Consequently ϕ
λ
t = 1 + 2 λ t
−N2
. Therefore,
E[exp −λ L
T
1
] = 1 + 2 λ
−N2
= E[exp−λ R
2 N
1], which proves the desired result.
4 About the process
R
t
R
2 N
s ds, t ≥ 0
4.1 A class of Sato processes
Let ℓ
t
, t ≥ 0 be the local time in 0 of a linear Brownian motion B
t
, t ≥ 0 starting from 0. We
denote, as usual, by τ
t
, t ≥ 0 the inverse of this local time:
τ
t
= inf{s ≥ 0; ℓ
s
t}.
Proposition 4.1. Let f x, u be a Borel function on R
+
× R
+
such that ∀t 0
Z Z
R
+
×[0,t]
| f x, u| dx du ∞. 2
Then the process A
f
defined by: A
f t
= Z
τ
t
f |B
s
|, ℓ
s
ds, t
≥ 0 is an integrable additive process. Furthermore,
E[A
f t
] = 2 Z Z
R
+
×[0,t]
f x, u dx du.
Proof
Assume first that f is nonnegative. Then, A
f t
= X
≤u≤t
Z
τ
u
τ
u −
f |B
s
|, u ds. By the theory of excursions Revuz-Yor [22, Chapter XII, Proposition 1.10] we have
E[A
f t
] = Z
t
du Z
nd ǫ
Z
V ǫ
ds f |ǫ
s
|, u
942
where n denotes the Itô measure of Brownian excursions and V ǫ denotes the life time of the
excursion ǫ. The entrance law under n is given by:
n ǫ
s
∈ dx; s V ǫ = 2πs
3 −12
|x| exp−x
2
2s dx. Therefore
E[A
f t
] = 2 Z
t
du Z
∞
dx f x, u. The additivity of the process A
f
follows easily from the fact that, for any t ≥ 0, B
τ
t
+s
, s ≥ 0 is a
Brownian motion starting from 0, which is independent of B
τ
t
where B
u
is the natural filtration of B.
Corollary 4.1.1. We assume that f is a Borel function on R
+
× R
+
satisfying 2 and which is m- homogeneous for m
−2, meaning that ∀a 0, ∀x, u ∈ R
+
× R
+
, f a x, au = a
m
f x, u. Then the process A
f
is a m + 2-Sato process.
Proof
This is a direct consequence of the scaling property of Brownian motion.
4.2 A particular case
Let N 0. We denote by A
N
the process A
f
with f x, u =
N
2
4 1
x≤
2 N
u
. By Proposition 4.1, A
N t
is an integrable process and E[A
N t
] = N t
2
2 .
We now consider the process Y
N
defined by Y
N
t = Z
t
R
2 N
s ds, t
≥ 0.
Theorem 4.2. The process A
N
is a 2-Sato process and Y
N
t
1.d
= A
N t
. 943
Proof
It is a direct consequence of Corollary 4.1.1 that A
N
is a 2-Sato process. By Mansuy-Yor [19, Theorem 3.4, p.38], the following extension of the Ray-Knight theorem holds:
For any u 0,
L
a −2uN
τ
u
, 0 ≤ a ≤ 2uN
d
= R
2 N
a, 0 ≤ a ≤ 2uN where L
x t
denotes the local time of the semi-martingale |B
s
| −
2 N
ℓ
s
, s ≥ 0 in x at time t.
We remark that s
∈ [0, τ
t
] =⇒ |B
s
| − 2
N ℓ
s
≥ − 2t
N .
Therefore, the occupation times formula entails: A
N t
= N
2
4 Z
−2tN
L
x τ
t
dx = N
2
4 Z
2t N
L
x −2tN
τ
t
dx. Thus, by the above mentioned extension of the Ray-Knight theorem,
A
N t
1.d
= N
2
4 Z
2t N
R
2 N
s ds .
The scaling property of R
N
also yields the identity in law: A
N t
1.d
= Z
t
R
2 N
s ds
, and the result follows from the definition of Y
N
.
We may now apply Proposition 2.3 to get:
Corollary 4.2.1. The process V
N
defined by: V
N
t = Y
N
t − N t
2
2 ,
t ≥ 0
is a PCOC and an associated martingale is M
N
defined by: M
N
t = A
N t
− N t
2
2 ,
t ≥ 0.
Moreover, M
N
is a centered 2-Sato process.
4.3 Representation of A
