MAPMO Romain Abraham
email: romain.abraham@univ-orleans.fr

Jean-Franc ¸ois Delmas
email: delmas@cermics.enpc.fr

WILLIAMS' DECOMPOSITION OF THE L ÉVY CONTINUUM RANDOM TREE AND SIMULTANEOUS EXTINCTION PROBABILITY FOR POPULATIONS WITH NEUTRAL MUTATIONS

Keywords: 2000 Mathematics Subject Classification. 60G55, 60J70, 60J80, 92D25 Continuous state branching process, immigration, continuum random tree, Williams' decomposition, probability of extinction, neutral mutation

Williams' decomposition of the Lévy continuous random tree and simultaneous extinction probability for populations with neutral mutations

Romain Abraham, Jean-

Introduction

We consider an initial Eve-population whose size evolves as a continuous state branching process (CB), Y 0 = (Y 0 t , t ≥ 0), with branching mechanism ψ Eve . We assume this population gives birth to a population of irreversible mutants. The new mutants population can be seen as an immigration process with rate proportional to the size of the Eve-population. We assume the mutations are neutral, so that this second population evolves according to the same branching mechanism as the Eve-population. This population of mutants gives birth also to a population of other irreversible mutants, with rate proportional to its size, and so on. In [START_REF] Abraham | Changing the branching mechanism of a continuous state branching process using immigration[END_REF], we proved that the distribution of the total population size Y = (Y t , t ≥ 0), which is a CB with immigration (CBI) proportional to its own size, is in fact a CB, whose branching mechanism ψ depends on the immigration intensity. The joint law of (Y 0 , Y ) is characterized by its Laplace transform, see Section 4.1.4. This model can also be viewed as a special case of multitype CB, with two types 0 and 1, the individuals of type 0 giving birth to offsprings of type 0 or 1, whereas individuals of type 1 only have type 1 offsprings, see [START_REF] Miermont | Invariance principles for spatial multitype Galton-Watson trees[END_REF][START_REF] Bertoin | The structure of the allelic partition of the total population for Galton-Watson processes with neutral mutations[END_REF] for recent related works.

In the particular case of Y being a sub-critical or critical CB with quadratic branching mechanism (ψ(u) = α 0 u + βu 2 , β > 0, α 0 ≥ 0), the probability for the Eve-population to disappear at the same time as the whole population is known, see [START_REF] Warren | Branching processes, the Ray-Knight theorem, and sticky Brownian motion[END_REF] for the critical case, α 0 = 0, or Section 5 in [START_REF] Abraham | Changing the branching mechanism of a continuous state branching process using immigration[END_REF] for the sub-critical case, α 0 > 0. Our aim is to extend those results for the large class of CB with unbounded total variation and a.s. extinction. Formulas given in [START_REF] Abraham | Changing the branching mechanism of a continuous state branching process using immigration[END_REF] could certainly be extended to a general branching mechanism, but first computations seem to be rather involved.

In fact, to compute those quantities, we choose here to rely on the description of the genealogy of sub-critical or critical CB introduced by Le Gall and Le Jan [START_REF] Gall | Branching processes in Lévy processes: The exploration process[END_REF] and developed later by Duquesne and Le Gall [START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF], see also Lambert [START_REF] Lambert | The genealogy of continuous-state branching processes with immigration[END_REF] for the genealogy of CBI with constant immigration rate. Le Gall and Le Jan defined via a Lévy process X the so-called height process H = (H t , t ≥ 0) which codes a continuum random tree (CRT) that describes the genealogy of the CB (see the next section for the definition of H and the coding of the CRT). Initially, the CRT was introduced by Aldous [START_REF] Aldous | The continuum random tree I[END_REF] in the quadratic case: ψ(λ) = λ 2 . Except in this quadratic case, the height process H is not Markov and so is difficult to handle. That is why they also introduce a measure-valued Markov process (ρ t , t ≥ 0) called the exploration process and such that the closed support of the measure ρ t is [0, H t ] (see also the next section for the definition of the exploration process).

We shall be interested in the case where a.s. the extinction of the whole population holds in finite time. The branching mechanism of the total population, Y , is given by: for λ ≥ 0, [START_REF] Abraham | Fragmentation associated with Lévy processes using snake[END_REF] ψ(λ) = α 0 λ + βλ 2 + (0,∞) π(dℓ) e -λℓ -1 + λℓ ,

where α 0 ≥ 0, β ≥ 0 and π is a Radon measure on (0, ∞) such that (0,∞) (ℓ ∧ ℓ 2 ) π(dℓ) < ∞.
We shall assume that Y is of infinite variation, that is β > 0 or (0,1) ℓπ(dℓ) = ∞. We shall assume that a.s. the extinction of Y in finite time holds, that is, see Corollary 1.4.2 in [START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF], we assume that

(2) +∞ dv ψ(v) < ∞.
We suppose that the process Y is the canonical process on the Skorokhod space D(R + , R + ) of càdlàg paths and that the pair (Y, Y 0 ) is the canonical process on the space D(R + , R + ) 2 . Let P x denote the law of the pair (Y, Y 0 ) (see [START_REF] Abraham | Changing the branching mechanism of a continuous state branching process using immigration[END_REF]) started at (Y 0 , Y 0 0 ) = (x, x). The probability measure P x is infinitely divisible and hence admits a canonical measure N: it is a σ-finite measure on

D(R + , R + ) 2 such that (Y, Y 0 ) (d) = i∈I (Y i , Y 0,i )
where ((Y i , Y 0,i ), i ∈ I) are the atoms of a Poisson measure on D(R + , R + ) 2 with intensity xN(dY, dY 0 ). In particular, we have

(3) E x [e -λYt ] = exp(-xN[1 -e -λYt ])
and u(λ, t) = N[1e -λYt ] is the unique non-negative solution of (4)

λ u(λ,t) dv ψ(v) = t, for t ≥ 0 and λ ≥ 0. Let τ Y = inf{t > 0; Y t = 0} be the extinction time of Y . Letting λ go to ∞ in the previous equalities leads to P x (τ Y < t) = exp -xN[τ Y ≥ t], where the positive function c(t) = N[τ Y ≥ t] solves (5) ∞ c(t) dv ψ(v) = t, for t > 0.
Let us consider the exploration process (ρ t , t ≥ 0) associated with this CB. We denote by N its excursion measure. Recall that the closed support of the measure ρ t is [0, H t ], where H is the height process. Let L a be the total local time at level a of the height process H (well-defined under N). Then, the process (L a , a ≥ 0) under N has the same distribution as the CB Y under N.

We decompose the exploration process, under the excursion measure, according to the maximum of the height process. In terms of the CRT, this means that we consider the longest rooted branch of the CRT and describe how the different subtrees are grafted along that branch, see Theorem 3.3. When the branching mechanism is quadratic, the height process H is a Brownian excursion and the exploration process ρ t is, up to a constant, the Lebesgue measure on [0, H t ]. In that case, this decomposition corresponds to Williams' original decomposition of the Brownian excursion (see [START_REF] Williams | Path decomposition and continuity of local time for one-dimensional diffusions[END_REF]). This kind of tree decomposition with respect to a particular branch (or a particular subtree) is not new, let us cite [START_REF] Geiger | The shape of large Galton-Watson trees with possibly infinite variance[END_REF][START_REF] Pitman | Growth of the Brownian forest[END_REF] for instance, or [START_REF] Serlet | The occupation measure of super-Brownian motion conditioned on non-extinction[END_REF][START_REF] Salisbury | On the conditioned exit measures of super Brownian motion[END_REF][START_REF] Etheridge | A decomposition of the (1 + β)-superprocess conditioned on survival[END_REF] for related works on superprocesses.

We present in the introduction a Poisson decomposition for the CB only, and we refer to Theorem 3.3 for the decomposition of the exploration process. Conditionally on the extinction time τ Y equal to m, we can represent the process Y as the sum of the descendants of the ancestors of the last individual alive. More precisely, let

N ′ (dℓ, dt) = i∈I δ (ℓ i ,t i ) (dℓ, dt) be a
Poisson point measure with intensity Let τ Y 0 = inf{t > 0; Y 0 t = 0} be the extinction time of the Eve-population. In the particular case where the branching mechanism of the Eve-population is given by a shift of ψ:

1 [0,m) (t) e -ℓc(m-t) ℓπ(dℓ)dt, and (6)
κ max (dt) = i∈I ℓ i δ t i (dt) + 2β1 [0,m) (t)
(7) ψ Eve (•) = ψ(θ + •) -ψ(θ),
for some θ > 0 and β = 0, the pruning procedure developed in [START_REF] Abraham | Fragmentation associated with Lévy processes using snake[END_REF] gives that the nodes of width ℓ i correspond to a mutation with probability 1e -θℓ i . As β = 0 there is no mutation on the skeleton of the CRT outside the nodes. In particular, we see simultaneous extinction of the whole population and the Eve-population if there is no mutation on the nodes in the ancestral lineage of the last individual alive. This happens, conditionally on κ max , with probability e -θ P i∈I ℓ i .

Integrating w.r.t. the law of N gives that the probability of simultaneous extinction, conditionally on {τ Y = m}, is under N, given by

N[τ Y 0 = m|τ Y = m] = exp -1 [0,m) (t) e -ℓc(m-t) ℓπ(dℓ)dt 1 -e -θℓ = exp - m 0 [ψ ′ (c(m -t) + θ) -ψ ′ (c(m -t))] dt = exp - m 0 φ ′ (c(t)) dt,
where φ = ψ Eveψ. Now, using that the distribution of (Y 0 , Y ) is infinitely divisible with canonical measure N, standard computations for Poisson measure yield that P x (τ

Y 0 = m|τ Y = m) = N[τ Y 0 = m|τ Y = m] that is P x (τ Y 0 = m|τ Y = m) = exp - m 0 φ ′ (c(t)) dt.
Notice that this formula is also valid for the quadratic branching mechanism (ψ(u) = α 0 u + βu 2 , β > 0, α 0 ≥ 0), see Remark 5.3 in [START_REF] Abraham | Changing the branching mechanism of a continuous state branching process using immigration[END_REF].

In fact this formula is true in a general framework. Following [START_REF] Abraham | Changing the branching mechanism of a continuous state branching process using immigration[END_REF], we consider the branching mechanisms of the total population and Eve-population are given by

ψ(λ) = α 0 λ + βλ 2 + (0,∞) π(dℓ)[e -λℓ -1 + λℓ], ψ Eve (λ) = α Eve λ + βλ 2 + (0,∞) π Eve (dℓ)[e -λℓ -1 + λℓ],
and the immigration function

φ(λ) = ψ Eve (λ) -ψ(λ) = α Imm λ + (0,∞) ν(dℓ)(1 -e -λℓ ),
where α Imm = α Eveα 0 -(0,∞) ℓν(dℓ) ≥ 0 and π = π Eve + ν, where π Eve and ν are Radon measures on (0, ∞) with (0,∞) ℓν(dℓ) < ∞. Notice the condition (0,∞) ℓν(dℓ) < ∞ is stronger than the usual condition on the immigration measure, (0,∞) (1 ∧ ℓ) ν(dℓ) < ∞, but is implied by the requirement that (1,∞) ℓν(dℓ) < (1,∞) ℓπ(dℓ) < ∞.

Inspired by Theorem 3.3, we consider N (dℓ, dt, dz)

= i∈I δ (ℓ i ,t i ,z i ) (dℓ, dt, dz) a Poisson point measure with intensity (8) 1 [0,m) (t) e -ℓc(m-t) ℓ [π Eve (dℓ)δ 0 (dz) + ν(dℓ)δ 1 (dz)] dt.
Intuitively, the mark z i indicates if the ancestor (of the last individual alive) alive at time t i had a new mutation (z i = 1) or not (z i = 0). Note however that if β > 0 we have to take into account mutation on the skeleton. More precisely, let T 1 = min{t i , z i = 1} be the first mutation on the nodes in the ancestral lineage of the last individual alive and let T 2 be an exponential random time with parameter α Imm independent of N . The time T 2 corresponds to the first mutation on the skeleton for the ancestral lineage of the last individual alive. We set (9)

T 0 = min(T 1 , T 2 ) if min(T 1 , T 2 ) < m, T 0 = +∞ otherwise.
In particular there is simultaneous extinction if and only if T 0 = +∞.

For t ≥ 0, let us denote by N t (dY 0 , dY ) the joint law of ((Y 0 (st), Y (st)), s ≥ t) under N. Recall κ max given by [START_REF] Bertoin | The structure of the allelic partition of the total population for Galton-Watson processes with neutral mutations[END_REF]. Conditionally on N and T 2 , let j∈J δ (t j ,Y 0,j ,Y j ) be a Poisson point measure, with intensity

κ max (dt)N t [(dY 0 , dY ), 1 {τ Y ≤m} ].
We set [START_REF] Lambert | The genealogy of continuous-state branching processes with immigration[END_REF] (

Y ′ 0 , Y ′ ) = t j <T 0 (Y 0,j , Y j ) + t j ≥T 0 (0, Y j ).
We write Q m for the law of (Y ′ 0 , Y ′ ) computed for a given value of m.

Theorem 1.2. Under Q m , (Y ′ 0 , Y ′ ) is distributed as (Y 0 , Y ) under N[•|τ Y = m] , or equiv- alently, under +∞ 0 |c ′ (m)|Q m (•)dm, (Y ′ 0 , Y ′ ) is distributed as (Y 0 , Y ) under N.
Let us remark that this Theorem is very close to Theorem 3.3 but only deals with CB and does not specify the underlying genealogical structure. This is the purpose of a forthcoming paper [START_REF] Abraham | Pruning a continuum random tree[END_REF] where the genealogy of multi-type CB is described.

Intuitively, conditionally on the last individual alive being at time m, until the first mutation in the ancestral lineage (that is for t j < T 0 ) , its ancestors give birth to a population with initial Eve type which has to die before time m, and after the first mutation on the ancestral lineage (that is for t j ≥ T 0 ), there is no Eve-population in the descendants which still have to die before time m. Now, using that the distribution of (Y 0 , Y ) is infinitely divisible with canonical measure N, standard computations for Poisson measure yield that P x (τ

Y 0 = m|τ Y = m) = N[τ Y 0 = m|τ Y = m]. As N[τ Y 0 = m|τ Y = m] = Q m (T 0 = +∞) = Q m (T 1 = +∞)Q m (T 2 ≥ m) = e - R m 0 dt R (0,∞) e -ℓc(m-t) ℓν(dℓ) e -α Imm m = e - R m 0 dt φ ′ (c(t))
, we deduce the following Corollary.

Corollary 1.3 (Probability of simultaneous extinction).

We have for almost every m > 0

P x (τ Y 0 = m|τ Y = m) = exp - m 0 φ ′ (c(t)) dt,
where c is the unique (non-negative) solution of (5).

The paper is organized as follows. In Section 2, we recall some facts on the genealogy of the CRT associated with a Lévy process. We prove a Williams' decomposition for the exploration process associated with the CRT in Section 3. We prove Theorem 1.2 in Section 4. Notice that Proposition 1.1 is a direct consequence of Theorem 1.2.

Notations

We recall here the construction of the Lévy continuum random tree (CRT) introduced in [START_REF] Gall | Branching processes in Lévy processes: The exploration process[END_REF][START_REF] Gall | Branching processes in Lévy processes: Laplace functionals of snake and superprocesses[END_REF] and developed later in [START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF]. We will emphasize on the height process and the exploration process which are the key tools to handle this tree. The results of this section are mainly extracted from [START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF].

2.1. The underlying Lévy process. We consider a R-valued Lévy process (X t , t ≥ 0) with Laplace exponent ψ (for λ ≥ 0 E e -λXt = e tψ(λ) ) satisfying ( 1) and [START_REF] Abraham | Changing the branching mechanism of a continuous state branching process using immigration[END_REF]. Let I = (I t , t ≥ 0) be the infimum process of X, I t = inf 0≤s≤t X s , and let S = (S t , t ≥ 0) be the supremum process, S t = sup 0≤s≤t X s . We will also consider for every 0 ≤ s ≤ t the infimum of X over [s, t]:

I s t = inf s≤r≤t X r .
The point 0 is regular for the Markov process X -I, and -I is the local time of X -I at 0 (see [START_REF] Bertoin | Lévy processes[END_REF], chap. VII). Let N be the associated excursion measure of the process X -I away from 0, and σ = inf{t > 0; X t -I t = 0} the length of the excursion of X -I under N. We will assume that under N, X 0 = I 0 = 0.

Since X is of infinite variation, 0 is also regular for the Markov process S -X. The local time, L = (L t , t ≥ 0), of S -X at 0 will be normalized so that

E[e -βS L -1 t ] = e -tψ(β)/β ,
where L -1 t = inf{s ≥ 0; L s ≥ t} (see also [START_REF] Bertoin | Lévy processes[END_REF] Theorem VII.4 (ii)).

2.2.

The height process and the Lévy CRT. For each t ≥ 0, we consider the reversed process at time t,

X(t) = ( X(t) s , 0 ≤ s ≤ t) by: X(t) s = X t -X (t-s)-if 0 ≤ s < t,
and X(t) t = X t . The two processes ( X(t) s , 0 ≤ s ≤ t) and (X s , 0 ≤ s ≤ t) have the same law. Let Ŝ(t) be the supremum process of X(t) and L(t) be the local time at 0 of Ŝ(t) -X(t) with the same normalization as L.

.3).

There exists a process H = (H t , t ≥ 0), called the height process, such that for all t ≥ 0, a.s. H t = L(t) t , and H 0 = 0. Because of hypothesis (2), the height process H is continuous.

The height process (H t , t ∈ [0, σ]) under N codes a continuous genealogical structure, the Lévy CRT, via the following procedure.

(i) To each t ∈ [0, σ] corresponds a vertex at generation

H t . (ii) Vertex t is an ancestor of vertex t ′ if H t = H [t,t ′ ] , where (11)
H [t,t ′ ] = inf{H u , u ∈ [t ∧ t ′ , t ∨ t ′ ]}.
In general H [t,t ′ ] is the generation of the last common ancestor to t and t

′ . (iii) We put d(t, t ′ ) = H t + H t ′ -2H [t,t ′ ] and identify t and t ′ (t ∼ t ′ ) if d(t, t ′ ) = 0.
The Lévy CRT coded by H is then the quotient set [0, σ]/ ∼, equipped with the distance d and the genealogical relation specified in (ii).

Let (τ s , s ≥ 0) be the right continuous inverse of -I: In what follows, we will use the notation N instead of N for the excursion measure to stress that we consider the genealogical structure of the branching process.

τ s = inf{t > 0; -I t > s}. Recall that -I is the local time of X -I at 0. Let L a t denote the local time at level a of H until time t, see Section 1.3 in [7].
2.3. The exploration process. The height process is not Markov. But it is a simple function of a measure-valued Markov process, the so-called exploration process.

If E is a Polish space, let B(E) (resp. B + (E)) be the set of real-valued measurable (resp. and non-negative) functions defined on E endowed with its Borel σ-field, and let M(E) (resp. M f (E)) be the set of σ-finite (resp. finite) measures on E, endowed with the topology of vague (resp. weak) convergence. For any measure µ ∈ M(E) and f ∈ B + (E), we write

µ, f = f (x) µ(dx).
The exploration process ρ = (ρ t , t ≥ 0) is a M f (R + )-valued process defined as follows: for every f ∈ B + (R + ),

ρ t , f = [0,t] d s I s t f (H s ),
or equivalently

(12) ρ t (dr) = 0<s≤t X s-<I s t (I s t -X s-)δ Hs (dr) + β1 [0,Ht] (r)dr.
In particular, the total mass of ρ t is ρ t , 1 = X t -I t .

For µ ∈ M(R + ), we set ( 13)

H(µ) = sup Supp µ,
where Supp µ is the closed support of µ, with the convention H(0) = 0. We have

Proposition 2.3 ([7], Lemma 1.2.2). Almost surely, for every t > 0, • H(ρ t ) = H t , • ρ t = 0 if and only if H t = 0, • if ρ t = 0, then Supp ρ t = [0, H t ].
In the definition of the exploration process, as X starts from 0, we have ρ 0 = 0 a.s. To state the Markov property of ρ, we must first define the process ρ started at any initial measure µ ∈ M f (R + ).

For a ∈ [0, µ, 1 ], we define the erased measure k a µ by

k a µ([0, r]) = µ([0, r]) ∧ ( µ, 1 -a), for r ≥ 0.
If a > µ, 1 , we set k a µ = 0. In other words, the measure k a µ is the measure µ erased by a mass a from the top of [0, H(µ)].

For ν, µ ∈ M f (R + ), and µ with compact support, we define the concatenation [µ, ν] ∈ M f (R + ) of the two measures by:

[µ, ν], f = µ, f + ν, f (H(µ) + •) , f ∈ B + (R + ).
Finally, we set for every µ ∈ M f (R + ) and every t > 0 ρ µ t = k -It µ, ρ t ]. We say that (ρ µ t , t ≥ 0) is the process ρ started at ρ µ 0 = µ, and write P µ for its law. Unless there is an ambiguity, we shall write ρ t for ρ µ t . Proposition 2.4 ([7], Proposition 1.2.3). The process

(ρ t , t ≥ 0) is a càd-làg strong Markov process in M f (R + ).
Notice that N is also the excursion measure of the process ρ away from 0, and that σ, the length of the excursion, is N-a.e. equal to inf{t > 0; ρ t = 0}.

2.4. The dual process and representation formula. We shall need the M f (R + )-valued process η = (η t , t ≥ 0) defined by

η t (dr) = 0<s≤t X s-<I s t (X s -I s t )δ Hs (dr) + β1 [0,Ht] (r)dr.
The process η is the dual process of ρ under N thanks to the following time reversal property: recall σ denotes the length of the excursion under N. It also enjoys the snake property: for all t ≥ 0, s ≥ 0

(ρ t , η t ) [0,H [t,s] ) = (ρ s , η s ) [0,H [t,s] ) ,
that is the measures ρ and η between two instants coincide up to the minimum of the height process between those two instants. We recall the Poisson representation of (ρ, η) under N. Let N * (dx dℓ du) be a Poisson point measure on [0, +∞) 3 with intensity

dx ℓπ(dℓ)1 [0,1] (u)du.
For every a > 0, let us denote by M a the law of the pair (µ a , ν a ) of finite measures on R + defined by: for

f ∈ B + (R + ) µ a , f = N * (dx dℓ du)1 [0,a] (x)uℓf (x), ν a , f = N * (dx dℓ du)1 [0,a] (x)ℓ(1 -u)f (x).
We finally set M = +∞ 0 da e -α 0 a M a .

Proposition 2.6 ([7], Proposition 3.1.3). For every non-negative measurable function F on

M f (R + ) 2 , N σ 0 F (ρ t , η t ) dt = M(dµ dν)F (µ, ν),
where σ = inf{s > 0; ρ s = 0} denotes the length of the excursion.

We can then deduce the following Proposition. Proposition 2.7. For every non-negative measurable function

F on M f (R + ) 2 , N σ 0 F (ρ t , η t ) dL a t = e -α 0 a M a (dµ dν)F (µ, ν),
where σ = inf{s > 0; ρ s = 0} denotes the length of the excursion.

Williams' decomposition

We work under the excursion measure. As the height process is continuous, its supremum

H max = sup{H r ; r ∈ [0, σ]} is attained. Let T max = inf{s ≥ 0; H s = H max }.
For every m > 0, we set T m (ρ) = inf{s > 0, H s (ρ) = m} the first hitting time of m for the height process. When there is no need to stress the dependence in ρ, we shall write T m for T m (ρ). Recall the function c defined by ( 5) is equal to

(14) c(m) = N[T m < ∞] = N[H max ≤ m].
We set ρ d = (ρ Tmax+s , s ≥ 0) and ρ g = (ρ (Tmax-s)+ , s ≥ 0), where x + = max(x, 0). For every finite measure with compact support µ, we write P * µ for the law of the exploration process ρ starting at µ and killed when it first reaches 0. We also set

P * µ := lim ε→0 P * µ ( • | H(µ) ≤ H max ≤ H(µ) + ε).
We now describe the probability measure P * µ via a Poisson decomposition. Let (α i , β i ), i ∈ I be the excursion intervals of the process X -I away from 0 (well defined under P * µ or under P * µ ). For every i ∈ I, we define h i = H α i and the measure-valued process ρ i by the formula

ρ i t , f = (h i ,+∞) f (x -h i )ρ (α i +t)∧β i (dx).
We then have the following result.

P * µa ( • | H max ≤ H(µ)).

Theorem 3.3 (Williams' Decomposition). (i)

The law of H max is characterized by N[H max ≤ m] = c(m), where c is the unique non-negative solution of ( 5). (ii) Conditionally on H max = m, the law of (ρ Tmax , η Tmax ) is under N the law of

i∈I v i r i δ t i + β1 [0,m] (t)dt, i∈I (1 -v i )r i δ t i + β1 [0,m] (t)dt , where δ (v i ,r i ,t i ) is a Poisson measure with intensity 1 [0,1] (v)1 [0,m] (t
) e -rc(m-t) dv rπ(dr) dt.

(iii) Under N, conditionally on H max = m, and (ρ Tmax , η Tmax ), (ρ d , ρ g ) are independent and ρ d (resp. ρ g ) is distributed as ρ (resp. η) under P * ρ Tmax (resp. P * η Tmax ).

Notice (i) is a consequence of [START_REF] Pitman | Growth of the Brownian forest[END_REF]. Point (ii) is reminiscent of Theorem 4.6.2 in [START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF] which gives the description of the exploration process at a first hitting time of the Lévy snake.

The end of this section is devoted to the proof of (ii) and (iii) of this Theorem. Let m > a > 0 be fixed. Let ε > 0. Recall T m = inf{t > 0; H t = m} is the first hitting time of m for the height process, and set L m = sup{t < σ; H t = m} for the last hitting time of m, with the convention that inf ∅ = +∞ and sup ∅ = +∞. We consider the minimum of H between T m and L m :

H [Tm,Lm] = min{H t ; t ∈ [T m , L m ]}.
We set ρ (d) = (ρ Tmax,a+t , t ≥ 0), with T max,a = inf{t > T max , H s = a}, the path of the exploration process on the right of T max after the hitting time of a, and ρ (g) = (ρ (Lmax,a-t)-, t ≥ 0), with L max,a = sup{t < T max ; H t = a}, the returned path of the exploration process on the left of T max before its last hitting time of a. Let us note that, by time reversal (see Proposition 2.5), the process ρ (g) is of the same type as η. This remark will be used later.

To prove the Theorem, we shall compute

A 0 = N F 1 (ρ (g) )F 2 (ρ (d) )F 3 (ρ Tmax |[0,a] )F 4 (η Tmax |[0,a] )1 {m≤Hmax<m+ε}
and let ε go down to 0. We shall see in Lemma 3.4, that adding 1 {H [Tm,Lm] >a} in the integrand does not change the asymptotic behavior as ε goes down to 0. Intuitively, if the maximum of the height process is between m and m + ε, outside a set of small measure, the height process does not reach level a between the first and last hitting time of m. So that we shall compute first ( 15)

A = N F 1 (ρ (g) )F 2 (ρ (d) )F 3 (ρ Tmax |[0,a] )F 4 (η Tmax |[0,a] )1 {H [Tm ,Lm] >a,m≤Hmax<m+ε} .
Notice that on {H [Tm,Lm] > a}, we have T max,a = T m,a := inf{s > T m , H s (ρ) = a} and, from the snake property,

ρ Tmax |[0,a] = ρ Tm |[0,a] and η Tmax |[0,a] = η Tm |[0,a] , so that A = N F 1 (ρ (g) )F 2 ((ρ Tm,a+t , t ≥ 0))F 3 (ρ Tm |[0,a] )F 4 (η Tm |[0,a] )1 {H [Tm,Lm] >a,m≤Hmax<m+ε} .
Let us remark that, we have 1 {H [Tm,Lm] >a,m≤Hmax<m+ε} = 1 {m≤sup{Hu,0≤u≤Tm,a}<m+ε} 1 {sup{Hu,u≥Tm,a}<m} .

By using the strong Markov property of the exploration process at time T m,a , we get

A=N F 1 (ρ (g) )F 4 (η Tm |[0,a] )1 {m≤sup{Hu,0≤u≤Tm,a}<m+ε} F 3 (ρ Tm |[0,a] )E * ρ Tm |[0,a] F 2 (ρ)1 {Hmax<m}
and so, by conditioning, we get

A = N F 1 (ρ (g) )F 4 (η Tm |[0,a] )G 2 (ρ Tm |[0,a] )1 {H [Tm,Lm] >a,m≤Hmax<m+ε} , where G 2 (µ) = F 3 (µ)E * µ [F 2 (ρ)|H max < m].
Using time reversibility (see Proposition 2.5) and the strong Markov property at time T m,a again, we have

A = N F 1 (ρ (d) )F 4 (ρ Tm |[0,a] )G 2 (η Tm |[0,a] )1 {H [Tm ,Lm] >a,m≤Hmax<m+ε} = N G 1 (ρ Tm |[0,a] )G 2 (η Tm |[0,a] )1 {H [Tm ,Lm] >a,m≤Hmax<m+ε} , where G 1 (µ) = F 4 (µ)E * µ [F 1 (ρ)|H max < m]
. Now, we use ideas from the proof of Theorem 4.6.2 of [START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF]. Let us recall the excursion decomposition of the exploration process above level a. We set τ a s = inf r,

r 0 du 1 {Hu≤a} > s .
Let E a be the σ-field generated by the process (ρ s , s ≥ 0) := (ρ τ a s , s ≥ 0). We also set ηs = η τ a s .

Let (α i , β i ), i ∈ I be the excursion intervals of H above level a. For every i ∈ I we define the measure-valued process ρ i by setting

ρ i s , ϕ = (a,+∞) ρ α i +s (dr)ϕ(r -a) if 0 < s < β i -α i , ρ s = 0 if s = 0 or s ≥ β i -α i ,
and the process η i similarly. We also define the local time at the beginning of excursion ρ i by ℓ i = L a α i . Then, under N, conditionally on E a , the point measure

i∈I δ (ℓ i ,ρ i ,η i ) is a Poisson measure with intensity 1 [0,L a σ ] (ℓ)dℓN[dρ dη].
In particular, we have

A = N   i∈I j =i 1 {Tm(ρ j )=+∞} G 1 (ρ α i )G 2 (η α i )1 {m≤Hmax(ρ i )<m+ε}   .
Let us denote by (τ a ℓ , ℓ ≥ 0) the right-continuous inverse of (L a s , s ≥ 0). Palm formula for Poisson point measures yields

A = N   N   i∈I j =i 1 {Tm(ρ j )=+∞} G 1 (ρ α i )G 2 (η α i )1 {m≤Hmax(ρ i )<m+ε} E a     = N   L a σ 0 dℓ G 1 (ρ τ a ℓ )G 2 (η τ a ℓ )N[m ≤ H max < m + ε]N   j∈I 1 {Tm(ρ j )=+∞} E a     .
A time-change then gives ( 16)

A = v(m -a, ε)N σ 0 dL a s G 1 (ρ s )G 2 (η s ) e -c(m-a)L a σ , where v(x, ε) = c(x) -c(x + ε) = N[x ≤ H max < x + ε]. We have A = v(m -a, ε)N σ 0 dL a s G 1 (ρ s )G 2 (η s ) e -c(m-a)L a s e -c(m-a)(L a σ -L a s ) = v(m -a, ε)N σ 0 dL a s G 1 (ρ s )G 2 (η s ) e -c(m-a)L a s e -ρs,N[1-e -c(m-a)L a-• σ ] ,
where we used for the last equality that the predictable projection of e -λ(L a σ -L a s ) is given by e -ρs,N[1-e -λL a-• σ ] . Notice that by using the excursion decomposition above level 0 < r < m, we have

c(m) = N[T m < ∞] = N[1 -e -c(m-r)L r σ ].
In particular, we get

A = v(m -a, ε)N σ 0 dL a s G 1 (ρ s )G 2 (η s ) e -c(m-a)L a s e -ρs,c(m-•) .
Using time reversibility, we have

A = v(m -a, ε)N σ 0 dL a s G 1 (η s )G 2 (ρ s ) e -c(m-a)(L a σ -L a s ) e -ηs,c(m-•) .

Similar computations as those previously done give

A = v(m -a, ε)N σ 0 dL a s G 1 (η s )G 2 (ρ s ) e -ηs+ρs,c(m-•) = v(m -a, ε)N σ 0 dL a s G 1 (ρ s )G 2 (η s ) e -ρs+ηs,c(m-•) .
Using Proposition 2.7, we get

A = v(m -a, ε) e -α 0 a M a (dµ dν)G 1 (µ)G 2 (ν) e -µ+ν,c(m-•) .
We can give a first consequence of the previous computation.

Lemma 3.4. We have 16), we deduce that

N[H [Tm,Lm] > a, m ≤ H max < m + ε] = c ′ (m) c(m -a) -c(m -a + ε) c ′ (m -a) . Proof. Taking F 1 = F 2 = F 3 = F 4 = 1 in (
N[H [Tm,Lm] > a, m ≤ H max < m + ε] = v(m -a, ε)N L a σ e -c(m-a)L a σ .
Let a 0 > 0 and let us compute B(a 0 , a) = N L a σ e -c(a 0 )L a σ . Thanks to Theorem 2.2, notice that

B(a 0 , a) = N Y a e -c(a 0 )Ya = ∂ a 0 N[1 -e -c(a 0 )Ya ] c ′ (a 0 ) .
On the other hand, we have

c(a + a 0 ) = N[Y a+a 0 > 0] = N[1 -E Ya [Y a 0 = 0]] = N 1 -e -Yac(a 0 ) ,
where we used the Markov property of Y at time a under N for the second equality and (3) with λ going to infinity for the last. Thus, we get B(a 0 , a) = c ′ (a 0 + a) c ′ (a 0 ) . We deduce that

N[H [Tm,Lm] > a, m ≤ H max < m + ε] = v(m -a, ε)B(a -m, a) = c ′ (m) c(m -a) -c(m -a + ε) c ′ (m -a) .
Since F 1 , F 2 , F 3 and F 4 are bounded, say by C, we have |A-

A 0 | ≤ C 4 N[H [Tm,Lm] < a, m ≤ H max < m + ε]. From Lemma 3.4, we deduce that lim ε→0 |A -A 0 | N[m ≤ H max < m + ε] ≤ C 4 1 -lim ε→0 N[H [Tm,Lm] > a, m ≤ H max < m + ε] N[m ≤ H max < m + ε] = 0.
We deduce that

lim ε→0 N F 1 (ρ (g) )F 2 (ρ (d) )F 3 (ρ Tmax |[0,a] )F 4 (η Tmax |[0,a] )1 {m≤Hmax<m+ε} N[m ≤ H max < m + ε] = c ′ (m -a) c ′ (m) e -α 0 a M a (dµ dν)G 1 (µ)G 2 (ν) e -µ+ν,c(m-•) = M a (dµ dν)G 1 (µ)G 2 (ν) e -µ+ν,c(m-•) M a (dµ dν) e -µ+ν,c(m-•) = Ma (dµ dν)G 1 (µ)G 2 (ν) = Ma (dµ dν)F 4 (ν)E * ν [F 1 (ρ (d) )|H max < m]F 3 (µ)E * µ [F 2 (ρ (d) )|H max < m],
where

µ(dt) = i∈I u i ℓ i δ t i + β1 [0,a] (t)dt ν(dt) = i∈I (1 -u i )ℓ i δ t i + β1 [0,a] (t)dt,
and i∈I δ (x i ,ℓ i ,t i ) is under Ma a Poisson point measure on [0, +∞) 3 with intensity

1 [0,a] (t)dt ℓ e -ℓc(m-t) π(dℓ)1 [0,1] (u)du.
Standard results on measure decomposition imply there exists a regular version of the probability measure N[ • |H max = m] and that, for almost every non-negative m,

N[ • |H max = m] = lim ε→0 N[ • |m ≤ H max < m + ε].
This gives (ii) and (iii) of Theorem 3.3 since F 1 , F 2 , F 3 , F 4 are arbitrary continuous functionals and by Remark 3.2.

Proof of Theorem 1.2

The proof of this Theorem relies on the computation of the Laplace transform for (Y ′ 0 , Y ′ ) and is given in the next three paragraphs. The next paragraph gives some preliminary computations.

Preliminary computations.

4.1.1. Law of T 0 . Recall the definition of Q m as the law of (Y ′ 0 , Y ′ ) defined by [START_REF] Lambert | The genealogy of continuous-state branching processes with immigration[END_REF] and T 0 defined by [START_REF] Geiger | The shape of large Galton-Watson trees with possibly infinite variance[END_REF] as the first mutation undergone by the last individual alive.

For r < m, we have

Q m (T 0 ∈ [r, r + dr], T 0 = T 2 ) = Q m (T 2 ∈ [r, r + dr])Q m (T 1 > r) = dr α Imm e -α Imm r exp - r 0 dt (0,∞)
e -ℓc(m-t) ℓν(dℓ)

= dr α Imm e - R r 0 φ ′ (c(m-t)) dt ,
and, with the notation φ 0 (λ) = φ(λ)α Imm λ,

Q m (T 0 ∈ [r, r + dr], T 0 = T 1 ) = Q m (T 2 > r)Q m (T 1 ∈ [r, r + dr]) = dr φ ′ 0 (c(m -r)) e -α Imm r exp - r 0 dt (0,∞)
e -ℓc(m-t) ℓν(dℓ)

= dr φ ′ 0 (c(m -r)) e - R r 0 φ ′ (c(m-t)) dt .
In particular, we have for r < m

Q m (T 0 ∈ [r, r + dr]) = dr φ ′ (c(m -r)) e - R r 0 φ ′ (c(m-t)) dt . and (17) Q m (T 0 > r) = e - R r 0 φ ′ (c(m-t)) dt . Notice we have Q m (T 0 = ∞) = exp - m 0 φ ′ (c(t)) dt.
4.1.2. Conditional law of N given T 0 . Recall N is under Q m a Poisson point measure with intensity given by [START_REF] Etheridge | A decomposition of the (1 + β)-superprocess conditioned on survival[END_REF]. Conditionally on {T 0 = r, T 0 = T 2 }, with m > r > 0, N is under Q m a point Poisson measure with intensity

1 [0,r) (t) e -ℓc(m-t) ℓπ Eve (dℓ)δ 0 (dz)dt+ 1 (r,m) (t) e -ℓc(m-t) ℓ [π Eve (dℓ)δ 0 (dz) + ν(dℓ)δ 1 (dz)] dt.
Conditionally on {T 0 = r, T 0 = T 1 }, with r < m, N is distributed under Q m as Ñ + δ (L,r,1) where Ñ is a point Poisson measure with intensity 1 [0,r) (t) e -ℓc(m-t) ℓπ Eve (dℓ)δ 0 (dz)dt + 1 (r,m) (t) e -ℓc(m-t) ℓ [π Eve (dℓ)δ 0 (dz) + ν(dℓ)δ 1 (dz)] dt, and L is a random variable independent of Ñ with distribution e -ℓc(m-r) ℓν(dℓ)

(0,∞) e -ℓ ′ c(m-r) ℓ ′ ν(dℓ ′ ) . Conditionally on {T 0 = ∞}, N is under Q m a point Poisson measure with intensity 1 [0,m) (t) e -ℓc(m-t) ℓπ Eve (dℓ)δ 0 (dz)dt.
4.1.3. Formulas. The following two formulas are straightforward: for all x, γ ≥ 0,

ψ ′ Eve (x + γ) -ψ ′ Eve (γ) = 2βx + (0,∞) e -ℓγ ℓπ Eve (dℓ)[1 -e -ℓx ], (18)
ψ ′ (x + γ) -ψ ′ (γ) = 2βx + (0,∞) e -ℓγ ℓπ(dℓ)[1 -e -ℓx ], (19)
= a 0 < a 1 < • • • < a n < a n+1 = ∞. For m ∈ (0, +∞) \ A, we consider w m (t) = N[1 -e - R Y 0 r-t µ Eve (dr)- R Y r-t µ Total (dr) 1 {τ Y <m-t} ], w * m (t) = N[1 -e - R Y r-t µ Total (dr) 1 {τ Y <m-t} ].
By noticing that N-a.e. 1 {τ Y <m-t} = lim λ→∞ exp -Y r-t µ λ (dr), where µ λ (dr) = λδ m (dr), we deduce from Lemma 3.1 in [START_REF] Abraham | Changing the branching mechanism of a continuous state branching process using immigration[END_REF] that (w m , w * m ) are right continuous and are the unique non-negative solutions of : for k ∈ {0, . . . , n}, m ∈ (a k , a k+1 ), t ∈ (-∞, m),

(21) w * m (t) + [t,a k ] ψ(w * m (r))dr = [t,a k ] µ Total (dr) + c(m -a k ), ( 22
) w m (t) + [t,a k ] ψ Eve (w m (r))dr = [t,a k ] µ Eve (dr) + [t,a k ] µ Total (dr) + c(m -a k ) + [t,a k ] φ(w * m (r))dr.
We define

(23) ām = max{a k ; a k < m, k ∈ {0, . . . , n}}. Notice that w m (t) = w * m (t) = c(m -t) for t ∈ (ā m , m). 4.2. Proof of Theorem 1.2.
4.2.1. Aim. Theorem 1.2 will be proved as soon as we check that the following equality

w(0) = ∞ 0 -c ′ (m)Q m [1 -e - R Y ′0 r µ Eve (dr)- R Y ′ r µ Total (dr) ]dm
holds for all the possible choices of measures µ Eve and µ Total satisfying the assumptions of Section 4.1.4, with w = w ∞ defined by (22). Notice the integrand of the right-hand side is null for m < a 1 . Let ∆ denote the right-hand side. We have for 0 < ε ≤ a

1 : ∆ = ∞ ε dm (-c ′ (m))Q m [1 -e - R Y ′0 r µ Eve (dr)- R Y ′ r µ Total (dr) ] = c(ε) + ∞ ε dm 1 A c (m)c ′ (m)Q m [Z],
with, thanks to the definition (6) of κ max ,

Z = exp - ām 0 κ max (dt) [n t 1 {t<T 0 } + n * t 1 {t≥T 0 } ]
and

n t = N[(1 -e - R Y 0 r-t µ Eve (dr)- R Y r-t µ Total (dr) )1 {τ Y ≤m-t} ] = w m (t) -c(m -t) n * t = N[(1 -e - R Y r-t µ Total (dr) )1 {τ Y ≤m-t} ] = w * m (t) -c(m -t)
, with (w m , w * m ) the non-negative solutions of (21) and ( 22). Notice that w m (t) = w * m (t) = c(mt) for t ∈ (ā m , m) and thus n t = n * t = 0 when t ∈ (ā m , m).

We set ∆ = c(ε)

+ ∞ ε 1 A c (m)(∆ 1 + ∆ 2 + ∆ 3 ) dm with ∆ 1 = c ′ (m)Q m [Z|T 0 > ām ]Q m (T 0 > ām ), ∆ 2 = c ′ (m) ām 0 Q m [Z|T 0 = r, T 0 = T 1 ]Q m (T 0 ∈ [r, r + dr], T 0 = T 1 ), ∆ 3 = c ′ (m) ām 0 Q m [Z|T 0 = r, T 0 = T 2 ]Q m (T 0 ∈ [r, r + dr], T 0 = T 2 ).
We shall assume m ∈ A.

4.2.2. Computation of ∆ 1 . We have, using formula [START_REF] Bertoin | The structure of the allelic partition of the total population for Galton-Watson processes with neutral mutations[END_REF],

∆ 1 = c ′ (m)Q m (T 0 > ām )Q m [e -

Definition 2 . 1 (

 21 [START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF], Definition 1.2.1 and Theorem 1.4

Theorem 2 . 2 (

 22 [START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF], Theorem 1.4.1). The process (L a τx , a ≥ 0) is under P (resp. N) defined as Y under P x (resp. N).

Proposition 2 . 5 (

 25 [START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF], Corollary 3.1.6). The processes ((ρ s , η s ); s ≥ 0) and ((η (σ-s)-, ρ (σ-s)-); s ≥ 0) have the same distribution under N.

Finally we deduce from ( 5 ) 4 . 1 . 4 .

 5414 that ψ(c) = -c ′ , ψ ′ (c)c ′ = -c ′′ and (20) ψ ′ (c) =log(c ′ ). Laplace transform. Recall τ Y = inf{t > 0; Y t = 0}is the extinction time of Y . Let µ Eve and µ Total be two finite measures with support a subset of a finite set A = {a 1 , . . . , a n } with 0

4 . 2 . 3 . 1 ∆ 2 = 1 ∆ 3 = 4 . 2 . 5 .

 4231213425 )nt |T 0 > ām ] = c ′ (m) e - R ām 0 φ ′ (c(m-t))dt exp -2β ām 0 (w m (t)c(mt)) dt -ām 0 dt e -ℓc(m-t) ℓπ Eve (dℓ)[1e -ℓ(wm(t)-c(m-t)) ] = c ′ (m) e - R ām 0 φ ′ (c(m-t))dt exp -ām 0 dt[ψ ′ Eve (w m (t))ψ ′ Eve (c(mt))] = c ′ (m) e R m m-ām ψ ′ (c(t))dt e - R ām 0 dt ψ ′ Eve (wm(t)) = c ′ (mām ) e - ψ ′ (c(t))dt = e -[log(c ′ (t))] m m-ām = c ′ (mām ) c ′ (m). Computation of ∆ 2 . Using Section 4.1.2, we getQ m [Z|T 0 = r, T 0 = T 1 ] = e -2β R r 0 (wm(t)-c(m-t)) dt-2β R ām r (w * m (t)-c(m-t)) dt exp(-r 0 dt e -ℓc(m-t) ℓπ Eve (dℓ)[1e -ℓnt ]) exp(-ām r dt e -ℓc(m-t) ℓπ(dℓ)[1e -ℓn * t ]) (0,∞) ν(dℓ ′ ) e -ℓ ′ c(m-r) ℓ ′ e -ℓ ′ n * Eve (w m (t))ψ ′ Eve (c(mt))]) exp(-ām r dt[ψ ′ (w * m (t))ψ ′ (c(mt))]) φ ′ 0 (w * m (r)) φ ′ 0 (c(mr)).We deduce from Section 4.1.c ′ (m)ām 0 Q m [Z|T 0 = r, T 0 = T 1 ]Q m (T 0 ∈ [r, r + dr], T 0 = T 1 ), = c ′ (m) (c(m-t)) dt exp(-r 0 dt[ψ ′ Eve (w m (t))ψ ′ Eve (c(mt))]) exp(-ām r dt[ψ ′ (w * m (t))ψ ′ (c(mt))]) = c ′ (m) e (w * m (t)) ,where we used (24) for the last equality. 4.2.4. Computation of ∆ 3 . Using Section 4.1.2, we getQ m [Z|T 0 = r, T 0 = T 2 ] = e -2β R r 0 (wm(t)-c(m-t)) dt-2β R ām r (w * m (t)-c(m-t)) dt exp -r 0 dt e -ℓc(m-t) ℓπ Eve (dℓ)[1e -ℓnt ] exp -ām r dt e -ℓc(m-t) ℓπ(dℓ)[1e -ℓn * t ] Eve (w m (t))ψ ′ Eve (c(mt))]) exp(-ām r dt[ψ ′ (w * m (t))ψ ′ (c(mt))]).We deduce from Section 4.1.c ′ (m)ām 0 Q m [Z|T 0 = r, T 0 = T 2 ]Q m (T 0 ∈ [r, r + dr], T 0 = T 2 ), = c ′ (m) ām 0 dr α Imm e - R r 0 φ ′ (c(m-t)) dt exp(-r 0 dt[ψ ′ Eve (w m (t))ψ ′ Eve (c(mt))]) exp(-ām r dt[ψ ′ (w * m (t))ψ ′ (c(mt))]) = c ′ (m) e R ām 0 ψ ′ (c(m-t)) dt ām 0 dr α Imm e - R r 0 dt ψ ′ Eve (wm(t))-R ām r dt ψ ′ (w * m (t)) = c ′ (mām ) Imm e - R r 0 dt ψ ′ Eve (wm(t))-R ām r dt ψ ′ (w * m (t)) ,where we used (24) for the last equality. Computation of ∆ 2 + ∆ 3 . We have∆ 2 + ∆ 3 = c ′ (mām )ām 0 dr φ ′ (w * m (r)) e - R r 0 dt ψ ′ Eve (wm(t))-R ām r dt ψ ′ (w * m (t)) . Differentiating (21) w.r.t. time and m, we get for t < m ∂ m (w * m ) ′ (t) -∂ m w * m (t)ψ ′ (w * m (t)) = 0.

  [dY, 1 {τ Y ≤m} ] denotes the restriction of the measure N t to the event {τ Y ≤ m}. The process j∈J Y j is distributed as Y under N, conditionally on {τ Y = m}.

	δ (t j ,Y j ) be, conditionally
	j∈J
	on N ′ , a Poisson point measure with intensity
	κ max (dt)N t [dY, 1 {τ Y ≤m} ]
	where N t The next result is a direct consequence of Theorem 3.3.
	Proposition 1.1.

dt.

Let N t (dY ) denote the law of (Y (st), s ≥ t) under N and

Acknowledgments. The authors wish to thank an anonymous referee for his numerous and useful comments which improved the presentation of the paper.

Notice also that for m > t ≥ ām , we have ∂ m w * (t) = c ′ (mt) and thus

.

Differentiating (22) w.r.t. time and m, we get for t < m

Notice also that for m > t ≥ ām one has This ends the proof of the Theorem.


Jean-Philippe Prost
email: jean-philippe.prost@univ-amu.fr

Integrating a Phrase Structure Corpus Grammar and a Lexical-Semantic Network: the HOLINET Knowledge Graph

Keywords: Knowledge Graph, Lexical-Semantic Network, Grammar, Phrase Structure, Constituency, JeuxDeMots

In this paper we address the question of how to integrate grammar and lexical-semantic knowledge within a single and homogeneous knowledge graph. We introduce a graph modelling of grammar knowledge which enables its merging with a lexical-semantic network. Such an integrated representation is expected, for instance, to provide new material for languagerelated graph embeddings in order to model interactions between Syntax and Semantics. Our base model relies on a phrase structure grammar. The phrase structure is accounted for by both a Proof-Theoretical representation, through a Context-Free Grammar, and a Model-Theoretical one, through a constraint-based grammar. The constraint types colour the grammar layer with syntactic relationships such as Immediate Dominance, Linear Precedence, and more. We detail a creation process which infers the grammar layer from a corpus annotated in constituency and integrates it with a lexical-semantic network through a shared POS tagset. We implement the process, and experiment with the French Treebank and the JeuxDeMots lexical-semantic network. The outcome is the HOLINET knowledge graph.

Introduction

While the pipeline software architecture, which steps from one linguistic dimension to the next, has been typical for decades for most Natural Language Processing (NLP) applications, it often prevents many potential interactions across dimensions from actually occurring. A variety of sentence-level ambiguities, for instance, require the full sentence to be parsed morphologically, then syntactically, then semantically, prior to being disambiguated through a pipeline. We hypothesize that a holistic modelling of the linguistic knowledge, which also approaches language as a whole rather than solely as a sum of its parts on various dimensions, would ease its processing and improve its performances in many respects. Knowledge graphs provide a convenient means for heterogeneous knowledge to interact rather seamlessly. They open perspectives with regard to holistic approaches to the modelling of knowledge, and linguistic knowledge in particular. Yet, the modelling of knowledge, in ways that make it suitable for Knowledge Graphs, is not always trivial and often constitutes a challenge as such. Concerning the representation of natural language syntax, the landscape of knowledge graphs has few resources, if any. Syntactic knowledge remains nearly exclusively represented through treebanks and other kinds of annotated corpora, and to a lesser extent through knowledge-based grammars, such as the HPSG grammars from the DELPH-IN consortium 1 , or meta-grammars such as FRMG [START_REF] De La Clergerie | From Metagrammars to Factorized TAG/TIG Parsers[END_REF]. Ultimately, our aim is to design and build a knowledge 1 http://www.delph-in.net graph which overcomes representational discrepancies among knowledge areas that are relevant to natural language processing and make natural language processable in a holistic manner. Assuming such a holistic knowledge graph, we may wonder whether the presentation of grammar knowledge as a knowledge graph would benefit applications down the road. For instance, would graph embeddings benefit from including syntactic relationships? Would knowledge graph completion enable the inference of syntactic relationships which were not observed in corpora? Etc. In this paper we focus on the graph modelling of phrase structure grammar knowledge in a way that makes it compatible with a lexical-semantic network. We introduce a base model, detail its acquisition based on corpus grammars, and describe the creation process from the grammar acquisition to the merging of the lexicalsemantic and grammar layers into a single structure. We implement the process and experiment with the French Treebank (FTB) and the JeuxDeMots (JDM) lexical-semantic network. The outcome is the HO-LINET Knowledge Graph which we introduce here. Section 2 reviews the literature. Section 3 describes the design of the graph model for the grammar layer, bearing in mind the requirements for compatibility with a lexical-semantic network. Section 4 details the creation process and its implementation with the FTB and JDM. Section 5 gives elements of evaluation. Section 6 discusses perspectives, while Section 7 concludes.

Literature Review

In their survey [START_REF] Ji | A Survey on Knowledge Graphs: Representation, Acquisition, and Applications[END_REF] propose to define a knowledge graph as G = {E, R, F}, where E,R and F are sets of entities, relations and facts, respectively.

Language-related data and Knowledge Bases The knowledge bases and resources for human language are rarely holistic. They are essentially found in the different linguistic dimensions. The Linguistic Linked Open Data (LLOD) initiative2 references resources which are linked with each other. It aims to address problems such as representation formats, federation of multiple data sources, or interoperability. The alignment of knowledge among resources is not the prime concern of the initiative, and often remains a challenge to be addressed. The Multiple Knowledge DB [START_REF] Faralli | Multiple knowledge graphdb (mkgdb)[END_REF], for instance, integrates 5 of these resources: ConceptNet [START_REF] Speer | Conceptnet 5.5: An open multilingual graph of general knowledge[END_REF], DBpedia, WebIsAGraph [START_REF] Faralli | Webisagraph: A very large hypernymy graph from a web corpus[END_REF], WordNet and the Wikipedia3 category hierarchy. Although many linguistic dimensions are still absent from the picture, such an integration goes in the direction of a holistic representation and processing of language knowledge.

Around syntax LLOD references annotated corpora for syntax, but as far as we know no knowledge base or graph is referenced which would account for grammar knowledge of human language. SAR-Graphs introduce a kind of resource inclusive of syntactic data with links to different lexicalsemantic resources such as WordNet, BabelNet, and YAGO. They are described as graphs of Semantically-Associated Relations (Krause et al., 2015a). They address the question of the integration of different aspects of linguistic knowledge within a homogeneous structure referred to as a language network [START_REF] Uszkoreit | From strings to things sar-graphs: A new type of resource for connecting knowledge and language[END_REF][START_REF] Krause | Sar-graphs: A linked linguistic knowledge resource connecting facts with language[END_REF]. They integrate lexical semantics with semantic relations about facts and events extracted from KBs such as Freebase [START_REF] Bollacker | Freebase: a collaboratively created graph database for structuring human knowledge[END_REF]. However, the grammar knowledge as such is contained in DARE [START_REF] Xu | A seed-driven bottom-up machine learning framework for extracting relations of various complexity[END_REF], a relation extraction system, through the use of the principle-based MINIPAR dependency parser [START_REF] Lin | Principar: an efficient, broadcoverage, principle-based parser[END_REF]. The gathering of grammar knowledge is a tedious and labour-intensive task, which probably explains why so few resources exist, or are made public. We have already mentioned the DELPH-IN project, which is dedicated to the development of NLP resources and applications, all based on Head-Driven Phrase Structure Grammar (HPSG) [START_REF] Pollard | Head-driven Phrase Structure Grammar[END_REF] for syntax, and Minimal Recursion Semantics (MRS) [START_REF] Copestake | Minimal recursion semantics: An introduction[END_REF] for semantics. A limited number of grammars are made available for a variety of languages [START_REF] Copestake | An open source grammar development environment and broad-coverage English grammar using HPSG[END_REF]. The French Meta-Grammar (FRMG) is another extensive source of computational grammar knowledge. Although not primarily designed as knowledge graphs, resources such as DELPH-IN and FRMG remain very valuable for the structured grammar knowledge they represent.

Complex Syntactic Networks

The past decade has seen the emergence of the use of complex networks [START_REF] Newman | Networks: An Introduction[END_REF] for studying the syntax of human languages, through Complex Syntactic Networks. However, they have been used exclusively for representing dependency relationships among words, or word forms ( Čech et al., 2016): the words (lemmas) are nodes, and the dependency relations are edges between nodes. The term dependency must be taken in the sense of the Dependency Grammar formalism [START_REF] Tesnière | Éléments de syntaxe structurale[END_REF][START_REF] Hudson | Language Networks: The New Word Grammar[END_REF]. Although nothing prevents the use of other grammar formalisms (e.g. phrase structure grammars) for syntactic network analysis, the literature shows no evidence of it ( Čech et al., 2016).

Lexical-semantic networks WordNet is a lexicalsemantic network of 150,000+ words, organised in 170,000+ synsets. Synsets can somehow be seen as concepts. EuroWordnet (Vossen, 1998), a multilingual version of WordNet, and WOLF [START_REF] Sagot | Construction d'un WordNet libre du franc ¸ais à partir de ressources multilingues[END_REF], a French version of WordNet, were built automatically through the crossing of WordNet and other lexical resources along with some manual checking. [START_REF] Navigli | BabelNet: Building a very large multilingual semantic network[END_REF] constructed automatically BabelNet, a large multilingual lexical network, from term co-occurrences in the Wikipedia encyclopedia. HowNet [START_REF] Dong | HowNet and the Computation of Meaning[END_REF]) is a hand-crafted lexical network based on concepts linking both English and Chinese. The Réseau Lexical du Franc ¸ais (RLF, French Lexical Network [START_REF] Polguère | From Writing Dictionaries to Weaving Lexical Networks[END_REF]) is a resource based on the notion of lexical function as defined by Igor Mel'čuk. The resource concerns about 10000 terms and is mainly manually populated with data. FrameNet [START_REF] Baker | The Berkeley FrameNet Project[END_REF]) is a lexical-semantic network dedicated to the Frame Semantics theoretical framework [START_REF] Fillmore | Cognitive Linguistics: Basic Readings, chapter Frame semantics[END_REF]. It links lexical units (i.e., the meaning of terms), semantic frames, and illustrative sentences annotated with predicate-argument structures. The frames are also related to each other through typed semantic relationships. JDM is a crowdsourced lexical-semantic network for French [START_REF] Lafourcade | Making people play for Lexical Acquisition[END_REF]. It is associated with a live ecosystem of semi-automatic processes for validation, evaluation, and knowledge completion. At time of writing, the network comprises4 16.5+ million nodes including 5.2+ million terms, 400+ million relations and 150+ relation types. The nodes are terms, concepts and symbolic information. The relations are lexical, morphological, pragmatic, logical, ontological, . . .

Graph modelling of a phrase structure grammar

The problem we are concerned with here is to design a graph structure which (i) can represent a phrase structure grammar, and (ii) can be connected to a lexical network. The grammar layer must share at least the POS nodes and the POS relationships (to relate the terms to their POS nodes) with the lexical layer.

3.1. What kind of grammar knowledge can be represented as a graph structure?

By "grammar knowledge" we do not refer to a collection of syntactic trees or graphs, which would each be specific to an utterance, even though such a reference would be a valid one. Foremost we refer to some form of knowledge that abstracts away from specific utterances, and provides some degree of generalisation.

In this paper, our intention is to model grammar knowledge in a way that makes it compatible with a knowledge graph. This means essentially that we can represent that grammar knowledge as a set of triples, that is, as a set of nodes related to each other by relations. We focus on phrase structure grammars. In other words, we need a way to represent descriptions of phrase structures as a set of objects (nodes), which are related to each other by syntactic relationships. While the POS categories occur quite obviously as good candidates for the nodes, the question of what the syntactic relationships should be is not as obvious.

We propose a model which combines the knowledge from a generative grammar and the knowledge from a constraint-based grammar. The generative grammar provides us with the means to specify tree structures and the objects of a tree domain. The constraint-based grammar provides us with the means to rely on various (constraint) types to further specify relationships among syntactic objects in the tree structure.

A preliminary model: the Alpha model

Let us start the description of the model with an illustration, and let us consider the example phrase structure in Fig. 1 as annotated in the FTB5 . The annotated tree structure can be modelled with the three rewrite rules from Fig. 2 (the feature structures have only been stripped out from this example for simplicity's sake). We can model these rules as illustrated in Fig. 3. In what follows, the node and edge types are given in true type font. For reference sake, we call this preliminary model the Alpha Model.

In this Alpha model, every rewrite rule is reified as a node, typed n g cfRule6 . The left-hand side of each rule is itself reified as a POS node (of type n pos), and every rule is connected to its left-hand side POS node with the r g rewrites relation.

Meanwhile, on the right-hand side (RHS) of each rule, every constituent POS is connected to its rule with the r g constitutes relation. In order to allow redundancy of POS, like here the AP, every constituent on the RHS is related to its POS node with an r g instantiates relation.

The Beta (and final) model

At this stage, we can observe that the Alpha model captures the relationship of Immediate Dominance among the constituents in the rewrite rules, but it fails to represent all the implicit knowledge covered by the rules.

For instance, we observe that the following relationships, which are implicit in rule 1, are not accounted for in the Alpha Model (Fig. 3):

• word order (Linear Precedence)

the DET precedes the NC the DET precedes the AP 1

the DET precedes the AP 2

• oriented co-occurrence a DET requires an NC These extra relationships can be modelled as such with properties of a Property Grammar [START_REF] Blache | Les Grammaires de Propriétés: des contraintes pour le traitement automatique des langues naturelles[END_REF]. To make it short, the PG properties are relational constraints over POS categories. Adding these relationships to the Alpha Model from Fig. 3 gives the Beta Model illustrated in Fig. 4. Next, we detail it further.

What semantics should define the grammar relationships?

Property Grammar (PG) is a formal framework for specifying constraint-based grammars. It provides us with well-defined semantics for a set of syntactic relationship types called Properties. We borrow the following properties from PG for our base model: Constituency (r g constitutes), Linear Precedence (r g precedes), and Requirement (r g requires). Their semantics are re-interpreted for our purpose, following the definitions in [START_REF] Duchier | A Model-Theoretic Framework for Grammaticality Judgements[END_REF]. Table 1 gives the semantics in use in HO-LINET for each type.

For the purpose of our work, a PG property can simply be seen as a triple (Node, relationship, Node).

What satisfaction value should be assigned to the constraint-based relationships?

A constraint (i.e., a property) from a constraint-based grammar can usually be seen as a statement which may be true (the constraint is satisfied) or false (the constraint is violated). Such a satisfaction value can be assigned to a given property, provided that we know the CFG rule that serves as an evaluation context. So, for instance, in the example above we used Rule 1 as the evaluation context to state that "the DET precedes the NC", that is, that property (DET, r g precedes, NC) is satisfied. Sometimes we may want to be specific and denote the satisfaction value with the type r g precedes+.

In the end, to build a grammar layer we first need:

• a CFG comprised of a set of rewrite rules,

(NP (DET##lem=un|cpos=D|g=f|n=s|s=ind## une) (NC##lem=bataille|cpos=N|g=f|n=s|s=c## bataille) (AP (ADJ##lem=politique|cpos=A|n=s|s=qual## politique)) (AP (ADV##lem=extrêmement|cpos=ADV|_## extrêmement) (ADJ##lem=ardu|cpos=A|g=f|n=s|s=qual## ardue)))
Figure 1: syntactic tree structure for the phrase une bataille politique extrêmement ardue (an extremely arduous political struggle), from the FTB.

Constituency (A, r g constitutes, B) the n g instance node A occurs in the rhs of the n g cfRule node B Precedence (A, r g precedes, B) the n g instance node A precedes the n g instance node B in read direction (implicitely in the context of the n g cfRule node they are both related to with r g constitutes) Requirement (A, r g requires, B)

the n g instance node A requires the n g instance node B to co-occur (implicitely in the context of the n g cfRule node they are both related to with r g constitutes)

Table 1: semantics of the PG property types in use in the HOLINET grammar layer

(1) NP -→DET NC AP AP

(2) AP -→ADJ

(3) AP -→ADV ADJ Figure 2: CFG for phrase structure in Fig. 1 • a set of consistent properties to complete the CFG.

Given these two resources, we can then move on to assess, for each rule in the CFG, the satisfaction value of each property that is relevant in this context. We call this process the characterisation of every rewrite rule, following the PG terminology. The outcome is a set of assessed properties associated with each rule, as illustrated in Fig. 2. As far as our modelling is concerned, we can see each assessed statement, that is, each evaluated PG property, as a relationship between (instances of) POS categories. Therefore, these assessed properties make as many relationships on our grammar layer. [START_REF] Vanrullen | Vers une analyse syntaxique à granularité variable[END_REF] proposed to model the PG theoretical framework as a graph structure. Representing assessed PG properties is what makes our graph modelling of PG significantly different from VanRullen's one, which represents the properties irrespective of their assessment. Said differently, and to mimic the terminology from Logic or Constraint Programming, we could say that VanRullen's model represents in intension the properties that comprise a PG grammar, while we represent them in extension. At this stage, we limit the model to representing only the satisfied properties. Further works will investigate the possible extension of the model with violated properties in order to cover degrees of grammaticality.

In the next section we will detail how to automate the entire creation process of a grammar layer on the basis of a constituency treebank, and how to connect it to JDM.

The creation process

We assume the existence of the following resources:

• a constituency treebank

• a (coloured) lexical-semantic network of typed nodes and typed relationships, where the lexical entries are terms, represented as nodes of type n term the Part-of-Speech (POS) labels are nodes of type n pos each term is related to one or several POS nodes through a relationship of type r pos

• if necessary, a conversion table between the treebank POS labels and the network POS labels.

The creation process, in short, is made up of the following steps:

1. extract the corpus CFG from the treebank 2. derive the PG from the CFG 3. characterise the CFG according to the PG 4. convert the corpus tagset as required 5. create the sets of nodes and edges for the grammar layer 6. merge the grammar and the lexical layers

DET#fsi          TYPE n pos CONTENT       CAT DET CPOS D GEND f NUM s SUBCAT ind                N#fsc          TYPE n pos CONTENT       CAT NC CPOS N GEND f NUM s SUBCAT c                ADJ#sq       TYPE n pos CONTENT     CAT ADJ CPOS A NUM s SUBCAT qual           ADJ#fsq          TYPE n pos CONTENT       CAT ADJ CPOS A GEND f NUM s SUBCAT qual                RULE1            TYPE n g cfRule RULE          LHS CAT NP RHS       CAT DET CPOS D GEND f NUM s SUBCAT ind       ,       CAT NC CPOS N GEND f NUM s SUBCAT c       , CAT AP , CAT AP                    
(idem for the other rules)

Implementation

In the following we work with the French Treebank (FTB) [START_REF] Abeillé | Building a Treebank for French, chapter Ch[END_REF] 7 , and the JeuxDeMots 7 version 1.0 2016, annotated with the Penn Treebank tagset. The functional tags have been removed.

(JDM) [START_REF] Mathieu Lafourcade | JeuxDeMots. Distributed by Laboratoire d'Informatique[END_REF] lexical-semantic network 8 , but the process is fairly basic and should easily apply to other resources with little changes. Given the respective nature and size of the resources involved (a fixed treebank and an ever-changing network), we choose to adapt the treebank tagset to the network one. The distribution of all the software involved in this section is detailed in Appendix 7.

Step 1. Extract CFG from FTB is straightforward.

Step 2. Derive PG from CFG Given an observed corpus CFG, it is possible to automate the derivation of the relationships we are interested in [START_REF] Prost | Compilation de grammaire de propriétés pour l'analyse syntaxique par optimisation de contraintes[END_REF]. In order to do so, each type of relationship, according to its semantics, corresponds to a derivation rule. We recall those derivation rules below. Let lhs(x) be the function that maps the non-terminal category x to the set of all the rewrite rules in the CFG that take x as its left-hand side.

Constituency The POS category B is in a Constituency relationship with the non-terminal category A iff at least one rewrite rule in lhs(A) can be found where B occurs in the right-hand side.

Linear Precedence

The two POS categories B and C are in a Linear Precedence relationship in the context of the non-terminal A (denoted A : B ≺ C for short), iff at least one rewrite rule in lhs(A) can be found where B ≺ C, and no rule can be found where C ≺ B.

Requirement The two POS categories B and C are in a Requirement relationship in the context of the nonterminal category A iff no rewrite rule can be found in lhs(A) where B occurs without C.

Step 3. Characterise the CFG The characterisation process can somehow be seen as checking the satisfaction of a constraint system against an assignment, where the constraint system is the grammar made up of a set of constraints/properties/relationships, and the assignment is the constituents of the tree structure represented here by one or more rewrite rules. In our case, since the relationships that make the PG grammar were observed in corpus, and since they were derived from the very tree structures described in the CFG we want to characterise, they are all deemed satisfied. Therefore, the characterisation process comes down to simply listing, for each rewrite rule, which relationships from the PG are relevant in this context. By "relevant" we mean that the relationship applies in this context and is not trivially satisfied, as would be, for instance, the relationship (NP,r g precedes,NC) in the context of the rewrite rule (AP -→ADJ), since the NP and AP categories mismatch. Table 2 reports a sample of the characterised corpus grammar that results from this step. The first two columns are for the rewrite rules, one per line. The right-hand side of rule is a comma-separated list of labels. Each of the next three columns is for a relationship type: Constituency is a comma-separated list of constituent POS; Linear Precedence and Requirement are comma-separated lists of semi-colonseparated pairs. The full grammar is available as part of the current release (see Appendix 7 for details).

Step 4. Convert the FTB tagset to JDM The mismatches encountered are mainly caused by the following reasons:

• A single FTB tag might correspond to a combination of several distinct JDM tags.

For instance, the FTB tag

P+PRO##cpos=P+PRO|g=f|n=p|p=3|s=rel##
should, in theory, correspond in JDM to Pre+Pro:Fem+PL+Rel. The full tag does not exist, that is, there exists no POS node labelled with this tag. Instead, there exist four distinct nodes: Pre:, Pro:Fem+PL, Pro:Rel, and Pre:. Hence, all the required information is present in JDM, but spread across several nodes, while a single one would be required by rewrite rules in the FTB.

• Since JDM adopted as a general strategy not to overload the network with data that has never been encountered, some features are absent from JDM. E.g, the definite/indefinite subcategorisation values for determiners.

• Some base categories in the FTB do not exist in JDM and had to be created. E.g., VPinf, VPpart.

Table 3 reports a small sample of the conversion table.

The full table is available as part of the current release (see Appendix 7 for details) 9 .

Step 5. Create the sets of nodes and edges for the grammar layer Algorithm 1 describes the procedure for creating a grammar layer. The algorithm must be adapted to the base lexical-semantic network in order to enforce consistency between the two layers. For instance, the pre-existence of POS nodes in the network may require that the existing nodes be listed prior to running the algorithm.

Step 6. Merge JDM and the grammar layer In short, the merging involves the creation of the JDM graph from a released dump, and the consistent merging of the grammar layer. By 'consistent' we mean, for instance, making sure that no duplicate nodes and triples are created. Detailing this step further goes beyond the scope of this paper, but the software is publicly available.

Our implementation of the entire process is available as a combination of open source projects, which are all made available in various public repositories (see Appendix 7). CREATE the r g instantiates edge, from the current instance node to the r pos node of its category 8:

CREATE an r g constitutes edge from the current instance node to the current compound 9:

end for 10:

CREATE an r g rewrites edge from the compound node to the n pos root node 11:

CREATE the edges for the PG properties for this rule 12: end for

Evaluation

We evaluate the adequacy of our implementation (of our graph model) with respect to the main goal we set initially: integrate a phrase structure grammar with a lexical-semantic network.

Integration we are primarily concerned with evaluating the integration of the grammar layer with the lexical-semantic layer. As emphasized in Section 4.1, many of the POS tags observed in the FTB go missing in JDM. Of course the corresponding missing nodes were created. But as of today, those newly created nodes, although connected to the grammatical layer through the rewrite rules they are involved in, remain disconnected from the lexical-semantic layer. Fixing the problem is quite straightforward at the scale of the FTB. But scaling up the fix to the rest of HOLINET is more challenging, since it would involve the appropriate POS tagging of all the terms in JDM. In order to estimate the extent of the problem, for each POS tag involved in a rewrite rule we check wether the required n pos node was pre-existing in JDM. 2019) among a list of possible criteria of evalutation of a knowledge graph, we measure the density of connexions through the number of occurrences of each POS node in rewrite rules. As expected, the number of instance edges by POS node shows a power law distribtuion, as illustrated in Fig. 6. 2019), we measure the node redundancy. We found two POS nodes that were created on the grammar layer, whereas they already existed in JDM. This can easily be fixed. Note that depending on the underlying data management system, future such redundancy can easily be avoided with integrity constraints being set on the POS nodes.

Perspectives

HOLINET opens the door to a large variety of future works, along several dimensions.

Extending the graph model

Various extensions of the base model can be considered in order to foster the holistic nature of HOLINET. Integration of dependency grammar The integration of dependency grammar knowledge should be eased by various works around theoretical frameworks for its constraint-based modelling [START_REF] Maruyama | Structural Disambiguation with Constraint Propagation[END_REF][START_REF] Debusmann | The XDG grammar development kit[END_REF]. The constraint-based nature of existing frameworks makes quite likely the possibility of integrating a new 'dependency' relationship type with a well-defined semantics. Integration of syntactic constructions A quite natural extension of a constraint-based phrase structure grammar is to evolve towards a construction Grammar (CxG) [START_REF] Goldberg | Constructions: A Construction Grammar Approach to Argument Structure[END_REF]. Recent works [START_REF] Müller | Head-Driven Phrase Structure Grammar: The handbook, chapter HPSG and Construction Grammar[END_REF] show, for instance, that Head-Driven Phrase Structure Grammar (HPSG) can be seen as a CxG. In order to do so, the question of the graph modelling of a formmeaning pair will have to be addressed. Weighting scheme The question of the assignment of weight to both nodes and edges in the grammar layer is a critical question for a variety of applications. Certain graph algorithms, for instance, such as the classical shortest path, rely on the edges to be associated with weights to perform well. Though we have left aside the question so far, annotated corpora such as the FTB provide with the basic statistical material to turn the underlying CFG into a Probabilistic CFG, hence assign initial weights to the grammar edges.

Implementation of the grammar layer

Our implementation leaves room for improvement. Different problems that have been stressed in Section 5 should be quite easily fixed (e.g. the 'null' values, the POS-tagging of all the terms in JDM). More relationship types can be extracted from corpus, such as Uniqueness, or the property for category to head a phrase. Other grammar resources, such as FRMG, could also be integrated. The question of maintaining consistency among grammar resources will likely be triggered.

Graph embeddings

The integration of syntactic knowledge along with lexical and semantic knowledge within the same embeddings is modern ground for investigation [START_REF] Limisiewicz | Syntax Representation in Word Embeddings and Neural Networks-A Survey[END_REF][START_REF] Al-Ghezi | Graph-based syntactic word embeddings[END_REF]. With the introduction in HOLINET of syntactic relationships within the same knowledge graph as lexical and semantic ones, one may wonder whether these relationships could be represented in graph embeddings, and, subsequently, whether such embeddings would provide the NLP applications down the road with a better modelling of the interactions between syntax and semantics.

Conclusion

In this paper we addressed the question of the integration of grammar knowledge and lexical-semantic knowledge within a homogeneous knowledge graph. We introduced a graph model of a phrase structure grammar and showed how to integrate it with a lexicalsemantic network through a shared POS tagset. This model relies on several types of syntactic relationships which are inspired from Property Grammar, a constraint-based theoretical framework for Model-Theoretic Syntax. We proposed a creation process which derives these relationships from a CFG, and integrates the grammar with a lexical-semantic network. We implemented the creation process with a corpus grammar extracted from the French Treebank and the lexical-semantic network JeuxDeMots. The outcome is the HOLINET knowledge graph [START_REF] Prost | HOLINET. Distributed by ORTOLANG (Open Resources and TOols for LANGuage) -www[END_REF]. The evaluation we performed of the quality of the data that we produced shows a rather weak integration of the grammar with JDM, with only a small 11% of the POS nodes that are connected to JDM. While it obviously leaves room for improvement implementationwise, the integration model as such is not to be blamed. Future works should most notably investigate the computation of syntactic-lexical-semantic graph embeddings and measure their impact on NLP applications.

Figure 3 :

 3 Figure3: Alpha model: the syntactic relationships from the CFG in Fig.2

Figure 4 :

 4 Figure 4: Beta Model: The PG relationships added to the Alpha model for RULE1

Fig. 5

 5 Figure 5: sample HOLINET graph for part of the utterance une bataille politique extrêmement ardue (an extremely arduous political struggle)

Figure 6 :

 6 Figure 6: power law distribution of the r g instantiates edges by POS node The grammar layer is comprised of 1,048,575 edges, and 180,479 rewrite rules. Redundancy Still suggested in Chen et al. (2019), we measure the node redundancy. We found two POS nodes that were created on the grammar layer, whereas they already existed in JDM. This can easily be fixed. Note that depending on the underlying data management system, future such redundancy can easily be avoided with integrity constraints being set on the POS nodes.

Table 3 :

 3 sample of the lookup table for converting the FTB tagset to the JDM one.

	Algorithm 1 Create the HOLINET Nodes and Edges
	1: for all CF rule do
	2:	CREATE an n pos node for this rule's root if it
		does not exist yet
	3:	CREATE n pos nodes for the rule's daughter
		labels (i.e. RHS) if they do not exist yet
	4:	CREATE the n g cfRule node for this rule
	5:	for all categories in this rule's RHS do
	6:	CREATE an n g instance node with this
		category
	7:

Table 4 :

 4 Table 4 reports the number of occurrences of each POS in the grammar, whether pre-existing or not. The proportion of

	Connected Disconnected Null Total
	Num. nodes 22,742	163,210 9,408 195,360
	% 11.6%	83.5%	4.8% 100

measures of integration connected POS at the scale of the FTB probably gives a rough estimate of the proportion of those at full scale. The 'Null' values are conversion errors. Density As suggested in Chen et al. (

http://linguistic-lod.org/

https://www.wikipedia.org/

As of January 2022, according to the values presented at http://www.jeuxdemots.org/jdm-about.php.

The functional tags have been removed.

n g cfRule and n g compound are synonym.

Since the JDM crew was notified about these issues, the live version of the network might partially fix them by the time this paper is published.

Appendix: data and software release notes • the grammar layer (nodes and edges) (CSV)

• the full lookup table for converting the FTB tagset to the JDM one (CSV)

• a human-readable version of the characterised grammar (CSV)

software The implementation described in Section 4.1 relies on the following pieces of software, all made available in various public repositories:

• rdb4jdm: a fork of the rdb4jdm project by Kevin Cousot, slightly modified for preprocessing the JDM dump. At sourceforge.net, version tag "LREC 2022-final" (https: //sourceforge.net/p/rdb4jdm/code/ ci/LREC_2022-final/tree/)