1.4.1 Marks on the skeleton
Let α
1
≥ 0. We want to construct a “Lévy Poisson snake” i.e. a Lévy snake with spatial motion a Poisson process, whose jumps give the marks on the branches of the CRT. More precisely, we set
W the space of killed càd-làg paths w : [0,
ζ → R where ζ ∈ 0, +∞ is called the lifetime of the path w
. We equip W with a distance d defined in [22] Chapter 4 and whose expression is not important
for our purpose such that W , d is a Polish space.
By Proposition 4.4.1 of [22] when H is continuous, or the results of the appendix in the general case, there exists a probability measure ˜
P on ˜ Ω = DR
+
, M
f
R
+
× W under which the canonical process
ρ
s
, W
s
satisfies 1. The process
ρ is the exploration process starting at 0 associated with a branching mechanism ψ,
2. For every s ≥ 0, the path W
s
is distributed as a Poisson process with intensity α
1
stopped at time H
s
:= H ρ
s
, 3. The process
ρ, W satisfies the so-called snake property: for every s s
′
, conditionally given ρ, the paths W
s
· and W
s
′
· coincide up to time H
s ,s
′
:= inf {H
u
, s ≤ u ≤ s
′
} and then are independent.
So, for every t ≥ 0, the path W
t
is a.s. càd-làg with jumps equal to one. Its derivative m
ske t
is an atomic measure on [0, H
t
; it gives the marks on the skeleton on the ancestral line of the individual labeled t.
We shall denote by ˜ N the corresponding excursion measure out of 0, 0.
1.4.2 Marks on the nodes
Let p be a measurable function defined on R
+
taking values in [0, 1] such that Z
0,+∞
ℓpℓ πdℓ ∞. 16
We define the measures π
1
and π
by their density: d
π
1
x = pxdπx and
d π
x = 1 − pxdπx. Let Ω
′
, A
′
, P
′
be a probability space with no atom. Recall that J , defined by 10, denotes the jumping times of the Lévy process X and that ∆
s
represents the height of the jump of X at time s
∈ J . As J is countable, we can construct on the product space ˜ Ω×Ω
′
with the product probability measure ˜
P ⊗P
′
a family U
s
, s ∈ J of random variables which are, conditionally on X , independent,
uniformly distributed over [0, 1] and independent of ∆
s
, s ∈ J and W
s
, s ≥ 0. We set, for every
s ∈ J :
V
s
= 1
{U
s
≤p∆
s
}
, so that, conditionally on X , the family V
s
, s ∈ J are independent Bernoulli random variables with
respective parameters p∆
s
. 1439
We set J
1
= {s ∈ J , V
s
= 1} the set of the marked jumps and J = J \ J
1
= {s ∈ J , V
s
= 0} the set of the non-marked jumps. For t
≥ 0, we consider the measure on R
+
, m
nod t
d r = X
s≤t, s∈J 1
X
s −
I
s t
I
s t
− X
s −
δ
H
s
d r. 17
The atoms of m
nod t
give the marked nodes of the exploration process at time t. The definition of the measure-valued process m
Kategori