertices. We view it as a real tree, the edges of τ n being intervals of length one, and we denote by (τ n , d n ) the resulting metric space. Denote by (C (n) t , t ∈ [0, 2(n -1)]) its contour function that is (informally) defined as follows. We let a particle explore the planar tree at speed one, from the left to the right, beginning at the root. We set C (n) t as the distance from the root of the particle at time t. It can be shown (see [START_REF] Gall | Random trees and applications[END_REF] Th. 1.17) that (C

(n) t , t ∈ [0, 2(n -1)]
) has the law of a simple random walk conditioned to be positive on [1, 2(n-1) -1] and null at 2(n-1). Using Donsker invariance principle, the rescaled contour function

(n -1/2 C (n) 2(n-1)t , t ∈ [0, 1]) converges in law towards the law of (e t , t ∈ [0, 1]) under N(• | ζ = 1)
. In terms of trees, (τ n , n -1/2 d n ) converges towards the CRT, that is the tree (T 1 , d) coded by e under N(• | ζ = 1). The latter convergence can be stated using the distance of Gromov-Hausdorff (see Evans, Pitman, Winter [START_REF] Evans | Rayleigh processes, real trees, and root growth with re-grafting[END_REF]).

Recalling definition (5), we get from [START_REF] Blumenthal | Excursions of Markov processes[END_REF] IV 1.1 that with our normalization,

∀a ∈ (0, ∞) N sup t∈[0,ζ] e t > a = N h(T ) > a = 1 a . (7)
In the paper, for a ∈ (0, ∞) we shall use the probability measure,

N a = N (• | h(T ) > a) = aN •1 {h(T )>a} . (8)
Recall that the a-level set of the Brownian tree is defined by

T (a) = {σ ∈ T : d(ρ, σ) = a} . (9)
As a consequence of Trotter's theorem on the regularity of Brownian local time ( [START_REF] Blumenthal | Excursions of Markov processes[END_REF] sec VI.3) there exists a [0, ∞)-valued process (L a t ) a,t∈[0,∞) such that N-a.e. the following holds true:

• (a, t) → L a t is continuous, • for all a ∈ [0, ∞), t → L a t is non-decreasing,
• for all a ∈ [0, ∞), for all t ∈ [0, ∞) and for all b ∈ (0, ∞),

lim ε→0 N 1 {sup e>b} sup 0 s t∧ζ 1 ε s 0 1 {a-ε<e(u) a} du -L a s = 0 . ( 10
)
We refer to [START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF], Proposition 1.3.3. for details in a more general setting.

The image by the projection p : [0, ζ] → T of those local times defines the collection of local time measures on the tree, (ℓ a (dσ), σ ∈ T , a ∈ (0, ∞)). More precisely, N-a.e. for all f :

T meas. → [0, ∞) ∀a ∈ (0, ∞) T f (σ)ℓ a (dσ) = ζ 0 f (p(t))dL a t . (11)
See [START_REF] Duquesne | Probabilistic and fractal aspects of Lévy trees[END_REF], Th. 4.2 for an intrinsic definition of the measure ℓ a (for fixed a). Let G a the σ-field generated by the excursion below level a (formal definitions and details on what follows are given in section 3.1). The approximation [START_REF] Gall | Brownian excursions, trees and measure-valued branching processes[END_REF] entails that for fixed a, ℓ a (T ) = L a ζ is G a measurable. Moreover, the Ray-Knight theorem ([4] VI 2.10) entails that under N a (•) conditionally on G a , the process ℓ a+a ′ (T ), a ′ 0 is a Feller diffusion started at ℓ a (T ). In particular, one has

∀a, λ ∈ (0, ∞) N 1 -e -λℓ a (T ) = λ 1 + aλ , (12)
which implies that under N a , ℓ a (T ) is exponentially distributed with mean a. The regularity of a → ℓ a (T ) is extended by Duquesne and Le Gall [START_REF] Duquesne | Probabilistic and fractal aspects of Lévy trees[END_REF] : they prove that N-a.e. the process a → ℓ a is continuous for the weak topology of measures. In the same work, the topological support of the level set measures is precised as follows. A vertex σ ∈ T is called an extinction point if there

exists ε ∈ (0, ∞) such that d(ρ, σ) = sup{d(ρ, τ ), τ ∈ B(σ, ε)}, where B(σ, ε) is the open ball in T with center σ and radius ε. For s ∈ [0, ζ], the vertex p(s) ∈ T is an extinction time iff s ∈ [0, ζ] is a local maximum of e.
As a consequence, the set of all extinction points, denoted E , is countable. Let us denote Supp(µ) for the topological support of the measure µ. The result states that

N-a.e. ∀a ∈ (0, ∞) \ E , Supp(ℓ a ) = T (a), and ∀a ∈ E , Supp(ℓ a ) = T (a) \ {σ a }, (13)
where σ a is the (unique) extinction point at level a (see Perkins [START_REF] Perkins | Polar sets and multiple points for super-Brownian motion[END_REF] for previous results on Super-Brownian motion).

Let us briefly introduce the construction of the Hausdorff measure. We set the gauge function g as g(r) = r log log 1/r, r ∈ (0, e -1 ).

In all the paper it will be assumed implicitly that g(r) is considered only for r ∈ (0, e -1 ). On that interval, g is an increasing continuous function. For any subset A of T , one can define

H g (A) = lim ε→0 inf i∈N g (diam(E i )) ; A ⊂ i∈N E i , diam(E i ) < ε . (15)
Standard results on Hausdorff measures (see e.g. [START_REF] Rogers | Hausdorff measures[END_REF]) ensure that H g defines a Borel-regular outer measure on T called the g-Hausdorff measure on T . The main result of the paper is the following.

Theorem 1 Let T be the Brownian tree, that is the tree encoded by the excursion e under N. Let (ℓ a (dσ), σ ∈ T , a ∈ (0, ∞)) the collection of local time measures and H g the g-Hausdorff measure on T , where g(r) = r log log 1/r. Then, the following holds :

N-a.e. ∀a ∈ (0, ∞) ℓ a (•) = 1 2 H g (• ∩ T (a)) . ( 16
)
Comment 1.1 Thanks to the scaling properties of the Brownian excursion, one can derive from Theorem 1 a similar statement for the tree coded by e under N(• | ζ = 1), that is Aldous CRT.

Comment 1.2 Our result seems close to a theorem of Perkins [START_REF] Perkins | The exact Hausdorff measure of the level sets of Brownian motion[END_REF] on linear Brownian motion. Let (L a t , t 0, a ∈ R) be the bi-continuous version of the local times for the process (X t , t 0) defined above. Those local times are given by an approximation of the type of [START_REF] Gall | Brownian excursions, trees and measure-valued branching processes[END_REF]. Perkins proves that almost surely, uniformly in a, one has L a t = H g ({s ∈ [0, t] : X s = x}), where H g stands for the Hausdorff measure on the line associated with the gauge g(r) = r log log 1/r (the result for fixed a had been obtain by Taylor and Wendel in [START_REF] Taylor | The exact Hausdorff measure of the zero set of a stable process[END_REF]). The Brownian tree being coded by the Brownian excursion, everything happens as if the projection mapping p : [0, ζ] → T is 1/2-HÃPlder and induces a strong "doubling", such that the entire gauge function is squared. Nevertheless, we don't see how to derive our result from [START_REF] Perkins | The exact Hausdorff measure of the level sets of Brownian motion[END_REF].

The paper is organised as follows. In section 2, we state some deterministic facts on the geometry of the level sets for a real tree. In particular, we provide two comparison lemmas with respect to Hausdorff measure on real trees. The second one, that is specific to our setting, seems new to us. In section 3, we recall basic facts on the Brownian tree and we establish some technical estimates. Section 4 is devoted to the proof of Theorem 1. As a first step, we prove Theorem 2, which gives an upper bound for the local time measures. To that end, we need to control the total mass of the balls that are "too large". The second step is the proof of Theorem 3, which requires a control of the number of balls that are "too small". Let us mention again that our strategy and many ideas in this work were borrowed from [START_REF] Perkins | A space-time property of a class of measure-valued branching diffusions[END_REF][START_REF] Perkins | The Hausdorff measure of the closed support of super-Brownian motion[END_REF].

2 Geometric properties of the level sets of real trees.

2.1

The balls of the level sets of real trees.

Let (T, d, ρ) be a compact rooted real tree as defined in the introduction. Recall that for any σ, σ ′ ∈ T , [[σ, σ ′ ]] stands for the unique geodesic path joining σ to σ ′ . We shall view T as a family tree whose ancestor is the root ρ and we then denote by σ ∧ σ ′ the most recent common ancestor of σ and σ ′ that is formally defined by

[[ρ, σ ∧ σ ′ ]] = [[ρ, σ]] ∩ [[ρ, σ ′ ]] . Observe that ∀σ, σ ′ ∈ T, d(σ, σ ′ ) = d(ρ, σ) + d(ρ, σ ′ ) -2d(ρ, σ ∧ σ ′ ) . ( 17
)
Let a ∈ [0, ∞). Recall that the a-level set of T is given by

T (a) = σ ∈ T : d(ρ, σ) = a .
j∈J b T o,b j = σ ∈ T : d(ρ, σ) > b .
Then for any j ∈ J b , there exists a unique point

σ j ∈ T (b) such that T b j := T o,b j ∪ {σ j } is the closure of T o,b j in T . Note that (T b j , d, σ j ) is a compact rooted real tree and that ∀j ∈ J b , ∀σ ∈ T b j , σ j ∈ [[ρ, σ]] .
Open balls in T (a). Recall that B(σ, r) stands for the open ball in T with center σ and radius r. We shall also denote by Γ(σ, r) the open ball with center σ and radius r in the level set of σ, namely Γ(σ, r) = B(σ, r) ∩ T (a) , where a = d(ρ, σ).

If σ ∈ T (a), then we call Γ(σ, r) a T (a)-ball with radius r; we denote by B a,r the set of all the T (a)-balls with radius r:

B a,r = Γ(σ, r); σ ∈ T (a) . (19)
The following proposition provides the geometric properties of T (a)-balls that we shall use.

Proposition 1 Let (T, d, ρ) be a compact rooted real tree. Let a, r ∈ (0, ∞) be such that a r/2. Then, the number of T (a)-balls with radius r is finite. We set

Z a,r = #B a,r and Γ i , 1 i Z a,r = B a,r . (20)
Then , the following holds true.

(i) Set b = a-1 2 r. Then, there are Z a,r distinct subtrees above b denoted by (T b j i , d, σ j i ), j i ∈ J b , 1 i Z a,r such that

Γ i = T (a) ∩ T b j i = {σ ′ ∈ T b j i : d(σ j i , σ ′ ) = r/2 .
Thus, the T (a)-balls with radius r are pairwise disjoint.

(ii) For all σ ∈ T (a), one has diam(Γ(σ, r)) r. If furthermore r ∈ (0, 2a), then diam(Γ(σ, r)) < r and ∀r ′ ∈ diam(Γ(σ, r)), r Γ(σ, r ′ ) = Γ(σ, r) .

Therefore, the set of all T (a)-balls is countable.

(iii) Two T (a)-balls are either contained one in the other or disjoint. Namely, for all r ′ < r and all σ, σ ′ ∈ T (a), either Γ(σ ′ , r ′ ) ⊂ Γ(σ, r) or Γ(σ ′ , r ′ ) ∩ Γ(σ, r) = ∅.

Proof. Let us prove (i). Let σ, σ ′ ∈ T (a) and set b = a-1 2 r. By [START_REF] Rogers | Hausdorff measures[END_REF],

d(σ, σ ′ ) = 2a -2d(ρ, σ ∧ σ ′ ). Thus, d(σ, σ ′ ) < r iff d(ρ, σ ∧ σ ′ ) > b. Let j ∈ J b be such that σ ∈ T b j ; namely, T b j is the unique subtree above b containing σ and σ j is the unique point γ ∈ [[ρ, σ]] such that d(ρ, γ) = b. Now observe that for all σ ′ ∈ T (a), d(ρ, σ ∧ σ ′ ) > b ⇐⇒ σ ∧ σ ′ ∈ ]]σ j , σ]] ⇐⇒ σ ′ ∈ T b j .
This proves that

Γ(σ, r) = T (a) ∩ T b j . (22)
Conversely, let j ∈ J b be such that h(T b j ) := max d(σ j , γ); γ ∈ T b j r/2. Let σ ∈ T (a) ∩ T b j ; then the previous arguments imply (22). Since T is compact, the set {j ∈ J b : h(T b j ) r/2} is finite, which completes the proof of (i).

Let us prove (ii):

let σ ∈ T (a), let r ∈ (0, 2a) and set δ = diam(Γ(σ, r)). Then (22) implies that Γ(σ, r) is compact and there are σ 1 , σ 2 ∈ Γ(σ, r) such that d(σ 1 , σ 2 ) = δ. Observe that it implies Γ(σ, r) = σ ′ ∈ T (a) : σ 1 ∧ σ 2 ∈ [[ρ, σ ′ ]] .
Thus, Γ(σ, r) = Γ(σ, δ), that is the closure of Γ(σ, δ), and it implies (21). The set of all T (a)-balls is therefore q∈Q∩[0,∞) B a,q , which is a countable set.

Let us prove (iii): r ′ < r and σ, σ ′ ∈ T (a) and suppose that Γ(σ ′ , r ′ ) ∩ Γ(σ, r) = ∅. Then (i) and (ii) implies that Γ(σ, r) = Γ(σ ′ , r), which implies that Γ(σ ′ , r ′ ) ⊂ Γ(σ, r).

Comparison lemmas for Hausdorff measures on real trees.

Let (T, d, ρ) be a compact real tree. We briefly recall the definition of Hausdorff measures on T and we state two comparison lemmas that are used in the proofs. Let r 0 ∈ (0, ∞) and let g : [0, r 0 ) → [0, ∞) be a function that is assumed to be increasing, continuous and such that g(0) = 0. For all ε ∈ (0, r 0 ) and all A ⊂ T , we set

H (ε) g (A) = inf n∈N g (diam(E n )) ; A ⊂ n∈N E n , diam(E n ) < ε and H g (A) = lim ε↓0 ↑ H (ε) g (A) .
Under our assumptions, H g is a Borel-regular outer measure : this is the g-Hausdorff measure on T (see Rogers [START_REF] Rogers | Hausdorff measures[END_REF]). The following comparison lemma was first stated for Euclidean spaces by Rogers and Taylor [START_REF] Rogers | Functions continuous and singular with respect to a Hausdorff measure[END_REF]. The proof can be easily adapted to general metric spaces (see Edgar [START_REF] Edgar | Centered densities and fractal measures[END_REF]).

We include a brief proof of it in order to make the paper self-contained.

Lemma 2 Let (T, d, ρ) be a compact rooted real tree. Let µ be a Borel measure on T . Let A be a Borel subset of T and let c ∈ (0, ∞). Assume that

∀σ ∈ A lim sup r→0 µ (B(σ, r)) g(r) < c .
Then, µ(A) cH g (A).

Proof. For any ε ∈ (0, r 0 ), set

A ε = σ ∈ A : sup r∈(0,ε) µ(B(σ, r)) g(r) < c .
Observe that for all

ε ′ < ε, A ε ⊂ A ε ′ ⊂ A and A = ε∈(0,r 0 ) A ε . Let (E n ) n∈N be a ε-covering of A ε : namely A ε ⊂ n∈N E n and diam(E n ) < ε, for all n ∈ N. Set I = {n ∈ N : E n ∩ A ε = ∅} and for all n ∈ I, fix σ n ∈ E n ∩ A ε . Since g is continuous, for all n ∈ I there exists r n ∈ (diam(E n ), ε) such that E n ⊂ B(σ n , r n ) and g(r n ) 2 -n-1 ε + g(diam(E n )) . Observe that µ(B(σ n , r n )) < cg(r n ) and that A ε ⊂ n∈I B(σ n , r n ). Thus, µ(A ε ) µ n∈I B(σ n , r n ) n∈I µ(B(σ n , r n )) n∈I c g(r n ) cε + n∈N c g(diam(E n )) .
Taking the infimum over all the possible ε-coverings of A ε yields

µ(A ε ) cε + cH (ε) g (A ε ) cε + cH g (A ε ) cε + H g (A) ,
which implies the desired result since µ(A) = lim ε↓0 ↑ µ(A ε ).

In the next comparison lemma, that seems new to us, we restrict our attention to the level sets of real trees. A more general variant of this result involves a multiplicative constant depending on the gauge function. It has been first stated in Euclidian spaces by Rogers and Taylor [START_REF] Rogers | Functions continuous and singular with respect to a Hausdorff measure[END_REF] (see also Perkins [START_REF] Perkins | A space-time property of a class of measure-valued branching diffusions[END_REF]) and in general metric spaces (see Edgar [START_REF] Edgar | Centered densities and fractal measures[END_REF]).

Lemma 3 Let (T, d, ρ) be a compact rooted real tree. Let a ∈ (0, ∞) be such that the a-level set T (a) is not empty. Let µ be a finite Borel measure on T such that µ(T \T (a)) = 0. Let A ⊂ T (a) be a Borel subset and let c ∈ (0, ∞). Assume that

∀σ ∈ A lim sup r→0 µ (B(σ, r)) g(r) > c .
Then, µ(A) cH g (A).

Proof. Let ε ∈ (0, (2a) ∧ r 0 ). Let U be an open set of T such that A ⊂ U . For all σ ∈ A, there exists r σ ∈ (0, ε) such that µ (Γ(σ, r σ )) = µ (B(σ, r σ )) > cg(r σ ) and Γ(σ, r σ ) ⊂ U .
Thus, A ⊂ σ∈A Γ(σ, r σ ) ⊂ U . Then, Proposition 1 (ii) asserts that the set of all T (a)-balls is countable and Proposition 1 (iii) asserts that two T (a)-balls are either contained one in the other or disjoint. Therefore, there exists I ⊂ N and σ n ∈ A, n ∈ I, such that the Γ(σ n , r σn ), n ∈ I, are pairwise disjoint and A ⊂ n∈I Γ(σ n , r σn ) ⊂ U . Moreover, by Proposition 1 (ii), diam(Γ(σ n , r σn )) r σn . Thus, we get

c H (ε) g (A) n∈I c g diam(Γ(σ n , r σn )) n∈I c g(r σn ) n∈I µ Γ(σ n , r σn ) = µ n∈I Γ(σ n , r σn ) µ(U ) . As ε → 0, it entails c H g (A) µ(U )
, for all open set U containing A. Since µ is a finite Borel measure, it is outer-regular for the open subsets, which implies the desired result.

3 Preliminary results on the Brownian tree.

Basic facts on the Brownian excursion.

We work under the excursion measure N defined in the introduction and e denote the canonical excursion whose duration is denoted by ζ (see ( 6)). We shall denote by (T , d, ρ) the compact rooted real tree coded by e.

The branching property. Fix b ∈ (0, ∞). We discuss here a decomposition of e in terms of its excursions above level b; this yields a decomposition of the Brownian tree called the branching property. To that end we first introduce the following time change: for all t ∈ [0, ∞), we set

τ b (t) = inf s ∈ [0, ∞) : s 0 1 {eu b} du > t and e b (t) = e(τ b (t)). (23)
Note that ( e b (t)) t∈[0,∞) codes the tree below b namely {σ ∈ T : d(ρ, σ) b} that is the closed ball with center ρ and radius b. We denote by G b , the sigma-field generated by ( e b (t)) t∈[0,∞) and completed with the N-negligible sets. The approximation [START_REF] Gall | Brownian excursions, trees and measure-valued branching processes[END_REF]

implies that L b ζ is G b -measurable. Then denote by (α j , β j ), j ∈ J b , the connected components of the time-set {s ∈ [0, ∞) : e(s) > b}. Namely, j∈J b (α j , β j ) = {s ∈ [0, ∞) : e(s) > b} ,
and we call (α j , β j ) the excursion intervals of e above level b. For all j ∈ J b , we next set

l b j = L b τ b (α j ) and ∀s ∈ [0, ∞), e b j (s) = e (α j +s)∧β j -b .
Then, the (e b j ) j∈J b are the excursions of e above level b. Recall from ( 7) and ( 8) the notation

N b = N( • | sup e > b)
, that is a probability measure. The branching property asserts the following: under N b and conditionally on G b , the measure

M b (dl, de) = j∈J b δ (l b j ,e b j ) (24) is a Poisson point measure on [0, L b ζ ]×C 0 with intensity 1 [0,L b ζ ] (l)dl N(de).
The following decomposition of e is interpreted in terms of the Brownian tree T as follows. Recall that p : [0, ζ] → T stands for the canonical projection. Then for all j ∈ J b , we set

σ j = p(α j ) = p(β j ) , T o,b j = p (α j , β j ) and T b j = p [α j , β j ] .
Then, we easily check that the T o,b j , j ∈ J b , are the connected components of the open subset {σ ∈ T : d(ρ, σ) > b} and that T o,b j = T b j \{σ j }. Namely, the (T j , d, σ j ), j ∈ J b are the subtrees above level b of T as introduced in Section 2.1. Moreover note that for all j ∈ J b , the rooted compact real tree (T j , d, σ j ) is isometric to the tree coded by the excursion e b j . We next use this and Proposition 1 to discuss the balls in a fixed level of T .

To that end, we fix a, r ∈ (0, ∞) such that a > r/2 and we conveniently set b = a-r/2. Recall that T (a) = {σ ∈ T : d(ρ, σ) = a} and that for all σ ∈ T (a), we have set Γ(σ, r) = T (a) ∩ B(σ, r) that is the ball in T (a) with center σ and radius r. We also recall that B a,r = {Γ(σ, r); σ ∈ T (a)} stands for the set of all T (a)-balls with radius r. By Proposition 1, B a,r is a finite set and that

B a,r = T (a) ∩ T b j ; j ∈ J b : h(T b j ) r/2 ,
where the trees (T b j , d, σ j ), j ∈ J b , are the subtrees of T above level b as previously defined; here h(T b j ) = sup σ∈T b j d(σ j , σ) stands for the total height of T b j . Note that h(T b j ) = sup e b j that is maximum of the excursion corresponding to T b j , as explained above. Then, we set Z a,r = #B a,r , that is the number of T (a)-ball with radius r. Assume that Z a,r 1. We then define the indices j 1 , . . . , j Za,r ∈ J b by {j 1 , . . . , j Za,r } = j ∈ J b : h(T b j ) r/2 and α j 1 < . . . < α j Za,r .

and we set ∀i ∈ {1, . . . , Z a,r },

Γ i := T (a) ∩ T b j i . ( 25
)
Namely B a,r = Γ i ; 1 i Z a,r is the set of the T (a)-balls with radius r listed in their order of visit by the excursion e coding T .

Lemma 4 Let a, r ∈ (0, ∞) such that a > r/2. Let Γ i ; 1 i Z a,r is the set of the T (a)-balls with radius r listed in their order of visit as explained above. Then the following holds true.

(i) Under N a = N( • | sup e > a), Z a,r has a geometric law with parameter 2a/r. Namely,

∀k 1, N a [Z a,r = k] = 1- r 2a k-1 r 2a . (ii) For all k 1, under N a ( • | Z a,r = k), the r.v. (ℓ a (Γ i ))
1 i k are independent and exponentially distributed with mean r/2.

Proof. Let a ∈ (0, ∞) and denote b = ar/2. Let k 1 and F 1 , . . . F k : C 0 → [0, ∞) be measurable functionals. Recall from [START_REF] Duquesne | The Hausdorff measure of stable trees[END_REF] that N(sup e r/2) = 2/r. Then, the definition of the j i combined with the branching property and basic results on Poisson point measures entail

N b 1 {Za,r=k} 1 i k F i (e b j i ) G b = ( 2 r L b ζ ) k k! e -2 r L b ζ 1 i k N r/2 F i (e) . (26)
Then recall [START_REF] Gall | Random trees and applications[END_REF] that implies that L b ζ under N b is exponentially distributed with mean b. Thus,

1 k! N b 2 r L b ζ k e -2 r L b ζ = ( 2 r b) k (1 + 2 r b) k+1 = r 2a 1 - r 2a k , because b = a-r/2 and (1 + 2 r b) -1 = r/(2a). It implies N b 1 {Za,r=k} 1 i k F i (e b j i ) = r 2a 1 - r 2a k 1 i k N r/2 F i (e) .
Next observe that N b -a.s. 1 {sup e>a} = 1 {Za,r 1} . Thus, we get

N a 1 {Za,r=k} 1 i k F i (e b j i ) = a b N b 1 {Za,r=k} 1 i k F i (e b j i ) = r 2a 1 - r 2a k-1 1 i k N r/2 F i (e) (27) because a/b = (1-r 2a ) -1 . Recall that (12) implies that under N r/2 , ℓ r/2 (T ) = L r/2 ζ is exponentially distributed with mean r/2. By taking F i (e) = f i (L r/2 ζ ) in (27) we then get N a 1 {Za,r=k} 1 i k f i ℓ a (Γ i ) = r 2a 1 - r 2a k-1 1 i k ∞ 0 f i (s)
2 r e -2 r s ds , with entails the desired result.

Ray-Knight theorem under N. We first recall the definition of Feller diffusion, namely a Continuous States space Branching Process (CSBP) with branching mechanism ψ(λ) = λ 2 . Let x ∈ [0, ∞) and let (Y x a ) a∈[0,∞) be a [0, ∞)-valued continuous process defined on the probability space (Ω, F, P). It is a Feller diffusion with branching mechanism ψ(λ) = λ 2 and initial value

Y x 0 = x if it is a Markov process such that E exp(-λY x a+a ′ ) Y x a = exp - Y x a λ 1 + a ′ λ , a, a ′ , λ ∈ [0, ∞) . Recall notation N a = N(• sup e > a)
and G a for the sigma-field generated by the excursion e a defined in (23). Recall that ℓ a (T ) = L a ζ , the total mass of the local-time measure at level a, is G a -measurable. We shall use the following statement of Ray-Knight theorem. Let a ∈ (0, ∞).

(i) N a [exp(-λℓ a (T ))] = 1 1+aλ .
(ii) Under N a and conditionally given G a , the process (ℓ a+a ′ (T )) a ′ ∈[0,∞) is a Feller diffusion with branching mechanism ψ(λ) = λ 2 and initial value ℓ a (T ).

This is an immediate consequence of the Ray-Knight theorem for standard Brownian motion and of the Markov property under N : see [4] III 3 and VI 2.10.

Combined with the branching property, the above Ray-Knight theorem, has the following consequence. Let us recall that we enumerate the T (a)-balls of B a,r as {Γ i , 1 i Z a,r } (see (25)). Let Γ such a T (a)-ball. For a ′ 0, we define

Γ a+a ′ = σ ∈ T (a + a ′ ) ∃σ ′ ∈ Γ : σ ′ ∈ ρ, σ , (28)
the set of vertices at level a + a ′ that have an ancestor in Γ (notice that Γ a = Γ). The following lemma is a straightforward consequence of Ray-Knight theorem.

Lemma 5 Let a ∈ (0, ∞), r ∈ [0, 2a]. Let {Γ i , 1 i Z a,r } the set of T (a)-balls of radius r.
Under N a conditionally on G a , the processes

ℓ a+a ′ (Γ a+a ′ i ), a ′ 0 , 1 i Z a,r , are independent Feller diffusions started at (ℓ a (Γ i )) , 1 i Z a,r .
Proof. Recalling for b = a-r/2 the decomposition (25), we see that

∀i ∈ {1, . . . , Z a,r }, Γ a+a ′ i := T (a + a ′ ) ∩ T b j i . (29)
Hence, one can use (26), and the Ray-Knight theorem (see (ii) above) to get the desired result.

Spinal decomposition. We recall another decomposition of the Brownian tree called spinal decomposition. This is a consequence of Bismut's decomposition of the Brownian excursion that we recall here. Let X be a real valued process defined on (Ω, F, P) such that

( 1 √ 2 X t ) t∈[0,∞)
is distributed as a standard Brownian motion with initial value 0. Let X ′ be an independent copy of X on (Ω, F, P). We fix a ∈ (0, ∞) and we set s = e (t-s) + and êt s = e t+s . Then the Bismut's identity (see [START_REF] Bismut | Last exit decompositions and regularity at the boundary of transition probabilities[END_REF] or [START_REF] Gall | The uniform random tree in a Brownian excursion[END_REF]) states that for any non-negative measurable functional

T a = inf{t ∈ [0, ∞) : X t = -a} and T ′ a = inf{t ∈ [0, ∞) : X ′ t = -a} . We next set for any s ∈ [0, ∞),
F on (C 0 ) 2 , N ζ 0 dL a t F ět ; êt = E F (a + X •∧Ta ; a + X ′ •∧T ′ a ) . (30)
We derive from (30) an identity involving the excursions above the infimum of êt and ět . To that end, we introduce the following. Let h : [0, ∞) → [0, ∞) with compact support. We define a point measure point N (h) as follows: set h(t) = inf [0,t] h and denote by (g i , d i ), i ∈ I(h) the excursion intervals of hh away from 0 that are the connected component of the open set {t 0 :

h(t) -h(t) > 0}. For any i ∈ I(h), set h i (s) = ((h -h)((g i + s) ∧ d i ) , s 0).
We then define N (h) as the point measure on [0, ∞) × C 0 given by

N (h) = i∈I(h) δ (h(g i ),h i ) .
Then, for any t, a ∈ (0, ∞),

N t := N (ě t ) + N (ê t ) =: j∈Jt δ (h t j ,e t,j ) (31)
and

N * a := N (a + X •∧Ta ) + N (a + X ′ •∧T ′ a ) =: j∈J * a δ (h * j ,e * j ) . (32)
We deduce from (30) that for all any a and for all nonnegative measurable function F on the set of positive measures on [0, ∞) × C 0 , one has

N ζ 0 dL a t F N t = E F (N * a ) (33
σ j ∈ [[ρ, σ]] such that T j = {σ j } ∪ T o j .
Recall notation (h t j , e t,j ), j ∈ J t from (31). The specific coding of T by e entails that for any j ∈ J there exists a unique j ′ ∈ J t such that d(ρ, σ j ) = h t j ′ and such that the rooted compact real tree (T j , d, σ j ) is isometric to the tree coded by e t,j ′ .

Recall that p(t) = σ. We fix r, r ′ ∈ [0, 2a) such that r ′ r. We now compute the mass of the ring B(σ, r) \ B(σ, r ′ ) in terms of N t . First, observe that for any s ∈ [0, ζ] such that e s = a, we have

r ′ d(s, t) < r ⇐⇒ a -(r ′ /2) inf u∈[s∧t,s∨t]
e u > a -(r/2) .

We then get

ℓ a B(σ, r)\B(σ, r ′ ) = j∈Jt 1 (a-r 2 , a-r ′ 2 ] (h t j ) L a-h t j ζ t j (t, j) , (34)
where L a-h t j ζ t j (t, j) stands for the local time at level ah t j of the excursion e t,j . Then, for any a ∈ (0, ∞) and any r, r ′ ∈ (0, 2a) such that r ′ r, we also set

Λ a r ′ ,r = j∈J * a 1 (a-r 2 , a-r ′ 2 ] (h * j ) L a-h * j ζ * j , (35)
where, L a-h * j ζ * j stands for the local time at level ah * j of the excursion e * j defined in (32). Then, (33) implies that for any a ∈ (0, ∞) and for all non-negative measurable F function

N T ℓ a (dσ) F ℓ a B(σ, r)\B(σ, r ′ ) ; 0 r ′ r 2a = E F Λ a r ′ ,r ; 0 r ′ r 2a (36)
On the right-hand-side, the dependency with respect to the level a is a bit artificial. Indeed, for a ∈ (0, ∞), the Poisson point measure N * a (dhde) has its law invariant under the transformation (h, e) → (ah, e). Thus, let us consider on (Ω, F, P) a new Poisson point measure M * = j∈I * δ (h * j ,e * j ) with intensity 2dhN(de) (we abuse notations and keep the notation (h * j , e * j ) for the atoms). We set

Λ * r ′ ,r = j∈I * 1 [ r ′ 2 , r 2 ) (h * j ) L h * j ζ * j , (37)
where

L h * j ζ * j
stands for the local time at height h * j for the excursion e * j . One can now rewrite (36) as

N T ℓ a (dσ) F ℓ a B(σ, r)\B(σ, r ′ ) ; 0 r ′ r 2a = E F Λ * r ′ ,r ; 0 r ′ r 2a . ( 38
)
The law of the Λ * r ′ ,r is quite explicit as shown by the following lemma.

Lemma 6 Let 0 r n r n-1 . . . r 1 2a. Then,

Λ * rn,r n-1 , Λ * r n-1 ,r n-2 , . . . , Λ * r 2 ,r 1
are independent. Moreover, for any 0 r ′ r 2a,

∀y ∈ (0, ∞) P Λ * r ′ ,r > y = 1 - r ′ r 2 2y
r e -2y/r + 1 -r ′ r 2 e -2y/r , and

P(Λ * r ′ ,r = 0) = (r ′ /r) 2 .
Proof. The intervals [r k+1 /2, r k /2) being pairwise disjoint, the independence of the increments is a straightforward consequence of the properties the Poisson point measure M * . Using Campbell formula and ( 12), we compute, for all λ 0,

E e -λΛ * 2r ′ ,2r = exp - r r ′ 2dhN 1 -e -λℓ h (T ) = exp - r r ′ 2dh λ 1 + hλ = 1 + r ′ λ 1 + rλ 2 . Thus, Λ * 2r ′ ,2r (law)
= X 1 + X 2 , where X 1 and X 2 are i.i.d random variables where

E e -λX 1 = r ′ r + 1 - r ′ r 1 1 + rλ .
Thus, X 1 = 0 with probability r ′ /r and conditionally on being non-zero, it is exponentially distributed with mean r. Thus, for y > 0,

P Λ * 2r ′ ,2r > y = 2P (X 1 = 0; X 2 > y) + P (X 1 > 0 ; X 2 > 0 ; X 1 + X 2 > y) = 2 r ′ r P(X 1 > y) + 1 - r ′ r 2 P (Z > y) ,
where Z has law Gamma(2, 1/r). The result proceeds now from elementary computations.

Estimates.

The following elementary computation is needed twice in our proofs.

Lemma 7 Let (X n ) n 1 a sequence of i.i.d real valued random variables on (Ω, F, P), with mean 0 and a moment of order 4. Let Z be a random variable taking its values in N, independent of the sequence (X n ). Then

E (X 1 + X 2 + • • • + X Z ) 4 3E[X 4 1 ]E Z 2 .
Moreover, the following holds :

E (X 1 -E[X 1 ]) 4 2E[X 4 1 ].
Proof. One has

E (X 1 + X 2 + • • • + X Z ) 4 | Z = 1 i 1 ,i 2 ,i 3 ,i 4 Z E [X i 1 X i 2 X i 3 X i 4 ] .
When (i 1 , i 2 , i 3 , i 4 ) contains an index that is distinct of the three others, then the contribution of the corresponding term will be null. Thus the latter mean equals ZE[X 4 1 ] (using Jensen's inequality). The second statement follows from

4 1 ] + 3Z(Z -1)E[X 2 1 ] 2 3Z 2 E[X
E (X 1 -E[X 1 ]) 4 = E (X 1 -E[X 1 ]) 4 1 {X 1 E[X 1 ]} + E (X 1 -E[X 1 ]) 4 1 {X 1 <E[X 1 ]} E X 4 1 + E[X 1 ] 4 ,
and using Jensen's inequality.

We explained in Section 3.1 the link between the process (ℓ a (T ), a ∈ (0, ∞)) and the Feller diffusion, for which we provide here some basic estimates.

Lemma 8 Let (Y x a ) a 0 be a Feller diffusion starting at x 0, defined on (Ω, F, P). For all x, y ∈ [0, ∞), for all a ∈ (0, ∞), the following inequalities hold :

(i) If y x, then P inf b∈[0,a] Y x b y exp -1 a ( √ x - √ y) 2 . (ii) If y x, then P sup b∈[0,a] Y x b y exp -1 a ( √ y - √ x) 2 .
Proof. Let us prove (i). Recall that for all x, b, λ ∈ [0, ∞), E e -λY x b = exp -λx 1+bλ . Thus, for fixed a ∈ (0, ∞), and for λ ∈ [0, 1 a ), we set

∀b ∈ [0, a], M (λ,x) b := exp - λY x b 1 -bλ . ( 39
)
We stress that for b ∈ [0, a], one has 1bλ 1aλ > 0, and one can compute

E[M (λ,x) b ] = exp - λ 1 -bλ x 1 + bλ 1 -bλ = e -λx .
Combined with the Markov property, this entails that (M The reader can check using elementary computations that the function λ → λy 1-aλλx has a negative minimum on (0, 1/a) at the value λ = 1 a 1 -y x , and this minimum is

(λ,x) b , b ∈ [0, a]) is a martingale. More- over, on { inf b∈[0,a]
-1 a ( √ x - √ y) 2 ,
which completes the proof.

In order to prove (ii), one could extend the definition of (M

(λ,x) b , b ∈ [0, a]) for λ ∈ (-1/a, 0).
In what follows, we use a simpler argument. Let us begin with the following remark: let b ∈ (0, ∞), let E be a r.v. on (Ω, F, P) that is exponentially distributed with mean b, then for all λ 0, E[e -λE ] = 1 1+bλ , and this Laplace transform remains finite for λ ∈ (-1/b, 0). Moreover, one can plainly check that for x, b ∈ (0, ∞), Y x b has the same law as

N i=1
E i , where the E i are independent copies of E and N is an independent Poisson r.v. with mean x/b. Thus, one has

∀µ ∈ (0, 1/b), E e µY x b = exp µx 1 -µb . (40)
The Feller diffusion (Y x b , b 0) is a martingale, so by convexity (e µY x b , b 0) is a submartingale. Thus, for all µ ∈ (0, 1/a), and y x 0, one has The next result is a corollary of Lemma 8 (ii).

P sup b∈[0,a]
Lemma 9 Let m ∈ (0, 1/2). For all y ∈ (0, ∞),

N sup b∈[m,m -1 ] ℓ b (T ) > y (2/m) exp (-my/2) .
Proof. Let m ∈ (0, 1/2) and recall from (23) the definition of G m . As recalled in Section 3.1, under N m , conditionally on G m , the process (ℓ b (T ), b m) is a Feller diffusion started at ℓ m (T ).

Hence, conditioning with respect to G m and using Lemma 8 (ii), we get

N m sup b∈[m,m -1 ] ℓ b (T ) > y N m exp -m √ y -ℓ m (T ) 2 .
Expanding ( u/2 -√ 2v) 2 , one shows that for all u, v 0, ( T ) . Recalling from ( 12) that under N m , ℓ m (T ) is exponentially distributed with mean m, we get N m e mℓ m (T ) = (1-m 2 ) -1 2, because m < 1/2. This entails the desired result, recalling that N m (•) = mN •1 {h(T )>m} and that the events {h(T ) > m} and {ℓ m (T ) > 0} are equal, up to a N negligible set.

√ u - √ v) 2 u/2 -v. Thus, N m sup b∈[m,m -1 ] ℓ m (T ) > y exp -m y 2 N m e mℓ m (
Estimates for small balls. We consider here a level a ∈ (0, ∞) and recall that T

We consider positive real numbers r 1 > r 2 > . . . > r n > 0, and ε 1 > . . . > ε n-1 > 0, where n ∈ N * . We set r = {r 1 , . . . , r n } and ε = {ε 1 , . . . , ε n-1 }. We shall say that Γ, a T (a)-ball of radius r n , is (r, ε)-small if and only if for all 1 k n-1, the enlarged ball of radius k has a local time smaller than ε k , namely

∀k ∈ {1, . . . , n-1} ℓ a (Γ[r k ]) ε k . ( 42
)
We denote by S a,r,ε the total number of such (r, ε)-small balls at level a: S a,r,ε :=

Γ∈Ba,r n 1 {Γ is (r,ε)-small} . ( 43
)
To control that number, we introduce

µ(r, ε) := N S r 1 /2,r,ε . (44)
Let us stress that its definition does not depend on a.

Lemma 10 Let a ∈ (0, ∞), r = {r 1 , . . . , r n }, and ε = {ε 1 , . . . , ε n-1 }, where r 1 > . . . > r n > 0, and ε 1 > . . . > ε n-1 > 0. There exists a constant c 0 ∈ (0, 10 4 ] such that if a/r 1 > 1 and r 1 /r n > 2,

N (S a,r,ε -µ(r, ε)ℓ a (T )) 4 c 0 a r 2 1 r 4 n .
Proof. Let a, r, ε as above. From Proposition 1 (iii), we know that the T (a)-balls of radius r n are disjoint and that for all Υ ∈ B a,rn , there exists a unique

T (a)-ball Γ ∈ B a,r 1 such that Υ ⊂ Γ. Let us enumerate B a,r 1 as {Γ i , 1 i Z a,r 1 }, and set ∀i ∈ {1 . . . Z a,r 1 }, B (i) a,rn = {Υ ∈ B a,rn : Υ ⊂ Γ i } and S (i) a,r,ε = # Υ ∈ B (i) a,rn : Υ is (r, ε)-small .
One has

S a,r,ε -µ(r, ε)ℓ a (T ) = Za,r 1 i=1 S (i) a,r,ε -µ(r, ε)ℓ a (Γ i ) =: Za,r 1 i=1 X i . ( 45
)
Let us denote b = ar 1 /2 and recall from (23) the definition of the sigma-field G b . Adapting the proof of Lemma 4, it is not difficult to see that under N b , conditionally on G b , and conditionally on {Z a,r 1 = k}, the r.v. X 1 , . . . X k are independent and have the same law as

S r 1 /2,r,ε -µ(r, ε)ℓ r 1 /2 (T ) under N r 1 /2 . Recalling from (12) that N ℓ r 1 /2 (T ) = (2/r 1 )N r 1 /2 ℓ r 1 /2 (T ) = 1, we see that N b [X 1 | G b ] = N r 1 /2 S r 1 /2,r,ε -µ(r, ε)ℓ r 1 /2 (T ) = 0,
which explains the definition (44). We thus apply Lemma 7 to get from (45):

N b (S a,r,ε -µ(r, ε)ℓ a (T )) 4 | G b 3N r 1 /2 X 4 1 N b Z 2 a,r 1 | G b . ( 46
)
The second assertion in Lemma 7 entails N r 1 /2 [X 4 1 ] 2N r 1 /2 S 4 r 1 /2,r,ε . Moreover, we can use that S r 1 /2,r,ε is smaller than Z r 1 /2 (r n ), the total number of T (r 1 /2)-balls of radius r n which has under N r 1 /2 a geometric distribution with success probability r n /r 1 < 1/2. Thus,

N r 1 /2 X 4 1 2N r 1 /2 S 4 r 1 /2,r,ε 2N r 1 /2 Z r 1 /2 (r n )) 4 48 1 -r n /r 1 r 1 r n 4 96 r 1 r n 4 . (47)
In addition, according to the branching property, under

N b , conditionally on G b , Z a,r 1 is a Poisson variable with mean N (h(T ) > r 1 /2) ℓ b (T ) = (2/r 1 )ℓ b (T ). Thus, N b Z 2 a,r 1 = (2/r 1 )N b ℓ b (T ) + (2/r 1 ) 2 N b ℓ b (T ) 2 . ( 48
)
Recalling that b = r 1 /2, we get We state now the main technical Lemma of the paper. Let us recall from (44) the definition of µ(r, ε). The proof of the lemma makes use of the spinal decomposition described in Section 3.1. In particular, a geometric argument allows to rely the problem to the variables introduced in (37).

(2/r 1 )N b ℓ b (T ) = 1 and (2/r 1 ) 2 N b ℓ b (T ) 2 = 8a 2 /r 2 1 . We assumed that a/r 1 > 1, thus N b Z 2 a,r 1
Lemma 11 Let r = {r 1 , . . . , r n }, where r 1 > . . . > r n > 0, and ε = {ε 1 , . . . , ε n-1 }, where ε 1 > . . . > ε n-1 > 0. The following inequality holds :

µ(r, ε) 5 r n n-1 k=1 P Λ * r k+1 ,r k ε k . ( 49
)
Proof. Let r = {r 1 , . . . , r n } and {ε 1 , . . . ε n-1 } as above. In that proof, we denote, for convenience, b = r 1 /2; hence, a dependency with respect to b is actually a dependency with respect to r. Let us consider Γ a T (b)-ball of radius r n and recall the notation (41). The ball Γ is (r, ε)-small iff ((42)) holds. But, for all σ ∈ Γ, k ∈ 1, n-1 ,

Γ[r k ] = Γ(σ, r k ) ⊃ Γ(σ, r k ) \ Γ(σ, r k+1 ).
Thus, if Γ is (r, ε)-small, then all the vertices in Γ belong to the set

S (r, ε) := σ ∈ T (b) : ∀k ∈ {1 . . . n-1} ℓ b (Γ(σ, r k ) \ Γ(σ, r k+1 )) ε k . ( 50
)
The last set is easy to handle using the spinal decomposition. Indeed, according to (38) and the independence stated in Lemma 6, one has

ν(r, ε) := N ℓ b (dσ)1 {σ∈S (r,ε)} = n-1 k=1 P Λ * r k+1 ,r k ε k (51)
To rely µ(r, ε) and ν(r, ε), one can write

1 {Γ is (r,ε)-small} 1 {ℓ b (Γ) rn √ ν(r,ε)} + ℓ b (Γ) r n ν(r, ε) 1 {Γ is (r,ε)-small} . (52)
Moreover, (50

) entails that ℓ b (Γ)1 {Γ is (r,ε)-small} Γ ℓ b (dσ)1 {σ∈S (r,ε)} .
Recall now from Proposition 1 (i) that the balls of the set B b,rn are pairwise disjoint. Summing in (52) over this set entails S b,r,ε Γ∈B b,rn

1 ℓ b (Γ) rn √ ν(r,ε) + ℓ b (dσ)1 {σ∈S (r,ε)} r n ν(r, ε) . (53)
Now, recalling Lemma 4, we compute

N b   Γ∈B b,rn 1 ℓ b (Γ) rn √ ν(r,ε)   = N b [Z b,rn ] 1 -exp -(2/r n )r n ν(r, ε) r 1 r n 2 ν(r, ε),
so the N-measure of the first term in (53) is smaller than 2b -1 r 1 rn ν(r, ε). Recalling that b = r 1 /2, we get that the latter equals 4 rn ν(r, ε). Moreover, by the mere definition (51), the N-measure of the second term in (53) equals 1 rn ν(r, ε), so the first inequality is checked.

Proof of Theorem 1.

The proof of Theorem 1 will combine the following two theorems.

Theorem 2 Let κ ∈ ( 1 2 , ∞) and m ∈ (0, 1 2 ). Then, there exists a Borel subset V = V(κ, m) ⊂ C 0 such that N C 0 \ V = 0 and such that on V, for all Borel subset A ⊂ T , ∀a ∈ [m, m -1 ], ℓ a (A) κH g (A ∩ T (a)) .
For all a, α ∈ (0, ∞), let us set

∆ α a := σ ∈ T (a) : lim sup r→0 ℓ a (B(σ, r)) g(r) < α . (54)
Theorem 3 Let α ∈ (0, 1 2 ) and m ∈ (0, 1/2). Then, there exists a Borel subset

V ′ = V ′ (α, m) ⊂ C 0 such that N C 0 \ V ′ = 0 and such that on V ′ , ∀a ∈ [m, m -1 ], H g (∆ α a ) = 0.
The proofs of Theorem 2 and 3 share a common strategy, taken from Perkins [START_REF] Perkins | A space-time property of a class of measure-valued branching diffusions[END_REF][START_REF] Perkins | The Hausdorff measure of the closed support of super-Brownian motion[END_REF]. We need to control the mass, or the number of "bad" T (a)-balls where "bad" means too large or too small. And we want to do it uniformly for all levels a. This problem will be linked with a discrete one using a finite grid, and the measure or the number of bad T (a)-balls will be compared with a convenient multiple of ℓ a (T ), the total mass at level a.

4.1 Proof of Theorem 2.

Large balls.

Let us fix a level a ∈ (0, ∞), and recall from section 3.1 the definition of the sigma-field G a , generated by the excursion below level a. We also recall the definition of T (a)-balls [START_REF] Rogers | Functions continuous and singular with respect to a Hausdorff measure[END_REF]. We fix a threshold y ∈ (0, ∞) and we consider the following set of "large" points on T (a) :

L a,r,y = {σ ∈ T (a) : ℓ a (Γ(σ, r)) > y}. (55)
According to Lemma 4, the "total large mass" ℓ a (L a,r,y ) = Γ∈Ba,r ℓ a (Γ)1 {ℓ a (Γ)>y} is G ameasurable.

Lemma 12 For all a, l, y, r, δ ∈ (0, ∞), for all c ∈ (1, ∞),

N ℓ a L a,r,y/c l ; sup b∈[a,a+δ] ℓ b L b,r,y > 4l 1 a exp (-l/δ) + 2 r exp -(1-c -1/2 ) 2 y/δ .
Proof. For all a ∈ (0, ∞), and all c ∈ (1, ∞), we define A 0 , a Borel subset of C 0 , as the event

A 0 = ℓ a L a,r,y/c l ; sup b∈[a,a+δ] ℓ b (L b,r,y ) > 4l . (56)
We recall from Proposition 1 that for r ∈ (0, ∞), B a,r = {Γ i , 1 i Z a,r } is the collection of T (a)-balls of radius r at level a. For Γ a T (a)-ball and b ∈ [a, ∞), we defined

Γ b = {σ ∈ T (b) : ∃σ ′ ∈ Γ, σ ′ ∈ ρ, σ
} as the set of vertices at level b having an ancestor in Γ (see (29) for details). Next we define A 1 a Borel subset of C 0 as the event

A 1 := ∃i ∈ {1, . . . , Z a,r }, ℓ a (Γ i ) y/c and sup b∈[a,a+δ] ℓ b Γ b i > y , (57)
and set

L b a,r,y/c := i∈{1,...,Za,r } ℓ a (Γ i ) y/c Γ b i ⊂ T (b), (58)
which is the set of all vertices at level b having an "large" ancestor at level a. We prove the following :

on C 0 \ A 1 , ∀b ∈ [a, a + δ] L b,r,y ⊂ L b a,r,y/c . (59)
Proof of (59). Let b a r/2 > 0 and let σ ∈ L b,r,y . Thus, the ball

Γ := Γ(σ, r) ∈ B b,r is such that ℓ b (Γ)
y. Let σ a the unique ancestor of σ at level a, namely d(ρ, σ a ) = a and σ a ∈ ρ, σ . We set Υ := Γ(σ a , r) ∈ B a,r and we first claim that Γ ⊂ Υ b . Indeed, let σ ′ ∈ Γ (so d(σ, σ ′ ) < r) and let σ ′ a the unique ancestor of σ ′ at level a. Recalling that σ ∧ σ ′ stands for the most recent common ancestor of σ and σ ′ , one get

d(ρ, σ ∧ σ ′ ) = 1 2 (2b -d(σ, σ ′ )). Then two cases may occur. First, if d(σ, σ ′ ) 2(b -a), then d(ρ, σ ∧ σ ′ ) a, thus σ a = σ ′ a and σ ′ ∈ Υ b . If d(σ, σ ′ ) ∈ (2(b -a), r). Then, one has d(ρ, σ ∧ σ ′ ) < a. We deduce from that inequality that σ a = σ ′ a and that σ a ∧ σ ′ a = σ ∧ σ ′ . Hence, d(σ a , σ ′ a ) = 2a -2d ρ, σ a ∧ σ ′ a = 2b -2d ρ, σ ∧ σ ′ + 2a -2b = d(σ, σ ′ ) -2(b -a) < r. Thus σ ′ a ∈ Υ and σ ′ ∈ Υ b , which ends the proof of the inclusion Γ ⊂ Υ b . We then get ℓ b (Υ b ) ℓ b (Γ) > y. On C 0 \ A 1 , one cannot have both ℓ a (Υ)
y/c and ℓ b (Υ b ) > y, which entails that here, ℓ a (Υ) > y/c. To sum up, on C 0 \ A 1 , a vertex σ, taken in L b,r,y , has an ancestor in a ball Υ, such that ℓ a (Υ) y/c. Thus, this ancestor belongs to L a,r,y/c and σ ∈ L b a,r,y/c . End of the proof of (59).

Let us finish the proof of the lemma. From (59), we see that

N(A 0 ) N (A 1 ) + N A 0 ∩ (C 0 \ A 1 ) N (A 1 ) + N (A 2 ) , (60)
where A 2 is defined by

A 2 := ℓ a L a,r,y/c l ; sup b∈[a,a+δ] ℓ b L b a,r,y/c 4l . (61)
We control N(A 1 ) and N(A 2 ) thanks to Lemma 8 (ii). Indeed, Lemma 5 states that under N a , conditionally on G a , the processes

ℓ a+a ′ (Γ a+a ′ i ), a ′ 0 ,1 i Z a,r are independent Feller diffusions started at ℓ a (Γ i ), 1 i Z a,r . Since ℓ b L b a,r,y/c = Za,r i=1 1 {ℓ a (Γ i )>y/c} ℓ b (Γ b i ), it implies that ℓ a+a ′ (L a+a ′
a,r,y/c ), a ′ 0 is a Feller diffusion started at ℓ a L a,r,y/c . Thus, on the one hand, sub-additivity and Lemma 8 (ii) entails

N (A 1 ) 1 a N a   Za,r i=1
1 {ℓ a (Γ i ) y/c} exp -δ -1 √ y -ℓ a (Γ i ) 2   1 a exp -(1-c -1/2 ) 2 δ -1 y N a [Z a,r ] = 2 r exp -(1-c -1/2 ) 2 δ -1 y . ( 62
)
On the other hand,

N (A 2 ) 1 a N a 1 {ℓ a (La,r,y/c) l} exp -δ -1 2 √ l -ℓ a L a,r,y/c 2 1 a exp (-l/δ) . (63)
Hence, the desired result follows from (60), (62), and (63).

Recall that g(r) = r log log(1/r). We fix κ ∈ ( 1 2 , ∞), and we shall apply the previous lemma with y = κg(r). The next lemma allows to control ℓ a L a,r,κg(r) uniformly for all levels a. Its proof involves a discrete grid : for m < 1/2 and r ∈ (0, ∞), we set

G(r, m) := m + kδ r , k ∈ N * ∩ [m, m -1 ], (64)
where δ r is the mesh of the grid, defined by

δ r = r 3/2 . ( 65
)
Note that G(r, m) contains less than (mδ r ) -1 points.

Lemma 13 Let m ∈ (0, 1/2). Let κ ∈ ( 1 2 , ∞) and β ∈ (1, ∞) such that 2κ -β > 0.
There exists a constant r 1 ∈ (0, ∞) only depending on κ, β, m, such that

∀r ∈ (0, r 1 ), N sup b∈[m,m -1 ] ℓ b L b,r,κg(r) > 4 log(1/r) -β log(1/r) -2 . (66)
Proof. In what follows, we denote T 0 the left-hand-side of (66). Let us consider c ∈ (1, ∞) such that 2κ/cβ > 0. Recall that G(r, m) stands for the grid defined by (64). Then we have T 0 T 1 + T 2 , where :

T 1 = N sup a∈G(r,m) ℓ a L a,r,κg(r)/c log(1/r) -β ; sup b∈[m,m -1 ] ℓ b L b,r,κg(r) 4 log(1/r) -β , T 2 = N sup a∈G(r,m)
ℓ a L a,r,κg(r)/c > log(1/r) -β .

Using sub-additivity and Lemma 12, one get

T 1 = N a∈G(r,m) ℓ a L a,
(mδ r ) -1 m -1 exp -δ -1 r log(1/r) -β + 2 r exp -(1-c -1/2 ) 2 κδ -1 r g(r)
One has δ -1 r log(1/r) -β r -1 and δ -1 r g(r) r -1/2 for all r sufficiently small. Thus, for example, T 1 exp(-r -1/4 ) (1/2) log(1/r) -2 for all r sufficiently small.

We set T 2 T 3 + T 4 , where

T 3 = N sup a∈G(r,m) ℓ a L a,r,κg(r)/c -λ(r, κ, c)ℓ a (T ) > 1 2 log(1/r) -β , T 4 = N sup a∈G(r,m) λ(r, κ, c)ℓ a (T ) > 1 2 log(1/r) -β .
By sub-additivity and a Markov inequality involving a moment of order 4, we get

T 3 (mδ r ) -1 sup a∈G(r,m) N ℓ a L a,r,κg(r)/c -λ(r, κ, c)ℓ a (T ) > 1 2 log(1/r) -β (mδ r ) -1 2 4 log(1/r) 4β sup a∈G(r,m)
a -1 N a ℓ a L a,r,κg(r)/cλ(r, κ, c)ℓ a (T ) 4 .

(69)

Recall notation B a,r = {Γ i , 1 i Z a,r } for the set of T (a)-balls with radius r. Then, consider the decomposition

ℓ a L a,r,κg(r)/c -λ(r, κ, c)ℓ a (T ) = Za,r i=1 X i ,
where X i := ℓ a (Γ i ) 1 {ℓ a (Γ i ) κg(r)/c}λ(r, κ, c) . Using Lemma 4, we see that under N a , conditionally on Z a,r , the random variables ℓ a (Γ 1 ), . . . ℓ a (Γ Za,r ) are independent and exponentially distributed with mean r/2. Thus, the definition (67) of λ(r, κ, c) entails that under N a , conditionally on Z a,r , the r.v. X 1 , . . . X Za,r are i.i.d., with mean 0 and a moment of order 4. Then, by Lemma 7,

N a     Za,r i=1 X i   4   3N a (X 4 1 )N a Z 2 a,r . (70)
From (68), we know that λ(r, κ, c) r→0 -→ 0, so for all sufficiently small r, λ(r, κ, c) 1/2 and |X 4 1 | ℓ a (Γ 1 ), which implies N a [X 1 ] N a [ℓ a (Γ 1 ) 4 ] = 3 2 r 4 for all sufficiently small r. Moreover, Z a,r is under N a a geometric r.v. with "success" probability p = r/2a (see Lemma 4), thus N a Z 2 a,r = (2p)/p 2 8a 2 /r 2 . Combining (69) and (70), we get, for all sufficiently small r,

T 3 3.2 4 .(mδ r ) -1 log(1/r) 4β sup a∈G(r,m) a -1 3r 4 2 8a 2 r 2 10 3 m -2 log(1/r) 4β r 1/2 , (71)
recalling that δ r = r 3/2 . Observe now that the right hand side is smaller than (1/4) log(1/r) -2 for all sufficiently small r.

For the term T 4 , Lemma 9 entails

T 4 N sup b∈[m,m -1 ] ℓ b (T ) > 1 2 λ(r, κ, c) -1 log(1/r) -β (2/m) exp -(m/4)λ(r, κ, c) -1 log(1/r) -β . (72)
By (68),

λ(r, κ, c) -1 log(1/r) -β ∼ r→0 c 2κ log(1/r) 2κ/c-β log log(1/r) -1 .
Recall that 2κ/c > 0 and take ε ∈ (0, 2κ/c-β). Thus, for all sufficiently small r,

T 4 (2/m) exp (-log(1/r) ε ) ,
which is smaller than (1/4) log(1/r) -2 for all sufficiently small r.

Moreover, log(1/r p ) -β = log(y) -β p -β , and recall that β > 1. Thus, (75) entails that there exists a Borel subset V y ⊂ C 0 , such that N(C 0 \ V y ) = 0, and on V y :

∀a ∈ [m, m -1 ], ∞ p=1 ℓ a L a,rp,κg(rp) = ∞ p=1 ℓ a ({σ : ℓ a (B(σ, r p )) > κg(r p )}) < ∞.
We can apply again the Borel-Cantelli Lemma, to the finite measures ℓ a to get that,

on V y ∀a ∈ [m, m -1 ], ℓ a (dσ)-a.e. ∃p 0 (a, σ), ∀p p 0 (a, σ), ℓ a (B(σ, r p )) g(r p ) κ. (76)
If u ∈ (r p+1 , r p ], one has ℓ a (B(σ,u))

g(u) < ℓ a (B(σ,rp)) g(r p+1 ) y ℓ a (B(σ,rp)) g(rp)
. Combined with (76), this entails that on V y , for all a in [m, m -1 ], for ℓ a -almost every σ in T (a), lim sup r→0 ℓ a (B(σ, r))/g(r) < yκ. This can be rewritten in

on V y , ∀a ∈ [m, m -1 ], ℓ a (T (a) \ ∆ yκ a ) = 0. (77)
Now set V = {V y ; y > 1; y ∈ Q}. Clearly, N(C 0 \ V) = 0 and by monotonicity, for all

κ ′ ∈ (κ, ∞), T (a) \ ∆ κ ′ a ⊂ y>1;y∈Q {T (a) \ ∆ yκ a }. It follows easily from (77) that on V, ∀a ∈ [m, m -1 ], ∀κ ′ ∈ (κ, ∞) ℓ a T (a) \ ∆ κ ′ a = 0. (78)
Thus, using Lemma 2, we get :

on V ∀A Borel subset of T ∀a ∈ [m, m -1 ] ∀κ ′ ∈ (κ, ∞) ℓ a (A) = ℓ a A ∩ ∆ κ ′ a κ ′ H g A ∩ ∆ κ ′ a κ ′ H g (A ∩ T (a)) .
This ends the proof of Theorem 2 letting κ ′ ց κ.

4.2 Proof of Theorem 3.

Small balls.

For given level a ∈ (0, ∞) and r ∈ (0, ∞) we recall the notation B a,r for the set of T (a)-balls of radius r. We recall from (41) that for r r

′ > 0, a ball Γ ∈ B a,r ′ is contained in a unique ball in B a,r , denoted Γ[r]. Let r = {r 1 > . . . > r n } and ε = {ε 1 > . . . > ε n-1 }. Recall from (42) that Γ, a T (a)-ball of radius r n is (r, ε)-small iff ∀k ∈ 1, n-1 ℓ a (Γ[r k ]) ε k .
The total number of (r, ε)-small balls at level a is denoted by S a,r,ε (see (43)). For u ∈ (0, ∞), we write ur for the set {ur 1 > . . . > ur n }. We recall from (28) the following notation : if Γ is a T

> . . . > ε n-1 } and c ∈ (1, ∞). Let α ∈ (0, 1/2) and α ∈ (α, 1/2). If δ < c-1 2c r n , then N sup b∈[a,a+δ] S b,r,αε > S a,c -1 r, αε 2n r n exp - √ α- √ α 2 ε n-1 /δ .
Proof. Let us denote B 0 = sup b∈[a,a+δ] S b,r,αε > S a,c -1 r, αε . Next, we define the event B 1 by

B 1 = ∃k ∈ {1 . . . , n-1}, ∃Γ ∈ B a,r k /c : ℓ a (Γ) αε k and inf b∈[a,a+δ] ℓ b (Γ b ) < αε k . (79)
We will prove that B 0 ⊂ B 1 , that is to say

on C 0 \ B 1 , sup b∈[a,a+δ] S b,r,αε S a,c -1 r, αε . (80)
Proof of (80). We work deterministically on C 0 \B 1 . The inequality (80) follows from the following claim.

For every b ∈ [a, a + δ], for every Γ a T (b)-ball of radius r n which is (r, αε)-small, there exists Υ a T (a)-ball of radius r n /c such that Υ is (c -1 r, αε)-small and Υ b ⊂ Γ.

Assume that the latter is true. Then, to any (r, αε)-small ball at level b corresponds a (c -1 r, αε)small ball at level a and the correspondence is injective. Summing over all T (b)-ball, we obtain (80). Now let b ∈ [a, a + δ] and Γ ∈ B b,rn such that Γ is (r, αε)-small. Let σ ∈ Γ and let σ a its unique ancestor at level a. Namely σ a ∈ T (a) and σ a ∈ ρ, σ . We denote Υ = Γ(σ a , r n /c) ∈ B a,rn/c the T (a)-ball of radius r n /c that contains σ a . We claim that Υ is (c -1 r, αε)-small and that Υ b ⊂ Γ.

To prove this, we show ∀k ∈ {1, . . . , n}

(Υ[r k /c]) b ⊂ Γ[r k ]. (81
d(σ, γ) = 2b -2d(ρ, σ ∧ γ) = 2a -2d(ρ, σ a ∧ γ a ) + 2b -2a d(σ a , γ a ) + 2δ < r k c + c -1 c r n r k ,
where we used in the last line that

σ a ∈ Γ ⊂ Γ[r k /c]. In both cases, d(σ, γ) < r k so γ ∈ Γ(σ, r k ) = Γ[r k ],
the last equality being a consequence of Proposition 1 (ii), and the definition of Γ = Γ(σ, r n ). Thus, (81) is proved and it implies

∀k ∈ {1 . . . n-1} ℓ b (Υ[r k /c]) b ℓ a (Γ[r k ]) αε k , which, on C 0 \ B 1 , implies ∀k ∈ {1 . . . n-1} ℓ a (Υ[r k /c]) αε k .
This entails that Υ is (c -1 r, αε)-small. The inclusion Υ b ⊂ Γ was proved at line (81

) with k = n because Υ = Υ[r n /c] ⊂ Γ[r n ] = Γ.
End of the proof of (80)

As in the proof of Lemma 12, we can use the fact that under N a , conditionally on G a , if Γ is a T (a)-ball, then the process ℓ a+a ′ (Γ a+a ′ ), a ′ 0 is a Feller diffusion started at ℓ a (Γ). Using sub-additivity and Lemma 8 (i), we get

N (B 1 ) n-1 k=1 1 a N a   Za,r k i=1 1 {ℓ a (Γ i ) αε k } exp -δ -1 ℓ a (Γ i ) - √ αε k 2   (82) 1 a exp -δ -1 αε k - √ αε k 2 n-1 k=1 E [Z a,r k ] (83)
The proof is completed recalling that for all k ∈ {1 . . . n-1}, ε k ε n-1 , and that, by Lemma 4,

N a [Z a,r k ] = 2a r k 2a rn . Let us introduce ∀j ∈ N, r j = 2 -j and ε j = g(r j ) (84)
and then ∀p ∈ N, j p = ⌊(4/3) p ⌋, r (p) = {r j , j p j j p+1 -1} and ε (p) = {ε j ; j p j < j p+1 -1}. (85)

Let m ∈ (0, 1/2), we also introduce the following discrete grid

G ′ (p, m) := m -1 + kδ p , k ∈ N * ∩ [m, m -1 ], (86)
where δ p is the mesh of the grid, given by δ p = r(j p+1 ) 5/4 . (87)

Note that G ′ (p, m) contains less than (mδ p ) -1 points.

Lemma 15 Let α ∈ (0, 1/2), m ∈ (0, 1/2). For p ∈ N, denote u p := g (r(j p+1 )) -1 p -2 . Then there exists p 0 ∈ N only depending on α, m such that for all p p 0 , (mδ p ) -1 2(j p+1j p ) r(j p+1 ) exp -√ α-√ α 2 δ -1 p g(r(j p+1 -2)) .

One has δ -1 p g(r(j p+1 -2)) δ -1 p g(r(j p+1 )) = r(j p+1 ) -1/4 log log 1/r(j p+1 ), which implies that T ′ 1 is smaller than (1/2)p -2 , for all p sufficiently large (it is obviously not a sharp bound).

Recalling the definitions (44), we set where q j = P Λ * r j+1 /c,r j /c > αr j log log(1/r j ) . Recalling that r j = 2 -j , it follows from Lemma 6 that q j = 1-1 2

2 2α r j log log 1/r j r j /c exp -2α r j log log 1/r j r j /c + 1-1 4 exp -2α r j log log 1/r j r j /c ∼ j→∞ αc 2 log log(1/r j )e -2 αc log log(1/r j ) ∼ j→∞ c ′ log(j)j -2 αc , where c ′ is a positive constant depending on α, α, c. We stress that the particular choice of c was made to ensure that χ := 1 -2αc is strictly positive, so that the following is true for all large p : Thus, for all p sufficiently large, j p+1 jp q j 2p which, combined with (95), entails that u -1 p µ p 5p 3 exp (-p). Thus, u p µ -1 p 5 -1 p -3 e p . Finally, we see from (93) that T ′ 3 is smaller than (1/4)p -2 for all p sufficiently large, which ends the proof.

Proof of Theorem 3.

Let α ∈ (0, 1/2). For a level a ∈ (0, ∞), we recall the definition (54) of ∆ α a . To show that the g-Hausdorff measure of ∆ α a is null, we need an efficient covering of this set. Let us recall the integer sequence j p = ⌊(4/3) p ⌋ and the radii r j = 2 -j . For p ∈ N, we recall the definition of the finite subsets r (p) = {r j , j p j j p+1 -1}, and ε (p) = {ε j , j p j < j p+1 -1} where ε j = g(r j ). Recalling the definition (42) for small balls, we set S a,r (p) ,αε (p) > u p < ∞,

29

where we recall the notation u p = g (r(j p+1 )) -1 p -2 . By Borel-Cantelli lemma there exists a subset V ′ ⊂ C 0 such that N C 0 \ V ′ = 0 and such that on V ′ , g (r(j p+1 )) sup

a∈[m,m -1 ]
S a,r (p) ,αε (p) p -2 , for all suff. large p.

Combined with (96), we deduce on V ′ , for a ∈ [m, m -1 ], one has

H g (∆ α a ) lim n→∞ ∞ p=n p -2 = 0,
which is the desired result.

Proof of Theorem 1.

Let κ ∈ ( 1 2 , ∞), α ∈ (0, 1 2 ), and m ∈ (0, 1/2). Theorem 2 entails that there exists a Borel subset V = V(κ, m) ⊂ C 0 such that N C 0 \ V = 0 and on V(κ, m), for all Borel subset A ⊂ T , ∀a ∈ [m, m -1 ], ℓ a (A) κH g (A ∩ T (a)) .

Clearly, Ṽ is a Borel subset of C 0 such that N C 0 \ Ṽ = 0. Moreover, combining (98) and

(102), we get that on Ṽ, for all Borel subset A ⊂ T , and for all level a ∈ (0, ∞), one has ℓ a (A) = 1 2 H g (A ∩ T (a)).

  Subtrees above level b. Let b ∈ [0, ∞) and denote by (T o,b j ) j∈J b the connected components of the open set {σ ∈ T : d(ρ, σ) > b}:

  ět

  ) and as consequence of Itô's decomposition of Brownian motion above its infimum, N * a is a Poisson point measure on [0, ∞) × C 0 with intensity 21 [0,a] (h)dh N(de). Let us interpret this decomposition in terms of the Brownian tree. Choose t ∈ (0, ζ) such that e t = a and set σ = p(t) ∈ T (namely σ ∈ T (a)). Then, the geodesic [[ρ, σ]] is interpreted as the ancestral line of σ. Let us 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

  µy , and the result follows by optimizing the same function as before.

  (a) is the a-level set of the Brownian tree T . If r ∈ [0, 2a], we recall from[START_REF] Rogers | Functions continuous and singular with respect to a Hausdorff measure[END_REF] the notation Γ(σ, r) for the T (a)-ball of radius r and center σ ∈ T (a), the set of T (a)-balls of radius r being denoted B a,r . Let Γ be a T (a)-ball of radius r ′ , where r ′ ∈ [0, 2a]. From Proposition 1 (iii), we know that if r ∈ [r ′ , 2a], there exists a unique T (a)-ball of radius r that contains Γ, and we shall denote this "enlarged" ball by Γ[r] := Υ where Υ ∈ B a,r and Γ ⊂ Υ.

9a 2 /r 2 1 .c 0 a 2 r 2 1 r 4 n,

 114 Combined with (46) and (47) it entails N b (S a,r,εµ(r, ε)ℓ a (T ))4 with c 0 a positive constant smaller than (1/2)10 4 . This implies the desired result, using that N (h(T ) > b) = 1/b 2/a.

  r,κg(r)/c log(1/r) -β ; sup b∈[a,a+δr] ℓ b L b,r,κg(r) 4 log(1/r) -β (mδ r ) -1 sup a∈G(r,m) N ℓ a L a,r,κg(r)/c log(1/r) -β ; sup b∈[a,a+δr ] ℓ b L b,r,κg(r) 4 log(1/r) -β

Let us bound T 2 .

 2 To that end, we set λ(r, κ, c) := (2/r)E E1 {E>κg(r)/c} , (67) where E is a r.v. on (Ω, F, P) exponentially distributed with mean r/2. For fixed κ and c, elementary computations entail λ(r, κ, c) = (2/r)P (E > κg(r)/c) E[κg(r)/c + E] = (2/r) exp (-2(κ/c) log log 1/r) ((κ/c)r log log 1/r + r/2) ∼ r→0 (2κ/c) log(1/r) -2κ/c log log 1/r.

4. 1 . 2

 12 Proof of Theorem 2.Let κ ∈ (1/2, ∞), and let m ∈ (0, 1/2). Let β ∈ (1, ∞) such that 2κβ > 0. For all a ∈ (0, ∞), y ∈ (1, ∞) recall from (54) the definition :∆ yκ a = σ ∈ T (a) : lim sup r→0 ℓ a (B(σ, r)) g(r) < yκ .(73)For any p ∈ N, set r p := y -p . By Lemma 13, for all sufficiently large p,N sup a∈[m,m -1 ] ℓ a L a,rp,κg(rp) > 4 log(1/r p ) -β log(1/r p ) -2 = log(y) -2 p -2 ,(74)whose sum over p is finite. By Borel Cantelli lemma, N-a.e., for all sufficiently large p, supa∈[m,m -1 ]ℓ a L a,rp,κg(rp) 4 log(1/r p ) -β .

  a)-ball, then, for all b a, Γ b is the subset of all the vertices in T (b) having an ancestor in Γ. Namely, Γ b = {σ ∈ T (b), ∃σ ′ ∈ Γ : σ ′ ∈ ρ, σ }. Lemma 14 Let a, δ ∈ (0, ∞), and n 2. Let r = {r 1 > . . . > r n } and ε = {ε 1

) 1 2

 1 Let k ∈ {1 . . . n} and let γ ∈ (Υ[r k /c]) b . Its unique ancestor at level a, denoted γ a , is such that γ a ∈ Υ[r k /c]. Two cases may occur. First, if d(σ, γ) 2(ba), then we have 2(ba) 2δ < c-1 c r n < r n r k . The other case corresponds to d(σ, γ) > 2(ba). Then d(ρ, σ ∧ γ) = (2bd(σ, γ)) < a. Thus, σ ∧ γ = σ a ∧ γ a and we have

N

  sup b∈[m,m -1 ] S b,r (p) ,αε (p) > u p p -2 . (88) Proof. Let α ∈ (α, 1/2) and c in (1, ∞) such that 2cα ∈ (0, 1). In what follows, we denote T ′ 0 the left-hand-side of (88). Observe that T ′ 0 T ′ 1 + T ′ 2 , where we have set T ′ 1 = N sup a∈G ′ (p,m) S a,c -1 r (p) , αε (p) u p ; sup b∈[m,m -1 ] S b,r (p) ,αε (p) > u p , T ′ 2 = N sup a∈G ′ (p,m) S a,c -1 r (p) , αε (p) > u p . Using sub-additivity and Lemma 14, we get r (p) ,αε (p) > S a,c -1 r (p) , αε (p)   (mδ p ) -1 sup a∈G ′ (p,m) N sup b∈[a,a+δp] S b,r (p) ,αε (p) > S a,c -1 r (p) , αε (p)

j p+1 -2 jp q j j p+1 - 2 jpj -2 αc j p+1 - 1 jpx - 2

 212 αc dx ∼ p→∞ χ -1 ((4/3) χ -1)4 3pχ .

C

  n := ∞ p=n Γ ∈ B a,r(j p+1 ) : Γ is (r(p) , αε(p) )-small .Observe that if σ ∈ ∆ α a , then the T (a)-ball Γ (σ, r(j p+1 )) is (r (p) , αε (p) )-small for all large p, thus for all n ∈ N, we have ∆ α a ⊂ C n . Let us recall the definition (15) of Hausdorff measures, and the fact that the diameter of a T (a)-ball is smaller than its radius. We get∀a ∈ [m, m -1 ] H (r(j p+1 )) r (p) ,αε (p) . g (r(j p+1 )) , (96) because ∆ α a ⊂ C n . Thus, ∀a ∈ [m, m -1 ] H g (∆ α a ) lim sup n→∞ ∞ p=n S a,r (p) ,αε (p) .g (r(j p+1 )) .(97)Now, let m ∈ (0, 1/2). Applying Lemma 15, we easily get that∞ p=1 N sup a∈[m,m -1 ]

  (98) Now, let us rewrite the definition (54)∆ α a = σ ∈ T (a) : lim sup r→0 ℓ a (B(σ, r)) g(r) < α .(99)According to Theorem 3, there exists a Borel subsetV ′ = V ′ (α, m) ⊂ C 0 such that N C 0 \ V ′ = 0 and on V ′ (α, m) ∀a ∈ [m, m -1 ] H g (∆ α a ) = 0. (100)Let α ′ < α and notice that T (a) \ ∆ α a ⊂ σ : lim sup r→0 ℓ a (B(σ,r)) g(r)> α ′ . Moreover, from (13), we know that N-a.e. for all a ∈ (0, ∞), ℓ a (T \ T (a)) = 0. Thus, on V ′ , for all Borel subset A ⊂ T , and for all a ∈ [m, m -1 ] and all α < α, Lemma 3 entailsℓ a (A) ℓ a (A ∩ (T (a) \ ∆ α a )) α ′ H g (A ∩ (T (a) \ ∆ α a )) = α ′ H g (A ∩ T (a)) ,(101)where we used (100) for the last equality. Letting α ′ → α, we get on V ′ (α, m) for all Borel subset A ⊂ T , ∀a ∈ [m, m -1 ] ℓ a (A) αH g (A ∩ T (a)) .

Acknowledgments. I would like to thank my advisor Thomas Duquesne for introducing this problem, as well as for his help and the many improvements he suggested.

µ p = µ(c -1 r (p) , αε (p) ) = N S r(jp)/(2c),c -1 r (p) , αε (p) .

(89)

We will prove that T ′ 2 T ′ 3 + T ′ 4 , where

By sub-additivity and a Markov inequality involving a moment of order 4, we get

We want to apply Lemma 10 with r =, c -1 r (p) and ε = αε (p) . Thus, recalling (84) and ( 85), we check that for all sufficiently large p, m/r(j p ) > 1 and r(j p )/r(j p+1 -1). Recalling that c 0 ∈ (0, 10 4 ] is the universal constant given by Lemma 10, we get from (90)

Recall that u p = g(r(j p+1 )) -1 p -2 , and by ( 84) and (85), we get log log(1/r(j p )) ∼ p→∞ p log(4/3).

Hence, u p p -3 r(j p+1 ) -1 and (91) implies

Now, one can plainly check that r(jp) 2 r(j p+1 ) 5/4 is smaller than r(j p ) 1/3 . Thus, T ′ 3 is smaller than (1/4)p -2 for all p sufficiently large.

For the term T ′ 4 , we use Lemma 9 to obtain

Recalling (89) and Lemma 11, we get that for all p,

We want to get an lower bound of u p µ -1 p , so we compute an upper bound for u -1 p µ p . Recalling that u p p -3 r(j p+1 ) -1 , one has


Laurent Houssin
email: laurent.houssin@laas.fr

A heap of pieces model for the cyclic job shop problem

Keywords:

Introduction

In classical scheduling, a set of tasks is executed once while the determined schedule optimizes objective functions such as the makespan or earliness-tardiness. In contrast, cyclic scheduling means performing a set of generic tasks infinitely often while minimizing the time between two occurrences of the same task. Cyclic scheduling has several applications, e.g. in robotic industry (see [START_REF] Kats | A strongly polynomial algorithm for no-wait cyclic robotic flowshop scheduling[END_REF]), in manufacturing systems (see [START_REF] Pinedo | Planning and Scheduling in Manufacturing and Services[END_REF] and [START_REF] Hillion | Performance evaluation of job-shop systems using timed event graphs[END_REF]) or multiprocessor computing (see Hanen and Munier [1995b]). It has been studied from multiple perspectives, since there exist several possible representations of the problem such as graph theory, mixed integer linear programming, Petri nets or (max, +) algebra. An overview about cyclic scheduling problems and the different approaches can be found in Hanen and Munier [1995a].

However, a less known tool is the heaps of pieces approach. A first paper considering this modelling for cyclic scheduling is [START_REF] Gaubert | Modeling and analysis of timed petri nets using heap of pieces[END_REF]. The authors introduce a method to represent safe Petri nets as particular automata which compute the height of heaps of pieces. This approach enables to compute efficiently the throughput of a cyclic schedule. In this paper, we use the heaps of pieces approach for solving cyclic scheduling problems with resource constraints.

Cyclic scheduling problems

The basic cyclic scheduling problem (BCS) involves generic tasks and precedence constraints between tasks but no resource constraints are considered. In this problem, a set of n generic operations are processed in parallel by an unbounded number of machines and there is a set of q precedence constraints. We denote this set A = {a 1 , . . . , a q } where a l corresponds to a constraint represented by a triple (i, j, h). More precisely, the uniform constraint (i, j, h) means that

s i (k) + p i ≤ s j (k + h), ∀k ≥ 1
where s i denotes the beginning of the operation i and p i is the processing time of i. In this framework, the asymptotic cycle time α(S) is usually minimized (with S a feasible schedule). Equivalently we can aim at maximizing the throughput r(S) = 1 α(S) . We can distinguish two categories of cyclic schedules: the 1-periodic schedules and the K-periodic schedules. The first category is characterized by a period α such that :

s i (k + 1) = α + s i (k), ∀i ∈ T , ∀k ≥ 1,
where T is the set of tasks to perform. Whereas K-periodic schedules are also defined by period which corresponds to a fixed interval time between any K consecutive occurences of i :

s i (k + K) = α K + s i (k), ∀i ∈ T , ∀k ≥ 0.
For our concern, we are interested in the cyclic job shop problem. In this case, tasks are a priori mapped onto machines and the number of machines is smaller than the number of tasks to perform .More precisely, a cyclic job shop is defined by :

• a set T of elementary tasks, • a set R of machines, • for each task t ∈ T , a processing time p t and a machine m t ∈ R on which the task has to be performed, • a set A of uniform constraints (as defined above),

• a set of jobs J corresponding to a production sequence of elementary tasks. More precisely, a job J 1 defines a sequence J 1 = t 11 . . . t 1k to be executed in this order.

Previous studies of this problem have shown that K-periodic schedules are dominant (Hanen and Munier [1995a]) and the problem is NP-hard [START_REF] Hanen | Study of a np-hard cyclic scheduling problem: The recurrent job-shop[END_REF]) for throughput maximization.

One can find the description of two classical methods of performance evaluation of cyclic job shop in Hanen and Munier [1995a]. The first one is based on graph theory. In this framework, the throughput computation problem of a cyclic job shop schedule becomes the search of maximum circuit ratio in a graph. The second approach considers the job shop as a (max, +)-linear system. Then, the spectrum of the evolution matrix of the system gives the cycle time. In both approaches, the complexity of the evaluation of the cycle time is the same (polynomial).

The heap of pieces model

We first recall some algebraic tools concerning (max, +) algebra. The (max, +) semiring is the set R ∪ {-∞} endowed with the max operator, written a ⊕ b = max(a, b), and the usual sum written a ⊗ b = a + b. The sum (resp. product) admits a neutral element denoted ε = -∞ (resp. e = 0), it leads to a ⊕ ε = a and a ⊗ e = a. For matrices, additions and products give

(A ⊕ B) ij = A ij ⊕ B ij and (A ⊗ B) ij = n k=1 A ik ⊗ B kj .
An exhaustive presentation of (max, +) algebra can be found in [START_REF] Baccelli | Synchronization and Linearity[END_REF]. We now present the heap model structure that was introduced in Gaubert and Mairesse [1999].

Definition 1 (Heap model). A heap model is composed by :

• P a finite set of pieces, • S a finite set of slots, • R gives the subset of slots occupied by a piece, • l : P × S → R ∪ {-∞} gives the lower contour of a piece, • u : P × S → R ∪ {-∞} gives the upper contour of a piece.

For each piece p (possibly non connected) of a heap model, we define the matrix M(p)

of dimension |S| × |S| by M(p) sr =      0 if s = r, r / ∈ R(p), u(p) r -l(p) r ifr ∈ R(p), s ∈ R(p), ε
otherwise. Calculus in the heap model are based on (max, +)-algebra. More precisely, if a piece p 1 is piled up in an empty heap, the upper contour of the heap is given by x(p 1 ) = I ⊗ M(p 1 ) where I is a 1 × |S| matrix defined by I j = e, ∀j ∈ {1, . . . , |S|}. In the same manner, the pile of two pieces p 1 and p 2 give the following upper contour of the heap :

x(p 1 p 2 ) = I ⊗ M(p 1 ) ⊗ M(p 2 ).

Application to cyclic job shop

In this section, we show how to model the cyclic job shop problem as a heap of pieces. We take the assumption that constraints a k ∈ A are such that h = 0 (see §2). As shown in [START_REF] Gaubert | Modeling and analysis of timed petri nets using heap of pieces[END_REF], the job shop problem admits a heap realization described under

P = T S = R ∪ J R(t) = m(t) ∪ {J i |t ∈ J i } l(t, r) = 0 if m(t) = r, l(t, r) = ε otherwise u(t, r) = p(t) if m(t) = r, l(t, r) = ε otherwise.
In this framework, each elementary task t is represented by a matrix M(t). Considering the objective of the cyclic job shop problem, the problem is to find the sequence of pieces piled up in the heap that maximize the throughput. The following theorem indicates how to compute the cycle time Theorem 1. The throughput of a job J i for the cyclic sequence p 1 . . . p n in a heap model representing a job shop system is given by

λ Ji = |p 1 . . . p n | Ji ⊗ (ρ(M(p 1 ) ⊗ . . . ⊗ M(p n ))) -1 .
(1)

where |p 1 . . . p n | Ji denotes the number of jobs J i completed under the sequence p 1 . . . p n and ρ(X) is the unique eigenvalue of the irreducible matrix X (see [Baccelli et al., 1992, chap.2]).

Proof. Considering the heap, the cyclic sequence p 1 . . . p n corresponds to a cyclic piled-up of these pieces. Let v = p 1 ⊗. . .⊗p n . The matrix M(v) can be seen as the matrix representing an unique piece p 1 . . . p n . The evolution of the heap is now given by the pile up of v. As seen in section 3, the upper contour of the heap is given by x

(v . . . v) = I ⊗ M(v) ⊗ . . . ⊗ M(v).
Since the job shop is connected (and we assume it is), the matrix M(v) is irreducible. After a certain number of iterations (that is called the transient time), a cyclic behaviour appears for an irreducible matrix X : X k+c = ρ(X) c ⊗ X k (see [START_REF] Cohen | Analyse du comportement périodique des systèmes de production par la théorie des dioïdes[END_REF]). This property hold for M(v) and after a certain number of iterations, the growth rate of the heap become ρ(M(v)) and the throughput is given by (1).

Considering the heap of pieces model, the cyclic job shop problem is equivalent to find a bounded sequence of pieces with the maximum throughput. Compared to classical approaches, this model takes benefit from K-cyclic schedules. Indeed, graph based approach have to consider K × n number of nodes. Regarding mixed integer linear programming, the number of variables grows substantially. To deal with this situation, the heap of pieces is the best model since the number of slots is independant of K. Nonetheless the drawback of this method is that we can only consider a work in progress of one and it can not represent a "nested" sechedule since it can only represent patern that can be piled up (no intertwine). We propose a branch and bound procedure to solve this problem. The branching rule chose the piece to pile up in the heap and the evaluation fonction is the cycle time computation of the pattern in the heap. We extend this model to consider several work in process inside the pattern (but not nested pattern) through an extension of the number of slots.