Williams’ decomposition of the L´evy continuum random tree and simultaneous extinction probability for

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www.elsevier.com/locate/spa

Williams’ decomposition of the L´evy continuum random tree and simultaneous extinction probability for

populations with neutral mutations

Romain Abraham

^a

, Jean-Franc¸ois Delmas

^b,^∗

aMAPMO, Fédération Denis Poisson, Université d’Orléans, B.P. 6759, 45067 Orléans cedex 2, France bCERMICS, Univ. Paris-Est, 6-8 av. Blaise Pascal, Champs-sur-Marne, 77455 Marne La Vallée, France

Received 4 May 2007; received in revised form 29 May 2008; accepted 2 June 2008 Available online 12 June 2008

Abstract

We consider an initial Eve-population and a population of neutral mutants, such that the total population dies out in finite time. We describe the evolution of the Eve-population and the total population with continuous state branching processes, and the neutral mutation procedure can be seen as an immigration process with intensity proportional to the size of the population. First we establish a Williams’

decomposition of the genealogy of the total population given by a continuum random tree, according to the ancestral lineage of the last individual alive. This allows us to give a closed formula for the probability of simultaneous extinction of the Eve-population and the total population.

c

MSC:60G55; 60J70; 60J80; 92D25

Keywords: Continuous state branching process; Immigration; Continuum random tree; Williams’ decomposition;

Probability of extinction; Neutral mutation

1. Introduction

We consider an initial Eve-population whose size evolves as a continuous state branching process (CB), Y⁰ = (Y_t⁰,t ≥ 0), with branching mechanism ψEve. We assume that this

∗Corresponding author.

E-mail addresses:[email protected](R. Abraham),[email protected](J.-F. Delmas).

doi:10.1016/j.spa.2008.06.001

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population gives birth to a population of irreversible mutants. The new mutants population can be seen as an immigration process with rate proportional to the size of the Eve-population. We assume that the mutations are neutral, so that this second population evolves according to the same branching mechanism as the Eve-population. This population of mutants gives birth also to a population of other irreversible mutants, with rate proportional to its size, and so on. In [2], we proved that the distribution of the total population sizeY =(Yt,t ≥0), which is a CB with immigration (CBI) proportional to its own size, is in fact a CB, whose branching mechanismψ depends on the immigration intensity. The joint law of(Y⁰,Y)is characterized by its Laplace transform, see Section4.1.4. This model can also be viewed as a special case of multitype CB, with two types 0 and 1, the individuals of type 0 giving birth to offsprings of type 0 or 1, whereas individuals of type 1 only have type 1 offsprings, see [13,6] for recent related works.

In the particular case of Y being a subcritical or critical CB with quadratic branching mechanism (ψ(u) = α0u +βu²,β > 0,α0 ≥ 0), the probability for the Eve-population to disappear at the same time as the whole population is known, see [17] for the critical case, α0=0, or Section 5 in [2] for the subcritical case,α0>0. Our aim is to extend those results for the large class of CB with unbounded total variation and a.s. extinction. Formulas given in [2]

could certainly be extended to a general branching mechanism, but first computations seem to be rather involved.

In fact, to compute those quantities, we choose here to rely on the description of the genealogy of subcritical or critical CB introduced by Le Gall and Le Jan [12] and developed later by Duquesne and Le Gall [7], see also Lambert [10] for the genealogy of CBI with constant immigration rate. Le Gall and Le Jan defined via a L´evy processXthe so-called height process H =(H_t,t ≥0)which codes a continuum random tree (CRT) that describes the genealogy of the CB (see the next section for the definition ofHand the coding of the CRT). Initially, the CRT was introduced by Aldous [4] in the quadratic case:ψ(λ) =λ². Except in this quadratic case, the height processHis not Markov and so is difficult to handle. That is why they also introduce a measure-valued Markov process(ρt,t ≥ 0)called the exploration process and such that the closed support of the measureρt is[0,H_t](see also the next section for the definition of the exploration process).

We shall be interested in the case where a.s. the extinction of the whole population holds in finite time. The branching mechanism of the total population,Y, is given by: forλ≥0,

ψ(λ)=α0λ+βλ²+ Z

(0,∞)π(d`)

e⁻^λ`−1+λ`

, (1)

whereα0≥0,β≥0 andπis a Radon measure on(0,∞)such thatR

(0,∞)(`∧`²) π(d`) <∞. We shall assume thatY is of infinite variation, that is β > 0 orR

(0,1)`π(d`) = ∞. We shall assume that a.s. the extinction ofY in finite time holds, that is, see Corollary 1.4.2 in [7], we assume that

Z +∞ dv

ψ(v)<∞. (2)

We suppose that the processY is the canonical process on the Skorokhod spaceD(R+,R+)of c`adl`ag paths and that the pair(Y,Y⁰)is the canonical process on the spaceD(R+,R+)². Let P_x denote the law of the pair(Y,Y⁰)(see [2]) started at(Y0,Y₀⁰)=(x,x). The probability measure P_x is infinitely divisible and hence admits a canonical measure N: it is aσ-finite measure on D(R⁺,R⁺)²such that

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(Y,Y⁰)⁽=^d⁾X

i∈I

(Yⁱ,Y⁰^,ⁱ),

where((Yⁱ,Y⁰^,ⁱ),i ∈ I)are the atoms of a Poisson measure onD(R+,R⁺)² with intensity xN(dY,dY⁰). In particular, we have

E_x[e⁻^λY^t] =exp(−xN[1−e⁻^λY^t]) (3)

andu(λ,t)=N[1−e⁻^λY^t]is the unique non-negative solution of Z _λ

u(λ,t)

dv

ψ(v) =t, fort≥0 andλ≥0. (4)

LetτY = inf{t > 0;Yt = 0}be the extinction time ofY. Lettingλgo to∞in the previous equalities leads to

P_x(τY <t)=exp−xN[τY ≥t],

where the positive functionc(t)=N[τY ≥t]solves Z ∞

c(t)

dv

ψ(v) =t, fort >0. (5)

Let us consider the exploration process(ρt,t ≥0)associated with this CB. We denote byN its excursion measure. Recall that the closed support of the measureρt is[0,H_t], where H is the height process. LetL^abe the total local time at levelaof the height processH(well defined underN). Then, the process(L^a,a≥0)underNhas the same distribution as the CBYunder N.

We decompose the exploration process, under the excursion measure, according to the maximum of the height process. In terms of the CRT, this means that we consider the longest rooted branch of the CRT and describe how the different subtrees are grafted along that branch, see Theorem 3.3. When the branching mechanism is quadratic, the height process H is a Brownian excursion and the exploration processρt is, up to a constant, the Lebesgue measure on[0,H_t]. In that case, this decomposition corresponds to Williams’ original decomposition of the Brownian excursion (see [18]). This kind of tree decomposition with respect to a particular branch (or a particular subtree) is not new, let us cite [9,14] for instance, or [16,15,8] for related works on superprocesses.

We present in the introduction a Poisson decomposition for the CB only, and we refer to Theorem 3.3for the decomposition of the exploration process. Conditionally on the extinction timeτY equal tom, we can represent the processY as the sum of the descendants of the ancestors of the last individual alive. More precisely, letN⁰(d`,dt) = P

i∈Iδ(`i,t_i)(d`,dt)be a Poisson point measure with intensity

1_[0,m)(t)e⁻^`^c⁽^m−t⁾`π(d`)dt, and

κmax(dt)=X

i∈I

`iδt_i(dt)+2β1_[0_,_m₎(t)dt. (6)

Let N_t(dY)denote the law of(Y(s−t),s ≥t)under N andP

j∈Jδ_(t_j_,Y^j₎ be, conditionally onN⁰, a Poisson point measure with intensity

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κmax(dt)N_t[dY,1{τY≤m}],

where N_t[dY,1{τY≤m}]denotes the restriction of the measureNt to the event{τY ≤m}.

The next result is a direct consequence ofTheorem 3.3.

Proposition 1.1. The processP

j∈JY^j is distributed as Y underN, conditionally on{τY =m}. LetτY⁰ =inf{t >0;Y_t⁰=0}be the extinction time of the Eve-population. In the particular case where the branching mechanism of the Eve-population is given by a shift ofψ:

ψEve(·)=ψ(θ+ ·)−ψ(θ), (7)

for someθ >0 andβ =0, the pruning procedure developed in [1] gives that the nodes of width

`i correspond to a mutation with probability 1−e⁻^θ`ⁱ. Asβ = 0 there is no mutation on the skeleton of the CRT outside the nodes. In particular, we see simultaneous extinction of the whole population and the Eve-population if there is no mutation on the nodes in the ancestral lineage of the last individual alive. This happens, conditionally onκmax, with probability

e

−θP

i∈I

`i

.

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

N[τ_Y⁰ =m|τY =m] =exp− Z

1_[0_,_m₎(t)e⁻^`^c⁽^m−t⁾`π(d`)dth

1−e⁻^θ`

i

=exp− Z m

0

[ψ⁰(c(m−t)+θ)−ψ⁰(c(m−t))]dt

=exp− Z m

0

φ⁰(c(t))dt,

whereφ = ψEve−ψ. Now, using that the distribution of (Y⁰,Y)is infinitely divisible with canonical measure N, standard computations for Poisson measure yield that P_x(τ_Y⁰ =m|τY = m)=N[τ_Y⁰ =m|τY =m]that is

P_x(τ_Y⁰ =m|τY =m)=exp− Z m

0

φ⁰(c(t))dt.

Notice that this formula is also valid for the quadratic branching mechanism (ψ(u)=α0u+βu², β >0,α0≥0), see Remark 5.3 in [2].

In fact this formula is true in a general framework. Following [2], we consider the branching mechanisms of the total population and Eve-population are given by

ψ(λ)=α0λ+βλ²+ Z

(0,∞)π(d`)[e⁻^λ`−1+λ`], ψEve(λ)=αEveλ+βλ²+

Z

(0,∞)πEve(d`)[e⁻^λ`−1+λ`], and the immigration function

φ(λ)=ψEve(λ)−ψ(λ)=αImmλ+ Z

(0,∞)ν(d`)(1−e⁻^λ`),

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whereαImm = αEve −α0−R

(0,∞)`ν(d`) ≥ 0 and π = πEve +ν, where πEve andν are Radon measures on(0,∞)withR

(0,∞)`ν(d`) <∞. Notice the conditionR

(0,∞)`ν(d`) <∞is stronger than the usual condition on the immigration measure,R

(0,∞)(1∧`) ν(d`) <∞, but is implied by the requirement thatR

(1,∞)`ν(d`) <R

(1,∞)`π(d`) <∞. Inspired byTheorem 3.3, we considerN(d`,dt,dz)= P

i∈Iδ_(`_i_,t_i,z_i)(d`,dt,dz)a Poisson point measure with intensity

1_[0_,_m₎(t)e⁻^`^c⁽^m−t⁾`[πEve(d`)δ0(dz)+ν(d`)δ1(dz)] dt. (8) Intuitively, the markz_iindicates if the ancestor (of the last individual alive) alive at timet_i had a new mutation (z_i =1) or not (z_i =0). Note however that ifβ >0 we have to take into account mutation on the skeleton. More precisely, letT1 =min{ti,zi = 1}be the first mutation on the nodes in the ancestral lineage of the last individual alive and letT2be an exponential random time with parameterαImm independent ofN. The timeT2corresponds to the first mutation on the skeleton for the ancestral lineage of the last individual alive. We set

T₀=min(T₁,T₂) if min(T₁,T₂) <m,

T₀= +∞ otherwise. (9)

In particular there is simultaneous extinction if and only ifT0= +∞.

Fort ≥0, let us denote by N_t(dY⁰,dY)the joint law of((Y⁰(s−t),Y(s−t)),s≥t)under N. Recallκmaxgiven by(6). Conditionally onN andT2, letP

j∈Jδ₍_t_j_,_Y⁰,j,Y^j)be a Poisson point measure, with intensity

κmax(dt)Nt[(dY⁰,dY), 1{τY≤m}]. We set

(Y⁰⁰,Y⁰)= X

t_j<T₀

(Y⁰^,^j,Y^j)+ X

t_j≥T₀

(0,Y^j). (10)

We writeQmfor the law of(Y⁰⁰,Y⁰)computed for a given value ofm.

Theorem 1.2. UnderQm,(Y⁰⁰,Y⁰)is distributed as(Y⁰,Y)underN[·|τY =m], or equivalently, underR+∞

0 |c⁰(m)|Qm(·)dm,(Y⁰⁰,Y⁰)is distributed as(Y⁰,Y)underN.

Let us remark that this theorem is very close toTheorem 3.3but only deals with CB and does not specify the underlying genealogical structure. This is the purpose of a forthcoming paper [3]

where the genealogy of multitype CB is described.

Intuitively, conditionally on the last individual alive being at timem, until the first mutation in the ancestral lineage (that is fortj < T0), its ancestors give birth to a population with initial Eve type which has to die before timem, and after the first mutation on the ancestral lineage (that is fort_j ≥T₀), there is no Eve-population in the descendants which still have to die before timem.

Now, using that the distribution of(Y⁰,Y)is infinitely divisible with canonical measure N, standard computations for Poisson measure yield that P_x(τY⁰ =m|τY =m)=N[τY⁰ =m|τY = m]. As

N[τ_Y⁰ =m|τY =m] =Qm(T₀= +∞)

=Qm(T1= +∞)Qm(T2≥m)

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=e⁻

Rm 0 dtR

(0,∞)e⁻^`^c⁽^m−t⁾`ν(d`)e⁻^α^Imm^m

=e⁻

Rm

0 dtφ⁰(c(t)), we deduce the following corollary.

Corollary 1.3 (Probability of Simultaneous Extinction). We have for almost every m>0 P_x(τ_Y⁰ =m|τY =m)=exp−

Z m 0

φ⁰(c(t))dt, where c is the unique (non-negative) solution of (5).

The paper is organized as follows. In Section2, we recall some facts on the genealogy of the CRT associated with a L´evy process. We prove a Williams’ decomposition for the exploration process associated with the CRT in Section3. We proveTheorem 1.2in Section4. Notice that Proposition 1.1is a direct consequence ofTheorem 1.2.

2. Notations

We recall here the construction of the L´evy continuum random tree (CRT) introduced in [12, 11] and developed later in [7]. We will emphasize on the height process and the exploration process which are the key tools to handle this tree. The results of this section are mainly extracted from [7].

2.1. The underlying L´evy process

We consider anR-valued L´evy process(X_t,t ≥ 0)with Laplace exponentψ (for λ ≥ 0 E

e⁻^λ^X^t

=e^t^ψ(λ)) satisfying(1) and(2). Let I = (It,t ≥ 0)be the infimum process ofX, It =inf0≤s≤tXs, and letS =(St,t ≥0)be the supremum process,St =sup_0≤s≤tXs. We will also consider for every 0≤s≤tthe infimum ofXover[s,t]:

I_t^s = inf

s≤r≤tX_r.

The point 0 is regular for the Markov processX−I, and−I is the local time ofX−I at 0 (see [5], chap. VII). LetNbe the associated excursion measure of the processX−I away from 0, andσ =inf{t >0;Xt−It =0}the length of the excursion ofX−IunderN. We will assume that underN,X0=I0=0.

SinceXis of infinite variation, 0 is also regular for the Markov processS−X. The local time, L=(L_t,t≥0), ofS−Xat 0 will be normalized so that

E[e⁻^β^S^L⁻¹^t ] =e^{−tψ(β)/β},

whereL⁻¹_t =inf{s≥0;Ls ≥t}(see also [5] Theorem VII.4(ii)).

2.2. The height process and the L´evy CRT

For eacht ≥0, we consider the reversed process at timet,Xˆ^(t)=(Xˆs⁽^t⁾,0≤s≤t)by:

Xˆ_s⁽^t⁾=Xt −X_(t−s)− if 0≤s<t,

andXˆ⁽_t^t⁾=X_t. The two processes(Xˆ_s⁽^t⁾,0≤s≤t)and(X_s,0≤s≤t)have the same law. Let Sˆ⁽^t⁾be the supremum process ofXˆ⁽^t⁾andLˆ⁽^t⁾be the local time at 0 ofSˆ⁽^t⁾− ˆX⁽^t⁾with the same normalization asL.

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Definition 2.1([7], Definition 1.2.1 and Theorem 1.4.3). There exists a process H =(H_t,t ≥ 0), called the height process, such that for allt ≥ 0, a.s.H_t = ˆL⁽_t^t⁾, and H0 = 0. Because of hypothesis(2), the height processHis continuous.

The height process(H_t,t ∈ [0, σ])underNcodes a continuous genealogical structure, the L´evy CRT, via the following procedure.

(i) To eacht ∈ [0, σ]corresponds a vertex at generationHt. (ii) Vertextis an ancestor of vertext⁰ifHt =H_[t,t⁰], where

H_[t_,_t⁰]=inf{Hu,u ∈ [t∧t⁰,t∨t⁰]}. (11) In generalH_[t,t⁰]is the generation of the last common ancestor totandt⁰.

(iii) We putd(t,t⁰)=Ht+H_t⁰−2H_[t_,_t⁰]and identifytandt⁰(t ∼t⁰) ifd(t,t⁰)=0.

The L´evy CRT coded byH is then the quotient set[0, σ]/∼, equipped with the distanced and the genealogical relation specified in (ii).

Let(τs,s ≥0)be the right-continuous inverse of−I:τs =inf{t >0; −I_t >s}. Recall that

−I is the local time of X−I at 0. LetL^a_t denote the local time at levelaof Huntil timet, see Section 1.3 in [7].

Theorem 2.2 ([7], Theorem 1.4.1).The process(L^a_τ_x,a ≥0)is underP(resp.N) defined as Y underP_x (resp.N).

In what follows, we will use the notationNinstead of N for the excursion measure to stress that we consider the genealogical structure of the branching process.

2.3. The exploration process

The height process is not Markov. But it is a simple function of a measure-valued Markov process, the so-called exploration process.

If E is a Polish space, letB(E)(resp.B+(E)) be the set of real-valued measurable (resp.

and non-negative) functions defined onEendowed with its Borelσ-field, and letM(E)(resp.

M_f(E)) be the set ofσ-finite (resp. finite) measures onE, endowed with the topology of vague (resp. weak) convergence. For any measureµ∈M(E)and f ∈B+(E), we write

hµ, fi = Z

f(x) µ(dx).

The exploration processρ =(ρt,t≥0)is aM_f(R+)-valued process defined as follows: for every f ∈B+(R⁺),

hρt, fi = Z

[0,t]

d_sI_t^sf(H_s), or equivalently

ρt(dr)= X

0<s≤t

Xs−<I_t^s

(I_t^s−Xs−)δH_s(dr)+β1_[0_,_H_t](r)dr. (12)

In particular, the total mass ofρt ishρt,1i = Xt−It.

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Forµ∈M(R+), we set

H(µ)=sup Suppµ, (13)

where Suppµis the closed support ofµ, with the conventionH(0)=0. We have Proposition 2.3 ([7], Lemma 1.2.2). Almost surely, for every t >0,

• H(ρt)=Ht,

• ρt =0if and only if H_t =0,

• if ρt 6=0, thenSuppρt = [0,Ht].

In the definition of the exploration process, as X starts from 0, we have ρ0 = 0 a.s. To state the Markov property ofρ, we must first define the processρ started at any initial measure µ∈M_f(R+).

Fora ∈ [0,hµ,1i], we define the erased measurekaµby kaµ([0,r])=µ([0,r])∧(hµ,1i −a), forr≥0.

Ifa > hµ,1i, we setk_aµ =0. In other words, the measurek_aµis the measureµerased by a massafrom the top of[0,H(µ)].

Forν, µ ∈ M_f(R+), andµ with compact support, we define the concatenation [µ, ν] ∈ M_f(R+)of the two measures by:

h[µ, ν], fi = hµ, fi + hν, f(H(µ)+ ·)i, f ∈B+(R+).

Finally, we set for every µ ∈ M_f(R+) and everyt > 0ρt^µ = [k_−I_tµ, ρt]. We say that (ρt^µ,t ≥ 0)is the processρ started atρ₀^µ = µ, and writePµ for its law. Unless there is an ambiguity, we shall writeρt forρt^µ.

Proposition 2.4 ([7], Proposition 1.2.3). The process(ρt,t ≥ 0)is a c`ad-l`ag strong Markov process inM_f(R+).

Notice thatNis also the excursion measure of the processρ away from 0, and thatσ, the length of the excursion, isN-a.e. equal to inf{t >0;ρt =0}.

2.4. The dual process and representation formula

We shall need theM_f(R+)-valued processη=(ηt,t ≥0)defined by ηt(dr)= X

0<s≤t

Xs−<It^s

(Xs−I_t^s)δH_s(dr)+β1_[0_,_H_t](r)dr.

The processηis the dual process ofρ underNthanks to the following time reversal property:

recallσ denotes the length of the excursion underN.

Proposition 2.5 ([7], Corollary 3.1.6).The processes((ρs, ηs);s≥0)and((η_(σ−s)−, ρ_(σ−s)−); s≥0)have the same distribution underN.

It also enjoys the snake property: for allt ≥0,s≥0 (ρt, ηt)[0,H[t,s])=(ρs, ηs)[0,H[t,s]),

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that is the measuresρ andηbetween two instants coincide up to the minimum of the height process between those two instants.

We recall the Poisson representation of(ρ, η)underN. LetN∗(dxd`du)be a Poisson point measure on[0,+∞)³with intensity

dx`π(d`)1_[0_,_1](u)du.

For everya > 0, let us denote byMa the law of the pair(µa, νa)of finite measures on R⁺ defined by: for f ∈B+(R+)

hµa, fi = Z

N∗(dxd`du)1_[0_,_a](x)u`f(x), hνa, fi =

Z

N∗(dxd`du)1_[0_,_a](x)`(1−u)f(x).

We finally setM=R+∞

0 dae⁻^α⁰^aMa.

Proposition 2.6 ([7], Proposition 3.1.3). For every non-negative measurable function F on M_f(R+)²,

N Z _σ

0

F(ρt, ηt)dt

= Z

M(dµdν)F(µ, ν),

whereσ =inf{s>0;ρs =0}denotes the length of the excursion.

We can then deduce the following proposition.

Proposition 2.7.For every non-negative measurable function F onM_f(R+)², N

Z _σ

0

F(ρt, ηt)dL^a_t

=e⁻^α⁰^a Z

Ma(dµdν)F(µ, ν), whereσ =inf{s>0;ρs =0}denotes the length of the excursion.

3. Williams’ decomposition

We work under the excursion measure. As the height process is continuous, its supremum H_max=sup{H_r;r ∈ [0, σ]}is attained. LetT_max=inf{s≥0;H_s =H_max}.

For everym >0, we setT_m(ρ) =inf{s>0,H_s(ρ) =m}the first hitting time ofmfor the height process. When there is no need to stress the dependence inρ, we shall writeT_mforT_m(ρ). Recall the functioncdefined by(5)is equal to

c(m)=N[Tm <∞] =N[Hmax≤m]. (14)

We setρd =(ρTmax+s,s≥0)andρg=(ρ₍Tmax−s)+,s≥0), wherex+=max(x,0).

For every finite measure with compact supportµ, we writeP^∗_µfor the law of the exploration processρstarting atµand killed when it first reaches 0. We also set

Pˆ^∗_µ:= lim

ε→0P^∗_µ(· |H(µ)≤ Hmax≤H(µ)+ε).

We now describe the probability measure ˆ

P^∗_µvia a Poisson decomposition. Let(αi, βi),i∈ Ibe the excursion intervals of the processX−I away from 0 (well defined underP^∗_µor under ˆ

P^∗_µ).

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For everyi∈ I, we defineh_i =H_α_i and the measure-valued processρⁱ by the formula hρⁱt, fi =

Z

(hi,+∞) f(x−hi)ρ(αi+t)∧βi(dx).

We then have the following result.

Lemma 3.1. Under the probability ˆ

P^∗_µ, the point measure P

i∈Iδ_(h_i_,ρⁱ₎ is a Poisson point measure with intensityµ(dr)N[·,H_max≤m−r].

Proof. We know (cf. Lemma 4.2.4 of [7]) that the point measureP

i∈Iδ₍h_i,ρⁱ) is underP^∗_µ a Poisson point measure with intensityµ(dr)N(dρ). The result follows then easily from standard results on Poisson point measures.

Remark 3.2. Lemma 3.1gives also that, for every finite measure with compact supportµ, if we writeµa=µ(· ∩ [0,a]),

Pˆ^∗_µ= lim

a→H(µ)P^∗_µ_a(· |H_max≤ H(µ)).

Theorem 3.3 (Williams’ Decomposition).

(i) The law of H_maxis characterized byN[H_max ≤ m] = c(m), where c is the unique non- negative solution of (5).

(ii) Conditionally on Hmax=m, the law of (ρT_max, ηT_max)is underNthe law of X

i∈I

vir_iδti +β1_[0_,_m](t)dt,X

i∈I

(1−vi)r_iδti +β1_[0_,_m](t)dt

! ,

wherePδ_(v_i_,r_i,t_i)is a Poisson measure with intensity 1_[0_,_1](v)1_[0_,_m](t)e^{−r c}⁽^m−t⁾dvrπ(dr)dt.

(iii) UnderN, conditionally on H_max=m, and(ρTmax, ηTmax),(ρd, ρg)are independent andρd

(resp.ρg) is distributed asρ(resp.η) under ˆ

P^∗_ρ_Tmax (resp.ˆ P^∗_η_Tmax).

Notice (i) is a consequence of(14). Point (ii) is reminiscent of Theorem 4.6.2 in [7] which gives the description of the exploration process at a first hitting time of the L´evy snake.

The end of this section is devoted to the proof of (ii) and (iii) of this theorem.

Letm > a > 0 be fixed. Letε > 0. Recall Tm = inf{t > 0;Ht = m}is the first hitting time ofmfor the height process, and setLm = sup{t < σ;Ht = m}for the last hitting time ofm, with the convention that inf∅ = +∞and sup∅ = +∞. We consider the minimum ofH betweenT_mandL_m:H_[T_m_,_L_m]=min{H_t;t ∈ [T_m,L_m]}.

We setρ^(d) =(ρT_max,a+t,t ≥0), with T_max_,_a=inf{t >T_max,H_s =a},

the path of the exploration process on the right of T_max after the hitting time of a, and ρ⁽^g⁾ = (ρ₍Lmax,a−t)−,t ≥ 0), with L_max,a = sup{t < T_max;H_t = a}, the returned path of the exploration process on the left ofTmaxbefore its last hitting time ofa. Let us note that, by time reversal (seeProposition 2.5), the processρ⁽^g⁾is of the same type asη. This remark will be used later.

(11)

To prove the theorem, we shall compute A₀=N

h

F₁(ρ⁽^g⁾)F₂(ρ⁽^d⁾)F₃(ρTmax|[0,a])F₄(ηTmax|[0,a])1_{m≤H_max_<_m+_ε}

i

and letεgo down to 0. We shall see inLemma 3.4, that adding1_{H_[_Tm,Lm_]_>a} in the integrand does not change the asymptotic behavior asεgoes down to 0. Intuitively, if the maximum of the height process is betweenmandm+ε, outside a set of small measure, the height process does not reach levelabetween the first and last hitting time ofm. So that we shall compute first

A=N h

F₁(ρ⁽^g⁾)F₂(ρ⁽^d⁾)F₃(ρTmax|[0,a])F₄(ηTmax|[0,a])1_{H_[_Tm_,Lm_]_>a,m≤H_max_<m+_ε}

i. (15) Notice that on{H[T_m,L_m]>a}, we haveTmax,a=Tm,a:=inf{s>Tm,Hs(ρ)=a}and, from the snake property,ρTmax|[0,a]=ρT_m|[0,a]andηTmax|[0,a]=ηT_m|[0,a], so that

A =N h

F₁(ρ^(g))F₂((ρTm,a+t,t≥0))F₃(ρTm|[0,a])F₄(ηTm|[0,a])

×1_{H_[_Tm_,Lm_]_>a,m≤H_max_<m+_ε}

i. Let us remark that, we have

1_{H_[_Tm_,Lm_]_>a,m≤H_max_<m+_ε}=1_{m≤sup{H_u_,0≤u≤T_m,a}<m+ε}1_{sup{H_u_,u≥T_m,a}<m}. By using the strong Markov property of the exploration process at timeT_m_,_a, we get

A =N h

F₁(ρ⁽^g⁾)F₄(ηTm|[0,a])1_{m≤sup{H_u_,_0≤u≤T_m_,_a}<m+ε}F₃(ρTm|[0,a])

× E^∗_ρ_Tm_|[0,a]

F2(ρ)1_{H_max_<m}i and so, by conditioning, we get

A=N h

F1(ρ⁽^g⁾)F4(ηTm|[0,a])G2(ρTm|[0,a])1_{H_[_Tm_,Lm_]_>a,m≤H_max_<m+_ε}

i,

whereG₂(µ)=F₃(µ)E^∗_µ[F₂(ρ)|H_max<m]. Using time reversibility (seeProposition 2.5) and the strong Markov property at timeT_m_,_aagain, we have

A =N h

F1(ρ⁽^d⁾)F4(ρTm|[0,a])G2(ηTm|[0,a])1_{H_[_Tm,Lm_]_>a,m≤H_max_<m+_ε}

i

=N h

G1(ρT_m|[0,a])G2(ηT_m|[0,a])1{H[Tm,Lm]>a,m≤H_max<m+ε}

i, whereG₁(µ)=F₄(µ)E^∗_µ[F₁(ρ)|H_max<m].

Now, we use ideas from the proof of Theorem 4.6.2 of [7]. Let us recall the excursion decomposition of the exploration process above levela. We setτs^a =inf

r, Rr

0 du1_{H_u_≤a}>s . LetEabe theσ-field generated by the process(ρ˜s,s≥0):=(ρτs^a, s≥0). We also setη˜s =ητs^a. Let(αi, βi),i ∈ I be the excursion intervals of H above levela. For everyi ∈ I we define the measure-valued processρⁱ by setting







hρsⁱ, ϕi = Z

(a,+∞)ραi+s(dr)ϕ(r−a) if 0<s< βi−αi, ρs =0 ifs=0 ors≥βi −αi,

(12)

and the processηⁱ similarly. We also define the local time at the beginning of excursionρⁱ by

`i =L^a_α

i. Then, underN, conditionally onE_a, the point measure X

i∈I

δ_(`_i_,ρⁱ_,ηⁱ₎

is a Poisson measure with intensity1_[0_,_La

σ](`)d`N[dρdη]. In particular, we have

A=N

"

X

i∈I

Y

j6=i

1_{T

m(ρ^j)=+∞}G1(ραi)G2(ηαi)1_{m≤H

max(ρⁱ)<m+ε}

# .

Let us denote by (τ_`^a, ` ≥ 0)the right-continuous inverse of (L^a_s,s ≥ 0). Palm formula for Poisson point measures yields

A =N

"

N

"

X

i∈I

Y

j6=i

1_{T

m(ρ^j)=+∞}G₁(ρ_αi)G₂(η_αi)1_{m≤H

max(ρⁱ)<m+ε}|E_a

##

=N

"

Z L^a_σ 0

d`G1(ρ_τ_`^a)G2(η_τ_`^a)N[m≤ Hmax<m+ε]N

"

Y

j∈I

1_{T

m(ρ^j)=+∞}|E_a

##

.

A time-change then gives A=v(m−a, ε)N

Z _σ

0

dL^a_sG₁(ρs)G₂(ηs)e^−c⁽^m−a⁾^L^a^σ

, (16)

wherev(x, ε)=c(x)−c(x+ε)=N[x≤ H_max<x+ε]. We have A =v(m−a, ε)NZ _σ

0

dLâ_sG₁(ρs)G₂(ηs)e^−c(m−a)Lâ^se^−c(m−a)(^Lâ^σ^−Lâ^s⁾

=v(m−a, ε)NZ _σ

0

dL^a_sG₁(ρs)G₂(ηs)e^−c(m−a)L^a^se^−h^ρ^s^,N[1−e^−c(m−a)L

a−·σ ]i

,

where we used for the last equality that the predictable projection of e⁻^λ(^L^a^σ^−L^a^s⁾ is given by e^−h^ρ^s^,N^[1−e⁻^λL

a−·σ ]i. Notice that by using the excursion decomposition above level 0 < r <m, we have

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

A=v(m−a, ε)NZ _σ

0

dL^a_sG₁(ρs)G₂(ηs)e^−c⁽^m−a⁾^L^a^se^−h^ρ^s^,^c⁽^m−·⁾ⁱ

.

Using time reversibility, we have A=v(m−a, ε)NZ _σ

0

dLâ_sG₁(ηs)G₂(ρs)e^{−c(m−a)(L}â^σ^−Lâ^s⁾e^−h^η^s^,c(m−·⁾ⁱ

.

(13)

Similar computations as those previously done give A =v(m−a, ε)NZ _σ

0

dL^a_sG1(ηs)G2(ρs)e^−h^η^s⁺^ρ^s^,^c⁽^m−·⁾ⁱ

=v(m−a, ε)NZ _σ

0

dL^a_sG₁(ρs)G₂(ηs)e^−h^ρ^s⁺^η^s^,c(m−·⁾ⁱ

.

UsingProposition 2.7, we get A=v(m−a, ε)e⁻^α⁰^a

Z

Ma(dµdν)G1(µ)G2(ν)e^−h^µ⁺^ν,^c⁽^m−·⁾ⁱ. We can give a first consequence of the previous computation.

Lemma 3.4. We have

N[H_[T_m_,_L_m]>a,m≤H_max<m+ε] =c⁰(m)c(m−a)−c(m−a+ε) c⁰(m−a) . Proof. TakingF1=F2=F3=F4=1 in(16), we deduce that

N[H_[T_m_,L_m]>a,m≤Hmax<m+ε] =v(m−a, ε)Nh

L^a_σe^−c⁽^m−a⁾^L^a^σi . Leta₀>0 and let us computeB(a₀,a)=N

Lâ_σe^−c⁽â⁰⁾^Lâ^σ

. Thanks toTheorem 2.2, notice that B(a₀,a)=Nh

Y_ae^−c⁽^a⁰⁾^Y^ai

= ∂a₀N[1−e^−c(a⁰^)Y^a] c⁰(a0) . On the other hand, we have

c(a+a0)=N[Ya+a₀ >0] =N[1−EY_a[Ya₀ =0]] =N h

1−e^−Y^a^c⁽^a⁰⁾ i,

where we used the Markov property ofY at timeaunder N for the second equality and(3)with λgoing to infinity for the last. Thus, we getB(a₀,a)= ^c⁰^(a⁰^+a)

c⁰(a0) . We deduce that N[H_[T_m_,_L_m]>a,m≤H_max<m+ε] =v(m−a, ε)B(a−m,a)

=c⁰(m)c(m−a)−c(m−a+ε) c⁰(m−a) .

SinceF₁,F₂,F₃andF₄are bounded, say byC, we have|A−A₀| ≤C⁴N[H_[T_m_,_L_m]<a,m≤ H_max<m+ε]. FromLemma 3.4, we deduce that

εlim→0

|A−A0|

N[m≤ Hmax<m+ε]≤C⁴

1−lim

ε→0

N[H[T_m,L_m]>a,m≤Hmax<m+ε] N[m≤ Hmax<m+ε]

=0. We deduce that

εlim→0

N h

F₁(ρ⁽^g⁾)F₂(ρ⁽^d⁾)F₃(ρTmax|[0,a])F₄(ηTmax|[0,a])1_{m≤H_max_<_m+_ε}

i

N[m≤Hmax<m+ε]

= c⁰(m−a) c⁰(m) e⁻^α⁰^a

Z

Ma(dµdν)G₁(µ)G₂(ν)e^−h^µ⁺^ν,^c⁽^m−·⁾ⁱ

(14)

=

R Ma(dµdν)G₁(µ)G₂(ν)e^−h^µ⁺^ν,^c⁽^m−·⁾ⁱ RMa(dµdν)e^−h^µ⁺^ν,^c⁽^m−·⁾ⁱ

= Z

M˜a(dµdν)G1(µ)G2(ν)

= Z

M˜a(dµdν)F4(ν)E^∗_ν[F1(ρ⁽^d⁾)|Hmax<m]F3(µ)E^∗_µ[F2(ρ⁽^d⁾)|Hmax<m], where

µ(dt)=X

i∈I

u_i`iδti +β1_[0_,_a](t)dt ν(dt)=X

i∈I

(1−u_i)`iδti +β1_[0_,_a](t)dt, andP

i∈Iδ₍xi,`i,ti)is under ˜

Maa Poisson point measure on[0,+∞)³with intensity 1_[0_,_a](t)dt`e⁻^`^c⁽^m−t⁾π(d`)1_[0_,_1](u)du.

Standard results on measure decomposition imply that there exists a regular version of the probability measureN[ · |Hmax=m]and that, for almost every non-negativem,

N[ · |H_max=m] =lim

ε→0N[ · |m≤H_max<m+ε].

This gives (ii) and (iii) ofTheorem 3.3sinceF₁,F₂,F₃,F₄are arbitrary continuous functionals and byRemark 3.2.

4. Proof ofTheorem 1.2

The proof of this theorem relies on the computation of the Laplace transform for(Y⁰⁰,Y⁰)and is given in the next three paragraphs. The next paragraph gives some preliminary computations.

4.1. Preliminary computations 4.1.1. Law of T₀

Recall the definition ofQm as the law of(Y⁰⁰,Y⁰)defined by(10)andT0defined by(9)as the first mutation undergone by the last individual alive.

Forr<m, we have

Qm(T0∈ [r,r+dr],T0=T2)=Qm(T2∈ [r,r+dr])Qm(T1>r)

=drαImme⁻^α^Imm^rexp− Z r

0

dt Z

(0,∞)e⁻^`c(m−t)`ν(d`)

=drαImme⁻

Rr

0 φ⁰(c(m−t))dt, and, with the notationφ0(λ)=φ(λ)−αImmλ,

Qm(T₀∈ [r,r+dr],T₀=T₁)

=Qm(T2>r)Qm(T1∈ [r,r+dr])

=drφ₀⁰(c(m−r))e⁻^α^Imm^rexp− Z r

0

dt Z

(0,∞)e⁻^`^c⁽^m−t⁾`ν(d`)

=drφ0⁰(c(m−r))e⁻

Rr

0φ⁰(c(m−t))dt.

(15)

In particular, we have forr <m

Qm(T₀∈ [r,r+dr])=drφ⁰(c(m−r))e⁻

Rr

0φ⁰(c(m−t))dt

and

Qm(T0>r)=e⁻

Rr

0φ⁰(c(m−t))dt. (17)

Notice we haveQm(T₀= ∞)=exp−Rm

0 φ⁰(c(t))dt. 4.1.2. Conditional law ofN given T₀

RecallN is underQm a Poisson point measure with intensity given by(8). Conditionally on {T0=r,T0=T2}, withm>r >0,N is underQma point Poisson measure with intensity

1_[0,r₎(t)e⁻^`^c⁽^m−t⁾`πEve(d`)δ0(dz)dt

+1₍_r_,_m₎(t)e⁻^`^c⁽^m−t⁾`[πEve(d`)δ0(dz)+ν(d`)δ1(dz)] dt.

Conditionally on{T0=r,T0=T1}, withr <m,N is distributed underQm asN˜ +δ₍L,r,1)

whereN˜ is a point Poisson measure with intensity 1_[0_,_r₎(t)e⁻^`^c⁽^m−t⁾`πEve(d`)δ0(dz)dt

+1₍_r_,_m₎(t)e⁻^`c(m−t)`[πEve(d`)δ0(dz)+ν(d`)δ1(dz)] dt, andLis a random variable independent ofN˜ with distribution

e⁻^`^c⁽^m−r⁾`ν(d`) R

(0,∞)e⁻^`⁰^c⁽^m−r⁾`⁰ν(d`⁰).

Conditionally on{T₀= ∞},N is underQma point Poisson measure with intensity 1_[0_,_m₎(t)e⁻^`^c⁽^m−t⁾`πEve(d`)δ0(dz)dt.

4.1.3. Formulas

The following two formulas are straightforward: for allx, γ ≥0, ψ_Eve⁰ (x+γ )−ψ_Eve⁰ (γ )=2βx+

Z

(0,∞)e⁻^`γ`πEve(d`)[1−e⁻^`^x], (18) ψ⁰(x+γ )−ψ⁰(γ )=2βx+

Z

(0,∞)e⁻^`γ`π(d`)[1−e⁻^`^x], (19)

Finally we deduce from(5)thatψ(c)= −c⁰,ψ⁰(c)c⁰= −c⁰⁰and Z

ψ⁰(c)= −log(c⁰). (20)

4.1.4. Laplace transform

Recall thatτY =inf{t >0;Y_t =0}is the extinction time ofY. LetµEveandµTotal be two finite measures with support a subset of a finite setA= {a1, . . . ,a_n}with 0=a0<a1<· · ·<

an<an+1= ∞. Form∈(0,+∞)\A, we consider