It is then easy to get, using h
3
and 35, that P-a.s φ
2
γ = sup
r ≥0
Z
0,∞
π
1
dℓ
1
Z
γℓ
1
e
−s
ds 1
− wℓ
1
− s
γ , r,
µ −
Z
0,∞
π
1
dℓ
1
1 − wℓ
1
, r, µ
converges to 0 as γ goes to infinity.
Recall that we assumed that lim
γ→∞
φ
1
γ = +∞. Thus, there exist ǫ 0 and a finite constant c 0
s.t. P-a.s for all ǫ ∈ 0, ǫ
], sup
r ≥0
A
3
r − Z
0,∞
π
1
dℓ
1
1 − wℓ
1
, r, µ
≤ c
φ
1
γ + φ
2
γ. 54
Using 53 and 54, we get that there exists a deterministic function R s.t. P-a.s
sup
r ≥0
φ
1
γ logK
ǫ
r, µ − α
1
vr ,
µ − Z
0,∞
π
1
dℓ
1
1 − wℓ
1
, r, µ
≤ Rǫ, where lim
ǫ→0
Rǫ = 0, thanks to 9 and 20.
The previous results allow us to compute the following limit. We keep the same notation as in Lemma 3.5.
Lemma 3.10. Let ϕ satisfying condition h
1
–h
3
. There exists a sub-sequence ǫ
j
, j ∈ N decreasing
to 0, s.t. P-a.s.
lim
j →∞
∞
Y
k= 1
K
ǫ
j
A
S
ǫ j k
, ρ
− S
ǫ k
= exp − Z
∞
du α
1
vu + Z
0,∞
π
1
dℓ 1 − wℓ, u, µ .
Proof. Notice that thanks to h
1
, the functions v and u, µ 7→ wℓ, u, µ are continuous and that
for r ≥ t, vr, µ = 0 and wℓ, r, µ = 1. The result is then a direct consequence of Corollary 3.8 and
Lemma 3.9.
3.6 Proof of Theorem 3.2
Now we can prove the special Markov property in the case lim
γ→∞
φ
1
γ = +∞. Let Z
∈ ˜ F
∞
non-negative such that E[Z] ∞. Let ϕ satisfying hypothesis of Theorem 3.2, h
1
– h
3
. We have, using notation of the previous sections E
Z exp
− X
i ∈I
ϕA
α
i
, ρ
α
i
−
, S
i
= lim
j →∞
E
Z exp −
∞
X
k= 1
ϕ
∗
A
S
ǫ j k
, ρ
− S
ǫ j k
, S
k ,
ǫ
j
= lim
j →∞
E
Z
∞
Y
k= 1
K
ǫ
j
A
S
ǫ j k
, ρ
− S
ǫ j k
= E Z
e
− R
∞
du α
1
vu , ˜
ρ
u
+ R
0,∞
π
1
dℓ 1−wℓ,u, ˜ ρ
u
, 1459
where we used Lemma 3.4 and dominated convergence for the first equality, Lemma 3.5 for the second equality, Lemma 3.10 and dominated convergence for the last equality. By monotone class
Theorem and monotonicity, we can remove hypothesis h
1
– h
3
. To ends the proof of the first part, notice that
R
t
du α
1
vu + R
0,∞
π
1
dℓ 1 − wℓ, u is ˜
F
∞
-measurable and so this is P- a.e. equal to the conditional expectation i.e. the left hand side term of 34.
4 Law of the pruned exploration process
Let ρ
be the exploration process of a Lévy process with Laplace exponent ψ
. The aim of this section is to prove Theorem 2.3.
4.1 A martingale problem for
˜ ρ
Let ˜ σ = inf{t 0, ˜
ρ
t
= 0}. In this section, we shall compute the law of the total mass process 〈 ˜
ρ
t ∧ ˜
σ
, 1 〉, t ≥ 0 under P
µ
= P
µ,0
, using martingale problem characterization. We will first show how a martingale problem for
ρ can be translated into a martingale problem for ˜ ρ, see also [1].
Unfortunately, we were not able to use standard techniques of random time change, as developed in Chapter 6 of [23] and used for Poisson snake in [7], mainly because t
−1
E
µ
[ f ρ
t
1
{m
t
=0}
] − f µ
may not have a limit as t goes down to 0, even for exponential functionals. Let F, K
∈ BM
f
R
+
be bounded. We suppose that N Z
σ
Kρ
s
ds
∞, that for any µ ∈ M
f
R
+
, E
∗ µ
Z
σ
Kρ
s
ds
∞ and that M
t
= F ρ
t ∧σ
− R
t ∧σ
K ρ
s
ds, for t ≥ 0, defines an F -martingale. In other words, if F belongs to the domain of the infinitesimal generator L of ρ, we
have K = L F . We will see in the proof of Corollary 4.2 that these assumptions on F and K are in
particular fulfilled for F
ν = e
−c〈ν,1〉
K ν = ψcF ν.
Notice that we have |M
t
| ≤ kF k
∞
+ Z
σ
Kρ
s
ds and thus E
∗ µ
sup
t ≥0
M
t
∞. Consequently, we can define for t ≥ 0,
N
t
= E
∗ µ
[M
C
t
| ˜ F
t
].
Proposition 4.1. The process N = N
t
, t ≥ 0 is an ˜
F -martingale. And we have the representation formula for N
t
: N
t
= F ˜ ρ
t ∧ ˜
σ
− Z
t ∧ ˜
σ
du ˜ K
˜ ρ
u
, 55
with ˜
K ν = Kν + α
1
N Z
σ
K[ ν, ρ
s
] ds
+ Z
0,∞
π
1
dℓ E
∗ ℓ
Z
σ
K[ ν, ρ
s
] ds
. 56
1460
Proof. Notice that N = N
t
, t ≥ 0 is an ˜
F -martingale. Indeed, we have for t, s ≥ 0, E
µ
[N
t+s
| ˜ F
t
] = E
µ
[E
µ
[M
C
t+s
| ˜ F
t+s
]| ˜ F
t
] = E
µ
[M
C
t+s
| ˜ F
t
] = E
µ
[E
µ
[M
C
t+s
|F
C
t
]| ˜ F
t
] = E
µ
[M
C
t
| ˜ F
t
], where we used the optional stopping time Theorem for the last equality. To compute E
µ
[M
C
t
| ˜ F
t
], we set N
′ t
= M
C
t
+ M
′ C
t
, where for u ≥ 0,
M
′ u
= Z
u ∧σ
K ρ
s
1
{m
s
6=0}
ds .
Recall that C = 0 P
µ
-a.s. by Corollary 2.2. In particular, we get N
′ t
= M
C
t
+ M
′ C
t
= F ρ
C
t
∧σ
− Z
C
t
∧σ
K ρ
s
1
{m
s
=0}
ds = F ˜
ρ
t ∧ ˜
σ
− Z
C
t
∧σ
K ρ
s
dA
s
= F ˜ ρ
t ∧ ˜
σ
− Z
t ∧ ˜
σ
K ˜
ρ
u
du, where we used the time change u = A
s
for the last equality. In particular, as ˜ σ is an ˜
F -stopping time, we get that the process N
′ t
, t ≥ 0 is ˜
F -adapted. Since N
t
= N
′ t
− E
µ
[M
′ C
t
| ˜ F
t
], we are left with the computation of E
µ
[M
′ C
t
| ˜ F
t
]. We keep the notations of Section 3. We consider
ρ
i
, m
i
, i ∈ I the excursions of the process ρ, m outside
{s, m
s
= 0} before σ and let α
i
, β
i
, i ∈ I be the corresponding interval excursions. In particular we can write
Z
C
t
∧σ
Kρ
s
1
{m
s
6=0}
ds = X
i ∈I
ΦA
α
i
, ρ
α
i
−
, ρ
i
, with
Φu, µ, ρ = 1
{ut}
Z
σρ
K[µ, ρ
s
] ds,
where σρ = inf{v 0; ρ
v
= 0}. We deduce from the second part of Theorem 3.2, that P
µ
-a.s. E
µ
Z
C
t
∧σ
Kρ
s
1
{m
s
6=0}
ds | ˜
F
∞
=
Z
˜ σ
1
{ut}
ˆ K
˜ ρ
u
du, 57
with, ˆ K
defined for ν ∈ M
f
R
+
by ˆ
K ν = α
1
N Z
σ
K[ν, ρ
s
] ds
+
Z
0,∞
π
1
dℓ E
∗ ℓ
Z
σ
K[ν, ρ
s
] ds
.
1461
Since E
µ
hR
C
t
∧σ
Kρ
s
1
{m
s
6=0}
ds i
≤ E
µ
hR
σ
Kρ
s
ds i
∞, we deduce from 57 that P
µ
-a.s. du
-a.e. 1
{u ˜ σ}
ˆ K
˜ ρ
u
is finite. We define ¯
K ∈ BM
f
R
+
for ν ∈ M
f
R
+
by ¯
K ν = α
1
N Z
σ
K[ ν, ρ
s
] ds
+ Z
0,∞
π
1
dℓ E
∗ ℓ
Z
σ
K[ ν, ρ
s
] ds
58 if ˆ
K ν ∞, or by ¯
K ν = 0 if ˆ
K ν = +∞. In particular, we have | ¯
K ν| ≤ ˆ
K ν and P
µ
-a.s. R
˜ σ
| ¯ K
˜ ρ
u
| du is finite. Using the special Markov property once again see 57, we get that P
µ
- a.s.,
E
µ
h M
′ C
t
| ˜ F
∞
i = E
µ
Z
C
t
∧σ
K ρ
s
1
{m
s
6=0}
ds | ˜
F
∞
=
Z
t ∧ ˜
σ
¯ K
˜ ρ
u
du. Finally, as N
t
= N
′ t
− E
µ
h M
′ C
t
| ˜ F
∞
i , this gives 55.
Corollary 4.2. Let µ ∈ M
f
R
+
. The law of the total mass process 〈 ˜ ρ
t
, 1 〉, t ≥ 0 under P
∗ µ,0
is the law of the total mass process of
ρ under P
∗ µ
. Proof.
Let X = X
t
, t ≥ 0 be under P
∗ x
a Lévy process with Laplace transform ψ started at x 0 and
stopped when it reached 0. Under P
µ
, the total mass process 〈ρ
t ∧σ
, 1 〉, t ≥ 0 is distributed as X un-
der P
∗ 〈µ,1〉
. Let c 0. From Lévy processes theory, we know that the process e
−cX
t
−ψc R
t
e
−cX
s
ds ,
for t ≥ 0 is a martingale. We deduce from the stopping time Theorem that M = M
t
, t ≥ 0 is an
F -martingale under P
µ
, where M
t
= F ρ
t ∧σ
− R
t ∧σ
K ρ
s
ds, with F, K ∈ BM
f
R
+
defined by F
ν = e
−c〈ν,1〉
for ν ∈ M
f
R
+
and K = ψcF . Notice K ≥ 0. We have by dominated convergence and monotone convergence.
e
−c〈µ,1〉
= lim
t →∞
E
µ
[M
t
] = E
µ
[e
−c〈ρ
σ
,1 〉
] − ψcE
µ
Z
σ
e
−c〈ρ
s
,1 〉
ds
.
This implies that, for any µ ∈ M
f
R
+
, E
µ
Z
σ
Kρ
s
ds
is finite. Using the Poisson representa- tion, see Proposition 1.9, it is easy to get that
N Z
σ
d t e
−c〈ρ
t
,1 〉
=
c ψc
. 59
In particular, it is also finite. From Proposition 4.1, we get that N = N
t
, t ≥ 0 is under P
µ
an ˜ F -martingale, where: for t ≥ 0,
N
t
= e
−c〈 ˜ ρ
t ∧ ˜
σ
,1 〉
− Z
t ∧ ˜
σ
˜ K
˜ ρ
u
du
1462
and ˜ K
given by 56. We can compute ˜ K
: ˜
K ν = ψc e
−c〈ν,1〉
1 + α
1
N Z
σ
e
−c〈ρ
s
,1 〉
ds
+ Z
0,∞
π
1
dℓ E
∗ ℓ
Z
σ
e
−c〈ρ
s
,1 〉
ds
= ψc e
−c〈ν,1〉
1 + α
1
c ψc
+ Z
0,∞
π
1
dℓ Z
ℓ
d r e
−cr
N Z
σ
e
−c〈ρ
s
,1 〉
ds
= e
−c〈ν,1〉
ψc + α
1
c + Z
0,∞
π
1
dℓ1 − e
−cℓ
= ψ c e
−c〈ν,1〉
, where we used 59 and the excursion decomposition for the second equality, and
ψ = ψ + φ
1
for the last one.
Thus, the process N
t
, t ≥ 0 with for t ≥ 0
N
t
= e
−c〈 ˜ ρ
t ∧ ˜
σ
,1 〉
−ψ c
Z
t ∧ ˜
σ
e
−c〈 ˜ ρ
u
,1 〉
du is under P
µ
an ˜ F -martingale.
Notice that ˜ σ = inf{s ≥ 0; 〈 ˜
ρ
s
, 1 〉 = 0}. Let X
= X
t
, t ≥ 0 be under P
∗ x
a Lévy process with Laplace transform
ψ started at x
0 and stopped when it reached 0. The two non-negative càd- làg processes
〈 ˜ ρ
t ∧ ˜
σ
, 1 〉, t ≥ 0 and X
solves the martingale problem: for any c ≥ 0, the process
defined for t ≥ 0 by
e
−cY
t ∧σ′
−ψ c
Z
t ∧σ
′
e
−cY
s
ds ,
where σ
′
= inf{s ≥ 0; Y
s
≤ 0}, is a martingale. From Corollary 4.4.4 in [23], we deduce that those two processes have the same distribution. To finish the proof, notice that the total mass process of
ρ under P
∗ µ
is distributed as X under P
∗ 〈µ,1〉
.
4.2 Identification of the law of
