Williams' decomposition of the L\'evy continuous random tree and
  simultaneous extinction probability for populations with neutral mutations
WILLIAMS’ DECOMPOSITION OF THE LÉVY CONTINUUM RANDOM
TREE AND SIMULTANEOUS EXTINCTION PROBABILITY FOR
POPULATIONS WITH NEUTRAL MUTATIONS
ROMAIN ABRAHAM AND JEAN-FRANÇOIS DELMAS
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 propor-
tional 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 an-
cestral 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.
1. Introduction
We consider an initial Eve-population whose size evolves as a continuous state branching
process (CB), Y 0 = (Y 0t , t ≥ 0), with branching mechanism ψEve. We assume this population
gives birth to a population of irreversible mutants. The new mutants population can be
seen as an immigration process with rate proportional to the size of the Eve-population. We
assume the mutations are neutral, so that this second population evolves according to the
same branching mechanism as the Eve-population. This population of mutants gives birth
also to a population of other irreversible mutants, with rate proportional to its size, and so
on. In [2], we proved that the distribution of the total population size Y = (Yt, t ≥ 0), which
is a CB with immigration (CBI) proportional to its own size, is in fact a CB, whose branching
mechanism ψ depends on the immigration intensity. The joint law of (Y 0, Y ) is characterized
by its Laplace transform, see Section 4.1.4. This model can also be viewed as a special case
of multitype CB, with two types 0 and 1, the individuals of type 0 giving birth to offsprings
of type 0 or 1, whereas individuals of type 1 only have type 1 offsprings, see [13, 6] for recent
related works.
In the particular case of Y being a sub-critical or critical CB with quadratic branching
mechanism (ψ(u) = α0u + βu
2, 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 sub-critical 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 sub-critical 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
Date: September 4, 2021.
2000 Mathematics Subject Classification. 60G55, 60J70, 60J80, 92D25.
Key words and phrases. Continuous state branching process, immigration, continuum random tree,
Williams’ decomposition, probability of extinction, neutral mutation.
http://arxiv.org/abs/0704.1475v2
2 ROMAIN ABRAHAM AND JEAN-FRANÇOIS DELMAS
constant immigration rate. Le Gall and Le Jan defined via a Lévy process X the so-called
height process H = (Ht, t ≥ 0) which codes a continuum random tree (CRT) that describes
the genealogy of the CB (see the next section for the definition of H and the coding of the
CRT). Initially, the CRT was introduced by Aldous [4] in the quadratic case: ψ(λ) = λ2.
Except in this quadratic case, the height process H is not Markov and so is difficult to
handle. That is why they also introduce a measure-valued Markov process (ρt, t ≥ 0) called
the exploration process and such that the closed support of the measure ρt is [0,Ht] (see also
the next section for the definition of the exploration process).
We shall be interested in the case where a.s. the extinction of the whole population holds
in finite time. The branching mechanism of the total population, Y , is given by: for λ ≥ 0,
(1) ψ(λ) = α0λ+ βλ
(0,∞)
π(dℓ)
e−λℓ−1 + λℓ
where α0 ≥ 0, β ≥ 0 and π is a Radon measure on (0,∞) such that
(0,∞)
(ℓ∧ ℓ2) π(dℓ) <∞.
We shall assume that Y is of infinite variation, that is β > 0 or
(0,1)
ℓπ(dℓ) = ∞. We shall
assume that a.s. the extinction of Y in finite time holds, that is, see Corollary 1.4.2 in [7],
we assume that
∫ +∞ dv
We suppose that the process Y is the canonical process on the Skorokhod space D(R+,R+) of
càdlàg paths and that the pair (Y, Y 0) is the canonical process on the space D(R+,R+)
2. Let
Px denote the law of the pair (Y, Y
0) (see [2]) started at (Y0, Y
0 ) = (x, x). The probability
measure Px is infinitely divisible and hence admits a canonical measure N: it is a σ-finite
measure on D(R+,R+)
2 such that
(Y, Y 0)
(Y i, Y 0,i)
where ((Y i, Y 0,i), i ∈ I) are the atoms of a Poisson measure on D(R+,R+)
2 with intensity
xN(dY, dY 0). In particular, we have
(3) Ex[e
−λYt ] = exp(−xN[1− e−λYt ])
and u(λ, t) = N[1 − e−λYt ] is the unique non-negative solution of
u(λ,t)
= t, for t ≥ 0 and λ ≥ 0.
Let τY = inf{t > 0;Yt = 0} be the extinction time of Y . Letting λ go to ∞ in the previous
equalities leads to
Px(τY < t) = exp−xN[τY ≥ t],
where the positive function c(t) = N[τY ≥ t] solves
= t, for t > 0.
Let us consider the exploration process (ρt, t ≥ 0) associated with this CB. We denote by
N its excursion measure. Recall that the closed support of the measure ρt is [0,Ht], where
H is the height process. Let La be the total local time at level a of the height process H
(well-defined under N). Then, the process (La, a ≥ 0) under N has the same distribution as
the CB Y under N.
WILLIAMS’ DECOMPOSITION 3
We decompose the exploration process, under the excursion measure, according to the
maximum of the height process. In terms of the CRT, this means that we consider the
longest rooted branch of the CRT and describe how the different subtrees are grafted along
that branch, see Theorem 3.3. When the branching mechanism is quadratic, the height
process H is a Brownian excursion and the exploration process ρt is, up to a constant, the
Lebesgue measure on [0,Ht]. 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 3.3 for the decomposition of the exploration process. Conditionally on the extinction
time τY equal to m, we can represent the process Y as the sum of the descendants of the
ancestors of the last individual alive. More precisely, let N ′(dℓ, dt) =
δ(ℓi,ti)(dℓ, dt) be a
Poisson point measure with intensity
1[0,m)(t) e
−ℓc(m−t) ℓπ(dℓ)dt,
(6) κmax(dt) =
ℓiδti(dt) + 2β1[0,m)(t)dt.
Let Nt(dY ) denote the law of (Y (s − t), s ≥ t) under N and
δ(tj ,Y j) be, conditionally
on N ′, a Poisson point measure with intensity
κmax(dt)Nt[dY,1{τY ≤m}]
where Nt[dY,1{τY ≤m}] denotes the restriction of the measure Nt to the event {τY ≤ m}.
The next result is a direct consequence of Theorem 3.3.
Proposition 1.1. The process
j∈J Y
j is distributed as Y under N, conditionally on {τY =
Let τY 0 = inf{t > 0;Y
t = 0} be the extinction time of the Eve-population. In the
particular case where the branching mechanism of the Eve-population is given by a shift of
(7) ψEve(·) = ψ(θ + ·)− ψ(θ),
for some θ > 0 and β = 0, the pruning procedure developed in [1] gives that the nodes of
width ℓi correspond to a mutation with probability 1− e
−θℓi . As β = 0 there is no mutation
on the skeleton of the CRT outside the nodes. In particular, we see simultaneous extinction
of the whole population and the Eve-population if there is no mutation on the nodes in
the ancestral lineage of the last individual alive. This happens, conditionally on κmax, with
probability
i∈I ℓi .
4 ROMAIN ABRAHAM AND JEAN-FRANÇOIS DELMAS
Integrating w.r.t. the law of N gives that the probability of simultaneous extinction, condi-
tionally on {τY = m}, is under N, given by
N[τY 0 = m|τY = m] = exp−
1[0,m)(t) e
−ℓc(m−t) ℓπ(dℓ)dt
1− e−θℓ
= exp−
[ψ′(c(m− t) + θ)− ψ′(c(m− t))] dt
= exp−
φ′(c(t)) dt,
where φ = ψEve − ψ. Now, using that the distribution of (Y
0, Y ) is infinitely divisible
with canonical measure N, standard computations for Poisson measure yield that Px(τY 0 =
m|τY = m) = N[τY 0 = m|τY = m] that is
Px(τY 0 = m|τY = m) = exp−
φ′(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+ βλ
(0,∞)
π(dℓ)[e−λℓ−1 + λℓ],
ψEve(λ) = αEveλ+ βλ
(0,∞)
πEve(dℓ)[e
−λℓ−1 + λℓ],
and the immigration function
φ(λ) = ψEve(λ)− ψ(λ) = αImmλ+
(0,∞)
ν(dℓ)(1 − e−λℓ),
where αImm = αEve − α0 −
(0,∞)
ℓν(dℓ) ≥ 0 and π = πEve + ν, where πEve and ν are
Radon measures on (0,∞) with
(0,∞)
ℓν(dℓ) <∞. Notice the condition
(0,∞)
ℓν(dℓ) <∞ is
stronger than the usual condition on the immigration measure,
(0,∞)
(1 ∧ ℓ) ν(dℓ) < ∞, but
is implied by the requirement that
(1,∞)
ℓν(dℓ) <
(1,∞)
ℓπ(dℓ) <∞.
Inspired by Theorem 3.3, we consider N (dℓ, dt, dz) =
δ(ℓi,ti,zi)(dℓ, dt, dz) a Poisson
point measure with intensity
(8) 1[0,m)(t) e
−ℓc(m−t) ℓ [πEve(dℓ)δ0(dz) + ν(dℓ)δ1(dz)] dt.
Intuitively, the mark zi indicates if the ancestor (of the last individual alive) alive at time ti
had a new mutation (zi = 1) or not (zi = 0). Note however that if β > 0 we have to take
into account mutation on the skeleton. More precisely, let T1 = min{ti, zi = 1} be the first
mutation on the nodes in the ancestral lineage of the last individual alive and let T2 be an
exponential random time with parameter αImm independent of N . The time T2 corresponds
to the first mutation on the skeleton for the ancestral lineage of the last individual alive. We
T0 = min(T1, T2) if min(T1, T2) < m,
T0 = +∞ otherwise.
In particular there is simultaneous extinction if and only if T0 = +∞.
WILLIAMS’ DECOMPOSITION 5
For t ≥ 0, let us denote by Nt(dY
0, dY ) the joint law of ((Y 0(s− t), Y (s− t)), s ≥ t) under
N. Recall κmax given by (6). Conditionally on N and T2, let
δ(tj ,Y 0,j ,Y j) be a Poisson
point measure, with intensity
κmax(dt)Nt[(dY
0, dY ), 1{τY ≤m}].
We set
(10) (Y ′
, Y ′) =
tj<T0
(Y 0,j, Y j) +
tj≥T0
(0, Y j).
We write Qm for the law of (Y
′0, Y ′) computed for a given value of m.
Theorem 1.2. Under Qm, (Y
′0, Y ′) is distributed as (Y 0, Y ) under N[·|τY = m] , or equiv-
alently, under
|c′(m)|Qm(·)dm, (Y
′0, Y ′) is distributed as (Y 0, Y ) under N.
Let us remark that this Theorem is very close to Theorem 3.3 but only deals with CB and
does not specify the underlying genealogical structure. This is the purpose of a forthcoming
paper [3] where the genealogy of multi-type CB is described.
Intuitively, conditionally on the last individual alive being at time m, until the first mu-
tation in the ancestral lineage (that is for tj < T0) , its ancestors give birth to a population
with initial Eve type which has to die before time m, and after the first mutation on the
ancestral lineage (that is for tj ≥ T0), there is no Eve-population in the descendants which
still have to die before time m.
Now, using that the distribution of (Y 0, Y ) is infinitely divisible with canonical measure
N, standard computations for Poisson measure yield that Px(τY 0 = m|τY = m) = N[τY 0 =
m|τY = m]. As
N[τY 0 = m|τY = m] = Qm(T0 = +∞)
= Qm(T1 = +∞)Qm(T2 ≥ m)
(0,∞)
e−ℓc(m−t) ℓν(dℓ)
e−αImmm
dt φ′(c(t)),
we deduce the following Corollary.
Corollary 1.3 (Probability of simultaneous extinction). We have for almost every m > 0
Px(τY 0 = m|τY = m) = exp−
φ′(c(t)) dt,
where c is the unique (non-negative) solution of (5).
The paper is organized as follows. In Section 2, we recall some facts on the genealogy
of the CRT associated with a Lévy process. We prove a Williams’ decomposition for the
exploration process associated with the CRT in Section 3. We prove Theorem 1.2 in Section
4. Notice that Proposition 1.1 is a direct consequence of Theorem 1.2.
2. Notations
We recall here the construction of the Lévy 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].
6 ROMAIN ABRAHAM AND JEAN-FRANÇOIS DELMAS
2.1. The underlying Lévy process. We consider a R-valued Lévy process (Xt, t ≥ 0) with
Laplace exponent ψ (for λ ≥ 0 E
e−λXt
= etψ(λ)) satisfying (1) and (2). Let I = (It, t ≥ 0)
be the infimum process of X, It = inf0≤s≤tXs, and let S = (St, t ≥ 0) be the supremum
process, St = sup0≤s≤tXs. We will also consider for every 0 ≤ s ≤ t the infimum of X over
[s, t]:
Ist = inf
s≤r≤t
The point 0 is regular for the Markov process X − I, and −I is the local time of X − I at
0 (see [5], chap. VII). Let N be the associated excursion measure of the process X − I away
from 0, and σ = inf{t > 0;Xt − It = 0} the length of the excursion of X − I under N. We
will assume that under N, X0 = I0 = 0.
Since X is of infinite variation, 0 is also regular for the Markov process S −X. The local
time, L = (Lt, t ≥ 0), of S −X at 0 will be normalized so that
t ] = e−tψ(β)/β ,
where L−1t = inf{s ≥ 0;Ls ≥ t} (see also [5] Theorem VII.4 (ii)).
2.2. The height process and the Lévy CRT. For each t ≥ 0, we consider the reversed
process at time t, X̂(t) = (X̂
s , 0 ≤ s ≤ t) by:
X̂(t)s = Xt −X(t−s)− if 0 ≤ s < t,
and X̂
t = Xt. The two processes (X̂
s , 0 ≤ s ≤ t) and (Xs, 0 ≤ s ≤ t) have the same law.
Let Ŝ(t) be the supremum process of X̂(t) and L̂(t) be the local time at 0 of Ŝ(t) − X̂(t) with
the same normalization as L.
Definition 2.1 ([7], Definition 1.2.1 and Theorem 1.4.3). There exists a process H = (Ht, t ≥
0), called the height process, such that for all t ≥ 0, a.s. Ht = L̂
t , and H0 = 0. Because of
hypothesis (2), the height process H is continuous.
The height process (Ht, t ∈ [0, σ]) under N codes a continuous genealogical structure, the
Lévy CRT, via the following procedure.
(i) To each t ∈ [0, σ] corresponds a vertex at generation Ht.
(ii) Vertex t is an ancestor of vertex t′ if Ht = H[t,t′], where
(11) H[t,t′] = inf{Hu, u ∈ [t ∧ t
′, t ∨ t′]}.
In general H[t,t′] is the generation of the last common ancestor to t and t
(iii) We put d(t, t′) = Ht +Ht′ − 2H[t,t′] and identify t and t
′ (t ∼ t′) if d(t, t′) = 0.
The Lévy CRT coded by H is then the quotient set [0, σ]/ ∼, equipped with the distance d
and the genealogical relation specified in (ii).
Let (τs, s ≥ 0) be the right continuous inverse of −I: τs = inf{t > 0;−It > s}. Recall that
−I is the local time of X − I at 0. Let Lat denote the local time at level a of H until time t,
see Section 1.3 in [7].
Theorem 2.2 ([7], Theorem 1.4.1). The process (Laτx , a ≥ 0) is under P (resp. N) defined
as Y under Px (resp. N).
In what follows, we will use the notation N instead of N for the excursion measure to stress
that we consider the genealogical structure of the branching process.
WILLIAMS’ DECOMPOSITION 7
2.3. The exploration process. The height process is not Markov. But it is a simple
function of a measure-valued Markov process, the so-called exploration process.
If E is a Polish space, let B(E) (resp. B+(E)) be the set of real-valued measurable (resp.
and non-negative) functions defined on E endowed with its Borel σ-field, and let M(E) (resp.
Mf (E)) be the set of σ-finite (resp. finite) measures on E, endowed with the topology of
vague (resp. weak) convergence. For any measure µ ∈ M(E) and f ∈ B+(E), we write
〈µ, f〉 =
f(x)µ(dx).
The exploration process ρ = (ρt, t ≥ 0) is a Mf (R+)-valued process defined as follows: for
every f ∈ B+(R+),
〈ρt, f〉 =
[0,t]
t f(Hs),
or equivalently
(12) ρt(dr) =
0<s≤t
Xs−<I
(Ist −Xs−)δHs(dr) + β1[0,Ht](r)dr.
In particular, the total mass of ρt is 〈ρt, 1〉 = Xt − It.
For µ ∈ M(R+), we set
(13) H(µ) = sup Supp µ,
where Supp µ is the closed support of µ, with the convention H(0) = 0. We have
Proposition 2.3 ([7], Lemma 1.2.2). Almost surely, for every t > 0,
• H(ρt) = Ht,
• ρt = 0 if and only if Ht = 0,
• if ρt 6= 0, then Supp ρt = [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 of ρ, we must first define the process ρ started at any initial measure
µ ∈ Mf (R+).
For a ∈ [0, 〈µ, 1〉], we define the erased measure kaµ by
kaµ([0, r]) = µ([0, r]) ∧ (〈µ, 1〉 − a), for r ≥ 0.
If a > 〈µ, 1〉, we set kaµ = 0. In other words, the measure kaµ is the measure µ erased by a
mass a from the top of [0,H(µ)].
For ν, µ ∈ Mf (R+), and µ with compact support, we define the concatenation [µ, ν] ∈
Mf (R+) of the two measures by:
[µ, ν], f
ν, f(H(µ) + ·)
, f ∈ B+(R+).
Finally, we set for every µ ∈ Mf (R+) and every t > 0 ρ
k−Itµ, ρt]. We say that
t , t ≥ 0) is the process ρ started at ρ
0 = µ, and write Pµ for its law. Unless there is an
ambiguity, we shall write ρt for ρ
Proposition 2.4 ([7], Proposition 1.2.3). The process (ρt, t ≥ 0) is a càd-làg strong Markov
process in Mf (R+).
Notice that N is also the excursion measure of the process ρ away from 0, and that σ, the
length of the excursion, is N-a.e. equal to inf{t > 0; ρt = 0}.
8 ROMAIN ABRAHAM AND JEAN-FRANÇOIS DELMAS
2.4. The dual process and representation formula. We shall need the Mf (R+)-valued
process η = (ηt, t ≥ 0) defined by
ηt(dr) =
0<s≤t
Xs−<I
(Xs − I
t )δHs(dr) + β1[0,Ht](r)dr.
The process η is the dual process of ρ under N thanks to the following time reversal property:
recall σ denotes the length of the excursion under N.
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 under N.
It also enjoys the snake property: for all t ≥ 0, s ≥ 0
(ρt, ηt)[0,H[t,s]) = (ρs, ηs)[0,H[t,s]),
that is the measures ρ and η between two instants coincide up to the minimum of the height
process between those two instants.
We recall the Poisson representation of (ρ, η) under N. Let N∗(dx dℓ du) be a Poisson point
measure on [0,+∞)3 with intensity
dx ℓπ(dℓ)1[0,1](u)du.
For every a > 0, let us denote by Ma the law of the pair (µa, νa) of finite measures on R+
defined by: for f ∈ B+(R+)
〈µa, f〉 =
N∗(dx dℓ du)1[0,a](x)uℓf(x),
〈νa, f〉 =
N∗(dx dℓ du)1[0,a](x)ℓ(1 − u)f(x).
We finally set M =
da e−α0aMa.
Proposition 2.6 ([7], Proposition 3.1.3). For every non-negative measurable function F on
Mf (R+)
F (ρt, ηt) dt
M(dµ dν)F (µ, ν),
where σ = 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 on Mf (R+)
F (ρt, ηt) dL
= e−α0a
Ma(dµ dν)F (µ, ν),
where σ = 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
Hmax = sup{Hr; r ∈ [0, σ]} is attained. Let Tmax = inf{s ≥ 0;Hs = Hmax}.
For every m > 0, we set Tm(ρ) = inf{s > 0,Hs(ρ) = m} the first hitting time of m for the
height process. When there is no need to stress the dependence in ρ, we shall write Tm for
Tm(ρ). Recall the function c defined by (5) is equal to
(14) c(m) = N[Tm <∞] = N[Hmax ≤ m].
WILLIAMS’ DECOMPOSITION 9
We set ρd = (ρTmax+s, s ≥ 0) and ρg = (ρ(Tmax−s)+, s ≥ 0), where x+ = max(x, 0).
For every finite measure with compact support µ, we write P∗µ for the law of the exploration
process ρ starting at µ and killed when it first reaches 0. We also set
µ := lim
µ( · |H(µ) ≤ Hmax ≤ H(µ) + ε).
We now describe the probability measure P̂∗µ via a Poisson decomposition. Let (αi, βi), i ∈ I
be the excursion intervals of the process X − I away from 0 (well defined under P∗µ or under
P̂∗µ). For every i ∈ I, we define hi = Hαi and the measure-valued process ρ
i by the formula
〈ρit, f〉 =
(hi,+∞)
f(x− hi)ρ(αi+t)∧βi(dx).
We then have the following result.
Lemma 3.1. Under the probability P̂∗µ, the point measure
δ(hi,ρi) is a Poisson point
measure with intensity µ(dr)N[·, H
≤ m− r].
Proof. We know (cf Lemma 4.2.4 of [7]) that the point measure
δ(hi,ρi) is under P
Poisson point measure with intensity µ(dr)N(dρ). The result follows then easily from standard
results on Poisson point measures. �
Remark 3.2. Lemma 3.1 gives also that, for every finite measure with compact support µ, if
we write µa = µ(· ∩ [0, a]),
µ = lim
a→H(µ)
( · |Hmax ≤ H(µ)).
Theorem 3.3 (Williams’ Decomposition).
(i) The law of H
is characterized by N[H
≤ m] = c(m), where c is the unique
non-negative solution of (5).
(ii) Conditionally on H
= m, the law of (ρTmax , ηTmax) is under N the law of
viriδti + β1[0,m](t)dt,
(1− vi)riδti + β1[0,m](t)dt
where
δ(vi,ri,ti) is a Poisson measure with intensity
1[0,1](v)1[0,m](t) e
−rc(m−t) dv rπ(dr) dt.
(iii) Under N, conditionally on H
= m, and (ρTmax , ηTmax), (ρd, ρg) are independent and
ρd (resp. ρg) is distributed as ρ (resp. η) under P̂
ρTmax
(resp. P̂∗ηTmax
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 Lévy snake.
The end of this section is devoted to the proof of (ii) and (iii) of this Theorem.
Let m > a > 0 be fixed. Let ε > 0. Recall Tm = inf{t > 0;Ht = m} is the first hitting
time of m for the height process, and set Lm = sup{t < σ;Ht = m} for the last hitting time
of m, with the convention that inf ∅ = +∞ and sup ∅ = +∞. We consider the minimum of
H between Tm and Lm: H[Tm,Lm] = min{Ht; t ∈ [Tm, Lm]}.
10 ROMAIN ABRAHAM AND JEAN-FRANÇOIS DELMAS
We set ρ(d) = (ρTmax,a+t, t ≥ 0), with
Tmax,a = inf{t > Tmax,Hs = a},
the path of the exploration process on the right of Tmax after the hitting time of a, and
ρ(g) = (ρ(Lmax,a−t)−, t ≥ 0), with Lmax,a = sup{t < Tmax;Ht = a}, the returned path of the
exploration process on the left of Tmax before its last hitting time of a. Let us note that, by
time reversal (see Proposition 2.5), the process ρ(g) is of the same type as η. This remark
will be used later.
To prove the Theorem, we shall compute
A0 = N
(g))F2(ρ
(d))F3(ρTmax |[0,a])F4(ηTmax |[0,a])1{m≤Hmax<m+ε}
and let ε go down to 0. We shall see in Lemma 3.4, that adding 1{H[Tm,Lm]>a}
in the integrand
does not change the asymptotic behavior as ε goes down to 0. Intuitively, if the maximum of
the height process is between m and m+ε, outside a set of small measure, the height process
does not reach level a between the first and last hitting time of m. So that we shall compute
first
(15) A = N
(g))F2(ρ
(d))F3(ρTmax |[0,a])F4(ηTmax |[0,a])1{H[Tm,Lm]>a,m≤Hmax<m+ε}
Notice that on {H[Tm,Lm] > a}, we have Tmax,a = Tm,a := inf{s > Tm,Hs(ρ) = a} and,
from the snake property, ρTmax |[0,a] = ρTm |[0,a] and ηTmax |[0,a] = ηTm |[0,a], so that
A = N
(g))F2((ρTm,a+t, t ≥ 0))F3(ρTm |[0,a])F4(ηTm |[0,a])1{H[Tm,Lm]>a,m≤Hmax<m+ε}
Let us remark that, we have
1{H[Tm,Lm]>a,m≤Hmax<m+ε}
= 1{m≤sup{Hu,0≤u≤Tm,a}<m+ε}1{sup{Hu,u≥Tm,a}<m}.
By using the strong Markov property of the exploration process at time Tm,a, we get
(g))F4(ηTm |[0,a])1{m≤sup{Hu,0≤u≤Tm,a}<m+ε}F3(ρTm |[0,a])E
ρTm |[0,a]
F2(ρ)1{Hmax<m}
and so, by conditioning, we get
A = N
(g))F4(ηTm |[0,a])G2(ρTm |[0,a])1{H[Tm,Lm]>a,m≤Hmax<m+ε}
where G2(µ) = F3(µ)E
µ[F2(ρ)|Hmax < m]. Using time reversibility (see Proposition 2.5)
and the strong Markov property at time Tm,a again, we have
A = N
(d))F4(ρTm |[0,a])G2(ηTm |[0,a])1{H[Tm,Lm]>a,m≤Hmax<m+ε}
G1(ρTm |[0,a])G2(ηTm |[0,a])1{H[Tm,Lm]>a,m≤Hmax<m+ε}
where G1(µ) = F4(µ)E
µ[F1(ρ)|Hmax < m].
Now, we use ideas from the proof of Theorem 4.6.2 of [7]. Let us recall the excursion decom-
position of the exploration process above level a. We set τas = inf
du1{Hu≤a} > s
Let Ea be the σ-field generated by the process (ρ̃s, s ≥ 0) := (ρτas , s ≥ 0). We also set
η̃s = ητas .
WILLIAMS’ DECOMPOSITION 11
Let (αi, βi), i ∈ I be the excursion intervals of H above level a. For every i ∈ I we define
the measure-valued process ρi by setting
〈ρis, ϕ〉 =
(a,+∞)
ραi+s(dr)ϕ(r − a) if 0 < s < βi − αi,
ρs = 0 if s = 0 or s ≥ βi − αi,
and the process ηi similarly. We also define the local time at the beginning of excursion ρi
by ℓi = L
. Then, under N, conditionally on Ea, the point measure
δ(ℓi,ρi,ηi)
is a Poisson measure with intensity 1[0,Laσ](ℓ)dℓN[dρ dη].
In particular, we have
A = N
j 6=i
1{Tm(ρj)=+∞}G1(ραi)G2(ηαi)1{m≤Hmax(ρi)<m+ε}
Let us denote by (τaℓ , ℓ ≥ 0) the right-continuous inverse of (L
s , s ≥ 0). Palm formula for
Poisson point measures yields
A = N
j 6=i
1{Tm(ρj)=+∞}G1(ραi)G2(ηαi)1{m≤Hmax(ρi)<m+ε}
∫ Laσ
dℓG1(ρτa
)G2(ητa
)N[m ≤ Hmax < m+ ε]N
1{Tm(ρj)=+∞}
A time-change then gives
(16) A = v(m− a, ε)N
dLasG1(ρs)G2(ηs) e
−c(m−a)Laσ
where v(x, ε) = c(x) − c(x+ ε) = N[x ≤ Hmax < x+ ε]. We have
A = v(m− a, ε)N
dLasG1(ρs)G2(ηs) e
−c(m−a)Las e−c(m−a)(L
= v(m− a, ε)N
dLasG1(ρs)G2(ηs) e
−c(m−a)Las e−〈ρs,N[1−e
−c(m−a)L
where we used for the last equality that the predictable projection of e−λ(L
s ) is given by
e−〈ρs ,N[1−e
σ ]〉. Notice that by using the excursion decomposition above level 0 < r < m,
we have
c(m) = N[Tm <∞] = N[1− e
−c(m−r)Lrσ ].
In particular, we get
A = v(m− a, ε)N
dLasG1(ρs)G2(ηs) e
−c(m−a)Las e−〈ρs,c(m−·)〉
Using time reversibility, we have
A = v(m− a, ε)N
dLasG1(ηs)G2(ρs) e
−c(m−a)(Laσ−L
s ) e−〈ηs ,c(m−·)〉
12 ROMAIN ABRAHAM AND JEAN-FRANÇOIS DELMAS
Similar computations as those previously done give
A = v(m− a, ε)N
dLasG1(ηs)G2(ρs) e
−〈ηs+ρs,c(m−·)〉
= v(m− a, ε)N
dLasG1(ρs)G2(ηs) e
−〈ρs+ηs,c(m−·)〉
Using Proposition 2.7, we get
A = v(m− a, ε) e−α0a
Ma(dµ dν)G1(µ)G2(ν) e
−〈µ+ν,c(m−·)〉 .
We can give a first consequence of the previous computation.
Lemma 3.4. We have
N[H[Tm,Lm] > a,m ≤ Hmax < m+ ε] = c
c(m− a)− c(m− a+ ε)
c′(m− a)
Proof. Taking F1 = F2 = F3 = F4 = 1 in (16), we deduce that
N[H[Tm,Lm] > a,m ≤ Hmax < m+ ε] = v(m− a, ε)N
Laσ e
−c(m−a)Laσ
Let a0 > 0 and let us compute B(a0, a) = N
Laσ e
−c(a0)L
. Thanks to Theorem 2.2, notice
B(a0, a) = N
−c(a0)Ya
∂a0N[1− e
−c(a0)Ya ]
c′(a0)
On the other hand, we have
c(a+ a0) = N[Ya+a0 > 0] = N[1− EYa[Ya0 = 0]] = N
1− e−Yac(a0)
where we used the Markov property of Y at time a under N for the second equality and (3)
with λ going to infinity for the last. Thus, we get B(a0, a) =
c′(a0 + a)
c′(a0)
. We deduce that
N[H[Tm,Lm] > a,m ≤ Hmax < m+ ε] = v(m− a, ε)B(a−m,a)
= c′(m)
c(m− a)− c(m− a+ ε)
c′(m− a)
Since F1, F2, F3 and F4 are bounded, say by C, we have |A−A0| ≤ C
4N[H[Tm,Lm] < a,m ≤
Hmax < m+ ε]. From Lemma 3.4, we deduce that
|A−A0|
N[m ≤ Hmax < m+ ε]
1− lim
N[H[Tm,Lm] > a,m ≤ Hmax < m+ ε]
N[m ≤ Hmax < m+ ε]
WILLIAMS’ DECOMPOSITION 13
We deduce that
(g))F2(ρ
(d))F3(ρTmax |[0,a])F4(ηTmax |[0,a])1{m≤Hmax<m+ε}
N[m ≤ Hmax < m+ ε]
c′(m− a)
c′(m)
e−α0a
Ma(dµ dν)G1(µ)G2(ν) e
−〈µ+ν,c(m−·)〉
Ma(dµ dν)G1(µ)G2(ν) e
−〈µ+ν,c(m−·)〉
Ma(dµ dν) e
−〈µ+ν,c(m−·)〉
M̃a(dµ dν)G1(µ)G2(ν)
M̃a(dµ dν)F4(ν)E
ν [F1(ρ
(d))|Hmax < m]F3(µ)E
µ[F2(ρ
(d))|Hmax < m],
where
µ(dt) =
uiℓiδti + β1[0,a](t)dt
ν(dt) =
(1− ui)ℓiδti + β1[0,a](t)dt,
i∈I δ(xi,ℓi,ti) is under M̃a a Poisson point measure on [0,+∞)
3 with intensity
1[0,a](t)dt ℓ e
−ℓc(m−t) π(dℓ)1[0,1](u)du.
Standard results on measure decomposition imply there exists a regular version of the
probability measure N[ · |Hmax = m] and that, for almost every non-negative m,
N[ · |Hmax = m] = lim
N[ · |m ≤ Hmax < m+ ε].
This gives (ii) and (iii) of Theorem 3.3 since F1, F2, F3, F4 are arbitrary continuous functionals
and by Remark 3.2.
4. Proof of Theorem 1.2
The proof of this Theorem relies on the computation of the Laplace transform for (Y ′
, Y ′)
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 of Qm as the law of (Y
′0, Y ′) defined by (10) and T0
defined by (9) as the first mutation undergone by the last individual alive.
For r < m, we have
Qm(T0 ∈ [r, r + dr], T0 = T2) = Qm(T2 ∈ [r, r + dr])Qm(T1 > r)
= dr αImm e
−αImmr exp−
(0,∞)
e−ℓc(m−t) ℓν(dℓ)
= dr αImm e
φ′(c(m−t)) dt,
14 ROMAIN ABRAHAM AND JEAN-FRANÇOIS DELMAS
and, with the notation φ0(λ) = φ(λ)− αImmλ,
Qm(T0 ∈ [r, r + dr], T0 = T1) = Qm(T2 > r)Qm(T1 ∈ [r, r + dr])
= dr φ′0(c(m− r)) e
−αImmr exp−
(0,∞)
e−ℓc(m−t) ℓν(dℓ)
= dr φ′0(c(m− r)) e
φ′(c(m−t)) dt .
In particular, we have for r < m
Qm(T0 ∈ [r, r + dr]) = dr φ
′(c(m− r)) e−
φ′(c(m−t)) dt .
(17) Qm(T0 > r) = e
φ′(c(m−t)) dt .
Notice we have Qm(T0 = ∞) = exp−
φ′(c(t)) dt.
4.1.2. Conditional law of N given T0. Recall N is under Qm a Poisson point measure with
intensity given by (8). Conditionally on {T0 = r, T0 = T2}, with m > r > 0, N is under Qm
a point Poisson measure with intensity
1[0,r)(t) e
−ℓc(m−t) ℓπEve(dℓ)δ0(dz)dt+
1(r,m)(t) e
−ℓc(m−t) ℓ [πEve(dℓ)δ0(dz) + ν(dℓ)δ1(dz)] dt.
Conditionally on {T0 = r, T0 = T1}, with r < m, N is distributed under Qm as Ñ + δ(L,r,1)
where Ñ is a point Poisson measure with intensity
1[0,r)(t) e
−ℓc(m−t) ℓπEve(dℓ)δ0(dz)dt
+ 1(r,m)(t) e
−ℓc(m−t) ℓ [πEve(dℓ)δ0(dz) + ν(dℓ)δ1(dz)] dt,
and L is a random variable independent of Ñ with distribution
e−ℓc(m−r) ℓν(dℓ)
(0,∞)
′c(m−r) ℓ′ν(dℓ′)
Conditionally on {T0 = ∞}, N is under Qm a point Poisson measure with intensity
1[0,m)(t) e
−ℓc(m−t) ℓπEve(dℓ)δ0(dz)dt.
4.1.3. Formulas. The following two formulas are straightforward: for all x, γ ≥ 0,
ψ′Eve(x+ γ)− ψ
Eve(γ) = 2βx+
(0,∞)
e−ℓγ ℓπEve(dℓ)[1 − e
−ℓx],(18)
ψ′(x+ γ)− ψ′(γ) = 2βx+
(0,∞)
e−ℓγ ℓπ(dℓ)[1− e−ℓx],(19)
Finally we deduce from (5) that ψ(c) = −c′, ψ′(c)c′ = −c′′ and
ψ′(c) = − log(c′).
WILLIAMS’ DECOMPOSITION 15
4.1.4. Laplace transform. Recall τY = inf{t > 0;Yt = 0} is the extinction time of Y . Let
µEve and µTotal be two finite measures with support a subset of a finite set A = {a1, . . . , an}
with 0 = a0 < a1 < · · · < an < an+1 = ∞. For m ∈ (0,+∞) \A, we consider
wm(t) = N[1− e
Y 0r−t µEve(dr)−
Yr−t µTotal(dr) 1{τY <m−t}],
w∗m(t) = N[1− e
Yr−t µTotal(dr) 1{τY <m−t}].
By noticing that N-a.e. 1{τY <m−t} = limλ→∞ exp−
Yr−t µ
λ(dr), where µλ(dr) = λδm(dr),
we deduce from Lemma 3.1 in [2] that (wm, w
m) are right continuous and are the unique
non-negative solutions of : for k ∈ {0, . . . , n}, m ∈ (ak, ak+1), t ∈ (−∞,m),
(21) w∗m(t) +
[t,ak]
ψ(w∗m(r))dr =
[t,ak]
µTotal(dr) + c(m− ak),
(22) wm(t) +
[t,ak]
ψEve(wm(r))dr
[t,ak]
µEve(dr) +
[t,ak]
µTotal(dr) + c(m− ak) +
[t,ak]
φ(w∗m(r))dr.
We define
(23) ām = max{ak; ak < m, k ∈ {0, . . . , n}}.
Notice that wm(t) = w
m(t) = c(m− t) for t ∈ (ām,m).
4.2. Proof of Theorem 1.2.
4.2.1. Aim. Theorem 1.2 will be proved as soon as we check that the following equality
w(0) =
−c′(m)Qm[1− e
r µEve(dr)−
Y ′r µTotal(dr)]dm
holds for all the possible choices of measures µEve and µTotal satisfying the assumptions of
Section 4.1.4, with w = w∞ defined by (22).
Notice the integrand of the right-hand side is null for m < a1. Let ∆ denote the right-hand
side. We have for 0 < ε ≤ a1:
dm (−c′(m))Qm[1− e
r µEve(dr)−
Y ′r µTotal(dr)]
= c(ε) +
dm 1Ac(m)c
′(m)Qm[Z],
with, thanks to the definition (6) of κmax,
Z = exp−
∫ ām
κmax(dt) [nt1{t<T0} + n
t1{t≥T0}]
nt = N[(1 − e
r−t µEve(dr)−
Yr−t µTotal(dr))1{τY ≤m−t}] = wm(t)− c(m− t)
n∗t = N[(1 − e
Yr−t µTotal(dr))1{τY ≤m−t}] = w
m(t)− c(m− t),
with (wm, w
m) the non-negative solutions of (21) and (22). Notice that wm(t) = w
m(t) =
c(m− t) for t ∈ (ām,m) and thus nt = n
t = 0 when t ∈ (ām,m).
16 ROMAIN ABRAHAM AND JEAN-FRANÇOIS DELMAS
We set ∆ = c(ε) +
1Ac(m)(∆1 +∆2 +∆3) dm with
∆1 = c
′(m)Qm[Z|T0 > ām]Qm(T0 > ām),
∆2 = c
∫ ām
Qm[Z|T0 = r, T0 = T1]Qm(T0 ∈ [r, r + dr], T0 = T1),
∆3 = c
∫ ām
Qm[Z|T0 = r, T0 = T2]Qm(T0 ∈ [r, r + dr], T0 = T2).
We shall assume m 6∈ A.
4.2.2. Computation of ∆1. We have, using formula (6),
∆1 = c
′(m)Qm(T0 > ām)Qm[e
R ām
κmax(dt)nt |T0 > ām]
= c′(m) e−
R ām
φ′(c(m−t))dt
∫ ām
(wm(t)− c(m− t)) dt−
∫ ām
dt e−ℓc(m−t) ℓπEve(dℓ)[1− e
−ℓ(wm(t)−c(m−t))]
= c′(m) e−
φ′(c(m−t))dt exp
∫ ām
dt[ψ′Eve(wm(t))− ψ
Eve(c(m− t))]
= c′(m) e
m−ām
ψ′(c(t))dt
dt ψ′Eve(wm(t))
= c′(m− ām) e
dt ψ′Eve(wm(t)),
where we used (20) for the last equality to get
(24) e
m−ām
ψ′(c(t))dt
= e−[log(c
′(t))]m
m−ām =
c′(m− ām)
c′(m)
4.2.3. Computation of ∆2. Using Section 4.1.2, we get
Qm[Z|T0 = r, T0 = T1] = e
(wm(t)−c(m−t)) dt−2β
R ām
(w∗m(t)−c(m−t)) dt
exp(−
dt e−ℓc(m−t) ℓπEve(dℓ)[1 − e
−ℓnt ])
exp(−
∫ ām
dt e−ℓc(m−t) ℓπ(dℓ)[1 − e−ℓn
(0,∞)
ν(dℓ′)
′c(m−r) ℓ′ e−ℓ
φ′0(c(m− r))
= exp(−
dt[ψ′Eve(wm(t))− ψ
Eve(c(m− t))])
exp(−
∫ ām
dt[ψ′(w∗m(t))− ψ
′(c(m− t))])
φ′0(w
m(r))
φ′0(c(m− r))
WILLIAMS’ DECOMPOSITION 17
We deduce from Section 4.1.1
∆2 = c
∫ ām
Qm[Z|T0 = r, T0 = T1]Qm(T0 ∈ [r, r + dr], T0 = T1),
= c′(m)
∫ ām
dr φ′0(w
m(r)) e
φ′(c(m−t)) dt exp(−
dt[ψ′Eve(wm(t)) − ψ
Eve(c(m − t))])
exp(−
∫ ām
dt[ψ′(w∗m(t))− ψ
′(c(m− t))])
= c′(m) e
ψ′(c(m−t)) dt
∫ ām
dr φ′0(w
m(r)) e
dt ψ′Eve(wm(t))−
dt ψ′(w∗m(t))
= c′(m− ām)
∫ ām
dr φ′0(w
m(r)) e
dt ψ′
(wm(t))−
R ām
dt ψ′(w∗m(t)),
where we used (24) for the last equality.
4.2.4. Computation of ∆3. Using Section 4.1.2, we get
Qm[Z|T0 = r, T0 = T2] = e
(wm(t)−c(m−t)) dt−2β
(w∗m(t)−c(m−t)) dt
dt e−ℓc(m−t) ℓπEve(dℓ)[1 − e
−ℓnt ]
∫ ām
dt e−ℓc(m−t) ℓπ(dℓ)[1 − e−ℓn
= exp(−
dt[ψ′Eve(wm(t)) − ψ
Eve(c(m − t))])
exp(−
∫ ām
dt[ψ′(w∗m(t))− ψ
′(c(m− t))]).
We deduce from Section 4.1.1
∆3 = c
∫ ām
Qm[Z|T0 = r, T0 = T2]Qm(T0 ∈ [r, r + dr], T0 = T2),
= c′(m)
∫ ām
dr αImm e
φ′(c(m−t)) dt exp(−
dt[ψ′Eve(wm(t))− ψ
Eve(c(m− t))])
exp(−
∫ ām
dt[ψ′(w∗m(t))− ψ
′(c(m− t))])
= c′(m) e
ψ′(c(m−t)) dt
∫ ām
dr αImm e
dt ψ′Eve(wm(t))−
dt ψ′(w∗m(t))
= c′(m− ām)
∫ ām
dr αImm e
dt ψ′
(wm(t))−
R ām
dt ψ′(w∗m(t)),
where we used (24) for the last equality.
4.2.5. Computation of ∆2 +∆3. We have
∆2 +∆3 = c
′(m− ām)
∫ ām
dr φ′(w∗m(r)) e
dt ψ′Eve(wm(t))−
dt ψ′(w∗m(t)) .
Differentiating (21) w.r.t. time and m, we get for t < m
′(t)− ∂mw
m(t)ψ
′(w∗m(t)) = 0.
18 ROMAIN ABRAHAM AND JEAN-FRANÇOIS DELMAS
Notice also that for m > t ≥ ām, we have ∂mw
∗(t) = c′(m− t) and thus
∗(ām) = c
′(m− ām).
We get
exp(−
∫ ām
dtψ′(w∗m(t))) =
∂mw∗m(ām)
c′(m− ām)
Differentiating (22) w.r.t. time and m, we get for t < m
m(t)− ∂mwm(t)ψ
Eve(wm(t)) = −∂mw
m(t)φ
′(w∗m(t)).
We deduce that
∆2 +∆3 =
∫ ām
dr∂mw
m(t)φ
′(w∗m(r)) e
dt ψ′
(wm(t))
∫ ām
dr[∂mw
m(r)− ∂mwm(r)ψ
Eve(wm(r))] e
dt ψ′Eve(wm(t))
∂mwm(r) e
dt ψ′Eve(wm(t))
= ∂mwm(0) − ∂mwm(ām) e
dt ψ′Eve(wm(t)) .
Notice also that for m > t ≥ ām one has ∂mw(t) = c
′(m − t), in particular ∂mw(ām) =
c′(m− ām). This implies that
∆2 +∆3 = ∂mwm(0) − c
′(m− ām) e
dt ψ′Eve(wm(t)) .
4.3. Conclusion. Thus, for m 6∈ A, we have
∆1 +∆2 +∆3 = ∂mwm(0),
∆ = c(ε) +
∂mwm(0) = c(ε) + w∞(0)− wε(0) = w(0).
This ends the proof of the Theorem.
Acknowledgments. The authors wish to thank an anonymous referee for his numerous
and useful comments which improved the presentation of the paper.
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Romain Abraham, MAPMO, Fédération Denis Poisson, Université d’Orléans, B.P. 6759, 45067
Orléans cedex 2, France.
E-mail address: romain.abraham@univ-orleans.fr
Jean-François Delmas, CERMICS, Univ. Paris-Est, 6-8 av. Blaise Pascal, Champs-sur-Marne,
77455 Marne La Vallée, France.
E-mail address: delmas@cermics.enpc.fr
	1. Introduction
	2. Notations
	2.1. The underlying Lévy process
	2.2. The height process and the Lévy CRT
	2.3. The exploration process
	2.4. The dual process and representation formula
	3. Williams' decomposition
	4. Proof of Theorem ??
	4.1. Preliminary computations
	4.2. Proof of Theorem ??
	4.3. Conclusion
	References
