Bernard Roynette

Pierre Vallois

Marc Yor

PENALIZING A BES(d) PROCESS (0 < d < 2) WITH A FUNCTION OF ITS LOCAL TIME, V

Keywords: penalization, Bessel process, local time, AMS 2000 subject classifications : 60 B 10, 60

We describe the limit laws, as t → ∞, of a Bessel process (R s , s ≤ t) of dimension d ∈ (0, 2) penalized by an integrable function of its local time L t at 0, thus extending our previous work of this kind, relative to Brownian motion.

1 Introduction 1) Let (Ω, (X t ) t≥0 , (F t ) t≥0 , F ∞ , P 0 ) denote the canonical real-valued Brownian motion, starting from 0. We denote by (L t ) t≥0 its local time at 0. Let h : R + → R + be a Borel function such that :

∞ 0 h(x)dx = 1. Define H(x) = x 0 h(y)dy , x ≥ 0 , (1.1)
the primitive of h such that H(0) = 0 . For any t ≥ 0 , we introduce the probability P (t) 0 on F t , which is defined by :

P (t) 0 (Λ t ) = E 0 (1 Λt h(L t )) E 0 [h(L t )] , Λ t ∈ F t . (1.2)
We have shown, in [START_REF] Roynette | Limiting laws associated with Brownian motion perturbed by its maximum, minimum and local time, II[END_REF] that the limit, as t → ∞ , of P (t) 0 (Λ s ) , for Λ s ∈ F s , and s fixed, exists :

Q (h) 0 (Λ s ) := lim t→∞ P (t) 0 (Λ s ) (s ≥ 0, Λ s ∈ F s ). (1.3)
This is a kind of "Brownian Gibbs measure", which induces a probability on (Ω, F ∞ ); in [START_REF] Roynette | Limiting laws associated with Brownian motion perturbed by its maximum, minimum and local time, II[END_REF], we described precisely the process (X t ) t≥0 under Q (h) 0 ; the pair (X t , L t ) t≥0 is Markov under Q (h) 0 , while (X t ) t≥0 is not, in general, Markov on its own.

2) The aim of the present work is to extend the above result for a d-dimensional Bessel process, 0 < d < 2. Denote by ν = d 2 -1 the index of this Bessel process, and let α = -ν = 1 -

d 2 ∈]0, 1[.
More precisely, let Ω + , (R t ) t≥0 , (F t ) t≥0 , F ∞ , (P

(-α) r
) r≥0 denote the canonical Bessel process of dimension d, or index ν = -α, with α ∈]0, 1[. Ω + denotes the set of continuous functions from R + to R + , (R t , t ≥ 0) is the coordinate process on Ω + , and (F t , t ≥ 0) its natural filtration. Finally, we denote : F ∞ = ∨ s≥0 F s . The probability P (-α) r makes the coordinate process (R t , t ≥ 0) a Bessel process with index (-α), starting from r. We denote by (L x t ; t ≥ 0, x ∈ R + ) the jointly continuous family of local times of the process (R t , t ≥ 0). We choose the normalization of this family such that (R 2α t -L 0 t , t ≥ 0) is a martingale. We note simply (L t ) t≥0 for (L 0 t ) t≥0 , and we consider a probability density h : R + → R + . Similarly as in (1.3), we are interested in the limit, as t → ∞, of :

P (t) 0 (Λ s ) = E (-α) 0 (1 Λs h(L t )) E (-α) 0 (h(L t ))
, Λ s ∈ F s ; s fixed. (1.4) Since, throughout this paper, the process of reference shall be the d-dimensional Bessel process with index (-α), we shall almost never again mention (-α) in our symbols, e.g., we shall write E 0 for E (-α) 0

. We shall prove : Theorem 1.1 Let h as in 1).

1. For every s ≥ 0, and Λ s ∈ F s ,

Q (h) 0 (Λ s ) := lim t→∞ E 0 (1 Λs h(L t ))
E 0 (h(L t )) exists.

(1.5)

Q

(h) 0 satisfies :

Q (h) 0 (Λ s ) = E 0 (1 Λs M h s ), (1.6)
with

M h s := h(L s )R 2α s + 1 -H(L s ). (1.7)
The process (M h s ) s≥0 is a ((F s ), P 0 ) positive martingale, which converges to 0, as s → ∞. In particular, it is not a uniformly integrable martingale.

The formula (1.6) induces a probability

Q (h) 0 on (Ω + , F ∞ ). Under Q (h)
0 , the canonical process (R t , t ≥ 0) satisfies :

(a) The random variable L ∞ is finite a.s., and it admits h as its probability density.

(b) Let g = sup{t ≥ 0 : R t = 0}. Then, Q (h) 0 (0 < g < ∞) = 1. (c) i. The two processes (R t , t ≤ g) and (R g+t , t ≥ 0) are independent;

ii. The process (R g+t , t ≥ 0) is a Bessel process with dimension (4d), starting from 0;

iii. Conditionally on L ∞ = l, the process (R t , t ≤ g) is a Bessel process of dimension d, starting from 0, stopped at τ l := inf{t > 0 : L t > l}.

Let :

A t := 4α 2 t 0 R 2(2α-1)

s ds t ≥ 0 , (1.8)
and denote its inverse by : ρ(u) := inf{t ≥ 0 : A t > u}.

(1.9)

Then, under Q (h) 0 , the process (R 2α ρu + L ρu , u ≥ 0) is a 3-dimensional Bessel process, starting from 0, which is independent from the random variable L ∞ .

Remark 1. [START_REF] Chaumont | Exercises in probability[END_REF] 1. We now remark that, for d = 1, i.e : α = 1/2, part 4. of Theorem 1.1 may be presented as follows :

(R t + L t , t ≥ 0) is a 3-dimensional Bessel process, independent from L ∞ .

(1.10)

2. Via Lévy's theorem : if (B t , t ≥ 0) denotes a Brownian motion, starting from 0, and if : S B t = sup s≤t B s , then the two processes (S B t -B t , S B t ; t ≥ 0) and (|B t |, L t ; t ≥ 0) have the same law , the result (1.10) has already been obtained in [START_REF] Roynette | Some extensions of Pitman's and Ray-Knight's theorems. for penalized Brownian motions and their local times, IV[END_REF] : thus, point 4. of Theorem 1.1 appears as a generalization of Pitman's theorem which asserts that :

(2S B t -B t ; t ≥ 0) (d) ≡ (|B t | + L t , t ≥ 0). (1.11)
is a 3-dimensional Bessel process.

3) Just as we did in [START_REF] Roynette | Limiting laws for long Brownian bridges perturbed by their one-sided maximum, III. Period[END_REF] concerning the 1-dimensional case, the above Theorem 1.1 invites to study the penalization with a function of the local time, not for the Bessel process itself, but for its "long bridges". Precisely, we shall be interested to show the existence of the limit, as t → ∞, of :

P 0 (Λ s |L t = y), y ≥ 0, Λ s ∈ F s , (1.12)
and even of :

P 0 (Λ s |R t = a, L t = y), a ≥ 0, y ≥ 0, Λ s ∈ F s . (1.13)
We obtain the following :

Theorem 1.3 1. The limit Q (y) 0 (Λ s ) := lim t→∞ P 0 (Λ s |L t = y) (1.14)
(with Λ s ∈ F s ) exists and satisfies :

Q (y) 0 (Λ s ) = p Ls (y)E 0 [1 Λs R 2α s |L s = y] + E 0 [1 Λs 1 (Ls<y) ] (1.15)
where p Ls is the density of L s .

2. The preceding formula (1.15) induces a probability Q (y) 0 on (Ω, F ∞ ). The probability Q (h) 0 defined in Theorem 1.1 admits the following disintegration :

Q (h) 0 (•) = ∞ 0 h(y)Q (y) 0 (•)dy.
(1.16)

Consequently, for any Λ ∈ F ∞ :

Q (h) 0 (Λ|L ∞ = y) = Q (y) 0 (Λ).
(1.17)

Thus, the conditional law of Q (h) 0

given L ∞ = y does not depend on h.

3. For every s ≥ 0, Λ s ∈ F s and every x, y ≥ 0,

Q (x,y) 0 (Λ s ) := lim t→∞ P 0 (Λ s |R t = x, L t = y) (1.18)
exists, and satisfies :

Q (x,y) 0 (Λ s ) = x x + yx 1-2α p Ls (y)E 0 (1 Λs R 2α s |L s = y) + 1 x + yx 1-2α E 0 1 Λs 1 (Ls<y) x + (y -L s + R 2α s )x 1-2α . (1.19)
4. For every x, y ≥ 0,

Q (x,y) 0 (•) = x x + yx 1-2α Q (y) 0 (•) + x 1-2α x + yx 1-2α y 0 Q (z) 0 (•)dz.
(1.20)

Note that formula (1.20) simplifies, in the case α = 1/2, to yield formula (1.13) of Theorem 1.3 in [START_REF] Roynette | Limiting laws for long Brownian bridges perturbed by their one-sided maximum, III. Period[END_REF] (via Lévy's Theorem) :

Q (x,y) 0 (•) = x x + y Q (y) 0 (•) + 1 x + y y 0 Q (z) 0 (•)dz.
(1.21)

4)

As a Corollary of Theorem 1.3, we now present Theorem 1.4, which describes the penalization of "long Bessel bridges" by an integrable function of their local times at 0 (see formula (1.23) below).

Theorem 1.4 Let Ω + , (R t ) t≥0 , (F t ) t≥0 , F ∞ , P 0 denote the canonical Bessel process starting from 0, with dimension d = 2(1 -α), 0 < α < 1. 1. Let h : R + → R + be a Borel function such that ∞ 0 h(y)dy = 1. Denote, for x > 0, h x := ∞ 0 h(y)(x + yx 1-2α )dy, (1.22)
assumed to be finite, and h * x = 1/h x . Then, for every s > 0, and Λ s ∈ F s ,

lim t→∞ E 0 ( 1 Λs h(L t )|R t = x E 0 h(L t )|R t = x exists and is equal to Q (hx) 0 (Λ s ), (1.23)
where :

h x (y) = h * x {xh(y) + x 1-2α (1 -H(y))} (y ≥ 0). (1.24)
2. Let f : R + × R + → R + be a Borel function such that :

f := R+×R+ f (x, y)(x + yx 1-2α )dx dy < ∞. (1.25)
Then, for every s ≥ 0 , and Λ s ∈ F s ,

lim t→∞ E 0 1 Λs f (R t , L t ) E 0 f (R t , L t )
exists and is equal to

Q ( f ) 0 (Λ s ), (1.26)
with :

f (y) := f * ∞ 0 xf (x, y)dx + ∞ 0 x 1-2α dx ∞ y f (x, z)dz (1.27)
and f * = 1/f . Note that, for both points 1. and 2. of Theorem 1.4, the main properties of the canonical process (R t ) t≥0 under the limit probabilities

Q (hx) 0 and Q ( f ) 0
are given by Theorem 1.1 : it suffices, in this Theorem 1.1, to replace h resp. by h a and f and to note that :

∞ 0 h a (y)dy = ∞ 0 f (y)dy = 1 .
5) Point 2. of Theorem 1.5 invites to study the penalization of (R t ) t≥0 by a function of L t and R t which is not integrable, i.e. which does not satisfy (1.25). This led us to the following :

Theorem 1.5 Let λ > 0, and h : R + → R + a Borel function such that :

∞ 0 h(y)dy < ∞ and ∞ 0 h(y)e -σ λ y dy = 1 , (1.28)
with

σ λ := λ 2 2α Γ(1 -α) Γ(1 + α) .
(1.29)

1. For every s ≥ 0 , and

Λ s ∈ F s , lim t→∞ E 0 [1 Λs h(L t ) exp(λR t )] E 0 [h(L t ) exp(λR t )] exists (1.30)
and is equal to :

Q (λ, h) 0 (Λ s ) := E 0 1 Λs M λ, h s (1.31) with M λ, h s := e -λ 2 s/2 R α s h(L s ) 2 λ α Γ(1 + α)I α (λR s ) + 1 -H(L s ) 2 λ -α Γ(1 -α)I -α (λR s ) , (1.32)
where I ν denotes the modified Bessel function with index ν (cf [START_REF] Lebedev | Special functions and their applications[END_REF]), and

h(y) := h(y) -σ λ e σ λ y ∞ y h(z)e -σ λ z dz, (1.33) 1 -H(y) := e σ λ y ∞ y h(z)e -σ λ z dz = ∞ y h(z)dz. (1.34) 2. (M λ, h s , s ≥ 0
) is a positive martingale, which tends to 0 a.s. as s → ∞.

3. Formula (1.31) induces a probability Q (λ, h) 0 on the canonical space (Ω + , F ∞ ), with respect to which the canonical process (R t , t ≥ 0) satisfies : (a) L ∞ is finite a.s. and its distribution function is :

Q (λ, h) 0 (L ∞ < c) = 1 -1 -H(c) e -cσ λ (1.35)
with σ λ given by (1.29).

(b) Let g = inf {t ≥ 0 : L t = L ∞ } = sup {t ≥ 0 : R t = 0}. Then :

Q (λ, h) 0 (0 < g < ∞) = 1.
(1.36) (c) i. The processes (R t , t < g) and (R g+t , t ≥ 0) are independent.

ii. The process (R g+t , t ≥ 0) is a diffusion process starting from 0, whose infinitesimal generator L ↑ satisfies :

L ↑ f (r) = 1 2 f ′′ (r) + 1 -2α 2r + λ I α-1 (λr) I α (λr) f ′ (r).
(1.37)

iii. Conditionally on L ∞ = l, the process (R t , t ≤ g) is a diffusion process starting from 0, whose infinitesimal generator L ↓ satisfies :

L ↓ f (r) = 1 2 f ′′ (r) + 1 -2α 2r -λ K α-1 (λr) K α (λr) f ′ (r) (1.38)
stopped when its local time at 0 reaches level l.

Remark 1.6 1. Let h 0 : R + → R + a Borel function such that ∞ 0 h 0 (y)dy < ∞. Note that h = h 0 /c verifies (1.28) where c = ∞ 0 h(y)e -σ λ y dy.
2. It is not difficult to check that, as λ → 0, Theorem 1.5 yields precisely Theorem 1.1, because

I ν (z) ∼ z→0 1 Γ(ν + 1) z 2 ν and K ν (z) ∼ z→0 1 2 Γ(ν) z 2 -ν
(cf [START_REF] Lebedev | Special functions and their applications[END_REF]).

3. Recall that the diffusions whose infinitesimal generators L ↑ and L ↓ are given in (1.37) and (1.38) are the Bessel processes with dimension d = 2(1α), and drift λ ↑ and λ ↓ respectively. These processes have been studied by Watanabe [START_REF] Watanabe | On time inversion of one-dimensional diffusion processes[END_REF] and Pitman-Yor [START_REF] Pitman | Bessel processes and infinitely divisible laws[END_REF]. They play an important role in , [START_REF] Matsumoto | An analogue of Pitman's 2M -X theorem for exponential Wiener functionals. II. The role of the generalized inverse Gaussian laws[END_REF]).

5)

Organization of the paper.

• In Section 2, we define precisely the normalization of the continuous family of the local times (L x t ; t ≥ 0; x ≥ 0) of the Bessel process (R t , t ≥ 0) of dimension d ∈]0, 2[, which we use throughout this paper.

• In Section 3, we prove Theorem 1.1.

• In Section 4 we prove Theorem 1.3, and we deduce Theorem 1.4 from Theorem 1.3 .

• Finally, Section 5 is devoted to the proof of Theorem 1.5.

6)

An overview of some penalization results. In our paper [START_REF] Roynette | Pénalisations et extensions du théorème de Pitman, relatives au mouvement brownien et à son maximum unilatère[END_REF], we propose a survey-without proofs-of most of the results obtained in our previous works [START_REF] Roynette | Limiting laws for long Brownian bridges perturbed by their one-sided maximum, III. Period[END_REF], [START_REF] Roynette | Limiting laws associated with Brownian motion perturbed by its maximum, minimum and local time, II[END_REF], [START_REF] Roynette | Some extensions of Pitman's and Ray-Knight's theorems. for penalized Brownian motions and their local times, IV[END_REF] on the subject. Acknowledgment : We thank the referee for a detailed list of suggestions which helped us to improve our paper.

2 Definition and properties of the local time at 0 1) The Bessel process (R t , t ≥ 0), with dimension d = 2(1α) ∈ (0, 2) which is being considered throughout this paper, is an R + -valued diffusion whose infinitesimal generator L is defined as :

Lf (r) = 1 2 d 2 f dr 2 + 1 -2α 2r df dr , (2.1)
on the domain

D = f : R + → R ; Lf ∈ C b (R + ); lim r→0 r 1-2α f ′ (r) = 0 (2.2)
2) The normalization we shall use for the local time at 0 , (L t , t ≥ 0) of (R t , t ≥ 0) is such that :

(N t := R 2α t -L t , t ≥ 0) is a martingale (2.3)
We note that the bracket of (N t ) equals :

A t :=< N > t = 4α 2 t 0 R 2(2α-1) s ds, (2.4)
and that there exists a reflecting Brownian motion (γ u , u ≥ 0) such that

R 2α t = γ At ; L t = ℓ At , (2.5)
where (ℓ u , u ≥ 0) is the local time at 0 of γ, chosen such that :

(γ u -ℓ u , u ≥ 0) is a (F γ u := σ{γ s , s ≤ u}, u ≥ 0) martingale. (2.6)
Note that the finiteness of A t , especially for α < 1/2, follows from :

E 0 R 2(2α-1) s = 2 α Γ(1 -α) s α-1 ∞ 0 x 2α-1 e -x 2 2s dx = 2 2α-1 Γ(α) Γ(1 -α) s 2α-1 , which implies that E 0 [A t ] < ∞.
3) With this normalization of (L t , t ≥ 0) (cf [START_REF] Donati-Martin | On constants related to the choice of the local time at 0, and the corresponding Itô measure for Bessel processes with dimension d = 2(1α), 0 < α < 1[END_REF]) , the occupation density formula writes :

t 0 g(R s )ds = 1 α ∞ 0 g(x)L x t x 1-2α dx , (2.7)
for every Borel function g : R + → R + , and {L x t } a jointly continuous family of local times, such that L 0 t = L t . 4) Under P 0 , the variable L t is distributed as

t α L 1 , (2.8)
and the law of L 1 is the Mittag-Leffler distribution of index α (see for instance [START_REF] Chaumont | Exercises in probability[END_REF]; for details see [START_REF] Donati-Martin | On constants related to the choice of the local time at 0, and the corresponding Itô measure for Bessel processes with dimension d = 2(1α), 0 < α < 1[END_REF] and [START_REF] Lukacs | Characteristic functions[END_REF], p. 142), with density p L1 :

p L1 (l) = 1 παl ∞ k=1 (-1) k+1 Γ(αk + 1) k! Γ(1 -α) Γ(1 + α) 2 -α l k sin(kπα) (l ≥ 0). (2.9)
In particular it satisfies :

p L1 (0) = lim l→0 p L1 (l) = 2 -α Γ(1 + α) .
(2.10)

5)

Define the right continuous inverse of L :

τ l := inf {t ≥ 0 : L t > l} . (2.11)
Then, (τ l , l ≥ 0) is a stable subordinator with index α; more precisely, its Laplace transform is given by :

E [exp(-λτ l )] = exp -l Γ(1 -α) Γ(1 + α) 2 -α λ α λ, l ≥ 0.
(2.12)

6) Let T 0 = inf {t ≥ 0 : R t = 0} denote the first hitting time of 0 for the process (R t , t ≥ 0).

Then, T 0 is, under P r , distributed as :

T 0 (d) = r 2 /2γ α , (2.13)
where γ α is a standard gamma variable with index α. Consequently,

P r (T 0 ∈ dt) = 2 -α Γ(α) t -α-1 r 2α exp(- r 2 2t ) 1 [0,∞[ (t)dt, (2.14) E r e -λ 2 2 T0 = 2 Γ(α) λr 2 α K α (λr).
(2.15) Identity (2.15) is found in [START_REF] Kent | Some probabilistic properties of Bessel functions[END_REF], see also Proposition (2.3) in [START_REF] Pitman | Bessel processes and infinitely divisible laws[END_REF]. For (2.14), which also extends to α = 1 (that is : d = 0, when (R t ) is the 0-dimensional Bessel process), see [START_REF] Getoor | The Brownian escape process[END_REF], and e.g ( [START_REF] Revuz | Continuous martingales and Brownian motion, volume 293 of Grundlehren der Mathematischen Wissenschaften[END_REF], ex 4.16, p321). In [START_REF] Donati-Martin | On constants related to the choice of the local time at 0, and the corresponding Itô measure for Bessel processes with dimension d = 2(1α), 0 < α < 1[END_REF], the reader will find a more detailed discussion of the various normalisations of the local time process (L t ) at level 0 for a Bessel process of dimension d ∈ (0, 2) which have been used in the literature. The results presented in this section may be considered as standard knowledge; see, e.g. Borodin-Salminen [START_REF] Borodin | Handbook of Brownian motion-facts and formulae[END_REF] for a more general presentation of diffusion local times. 7) In Section 5, the role of the following martingale will be crucial :

M λ 2 2 ↓ t := R α t K α (λR t ) exp σ λ L t - λ 2 t 2 , t ≥ 0, (2.16)
where σ λ is defined in (1.29). That this process is indeed a martingale follows from the computation relative to a general diffusion (R t ), its local time (L t ), and inverse local time (τ l ) :

E 0 e -µτ l F t = e -µt f (R t , l -L t ), on {t < τ l } = {L t < l}, (2.17)
where f (r, λ) = E r A(λ) , with A(λ) = e -µτ λ . Using the strong Markov property we obtain :

f (r, λ) = E r e -µT0(R) e -λψ(µ) ,
where ψ(µ) denotes the Lévy exponent for (τ l ).

Using the Laplace transform given in (2.15), for µ = λ 2 /2, and the fact that : ψ λ 2 2 = σ λ in this particular case (cf (2.12) above), we get :

R α t K α (λR t ) exp σ λ L t - λ 2 t 2 = C λ e σ λ l E 0 e -µτ l F t , on {t < τ l }, (2.18)
where C λ is a positive constant. Now property (2.16) is a direct consequence of (2.18) together with the following calculations :

E 0 M λ 2 2 ↓ t 1 Λs = lim l→∞ E 0 M λ 2 2 ↓ t 1 Λs 1 {s<τ l } = lim l→∞ E 0 C λ e σ λ l E 0 e -µτ l F s 1 Λs 1 {s<τ l } = lim l→∞ E 0 M λ 2 2 ↓ s 1 Λs 1 {s<τ l } = E 0 M λ 2 2 ↓ s 1 Λs ,
for any s ≤ t and Λ s ∈ F s .

3 Proof of Theorem 1.1

Proof of Theorem 1.1 The proof of Theorem 1.1 will be divided into eleven steps. 1) We prove the existence of : lim t→∞ P (t) 0 (Λ s ). Let s ≥ 0, and Λ s ∈ F s . By conditioning h(L t ) with respect to F s , we get :

E 0 1 Λs h(L t ) E 0 [h(L t )] = E 0 1 Λs θ(R s , L s , t -s) θ(0, 0, t) , (3.1)
with θ(r, y, u)

:= E r [h(y + L u )], r, y, u ≥ 0. (3.2)
Thus we are led to estimate θ(r, y, u) when u tends to +∞. We denote by T 0 the first hitting time of 0 by the process (R t , t ≥ 0) :

T 0 := inf{t ≥ 0, R t = 0}. (3.3)
Thus, we obtain :

θ(r, y, t) = θ 1 (r, y, t) + θ 2 (r, y, t), (3.4)
with :

θ 1 (r, y, t) = h(y)P r (T 0 > t), θ 2 (r, y, t) = E r 1 {T0<t} h(y + L T0+(t-T0) ) . (3.5)
We examine separately the two terms θ 1 (r, y, t) and θ 2 (r, y, t) featured in (3.5). From (2.13), the first term θ 1 (r, y, t) is equal to :

θ 1 (r, y, t) = h(y)P γ α < r 2 2t = h(y) Γ(α) r 2 2t 0 x α-1 e -x dx ∼ t→∞ h(y) Γ(α + 1) r 2 2t α .
(3.6)

As to the second term θ 2 (r, y, t), we find it to be equal, thanks to the scaling property (2.8), and after conditioning with respect to F T0 , to :

θ 2 (r, y, t) = E r [1 {T0<t} θ 3 (y, t -T 0 )], with θ 3 (y, u) = E 0 h(y + u α L 1 )) .
Hence, denoting by p L1 the density of L 1 , under P 0 :

θ 2 (r, y, t) = ∞ 0 E r 1 {T0<t} h(y + (t -T 0 ) α x) p L1 (x)dx,
so that, after making the change of variable : (t -T 0 ) α x = z, we get :

θ 2 (r, y, t) = E r 1 {T0<t} 1 (t -T 0 ) α ∞ 0 h(y + z)p L1 z (t -T 0 ) α dz . (3.7)
Consequently (2.10) implies :

θ 2 (r, y, t) ∼ t→∞ p L1 (0) t α ∞ 0 h(y + z)dz = 2 -α t α Γ(1 + α) [1 -H(y)]. (3.8)
Bringing together (3.6), (3.8) and (3.4), we get :

θ(r, y, t) ∼ t→∞ 1 t α 2 -α Γ(α + 1) h(y)r 2α + 1 -H(y) . (3.9)
We then deduce from (3.1) and (3.9) :

lim t→∞ E 0 1 Λs h(L t ) E 0 [h(L t )] = E 0 1 Λs h(L s )R 2α s + 1 -H(L s ) . (3.10)
We note that, in (3.10), exchanging the order of taking either the limit or the expectation does not make any problem, since it is justified by Lebesgue's dominated convergence theorem, once it has been noted that, with the help of (3.6) and (3.7) :

t α θ(r, y, t) ≤ C[h(y) + 1]r 2α . 2) We now show that (M h s := h(L s )R 2α s + 1 -H(L s ), s ≥ 0) is a martingale. For h in C 1 ,
it easily follows from Itô's formula using (2.3), that (M h s , s ≥ 0) is a local martingale. Moreover it writes :

M h s = 1 + s 0 h(L u )dN u , s ≥ 0.
Now, to obtain the general case, it remains to apply the monotone class theorem. We might also have used a balayage argument, see e.g. [START_REF] Revuz | Continuous martingales and Brownian motion, volume 293 of Grundlehren der Mathematischen Wissenschaften[END_REF], Chap. VI. Thus, in particular,

M h t = 1 + t 0 h(L s )dN s , t ≥ 0, (3.11)
where (N t , t ≥ 0) is the martingale defined by (2.3) is a local martingale. Since M h t ≥ 0, for any t ≥ 0, (M h t ) is a positive supermartingale. In order to prove that (M h t , t ≥ 0) is a martingale, it suffices to show :

E 0 (M h t ) = 1, for every t ≥ 0. (3.12) Now, for n ∈ N, let h n (x) = (h(x) ∧ n)1 (x≤n) . It is clear that (M hn t
) is a martingale, therefore (3.12) implies that : E 0 M hn t = 1. Then, with the help of Beppo-Levi's Theorem, we obtain :

1 = lim n→∞ E 0 h n (L t )R 2α t + (1 -H n (L t ) = E 0 h(L t )R 2α t + 1 -H(L t ) = E 0 [M h t ].
3) We now prove that : M h t → 0 as t → ∞. Since (M h t , t ≥ 0) is a positive martingale, it converges a.s. as t → ∞. Let τ l = inf {s > 0 : L s > l} denote the inverse local time. Then :

M h τ l = h(L τ l )R 2α τ l + 1 -H(L τ l ) = 1 -H(l) → 0, as l → ∞. (3.13) Hence M h t → t→∞ 0 a.s. In particular, the martingale (M h t , t ≥ 0) is not uniformly integrable. 4) We now establish that Q (h) 0 (L ∞ ∈ dl) = h(l)dl .
Indeed, for every t ≥ 0, using (1.6), Doob's optional stopping theorem and (3.13), we have :

Q (h) 0 (L t > c) = Q (h) 0 (t > τ c ) = E 0 [1 {τc<t} M h t ] = E 0 [1 {τc<t} M h τc ] = 1 -H(c) P 0 (t > τ c
). Consequently, letting t → +∞, we obtain :

Q (h) 0 (L ∞ > c) = 1 -H(c).

5) An auxiliary result.

In the sequel, we shall use a general result about continuous positive martingales, which is stated and proven in [START_REF] Roynette | Limiting laws associated with Brownian motion perturbed by its maximum, minimum and local time, II[END_REF], and which we shall then apply to M = M h . Thus, we present this result without proof.

Proposition 3.1 (Theorem 4.2 in [19])

Let Ω, (F t ) t≥0 , F ∞ , P denote a given filtered probability space, and consider a strictly positive continuous martingale (M t ), with respect to (F t ) t≥0 , P , such that M 0 = 1 and M ∞ = 0 a.s. We then define the probability Q on (Ω, F ∞ ) via :

Q(Λ t ) = E 1 Λt M t t ≥ 0, Λ t ∈ F t . (3.14)
We also define :

M t = inf s≤t M s . (3.15)
Then, under Q, the following holds :

1. M ∞ is uniformly distributed on [0, 1]. 2. Let g := sup {t ≥ 0, M t = M ∞ }. Then, Q(0 < g < ∞) = 1. (3.16)
3. Let :

Z t = Q(g > t|F t ). (3.17)
Then :

(a) Z t = M t /M t . (b) (Z t , t ≥ 0) is a (F t ) t≥0
, Q positive supermartingale with additive decomposition :

Z t = 1 - t 0 M u M 2 u d M u + ln(M t ). (3.18)
where

M t := M t - t 0 d < M > u M u
is the martingale part of (M t ) under Q, from Girsanov's theorem.

6)

We now remark that :

M h t = 1 -H(L t ). (3.19) Indeed, M h s = h(L s )R 2α s + 1 -H(L s ) ≥ 1 -H(L s ) ≥ 1 -H(L t ), for any 0 ≤ s ≤ t. Moreover M h gt = 1 -H(L t )
where g t = sup{s ≤ t; R s = 0}. This implies (3.19). We also note that (3.19), together with point 1. of Proposition 3.1, allows to rediscover the fact, obtained in 4), that Q

(h) 0 (L ∞ ∈ dl) = h(l)dl. Indeed, we get : Q (h) 0 (L ∞ < c) = Q (h) 0 H(L ∞ ) < H(c) = Q (h) 0 1 -H(L ∞ ) > 1 -H(c) = Q (h) 0 M h ∞ > 1 -H(c) = H(c).
7) Another definition of g. Let g be defined as in point 2. of Proposition 3.1, but we now replace M by M h , ie :

g = sup t ≥ 0; M h t = M h ∞ . (3.20)
Then, under

Q (h) 0 : g = sup {s ≥ 0, R s = 0} = inf {s ≥ 0, L s = L ∞ } . (3.21)
Indeed, (3.21) follows from (3.19) and the fact that M h t = 1 -H(L t ) is constant after g (hence, so is L t ). 8) A preliminary step to prove point 3. (c) of Theorem 1.1. We shall use the technique of progressive enlargement of filtrations (see [START_REF] Jeulin | Semi-martingales et grossissement d'une filtration[END_REF], [START_REF] Yor | Grossissement de filtrations et absolue continuité de noyaux[END_REF], [START_REF] Jeulin | Grossissement de filtrations : exemples et applications[END_REF] and [START_REF] Mansuy | In Random Times and Enlargements of Filtrations in a Brownian Setting[END_REF]). We denote by (G t , t ≥ 0) the smallest filtration which contains (F t , t ≥ 0), and which makes g, defined by (3.21), a (G t , t ≥ 0) stopping time. a) Recall that (N t ) is the (F t ) t≥0 , P 0 martingale defined by (2.3), whose bracket is given by (2.4). Hence, from Girsanov's theorem, and (3.11), the process :

N t := N t - t 0 h(L u ) M h u d < N > u (3.22) is a (F t ) t≥0 , Q (h)
0 ) martingale, so that :

R 2α t = L t + N t + t 0 h(L u ) M h u d < N > u . ( 3
Z t = Q (h) 0 (g > t|F t ) = 1 - t 0 M h u (M h u ) 2 d M h u + ln(M h t ), (3.24)
with

M h t = M h t - t 0 d < M h > u M h u . (3.25) a (F t ) t≥0 , Q (h)
< N , • 0 M h u (M h u ) 2 d M h u > t =< N, • 0 M h u (M h u ) 2 dM h u > t = t 0 M h u (M h u ) 2 h(L u )d < N > u .
We deduce, after Jeulin [START_REF] Jeulin | Semi-martingales et grossissement d'une filtration[END_REF] and Yor [START_REF] Yor | Some aspects of Brownian motion[END_REF], that in the filtration (G t ) t≥0 , under Q (h) 0

:

N t = N (2) t - t∧g 0 1 Z u M h u (M h u ) 2 h(L u )d < N > u + t t∧g 1 1 -Z u M h u (M h u ) 2 h(L u )d < N > u . (3.26)
where ( N (3.23), we obtain :

t , t ≥ 0) is a (G t ) t≥0 , Q (h) 0 ) local martingale. Plugging (3.26) in (2)
R 2α t = L t + t 0 h(L u ) M h u d < N > u + N (2) t - t∧g 0 1 Z u M h u (M h u ) 2 h(L u )d < N > u + t t∧g 1 1 -Z u M h u (M h u ) 2 h(L u )d < N > u . ( 3
R 2α t = L t + N (2) t + t t∧g 1 M h u -M h u h(L u )d < N > u . (3.28)
But, since, from (3.19), we have :

M h u -M h u = h(L u )R 2α u + 1 -H(L u ) -(1 -H(L u )) = h(L u )R 2α u ,
then (2.4) implies :

R 2α t = L t + N (2) t + 4α 2 t t∧g R 2(α-1) u du. (3.29)
We note that, despite the different changes of probability, or of filtration, which we have made, the brackets of N and N (2) are equal, hence :

< N (2) > t = 4α 2 t 0 R 2(2α-1) u du. (3.30) 9) Description of the Q (h) 0
process, after g. From (3.29) and because R g = 0 and L t+g = 0, t ≥ 0, we have :

R 2α g+t = N (3) t + 4α 2 t 0 R 2(α-1) g+s ds, (3.31)
where

N (3) t = N (2) g+t -N (2) g , t ≥ 0. (3.32) Note that g is a (G t ) t≥0 stopping time, therefore ( N (3) t ) is a (G g+t ) t≥0 , Q (h) 0 continuous local mar- tingale.
We then apply Itô's formula to compute f (R 2α g+t ), with f (x) := x 1/2α ; we get, from (3.30) :

R g+t = 1 2α t 0 R 1-2α g+s d N (3) s + 2α t 0 R 1-2α g+s R 2(α-1) g+s ds + 1 -2α 2 t 0 R 1-4α g+s R 2(2α-1) g+s ds = 1 2α t 0 R 1-2α g+s d N (3) s + 1 + 2α 2 t 0 ds R g+s . (3.33) But, from (3.30), the (G g+t ) t≥0 , Q (h) 0 local martingale B t := 1 2α t 0 R 1-2α g+s d N (3)
s , t ≥ 0 admits as bracket :

1 4α 2 t 0 R 2(1-2α) g+s (4α 2 )R 2(2α-1) g+s ds = t. This implies that (B t , t ≥ 0) is a (G g+t ) t≥0 , Q (h) 0
Brownian motion and is therefore independent from G g . Finally (R g+t ) solves :

R g+t = B t + 1 + 2α 2 t 0 ds R g+s . (3.34)
This proves that (R g+t , t ≥ 0) is a Bessel process starting from 0, with dimension δ = 2 + 2α = 4d.

The solution of (3.34) being strong, the processes (R t , t ≤ g) and (R g+t , t ≥ 0) are independent under

Q (h) 0 . 10) Description of the Q (h) 0
process before g. Before g, we have, from (3.29) and (3.30) :

R 2α t∧g = L t∧g + N (2) t∧g , with < N (2) > t = 4α 2 t 0 R 2(2α-1) s ds. (3.35)
Let us introduce :

β t := 1 2α t 0 1 R 2α-1 u d N (2) u , t ≥ 0. It is clear that (3.35) implies that (β t , t ≥ 0) is a (G t ) t≥0 , Q (h) 0
Brownian motion and :

R 2α t∧g = L t∧g + 2α t∧g 0 R 2α-1 s dβ s . (3.36)
Then applying Itô's formula to (3.36), to compute R 2 t∧g = (R 2α t∧g ) 1/α , we obtain (since 1/α > 1) :

R 2 t∧g = t∧g 0 1 α R 2(1-α) s 2αR 2α-1 s dβ s + 1 -α 2α 2 t∧g 0 R 2(1-2α) s (4α 2 )R 2(2α-1)
s ds.

(note that, because 1 α > 1, the term in dL s disappears) hence : [START_REF] Mansuy | In Random Times and Enlargements of Filtrations in a Brownian Setting[END_REF] We now prove point 4. of Theorem 1.1. We first show that (R 2α ρ(u) + L ρ(u) , u ≥ 0) is a 3-dimensional Bessel process, starting from 0 (recall that ρ(u) is defined by (1.9)). a) Let us start by studying the process (R t ) before g. It is clear that (3.36) implies :

R 2 t∧g = 2 t∧g 0 R 2 s dβ s + 2(1 -α)(t ∧ g). (3.37) which proves that R 2 t∧g is the square of a Bessel process with dimension d = 2(1 -α) stopped at time g = inf{t ≥ 0; L t = L ∞ }.
R 2α ρ(u) = L ρ(u) -W u , u ≤ A g , (3.38)
where

W u := -2α ρ(u) 0 R 2α-1 s dβ s , u ≥ 0 is (G ρ(u) ) u≥0 , Q (h) 0
Brownian motion.

From Skorokhod's reflection lemma ( [START_REF] Revuz | Continuous martingales and Brownian motion, volume 293 of Grundlehren der Mathematischen Wissenschaften[END_REF], Chap. VI) we have :

L ρ(u) = sup s≤u W s , u ≥ 0.
According to Pitman's theorem (cf [START_REF] Pitman | One-dimensional Brownian motion and the three-dimensional Bessel process[END_REF]), the process 2 sup u≤t W u -W t , t ≥ 0 is a 3-dimensional Bessel process, started at 0. Finally R 2α ρ(u) + L ρ(u) , u ≤ A g is a three dimensional Bessel process, started at 0, stopped at the stopping time A g . b) We consider now (R g+t ). We first observe : R 2α u := R 2α ρ(u+Ag) = R 2α g+ ρ(u) , u ≥ 0, where ( ρ(u)) u≥0 is the right-inverse of :

A t = A g+t -A g = 4α 2 t 0 R 2(2α-1) s+g ds, t ≥ 0,
(recall that (A t ) t≥0 is the process defined by (1.8)). Then (3.31) may be written as :

R 2α u = N (3) ρ(u) + ρ(u) 0 d A s R 2α s+g . Since W ′ u := N (3) ρ(u) , u ≥ 0 is a (G ρ(u) ) u≥0 , Q (h) 0
martingale, with bracket A ρ(u) = u, we obtain after making the change of variables A s = v :

R 2α t = W ′ t + t 0 dv R 2α v . Note that L ρ(t+Ag) = L g+ ρ(t) = L g is a G ρ(0) measurable r.v. Consequently, R 2α t + L ρ(t+Ag)
, t ≥ 0 is a 3-dimensional Bessel process, starting from L g . This result, together with point a) proves that (R 2α ρ(t) +L ρ(t) t ≥ 0) is a three-dimensional Bessel process started at 0. The independence of L ∞ and of (R 2α ρ(u) + L ρ(u) ; u ≥ 0) follows from the fact that the law of (R 2α ρ(u) + L ρ(u) ; u ≥ 0), conditionally on L ∞ = y, does not depend on y. Remark 3.2 1. Replacing in step 4) of the above proof , the event {L t > c} by {L t1 > c 1 , . . . , L tn > c n } with t 1 > . . . > t n > 0 and c 1 ≥ . . . ≥ c n > 0, proves that the law of the process

(L t , t ≥ 0) under Q (h) 0
is the same as that of the process L t ∧ ξ, t ≥ 0 under P 0 , where ξ is a random variable of density h and independent from (L t , t ≥ 0) (under P 0 ).

2. We now present a heuristic method to obtain the distribution of L ∞ under Q (h) 0 . We write, for every function g, bounded and continuous :

E 0 g(L t )h(L t ) E 0 [h(L t )] = E 0 g(t α L 1 )h(t α L 1 ) E 0 [h(t α L 1 )] (by scaling) = ∞ 0 (gh)(t α x)p L1 (x)dx ∞ 0 h(t α x)p L1 (x)dx = ∞ 0 gh(y)p L1 ( y t α )dy ∞ 0 h(y)p L1 ( y t α )dy .
Property (2.10) implies :

lim t→∞ E 0 g(L t )h(L t ) E 0 [h(L t )] = p L1 (0) ∞ 0 (gh)(y)dy p L1 (0) ∞ 0 h(y)dy = ∞ 0 g(y)h(y)dy.
However, this computation is not, at least without any further justification, "licit". The correct manner to obtain the law of

L ∞ under Q (h) 0 is to first study E 0 (g(L s )h(L t )) E 0 (h(L t )) for a fixed s < t,
then to first let t tend to +∞, and finally to let s tend to ∞.

4 Proofs of Theorems 1.3 and 1.4

Recall that under P 0 , (R t ) is a d-dimensional Bessel process started at 0, with d = 2(1α) and α ∈]0, 1[. To prove Theorems 1.3 and 1.4, it is convenient to introduce the following notation :

1. p L,t (l) is the density function of L t , under P 0 , 2. p R,L,t (r, l) is the density function of the couple (R t , L t ), under P 0 , 3. {Π t } denotes the semigroup of (R t , L t ),

t (x, y) denotes the density of the transition semigroup of the Bessel process with index µ, at time t > 0.

Proof of Theorem 1.3

We begin the proof of Theorem 1.3 with two preliminary results : Lemma 4.1 and 4.2 below, in which we compute the conditional expectation of an event Λ s in F s , given L t , resp. given X t , L t with t > s.

Lemma 4.1 For every s and t such that 0 ≤ s ≤ t, Λ s ∈ F s , and y ≥ 0, one has :

E 0 [1 Λs |L t = y] = p L,s (y) p L,t (y) E 0 [1 Λs ϕ 1 (t -s, R s )|L s = y] + 1 p L,t (y) E 0 1 Λs 1 (Ls<y) ϕ 2 (t -s, R s , y -L s ) , (4.1)
with

ϕ 1 (u, r) := P r (T 0 > u), r, u ≥ 0, (4.2)
ϕ 2 (u, r, l) := E r 1 {T0<u} p L,u-T0 (l) , r, l, u ≥ 0, (4.3)
and T 0 := inf{s ≥ 0, R s = 0}.

Proof. Let f : R + → R + be a positive, Borel function. We compute in two different manners the quantity

E 0 [1 Λs f (L t )].
On one hand, by conditioning with respect to L t = y, we obtain :

E 0 (1 Λs f (L t )) = ∞ 0 E 0 (1 Λs |L t = y)f (y)p L,t (y)dy. (4.4)
On the other hand, by conditioning with respect to F s , we obtain :

E 0 1 Λs f (L t ) = E 0 1 Λs E(f (L t )|F s ) = E 0 1 Λs Π t-s f (R s , L s ) . (4.5)
Let us introduce :

ψ(t, l) = E 0 [f (l + L t )] = ∞ 0 f (y)1 {y>l} p L,t (y -l)dy. (4.6)
Then, using the strong Markov property at time T 0 , we get :

Π u f (r, l) = E r [f (l + L u )] = f (l)P r (T 0 > u) + E r 1 {T0<u} ψ(u -T 0 , l) = f (l)ϕ 1 (u, r) + ∞ 0 f (y)ϕ 2 (u, r, y -l)1 {y>l} dy. (4.7)
Now, plugging (4.7) into (4.5), and then comparing (4.4) and (4.5), for an arbitrary function f , yields Lemma 4.1.

Lemma 4.2 For every a, y ≥ 0, s ≥ 0, Λ s ∈ F s , and t ≥ s,

E 0 (1 Λs |R t = x, L t = y) = p L,s (y)x -2α p R,L,t (x, y) E 0 1 Λs R 2α s p (α) t-s (R s , x)|L s = y + 1 p R,L,t (x, y) E 0 1 Λs ϕ 3 (t -s, R s , x, y -L s )1 {y>Ls} (4.8)
with ϕ 3 (u, r, x, y) := E r 1 {T0<u} p R,L,u-T0 (x, y) , r, u, x, y ≥ 0, (4.9)

Proof. Let g : R + × R + be a positive Borel function. We shall compute in two different manners the quantity E 0 [1 Λs g(R t , L t )]. First, by conditioning with respect to R t = x and L t = y, we obtain :

E 0 [1 Λs g(R t , L t )] = R 2 + E 0 [1 Λs |R t = x, L t = y]g(x, y)p R,L,t (x, y)dxdy. (4.10)
Secondly, by conditioning with respect to F s , we obtain :

E 0 1 Λs g(R t , L t ) = E 0 1 Λs Π t-s g(R s , L s ) . (4.11)
We note that :

Π u g(r, l) = E r [g(R u , L u + l)). (4.12)
We proceed as in the proof of Lemma 4.1, decomposing the right-hand side of (4.12) in two parts A 1 , resp. A 2 depending upon whether u is smaller, or greater than T 0 = inf{s ≥ 0; R s = 0}. Thus, we obtain :

Π u g(r, l) = A 1 + A 2 , (4.13)
where :

A 1 := E r [g(R u , l)1 (T0>u) ], A 2 := E r [g(R u , L u + l)1 (T0<u) ]. (4.14)
We shall study successively A 1 and A 2 . a) Recall the absolute continuity relationship between : P -α r |Fu∩{u<T0} and P (+α) r

| Fu (see, [START_REF] Revuz | Continuous martingales and Brownian motion, volume 293 of Grundlehren der Mathematischen Wissenschaften[END_REF], Chap. XI or [START_REF] Gradinaru | Abel transform and integrals of Bessel local times[END_REF], section 1.2):

P -α r |Fu∩{u<T0} = r R u 2α P α r |Fu . ( 4

.15)

Consequently :

A 1 = E (α) r r 2α R 2α u g(R u , l) = r 2α ∞ 0 g(x, l) x 2α p (α) u (r, x)dx. (4.16)
b) Next we compute A 2 . Conditioning with respect to F T0 , we get :

A 2 = E r 1 {T0<u} ψ 2 (u -T 0 , l) , with ψ 2 (v, l) = E 0 g(R v , L v + l) = R 2 + g(x, y)p R,L,v (x, y -l)1 {y>l} dxdy.
Using (4.9) we have :

A 2 = R 2 + g(x, y)ϕ 3 (u, r, x, y -l)1 {y>l} dxdy. (4.17)
Combining (4.13), (4.14), (4.16) and (4.17), we get :

E 0 1 Λs g(R t , L t ) = E 0 1 Λs R 2α s ∞ 0 g(x, L s ) x 2α p (α) t-s (R s , x)dx + R 2 + g(x, y)E 0 1 Λs ϕ 3 (t -s, R s , x, y -L s )1 {y>Ls} dxdy.
It is then easy to conclude since the function g is arbitrary.

To prove the existence of the limit (1.14) (resp. (1.18)) of Theorem 1.3, we need to obtain an asymptotic estimate of p L,t (y) (resp. p R,L,t (x, y)) as t → ∞. The first result may be obtained directly. As for p R,L,t (x, y), we first prove in Lemma 4.3 below that this function can be written as a convolution of two functions, having a decay rate of the type Ct -(1+α) (t → ∞). Then Lemma 4.4 allows to prove in Lemma 4.5 that t → p R,L,t (x, y) enjoys an analogous polynomial decay.

Lemma 4.3 Let γ 1 denote the density of τ 1 .

1. For every x, y, q ≥ 0 there is the identity :

∞ 0 e -qt p R,L,t (x, y)dt = 2 -α Γ(1 + α) ∞ 0 e -qt-x 2 2t x t 1+α dt ∞ 0 e -qt γ 1 t y 1/α dt y 1/α . (4.18) 2. Let β 1 , β 2 :]0, ∞[×]0, ∞[→
R be the two functions :

β 1 (x, t) := 2 -α Γ(1 + α) x t α+1 e -x 2 2t , x, t > 0, (4.19)
and

β 2 (y, t) := 1 y 1/α γ 1 t y 1/α , y, t > 0. ( 4

.20)

Then :

p R,L,t (x, y) = (β 1 (x, •) * β 2 (y, •)) (t). (4.21)
Proof. Let Θ denote an exponential variable with parameter q > 0, independent from (R t , t ≥ 0). Let χ Θ denote the last zero of (R t , t ≥ 0) before Θ. It is well known, from the last exit decomposition results, that (R t , t ≤ χ Θ ) and (R χΘ+u , u ≤ Θχ Θ ) are two independent processes. Since R XΘ = 0, and L Θ = L χΘ , it follows that R Θ and L Θ are independent. As a consequence, we obtain, for every pair f , g of R + valued Borel functions :

E 0 [f (R Θ )g(L Θ )] = R 3 + qe -qt f (λ)g(y)p R,L,t (λ, y)dλdydt = E 0 [f (R Θ )] E 0 [g(L Θ )] . (4.22) a) We first compute E 0 [f (R Θ )]. Recall : p (-α) t (λ) = 2 α Γ(1 -α) t α-1 λ 1-2α e -λ 2 2t 1 {λ>0} , (4.23)
then :

E 0 [f (R Θ )] = ∞ 0 qe -qt dt ∞ 0 f (λ)p (-α) t (λ)dλ = q ∞ 0 dt ∞ 0 f (λ) 2 α Γ(1 -α) t α-1 λ 1-2α exp - λ 2 2t -qt dλ = 2 α q Γ(1 -α) ∞ 0 f (λ)λ 1-2α dλ ∞ 0 t α-1 exp - λ 2 2t -qt dt. (4.24)
Setting t = λ 2 /(2qs) we obtain :

E 0 [f (R Θ )] = q 1-α Γ(1 -α) ∞ 0 f (λ)dλ ∞ 0 λ s α+1 exp - λ 2 2s -qs ds (4.25) = 2 1+α/2 q 1-α/2 Γ(1 -α) ∞ 0 f (λ)K -α (λ 2q) dλ λ α b) As for the computation of E 0 [g(L Θ )] , we observe that L Θ is exponentially distributed with para- meter Γ(1 -α) Γ(1 + α) q 2 α
, since from (2.12) we have :

P 0 (L Θ > l) = P 0 (Θ > τ l ) = E 0 (e -qτ l ) = exp -l Γ(1 -α) Γ(1 + α) q 2 α , l > 0.
Hence :

E 0 [g(L Θ )] = Γ(1 -α) Γ(1 + α) q 2 α ∞ 0 g(l) exp -l Γ(1 -α) Γ(1 + α) q 2 α dl. (4.26)
Applying (2.12) with (l, λ) changed into (1, ql 1/α ) yields to :

exp -l Γ(1 -α) Γ(1 + α) q 2 α = E[e -ql 1/α τ1 ] = ∞ 0 e -ql 1/α s γ 1 (s)ds = ∞ 0 e -qt γ 1 t l 1/α dt l 1/α .
Consequently : Lemma 4.4 Let β 0 1 and β 0 2 be two integrable functions from R + to R + , such that :

E 0 [g(L Θ )] = Γ(1 -α) Γ(1 + α) q 2 α ∞ 0 g(l)dl ∞ 0 e -qt γ 1 t l 1/α dt l 1/α . ( 4
β 0 1 (t) ∼ t→∞ k 1 t 1+α , β 0 2 (t) ∼ t→∞ k 2 t 1+α .
(4.28)

Then :

β 0 1 * β 0 2 (t) ∼ t→∞ 1 t 1+α k 1 ∞ 0 β 0 2 (u)du + k 2 ∞ 0 β 0 1 (u)du . ( 4

.29)

Proof. Let us write :

β 0 1 * β 0 2 (t) = t 0 β 0 1 (u)β 0 2 (t -u)du = εt 0 β 0 1 (u)β 0 2 (t -u)du + (1-ε)t εt • • • du + t (1-ε)t • • • du = I 1 + I 2 + I 3 .
For t large enough, one has :

I 2 = t 1-ε ε β 0 1 (tu)β 0 2 (t(1 -u))du ≤ k ′ 1 k ′ 2 t 1+2α 1-ε ε 1 u 1+α 1 (1 -u) 1+α dv.
This implies that I 2 = o 1 t 1+α , t → ∞ and this term does not contribute to the limit. On the other hand, for any 0 < δ < k 2 , there exists r 0 > 0 such that :

k 2 -δ r 1+α ≤ β 0 2 (r) ≤ k 2 + δ r 1+α , r ≥ r 0 .
Let 0 < ε < 1/2 and t ≥ 2r 0 . For any u ≤ εt we have : t ≥ tu ≥ t(1ε) ≥ t/2 ≥ r 0 . Therefore replacing r by tu in the previous inequality we get :

k 2 -δ t 1+α ≤ β 0 2 (t -u) ≤ k 2 + δ t 1+α 1 (1 -ε) 1+α , t ≥ 2r 0 .
(4.30)

Integrating (4.30) over [0, εt] with respect to β 0 1 (u)du, we obtain :

(k 2 -δ) εt 0 β 0 1 (u)du ≤ t 1+α I 1 ≤ k 2 + δ (1 -ε) 1+α ∞ 0 β 0 1 (u)du.
Taking the limit t → ∞ we have :

(k 2 -δ) ∞ 0 β 0 1 (u)du ≤ lim inf t→∞ t 1+α I 1 ≤ lim sup t→∞ t 1+α I 1 ≤ k 2 + δ (1 -ε) 1+α ∞ 0 β 0 1 (u)du.
Taking the limit δ, ε → 0 implies that

I 1 ∼ t→∞ k 2 t (1+α) ∞ 0 β 0 1 (u)du. Since I 3 = t (1-ε)t β 0 1 (u)β 0 2 (t -u)du = εt 0 β 0 2 (u)β 0 1 (t -u)du,
we can apply the previous result, with β 0 1 and β 0 2 interchanged, to obtain :

I 3 ∼ t→∞ k 1 t (1+α) ∞ 0 β 0 2 (u)du.
Thanks to Lemmas 4.3 and 4.4, we are able to determine the asymptotic behavior of p R,L,t (x, y) as t → ∞. Observe that we may not deduce it from (4.18), since we do not know that t → p R,L,t (x, y) is monotone, hence the Tauberian theorem may not be applied.

Lemma 4.5

The following equivalence holds :

p R,L,t (x, y) ∼ t→∞ 2 -α Γ(1 + α) x + yx 1-2α t 1+α , x, y > 0. ( 4
β 1 (x, t) ∼ t→∞ 2 -α Γ(1 + α) x t α+1 . (4.32)
Recall that from (2.12), we have :

E 0 (e -q τ1 ) = exp - Γ(1 -α) Γ(1 + α) q 2 α , q ≥ 0.
Then we deduce, by differentiating both sides of this identity with respect to q and using the Tauberian theorem that :

γ 1 (t) ∼ t→∞ α2 -α Γ(1 + α) 1 t 1+α .
Hence :

β 2 (y, t) ∼ t→∞ α2 -α Γ(1 + α) y t 1+α . (4.33) b) We have : ∞ 0 β 1 (x, t)dt = x 1-2α Γ(1 + α) Γ(α) = x 1-2α α (4.34) ∞ 0 β 2 (y, t)dt = 1. ( 4
P r (T 0 > t -s) = P (γ α < r 2 2(t -s) ) ∼ t→∞ 1 Γ(1 + α) r 2 2t α . ( 4
p Lt (y) = 1 t α p L1 ( y t α ) ∼ t→∞ 1 t α p L1 (0) = 1 t α 2 -α Γ(1 + α) . ( 4
∞ 0 Q (y) 0 (Λ s )h(y)dy = ∞ 0 p Ls (y)E 0 1 Λs R 2α s |L s = y h(y)dy + ∞ 0 E 0 1 Λs 1 {Ls<y} h(y)dy = ∞ 0 E 0 1 Λs h(L s )R 2α s |L s = y p Ls (y)dy + E 0 1 Λs ∞ 0 h(y)1 {Ls<y} dy = E 0 1 Λs h(L s )R 2α s + 1 -H(L s ) = E 0 [1 Λs M h s ] = Q (h) 0 (Λ s ) = ∞ 0 Q (h) 0 (Λ s |L ∞ = y)h(y)dy,
the latter relation following from the fact that L ∞ admits h as its probability density.

Therefore the two probability measures on (Ω, F ∞ ),

∞ 0 Q (y) 0 (•)h(y)dy and Q (h) 0 coincide on F s , for
any s ≥ 0, hence they are equal :

Q (h) 0 (•) = ∞ 0 Q (y) 0 (•)h(y)dy.
On the other hand, from the definition (1.15) of Q (y) 0 , we easily deduce that Q (y) 0 is carried by L ∞ = y. Indeed, for every ε > 0,

Q (y) 0 (L ∞ ≤ y -ε) = lim s→∞ Q (y) 0 (L s ≤ y -ε) = lim s→∞ p Ls (y)E 0 1 {Ls≤y-ε} R 2α s |L s = y + P 0 (L s ≤ y -ε) = lim s→∞ P 0 (L s ≤ y -ε) = 0 since L ∞ = ∞, P 0 a.s. A similar computation shows that Q (y) 0 (L ∞ ≥ y + ε) = 0. Consequently : Q (h) 0 (•|L ∞ = y) = Q (y) 0 (•).
2) Proof of point 3. of Theorem 1.3. To prove that the limit in (1.18) exists, we start with Lemma 4.2 :

E 0 (Λ s |R t = x, L t = y) = Θ 1 + Θ 2 , Λ ∈ F s , t > s,
where

Θ 1 = p Ls (y)x -2α p R,L,t (x, y) E 0 1 Λs R 2α s p (α) t-s (R s , x)|L s = y Θ 2 = 1 p R,L,t (x, y) E 0 1 Λs ϕ 3 (t -s, R s , x, y -L s )1 {y>Ls} , (4.38)
the function ϕ 3 being defined by (4.9). We study successively the limits of Θ 1 , Θ 2 , as t → ∞. a) From ( [START_REF] Revuz | Continuous martingales and Brownian motion, volume 293 of Grundlehren der Mathematischen Wissenschaften[END_REF], Chap. 10), we have :

p (α) t (r, a) = a t a r α I α ar t exp - a 2 + r 2 2t . Since I α (z) ∼ z→0 1 Γ(α + 1) z 2 α an equivalent for p (α) t (r, a) as t → ∞ is easily deduced : p (α) t (r, a) ∼ t→∞ 2 -α Γ(α + 1) a 1+2α t 1+α .
(4.39)

Consequently using moreover Lemma 4.5, we obtain :

lim t→∞ Θ 1 = p Ls (y)x x + yx 1-2α E 0 1 Λs R 2α s |L s = y . (4.40)
b) Next we study the limit of Θ 2 , as t → ∞.

It is clear that (4.9) may be interpreted as :

ϕ 3 (u, r, x, y) = µ(r, •) ⋆ p R,L,• (x, y) (u), (4.41)
where µ is the density function of T 0 under P r . Thanks to (2.14), we have :

µ(r, t) ∼ t→∞ α2 -α Γ(1 + α) r 2α t 1+α .
Taking q = 0 in (4.18) we have :

∞ 0 p R,L,t (x, y)dt = x 1-2α α .
Hence, applying (4.41) together with Lemmas 4.5 and 4.4 leads to :

ϕ 3 (u, r, x, y) ∼ u→∞ α2 -α r 2α Γ(1 + α) x 1-2α α + 2 -α Γ(1 + α) x + yx 1-2α 1 u 1+α ∼ u→∞ 2 -α Γ(1 + α) x + (r 2α + y)x 1-2α 1 u 1+α . (4.42)
Plugging this expression in (4.38), and using again (4.31), we deduce that :

lim t→∞ Θ 2 = 1 x + yx 1-2α E 0 1 Λs 1 (Ls<y) x + R 2α s + y -L s a 1-2α .
This result together with (4.40), proves that the limit in (1.18) exists and has the form given in (1.19).

3) We end the proof of Theorem 1. and Q (y) 0 , we have :

x + yx 1-2α Q (x,y) 0 (Λ s ) = x Q (y) 0 (Λ s ) -E 0 1 Λs 1 {Ls<y} +E 0 1 Λs 1 {Ls<y} x + (y -L s + R 2α s )x 1-2α = xQ (y) 0 (Λ s ) + x 1-2α E 0 1 Λs 1 {Ls<y} (y -L s + R 2α s ) . (4.43)
For a given y > 0, let h y be the function :

h y (x) = 1 y 1 [0,y] (x),
x ≥ 0, and H y the primitive of h y , vanishing at 0; hence :

1 -H y (x) = ∞ x h y (z)dz = 1 {x≤y} 1 - x y . (4.44)
Thanks to (1.16), (1.7) and (1.6), we have :

∞ 0 Q (z) 0 (Λ s )h y (z)dz = E 0 1 Λs M h y s = E 0 1 Λs h y (L s )R 2α s + 1 {Ls<y} (1 - L s y ) = 1 y E 0 1 Λs R 2α s + y -L s 1 {Ls<y} . (4.45)
Plugging (4.45) in (4.43), we get :

Q (x,y) 0 (Λ s ) = x x + yx 1-2α Q (y) (Λ s ) + x 1-2α x + yx 1-2α y 0 Q (z) 0 (Λ s )dz.
This ends the proof of Theorem 1.3. Remark 4.6 Suppose that α = 1/2 (i.e : d = 1).

1. Several of the above computations become easier, in particular, that of the function ϕ 3 introduced in (4.9). From Lévy's theorem :

((S t -X t , S t ) , t ≥ 0) (d) = ((R t , L t ) , t ≥ 0) (4.46)
where, on the left-hand side (X t ) is a standard Brownian motion started at 0, (S t ) its unilateral maximum, i.e. S t = max u≤t X u , and the right-hand side (R t , t ≥ 0) is a reflected Brownian motion (i.e. a Bessel process with index (-1/2)), and (L t ) its local time at level 0, and ( [START_REF] Revuz | Continuous martingales and Brownian motion, volume 293 of Grundlehren der Mathematischen Wissenschaften[END_REF] section III.3 p105), we have :

p R,L,t (x, y) = 2 πt 3 (x + y)e -(x+y) 2 2t = 2 P 0 T x+y (X) ∈ dt dt , x, y > 0, ( 4

.47)

since :

P 0 (T r (X) ∈ dt) = 1 2πt 3 re -r 2 2t 1 {t>0} dt, r > 0,
where T r (X) denotes the first hitting time of level r for the Brownian motion (X t ).

Since under P r , T 0 is distributed as T r (X) and T r (X)+ T r ′ (X) is distributed as T r+r ′ (X), where

T r ′ (X) is independent from T r (X) and T r ′ (X) (d)
= T r ′ (X), then using (4.41), we have :

ϕ 3 (u, r, x, y) = 2 P (T x+y+r (X) ∈ du) du .
This implies that ϕ 3 (u, r, x, y) ∼ u→∞ 2 πu 3 x + y + r . We recover (4.42).

2. We keep the notation relative to Brownian motion introduced above. We have proven in ( [18], Theorems 1.2, 1.3 and the proof of Theorem 1.3) that, for a < y, y ≥ 0 and Λ u ∈ F u :

lim t→∞ E 1 Λu |X t = a, S t = y = y -a 2y -a p Su (y)E 1 Λu (y -X u )|S u = y + 1 2y -a E 1 Λu 1 {Su<y} (2y -a-X u ) ,
where p Su denotes the density function of S u .

Obviously, this result is equivalent to :

lim t→∞ E 1 Λu |X t = y -x, S t = y = x x + y p Su (y)E 1 Λu (y -X u )|S u = y + 1 x + y E 1 Λu 1 {Su<y} (x + y -X u ) .
Therefore, from Lévy's theorem (4.46), we obtain :

lim t→∞ E 0 1 Λu |R t = x, L t = y = x x + y p Lu (y)E 0 1 Λu R u |L u = y + 1 x + y E 0 1 Λu 1 {Lu<y} (x + y -L u + R u ) .
which is indeed (1.19) of our Theorem 1.3 for α = 1/2.

Proof of Theorem 1.4

Let h : R + → R + be a probability density. Recall that hx =

∞ 0 h(y)(x + yx 1-2α )dy < ∞ where x > 0. Define H(z) = z 0 h(y)dy, z ≥ 0. 1) We first prove (1.23).
It is clear that :

P 0 (L t ∈ dy|R t = x) = p R,L,t (x, y) p (-α) t (0, x) dy = 2 -α Γ(1 -α) e x 2 /2t x 1-2α 1 t α-1 p R,L,t (x, y)dy.
Applying Lemma 4.5, formula (4.31), we obtain :

P 0 (L t ∈ dy|R t = x) dy ∼ t→∞ 2 -2α Γ(1 -α) Γ(1 + α) x + yx 1-2α x 1-2α 1 t 2α . (4.48) Since E 0 1 Λs h(L t )|R t = x E 0 h(L t )|R t = x = ∞ 0 E 0 [1 Λs |R t = a, L t = y] h(y)P 0 (L t ∈ dy|R t = x) ∞ 0 h(y)P 0 (L t ∈ dy|R t = x)
, we deduce from (4.48) and point 3. of Theorem 1.3 that :

lim t→∞ E 0 1 Λs h(L t )|R t = x E 0 h(L t )|R t = x = h * x ∞ 0 Q (x,y) 0 (Λ s )h(y)(x + yx 1-2α )dy,
where h * x = 1/ hx . According to (1.20), we may write the right-hand side as follows :

h * x ∞ 0 Q (y) 0 (Λ s )xh(y)dy + ∞ 0 x 1-2α h(y)dy y 0 Q (z) 0 (Λ s )dz .
Applying Fubini's theorem, the previous term equals :

h * x ∞ 0 Q (y) 0 (Λ s ) xh(y) + x 1-2α (1 -H(y)) dy = Q hx (Λ s ).
This proves (1.23).

2) Next we prove point 2. of Theorem 1.4. Let f : R + × R + → R + be Borel and such that (1.25) holds. We proceed as above. Using Lemma 4.5 and point 3. of Theorem 1.3 we have :

E 0 1 Λs f (R t , L t ) E 0 f (R t , L t ) = R+×R+ E 0 1 Λs |R t = x, L t = y f (x, y)p R,L,t (x, y)dxdy R+×R+ f (x, y)p R,L,t (x, y)dxdy .
Consequently :

lim t→∞ E 0 1 Λs f (R t , L t ) E 0 f (R t , L t ) = f * R+×R+ Q (x,y) 0 (Λ s )(x + yx 1-2α )f (x, y) dy. E 0 1 Λs f (R t , L t ) E 0 f (R t , L t ) = R+×R+ E 0 1 Λs |R t = x, L t = y f (x, y)p R,L,t (x, y)dxdy R+×R+ f (x, y)p R,L,t (x, y)dxdy
.

lim t→∞ E 0 1 Λs f (R t , L t ) E 0 f (R t , L t ) = f * R+×R+ Q (x,y) 0 (Λ s )(x + yx 1-2α )f (x, y) dy.
Hence, from (1.20) and Fubini's theorem, this limit equals :

f * R+×R+ xQ (y) 0 (Λ s ) + x 1-2α y 0 Q (z) 0 (Λ s )dz f (x, y) dxdy = f * ∞ 0 Q (y) 0 (Λ s )dy ∞ 0 xf (x, y)dx + ∞ 0 x 1-2α dx ∞ y f (x, z) dz = ∞ 0 Q (y) 0 (Λ s ) f (y)dy = Q ( f ) 0 (Λ s ).
This ends the proof of Theorem 1.4.

Proof of Theorem 1.5

Let h : R + → R + satisfy (1.28). 1) We first prove point 1. of Theorem 1.5. Let t > s ≥ 0 and Λ s ∈ F s .

The Markov property at time s allows to write :

E 0 1 Λs h(L t )e λRt E 0 [h(L t )e λRt ] = N (s, t) D(t) , (5.1)
where :

D(t) = E 0 h(L t )e λRt = R+×R+ h(y)e λx p R,L,t (x, y)dxdy, (5.2)
N (s, t) = E 0 1 Λs h(L t )e λRt = E 0 [1 Λs N 1 (R s , L s , t -s)] , (5.3)
N 1 (x, y, u) = E x h(y + L u )e λRu , x, y, u ≥ 0. (5.4)
Recall that p R,L,t (x, y) denotes the density function of (R t , L t ) under P 0 . We study successively the asymptotic behaviors of D(t) and of N 1 (x, y, t) as t → ∞. Note that we cannot apply Lemma 4.5 since f (x, y) = h(y)e λx does not satisfy (1.25). 1.a) Let us determine the rate of decay of D(t), as t → ∞. Since p R,L,t satisfies (4.21), then

D(t) = 2 -α Γ(1 + α) t 0 du (t -u) α+1 ∞ 0 dyh(y)β 2 (y, u) ∞ 0 xe λx-x 2 2(t-u) dx.
Setting x = λ(tu) + z √ tu in the integral with respect to dx, we obtain :

∞ 0 xe λx-x 2 2(t-u) dx = (t -u) 3/2 e λ 2 (t-u) 2 ∞ -λ √ t-u z √ t -u + λ e -z 2 /2 dz ∼ t→∞ λ √ 2π(t -u) 3/2 e λ 2 (t-u) 2
.

Consequently :

D(t) ∼ t→∞ 2 -α λ √ 2π Γ(1 + α) t 0 e λ 2 (t-u) 2 (t -u) α-1/2 du ∞ 0 h(y)β 2 (y, u)dy ∼ t→∞ 2 -α λ √ 2π Γ(1 + α) t 1 2 -α e λ 2 t 2 t 0 e -λ 2 u 2 du ∞ 0 h(y)β 2 (y, u)dy.
Next, using the definition (4.20) of β 2 , we get :

D(t) ∼ t→∞ 2 -α λ √ 2π Γ(1 + α) t 1 2 -α e λ 2 t 2 ∞ 0 h(y) y 1/α dy t 0 e -λ 2 u 2 γ 1 u y 1/α du ∼ t→∞ 2 -α λ √ 2π Γ(1 + α) t 1 2 -α e λ 2 t 2 ∞ 0 h(y)dy t/y 1/α 0 γ 1 (v)e -λ 2 y 1/α 2 v dv. ∼ t→∞ 2 -α λ √ 2π Γ(1 + α) t 1 2 -α e λ 2 t 2 ∞ 0 h(y)dy ∞ 0 γ 1 (v)e -λ 2 y 1/α 2 v dv.
γ 1 being the density function of τ 1 , applying identity (2.12) (with l = 1 and λ replaced by λ 2 y 1/α 2 ) leads to :

D(t) ∼ t→∞ 2 -α λ √ 2π Γ(1 + α) ∞ 0 h(y)e -σ λ y dy t 1 2 -α e λ 2 t 2 , (5.5)
since σ λ is defined by (1.29).

We now consider the function N 1 (x, y, u) defined by (5.4), where x, y ≥ 0 are fixed and u → ∞. We decompose N 1 (x, y, u) as the sum of two terms :

N 1 (x, y, u) = N 1,1 (x, y, u) + N 1,2 (x, y, u), (5.6)
where :

N 1,1 (x, y, u) = E x [h(y + L u )e λRu 1 {u<T0} ] = h(y)E x [e λRu 1 {u<T0} ] N 1,2 (x, y, u) = E x [h(y + L u )e λRu 1 {u≥T0} ].
(5.7)

1. b) We look for an equivalent of N 1,1 (x, y, u), as u → ∞.

From the absolute continuity relationship between Bessel laws (cf ex 1.22, chap XI in [START_REF] Revuz | Continuous martingales and Brownian motion, volume 293 of Grundlehren der Mathematischen Wissenschaften[END_REF])) we get :

N 1,1 (x, y, u) = h(y)E (α) x x 2α R 2α u e λRu ,
where as in the Introduction, under P (α)

x , the process (R t ) is a Bessel process with index α, starting at x. Since (cf for instance section 1. p446 of [START_REF] Revuz | Continuous martingales and Brownian motion, volume 293 of Grundlehren der Mathematischen Wissenschaften[END_REF]) :

P (α) x (R u ∈ da) = a u a x α I α ax u exp - x 2 + a 2 2u da, (5.8)
we get :

N 1,1 (x, y, u) = h(y) x α u e -x 2 2u + λ 2 u 2 ∞ 0 a 1-α I α ax u e -1 2u (a-λu) 2 da.
Setting a = λu + √ ub, we get :

N 1,1 (x, y, u) = h(y)x α u 1 2 -α e -x 2 2u + λ 2 u 2 ∞ -λ √ u λ + b √ u 1-α I α (λ + b √ u )x e -b 2 /2 db ∼ u→∞ h(y)x α λ 1-α I α (λx) √ 2π u 1 2 -α e λ 2 u 2 .
(5.9)

1. c) We now find an equivalent of N 1,2 (x, y, u), u → ∞.

Conditioning with respect to T 0 , we get :

N 1,2 (x, y, u) = E x ψ(u -T 0 )1 {u≥T0} ,
where :

ψ(v) = E 0 h(y + L v )e λRv .
Applying (5.5), we have :

ψ(v) ∼ v→∞ 2 -α √ 2πλ Γ(1 + α) ∞ 0 h(y + z)e -σ λ z dz v 1 2 -α e λ 2 v 2 .
Consequently :

N 1,2 (x, y, u) ∼ u→∞ 2 -α √ 2πλe σ λ y Γ(1 + α) ∞ y h(z)e -σ λ z dz E x e -λ 2 2 T0 u 1 2 -α e λ 2 u
2 .

(5.10) Using (2.15), we get finally :

N 1,2 (x, y, u) ∼ u→∞ √ 2π 2 1-2α λ α+1 Γ(α)Γ(α + 1) x α K α (λx)e λ 2 u 2 u 1 2 -α (1 -H(y)), (5.11)
where we denoted :

1 -H(y) := e yσ λ ∞ y h(z)e -σ λ z dz.

d)

We now compute : lim

t→∞ E 0 1 Λs h(L t )e λRt E 0 [h(L t )e λRt ] .
From (5.5), (5.6), (5.7), (5.9) and (5.11), we have :

E 0 1 Λs h(L t )e λRt E 0 [h(L t )e λRt ] ∼ t→∞ A t B t
where :

A t := E 0 1 Λs √ 2π h(L s )R α s λ 1-α I α (λR s )(t -s) 1 2 -α e λ 2 2 (t-s) + √ 2π2 1-2α Γ(α)Γ(1 + α) λ α+1 R α s K α (λR s )e λ 2 2 (t-s) (t -s) 1 2 -α , B t = 2 -α √ 2π Γ(1 + α) λe λ 2 t 2 t 1 2 -α ∞ 0 h(ℓ)e -ℓσ λ dℓ. (5.12)
Hence :

lim t→∞ E 0 1 Λs h(L t )e λRt E 0 [h(L t )e λRt ] = E 0 [1 Λs M s ], (5.13)
with

M s = e -λ 2 s 2 R α s 2 λ α Γ(1 + α)h(L s )I α (λR s ) + λ 2 α 2 Γ(α) (1 -H(L s ))K α (λR s ) .
(5.14)

2) We now prove that M s = M λ, h s , s ≥ 0, where (M λ, h s ) is the process defined by (1.32). From (1.33) and (1.34), the function h can be written as a linear combination of h and 1 -H :

h(y) = h(y) + σ λ e σ λ y ∞ y h(z)e -σ λ z dz = h(y) + σ λ 1 -H(y) .

Consequently :

M s = e -λ 2 s 2 R α s 2 λ α Γ(1 + α) h(L s )I α (λR s ) + 1 -H(L s ) ξ s ,
where :

ξ s = 2 λ α Γ(1 + α)σ λ I α (λR s ) + λ 2 α 2 Γ(α) K α (λR s ).
Given the relations (see [START_REF] Lebedev | Special functions and their applications[END_REF] p3 and p108) :

Γ(1 -α) = π sin(πα) 1 Γ(α)
, K α (r) = π 2 sin(πα) (I α (r) -I -α (r)) (5.15) then the definition (1.29) of σ λ implies :

ξ s = λ 2 α Γ(1 -α)I α (λR s ) + λ 2 α Γ(1 -α) (I α (λR s ) -I -α (λR s )) = λ 2 α Γ(1 -α)I -α (λR s ).
This proves M s = M λ, h s . The local behavior of I β (z), z → 0 is known (see [START_REF] Lebedev | Special functions and their applications[END_REF] formula (5.7.1) p108) :

I β (z) = 1 Γ(1 + β) z 2 β + 0(z β+2 ) (z → 0), (5.16)
In particular : lim r→0 r α I α (λr) = 0, and lim

r→0 λ 2 α Γ(1 -α)r α I -α (λr) = 1.
(5.17)

This implies :

lim s→0 R α s I α (λR s ) = 0, lim s→0 R α s ξ s = 1.
It is clear that (1.28) and (1.34] imply that H(0) = 0. As a result : M λ, h 0 = 1. This ends the proof of point 1. of Theorem 1.5.

3) We verify that (M λ, h s , s ≥ 0) is a martingale. We shall show that (M λ, h t , t ≥ 0) is a local martingale. It will suffice to assume that h is of class C 1 to prove that (M λ, h t ) is a martingale (cf. point 2) of the proof of Theorem 1.1). It is clear that (M λ, h t ) can be decomposed as follows :

M λ, h t = e -λ 2 t 2 2 λ α Γ(1 + α) h(L t )Ψ 1 (R t ) + λ 2 α Γ(1 -α) 1 -H(L t ) Ψ 2 (R t ) , (5.18)
where Ψ 1 , Ψ 2 : R + → R + are defined by : Ψ 1 (r) := r α I α (λr) Ψ 2 (r) := r α I -α (λr).

(5.19)

a)

In a first step, we prove that (Ψ i (R t )), i = 1, 2 are two semimartingales, and we determine their decompositions.

Using [START_REF] Lebedev | Special functions and their applications[END_REF] p110, we have :

Y ′′ (r) + 1 r Y ′ (r) = 1 + α 2 r 2 Y (r), (5.20)
where Y (r) denotes either I α (r) or K α (r). Then,

LΨ i (r) = λ 2 2 Ψ i (r) (i = 1, 2), (5.21)
where L denotes the infinitesimal generator of (R t , t ≥ 0) (cf. (2.1)). We deduce from property (5.16) (with β = -α) and the definition (2.2) of the domain D of L that Ψ 2 ∈ D, and :

Ψ 2 (R t ) = R α t I -α (λR t ) = t 0 LΨ 2 (R s )ds + M 2 (t) = λ 2 2 t 0 R α s I -α (λR s )ds + M 2 (t), (5.22)
where (M 2 (t), t ≥ 0) is a P 0 local martingale.

As Ψ 1 ∈ D, we write :

Ψ 1 (r) = r α I α (λr) = λ 2 α 1 Γ(1 + α) r 2α + Ψ 1 (r), r > 0.
(5.23)

Relation (5.16) implies that Ψ 1 ∈ D. Moreover LΨ 1 = L Ψ 1 , then Ψ 1 (R t ) - t 0 LΨ 1 (R s )ds is a local martingale.
Since from (2.3), (R 2α t -L t , t ≥ 0) is a martingale, then, with the help of (5.21), we get :

Ψ 1 (R t ) = R α t I α (λR t ) = λ 2 α 1 Γ(1 + α) L t + λ 2 2 t 0 R α s I α (λR s )ds + M 1 (t), (5.24)
where (M 1 (t), t ≥ 0) is a P 0 local martingale.

b)

We are now able to prove that (M λ, h t ) is a P 0 local martingale. With the help of (1.34), (5.18),(5.22) and (5.24), we deduce, from Itô's formula, that :

dM λ, h t = - λ 2 2 M λ, h t dt + e -λ 2 t 2 2 λ α Γ(1 + α) h′ (L t )Ψ 1 (R t )dL t + 2 λ α Γ(1 + α) h(L t ) λ 2 α 1 Γ(1 + α) dL t + λ 2 2 R α t I α (λR t )dt + e -λ 2 t 2 λ 2 α Γ(1 -α) -h(L t )Ψ 2 (R t ))dL t + 1 -H(L t ) R α t I -α (λR t )dt +dM 3 (t), (5.25)
where (M 3 (t)) is a local martingale. It is clear that (5.17) implies that :

Ψ 1 (0) = 0, Ψ 2 (0) = 2 λ α 1 Γ(1 -α) .
(5.26)

Using moreover (5.14), it is easy to verify that, in (5.25), both the terms in (dt) and those in (dL t ) are equal to 0. Note that the relations (5.26) and (5.18) force M λ, h 0 = 1. 4) We now prove point 3. (a) of Theorem 1.5. Indeed, for every t and c > 0, one has :

Q (λ, h) 0 (L t > c) = Q (λ, h) 0 (τ c < t) = E 0 1 (τc<t) M λ, h t .
Using successively Doob's optional stopping theorem and the property :

M λ, h τc = e -λ 2 2 τc (1 -H(c)), we obtain : Q (λ, h) 0 (L t > c) = E 0 1 (τc<t) M λ, h τc = 1 -H(c) E 0 1 (τc<t) e -λ 2 2 τc .
Letting t → ∞ in the expression above and using (2.12) and (1.29) leads to :

Q (λ, h) 0 (L ∞ > c) = 1 -H(c) E 0 e -λ 2 2 τc = 1 -H(c) e -cσ λ .
In order to end the proof of Theorem 1.5 (i.e. points 3. (b) and (c)), we shall use the technique of progressive enlargement of filtrations, with respect to g = sup {t ≥ 0 : R t = 0}. Thus, we define (G t , t ≥ 0) to be the smallest filtration which contains (F t , t ≥ 0) and which makes g a (G t ) t≥0 stopping time. In order to use the enlargement formulae (see for instance [START_REF] Jeulin | Semi-martingales et grossissement d'une filtration[END_REF] or [START_REF] Yor | Grossissement de filtrations et absolue continuité de noyaux[END_REF]), it is necessary (cf (5.50))

to compute the (F t ), Q (λ, h) 0 supermartingale :

Z t := Q (λ, h) 0 (g > t|F t ).
(5.27)

We determine Z t in the next Lemma 5.1.

Lemma 5.1 We have :

Z t = 2 1-α Γ(α) e -λ 2 t 2 1 -H(L t ) (λR t ) α K α (λR t ) 1 M λ, h t ,
(5.28)

Q (λ, h) 0 (g < ∞) = 1.
(5.29)

Proof of Lemma 5.1 1) For any Γ t ∈ F t we compute :

E Q (λ, h) 0 1 Γt 1 {g>t} = E Q λ, h 0 1 Γt 1 {dt<∞} = E 0 1 Γt 1 {dt<∞} M λ, h dt
, where d t = inf{s > t; R s = 0} is the first time of visit of 0 after time t.

Since M dt = 1 -H(L t ) e -λ 2
2 dt , then according to Doob's optional stopping theorem we have :

E Q (λ, h) 0 1 Γt 1 {g>t} = E 0 1 Γt 1 {dt<∞} 1 -H(L t ) e -λ 2 2 dt .
Applying the Markov property at time t, we get :

E Q (λ, h) 0 1 Γt 1 {g>t} = e -λ 2 t 2 E 0 1 Γt 1 -H(L t ) E Rt e -λ 2 2 T0 = e -λ 2 t 2 E Q (λ, h) 0 1 Γt 1 -H(L t ) E Rt e -λ 2 2 T0 1 M λ, h t .
(5.30) Formula (5.28) now follows immediately from (2.15).

2) Taking Γ t = Ω in (5.30), we have :

Q (λ, h) 0 (g > t) = e -λ 2 t 2 E 0 1 -H(L t ) E Rt e -λ 2 2 T0 ≤ e -λ 2 2 t . Thus, Q (λ, h) 0 (g < ∞) = 1; and it is clear that Q λ, h 0 (g > 0) = 1, since the probabilities Q (λ, h) 0
and P 0 are equivalent on each σ-algebra F t .

To obtain the laws of (R t , t ≤ g) and (R g+t , t ≥ 0), the following lemma constitutes a main step. Lemma 5.2 There exists a (G t , t ≥ 0), Q (λ, h) 0

Brownian motion (W t , t ≥ 0), starting from 0, such that :

R 2α t = 2α t 0 R 2α-1 s dW s +L t -2αλ t∧g 0 R 2α-1 s K α-1 K α (λR s )ds+2αλ t t∧g R 2α-1 s I α-1 I α (λR s )ds. (5.31)
Proof of Lemma 5.2 We proceed in 4 steps.

i) In order to simplify our notation, we define :

A 1 := 2 λ α Γ(1 + α)I α-1 (λR s ) A 2 := λ 2 α Γ(1 -α)I 1-α (λR s ) A 3 := 2 λ α Γ(1 + α)I α (λR s ) A 4 := λ 2 α Γ(1 -α)I -α (λR s ) A 5 := 2 1-α Γ(α) λ α+1 K α-1 (λR s ) A 6 := 2 1-α Γ(α) λ α K α (λR s ) h := h(L s ) H := H(L s ).
(5.32)

Hence, with the help of these notation :

M λ, h s = e -λ 2 s 2 R α s A 3 h + A 4 (1 -H) , (5.33)
Z s = A 6 (1 -H) A 3 h + A 4 (1 -H) .
(5.34)

ii) With the help of (2.3), we know that there exists a ((F t ) t≥0 , P 0 ) Brownian motion (B t , t ≥ 0) such that :

R 2α t = 2α t 0 R 2α-1 s dB s + L t . (5.35)
This leads us to write the following function Ψ 1 (r) as a function of r 2α : Ψ 1 (r) := r α I α (λr) = Ψ 1 (r 2α ), r > 0, (5.36) where :

Ψ 1 (x) = √ xI α (λx 1 2α
), x > 0, (5.37)

Using the first formula of (5.7.9) in ( [START_REF] Lebedev | Special functions and their applications[END_REF], p.110) :

αI α (r) + rI ′ α (r) = rI α-1 (r), (5.38)
it is easy to prove that :

Ψ ′ 1 (x) = λ 2α x 1-α 2α I α-1 (λx 1 2α
), x > 0.

(5.39)

Similarly, introducing :

Ψ 2 (r) = r α I -α (λr) = Ψ 2 (r 2α ), r > 0, (5.40)
and using the second identity of (5.7.9) in ( [START_REF] Lebedev | Special functions and their applications[END_REF], p.110) :

αI -α (r) + rI ′ -α (r) = rI 1-α (r), (5.41)
we get :

Ψ ′ 2 (x) = λ 2α x 1-α 2α I 1-α (λx 1 2α
), x > 0.

(5.42)

Recall that (M λ, h t ) is a ((F t ) t≥0 , P 0 ) martingale and is given by (5.18). Using the above notation, we have :

M λ, h s = e -λ 2 s 2 2 λ α Γ(1 + α) hΨ 1 (R 2α s ) + 2 λ -α Γ(1 -α)(1 -H)Ψ 2 (R 2α s ) ,
then Itô's formula and (5.35) imply :

dM λ, h s = e -λ 2 s 2 λ 2 λ α Γ(1 + α) h(L s )R α s I α-1 (λR s ) + λ 2 λ -α Γ(1 -α)(1 -H(L s ))R α s I 1-α (λR s ) dB s = λe -λ 2 s 2 R α s A 1 h + A 2 (1 -H) dB s .
(5.43)

Next, we consider N s , the numerator of Z s , in (5.28), i.e :

N s := 2 1-α Γ(α) e -λ 2 s 2 (1 -H(L s )) (λR s ) α K α (λR s ) = e -λ 2 s 2 (1 -H)R α s A 6 .
(5.44) Since αK α (r) + rK ′ α (r) = -rK α-1 (r), (5.45) reasoning as previously, we can prove that, under P 0 , (N s , s ≥ 0) is a semi-martingale, whose martingale (M(N ) s ) part satisfies : iii) According to Girsanov's Theorem , we deduce from (5.43), (5.33), the existence of a (F t ) t≥0 , Q (λ, h) 0

dM(N ) s = - 2 1-α Γ(α) λ(1 -H(L s ))e -
Brownian motion B t , t ≥ 0 such that :

B t = B t + λ t 0 A 1 h + A 2 (1 -H) A 3 h + A 4 (1 -H) ds.
(5.48)

Consequently, from (5.47), under Q (λ, h) 0

, the martingale part M Q (Z) t of (Z t ) is given by :

dM Q (Z) s = -(1 -H) A 5 A 3 h + A 4 (1 -H) + λA 6 A 1 h + A 2 (1 -H) A 3 h + A 4 (1 -H) 2 d B s .
(5.49) iv) In this last step, we now use the technique of progressive enlargement of filtrations (see for instance [START_REF] Jeulin | Semi-martingales et grossissement d'une filtration[END_REF], [START_REF] Jeulin | Grossissement de filtrations : exemples et applications[END_REF], [START_REF] Mansuy | In Random Times and Enlargements of Filtrations in a Brownian Setting[END_REF] or [START_REF] Yor | Grossissement de filtrations et absolue continuité de noyaux[END_REF]) under the probability Q (λ, h) 0

. With respect to (G t ), the smallest filtration containing (F t ) and such that g is a (G t ) stopping time, there exists a (G t ), Q (λ, h) 0

Brownian motion (W t , t ≥ 0) starting from 0, such that : (5.54) Using (5.15) we have :

B t = W t + t∧g 0 1 Z u d < Z, B > u - t t∧g 1 1 -Z u d < Z, B > u . ( 5
A 4 -A 6 = λ 2 α Γ(1 -α)I -α (λR s ) - 2 1-α Γ(α) λ α K α (λR s ) = λ 2 α Γ(1 -α)I α (λR s ).
Similarly :

λA 2 + A 5 = λ λ 2 α Γ(1 -α)I 1-α (λR s ) + 2 1-α Γ(α) λ α+1 K α-1 (λR s ) = λ λ 2 α Γ(1 -α)I α-1 (λR s ) = λ(A 4 -A 6 ) A 1 A 3 .
Hence :

a)

First, we study (R t , t ≤ g) under Q

(λ,h) 0 . Let f : R + → R + be a function of class C 2 , whose support does not contain 0. We apply Itô's formula to (5.31) and with the function g(x) = f (x 1/2α ). We compute the two first derivatives of g in terms of those of f :

g ′ (x) = f ′ (x 1/2α ) 1 2α x 1-2α 2α g ′′ (x) = f ′′ (x 1/2α ) 1 4α 2 x 2(1-2α) 2α + 1 -2α 4α 2 f ′ (x 1/2α )x 1-4α 2α .
Consequently, we get, for t < g :

f (R t ) = g(R 2α t ) = t 0 f ′ (R s ) 1 2α R 1-2α s 2αR 2α-1 s dW s - λK α-1 K α (λR s )ds + 1 2 t 0 f ′′ (R s ) 1 4α 2 R 2(1-2α) s + 1 -2α 4α 2 f ′ (R s )R 1-4α s 4α 2 R 2(2α-1) s ds = t 0 f ′ (R s )dW s + t 0 1 2 f ′′ (R s ) + f ′ (R s ) 1 -2α 2R s - λK α-1 K α (λR s ) ds.
This proves that the process (R t , t ≤ g) admits the infinitesimal generator :

L ↓ f (r) = 1 2 f ′′ (r) + 1 -2α 2r - λK α-1 K α (λr) f ′ (r).

b)

In this last step we focus on (R g+t , t ≥ 0). It can be proved analogously, that the infinitesimal generator of this process is L ↑ (this operator is defined by (1.37)). The independence of the processes (R t , t ≤ g) and (R g+t , t ≥ 0) follows from the fact that the stochastic differential equation : Then, the process (R g+t , t ≥ 0) behaves, near 0, as a Bessel process with dimension : δ = 2(1 + α) = 4d. In particular, starting from the origin, it immediately leaves 0, and never comes back to it (since α > 0, hence δ > 2).

R
Remark 5.3 We note that, most likely, the description we have just given of the Q (λ,h) 0 -process may be reproduced for a "general" diffusion. The role of the functions r α I α (λr) and r α I -α (λr) being then played by two linearly independent eigenfunctions of the infinitesimal generator. (See, e.g, [START_REF] Pitman | Bessel processes and infinitely divisible laws[END_REF] for a general framework).

  .23) b) From Proposition 3.1, 3. (b), we have :

0

  ) martingale. Due to (3.22), (3.25) and (3.11) we have :

  .27) Using 3. (a) of Proposition 3.1 the relation (3.27) simplifies, and becomes :

λ 2 s 2 ( 2 =

 22 λR s ) α K α-1 (λR s )dB s = -(1 -H)e -λ 2 s 2 R α s A 5 dB s .(5.46)Thus, denoting (M(Z) s ) the martingale part of the semi-martingale (Z s , s ≥ 0) under P 0 , and using (5.33), (5.46), (5.44) and (5.43), we get :dM(Z) s = M λ, h s dM(N ) s -N s dM λ, -(1 -H) A 5 A 3 h + A 4 (1 -H) + λA 6 A 1 h + A 2 (1 -H) A 3 h + A 4 (1 -H)

ϕ 2 (

 2 s) = λA 1 h + (1 -H)λ(A 4 -A 6 ) A1 A3 A 3 h + (A 4 -A 6 )(1 -H) = λA 1 A 3 = λ I α-1 (λR s ) I α (λR s ) . (5.55)It is clear that plugging (5.53) and (5.55) in (5.52) proves Lemma 5.2.

4 )

 4 Proof of point 3. (c) of Theorem 1.5.

  .50) It is clear that < Z, B >=< M(Z), B >. Consequently ,relations (5.48) and (5.49) imply that :d < Z, B > s = -(1 -H) A 5 A 3 h + A 4 (1 -H) + λA 6 A 1 h + A 2 (1 -H) A 3 h + A 4 (1 -H) A 3 h + A 4 (1 -H) + λA 6 A 1 h + A 2 (1 -H) A 6 (A 3 h + A 4 (1 -H)) R 2α-1 (A 3 h + A 4 (1 -H)) + λA 6 (A 1 h + A 2 (1 -H)) A 3 h + A 4 (1 -H) A 3 h + A 4 (1 -H)) -A 6 (1 -H) R 2α-1

												2	ds.	(5.51)
	Combining (5.35), (5.48) and (5.50), we get :
	R 2α t = L t + 2α -2α	0 0	t t∧g R 2α-1 s A 5 s dW s + 2αλ t 0 A 1 h + A 2 (1 -H) R 2α-1 s ds A 3 h + A 4 (1 -H)	ds
	+ 2α	t t∧g	(1 -H)	A 5 s	ds,
	where we have used (5.34) and :
	1
	1 -Z Hence :
				t						t∧g		t
	R 2α t = L t + 2α	0	R 2α-1 s	dW s + 2α	0	R 2α-1 s	ϕ 1 (s)ds) + 2α	t∧g	R 2α-1 s	ϕ 2 (s)ds,	(5.52)
	with :
	= -	A 5 A 6	= -λ	K α-1 (λR s ) K α (λR s )	,		(5.53)
	and :
	ϕ 2 (s) = λ	A

s = A 3 h + A 4 (1 -H) A 3 h + A 4 (1 -H) -A 6 (1 -H) . ϕ 1 (s) = λ A 1 h + A 2 (1 -H) A 3 h + A 4 (1 -H) -A 5 A 3 h + A 4 (1 -H) + λA 6 A 1 h + A 2 (1 -H) A 6 A 3 h + A 4 (1 -H) 1 h + A 2 (1 -H) A 3 h + A 4 (1 -H) + (1 -H) A 5 (A 3 h + A 4 (1 -H)) + λA 6 (A 1 h + A 2 (1 -H)) (A 3 h + A 4 (1 -H))(A 3 h + A 4 (1 -H)) -A 6 (1 -H) = λA 1 h + (1 -H)(λA 2 + A 5 ) A 3 h + (A 4 -A 6 )(1 -H) .