MAPMO Guillaume Voisin
email: guillaume.voisin@univ-orleans.fr

DISLOCATION MEASURE OF THE FRAGMENTATION OF A GENERAL L ÉVY TREE

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Introduction

Continuous state branching processes (CSBP) have been introduced by Jirina [START_REF] Jirina | Stochastic branching processes with continuous state space[END_REF] and it is known since Lamperti [START_REF] John | The limit of a sequence of branching processes[END_REF] that these processes are the scaling limits of Galton-Watson processes. They model the evolution of a large population on a long time interval. The law of a CSBP is characterized by the so-called branching mechanism, which is the Laplace exponent of a spectrally positive Lévy process, ans is usually denoted by ψ. When the CSBP is critical or sub-critical, one can associate a continuum random tree (CRT) which describes the genealogy of the CSBP. Duquesne and Winkel [START_REF] Thomas | Growth of Lévy trees[END_REF] has constructed genealogical trees associated with super-critical branching processes, we also cite Delmas [START_REF] Delmas | Height process for super-critical continuous state branching process[END_REF] for the construction of the height process when the branching process is super-critical. Aldous and Pitman [START_REF] David | Inhomogeneous continuum trees and the entrance boundary of the additive coalescent[END_REF][START_REF] David | The standard additive coalescent[END_REF] did a pioneering work in fragmentation processes involving discrete and continuum trees. The construction of fragmentation processes from CRTs have been studied by Abraham and Serlet [START_REF] Romain | Poisson snake and fragmentation[END_REF] for the Brownian CRT (in the case where the Lévy measure of ψ is null) and by Abraham and Delmas [START_REF] Romain | Fragmentation associated with Lévy processes using snake[END_REF] for the CRT without Brownian part (in the case where ψ has no quadratic part). In these works, Lévy Poisson snakes are used to create marks on the CRT and to obtain a fragmentation process. In the first case, the marks are built on the skeleton of the CRT, in the second, they are placed on the nodes. Abraham, Delmas and Voisin [START_REF] Romain | Pruning a Lévy random continuum tree[END_REF] constructed a general pruning of a CRT where the marks are placed on the whole CRT, skeleton and nodes. In this work, they study the law of the sub-tree obtained after the pruning according to the marks.

The aim of this article is to study the fragmentation process associated with a general CRT and more precisely the dislocation measure associated with this CRT. Note that this measure has been studied in the Brownian case and in the case without Brownian part (see [START_REF] Romain | Poisson snake and fragmentation[END_REF] and [START_REF] Romain | Fragmentation associated with Lévy processes using snake[END_REF]).

The three following parts give a brief presentation of the mathematical objects and give the main results.

1.1. The exploration process. The coding of the CRT by its height process is well known. The height process of Aldous' CRT [START_REF] David | The continuum random tree III[END_REF] is a normalized Brownian excursion. In [START_REF] Gall | Branching processes in Lévy processes: The exploration process[END_REF], Le Gall and Le Jan associated with a Lévy process with no negative jumps that does not drift to infinity, X = (X t , t ≥ 0), a CSBP and a Lévy CRT which keeps track of the genealogy of the Date: February 25, 2010.

1

CSBP. Let ψ be the Laplace exponent of the process X. By the Lévy-Khintchine formula (and some additional assumptions on X, see Section 2.1), ψ is such that E e -λXt = e tψ (λ) and can be expressed by ψ(λ) = αλ + βλ 2 + (0,∞) (e -λl -1 + λl)π(dl) with α ≥ 0, β ≥ 0 and the Lévy measure π is a positive σ-finite measure on (0, ∞) such that (0,∞) (l ∧ l 2 )π(dl) < ∞. Following [START_REF] Thomas | Random trees, Lévy processes and spatial branching processes[END_REF], we assume that X is of infinite variation, which implies that β > 0 or (0,1) lπ(dl) = +∞. The term α is a drift term (if ψ(λ) = αλ, X is a Cauchy process), β is the quadratic term (if ψ(λ) = βλ 2 , X is a Brownian motion) and π gives the law of the jumps of X.

We first construct the height process H = (H t , t ≥ 0) associated with the process X (see Section 2.2). This process codes for a continuum random tree : each individual t is at distance H t from the root and the last common ancestor of the individuals s and t (s < t) is at distance:

H s,t = inf{H u ; u ∈ [s, t]}
(see Section 2.2 for a formal definition of a continuum random tree and its coding by the height process). This height process is an important object but is not a Markov process in general. Thus we introduce the exploration process ρ = (ρ t , t ≥ 0) which is a càd-làg, strong Markov process taking values in M f (R + ), the set of finite measures on R + endowed with the topology of weak convergence. It is defined by :

ρ t (dr) = β1 [0,Ht] (r) dr + 0<s≤t X s-<I s t (I s t -X s-)δ Hs (dr)
where

I s t = inf s≤u≤t X u .
The height process can easily be recovered from the exploration process as H t = H(ρ t ) where H(µ) is the supremum of the closed support of the measure µ (with the convention that H(0) = 0). Informally, ρ t can be seen as a measure on the branch from the root to the individual t which gives the intensity of the branching points (associated with individuals situated "on the right" of t) along that branch (see Bismut decomposition of Proposition 2.5 and the Poisson representation of the process of Lemma 3.5). We can hence see that the regular part of the measure ρ t gives birth to binary branching points whereas the atoms of the measure (which correspond to jumps of the Lévy process X) lead to nodes of infinite index.

1.2. The fragmentation. A fragmentation process is a Markov process which describes how an object with given total mass evolves as it breaks into several fragments randomly as time passes. This kind of processes has been widely studied in [START_REF] Jean | Random fragmentation and coagulation processes[END_REF]. To be more precise, the state space of a fragmentation process is the space of non-increasing sequences of masses with finite total mass

S ↓ = {s = (s 1 , s 2 , . . . ); s 1 ≥ s 2 ≥ • • • ≥ 0 and ∞ k=1 s k < ∞}.
We denote by P s the law of a S ↓ -valued process Λ = (Λ θ , θ ≥ 0) starting at s = (s 1 , s 2 , . . . ) ∈ S ↓ . For fixed θ > 0, we write (Λ θ ) = (Λ θ 1 , Λ θ 2 , . . . ) ∈ S ↓ and (Λ θ ) = i≥1 Λ θ i for the sum of the masses of the elements at time θ. We say that Λ is a fragmentation process if it is a Markov process such that θ → (Λ θ ) is decreasing and if it fulfills the fragmentation property : the law of (Λ θ , θ ≥ 0) under P s is the non-increasing reordering of the fragments of independent processes of respective laws P (s 1 ,0,... ) , P (s 2 ,0,... ) ,. . . In other words, each fragment behaves independently of the others, and its evolution depends only of its initial mass. Hence, it suffices to study the laws P r := P (r,0,... ) that is the law of the fragmentation process starting with a single mass r ∈ (0, ∞).

We want to construct a fragmentation process by cutting a Lévy CRT into several subtrees. The lengths of the height processes that code each subtrees, ranked in decreasing order, form an element of S ↓ . In order to construct our fragmentation process, we need to place marks on the CRT which give the different cut points and the number of marks must increase as time passes.

There will be two sort of marks : some are lying on the nodes of infinite index whereas the others are "uniformly" distributed on the skeleton of the tree.

The nodes of the tree are marked independently and, at time θ, a node with mass m is marked with probability 1e -mθ . To have a consistent construction as θ varies, we use a coupling construction so that the marks present at time θ are still marks at a further time.

For the marks on the skeleton of the CRT, we use a Lévy Poisson snake similar to those of [START_REF] Thomas | Random trees, Lévy processes and spatial branching processes[END_REF] but we must introduce the new parameter θ. At fixed time θ, the marks on the lineage of an individual t will be distributed as a Poisson process with intensity 2βθ1 [0,Ht] (r)dr, but the marks on two common lineages must be the same and a coupling construction must also apply.

By cutting according to these marks, we obtain a set of fragments. Let s 1 , s 2 , . . . be the "sizes" of these fragments ranked by non-increasing order completed with 0 if necessary so that (s 1 , s 2 , . . . ) ∈ S ↓ . When time θ increases, the number of marks increases and the fragments break again. Thus we obtain a process (Λ θ , θ ≥ 0), Theorem 4.1 checks that this process is a fragmentation process.

The choice of the parameters for the marks can be surprising as the pruning of [START_REF] Romain | Pruning a Lévy random continuum tree[END_REF] is much more general but the particular pruning considered here leads to a pruned exploration process that fulfills Lemma 3.4 which is necessary for getting a fragmentation process. We don't know if other pruning give such a property; one may conjecture that it is the only one.

1.3. The dislocation measure. The evolution of the process Λ is described by a family (ν r , r ≥ 0) of σ-finite measures called dislocation measures. ν r describes how a fragment of size r breaks into smaller fragments. In the case of self-similar fragmentations (with no loss of mass), the dislocation measure characterizes the law of the fragmentation process. In the general case, the characterization is an open problem.

To be more precise, we define T = {θ ≥ 0; Λ θ = Λ θ-} the set of jumping times of the process Λ. The dislocation process of the CRT fragmentation θ∈T δ θ,Λ θ is a point process with intensity dθ νΛ θ-(ds), where (ν x , x ∈ S ↓ ) is a family of σ-finite measure on S ↓ . There exists a family (ν r , r > 0) of σ-finite measures on S ↓ such that for any x = (x 1 , x 2 , . . . ) ∈ S ↓ and any non-negative function F , defined on S ↓ ,

F (s)ν x (ds) = i≥1,x i >0 F (x i,s )ν x i (ds)
where x i,s is the decreasing reordering of the merging of the sequences s and x, where x i has been removed of the sequence of x.

We will show in Section 4.2 that the measure ν r can be written as ν r = ν nod r + ν ske r where ν nod corresponds to a mark that appears on the node whereas ν ske to a mark on the skeleton.

The expression of the measure ν ske r is the main result of this article :

Theorem 1.1. Let S be a subordinator with Laplace exponent ψ -1 , let π * be its Lévy measure.

(1) For all non negative measurable function F on S ↓ ,

R + ×S ↓ F (x)ν nod r (dx)π * (dr) = π(dv)E [S v F ((∆S u , u ≤ v))]
where (∆S u , u ≤ v) ∈ S ↓ represents the jumps of S before time v, ranked by decreasing order.

(2) The measure ν ske r charges only the set of elements of S ↓ of the form (x 1 , x 2 , 0, . . .) with x 1 ≥ x 2 and x 1 + x 2 = r. It is the "distribution" of the non-increasing reordering of the lengths given by the measure νske r defined by

R + ×S ↓ 1 x 2 (1 -e -λ 1 x 1 )(1 -e -λ 2 x 2 )ν ske r (dx)π * (dr) = 2βψ -1 (λ 1 )ψ -1 (λ 2 ).
Remark 1.2. Under νske r (dx)π * (dr), the lengths of the two fragments are "independent". Remark 1.3. We will see in Section 4.2 that the measure ν nod is the same as the measure ν in the case of a tree without Brownian part (β = 0). Thus the proof of Part 1 of Theorem 1.1 is the same as in [START_REF] Romain | Fragmentation associated with Lévy processes using snake[END_REF]. Only Part 2 needs a proof.

The Lévy snake : notations and properties

2.1. The Lévy process. We consider a R-valued Lévy process (X t , t ≥ 0) with no negative jumps, starting from 0 characterized by its Laplace exponent ψ given by

ψ(λ) = α 0 λ + βλ 2 + (0,+∞) π(dℓ) e -λℓ -1 + 1 ℓ<1 λℓ ,
with β ≥ 0 and the Lévy measure π is a positive, σ-finite measure on (0, +∞) such that

(0,+∞) (1 ∧ ℓ 2 )π(dℓ) < ∞. We also assume that X • has first moments (i.e. (0,+∞) (ℓ ∧ ℓ 2 )π(dℓ) < ∞),
• is of infinite variation (i.e. β > 0 or (0,1) ℓπ(dℓ) = +∞),

• does not drift to +∞. With the first assumption, the Lévy exponent can be written as

ψ(λ) = αλ + βλ 2 + (0,+∞) π(dℓ) e -λℓ -1 + λℓ ,
with α ≥ 0 thanks to the third assumption.

Let J = {t ≥ 0; X t = X t-} be the set of jumping times of the process X.

For λ ≥ 1 ǫ > 0, we have e -λl -1 + λl ≥ 1 2 λl1 l≥2ǫ this implies that λ -1 ψ(λ) ≥ α + β 1 ǫ + (2ǫ,∞) lπ(dl). We deduce that lim λ→∞ λ ψ(λ) = 0.
Let I = (I t , t ≥ 0) be the infimum process of X, I t = inf 0≤s≤t X s . We also denote for all 0 ≤ s ≤ t, the minimum of X on [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] Jean | Lévy processes[END_REF], Chap. VII). Let N be the excursion measure of the process X -I away from 0, and let σ = inf{t > 0; X t -I t = 0} be the lengths of the generic excursion of X -I under N.

Notice that, under N, X 0 = I 0 = 0.

Thanks to [START_REF] Jean | Lévy processes[END_REF], Theorem VII.1, the right-continuous inverse of the process -I is a subordinator with Laplace exponent ψ -1 . We have already seen that this exponent has no drift, because lim λ→∞ λψ(λ) -1 = 0. We denote by π * its Lévy measure : for all λ ≥ 0

ψ -1 (λ) = (0,∞) π * (dl)(1 -e λl ).
Under N, π * is the "law" of the length of the excursions, σ. By decomposing the measure N w.r.t. the distribution of σ, we get that N(dE) = (0,∞) π * (dr)N r (dE), where (N r , r ∈ (0, ∞)) is a measurable family of probability measures on the set of excursions (that is to say for all A, r → N r (A) is B(R + )-measurable) and such that N r [σ = r] = 1 for π * -a.e. r > 0. (see [START_REF] Rangachari | Probability measures on metric spaces[END_REF] for more details for the existence of such a decomposition) 2.2. The height process and the Lévy CRT. We first define a continuum random tree (CRT) using the definition of Aldous [START_REF] Aldous | The continuum random tree II: an overview[END_REF][START_REF] Aldous | The continuum random tree I[END_REF][START_REF] David | The continuum random tree III[END_REF]. Definition 2.1. We say that a metric space (T , d) is a real tree if : for u, v ∈ T ,

• there exists a unique isometry

ψ u,v : [0, d(u, v)] → T such that ψ u,v (0) = u and ψ u,v (d(u, v)) = v, • if (w s , 0 ≤ s ≤ 1) is an injective path on T such that w 0 = u and w 1 = v then (w s , 0 ≤ s ≤ 1) = ψ u,v ([0, d(u, v)]).
A CRT is a random variable (T (ω), d(ω)) on a probability space (Ω, A, P) such that (T (ω), d(ω)) is a real tree for all ω ∈ Ω.

We can use a height function to define a genealogical structure on a CRT (see Aldous [START_REF] David | The continuum random tree III[END_REF]). Let g : R + → R + be a function with compact support, non trivial and such that g(0) = 0. For s, t ∈ T , we say that g(s) is the generation of the individual s and that s is an ancestor of t if g(t) = g s,t where g s,t = inf{g(u), s ∧ t ≤ u ≤ s ∨ t} is the generation of the last common ancestor of the individuals s and t.

We define an equivalence relation between two individuals:

t ∼ t ′ ⇐⇒ d(t, t ′ ) := g(t) + g(t ′ ) -2g t,t ′ = 0.
That is to say g(t) = g t,t ′ = g(t ′ ). The quotient set [0, σ]/ ∼ equipped with the distance d and the genealogical relation is then a CRT coded by g.

Let us now define a height process H associated with the Lévy process X, see Part 1.2 of Duquesne and Le Gall [START_REF] Thomas | Random trees, Lévy processes and spatial branching processes[END_REF]. For all t ≥ 0, we consider the reversed process at time t,

X(t) = ( X(t) s , 0 ≤ s ≤ t) defined by : X(t) s = X t -X (t-s)-if 0 ≤ s < t,
and X(t) t = X t . We denote by Ŝ(t) the supremum process of X(t) and L(t) the local time at 0 of Ŝ(t) -X(t) with the same normalization as in [START_REF] Romain | Fragmentation associated with Lévy processes using snake[END_REF]. Definition 2.2. There exists a [0, ∞]-valued lower semi-continuous process, called the height process such that, under N, We say that a CRT coded by its associated height process H is a Lévy CRT.

H 0 = 0, for all t ≥ 0,
2.3. The exploration process. The height process is not a Markov process in general. But it is a very simple function of a measure-valued Markov process, the exploration process. If E is a locally compact polish space, we denote by B(E) (resp. B + (E)) the set of R-valued measurable (resp. and non-negative) functions defined on E endowed with its Borel σ-field, and by M(E) (resp. M f (E)) 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 any function 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 by, for every f ∈ B + (R + ), ρ t , f = [0,t] d s I s t f (H s ), or equivalently ρ t (dr) = β1 [0,Ht] (r) dr + 0<s≤t X s-<I s t (I s t -X s-)δ Hs (dr).
In particular, the total mass of ρ t is ρ t , 1 = X t -I t .

The exploration process also codes the Lévy CRT. Indeed, we can recover the height process H from the exploration process. For µ ∈ M(R + ), we put

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

To better understand what the exploration process is, let us give some of its properties. For every t ≥ 0 such that ρ t = 0, the support of the exploration process at time t is [0, H t ]: Supp ρ t = [0, H t ]. We also have ρ t = 0 if and only if H t = 0. We can finally describe the jumps of the exploration process using the jumps of the Lévy process: ρ t = ρ t -+ ∆ t δ Ht , where ∆ t = 0 if t ∈ J . See [START_REF] Thomas | Random trees, Lévy processes and spatial branching processes[END_REF], Lemma 1.2.2 and Formula (1.12) for more details.

In the definition of the exploration process, as X starts from 0, we obtain ρ 0 = 0 a.s. To state the Markov property of ρ, we must first define the process ρ starting at any initial measure µ ∈ M f (R + ). We recall the notations given in [START_REF] Thomas | Random trees, Lévy processes and spatial branching processes[END_REF].

For a ∈ [0, µ, 1 ], we write k a µ for the erased measure which is the measure µ erased by a mass a backward from H(µ), that is to say:

k a µ([0, r]) = µ([0, r]) ∧ ( µ, 1 -a), for r ≥ 0.
In particular, k a µ, 1 = µ, 1a.

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

[µ, ν], f = µ, f + ν, f (H(µ) + •) , f ∈ B + (R + ).
Finally, we put for all µ ∈ M f (R + ) and for all t > 0,

ρ µ t = k -It µ, ρ t ].
We say that (ρ µ t , t ≥ 0) is the process ρ starting from 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 . We also denote by P * µ the law of ρ µ killed when it first reaches 0. Then we can state a useful property of the exploration process: the process (ρ t , t ≥ 0) is a càd-làg strong Markov process in M f (R + ). See [START_REF] Thomas | Random trees, Lévy processes and spatial branching processes[END_REF], Proposition 1.2.3 for a proof.

Remark 2.3. As in [START_REF] Romain | Fragmentation associated with Lévy processes using snake[END_REF], 0 is also a regular point for ρ. 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}.

The exponential formula for the Poisson point process of jumps of τ , the inverse subordinator of -I, gives (see also the beginning of the Section 3.2.2 [START_REF] Thomas | Random trees, Lévy processes and spatial branching processes[END_REF]) that for λ > 0

N 1 -e -λσ = ψ -1 (λ).
2.4. The dual process and the representation formula. We shall need the M f (R + )valued process η = (η t , t ≥ 0) defined by

η t (dr) = β1 [0,Ht] (r) dr + 0<s≤t X s-<I s t (X s -I s t )δ Hs (dr).
This process is called the dual process of ρ under N (see Corollary 3.1.6 of [START_REF] Thomas | Random trees, Lévy processes and spatial branching processes[END_REF]). We also denote, for s ∈ [0, σ] fixed, κ s = ρ s + η s . Recall the Poisson representation of (ρ, η) under N. Let N (dx dl du) be a point Poisson measure on [0, +∞) 3 with intensity

dx lπ(dl) 1 [0,1] (u)du.
For all a > 0, we denote by M a the law of the pair (µ a , ν a ) of measures on R + with finite mass defined by, for any

f ∈ B + (R + ) µ a , f = N (dx dl du)1 [0,a] (x)ulf (x) + β a 0 f (r)dr, ν a , f = N (dx dl du)1 [0,a] (x)(1 -u)lf (x) + β a 0 f (r)dr. We also put M = ∞ 0 dae -αa M a . Proposition 2.4. ([13], 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 we recall that σ = inf{s > 0; ρ s = 0} is the length of the excursion.

We also give the Bismut formula for the height process of the Lévy process which gives a spinal decomposition of the tree from a branch "uniformly randomly" chosen. Proposition 2.5. ( [START_REF] Thomas | Probabilistic and fractal aspects of Lévy trees[END_REF], Lemma 3.4.) For every non negative function

F defined on B + ([0, ∞]) 2 N σ 0 dsF ((H (s-t) + , t ≥ 0), (H (s+t)∧σ , t ≥ 0)) = M(dµdν) P * µ (dρ)P * ν (dρ)F (H(ρ), H(ρ)).

The Lévy Poison snake

As in [START_REF] Romain | Pruning a Lévy random continuum tree[END_REF], we construct a Lévy Poisson snake which marks the Lévy CRT on its nodes and on its skeleton. The aim is to fragment the CRT in several fragments using point processes whose intensities depend on a parameter θ such that, if θ = 0, there is no marks on the CRT and the number of marks increases with θ.

3.1. Marks on the skeleton. In order to mark the continuous part of the CRT and to keep track of marks along the lineage of each individual, we construct a snake on E = M(R 2 + ) where the parameter θ appears. To obtain a Polish space, we separate the space of the parameter θ in bounded intervals. We fix i ∈ N, thanks to [START_REF] Dawson | Measure-valued Markov processes[END_REF] Section 3.1,

E i = M f (R + × [i, i + 1)) the set of finite measures on R + × [i, i + 1
) is a Polish space for the topology of weak convergence. Thanks to [START_REF] Thomas | Random trees, Lévy processes and spatial branching processes[END_REF], Chap. 4, there exists a E i -valued process (W i t , t ≥ 0) such that conditionally on X,

each s ∈ [0, σ], W i s is a Poisson measure on [0, H s ] × [i, i + 1) with intensity 2β1 [0,Ht] (r)dr 1 [i,i+1) (θ)dθ, (2) For every s < s ′ , W i s ′ (dr, dθ)1 [0,H s,s ′ ] (r) = W i s (dr, dθ)1 [0,H s,s ′ ] (r), where we recall that H s,s ′ = inf [s,s ′ ] H. (1) For
We take the processes W i independently and we set m ske t = i∈N W i t . If β = 0, the CRT has no Brownian part, in this case, there is no mark on the skeleton and we set m ske = 0. For t ≥ 0 fixed, conditionally on H t , m ske t is Poisson point process with intensity 2β1 [0,Ht] (r)drdθ.

The process (ρ, m ske ) takes values in the space Mf := M f (R + ) × M(R 2 + ). We denote by (F s , s ≥ 0) the canonical filtration on the space of càd-làg trajectories on the space Mf . Using Theorem 4.1.2 of [START_REF] Thomas | Random trees, Lévy processes and spatial branching processes[END_REF] when H is continuous or the adapted result when H is not continuous (Prop. 7.2, [START_REF] Romain | Pruning a Lévy random continuum tree[END_REF]), we get the following result Proposition 3.1. (ρ, m ske ) is a strong Markov process with respect to the filtration (F s+ , s ≥ 0).

3.2.

Mark on the nodes. We mark every jump of the process X, say s such that ∆ s > 0, with an independent Poisson measure with intensity ∆ s 1 u>0 du, and this point Poisson measure is denoted by u>0 δ Vs,u .

When the Lévy measure of X is non trivial, we define the mark process on the nodes of the CRT as in [START_REF] Romain | Fragmentation associated with Lévy processes using snake[END_REF]. We use a Poisson point measure to introduce the parameter θ. Conditionally on X, we set

m nod t (dr, dθ) = 0<s≤t X s-<I s t (I s t -X s-) u>0
δ Vs,u (dθ) δ Hs (dr).

If π = 0, it is the Brownian case and there is no mark on the nodes, thus we set m nod = 0.

3.3. The snake. We join the marks on the skeleton and the marks on the nodes of the CRT in a mark process m = (m nod , m ske ). We write S = (ρ, m) the marked snake starting from ρ 0 = 0 and m 0 = 0. Let us recall the construction made in [START_REF] Romain | Pruning a Lévy random continuum tree[END_REF] to obtain a snake starting from an initial value and then to write a strong Markov property for the snake. We consider the set S of triplets (µ, Π nod , Π ske ) such that

• µ ∈ M f (R + ),
• Π nod can be written as Π nod (dr, dx) = µ(dr)Π nod r (dx) where (Π nod r , r > 0) is a family of σ-finite measures on R + and for every θ > 0,

Π nod (R + × [0, θ]) < ∞, • Π ske ∈ M(R 2 + ) and -Supp(Π ske (., R + )) ⊂ Supp(µ) -for every x < H(µ) and every θ > 0, Π ske ([0, x] × [0, θ]) < ∞, -if µ(H(µ)) > 0, then for every θ > 0, Π ske (R + × [0, θ]) < ∞
Then we define the snake S starting from an initial value (µ, Π) ∈ S, where Π = (Π nod , Π ske ). That is to say

S (µ,Π) 0 := (ρ µ 0 , (m nod ) (µ,Π) 0 , (m ske ) (µ,Π) 0 ) = (µ, Π).
We write

H µ t = H(k -It µ) and H µ 0,t = inf{H µ u ; u ∈ [0, t]}. We define (m nod ) (µ,Π) t = Π nod 1 [0,H µ t ) + 1 µ({H µ t })>0 k -It µ({H µ t })Π nod ({H µ t }, .) µ({H µ t }) δ H µ t Π nod H µ t , m nod t and (m ske ) (µ,Π) t = Π ske 1 [0,H µ 0,t ) , m ske t .
Notice that these definitions are coherent with the previous definitions of the processes m nod and m ske . By using the strong Markov property for the process (ρ, m nod ) (see [START_REF] Romain | Fragmentation associated with Lévy processes using snake[END_REF], Proposition 3.1) and Proposition 3.1, we obtain that the snake S is a càd-làg strong Markov process. See Proposition 2.5 of [START_REF] Romain | Pruning a Lévy random continuum tree[END_REF]. We write m (θ) (dr) = m ske (dr, [0, θ]) + m nod (dr, [0, θ]). Due to the properties of the Poisson point measures, we obtain the following result.

Proposition 3.2. m (θ+θ ′ ) t -m (θ) t is independent of m (θ) t
and has the same law as m

(θ ′ ) t .
We still denote by P µ (resp. P * µ ) the law of the snake (ρ, m nod , m ske ) starting from (µ, 0, 0) (resp. and killed when it reaches 0). We also denote by N the law of the snake S when ρ is distributed under N.

We define ψ (θ) by, for any θ ∈ R,

ψ (θ) (λ) = ψ(θ + λ) -ψ(θ) = α (θ) λ + β (θ) λ 2 + (0,+∞) (e -λl -1 + λl)π (θ) (dl) with      α (θ) = α + 2βθ + (0,+∞) (1 -e -θl )lπ(dl) β (θ) = β π (θ) (dl) = e -θl π(dl).
For fixed θ ≥ 0 and t ∈ [0, σ], we define the set A (θ) t of individuals of the Lévy CRT without marks on their lineage and its right-continuous inverse C (θ) t

given by the formulas:

A (θ) t = t 0 1 m (θ) s =0 ds and C (θ) t = inf{s > 0; A (θ) s > t}.
We define the exploration process ρ (θ) which describes the sub tree under the first marks given by m (θ) : ρ

(θ) t = ρ C (θ) t . Let F (θ) = (F (θ)
t , t ≥ 0) be the filtration generated by pruned Lévy Poisson snake S (θ) = (ρ (θ) , m (θ) ) completed the usual way. We also denote σ(θ) = inf{t > 0; ρ (θ) t = 0} and X (θ) the Lévy process with Laplace exponent ψ (θ) . We can write the key property of ρ (θ) proved by Abraham, Delmas and Voisin [START_REF] Romain | Pruning a Lévy random continuum tree[END_REF]. Proposition 3.3 (Theorem 1.1 [START_REF] Romain | Pruning a Lévy random continuum tree[END_REF]). The exploration process ρ (θ) is associated with a Lévy process with Laplace exponent ψ (θ) .

The next Lemma is also crucial for getting a fragmentation process and explains the choice of the parameters of the pruning. It has been proved by Abraham and Delmas [START_REF] Romain | Fragmentation associated with Lévy processes using snake[END_REF], see the comments under their Lemma 1.6. Notice that the proof of Abraham and Delmas is established in the general case when the quadratic coefficient β is nonnegative. Lemma 3.4. For π * (dr) a.e. r, the "law" of ρ (θ) under N, conditionally on σ (θ) = r is the same as the "law" of ρ under N, conditionally on σ = r.

3.4.

Poisson representation of the snake. We decompose the process ρ under P * µ according to excursions of the total mass of ρ above its past minimum. More precisely, let (α i , β i ), i ∈ J be the excursion intervals of X -I above 0 under P * µ . For i ∈ J, we define h i = H α i and ρ i by the formula : for t ≥ 0 and f ∈ B + (R + ),

ρ i t , f = (h i ,+∞) f (x -h i )ρ (α i +t)∧β i (dx).
We write σ i = inf{s > 0; ρ i s , 1 = 0}. We also define the mark process m above the intervals (α i , β i ). For every t ≥ 0 and f ∈ B + (R 2 + ), we set

m i,a t , f = (h i ,+∞) f (x -h i , θ)m a (α i +t)∧β i (dx, θ)
with a = ske, nod. We set for all i ∈ J, m i = (m i,nod , m i,ske ). It is easy to adapt the proof of Lemma 4.2.4 of [START_REF] Thomas | Random trees, Lévy processes and spatial branching processes[END_REF] to get the following Poisson representation.

Lemma 3.5. Let µ ∈ M f (R + ). The point measure i∈J δ (h i ,S i ) is under P * µ a Poisson point
measure with intensity µ(dr)N(dS).

3.5. Special Markov property. We fix θ ≥ 0. We define O (θ) as the interior of the set {s ≥ 0, m (θ) s = 0}. We write O (θ) = i∈ Ĩ (a i , b i ) and we say that (a i , b i ) are the excursions intervals of the Lévy marked snake S (θ) = (ρ (θ) , m (θ) ) away from {s ≥ 0; m (θ) s = 0}. We set h i = H a i and we define the process S (θ),i = (ρ (θ),i , m (θ),i ) above the excursion intervals ((a i , b i ), i ∈ Ĩ) as previously.

If Q is a measure on S and ϕ is a non-negative measurable function defined on the measurable space R + × Ω × S, we denote by

Q[ϕ(u, ω, •)] = S ϕ(u, ω, S)Q(dS).
We now recall the special Markov property proved by Abraham, Delmas and Voisin [START_REF] Romain | Pruning a Lévy random continuum tree[END_REF]. It gives the distribution of the Lévy snake "above" the "first" marks of the marked CRT knowing the part of the pruned CRT where the root belongs to.

Theorem 3.6 ([2], Theorem 4.2 ). (Special Markov property)

We fix θ > 0. Let φ be a non-negative measurable function defined on

R + × S such that t → φ(t, ω, S) is progressively F (θ)
∞ -measurable for any S ∈ S. Then, we have P-a.s.

(1) E   exp   - i∈ Ĩ ϕ(A (θ) a i , ω, S (θ),i )   F (θ) ∞   = exp - ∞ 0 du 2βθN 1 -e -ϕ(u,ω,•) exp - ∞ 0 du (0,∞) (1 -e -θℓ )π(dℓ) 1 -E * ℓ [e -ϕ(u,ω,•) ] .
Furthermore, the law of the excursion process

i∈ Ĩ δ (A (θ) a i ,ρ (θ)
a i -,S (θ),i ) , given F (θ)
∞ , is the law of a Poisson point measure with intensity

1 u≥0 du δ ρ (θ) u (dµ) 2βθN(dS) + (0,∞)
(1e -θℓ )π(dℓ)P * ℓ (dS) .

4. Links between the snake and the fragmentation 4.1. Construction of the fragmentation process. We are interested in the fragments of the tree given by the marks process. We do the same construction as in [START_REF] Romain | Fragmentation associated with Lévy processes using snake[END_REF], Section 4.1.

For fixed θ ≥ 0, we first construct an equivalence relation,R θ , on [0, σ] under N or under N σ by :

sR θ t ⇔ m (θ) s ([H s,t , H s ]) = m (θ)
t ([H s,t , H t ]) = 0. Two individuals, s and t, belong to the same equivalence class if they belong to the same fragment, that is to say if there is no mark on their lineage down to their most recent common ancestor. From the equivalence relation R θ , we get the family of sets G j of individuals with j marks in their lineage.

As we put marks on infinite nodes of the CRT, for θ > 0, for fixed j ∈ N, the set G j can be written as an infinite union of sub-intervals of [0, σ]. We get

G j = k∈J j R j,k
such that R j,k has positive Lebesgue measure. For j ∈ N and k ∈ J j , we set

A j,k t = t 0 1 s∈R j,k ds and C k,j t = inf{u ≥ 0; A j,k u > t},
with the convention inf ∅ = σ. We also construct the process Sj,k = (ρ j,k , mj,k ) by : for every

f ∈ B + (R + ), ϕ ∈ B + (R + , R + ) and t ≥ 0, ρj,k t , f = (H C j,k 0 ,+∞) f (x -H C j,k 0 )ρ C j,k t (dx) mj,k t , ϕ = (H C j,k 0 ,+∞)×(θ,+∞) ϕ(x -H C j,k 0 , v -θ)m C j,k t (dx, dv) σj,k corresponds to the Lebesgue measure of R j,k . We denote L (θ) = (ρ j,k ; j ∈ N, k ∈ J j ) = (ρ i ; i ∈ I (θ)
). We also define L (θ-) = (ρ i ; i ∈ I (θ-) ) the set defined similarly but using the equivalence relation R θ-which gives the fragments just before time θ.

We now define the process Λ θ = (Λ θ 1 , Λ θ 2 , . . . ) as the sequence of non trivial Lebesgue measure of the equivalence classes of R θ , (σ j,k , j ∈ N, k ∈ J j ), ranked in decreasing order. Notice that, when θ > 0, this sequence is infinite. When θ = 0, Λ 0 is the entire tree and we denote Λ 0 = (Λ 0 , 0, . . . ). Then we have that N-a.s. and N σ -a.e. Λ θ ∈ S ↓ .

We write P σ the law of (Λ θ , θ ≥ 0) under N σ and by convention P 0 is the Dirac mass at (0, 0, . . . ) ∈ S ↓ . Theorem 4.1. For π * (dr)-almost every r, under P r , (Λ θ , θ ≥ 0) is a S ↓ -valued fragmentation process.

Sketch of proof.

As the proof is exactly the same as for [START_REF] Romain | Fragmentation associated with Lévy processes using snake[END_REF], Theorem 1.1., we only give the main ideas of the proof and refer to [START_REF] Romain | Fragmentation associated with Lévy processes using snake[END_REF] for precise details.

1st step. Thanks to Proposition 3.3, the first sub-excursion ρ0,0 is "distributed" under N as ρ (θ) . Moreover, by Lemma 3.4 and the construction of the Poisson snake conditionally given ρ, the law of S0,0 conditionally on {σ 0,0 = s} is N s .

2nd step. By the special Markov property, Theorem 3.6, using the same notations as in this theorem, we have that, under N conditionally on σ0,0 = s, the processes (S (θ),i , i ∈ Ĩ) are given by a Poisson measure with intensity s2βθN(dS) + s (0,+∞)

(1e -θℓ )π(dℓ)P * ℓ (dS)

and are independent of S0,0 . Moreover, by the Poisson representation of the probability measure P * ℓ (Lemma 3.5), we get that the excursions of the snake S "above the first mark", that we denote (S 1,k , k ∈ J 1 ), form under N an i.i.d. family of processes with "distribution" N.

3rd step By induction on the number of marks, we get the following lemma Lemma 4.2. Under N, the law of the family ( Sj,k , j ∈ N, k ∈ J j ), conditionally on (σ j,k , j ∈ N, k ∈ J j ), is the law of independent Lévy Poisson snakes distributed respectively as N σj,k .

The theorem then follows easily. Conditionally on the process H (or equivalently on ρ), we set a Poisson point process Q(dθ, ds, da) under the epigraph of H with intensity dθ q ρ (ds, da) where q ρ (ds, da) = ds κ s (da) d s,ag s,a

= q ske ρ (ds, da) + q nod ρ (ds, da) For an individual t, marks on the skeleton are uniform from the height h 1 to the height h 2 . Thanks to snake property, if a mark appears at height a on the lineage of the individual t, it also appears for all children of t, that is to say the mark appears from g t,a to d t,a . We have

with            q nod ρ (ds, da) = ds d s,a -g s,a 0<u≤s X u-<I u s (X u -X u-)δ Hu (da) q ske ρ (ds, da) = 2β ds 1 [0,Hs] (a)da d s,a -g s,a with d s,a = sup{u ≥ s, min{H v , v ∈ [s, u]} ≥ a} and g s,a = inf{u ≤ s, min{H v , v ∈ [s, u]} ≥ a}. [g s,a , d s,a ] is
D 2βθda ds d s,a -g s,a = 2β(h 2 -h 1 )
where

D = {(a, s) ∈ [h 1 , h 2 ] × [g t,a , d t,a ]} because for a ∈ [h 1 , h 2 ] fixed, ∀s ∈ [g t,h , d t,h ],
g s,a = g t,a and d s,a = d t,a . (See Figure 1). Thus the point process q ske give the same marks as the process m ske .

Conditionally on H, under N, for fixed t ∈ [0, σ], m nod t puts marks on a node at height H u < t proportionally to the size ∆ u of the node. By construction of m nod , if a mark appears at height H u = a on the lineage of an individual t, it also appears from g t,a to d t,a . We have

D ∆ u θδ Hu (da)ds d s,a -g s,a = θ∆ u where D = {(a, s) ∈ [0, H t ] × [g t,a , d t,a ]}.
Thus the point process q nod give the same marks as the process m nod .

The equality m = m ske + m nod ends the proof. We use a notation for the fragments of the CRT obtained from a mark (s, a) under the epigraph of H. For s and a such that s ∈ [0, σ] and 0 ≤ a ≤ H s , we denote the fragments of the Lévy snake (ρ i , i ∈ Ĩ) by :

• the open intervals of the excursion of H after s and above a : ((α i , β i ), i ∈ Ĩ+ ) which are such that α i > s, H α i = H β i = a and for every s ′ ∈ (α i , β i ), H s ′ > a and H s,s ′ = a. • the open intervals of the excursion of H before s and above a : ((α i , β i ), i ∈ Ĩ-) which are such that β i < s, H α i = H β i = a and for every s ′ ∈ (α i , β i ), H s ′ > a and H s,s ′ = a. • the excursion i s , of H above a and which contains s

: (α is , β is ) such that α is < s < β is , H α is = H β is = a and for every s ′ ∈ (α is , β is ), H s ′ > a and H s,s ′ > a. • the excursion i 0 of H without the mark (s, a) : {s ∈ [0, σ]; H s,s ′ < a} = [0, α i 0 ) ∪ (β i 0 , σ]
We write Ĩ = Ĩ-∪ Ĩ+ ∪ {i s , i 0 } (see Figure 2). Then the family (ρ i , i ∈ Ĩ) contains the exploration processes of the fragments obtained when a single cutpoint is selected. We are interested in the computation of νρ , the "law" of (ρ i , i ∈ Ĩ) under N(dρ)q ρ (ds, da).

Figure 2. Fragments of the Lévy snake obtained from a mark (s, a)

α i0 α is s β is β i0 σ Ĩ+ Ĩ- a

The dislocation process

Let T be the set of jumping times of the Poisson process Q. For θ ∈ T , we consider the processes L (θ) = (ρ i ; i ∈ I (θ) ) and L (θ-) = (ρ i ; i ∈ I (θ-) ) defined in the Section 4.1. The life times (σ(ρ i ); i ∈ I (θ) ) (resp. (σ(ρ i ); i ∈ I (θ-) ) ), ranked by decreasing order, of these Lévy snakes correspond to the "sizes" of the fragments at time θ (resp. before time θ). Notice that, for θ ∈ T fixed, the families L (θ) and L (θ-) change in one family : the snake ρ i θ breaks in one family (ρ i , i ∈ Ĩ(θ) ) ∈ L (θ) . Thus we get

L (θ) = L (θ-) \ {ρ i θ } ∪ {ρ i ; i ∈ Ĩ(θ) }.
Let ν r be the distribution of the decreasing lengths of Lévy snakes under νρ , integrated w.r.t. the law of ρ conditionally on σ(ρ) = r, that is to say, for any non-negative measurable function

F defined on S ↓ S ↓ F (x)ν r (dx) = N r F ((σ i , i ∈ Ĩ))ν ρ (d(ρ i , i ∈ Ĩ))
where the (σ i , i ∈ Ĩ) are the lengths of the fragments (ρ i , i ∈ Ĩ) ranked in decreasing order. The family of measures (ν r , r > 0) is then the family of dislocation measures defined in the Section 1.3. Indeed, the formula above means that ν r gives the distribution of the lengths of the fragments (ranked by decreasing order) coming from the dislocation of one fragment of size r. For x = (x i , i ∈ I (θ) ) ∈ S ↓ , if we consider the dislocation of all the fragments of x with respective sizes x i > 0, we found the formula of the dislocation measure ν x i given in Section 1.3:

F (s)ν x (ds) = i≥1,x i >0 F (x i,s )ν x i (ds)
where νx is defined as the intensity of a Poisson point process and is the law of the lengths (x i , i > 0). 5.1. Computation of dislocation measure. We are interested in the family of dislocation measures (ν r , r > 0). Recall that N(.) = R + π * (dr)N r (.). The computation is easier under N, then we compute for any λ ≥ 0 : R + ×S ↓ F (x)ν r (dx)π * (dr) = N q ρ (ds, da)F ((σ i , i ∈ Ĩ))

= N q nod ρ (ds, da)F ((σ i , i ∈ Ĩ)) + N q ske ρ (ds, da)F (σ i 0 , σ is )

where we use the decomposition of q ρ for the second equality. The first part has already been computed in [START_REF] Romain | Fragmentation associated with Lévy processes using snake[END_REF]. Jumping times of the process ρ are represented by a subordinator W with Laplace exponent ψ ′α. Then we construct the length of the excursions of the snake by S W where S is a subordinator with exponent ψ -1 , independent of W . Then we have :

N e -λσ q nod ρ (ds, da)F ((σ i , i ∈ Ĩ)) = π(dv)E S v e -λSv F ((∆S u , u ≤ v)) .

We now compute the second part. Thanks to the definition of the snake, ρ ske = 0 if and only if β = 0 and in this case, we don't put mark on the skeleton of the tree. We assume that β > 0 and we write the key lemma of this article which prove the Part 2 of the Theorem 1.1.

A 2 = 2β M(dµdν) 1 0≤a≤H(µ) daE * µ   exp   -(λ 1 + λ 2 ) h i ≥a σ i -λ 2 h i <a σ i     E * ν   exp   -(λ 1 + λ 2 ) h i ≥a σ i -λ 2 h i <a σ i     .
Using standard properties of Poisson point measures, the atoms above level a are independent of the atoms below, the expectations can be separated.

E * µ e -(λ 1 +λ 2 ) h i ≥a σ i -λ 2 h i <a σ i = E * µ e -(λ 1 +λ 2 ) h i ≥a σ i E * µ e
-λ 2 h i <a σ i .

We use Lemma 3.5, and the equality ψ -1 (λ) = N 1e -λσ , we get

E * µ e
-(λ 1 +λ 2 ) h i ≥a σ i = e -µ([a,H(µ)])N[1-e -(λ 1 +λ 2 )σ ] = e -µ([a,b])ψ -1 (λ 1 +λ 2 ) .

And we do the same for the second expectation. = e αb e -(b-a)ψ ′ ψ -1 (λ 1 +λ 2 )-aψ ′ ψ -1 (λ 2 ) . We recall the expression of A 2

A 2 = 2β ∞ 0 db e -bψ ′ ψ -1 (λ 2 ) -e -bψ ′ ψ -1 (λ 1 +λ 2 ) ψ ′ ψ -1 (λ 1 + λ 2 ) -ψ ′ ψ -1 (λ 2 ) = 2β ψ ′ ψ -1 (λ 1 + λ 2 ) -ψ ′ ψ -1 (λ 2 ) 1 ψ ′ ψ -1 (λ 2 ) - 1 ψ ′ ψ -1 (λ 1 + λ 2 ) = 2β ψ ′ ψ -1 (λ 2 )ψ ′ ψ -1 (λ 1 + λ 2 )
.

We use the equality N q ske ρ (ds, da)σ is G(σ is , σ) = N q ske ρ (ds, da)σ is e -(λ 1 +λ 2 )σ is -λ 2 σ i 0 , we finally get the result.

Brownian case.

A similar result has been obtained by Abraham and Serlet [START_REF] Romain | Poisson snake and fragmentation[END_REF] in the Brownian case and conditionally on σ = 1. They use the same construction of the marks on the skeleton given by Aldous and Pitman [START_REF] David | The standard additive coalescent[END_REF].

We consider a standard Brownian motion with Laplace exponent ψ(λ) = λ 2 2 and we denote by Γ(de) the law of the Brownian excursion e. Thanks to [START_REF] Gall | Spatial branching processes, random snakes and partial differential equations[END_REF], Section VIII.3, the height

4. 2 .

 2 Another representation of the fragmentation. We give an another representation of the fragmentation by using a Poisson point measure under the epigraph of the height process. Recall that for every t ∈ [0, σ], κ t (dr) = 2β1 [0,Ht] (r)(dr) + 0<s≤t X s-<I s t (X s -X s-)δ Hs (dr).

Proposition 4 . 3 .

 43 the set of individuals of the CRT with common ancestor s after generation a. Conditionally on the process H, the mark process m and the Poisson point process Q have same distribution Proof. Conditionally on H and under N, for fixed t ∈ [0, σ], m ske t is a Poisson point process with intensity 2β1 [0,Ht] (a)da dθ.

Figure 1 .

 1 Figure 1. Marks under the epigraph of H

A 2

 2 b e -(µ+ν)([a,b])ψ -1 (λ 1 +λ 2 ) e -(µ+ν)([0,a))ψ -1 (λ 2 ) . Then, M b e -((µ+ν)([a,b])ψ -1 (λ 1 +λ 2 ) e -(µ+ν)([0,a))ψ -1 (λ 2 ) = M b e -((µ+ν)([a,b])ψ -1 (λ 1 +λ 2 ) M b e -(µ+ν)([0,a))ψ -1 (λ 2 ) = e -2(b-a)βψ -1 (λ 1 +λ 2 ) exp -)(1e -lψ -1 (λ 1 +λ 2 ) )e -2aβψ -1 (λ 2 ) exp -)(1e -lψ -1 (λ 2 ) )

  And a.s. for all s < t such that X s-≤ I s t and for s = t, if ∆ t > 0 then H s < ∞ and for all t ′ > t ≥ 0, the process H takes all the values between H t and H t ′ on the time interval [t, t ′ ].

	a.s. H t =	L(t)

t .

Acknowledgement

I am grateful to my PhD. advisor Romain Abraham for his helpful discussions.

Lemma 5.1. We set λ 1 > 0 and λ 2 > 0.

N

q ske ρ (ds, da)σ is e -λ 1 σ is -λ 2 σ i 0 = 2β ψ ′ ψ -1 (λ 1 )ψ ′ ψ -1 (λ 2 ) .

We recall that the measure νske r gives the law of the non-reordering of the two lengths given by the fragmentation from ν ske r . Proof of Part 2 of Theorem 1.1. We use Lemma 5.1, let x 1 and x 2 be the lengths of the fragments from ν ske ρ ranked by decreasing order among the elements of x ∈ S ↓ , we get

.

We integrate w.r.t. λ 1 and we take the primitive which vanishes in 0, and we do the same with λ 2 . We get that, for λ 1 > 0 and λ 2 > 0,

Thus, under νske r (dx)π * (dr), the lengths of the two fragments are independent. Proof of Lemma 5.1. In order to prove the lemma, we compute

We denote for 0 ≤ s ≤ σ and 0 ≤ a ≤ H s fixed

We use the generalization for Lévy processes of Bismut formula, Proposition 2.5.

where J µ (a) is the first passage time of the process H (µ) at level a. By the Poissonnian decomposition of ρ under P * µ w.r.t. the excursions of ρ above its minimum, under P * µ , we replace respectively J µ (0) and J µ (a) by i∈I σ i and h i ≥a σ i . We separate i∈I σ i =

process of the Brownian motion is given by H t = 2(X t -I t ). We resume the computation of [START_REF] Romain | Poisson snake and fragmentation[END_REF] by taking marks under the epigraph of H, we get

where ν is the dislocation measure of [START_REF] Romain | Poisson snake and fragmentation[END_REF]. The computation of [START_REF] Romain | Poisson snake and fragmentation[END_REF] uses the law the two independent 3-dimensional Bessel processes, then we get

As before, we compute with F (x, y) = xe -λ 1 x-λ 2 y .

For the end of this computation, we use the two changes of variable : z ↔ sin 2 x and then t ↔ tanx.

We integrate a last time, we get the same result as in Lemma 5.1 :

.


Tewfik Sari

GLOBAL DYNAMICS OF THE CHEMOSTAT WITH VARIABLE YIELDS

Keywords: . 2000 Mathematics Subject Classification. 92A15, 92A17, 34C15, 34C35 chemostat, competitive exclusion principle, Lyapunov function, global asymptotic stability, variable yield model. 1

In this paper, we consider a competition model between n species in a chemostat including both monotone and non-monotone response functions, distinct removal rates and variable yields. We show that only the species with the lowest break-even concentration survives, provided that additional technical conditions on the growth functions and yields are satisfied. LaSalle's extension theorem of the Lyapunov stability theory is the main tool.

Introduction

In this paper we study the global dynamics of the following model of the chemostat in which n populations of microorganisms compete for a single growth-limiting substrate:

S ′ (t) = D[S 0 -S(t)] -n i=1 f i (S(t))x i (t) x ′ i (t) = [p i (S(t)) -D i ]x i (t), i = 1 • • • n, (1.1)
where S(0) ≥ 0 and x i (0) > 0, i = 1 • • • n and S 0 , D and D i are positive constants. In these equations, S(t) denotes the concentration of the substrate at time t; x i (t) denotes the concentration of the ith population of microorganisms at time t; f i (S) represents the uptake rate of substrate of the ith population; p i (S) represents the per-capita growth rate of the ith population and so the function y i (S) defined by

y i (S) = p i (S) f i (S)
is the growth yield; S 0 and D denote, respectively, the concentration of substrate in the feed bottle and the flow rate of the chemostat; each D i represents the removal rate of the ith population. We make the following assumptions on the functions p i and f i :

• p i , f i : R + → R + are continuous,
• p i (0) = f i (0) = 0 and for all S > 0, p i (S) > 0 and f i (S) > 0,

For general background on model (1.1), in the constant yield case y i (S) = Y i , the reader is referred to the monograph of Smith and Waltman [START_REF] Smith | The Theory of the Chemostat, Dynamics of Microbial Competition[END_REF]. Following Butler and Wolkowicz [START_REF] Butler | A mathematical model of the chemostat with a general class of functions describing nutrient uptake[END_REF], we make the following assumptions on the form of the response functions p i : there exist positive extended real numbers λ i and µ i with

λ i ≤ µ i ≤ +∞ such that p i (S) < D i if S / ∈ [λ i , µ i ] p i (S) > D i if S ∈]λ i , µ i [.
Hence there are at most two values of S, S = λ i and S = µ i , called the breakeven concentrations, satisfying the equation p i (S) = D i . We adopt the convention µ i = ∞ if this equation has only one solution and λ i = ∞ if it has no solution.

The global analysis of this model was considered by Hsu, Hubbell and Waltman [START_REF] Hsu | A mathematical theory for single nutrient competition in continuous culture of micro-organisms[END_REF], in the Monod case [START_REF] Monod | La technique de culture continue[END_REF] when the response functions are of Michaelis-Menten form,

p i (S) = a i S b i + S , (1.2)
and the yields are constant y i (S) = Y i , and

D i = D for i = 1 • • • n.
The authors showed that only the species with the lowest break-even concentration survives. Thus the competitive exclusion principle (CEP) holds: only one species survives, namely the species which makes optimal use of the resources. Hsu [START_REF] Hsu | Limiting behavior for competing species[END_REF] applied a Lyapunov-LaSalle argument to give a simple and elegant proof of the result in [START_REF] Hsu | A mathematical theory for single nutrient competition in continuous culture of micro-organisms[END_REF] for the case of different removal rates D i . The Lyapunov function V H used by Hsu is

V H = S λ1 σ -λ 1 σ dσ + c 1 x1 x * 1 ξ -x * 1 ξ dξ + n i=2 c i x i , (1.3)
where

c i = 1 Y i a i a i -D i , i = 1 • • • n, and x * 1 = DY 1 S 0 -λ 1 D 1 .
CEP has been proved under a variety of hypothesis [START_REF] Armstrong | Competitive exclusion[END_REF][START_REF] Butler | A mathematical model of the chemostat with a general class of functions describing nutrient uptake[END_REF][START_REF] Li | Global asymptotic behavior of the chemostat : general response functions and differential removal rates[END_REF][START_REF] Wolkowicz | Global dynamics of a mathematical model of competition in the chemostat : general response functions and differential death rates[END_REF][START_REF] Wolkowicz | Global asymptotic behavior of a chemostat model with discrete delays[END_REF]. For a survey on the contribution of each paper the reader may consult the introduction of the papers [START_REF] De Leenheer | Competition in the chemostat: some remarks[END_REF][START_REF] Li | Global asymptotic behavior of the chemostat : general response functions and differential removal rates[END_REF][START_REF] Sari | Global dynamics of the chemostat with different removal rates and variable yields[END_REF]. The variable yield case was considered, for n = 1 and n = 2 by Pilyugin and Waltman [START_REF] Pilyugin | Multiple limit cycles in the chemostat with variable yields[END_REF], with a particular interest to linear and quadratic yields, and by Huang, Zhu and Chang [START_REF] Huang | Limit cycles in a chemostat with variable yields and growth rates[END_REF]. The model (1.1) was considered by Arino, Pilyugin and Wolkowicz [START_REF] Arino | Considerations on yield, nutrient uptake, cellular growth, and competition in chemostat models[END_REF] and Sari and Mazenc [START_REF] Sari | Global dynamics of the chemostat with different removal rates and variable yields[END_REF].

In this paper we generalize [START_REF] Wolkowicz | Global dynamics of a mathematical model of competition in the chemostat : general response functions and differential death rates[END_REF] by allowing variable yields and we generalize [START_REF] Arino | Considerations on yield, nutrient uptake, cellular growth, and competition in chemostat models[END_REF] by allowing multi species. We further extend [START_REF] Sari | Global dynamics of the chemostat with different removal rates and variable yields[END_REF] by providing less restrictive assumptions on the system. For biological motivations concerning the dependance of the yields on the substrate, the reader is refeered to [START_REF] Arino | Considerations on yield, nutrient uptake, cellular growth, and competition in chemostat models[END_REF][START_REF] Pilyugin | Multiple limit cycles in the chemostat with variable yields[END_REF] and the references therein.

Analysis of the model

It is known (see Theorem 4.1 [START_REF] Arino | Considerations on yield, nutrient uptake, cellular growth, and competition in chemostat models[END_REF]) that the non-negative cone is invariant under the flow of (1.1) and all solutions are defined and remain bounded for all t ≥ 0. System (1.1) can have many equilibria: the washout equilibrium

E 0 = (S 0 , 0, • • • , 0), which is locally exponentially stable if and only if for all i = 1 • • • n, S 0 / ∈ [λ i , µ i ] and the equilibria E * i and E * *
i where all component of E * i and E * * i vanish except for the first and the (i + 1)th, which are

S = λ i , x i = x * i := F i (λ i ), for E * i and S = µ i , x i = x * * i := F i (µ i ), for E * * i respectively,
where

F i (S) = D S 0 -S f i (S) .
The equilibrium E * i lies in the non-negative cone if and only if λ i ≤ S 0 . If λ i < λ j for all i = j and F ′ i (λ i ) < 0 then it is locally exponentially stable. It coalesces with E 0 when λ i = S 0 . The equilibrium E * * i lies in the non-negative cone if and only if µ i ≤ S 0 and is locally exponentially unstable if it exists. Its coalesces with E 0 when µ i = S 0 . Besides these equilibria, the system (1.1) can have a continuous set of non-isolated equilibria in the non-generic cases where two or more of the break-even concentrations are equal. In what follows we assume, that

λ 1 < λ 2 ≤ • • • ≤ λ n ,
and

λ 1 < S 0 < µ 1 . (2.1)
Hence E 0 is locally exponentially unstable and the equilibrium E * 1 lies in the nonnegative cone. It is locally exponentially stable if and only if

F ′ 1 (λ 1 ) < 0 ⇐⇒ f 1 (λ 1 ) + f ′ 1 (λ 1 )(S 0 -λ 1 ) > 0. (2.2)
We consider the global asymptotic stability of E * 1 . Before presenting the results, we need the following lemma, Lemma 2.1. The solutions S(t), x i (t), i = 1 • • • n of (1.1) with positive initial conditions are positive and bounded, and if λ i < S 0 < µ i for some i = 1 • • • n, then S(t) < S 0 for all sufficiently large t.

Proof. The proof is similar to the proof of Lemma 2.1 in [START_REF] Wolkowicz | Global dynamics of a mathematical model of competition in the chemostat : general response functions and differential death rates[END_REF] obtained for the model (1.1) in the case where the yields are constant. Theorem 2.2. Assume that

(1) λ 1 < λ 2 ≤ • • • ≤ λ n , and λ 1 < S 0 < µ 1 .
(2) There exist constants α i > 0 for each i ≥ 2 satisfying λ i < S 0 such that

max 0<S<λ1 g i (S) ≤ α i ≤ min λi<S<ρi g i (S), i ≥ 2. (2.3)
where

g i (S) = f i (S) f 1 (λ 1 ) p 1 (S) -D 1 p i (S) -D i S 0 -λ 1 S 0 -S . (2.4)
(3) The function

F (S) = f 1 (S) S 0 -S . (2.5)
satisfies

F (S) < F (λ 1 ) if S ∈]0, λ 1 [, F (S) > F (λ 1 ) if S ∈]λ 1 , S 0 [. (2.6)
Then the equilibrium E * 1 is globally asymptotically stable for system (1.1) with respect to the interior of the positive cone.

Proof. From Lemma 2.1 it follows that there is no loss of generality in restricting our attention to 0 ≤ S < S 0 . Consider the function

V = V (S, x 1 , • • • , x n ) as follows V = S λ1 (p 1 (σ) -D 1 )(S 0 -λ 1 ) f 1 (λ 1 )(S 0 -σ) dσ + x1 x * 1 ξ -x * 1 ξ dξ + n i=2 α i x i (2.7)
where α i are the positive constants satisfying (2.3). The function V is continuously differentiable for 0 < S < S 0 and x i > 0 and positive except at the point E * 1 . The derivative of V along the trajectories of (1.1) is given by

V ′ = x 1 [p 1 (S) -D 1 ] F (λ 1 ) -F (S) F (λ 1 ) + n i=2 x i h i (S)
where

h i (S) = [p i (S) -D i ][α i -g i (S)]
and g i (S) and F (S) are given by (2.4) and (2.5) respectively. First, note that, using (2.1) and (2.6), the first term of the above sum is always non-positive for 0 < S < S 0 and equals 0 for S ∈]0, S 0 [ if and only if S = λ 1 or x 1 = 0. If S ∈ [λ 1 , λ i ] then p i (S) < D i and p 1 (S) > D 1 so that g i (S) < 0 < α i for any choice of α i > 0. Similarly if µ i < S 0 and S ∈ [µ i , S 0 ] then p i (S) < D i and p 1 (S) > D 1 so that g i (S) < 0 < α i for any choice of α i > 0. On the other hand, if S ∈ [0, λ 1 ] then p i (S) < D i and, using (2.3), g i (S) ≤ α i so that h i (S) < 0. Finally, if S ∈ [λ i , ρ i ] then p i (S) > D i and g i (S) ≥ α i so that h i (S) < 0. Thus h i (S) < 0 for every S ∈]0, S 0 [, provided that the numbers α i satisfy (2.3). Hence V ′ ≤ 0 and V ′ = 0 if and only if

x i = 0 for i = 1 • • • n or S = λ 1 and x i = 0 for i = 2 • • • n.
We use LaSalle theorem (the details are as in the proof of Theorem 2.3 in [START_REF] Wolkowicz | Global dynamics of a mathematical model of competition in the chemostat : general response functions and differential death rates[END_REF]), the ω-limit set of the trajectory is E * 1 .

Condition (2.6) is satisfied in the particular case where F ′ (S) > 0 for 0 < S < S 0 . For S = λ 1 we obtain the condition (2.2) of local exponential stability of E * 1 .

Applications

In this section we show how Theorem 2.2 extends results in [START_REF] Arino | Considerations on yield, nutrient uptake, cellular growth, and competition in chemostat models[END_REF][START_REF] Sari | Global dynamics of the chemostat with different removal rates and variable yields[END_REF][START_REF] Wolkowicz | Global dynamics of a mathematical model of competition in the chemostat : general response functions and differential death rates[END_REF].

3.1. One species. In the case n = 1 the equations take the form

S ′ = D(S 0 -S) -x 1 f 1 (S) x ′ 1 = [p 1 (S) -D 1 ]x 1 (3.1)
Corollary 3.1. If λ 1 < S 0 < µ 1 and 1 -f1(S)(S 0 -λ1) f1(λ1)(S 0 -S) has exactly one sign change for S ∈ (0, S 0 ) then E * 1 is globally asymptotically stable with respect to the interior of the positive quadrant.

Proof. The result follows from Theorem 2.2 since the hypothesis on the change of sign is is equivalent to (2.6). In the case when n = 1 the condition (2.3) is obviously satisfied. Corollary 3.1 was obtained by Arino, Pilyugin and Wolkowicz (see [START_REF] Arino | Considerations on yield, nutrient uptake, cellular growth, and competition in chemostat models[END_REF], Theorem 2.11). These authors used the following Lyapunov function

V AP W = S λ1 (p 1 (σ) -D 1 )(S 0 -λ 1 ) f 1 (λ 1 )(S 0 -σ) dσ + x1 x * 1 ξ -x * 1 ξ dξ.
Notice that the Lyapunov function (2.7) we use reduces to the function V AP W in the case when n = 1.

Constant yields.

In the case when the yields are constant, y i (S) = Y i , the equations take the form

S ′ = D[S 0 -S] - n i=1 pi(S) Yi x i x ′ i = [p i (S) -D i ]x i , i = 1 • • • n, (3.2) Corollary 3.2. Assume that (1) λ 1 < λ 2 ≤ • • • ≤ λ n , and λ 1 < S 0 < µ 1 .
(2) There exist constants α i > 0 for each i ≥ 2 satisfying λ i < S 0 such that

max 0<S<λ1 g W L i (S) ≤ α i ≤ min λi<S<ρi g W L i (S), i ≥ 2. (3.3)
where

g W L i (S) = p i (S) D 1 p 1 (S) -D 1 p i (S) -D i S 0 -λ 1 S 0 -S . (3.4)
Then the equilibrium E * 1 is globally asymptotically stable for system (3.2) with respect to the interior of the positive cone.

Proof. In the case when the yields are constant, we have

g i (S) = Y 1 Y i g W L i (S).
Hence the conditions (3.3) imply the conditions (2.3) with constants α i replaced by α i Y 1 /Y i . On the other hand F (S) = p1(S) (S 0 -S)Y1 . Thus, assumption (2.6) follows from hypothesis λ 1 < S 0 < µ 1 . The result follows from Theorem 2.2. Corollary 3.2 was obtained by Wolkowicz and Lu (see [START_REF] Wolkowicz | Global dynamics of a mathematical model of competition in the chemostat : general response functions and differential death rates[END_REF], Theorem 2.3). These authors used the following Lyapunov function

V W L = S λ1 (p 1 (σ) -D 1 )(S 0 -λ 1 ) D 1 (S 0 -σ) dσ + 1 Y 1 x1 x * 1 ξ -x * 1 ξ dξ + n i=2 α i Y i x i .
Notice that the Lyapunov function (2.7) we use satisfies V = Y 1 V W L in the case when the yields are constant.

3.3. Variable yields. In [START_REF] Sari | Global dynamics of the chemostat with different removal rates and variable yields[END_REF], another Lyapunov function is proposed in the case when the yields are variable, leading to the following result.

Corollary 3.3. Assume that (1) λ 1 < λ 2 ≤ • • • ≤ λ n , and λ 1 < S 0 < µ 1 .
(2) There exist constants α i > 0 for each i ≥ 2 such that

max 0<S<λ1 g SM i (S) ≤ α i ≤ min λi<S<ρi g SM i (S), i ≥ 2. ( 3.5)
where

g SM i (S) = f i (S) f 1 (S) p 1 (S) -D 1 p i (S) -D i . (3.6)
(3) The function

F (S) = f 1 (S) S 0 -S satisfies
F (S) < F (λ 1 ) if S ∈]0, λ 1 [, F (S) > F (λ 1 ) if S ∈]λ 1 , S 0 [. (3.7)
Then the equilibrium E * 1 is globally asymptotically stable for system (1.1) with respect to the interior of the positive cone.

Proof. The condition (3.7) is the same as the condition (2.6). On the other hand we have

g i (S) = S 0 -λ 1 f 1 (λ 1 ) F (S)g SM i (S).
By (3.5) and (3.7) we have (2.3) with appropriate constants α i . The result follows from Theorem 2.2.

Corollary 3.3 was obtained by Sari and Mazenc (see [START_REF] Sari | Global dynamics of the chemostat with different removal rates and variable yields[END_REF], Theorem 2.2). These authors used the following Lyapunov function

V SM = S λ1 p 1 (σ) -D 1 f 1 (σ) dσ + x1 x * 1 ξ -x * 1 ξ dξ + n i=2 α i x i
where α i , i = 2 • • • n are positive constants satisfying (3.5). In the case when the response functions are of Michaelis-Menten form (1.2) and the yields are constant the Lyapunov function V SM reduces to the Lyapunov function V H (1.3) useb by Hsu [START_REF] Hsu | Limiting behavior for competing species[END_REF]. Notice that the Lyapunov function (2.7) we use is not proportional to the function V SM .

Discussion

In this paper we considered a mathematical model (1.1) of n species of microorganisms in competion in a chemostat for a single resource. The model incorporates both monotone and non-monotone response functions, distinct removal rates and variable yields. We demonstrated that the CEP holds for a large class of growth functions and yields.

Even with constant yields, the problem is not yet completely solved: the CEP holds for a large class of growth functions [START_REF] Armstrong | Competitive exclusion[END_REF][START_REF] Butler | A mathematical model of the chemostat with a general class of functions describing nutrient uptake[END_REF][START_REF] Hsu | Limiting behavior for competing species[END_REF][START_REF] Li | Global asymptotic behavior of the chemostat : general response functions and differential removal rates[END_REF][START_REF] Wolkowicz | Global dynamics of a mathematical model of competition in the chemostat : general response functions and differential death rates[END_REF][START_REF] Wolkowicz | Global asymptotic behavior of a chemostat model with discrete delays[END_REF] but an important open question remains: is the CEP true assuming only that the f i are monotone with no restriction on the D i ? This major open problem remains unresolved after more than thirty years [START_REF] De Leenheer | Competition in the chemostat: some remarks[END_REF]. However, in the case of constant yields numerical simulations of model (1.1) have only displayed competitive exclusion.

In the case where the yields are constant, it is known [START_REF] Butler | A mathematical model of the chemostat with a general class of functions describing nutrient uptake[END_REF] that the CEP holds provided that D i = D for all i, the set Q = i∈N ]λ i , µ i [ is connected, and S 0 ∈ Q, where N = {i : λ i < S 0 }. Wolkowicz and Lu [START_REF] Wolkowicz | Global dynamics of a mathematical model of competition in the chemostat : general response functions and differential death rates[END_REF] conjectured that this result can be extented to the case of different removal rates. Under condition (2.1), it is clear that the set Q is connected, and S 0 ∈ Q. The condition λ 1 < λ i for i = 1 can be stated without loss of generality, by labelling the populations such that the index i = 1 corresponds to the lowest break-even concentration, but the condition λ 1 < S 0 < µ 1 in (2.1) cannot be stated without loss of generality. If µ 1 < S 0 , it is not possible to show the CEP by the methods that we used. To the best of our knowledge, in the case of different removal rates and non-monotone response functions, the CEP has been proved only under the assumption S 0 < µ 1 [START_REF] Li | Global asymptotic behavior of the chemostat : general response functions and differential removal rates[END_REF][START_REF] Wolkowicz | Global dynamics of a mathematical model of competition in the chemostat : general response functions and differential death rates[END_REF][START_REF] Wolkowicz | Global asymptotic behavior of a chemostat model with discrete delays[END_REF]. However, Rapaport and Harmand [START_REF] Rapaport | Biological control of the chemostat with nonmonotone response and different removal rates[END_REF] considered the case of two populations and proposed conditions on the growth functions such that the CEP holds under the condition µ 1 < S 0 . It should be interesting to extend their methods to more general cases. We leave this problem for future investigations.

Our result concern also the case of variable yields, for which it is known [START_REF] Arino | Considerations on yield, nutrient uptake, cellular growth, and competition in chemostat models[END_REF][START_REF] Huang | Limit cycles in a chemostat with variable yields and growth rates[END_REF][START_REF] Pilyugin | Multiple limit cycles in the chemostat with variable yields[END_REF]] that more exotic dynamical behaviours, including limit cycles, are possible. Thus in the case of variable yields, it is of great importance to have criteria ensuring the global convergence to an equilibrium with at most one surviving species.

  Laboratoire de Mathématiques, Informatique et Applications, Université de Haute Alsace, 4 rue des frères Lumière, 68093 Mulhouse, and EPI MERE INRIA-INRA, UMR MISTEA, 2, pl. Viala, 34060 Montpellier, France E-mail address: Tewfik.Sari@uha.fr


Oleg Davidyuk
email: oleg.davidyuk@inria.fr

Iván Sánchez

Jon Imanol Durán
email: jon.duran@esi.es

Jukka Riekki

CADEAU: Collecting and Delivering Multimedia Content in Ubiquitous Environments

The aim of our research is to support users in composing applications from various multimedia resources provided by the ubiquitous environment. In order to approach this goal, we present CADEAU, a prototype for Collecting And DElivering multimediA content in Ubiquitous environments. This paper introduces the user control techniques, basic technologies and a concrete application which demonstrates how the CADEAU prototype can be used.

Introduction

During the past years, we have witnessed that our public and private environments are increasingly becoming ubiquitous, i.e. being populated with various 'smart' resources. Ubiquitous environments assist users to accomplish their everyday activities in an intelligent, adaptable, continuous and secure way. However, this vision also increases complexity and sometimes requires users to spend considerable efforts to accomplish even a simple task. Consequently, the users are still reluctant to accept and use these environments.

Our research aims to increase the usability of ubiquitous environments. For this purpose, we develop new mechanisms to allow users to accomplish and control their tasks easily. In our earlier work [START_REF] Davidyuk | Autonomic Composition of Ubiquitous Multimedia Applications in REACHES[END_REF], we presented a system that automatically assembles user applications according to their preferences and needs. We also conducted a user evaluation study on the system's prototype and found out that users need optional mechanisms to control the application composition, because they do not trust an autonomic system that undertakes all actions on their behalf.

The main contribution of this paper is the CADEAU prototype, which allows users to control the application composition using three user interaction methods. In this paper, we present the implemented prototype and a concrete multimedia application, which demonstrates how CADEAU can be used.

Application composition in CADEAU

The goal of CADEAU is to make the realization of tasks in the ubiquitous environments easy for users. A user only needs to specify a high-level description of the desired task to CADEAU (running on a server), which will accomplish it as required. CADEAU realizes user tasks by composing appropriate applications from software components that are allocated onto resources in the environment. These resources may provide, for example, multimedia, sensing, computational or data processing capabilities.

Initially, the prototype supported the autonomic application composition only. However, as we found out during the user evaluation, users may prefer to control the application composition manually (see [START_REF] Davidyuk | Autonomic Composition of Ubiquitous Multimedia Applications in REACHES[END_REF] for details). Thus, we additionally developed the manual and the semi-autonomic control methods in CADEAU. These methods are based on the NOKIA 6131 mobile phone, which plays the role of the interaction tool between the user, CADEAU and the real world. Figure 1 (left) demonstrates how a user is interacting with the prototype. These methods differ from each other on how much the user is involved in the control. The prototype offers these methods so that the users can choose the appropriate method on the account of their needs, situation and the application, as users requested during the evaluation. We believe that one method cannot always be better than the others.

The manual method. The user has full control over the application composition and is able to choose resources by simply touching them using his mobile phone. This method is based on the RFID technology. Thus, each resource has an RFID tag attached; these tags contain resource identity information that is accessed when touching tags. The environment provides a control panel with RFID tags for non-visual resources (e.g. a Web service) or for resources located in a hard to reach places (e.g. on the ceiling). An example of a visual resource and a control panel are shown in Figure 1 (the middle and the right pictures respectfully). The mobile phone's UI informs the user about the resources he has already chosen and reminds about the ones he needs to accomplish his task (Figure 2, left). The user can also replace the chosen resources by touching other ones of the same type. When all necessary resources are chosen, the user requests the CADEAU prototype to deploy and start the application.

The semi-autonomic method. The user's control is restricted to the options suggested by the prototype. CADEAU dynamically produces an ordered (best-first) list of application configurations and presents it to the user. These configurations have been optimized according to the criteria specified by the user before. The user can browse the application configuration list and identify the resources used in each configuration by clicking the phone's middle button. In this case, the resources providing display capabilities respond to the user by showing the 'splash screen' while the audio resources play a welcoming audio note. This optional feature helps the user to locate and identify the physical resources that are associated to each application configuration. The user may also omit this step (e.g. if he is familiar with the environment) and start the application without identifying the resources. Due to space limitations, we do not include figures to demonstrate the GUI of this method.

The autonomic method. This method supposes that the user is not willing to control the application composition, thus CADEAU is given full autonomy. The application is composed according to user specified criteria (e.g. the nearest, the fastest or the cheapest option), his preferences (e.g. fidelity requirements) and situation in the environment, and is started without distracting the user. At first, CADEAU searches for resources according to their capabilities, location and quality. Next, an application configuration is produced: the application is allocated onto the resources according to the aforementioned criteria. More about the optimization criteria and the application allocation algorithms used in CADEAU can be found in [START_REF] Davidyuk | Algorithms for Composing Pervasive Applications[END_REF]. The user interacts with the application when the resources, specified in the application configuration, are locked (to prevent other users and application accessing them) and the application components are deployed onto them.

Application

We have implemented an application to demonstrate how the CADEAU prototype can be used. Consider the following scenario: John is reading a newspaper in a cafeteria. The newspaper has some hidden multimedia content (audio, video and images) that can be accessed when touching the newspaper with a RFIDenabled mobile phone. John touches the newspaper titles and, shortly after, the system prompts John to browse the hidden content on a nearby wall display. John browses the list of multimedia files and selects the most interesting ones by pressing the mobile phone's middle button. When files are chosen, the system stores files' links in John's mobile phone. Later this day, John decides to watch the videos and listen to the audios in a conference room at work. John decides to use a semi-autonomic method to choose appropriate resources. The system suggests John several combinations of a display, an audio system and Web servers that host the multimedia files. John chooses the combination named 'nearest resources' and starts the application that plays the multimedia files using these resources. John can control the playback (stop, pause, next/previous) by pressing the phone's middle button (Figure 2, right).

The collection of content is initiated when the user touches RFID tags in the newspaper (shown in Figure 2, middle). The information from the tags is communicated to CADEAU, that is running on a server, via the HTTP requests/responses sent over GPRS. Content browsing is based on a Web browser and Javascripts which are controlled with the mobile phone via HTTP/GPRS protocol. The mobile phone's UI is implemented using Java MIDlets. The content delivery and the playback are based on two flash players, the JWMediaPlayer and the JWImageIterator (available from www.joerenwijering.com).

Conclusion

This paper has briefly introduced the CADEAU prototype that supports the autonomic (solely based on the user specified criteria) or user controlled application composition. The latter provides two control methods (the manual and the semiautonomic) that use the mobile phone as an interaction tool. The user chooses these methods depending on the application, the situation or his needs. We also implemented an example application which enables collecting and delivering of multimedia content across multiple environments.

Fig. 1 .

 1 Fig. 1. User interacting with the prototype (left), a public display resource (middle) and a control panel (right).

Fig. 2 .

 2 Fig.2. The manual method's GUI (left), the newspaper (middle) and the controller's GUI (right) used with the application.


Mohammad Wehbe

Spinor algebra and null solutions of the wave equation

Keywords: numbers: 47.60.Dx, 47.57.E-, 82.56.Lz

In this paper we exploit the ideas and formalisms of twistor theory, to show how, on Minkowski space, given a null solution of the wave equation, there are precisely two null directions in ker df , at least one of which is a shear-free ray congruence.

Introduction

Harmonic morphisms are mappings which pull back local harmonic functions to harmonic functions. On a Riemannian or semi-Riemannian manifold ,they can be characterized as harmonic maps which enjoy an extra property called horizontal weak conformality or semi-conformality [START_REF] Fuglede | Harmonic morphisms between riemannian manifolds[END_REF], [START_REF] Fuglede | Harmonic morphism between semi-riemannian manifolds[END_REF].

If ϕ : (M, g) → (N, h) is a mapping between Riemannian manifolds, then ϕ is semi-conformal if and only if for each x ∈ M , there exists a function λ : M → R( = 0) such that dϕ • dϕ * : T ϕ(x) N → T ϕ(x) N is λ 2 (x)id, where dϕ * denotes the adijoint of dϕ. In the case of a map ϕ : U ⊂ R n → C, this is equivalent to the equation

m i=1 ∂ϕ ∂x i 2 = 0.
On Minkwoski space M endowed with standard coordinates (t, x 1 , x 2 , x 3 ), a harmonic morphism f : U ⊂ M → C corresponds to a null solution of the wave equation:

(∂ t f ) 2 -(∂ 1 f ) 2 -(∂ 2 f ) 2 -(∂ 3 f ) 2 = 0 ∂ 2 tt f -∂ 2 11 f -∂ 2 22 f -∂ 2 33 f = 0.
The first equation can be interpreted as the semi-conformality of f with respect to the Minkowski metric g = -dt 2 + dx 2 1 + dx 2 2 + dx 2 3 , and the second its harmonicity.

Our aim in this paper is to give a direct proof that such a mapping determines a shear-free ray congruence. By the Kerr theorem, thus latter object corresponds to a complex analytic surface in twistor space [START_REF] Penrose | Spinors and Space-Time[END_REF]. In fact, our proof provides more precise information, specifically, we show there are exactly two null directions in ker df , at least one of which is geodesic and shear free. Furthermore, we only require that f be of class C 2 . The correspondence between a null solution of the wave equation and a shear-free ray congruence has already been noted in [START_REF] Baird | Harmonic morphisms, conformal foliations and shear-free ray congruences[END_REF], their proof relies on the analyticity of f , there by analytically countinuing f to Riemannian R 4 , one can then apply a theorem of J.C. Wood [START_REF] Wood | Harmonic morphisms and hermitien structures on Einstein 4-manifols[END_REF] to show that such a mapping determines an integrable complex structure, which one more corresponds to a complex analytic surface in twistor space.

In more generality, Wood shows that, if (M 4 , g) is an oriented Einstein 4-manifold and ϕ : (M 4 , g) → (N 2 , h) a harmonic morphism with a condition on the set of critical points, then ϕ determines two almost Hermitien structures on (M 4 , g), at least one of which is integrable. Conversely, if (M 4 , g) is also anti-self-dual, then any Hermitien structure determines a harmonic morphism in a surface.

Our result should be seen as the direct analogue of this, with a shear-free ray congruence on Minkowski space is precisely an integrable complex structure on a domain of R 4 . Our perspective will provide some insight into the case of more general curved space times, where the methods of twistor theory no larger apply.

(t = x 0 , x 1 , x 2 , x 3 ) ↔ 1 √ 2 t + x 1 x 2 + ix 3 x 2 -ix 3 t -x 1 = (x AA ′ );
the spinor covariant derivatives ∇ AA ′ are then given by

∇ AA ′ = 1 √ 2 ∂ 0 + ∂ 1 ∂ 2 -i∂ 3 ∂ 2 + i∂ 3 ∂ 0 -∂ 1 = ∇ 00 ′ ∇ 01 ′ ∇ 10 ′ ∇ 11 ′ ,
where ∂ a = ∂ ∂x a (Indices are raised and lowered using the skew forms ǫ AB = ǫ AB = 0 1 -1 0 ). Note that (x a ) is a real vector if and only if the matrix (x AA ′ ) is Hermitian. Now let ϕ : M → C be a C 2 -mapping, then ϕ is semi-conformal if and only if

- ∂ϕ ∂x 0 2 + 3 a=1 ∂ϕ ∂x a 2 = 0, ( 1
)
and ϕ is harmonic, equivalently, ϕ satisfies the wave equation, if and only if

- ∂ 2 ϕ (∂x 0 ) 2 + 3 a=1 ∂ 2 ϕ (∂x a ) 2 = 0. (2)
Equation ( 1) is equivalent to the condition det ∇ AA ′ ϕ = 0, so we deduce that ϕ is semi-conformal if and only if

∇ AA ′ ϕ = ξ A η A ′ , (3)
for some spinor fields ξ A , η A ′ defined on M. One can now show that (1) and ( 2) are equivalent to the pair of spinor equations [START_REF] Baird | A spinor formulation for harmonic morphisms[END_REF], [START_REF] Baird | Monopoles, harmonic morphisms and spinor fields[END_REF]:

∇ AA ′ ξ A η B ′ = 0 ∇ AB ′ ξ C η B ′ = 0.
Note that these equations imply the integrability of the spinor field ξ A η A ′ ; thus there is locally defined a mapping ϕ such that

∇ AA ′ ϕ = ξ A η A ′ . In Minkowski space, a vector v = (v 0 , v 1 , v 2 , v 3 ) ∈ M is null if: -v 0 2 + v 1 2 + v 2 2 + v 3 2 = 0, in which case, v AA ′ = λρ A ρA ′ ,
for some spinor ρ A and for some real number λ. The following lemma is useful in what follows.

Lemma I.1. Let ξ A non-zero spinor, (x a ) an non-zero vector in M and (x AA ′ ) its corresponding spinor expression, then: ξ A x AA ′ = 0 if and only if there exits a dual spinor σ A ′ such that

x AA ′ = ξ A σ A ′ .
Proof We know that ξ A x AA ′ = 0 if and only if

ξ 0 x 00 ′ + ξ 1 x 10 ′ = 0 ξ 0 x 01 ′ + ξ 1 x 11 ′ = 0
Since (x AA ′ ) = 0, we can assume for example that the component x 10 ′ = 0, hence

x 00 ′ x 10 ′ = x 01 ′ x 11 ′ = - ξ 1 ξ 0 = ξ 0 ξ 1 .

It follows that

x 00 ′ x 10 ′ =σ 0 ′ ξ 0 ξ 1 and

x 01 ′ x 11 ′ =σ 1 ′ ξ 0 ξ 1 for two numbers σ 0 ′ , σ 1 ′ ∈ C, hence x AA ′ = ξ A σ A ′ .
Recall that twistor space is the space T = C 4 equipped with coordinate (ξ A , η A ′ ). The incidence relation [START_REF] Penrose | Twistor algebra[END_REF] ξ

A = ix AA ′ η A ′ (4)
describes the correspondence between points of T and points of Minkowski space M. The dual space T * = C 4 has coordinates (ρ A , σ A ′ ) and given

X α = (ξ A , η A ′ ) ∈ T, we can associate its conjugate Xα = (η A , ξ A ′ ) ∈ T * , where η0 = η 0 ′ , η1 = η 1 ′ , ξ0 ′ = ξ 0 , and ξ1 ′ = ξ 1 thus X0 = X 2 , X1 = X 3 , X2 = X 0 , X3 = X 1 (5)
There is a natural product between an element of the twistor space

X α = (ξ A , η A ′ ) ∈ T and L α = (λ A , ν A ′ ) ∈ T
given by:

X α L α = ξ A ν A + η A ′ λ A ′ .
The correspondence between a light ray in Minkowski space and a point in twistor space is then given by the following lemma. The proof is standard linear algebra.

Lemma I.2. [START_REF] Wehbe | Aspects twistoriels des applications semi-conformes[END_REF] Let X α = (ξ A , η A ′ ) be a twistor such that η A ′ = 0, then X α defines a light ray if and only if

X α X α = 0. ( 6
)
We can give a more invariant picture as follows. Let π be the Hopf fibration defined on CP 3 with valued in HP 1 ∼ = S 4 , where H is the space of the quaternions {z

+ wj : z, w ∈ C, ij = -ji, i 2 = j 2 = -1}. Then π([f, g, h, k]) = [f + gj, h + kj]
Let S 3 ⊂ S 4 be the equatorial space given by S 3 = {(0, x 1 , x 2 , x 3 ) ⊂ S 4 }, then N 5 ⊂ CP 3 can be identified with S 3 by the Hopf fibration:

N 5 = π -1 (S 3 ), in particular [f, g, h, k] ∈ N 5 if and only if Re{ hf + kḡ + ( hg -k f )j} = 0, i.e. if and only if hf + kḡ + h f + kg = 0. ( 7
)
To complete the correspondence given by lemma I.2 by admitting a twistor of type X α = (ξ A , 0), one comptifies Minkowski space by adding a light cone at infinity to obtain the space M C homeomorphic to S 1 × S 3 [START_REF] Penrose | Twistor algebra[END_REF]. With this compactification, noting that the equation ( 6) is none other than condition [START_REF] Fuglede | Harmonic morphisms between riemannian manifolds[END_REF], we obtain a correspondence between a point of N 5 ⊂ CP 3 and a light ray in M C , i.e.

CP 3 ⊃ N 5 = {light of rays in M C }.
Let S ⊂ CP 3 a regular complex surface, i.e. S is locally parameterized in the form (z, w)

→ [ξ A (z, w), η A ′ (z, w)],
where ξ A (z, w), η A ′ (z, w) are holomorphic in (z, w) and the vectors (

∂ξ A ∂z , ∂η A ′ ∂z ), ( ∂ξ A ∂w , ∂η A ′ ∂w ) ∈ C 4 are independent
for each (z, w). We call such a surface a twistorial surface.

Let ℓ be a congruence of null curves on a domain A ⊂ M, that is ℓ is a family of null curves which give a C ∞ foliation of A. Let ξ A be the corresponding spinor field on A. Thus at each point x ∈ A, ξ A (x) determines the null direction via ℓ passing thought x via the corresponding v a ↔ v AA ′ = ξ A ξA ′ . Then the congruence is said to be a shear-free ray congruence (SFR) if and only if each null curve of ℓ is geodesic and lie transport of vectors in a 2-dimensional space like complement of v called the screen space, is conformal, see [START_REF] Penrose | Spinors and Space-Time[END_REF]. Then ℓ is an SFR if and only if

ξ A ξ B ∇ AA ′ ξ B = 0. ( 8
)
We note that this equation depends only on the direction ξ 0 /ξ 1 . Indeed, writing ξ = ξ 0 /ξ 1 , the equation ( 8) is equivalent to:

ξ∇ 00 ′ ξ + ∇ 10 ′ ξ = 0 ξ∇ 01 ′ ξ + ∇ 11 ′ ξ = 0. Lemma I.3. Let (z, w) → [ξ A (z, w), η A ′ (z, w)
] be a parameterization of a twistorial surface S, with η 1 ′ (z, w) = 0 for each z, w. Then there exists a local parametrisation of the form z = z(z, w) and w = w(z, w) with respect which S is given by

( z, w) → [ξ 0 ( z, w), ξ 1 ( z, w), z, 1].
Proof First, dividing by η 1 ′ , we obtain a parametrization of S in the form

(z, w) → [ξ 0 (z, w), ξ 1 (z, w), η 0 ′ (z, w), 1].
Then we look for a biholomorphic transformation z = z(z, w), w = w(z, w) such that

∂η 0 ′ ∂ w = ∂η 0 ′ ∂z ∂z ∂ w + ∂η 0 ′ ∂w ∂w ∂ w = 0.
By the assumption of regularity

rank      ∂ξ 0 ∂z ∂ξ 1 ∂z ∂η 0 ′ ∂z ∂ξ 0 ∂w ∂ξ 1 ∂w ∂η 0 ′ ∂w      = 2
so that one of the following two conditions is satisfied:

∂η 0 ′ ∂z ∂η 0 ′ ∂w ∂ξ0 ∂z ∂ξ0 ∂w = 0, or ∂η 0 ′ ∂z ∂η 0 ′ ∂w ∂ξ1 ∂z ∂ξ1 ∂w = 0.
Suppose that ∂η 0 ′ ∂z ∂η 0 ′ ∂w ∂ξ0 ∂z ∂ξ0 ∂w = 0, the other case being similar.

Define the map ψ : (z, w) → ( z = η 0 ′ (z, w), w = ξ 0 (z, w)). Then, since ∂η 0 ′ ∂z ∂η 0 ′ ∂w ∂ξ0 ∂z ∂ξ0 ∂w = 0, one can locally find an inverse and so express z = z( z, w), w = w( z, w). By construction

∂η 0 ′ ∂ w = ∂η 0 ′ ∂z ∂z ∂ w + ∂η 0 ′ ∂w ∂w ∂ w = ∂ z ∂z ∂z ∂ w + ∂ z ∂w ∂w ∂ w = ∂ z ∂ w = 0.
Finally, once more by the regularity, η 0 ′ ( z) is not constant and can be replaced by z in a neighborhood a point where η 0 ′ ( z) = 0.

Proposition I.1. Let S ⊂ CP 3 a twistorial surface equipped with a parameterization (z, w) → [ξ A (z, w), η A ′ (z, w)], and let

ξ A = ix AA ′ η A ′ (9)
be the incidence relation. Then, any solution z = z(x) (x ∈ M) of the equation ( 9) is a null solution of the wave

equation z = 0 if and only if ∂ ∂w η 0 ′ η 1 ′ = 0.
In particular, any twistor surface admits such parameterizations and therefore determines an null solution of the wave equation.

Proof Write u = x 0 + x 1 , v = x 0x 1 and q = x 2 + ix 3 . Then the incidence relation [START_REF] Penrose | Twistor algebra[END_REF] takes the form: r(z, w, x) := uη 0 ′ + qη 1 ′ + iξ 0 = 0 s(z, w, x) := qη 0 ′ + vη 1 ′ + iξ 1 = 0 One taking the derivative of these two equations, and writing r w = ∂r ∂w

, {r, s} = r z s ws z r w , we obtain:

∂z ∂u = - s w η 0 ′ {r, s} , ∂z ∂v = r w η 1 ′ {r, s} , ∂z ∂q = - s w η 1 ′ {r, s} , ∂z ∂ q = r w η 0 ′ {r, s} .
It follows that ∂z ∂u ∂z ∂v -∂z ∂q ∂z ∂ q = 0.

A similar calculation of the second derivatives shows that wave equation is satisfied if and only if {r, s}(η 1 ′ ∂ w η 0 ′η 0 ′ ∂ w η 1 ′ ) = 0, from which the result follows.

II. NULL SOLUTIONS OF THE WAVE EQUATION

We study the functions f :

A ⊂ M → C, A open in M, satisfying the two equations (∂ t f ) 2 -(∂ 1 f ) 2 -(∂ 2 f ) 2 -(∂ 3 f ) 2 = 0 ∂ 2 tt f -∂ 2 11 f -∂ 2 22 f -∂ 2 33 f = 0. ( 10
)
We call such a function a "null solution" of the wave equation.

Theorem II.1.

[4] Let f : A ⊂ M → C be a C 2 null solution of the wave equation [START_REF] Ward | Twistor Geometry and Field Theory[END_REF]. Then there is a pair of spinor field ξ A , η

A ′ such that ∇ AA ′ f = ξ A η A ′ which verify ∇ AA ′ ξ B η A ′ = ∇ AA ′ ξ A η B ′ = 0. ( 11
)
Conversely, any pair of spinor fields ξ A , η A ′ satisfying [START_REF] Ward | Twistor Geometry and Field Theory[END_REF], determines a solution of [START_REF] Ward | Twistor Geometry and Field Theory[END_REF].

Theorem II.2. Let f : A ⊂ M → C be a solution of [START_REF] Ward | Twistor Geometry and Field Theory[END_REF], satisfying df = 0 at each point of A, then the only null fields in ker df are given by

λ → λξ A ξA ′ et µ → µη A η A ′ (λ, µ ∈ R),
and at least one of these fields are tangent to an SFR. Conversely, let ξ be tangent to an SFR. We define ξ 0 = ξ, ξ 1 = 1, η 0 ′ = -∇ 01 ′ ξ, η 1 ′ = ∇ 00 ′ ξ. Then ξ A η A ′ determines a solution of [START_REF] Ward | Twistor Geometry and Field Theory[END_REF].

To prove our theorem we require the following lemmas. We note that for any light ray v ∈ (M ), there is a spinor ρ A with v (AA ′ ) = λρ A ρA ′ , where v (AA ′ ) is the spinor expression of v.

Lemma II.1. Let f : A ⊂ M → C be a C 2 solution of [START_REF] Ward | Twistor Geometry and Field Theory[END_REF]. Then there is a pair of spinor fields ξ A , η A ′ such that

∇ AA ′ f = ξ A η A ′ . Moreover, if v AA ′ = λρ A ρA ′ is
a null direction in ker df and df = 0, then, at each point

• either ρ A = αξ A , • or ρ A = βη A ′ (= β ηA ), with α, β ∈ C.
Proof The function f is a null solution of the wave equation [START_REF] Ward | Twistor Geometry and Field Theory[END_REF], and in particular, a semi-conformal map: thus det(∇ AA ′ f ) = 0, and we conclude that there exists two spinor field ξ A , η A ′ such that ∇ AA ′ f = ξ A η A ′ . Suppose that the light ray v AA ′ is in ker df , then:

∇ AA ′ f v AA ′ = 0 ⇔ ξ A η A ′ ρ A ρA ′ = 0 ⇔ ξ 0 η 0 ′ ρ 0 ρ0 ′ + ξ 0 η 1 ′ ρ 0 ρ1 ′ + ξ 1 η 0 ′ ρ 1 ρ0 ′ + ξ 1 η 1 ′ ρ 1 ρ1 ′ = 0 ⇔ ξ 0 η 0 ′ |ρ 1 | 2 -ξ 0 η 1 ′ ρ 1 ρ 0 -ξ 1 η 0 ′ ρ 0 ρ 1 + ξ 1 η 1 ′ |ρ 0 | 2 = 0 ⇔ η 0 ′ (ξ 0 |ρ 1 | 2 -ξ 1 ρ 0 ρ 1 ) + η 1 ′ (ξ 1 |ρ 0 | 2 -ξ 0 ρ 1 ρ 0 ) = 0 ⇔ η 0 ′ η 1 ′ = ξ 0 ρ 1 ρ 0 -ξ 1 |ρ 0 | 2 ξ 0 |ρ 1 | 2 -ξ 1 ρ 0 ρ 1 = [ ξ0 ξ1 ρ 1 -ρ 0 ]ρ 0 [ ξ0 ξ1 ρ 1 -ρ 0 ]ρ 1 ⇔ η 0 ′ η 1 ′ [ ξ 0 ξ 1 - ρ 0 ρ 1 ] = ρ 0 ρ 1 [ ξ 0 ξ 1 - ρ 0 ρ 1 ].
Then either 1.

[

ξ 0 ξ 1 - ρ 0 ρ 1 ] = 0, i.e. ρ A = αξ A , 2. η 0 ′ η 1 ′ = ρ 0 ρ 1 , i.e. ρ A = βη A ′ . One writing v = v 0 ∂ 0 + v 1 ∂ 1 + v 2 ∂ 2 + v 3 ∂ 3
for vector field in T M, we have the corresponding 1-form and its derivative:

v a = v 0 dx 0 + v 1 dx 1 + v 2 dx 2 + v 3 dx 3 . dv a = (∂ 1 v 0 -∂ 0 v 1 )dx 0 ∧ dx 1 + (∂ 2 v 0 -∂ 0 v 2 )dx 0 ∧ dx 2 + (∂ 3 v 0 -∂ 0 v 3 )dx 0 ∧ dx 3 + (∂ 2 v 1 -∂ 1 v 2 )dx 1 ∧ dx 2 + (∂ 3 v 1 -∂ 1 v 3 )dx 1 ∧ dx 3 + (∂ 3 v 2 -∂ 2 v 3 )dx 2 ∧ dx 3 .
Generally, for a 1-form θ = θ i dx i in a Riemannian manifold (M, g) we have:

div θ = -d * θ = g ij ∂ i θ j ,
where g ij is the metric tensor componants.

In the Minkowski space M:

div θ = ∂ 0 θ 0 -∂ 1 θ 1 -∂ 2 θ 2 -∂ 3 θ 3 .
This means that for v a = df = ∂ i f dx i , we have:

div df = ∂ 2 00 f -∂ 1 11 f -∂ 2 22 f -∂ 2 33 f = ∂ 0 v 0 -∂ 1 v 1 -∂ 2 v 2 -∂ 3 v 3 .
By the Poincar Lemma, if v a is a 1-form in an open connected set A ⊂ M, then: dv a = 0 if and only if there exist a function f :

A ⊂ M → C such that ∇ AA ′ f = v AA ′ .
The following Lemma is easily established.

Lemma II.2. The following two conditions are equivalent: a) dv a = 0 and div v a = 0 b)

∇ AA ′ v BA ′ = 0 and ∇ AA ′ v AB ′ = 0.
Proof of Theorem II.1. In fact, if f is a null solution of the wave equation then from lemma II.1, there exist two spinor fields ξ A and η A ′ such that ∇ AA ′ f = ξ A η A ′ . The fact that these two fields satisfy [START_REF] Wood | Harmonic morphisms and hermitien structures on Einstein 4-manifols[END_REF] is a consequence of lemma II.2. Conversely, if ξ A , η A ′ satisfies [START_REF] Wood | Harmonic morphisms and hermitien structures on Einstein 4-manifols[END_REF], then form lamma II.2, there exists a function f such that [START_REF] Ward | Twistor Geometry and Field Theory[END_REF].

∇ AA ′ f = ξ A η A ′ which satisfies
Proof of Theorem II.2. However, the first affirmation of this Theorem is a consequence of Lemma II.1. We are therefore required to show that at least one of these null fields is an SFR. However, according to the Theorem II.1, we have the two following equations:

∇ AA ′ ξ B η A ′ = ∇ AA ′ ξ A η B ′ = 0 which are equivalent to:                                  ξ 0 ∇ 00 ′ η 0 ′ + η 0 ′ ∇ 00 ′ ξ 0 + ξ 0 ∇ 01 ′ η 1 ′ + η 1 ′ ∇ 01 ′ ξ 0 = 0 (12) ξ 1 ∇ 00 ′ η 0 ′ + η 0 ′ ∇ 00 ′ ξ 1 + ξ 1 ∇ 01 ′ η 1 ′ + η 1 ′ ∇ 01 ′ ξ 1 = 0 (13) ξ 0 ∇ 10 ′ η 0 ′ + η 0 ′ ∇ 10 ′ ξ 0 + ξ 0 ∇ 11 ′ η 1 ′ + η 1 ′ ∇ 11 ′ ξ 0 = 0 (14) ξ 1 ∇ 10 ′ η 0 ′ + η 0 ′ ∇ 10 ′ ξ 1 + ξ 1 ∇ 11 ′ η 1 ′ + η 1 ′ ∇ 11 ′ ξ 1 = 0 (15) ξ 0 ∇ 00 ′ η 0 ′ + η 0 ′ ∇ 00 ′ ξ 0 + ξ 1 ∇ 10 ′ η 0 ′ + η 0 ′ ∇ 10 ′ ξ 1 = 0 (16) ξ 0 ∇ 00 ′ η 1 ′ + η 1 ′ ∇ 00 ′ ξ 0 + ξ 1 ∇ 10 ′ η 1 ′ + η 1 ′ ∇ 10 ′ ξ 1 = 0 (17) ξ 0 ∇ 01 ′ η 0 ′ + η 0 ′ ∇ 01 ′ ξ 0 + ξ 1 ∇ 11 ′ η 0 ′ + η 0 ′ ∇ 11 ′ ξ 1 = 0 (18) ξ 0 ∇ 01 ′ η 1 ′ + η 1 ′ ∇ 01 ′ ξ 0 + ξ 1 ∇ 11 ′ η 1 ′ + η 1 ′ ∇ 11 ′ ξ 1 = 0. ( 19
)
One multiplying ( 12) by ξ 1 and (13) by ξ 0 and subtracting yields:

η 0 ′ ∇ 00 ′ ξ 0 ξ 1 + η 1 ′ ∇ 01 ′ ξ 0 ξ 1 = 0.
Similary combining the other equations in pairs, one obtains:

η 0 ′ ∇ 10 ′ ξ 0 ξ 1 + η 1 ′ ∇ 11 ′ ξ 0 ξ 1 = 0 ξ 0 ∇ 00 ′ η 0 ′ η 1 ′ + ξ 1 ∇ 10 ′ η 0 ′ η 1 ′ = 0 ξ 0 ∇ 01 ′ η 0 ′ η 1 ′ + ξ 1 ∇ 11 ′ η 0 ′ η 1 ′ = 0.
We write η = η 0 ′ /η 1 ′ and ξ = ξ 0 /ξ 1 , then the above equations are equivalent to:

         η∇ 00 ′ ξ + ∇ 01 ′ ξ = 0 (20) η∇ 10 ′ ξ + ∇ 11 ′ ξ = 0 (21) ξ∇ 00 ′ η + ∇ 10 ′ η = 0 (22) ξ∇ 01 ′ η + ∇ 11 ′ η = 0 (23)
On the other hand, ξ A is tangent to an SFR if and only if:

ξ∇ 00 ′ ξ + ∇ 10 ′ ξ = 0 ξ∇ 01 ′ ξ + ∇ 11 ′ ξ = 0
and η A ′ is tangent to an SFR if and only if:

η∇ 00 ′ η + ∇ 01 ′ η = 0 η∇ 10 ′ η + ∇ 11 ′ η = 0
We take the derivative of (20) with respect to ∇ 11 ′ : ∇ 11 ′ η∇ 00 ′ ξ + η∇ 11 ′ ∇ 00 ′ ξ + ∇ 01 ′ ∇ 11 ′ ξ = 0 (by switching the derivatives) ⇒ -ξ∇ 01 ′ η∇ 00 ′ ξ + η∇ 11 ′ ∇ 00 ′ ξ + ∇ 01 ′ (-η∇ 10 ′ ξ) = 0 (by substituting (23) et ( 21

)) ⇒ -∇ 01 ′ η(ξ∇ 00 ′ ξ + ∇ 10 ′ ξ) + η(∇ 11 ′ ∇ 00 ′ ξ -∇ 01 ′ ∇ 10 ′ ξ) = 0
We take the derivative of (20) with respect to ∇ 10 ′ :

∇ 10 ′ η∇ 00 ′ ξ + η∇ 00 ′ ∇ 10 ′ ξ + ∇ 01 ′ ∇ 10 ′ ξ = 0 (by switching the derivatives) ⇒ -ξ∇ 00 ′ η∇ 00 ′ ξ -η∇ 00 ′ ( 1 η ∇ 11 ′ ξ) + ∇ 01 ′ ∇ 10 ′ ξ = 0 (by substituting (22) et (21)) ⇒ ∇ 00 ′ η(-ξ∇ 00 ′ ξ + 1 η ∇ 11 ′ ξ) -∇ 00 ′ ∇ 11 ′ ξ + ∇ 01 ′ ∇ 10 ′ ξ = 0 ⇒ ∇ 00 ′ η(ξ∇ 00 ′ ξ + ∇ 10 ′ ξ) + ∇ 00 ′ ∇ 11 ′ ξ -∇ 01 ′ ∇ 10 ′ ξ = 0 (par (21)).
Thus:

-∇ 01 ′ η(ξ∇ 00 ′ ξ + ∇ 10 ′ ξ) + η(∇ 11 ′ ∇ 00 ′ ξ -∇ 01 ′ ∇ 10 ′ ξ) = 0 (24) ∇ 00 ′ η(ξ∇ 00 ′ ξ + ∇ 10 ′ ξ) + ∇ 11 ′ ∇ 00 ′ ξ -∇ 01 ′ ∇ 10 ′ ξ = 0 (25)
In particular:

either -∇ 01 ′ η η ∇ 00 ′ η 1 = -∇ 01 ′ η -η∇ 00 ′ η = 0 or either ξ∇ 00 ′ ξ + ∇ 10 ′ ξ = ∇ 11 ′ ∇ 00 ′ ξ -∇ 01 ′ ∇ 10 ′ ξ = 0.
Now we take the derivative of (21) with respect to ∇ 00 ′ :

∇ 00 ′ η∇ 10 ′ ξ + η∇ 10 ′ ∇ 00 ′ ξ + ∇ 11 ′ ∇ 00 ′ ξ = 0 (by switching the derivatives)

⇒ - 1 ξ ∇ 10 ′ η∇ 10 ′ ξ + η∇ 10 ′ (- 1 η ∇ 01 ′ ξ) + ∇ 11 ′ ∇ 00 ′ ξ = 0 (by substituting (22) et (20)) ⇒ ∇ 10 ′ η(- 1 ξ ∇ 10 ′ ξ + 1 η ∇ 01 ′ ξ) -∇ 10 ′ ∇ 01 ′ ξ + ∇ 11 ′ ∇ 00 ′ ξ = 0 ⇒ ∇ 10 ′ η η ( 1 ξ ∇ 11 ′ ξ + ∇ 01 ′ ξ) + ∇ 11 ′ ∇ 00 ′ ξ -∇ 10 ′ ∇ 01 ′ ξ = 0 par (21).
We take the derivative of (20) with respect to ∇ 11 ′ (in another way):

∇ 11 ′ (- 1 η ∇ 01 ′ ξ) + η∇ 11 ′ ∇ 00 ′ ξ + ∇ 01 ′ (-η∇ 10 ′ ξ) = 0 ⇒ - 1 η ∇ 11 ′ η(∇ 01 ′ ξ) + η(∇ 11 ′ ∇ 00 ′ ξ -∇ 01 ′ ∇ 10 ξ) + 1 ξ ∇ 11 ′ η∇ 10 ′ ξ = 0 ⇒ - 1 η ∇ 11 ′ η(∇ 01 ′ ξ - η ξ ∇ 10 ′ ξ) + η(∇ 11 ′ ∇ 00 ′ ξ -∇ 01 ′ ∇ 10 ′ ξ) = 0 ⇒ - 1 η ∇ 11 ′ η(∇ 01 ′ ξ - 1 ξ ∇ 11 ′ ξ) + η(∇ 11 ′ ∇ 00 ′ ξ -∇ 01 ′ ∇ 10 ′ ξ) = 0.
Hence:

       - ∇ 10 ′ η η ( 1 ξ ∇ 11 ′ ξ + ∇ 01 ′ ξ) + ∇ 11 ′ ∇ 00 ′ ξ -∇ 10 ′ ∇ 01 ′ ξ = 0 (26) - 1 η ∇ 11 ′ η( 1 ξ ∇ 11 ′ ξ + ∇ 01 ′ ξ) + η(∇ 11 ′ ∇ 00 ′ ξ -∇ 01 ′ ∇ 10 ′ ξ) = 0 (27) and:
either

∇ 10 ′ η η 1 -1 η ∇ 11 ′ η 1 = ∇ 10 ′ η + 1 η ∇ 11 ′ η = 0
or either ξ∇ 01 ′ ξ + ∇ 11 ′ ξ = ∇ 11 ′ ∇ 00 ′ ξ -∇ 10 ′ ∇ 01 ′ ξ = 0.

Then if η∇ 00 ′ η + ∇ 01 ′ η = 0, we have ξ∇ 00 ′ ξ + ∇ 10 ′ ξ = 0 = ∇ 00 ′ ∇ 11 ′ ξ -∇ 01 ′ ∇ 10 ′ ξ = 0 and then (26) et (27) imply that, either ξ∇ 01 ′ ξ + ∇ 11 ′ ξ = 0, or ∇ 10 ′ η = ∇ 11 ′ η = 0. But in the latter case, equations ( 22) and (23) show that ∇ 00 ′ η = ∇ 01 ′ η = 0, which contradicts the assumption η∇ 00 ′ η + ∇ 01 ′ η = 0. Thus ξ∇ 00 ′ ξ + ∇ 10 ′ ξ = ξ∇ 01 ′ ξ + ∇ 11 ′ ξ = 0 and ξ is tangent to an SFR. On the other hand, if ξ∇ 00 ′ ξ + ∇ 10 ′ ξ = 0, we obtain in the same manner that η is tangent to an SFR. Conversely, suppose that ξ is tangent to an SFR, thus:

ξ∇ 00 ′ ξ + ∇ 10 ′ ξ = 0 ξ∇ 01 ′ ξ + ∇ 11 ′ ξ = 0.
We set ξ 0 = ξ, ξ 1 = 1, η 0 ′ = -∇ 01 ′ ξ, η 1 ′ = ∇ 00 ′ ξ. Then one can easily check that equations [START_REF] Wehbe | Aspects twistoriels des applications semi-conformes[END_REF] to (19) are satisfied, thus ξ A η A ′ satisfy [START_REF] Wood | Harmonic morphisms and hermitien structures on Einstein 4-manifols[END_REF] and therefore there is a function f satisfying

∇ AA ′ f = ξ A η A ′
which is solution of [START_REF] Ward | Twistor Geometry and Field Theory[END_REF]. The proof is completed. In this case, one can easily check that z is a solution of the equation [START_REF] Ward | Twistor Geometry and Field Theory[END_REF].

Example II. 1 .

 1 Suppose that z = z(x) is a solution of the equation ξ(z).x = 1 (x ∈ M) (28)where ξ = (ξ 1 , ξ 2 , ξ 3 , ξ 4 ) is holomorphic in z, with ξ 1 2 -

We can parameterize such ξ in the form [START_REF] Baird | Harmonic Morphisms Between Riemannian Manifolds[END_REF] (iξ 1 , ξ 2 , ξ 3 , ξ

where f , g et h are meromorphic functions on z.

On differentiating (28) , we obtain ξ ′ (z).x.(∂z/∂x j ) + ξ j = 0, so that

x , which gives:

Then we find that:

We can easily check that ξ A et η A ′ satisfy the conditions

In this case, ξ A and η A ′ are both an SFR.

Remarque II.1. To extend the proof to varieties more general manifolds than M, when the derivatives commute, we must introduce curvature terms. In general then, we can not hoped that the theorem remains true, however, by analogy with the Riemannian case, we conjecture that the result is true for Einstein manifold.