Denis Villemonais

Approximation of quasi-stationary distributions for 1-dimensional killed diffusions with unbounded drifts

Keywords: quasi-stationary distribution, interacting particle system, empirical process, Yaglom limit, diffusion process. MSC 2000 subject : Primary 65C50, 60K35; secondary 60J60

Approximation of quasi-stationary distributions for 1-dimensional killed diffusions with unbounded drifts

Introduction

Let (X t ) be a killed Markov process with law P, taking its values in E ∪ {∂}, where ∂ is a cemetery point. We denote by τ ∂ = inf{t ≥ 0, X t = ∂} the killing time of (X t ). A probability measure ν on E is called a quasi-stationary distribution (QSD) if, for all t ≥ 0, the distribution of the process X, initially distributed with respect to ν and conditioned to be not killed before time t, is still ν at time t, that is P ν (X t ∈ A|τ ∂ > t) = ν(A) for every A ⊂ E and t ≥ 0. Without loss of generality, we suppose that ∂ is an absorbing point, so that {τ ∂ > t} = {X t = ∂}.

Let µ be a probability measure on E. If it exists and provided it is a probability, the limiting conditional distribution lim t→+∞ P x (X t ∈ .|X t = ∂) is called the Yaglom limit for µ, from the Russian Mathematician A.M. Yaglom. He showed in [START_REF] Yaglom | Certain limit theorems of the theory of branching processes (in Russian)[END_REF] that the limiting conditional distribution of the number of descendants in the n th generation of a Galton-Watson process always exists in the subcritical case.

The existence or uniqueness of such invariant conditional distributions have been proved in a host of contexts. When E is finite, it is proved in [START_REF] Darroch | On quasi-stationary distributions in absorbing discrete-time Markov chains[END_REF] that there exists a unique QSD ν and that the Yaglom limit converges to ν independently of the initial distribution. In [START_REF] Cavender | Quasi-stationary distributions of birth and death processes[END_REF], the case of a birth and death process on N is studied. For this process, the set of QSDs is either empty, or a singleton, or a continuum indexed by a real parameter and given by an explicit recursive formula. This is an exception : most of the known results on QSDs are related with existence or uniqueness problems. In [START_REF] Ferrari | Existence of quasi-stationary distributions. A renewal dynamical approach[END_REF], the existence of a quasi-stationary distribution for a continuous time Markov chain on N killed at 0 is proved under conditions on moments of the killing time, using an original renewal dynamical approach. In [START_REF] Collet | Asymptotic laws for one-dimensional diffusions conditioned to nonabsorption[END_REF], the case of 1-dimensional diffusion on [0, + ∞[ with C 1 drift and killed at 0 are studied, with the assumption that +∞ is a natural boundary. The dependence between the initial measure and the Yaglom limit is explored in [START_REF] Martinez | Domain of attraction of quasi-stationary distributions for the Brownian motion with drift[END_REF] (for a Brownian motion with constant drift killed at 0) and [START_REF] Lladser | Domain of attraction of the quasi-stationary distributions for the Orstein-Uhlenbeck process[END_REF] (for the Orstein-Uhlenbeck process killed at 0). In [START_REF] Steinsaltz | Quasistationary distributions for onedimensional diffusions with killing[END_REF], the case of 1-dimensional diffusions with general killing on the interior of a given interval is investigated. In [START_REF] Cattiaux | Quasi-stationarity distributions and diffusion models in population dynamics[END_REF], the authors study the existence and uniqueness of the QSD for 1-dimensional diffusions killed at 0 and whose drift is allowed to explode at the boundary, which is the case under study in the present paper. See [START_REF] Polett | Quasi-stationary distributions : a bibliography[END_REF] for a regularly updated extensive bibliography on QSD.

In this paper we are concerned with 1-dimensional Itô diffusions with values in ]0, + ∞[∪{∂} killed at 0 and defined by the stochastic differential equation

dX t = dB t -q(X t )dt, X 0 = x > 0,
where B is a standard 1-dimensional Brownian motion and q ∈ C 1 (]0, + ∞[). In [START_REF] Cattiaux | Quasi-stationarity distributions and diffusion models in population dynamics[END_REF], the Yaglom limit of this process is studied and the authors give some conditions on the drift q, which are sufficient for the existence and the uniqueness of the QSD. In particular, they allow the drift to explode at the origin. As explained in the paper, this diffusion is closely related with some Markov mortality models. Such applications need the computation of the process QSD, but the tools used in [START_REF] Cattiaux | Quasi-stationarity distributions and diffusion models in population dynamics[END_REF] are based on spectral theory's arguments and don't allow us to get explicit values. Our aim is to give an easily simulable approximation's method of this QSD.

The problem of QSD's approximation has been already explored in [2], [START_REF] Grigorescu | Hydrodynamic limit for a Fleming-Viot type system[END_REF] when E is a bounded open set of R d and X is a Brownian motion killed at the boundary of E. The diffusion with values in a bounded interval. In a second part, we prove an approximation method for each probability measure ν ǫ , based on the interacting process introduced in [START_REF] Burdzy | Configurational transition in a Fleming-Viot-type model and probabilistic interpretation of Laplacian eigenfunctions[END_REF]. Fix ǫ > 0 and let us describe the interacting particle process of size N ≥ 2: each particle moves independently in ]ǫ,1/ǫ[, each one with law P ǫ until one of them hits the boundary. At this time, the killed particle jumps on the position of an other particle, chosen uniformly between the N -1 remaining one. Then the particles evolve independently, until one of them is killed and so on (see Figure 1). One has to prove that the particles don't degenerate

1 τ 4 τ 3 τ 2 0 t τ 1 X 2 X 1
Figure 1: The interacting particle system (X 1 ,X 2 ) at the boundary. In [2], the authors prove a non-degeneracy result with arguments based on a construction of the d-dimensional Brownian motion due to Itô, where d ≥ 2. It seems that this tool can't be easily generalized to other diffusions. To prove such results under our settings, we build an original coupling between the interacting particle process and an independent particle system of the same size. This coupling is valid for all drifted Brownian motions with continuous bounded drift, killed at the boundary of a bounded interval of R + . It will be used in each step of the proof.

We conclude the paper by two numerical applications. At first, we treat the case of the logistic Feller diffusion introduced in [START_REF] Lambert | The branching process with logistic growth[END_REF] and studied in [START_REF] Cattiaux | Quasi-stationarity distributions and diffusion models in population dynamics[END_REF] with values in ]0, + ∞[, driven by the stochastic differential equation

dZ t = Z t dB t + (rZ t -cZ 2 t )dt, Z 0 = z > 0,
where B is a 1-dimensional Brownian motion and r, c are two positive constants. Clearly, 0 is an absorbing state for this diffusion. In a second time we study in detail the case of the Wright-Fisher diffusion on ]0,1[ conditioned to be killed at 1 (see [START_REF] Huillet | On Wright-Fisher diffusion and its relatives[END_REF]). This diffusion takes values in ]0,1[, and is defined by

dZ t = Z t (1 -Z t )dB t + (1 -Z t )dt, Z 0 = z ∈]0,1[,
where B is a 1-dimensional Brownian motion. This diffusion is absorbed at 1.

From unbounded drift to bounded drift

Let P 0 x be the law of a diffusion process taking its values in ]0, + ∞[∪{∂}, killed when it hits 0 and defined by the stochastic differential equation (SDE)

dX t = dB t -q(X t )dt, X 0 = x > 0,
where B is a 1-dimensional Brownian motion. The drift q is taken in the set of real valued continuously differentiable functions C 1 (]0, + ∞[). We denote by L 0 the infinitesimal generator associated with P 0 .

We define, ∀x ∈]0, + ∞[,

Q(x) = x 1 q(y)dy, dµ(x) = e -2Q(x) dx, and
W (x) = q(x) 2 -q ′ (x).
For all ǫ ∈]0,1/2[, we define P ǫ x as the law of the diffusion taking its values in ]ǫ,1/ǫ[, defined by the SDE

dX t = dB t -q(X t )dt, X 0 = x ∈]ǫ,1/ǫ[
and killed when it hits the boundary {ǫ,1/ǫ}. Let L ǫ be the infinitesimal generator 1/2∆-q∇ with the Dirichlet boundary condition on {ǫ,1/ǫ}. -L ǫ has a simple real eigenvalue λ ǫ (see [22, Theorem KR]) at the bottom of its spectrum. The corresponding eigenfunction η ǫ is positive and belongs to C 2 ([ǫ,1/ǫ]). We choose it so that

1/ǫ ǫ η ǫ (x) 2 dµ(x) = 1. (1)
Let us recall some results of [START_REF] Pinsky | On the convergence of diffusion processes conditioned to remain in a bounded region for large time to limiting positive recurrent diffusion processes[END_REF]:

Theorem [START_REF] Pinsky | On the convergence of diffusion processes conditioned to remain in a bounded region for large time to limiting positive recurrent diffusion processes[END_REF]) The Yaglom limit associated with P ǫ exists for all initial distributions δ x , x ∈]ǫ,1/ǫ[, and doesn't depend on x. This limit is a QSD, which we denote by ν ǫ . Furthermore, we have

dν ǫ (x) = η ǫ (x)dµ(x) 1/ǫ ǫ η ǫ (x)dµ(x) . (2)
In fact, ν ǫ is the unique QSD of the process, as proved in Lemma 19 below, but we won't use it in this section. The aim of this section is to study the asymptotic behaviour of (ν ǫ ) when ǫ goes to 0.

From now, M 1 (]0,+∞[) denotes the space of probability measures on ]0,+∞[ equipped with the weak topology. The following hypotheses arise naturally in the proof of Theorem 1, which is based on a compactness-uniqueness method.

Hypothesis 1 (H1) W is bounded below by -C, where C is a positive constant. Moreover, W (x) → +∞ when x → ∞.

Hypothesis 2 (H2)

+∞ 1 e -2Q(x) dx < +∞ and

1 0 1 W (x) + C + 1 µ(dx) < +∞.

Hypothesis 3 (H3)

+∞ 1 e -2Q(x) dx < +∞ and 1 0

x e -Q(x) dx < +∞.

Theorem 1 Assume that hypotheses (H1) and (H2 or H3) are satisfied. Then

ν ǫ -→ ǫ→0 ν ∈ M 1 (]0, + ∞[),
where ν is a QSD for P 0 , which is equal to the Yaglom limit

lim t→+∞ P 0 x (X t ∈ .|t < τ ∂ ), ∀x ∈]0, + ∞[.

Remark 1

The hypotheses (H1) and (H2 or H3) are the assumptions that are made in [START_REF] Cattiaux | Quasi-stationarity distributions and diffusion models in population dynamics[END_REF] to prove the existence of the Yaglom limit.

Remark 2 If a process satisfies the hypotheses of Theorem 1, then it is killed in finite time a.s or it is never killed a.s. Indeed, assume that the process can be killed in finite time with a positive probability. Then

0 1 e Q(x) x
1 e -Q(y) dy dx < +∞ (see [START_REF] Ikeda | Stochastic differential equations and diffusion processes[END_REF]) and

+∞ 1

e Q(x) dx = +∞ (as a consequence of (H1) and (H2 or H3)). But this two conditions are fulfilled if and only if the process is killed in finite time almost surely (see [14, Theorem 3.2 p.450]).

Remark 3

The existence of a QSD for P 0 can be seen as a consequence of Theorem 1. The existence of the Yaglom limit is proved in [START_REF] Cattiaux | Quasi-stationarity distributions and diffusion models in population dynamics[END_REF]Theorem 5.2].

Remark 4

In Part 2.3, we give the counterpart of Theorem 1 for diffusions with values in a bounded interval.

The end of the section is devoted to the proof of Theorem 1.

Tightness of the family

(ν ǫ ) 0<ǫ<1/2
This part is devoted to the proof of the following result, Proposition 2 Assume that the hypotheses (H1) and (H2 or H3) are satisfied. Then the family (ν ǫ ) 0<ǫ≤1/2 is tight. Moreover, every limit point is absolutely continuous with respect to the Lebesgue measure.

We know that L ǫ η ǫ = -λ ǫ η ǫ , η ǫ ∈ C 2 ([ǫ,1/ǫ]), and η ǫ satisfies the differential equation

1/2η ′′ ǫ (x) -q(x)η ′ ǫ (x) = -λ ǫ η ǫ (x)
with the boundary conditions η ǫ (ǫ) = η ǫ (1/ǫ) = 0.

Define v ǫ = η ǫ e -Q . By (1), we know that:

1/ǫ ǫ v ǫ (x) 2 dx = 1.
We have

v ǫ (x)W (x) -v ′′ ǫ (x) = 2λ ǫ v ǫ (x), with the boundary conditions v ǫ (ǫ) = v ǫ (1/ǫ) = 0. ( 3
)
Lemma 3 Assume that the hypothesis (H1) is fulfilled. Then (v ǫ ) 0<ǫ<1/2 is uniformly bounded above and the family (v 2 ǫ (x)dx) 0<ǫ<1/2 is tight.

Proof of Lemma 3 : From the differential equation satisfied by v ǫ , we have

v ǫ (x) 2 W (x) -v ′′ ǫ (x)v ǫ (x) = 2λ ǫ v ǫ (x) 2 .
Integrating by parts and looking at the boundary conditions (3),

1/ǫ ǫ v ′′ ǫ (x)v ǫ (x)dx = - 1/ǫ ǫ v ′ ǫ (x) 2 dx,
where v ǫ is normalized in L 2 (dx). That implies

1/ǫ ǫ v ′ ǫ (x) 2 dx + 1/ǫ ǫ v ǫ (x) 2 W (x)dx = 2λ ǫ .
The eigenvalue λ ǫ of -L ǫ is given by (see for instance [START_REF] Yosida | Functional analysis[END_REF]chapter XI,part 8])

λ ǫ = inf φ∈C ∞ 0 (]ǫ,1/ǫ[) (L ǫ φ,φ) µ , = inf φ∈C ∞ 0 (]ǫ,1/ǫ[) (L 0 φ,φ) µ , (4)
where C ∞ 0 (]ǫ,1/ǫ[) is the vector space of infinitely differentiable functions with compact support in ]ǫ,1/ǫ[ and (f,g) µ = +∞ 0 f (u)g(u)dµ(u). We deduce from it that λ ǫ increases with ǫ and is uniformly bounded above by λ 1/2 .

We have then

0 ≤ 1/ǫ ǫ v ′ ǫ (x) 2 dx + 1/ǫ ǫ v ǫ (x) 2 (W (x) + C + 1)dx ≤ 2λ 1/2 + C + 1. ( 5
)
Looking at the boundary conditions (3), we obtain, for all x ∈]ǫ,1/ǫ[,

v 2 ǫ (x) = -2 1/ǫ x v ′ ǫ (y)v ǫ (y)dy ≤ - 2 min [x,1/ǫ[ √ W + C + 1 1/ǫ x v ′ ǫ (y)v ǫ (y) W (y) + C + 1dy.
Then, applying the Cauchy-Schwarz inequality to the right term above,

v 2 ǫ (x) ≤ 2 min [x,1/ǫ[ √ W + C + 1 1/ǫ x v ′ ǫ (y) 2 dy 1/ǫ x v ǫ (y) 2 (W (y) + C + 1)dy.
From [START_REF] Chaleyat-Maurel | Un problème de réflexion et ses applications au temps local et aux équations différentielles stochastiques sur R. Cas continu[END_REF], the integral product is bounded by 2λ 1/2 + C + 1, thus ∃A > 0, independent from ǫ, such that

v 2 ǫ (x) ≤ A min [x,1/ǫ[ √ W + C + 1 ≤ A min [x,+∞[ √ W + C + 1 , (6)
where W (x) + C + 1 ≥ 1 for all x ∈]0, + ∞[, thanks to Hypothesis (H1). That implies the first part of Lemma 3.

Let us prove that the family (v 2 ǫ dx) 0<ǫ<1/2 is tight. Fix δ > 0. We have to find a compact subset

K δ in ]0, + ∞[ such that ]0,+∞[\K δ v 2 ǫ (x)dx ≤ δ, (7)
for all ǫ ∈]0,1/2[. Thanks to (6), we have

v 2 ǫ ≤ A, then δ/(2A) 0 v 2 ǫ ≤ δ/2.
From the second part of Hypothesis (H1), ∃M δ > 0 such that W (x) + C + 1 > 2(2λ + C + 1)/δ for all x ≥ M δ . That implies

+∞ M δ v ǫ (x) 2 dx ≤ 1/ǫ ǫ v ǫ (x) 2 δ(W (x) + C + 1) 2(2λ + C + 1) dx ≤ δ/2,
where the last inequality is due to [START_REF] Chaleyat-Maurel | Un problème de réflexion et ses applications au temps local et aux équations différentielles stochastiques sur R. Cas continu[END_REF]. Finally, the compact set K δ = [δ/(2A),M δ ] satisfies [START_REF] Darroch | On quasi-stationary distributions in absorbing discrete-time Markov chains[END_REF]. 2

Lemma 4 Assume that (H1) is satisfied. Then

1/ǫ ǫ η ǫ (y)dµ(y) is uniformly bounded below by a constant B > 0.
Proof of Lemma 4 : Assume that 1/ǫ ǫ η ǫ (y)dµ(y) isn't uniformly bounded below : one can find a sub-sequence

1/ǫ k ǫ k v ǫ k (y)e -Q(y) dy = 1/ǫ k ǫ k η ǫ k (y)dµ(y)
, where ǫ k → 0, which tends to 0. From Lemma 3, (v ǫ ) 0<ǫ≤1/2 is uniformly bounded, so that

1/ǫ k ǫ k v ǫ k (y) 2 e -Q(y) dy → 0.
The family (v ǫ (x) 2 dx) being tight, one can find (after extracting a sub-sequence) a positive map m such that, for all continuous and bounded φ : R + → R,

1/ǫ k ǫ k v ǫ k (y) 2 φ(y)dy → +∞ 0 m(y)φ(y)dy. (8)
Indeed, (v 2 ǫ ) being uniformly bounded, all limit measure is absolutely continuous with respect to the Lebesgue measure. In particular, But min (e -Q(.) ,1) is continuous and positive on R + , so that m vanishes almost every where. Finally, by the convergence property [START_REF] Down | Exponential and uniform ergodicity of Markov processes[END_REF] applied to φ equal to 1 almost everywhere, we have

1/ǫ k ǫ k v ǫ k (y) 2 min (e -Q(y) ,
1 = 1/ǫ k ǫ k v 2 ǫ k dx → 0,
what is absurd. Thus, one can define

B = inf ǫ 1/ǫ ǫ η ǫ (y)dµ(y)/A > 0. 2
Lemma 5 Assume that (H1) and (H2) are satisfied. Then the family (η ǫ (x)dµ(x)) 0<ǫ<1/2 is tight.

Proof of Lemma 5 : By (5), we have

1/ǫ ǫ η 2 ǫ (y)(W (y) + C + 1)dµ(y) = 1/ǫ ǫ v 2 ǫ (y)(W (y) + C + 1)dy ≤ λ 1/2 + C + 1.
For all δ,M > 0, using Cauchy-Schwarz inequality, we get on one hand

δ 0 η ǫ (y)dµ(y) ≤ δ 0 η ǫ (y) 2 (W (y) + C + 1)dµ(y) 1 2 δ 0 1 W (y) + C + 1 dµ(y) 1 2 (9)
≤ λ 1/2 + C + 1 1 2 δ 0 1 W (y) + C + 1 dµ(y) 1 2 . ( 10
)
On the other hand,

+∞ M η ǫ (y)dµ(y) ≤ +∞ M η 2 ǫ (y)dµ(y) 1 2 +∞ M dµ(y) 1 2 (11)
≤ +∞ M dµ(y) 1 2 . ( 12
)
Thanks to (H2), both terms are going to 0 uniformly in ǫ, when δ and M tend respectively to 0 and +∞. As a consequence, the family (η ǫ (x)dµ(x)) 0<ǫ<1/2 is tight. 2 Lemma 6 Assume that (H1) and (H3) hold. Then the family (η ǫ (x)dµ(x)) 0<ǫ<1/2 is tight.

Proof of Lemma 6 :

The first part of the hypothesis (H3) is the same as (H2)'s one, then +∞ M η ǫ (y)dµ(y) → 0 when M goes to infinity, uniformly in ǫ.

Moreover, there exists a constant K > 0 such that, for any x ∈]0,1] and any ǫ ∈]0,1/2],

η ǫ (x) ≤ Kxe Q(x) .
This is a consequence of [3, Proposition 4.3] whose proof is still available under our settings. This inequality allows us to conclude the proof of Lemma 6. 2 Thanks to equality (2) and Lemmas 4, 5 and 6, the first part of Proposition 2 is proved. Moreover, ν ǫ has a density with respect to the Lebesgue measure which is bounded on every compact set, uniformly in ǫ > 0. Thus every limit point is absolutely continuous with respect to the Lebesgue measure.

The limit points of the family

(ν ǫ ) 0<ǫ<1/2
Proposition 7 Assume that Hypotheses (H1) and (H2 or H3) are fulfilled and let ν be a probability measure which is the limit of a sub-sequence (ν ǫ k ) k∈N , where ǫ k → 0 when k → ∞. Then ν is a QSD with respect to P 0 . Proof of Proposition 7 : From Proposition 2, the family (ν ǫ ) 0<ǫ<1/2 is tight. Let ν be a limit point of the family (ν ǫ ) 0<ǫ<1/2 . There exists a sub-sequence (ν ǫ k ) k which converges to ν, where (ǫ k ) k∈N is a decreasing sequence which tends to 0. We already know that ν is absolutely continuous with respect to the Lebesgue measure. That implies that, for all open intervals

D =]c,d[⊂ R + , ν ǫ k (D) → ν(D), (13)
and, for all bounded maps φ continuous on R + ,

R + φ(x)dν ǫ k (x) → R + φ(x)dν(x). (14)
Let ν t (resp. ν ǫ,t ) be the distribution at time t of a diffusion with law P 0 ν (resp. P ǫ νǫ ), conditioned to be not killed until time t, that is

ν t (dx) = P 0 ν (ω t ∈ dx|τ ∂ > t) and ν ǫ,t (dx) = P ǫ νǫ (ω t ∈ dx|τ ∂ > t)
The probability measure ν ǫ being a QSD for P ǫ , we have ν ǫ,t = ν ǫ for all t > 0. We want to show that ν = ν t for all t > 0, we have then to prove the following convergence result:

∀t > 0, ∀D =]c,d[⊂ R + , ν ǫ k ,t (D) -→ k→+∞ ν t (D). ( 15
)
Indeed, suppose that (15) holds, then on the one hand,

ν ǫ k (D) = ν ǫ k ,t (D) → ν t (D). In the other hand ν ǫ k (D) → ν(D). We have then ν t (]c,d[) = ν(]c,d[), ∀]c,d[⊂ R + and this conclude the proof of Proposition 7.
Let us prove [START_REF] Jacod | Limit Theorems for Stochastic Processes[END_REF]. By definition,

ν ǫ k ,t (D) = ]ǫ k ,1/ǫ k [ P ǫ k x (ω t ∈ D)dν ǫ k (x) ]ǫ k ,1/ǫ k [ P ǫ k x (τ ∂ > t)dν ǫ k (x)
.

The numerator is equal to

]ǫ k ,1/ǫ k [ P ǫ k x (ω t ∈ D)dν ǫ k (x) = ]ǫ k ,1/ǫ k [ P 0 x (ω t ∈ D)dν ǫ k (x) + ]ǫ k ,1/ǫ k [ P ǫ k x (ω t ∈ D) -P 0 x (ω t ∈ D) dν ǫ k (x)
For all t > 0, the map x → P 0 x (ω t ∈ D) is continuous and bounded, then, by the convergence property ( 14),

]ǫ k ,1/ǫ k [ P 0 x (ω t ∈ D)dν ǫ k (x) → ]0,∞[ P 0 x (ω t ∈ D)dν(x),
Assume that (H2) is fulfilled, then, similarly to ( 9) and ( 11), we have, for all bounded continuous functions

f :]0, + ∞[→ R and all M > 0, +∞ M f (x)η ǫ k (x)dµ(x) ≤ +∞ M |f (x)| 2 dµ(x) 1 2
, and, for all m > 0,

m 0 f (x)η ǫ k (x)dµ(x) ≤ λ 1/2 + C + 1 1 2 m 0 |f (x)| 2 W (x) + C + 1 dµ(x) 1 2
.

Replacing f (x) by P 0 x (ω t ∈ D) -P ǫ k x (ω t ∈ D), which is decreasing to 0 when k → ∞, and by monotone convergence theorem, we have ]0,m[

P ǫ k x (ω t ∈ D) -P 0 x (ω t ∈ D) dν ǫ k (x) -→ k→+∞ 0 and ]M,+∞[ P ǫ k x (ω t ∈ D) -P 0 x (ω t ∈ D) dν ǫ k (x) -→ k→+∞ 0.
Finally, the density of ν ǫ k being bounded above in every compact set [m,M], uniformly in ǫ k , the same argument of monotone convergence gives us

]ǫ k ,1/ǫ k [ P ǫ k x (ω t ∈ D) -P 0 x (ω t ∈ D) dν ǫ k (x) -→ k→+∞ 0.
With similar arguments, the same holds under (H3). Finally, we obtain

]ǫ k ,1/ǫ k [ P ǫ k x (ω t ∈ D)dν ǫ k (x) → ]0,+∞[ P 0 x (ω t ∈ D)dν(x).
Thanks to [3, Lemma 5.3 and Theorem 2.3], the map

x → P 0 x (τ ∂ > t) = P 0 x (ω t ∈ ]0, + ∞[) is continuous, and P 0 x (τ ∂ > t) -P ǫ k x (τ ∂ > t) is increasing to 0 when k → ∞.
Thus the denominator can be treated in the same way. 2

We can now conclude the proof of Theorem 1:

Proposition 8 Assume that (H1) and (H2 or H3) hold. The limit measure ν in the statement of Proposition 7 is unique. Moreover ν is the Yaglom limit associated with P 0 x , ∀x ∈]0, + ∞[.

Proof of Proposition 8 :

The proof of Proposition 7 implies that

P ǫ νǫ (τ ∂ > t)-→ ǫ→0 P 0 ν (τ ∂ > t), ∀t > 0.
The probability measure ν ǫ being a QSD for P ǫ , we have

P ǫ νǫ (τ ∂ > t) = e -λǫt , ∀t > 0.
Thanks to (4), λ ǫ is decreasing to λ 0 = inf φ∈C ∞ 0 (]0,+∞[) (L 0 φ,φ) µ when ǫ goes to 0. As a consequence, P 0 ν (τ ∂ > t) = e -λ 0 t , ∀t > 0. In this case, the density of ν with respect to dµ is an eigenfunction of L 0 with eigenvalue -λ 0 < 0, where (L 0 ) * is the adjoint operator of L 0 (this is a consequence of the spectral decomposition proved in [3, Theorem 3.2]). As defined, -λ 0 is at the bottom of the spectrum of (L 0 ) * . Thanks to [3, Theorem 3.2], this eigenvalue is simple. Moreover, [3, Theorem 5.2] states that this QSD is equal to lim t→+∞ P 0

x (X t ∈ .|τ ∂ > t), what concludes the proof. 2

Diffusions with values in a bounded interval

Theorem 1 is stated for 1-dimensional diffusions with values in ]0, + ∞[. However, most of the proofs can be easily adapted to diffusions with values in a bounded interval ]a,b[, where -∞ < a < b < +∞, defined by the SDE

dX t = dB t -q(X t )dt, X 0 = x ∈]a,b[,
and killed when it hits a or b. Here B is a standard Brownian motion and q ∈ C 1 (]a,b[).

More precisely, let us denote by P 0 the law of such a diffusion. For each ǫ > 0, define P ǫ as the law of a diffusion with values in ]a + ǫ,b -ǫ[, driven by the SDE

dX t = dB t -q(X t )dt, X 0 = x ∈]a + ǫ,b -ǫ[,
and killed when it hits a + ǫ or b -ǫ. As proved in [START_REF] Pinsky | On the convergence of diffusion processes conditioned to remain in a bounded region for large time to limiting positive recurrent diffusion processes[END_REF], there exists a unique QSD ν ǫ associated with P ǫ .

We define Q(x) =

x (a+b)/2 q(y)dy and W (x) = q(x) 2 -q ′ (x). The counterpart of Theorem 1 under these settings is Theorem 9 Assume that the following hypotheses are fulfilled: Hypothesis 4 (HH1) W is uniformly bounded below by -C, where C is a positive constant.

Hypothesis 5 (HH2) x → 1 W (x)+C+1 e -2Q(x) or x → (x -a)e -Q(x) is integrable on a neighbourhood of a. Hypothesis 6 (HH3) x → 1 W (x)+C+1 e -2Q(x) or x → (b -x)e -Q(x) is integrable on a neighbourhood of b.
Then the family of QSD (ν ǫ ) is tight as family of measures on ]a,b[. Moreover, every limit point of the family (ν ǫ ) 0<ǫ<1/2 is a QSD for P 0 . Remark 5 Our aim isn't to develop this part, but we point out that to show that Proposition 8 remains valid, most of the arguments used to prove the key results [3, Theorem 3.2] and [3, Theorem 5.2] can be adapted to these settings.

3 Approximation of ν ǫ , QSD for P ǫ We are interested in proving an approximation method for the QSD associated with P ǫ . It will be sufficient to prove it for any diffusion (X t ) taking its values in ]0,1[ in place of ]ǫ,1/ǫ[, defined by the stochastic differential equation (SDE)

dX t = dB t -q(X t )dt, X 0 = x > 0, (16)
and killed when X t hits the boundary {0,1}. Here B is a real Brownian motion and q ∈ C 1 ([0,1]). The law of X will be denoted by P.

From [START_REF] Pinsky | On the convergence of diffusion processes conditioned to remain in a bounded region for large time to limiting positive recurrent diffusion processes[END_REF], the QSD of X is unique and equals the Yaglom limit. It will be denoted by ν. For notational convenience, new notations have been defined for this section, which is totally independent of the previous one.

Fix N ≥ 2 and let us define formally the interacting particle process with N particles described in the introduction. Let B 1 ,...,B N be N independent Brownian motions and (X 1 0 ,...,X N 0 ) ∈]0,1[ N be the starting point of the process. For each i ∈ {1,...,N}, the particle X i evolves in ]0,1[ and satisfies the SDE dX i t = dB i t -q(X i t )dt (and then it is independent of the others) until τ i 1 = inf{t ≥ 0, X i t-= 0 or 1}.

At time τ 1 = min{τ 1 1 ,...,τ 1 N }, the path of a particle, denoted by i 1 (it is unique), has a left limit equal to 0 or 1.

A particle j 1 is chosen in {1,...,N} \ {i 1 }. The particle i 1 jumps on the position of the particle j 1 : we set X i 1 τ 1 := X j 1 τ 1 . After time τ 1 , each particle X i evolves in ]0,1[ with respect to the SDE dX i t = dB i t -q(X i t )dt until τ i 2 = inf{t > τ 1 , X i t-= 0 or 1}. At time τ 1 , all the particles are in ]0,1[, so that we have τ i 2 > τ 1 for all i ∈ {1,...,N} almost surely. At time τ 2 = min{τ 1 2 ,...,τ N 2 } (which is then strictly bigger than τ 1 ), a unique particle i 2 has a path whose left limit is equal to 0 or 1.

A particle j 2 is chosen in {1,...,N} \ {i 2 }. The particle i 2 jumps on the position of the particle j 2 : we set X i 2 τ 2 := X j 2 τ 2 . After time τ 2 , the particles evolve independently from each other and so on.

Following this way, we define the strictly increasing sequence of stopping times 0 < τ 1 < τ 2 < τ 3 < ..., the time τ ∞ = lim n→∞ τ n and the interacting particle system (X 1 t ,...,X N t ) for all t ∈ [0,τ ∞ [. The law of (X 1 ,...,X N ) will be denoted by P ipp .

We can now state the main result of this section:

Theorem 10 (X 1 ,...,X N ) is well defined, that means τ ∞ = +∞ almost surely. It is geometrically ergodic, with unique stationary distribution M N . Let X N be the empirical stationary measure of the interacting particle process with N particles, that is the empirical measure of a random vector (x 1 ,...,x N ) ∈]0,1[ N distributed with respect to the stationary measure M N of the process (X 1 ,...,X N ). The sequence of random measures (X N ) N ≥2 converges in law to the deterministic measure ν, QSD of the process X. Subsection 3.1 is devoted to prove a coupling which ensures the non-degeneracy of the particles at the boundary. A consequence will be that τ ∞ = +∞ almost surely. In Subsection 3.2, the process is studied in finite time. We prove that the empirical measure of the process (X 1 ,...,X N ) at time t converges, when N goes to infinity, to the distribution of X t conditioned to not be killed until time t, which is P µ 0 (X t ∈ .|X t = ∂) (µ 0 denotes the limit of the empirical measure of the process at time 0). We prove in Subsection 3.3 that the interacting particle system (X 1 ,...,X N ) is geometrically ergodic. We conclude by showing the convergence of the empirical stationary measure to the QSD ν.

Existence and non-degeneracy at the boundary

One of the most important fact when studying the interacting particle system is that the particles don't degenerate at the boundary. This is an evidence if the particles are independent. In our case, we will prove a coupling between (X 1 ,...,X N ) and an other process (Y 1 ,...,Y N ) whose components are independent identically distributed and don't degenerate at the boundary and such that, for all i ∈ {1,...,N},

0 ≤ Y i ≤ X i ≤ 1 -Y i ≤ 1 a.s. ( 17
)
With this construction, the process (X 1 ,...,X N ) doesn't degenerate at the boundary, because each of its particles

X i is contained in [Y i ,1 -Y i ]
. This coupling will be useful in each step of the proof.

dY i t = dW i t -Qdt, Y i (0) = min X i (0),1 -X i (0),1/3 ,
with 0 and 1/3 as reflecting boundary (see [START_REF] Chaleyat-Maurel | Un problème de réflexion et ses applications au temps local et aux équations différentielles stochastiques sur R. Cas continu[END_REF] for the definition of a reflected diffusion).

The coupling inequality ( 17) is fulfilled at time t = 0. The Brownian motion W i will depend on B i and on the position of X i . If X i t belongs to [0,1/3], then X i t ≤ 1/3 and the second part of the coupling inequality is satisfied, independently of the choice of W i . We only need to ensure that X i t stays bigger than Y i t . If X i t belongs to [2/3,1[, we only need to ensure that it is smaller than 1 -Y i t . If it belongs to ]1/3,2/3[, the coupling inequality is obviously fulfilled, thanks to the reflection of Y i on 1/3. Assume that X i t is in ]0,1/3] and that the coupling inequality is fulfilled at time t. We have d(

X i t -Y i t ) = dB i t -dW i t + (Q -q(X t ))dt, with Q -q(X t ) ≥ 0. If we choose W i so that dB i t -dW i t = 0, then d(X i t -Y i t ) = (Q -q(X t ))
dt is increasing with time almost surely and the coupling inequality remains fulfilled. In a similar way, if X i t is in [2/3,1[, we have to choose W i so that dB i t + dW i t = 0. We will see in the proof of the coupling inequality that the jumps of X i and the reflexion of Y i on 0 do not play any role in the coupling inequality (see Figure 2). Let us define the Brownian motion W i in an explicit form. First, we build a sequence of strictly increasing times (t i n ) n≥0 such that, for each n ≥ 0,

1 0 t 2 3 1 3 t 0 t 1 t 2 t 3 Y 1 X 1 1 -Y 1
X i t ∈]0,2/3[ for all t ∈ [t i 2n ,t i 2n+1 [ and X i t ∈]1/3,1[ for all t ∈ [t i 2n+1 ,t i 2n+2 [. Define t i 0 = inf {t ∈ [0, + ∞], X i t ∈]0,1/3]}, t i 1 = inf {t ∈ [t 0 , + ∞], X i t ∈ [2/3,1[},
and, for n ≥ 1,

t i 2n = inf {t ∈ [t i 2n-1 , + ∞], X i t ∈]0,1/3]}, t i 2n+1 = inf {t ∈ [t i 2n , + ∞], X i t ∈]2/3,1]}.
Depending on the position of X i , which is stated by the sequence (t n ) n≥2 , we define W i by

W i t = -B i t for t ∈ [0,t 0 ],
and, for all n ≥ 0,

W i t = W i t 2n + (B i t -B i t 2n ) for t ∈ [t 2n ,t 2n+1 ] W i t = W i t 2n+1 -(B i t -B i t 2n+1 ) for t ∈ [t 2n+1 ,t 2n+2 ].
If lim n→∞ t n < +∞, then W i can be extended by continuity on [0, + ∞[, so that Y i t is well defined for all t ∈ [0, + ∞[.

The sequence (t i n ) n∈N is a sequence of stopping times for the natural filtration of the process (X 1 ,...,X N ). Conditionally to the sequence (j n ) n≥1 (the particles which are chosen at each successive jump), (X 1 ,...,X N ) only depends on (B 1 ,...,B N ). One can then apply the strong Markov property to (B 1 ,...,B N ) at time t i n , ∀i ∈ {1,...,N} and ∀n ∈ N. As a direct consequence of the symmetry of a Brownian motion's law and of the strong Markov property applied to the N-dimensional Brownian motion (B 1 ,...,B i ,...,B N ), W i is a Brownian motion. Note that the sequence of stopping times depends clearly on the position of other particles and then on the B j 's, for j ∈ {1,...,N}. However, the Brownian motions {W j } j=1,...,N are independent processes. Indeed, the sequence (t n ) n≥0 doesn't play any role in the law of W i and, to be convinced of that, one can compute the covariance matrix of the Gaussian variables W j t , for any t ∈ [0, + ∞[, which is clearly a diagonal one.

i , Y i ζ = 0, so that X i ζ = 0, what is impossible. Therefore X i ζ-= 1 > 1 -Y i ζ-
, and, by existence of left limits for the two processes, ∃t < ζ such that

X i t > 1 -Y i t . Define ζ ′ = inf {t ≥ 0, 1 -Y i t < X i t }. We have then, conditionally to ζ < τ ∞ , ζ ′ < ζ.
By symmetry, conditionally to the event

ζ ′ < τ ∞ , we have ζ < ζ ′ , then ζ < τ ∞ and therefore ζ < ζ ′ . Finally, ζ = ζ ′ = τ ∞ almost surely. 2

Existence of the interacting particle process

Proposition 12 For all N ≥ 2, the interacting particle system (X 1 ,...,X N ) is well defined, that is τ ∞ = +∞ almost surely.

Proof of Proposition 12 :

Let N ≥ 2 be the size of the interacting particle system and fix arbitrarily its starting point x ∈]0,1[ N . We define the event C x = {τ ∞ < +∞}.

Conditionally to C x , the total number of jumps is equal to +∞. There is a finite number of particles, then at least one particle, say i 0 , makes an infinite number of jump before τ ∞ . At each jump of i 0 , a particle is uniformly chosen in {1,...,N}. By the law of large numbers, each particle is chosen infinitely often before τ ∞ . Assume that a particle, say j 0 , remains all the time in ]ǫ,1 -ǫ[, with ǫ > 0. i 0 will jump on the position of j 0 infinitely often. Then it will come back from ]ǫ,1 -ǫ[ to the boundary infinitely often in finite time, what is impossible. We deduce that, conditionally to C x , all particles of the interacting particle system are going near to the boundary, that is

C x ⊂ lim t→τ∞ min (X i t ,1 -X i t ) = 0 , (18)
for each particle i ∈ {1,2,...,N}.

Using the coupling inequality of Proposition 11, we deduce from (18) that

C x ⊂ lim t→τ∞ (Y 1 t ,...,Y N t ) = 0 .
Then, conditionally to C x , Y 1 and Y 2 are independent reflected diffusions with bounded drifts, which hit 0 at the same time. This occurs for two independent reflected Brownian motions with probability 0, and then for Y 1 and Y 2 too, by the Girsanov's Theorem. That implies P (C x ) = 0. Finally, we have τ ∞ = +∞ almost surely.

Remark 6 One could hope to apply this method directly to the process with law P 0 studied in the first part of this paper. Unfortunately, it can be very difficult to show the existence of the process or the non-degeneracy at the boundary: the drift being unbounded, the law of the reflected diffusion used in this proof isn't absolutely continuous with respect to the law of the reflected Brownian motion and stay at 0 all the time after hitting it.

Non-degeneracy at the boundary

For all r > 0, we define the open set D r =]r,1 -r[. Let µ N (t,dx) (resp. µ ′N (t,dx)) be the empirical measure of the system of particles (X i t ) i=1,...,N (resp. (Y i t ) i=1,...,N ), that is

µ N (t,dx) = 1 N N i=1 δ X i t (dx) and µ ′N (t,dx) = 1 N N i=1 δ Y i t (dx).
We will suppose that, at time 0, the sequence of empirical measures (µ N (0,dx)) N ≥2 satisfies the following non-degeneracy property, which ensures that the mass of µ N (0,dx) doesn't degenerate at the boundary, uniformly in N:

Definition 1 The family of random probabilities {µ N (dx)} is said to verify the nondegeneracy property if, for any ǫ > 0, lim r→0 lim sup

N →∞ P µ N (D c r ) > ǫ = 0, ( 19
)
where

D c r =]0,r] ∪ [1 -r,1[.
From definition of (Y i 0 ) i∈{1,...,N } , the non-degeneracy of µ ′N (0,dx) is the consequence of the non-degeneracy of µ N (0,dx). The end of the section is devoted to prove the following Proposition, which states that the non-degeneracy property is maintained over time:

Proposition 13 Assume that (µ N (0,dx)) N ≥2 satisfies the non-degeneracy property, then, for all T > 0 and all ǫ > 0, lim r→0 lim sup

N →∞ P ( sup t∈[0,T ] µ N (t,D c r ) > ǫ) = 0.
Proof of Proposition 13 : Fix T > 0 and ǫ > 0. Because of the non-degeneracy of µ ′N (0,dx), one can find a > 0 such that

P 1 N N i=1 1 Y i 0 ≤a ≥ ǫ/2 → 0 ( 20
)
when N goes to ∞.

We want to apply a law of large numbers, but the Y i 0 aren't independent and depend on the number of particles N. Let us define the diffusion Z i with values in [0,1/3], defined by the SDE

dZ i t = dW ′i t -Qdt, Z i (0) = a,
with 0 and 1/3 as reflecting boundary. Here the W ′i are independent Brownian motions. The random processes 1 Z i . ∈D c r are independent, identically distributed with values in D([0,T ],R). One can apply to them the following law of large numbers, proved in [START_REF] Rao | The law of large numbers for D[0,1]-valued random variables[END_REF]:

sup t∈[0,T ] 1 N N i=1 1 Z i t ∈D c r -E 1 Z i t ∈D c r P rob -→ N →∞ 0. We have E 1 Z i . ∈D c r = P (Z i
. ∈ D c r ), which tends uniformly to 0 when r → 0. We deduce from it that lim r→0 lim

N →∞ P sup t∈[0,T ] 1 N N i=1 1 Z i t ∈D c r ≥ ǫ/2 = 0. ( 21
)
For each number of particles N ≥ 2, one can easily find a coupling between Z i and Y i , where

Z i 0 ≤ Y i 0 implies Z i t ≤ Y i t for all t ∈ [0,T ].
With such a coupling, we have

1 N N i=1 1 Y i t ∈D c r 1 Y i 0 ≥a ≤ 1 N N i=1 1 Z i t ∈D c r .
By adding the contribution of the Y i which starts in ]0,a[, we get

1 N N i=1 1 Y i t ∈D c r ≤ 1 N N i=1 1 Z i t ∈D c r + 1 N N i=1 1 Y i 0 <a .
The limits [START_REF] Meyn | Markov Chains and Stochastic Stability[END_REF] and ( 21) allow us to conclude the proof.

Convergence in finite time

Fix T > 0. This section is devoted to the proof of the following proposition, which states that the empirical measure process converges to the distribution of the process X conditioned to not be killed.

Proposition 14 Assume that (µ N (0,dx)) N ∈N converges in law to the random probability measure µ(0,dx) with respect to the weak topology and satisfies the non-degeneracy property.

Then, ∀T > 0, the measure processes (µ N (t,dx)) t∈[0,T ] converge in law to (P(X t ∈ dx|X t = ∂)) t∈[0,T ] in the Skorokhod space D([0,T ],M 1 (]0,1[)) when N → ∞. Here M 1 (]0,1[) denotes the space of probability measures on ]0,1[ equipped with the weak topology.

Proof of Proposition 14 : For all maps ψ ∈ C 2 0 ([0,1]) vanishing on {0,1}, one can apply the Itô's formula to the semimartingale ψ(X i t ) (see [START_REF] Métivier | Semimartingales, a Course on Stochastic Processes[END_REF]Theorem 27.1]), whose number of jumps in [0,T ] is finite almost surely :

ψ(X i t ) = ψ(X i 0 ) + t 0 ψ ′ (X i s-)dB i s + t 0 ψ ′ (X i s-)q(X i s-) + 1 2 ψ ′′ (X i s-) ds + 0≤s≤t ψ(X i s ) -ψ(X i s-) (22)
Let us denote by (τ i n ) 1≤n the increasing sequence of jump times of the particle i. We have

0≤s≤t ψ(X i s ) -ψ(X i s-) = 0≤τ i n ≤t ψ(X i τn ) -ψ(X i τn-) (23)
= 0≤τ i n ≤t ψ(X i τn ), (24)
because

X i τ i n -∈ {0,1} and ψ(0) = ψ(1) = 0. That implies 0≤s≤t ψ(X i s ) -ψ(X i s-) = 0≤τ i n ≤t ψ(X i τ i n ) - 1 N -1 N j=1 ψ(X j τ i n -) (25)
+ 1 N -1 N j=1 0≤τ i n ≤t ψ(X j τ i n -). (26)
By summing over i ∈ {1,...,N}, we obtain

µ N (t,ψ) = µ N (0,ψ) + t 0 µ N (s-,ψ ′ q + 1 2 ψ ′′ )ds + M c (ψ,t) + M j (ψ,t) + 1 N -1 0≤τn≤t µ N (τ n -,ψ), ( 27
)
where

M c (ψ,t) is the continuous martingale 1 N N i=1 t 0 ψ ′ (X i s-)dB i s and M j (ψ,t) is the pure jump martingale M j (ψ,t) = 1 N N i=1 0≤τ i n ≤t ψ(X i τ i n ) - 1 N -1 N j=1 ψ(X j τ i n -) . (28)
Now, we interpret each jump as a killing. Then we introduce a loss of 1/N of the total mass at each jump: we look at the measure process µ N decreased by a factor N -1 N at each jump. More precisely, we set

ν N (t,dx) = N -1 N A N t µ N (t,dx),
where A N (t) = # N i=1 {τ i n , 0 ≤ τ i n ≤ t} denotes the total number of jumps before time t. Lemma 15 The sequence of measure processes ν N (.,dx) N converges in law to P µ 0 (X. ∈ dx) in the Skorokhod topology D([0,T ],M(]0,1[)).

Proof of Lemma 15 :

Applying the Itô's formula to the semimartingale ν N (t,ψ), we deduce from ( 27) that

ν N (t,ψ) = ν N (0,ψ) + t 0 ν N (s-,ψ ′ q + ψ ′′ )ds + t 0 N -1 N A N s- dM c (ψ,s) + 0≤τn≤t (ν N (τ n ,ψ) -ν N (τ n -,ψ)),
Where we have

ν N (τ n ,ψ) -ν N (τ n -,ψ) = N -1 N A N τn µ N (τ n ,ψ) -µ N (τ n -,ψ) + µ N (τ n -,ψ) N -1 N A N τn - N -1 N A N τn- , with µ N (τ n ,ψ) -µ N (τ n -,ψ) = 1 N -1 µ N (τ n -,ψ) + M j (ψ,τ n ) -M j (ψ,τ n -)
and

N -1 N A N τn - N -1 N A N τn- = - 1 N -1 N -1 N A N τn , then ν N (τ n ,ψ) -ν N (τ n -,ψ) = N -1 N A N τn M j (ψ,τ n ) -M j (ψ,τ n -) . = N -1 N N -1 N A N τn- M j (ψ,τ n ) -M j (ψ,τ n -) That implies ν N (t,ψ) -ν N (0,ψ) - t 0 ν N (s-,ψ ′ q + 1 2 ψ ′′ )ds = t 0 N -1 N A N s- dM c (ψ,s) + N -1 N 0≤τn≤t N -1 N A N τn- M j (ψ,τ n ) -M j (ψ,τ n -)
We deduce that for all smooth functions Ψ(t,x) vanishing at the boundary

ν N (t,Ψ(t,.)) -ν N (0,Ψ(0,.)) - t 0 ν N (s-, ∂Ψ(s,.) ∂s + ∂Ψ(s,.) ∂x q + 1 2 ∂ 2 Ψ(s,.) ∂x 2 )ds = t 0 N -1 N A N s- dM c (Ψ(s,.),s) + N -1 N 0≤τn≤t N -1 N A N τn- M j (Ψ(τ n ,.),τ n ) -M j (Ψ(τ n -,.),τ n -) (29) Because N -1 N A N
s-≤ 1 a.s. and by the Doob's inequality , we have

E   sup t∈[0,T ] t 0 N -1 N A N s- dM c (Ψ(t,.),s) 2   ≤ 1 N T ∂Ψ ∂x 2 ∞ . (30)
Note that the jumps of the martingale M j are smaller than 2 N Ψ ∞ , then

E 0≤τn≤T N -1 N 2Aτ n- M j (Ψ(τ n ,.),τ n ) -M j (Ψ(τ n -,.),τ n -) 2 ≤ 4 N 2 Ψ 2 ∞ E 0≤τn≤T N -1 N 2Aτ n- ≤ 4 N Ψ 2 ∞
By the Doob's inequality, we have then

E   sup t∈[0,T ] 0≤τn≤t N -1 N A N τn- M j (Ψ(τ n ,.),τ n ) -M j (Ψ(τ n -,.),τ n -) 2   ≤ 4 N Ψ 2 ∞ , (31)
Define Ψ(s,x) = P t-s f (x), where f ∈ C ∞ ([0,1]) vanishes on {0,1}, and (P t ) is the semigroup associated with the diffusion X defined by [START_REF] Jakubowski | Tightness criteria for random measures with applications to the principle of conditioning in Hilbert spaces[END_REF]. From Kolmogorov's equation (see [9, Proposition 1.5 p.9]),

∂ ∂s Ψ(s,x) + 1 2 ∆Ψ(s,x) + q(x)∇Ψ(s,x) = 0.
We deduce from (29), (30) and (31), that

E sup t∈[0,T ] ν N (t,f ) - 1 0 P t f (x)dµ N (0,x) 2 ≤ 1 N C(f ),
where C(f ) is a positive constant, which only depends on f . For each map g ∈ C ∞ ([0,1]), one can set f r (x) = γ r (x)g(x), with r > 0, where γ r ∈ C ∞ ([0,1]) is equal to 1 on ]2r, 1-2r[ and vanishes on ]0,r[∪]1 -r,1[. Then ν N (t,g) -

1 0 P t g(x)dµ N (0,x) ≤ ν N (t,f r ) - 1 0 P t f r (x)dµ N (0,x) + ν N (t,(1 -γ r )g) (32) + 1 0 P t ((1 -γ r )g) (x)dµ N (0,x) , (33)
where (32) (see Proposition 13) and (33) are going to 0 when r tends to 0, uniformly in N, and

f r ∈ C ∞ ([0,1]) vanishes on {0,1}. Then sup t∈[0,T ]
ν N (t,g) -

1 0 P t g(x)dµ N (0,x) L 2 -→ N →∞ 0.
In particular, ν N (.,dx) converges in law to P µ 0 (X. ∈ dx). 2

Let us conclude the proof of Proposition 14. From Lemma 15,

(ν N (t,]0,1[),ν N (t,dx)) t∈[0,T ] law -→ N →∞ (P µ(0,dx) (X t ∈]0,1[),P µ(0,dx) (X t ∈ dx)) t∈[0,T ] in the Skorokhod topology D([0,T ],R × M 1 (]0,1[)). That means N -1 N A N t ,ν N (t,dx) t∈[0,T ] law -→ N →∞ (P µ(0,dx) (X t ∈]0,1[),P µ(0,dx) (X t ∈ dx)) t∈[0,T ] .
The process P µ(0,dx) (X. ∈]0,1[) never vanishing almost surely and the limit process being continuous almost surely, we have

N -1 N A N t ν N (t,dx) t∈[0,T ] law -→ N →∞ P µ(0,dx) (X t ∈ dx)/P µ(0,dx) (X t ∈]0,1[) t∈[0,T ] in the Skorokhod topology D([0,T ],M 1 (]0,1[)). That means (µ N (t,dx)) t∈[0,T ] law -→ N →∞ (P µ(0,dx) (X t ∈ dx|X t = ∂)) t∈[0,T ]
in the Skorokhod topology D([0,T ],M 1 (]0,1[)). The proof of Proposition 14 is then complete.

Existence and convergence of the empirical stationary measures X N

For each N ≥ 2, we say that the interacting particle process (X 1 ,...,X N ) is exponentially ergodic, if there exists a probability measure M N on ]0,1[ N such that,

||P x ((X 1 t ,...,X N t ) ∈ .) -M N || T V ≤ C(x)ρ t , ∀x ∈]0,1[ N , ∀t ∈ R + , (34)
where C(x) is finite, ρ < 1 and ||.|| T V is the total variation norm. In particular, M N is a stationary measure for the process (X 1 ,...,X N ). When M N exists, we denote by X N the empirical stationary measure associated with M N , that is a random probability which is distributed as 1 N N i=1 δ x i , where (x 1 ,...,x N ) is a random vector in ]0,1[ N distributed with respect to M N .

In a first time, we prove that for all N ≥ 2, the interacting particle process with N particles (X 1 ,...,X N ) associated with X is exponentially ergodic. We conclude by proving that (X N ) N converges in law to the unique QSD of X.

Exponential ergodicity

Here N ≥ 2 is fixed. We are interested in proving the following result, which is the first part of Theorem 10 Proposition 16 The interacting particle process (X 1 ,...,X N ) with law P ipp is exponentially ergodic.

Proof of Proposition 16 : We focus on the 1-skeleton of the interacting particle process with N particles, which is the Markov chain (X 1 n ,...,X N n ) n∈N . Thanks to [8, Theorem 5.3 p.1681], exponential ergodicity of (X 1 ,...,X N ) will be obtained as soon as the associated 1-skeleton is geometrically ergodic, which means that it exists a probability measure π N on ]0,1[ N such that

||P x ((X 1 n ,...,X N n ) ∈ .) -π N || T V ≤ C 0 (x)ρ n 0 , ∀x ∈]0,1[ N , ∀n ∈ N,
where C 0 (x) is finite and ρ 0 < 1.

To prove the geometrical ergodicity of the 1-skeleton, let us introduce the following definition:

Definition 2 C ⊂]0,1[ N is said to be a small set for the Markov chain (X 1 n ,...,X N n ) n∈N if, for some nontrivial probability measure ϑ and some n ≥ 1, ǫ > 0, the n-step transition probability kernel P n (x,A) := P ipp x ((X 1 n ,...,X N n ) ∈ A) satisfies, for all x ∈ C,

P n (x,A) ≥ ǫϑ(A), A ∈ B(]0,1[ N ).
Lemma 17 All compact set C = [r,1 -r] N (with r > 0) is a small set for the 1-skeleton. Moreover, ∃κ > 0 so that

sup x∈C E x (κ τ ′ C ) < ∞, ( 35
)
where τ ′ C is the return time to C.

Proof of Lemma 17: Fix r > 0 and let F be the event "the process (X 1 ,...,X N ) has no jumps between times 0 and 1". Define p = inf x∈[r,1-r] N P ipp x (F ). Thanks to the coupling with (Y 1 ,...,Y N ), we have p > 0. Conditionally to the event F , the particles of the interacting particle process are independent from each other.

Let us study ϑ 1 (dx) = P ipp (x 1 ,...,x N ) (X 1 1 ∈ dx and F ). The law of X 1 conditionally to F is the same as the law of X conditioned to not jump, because, given this last event, the process X 1 doesn't depend on the other particles. Thus the probability of "X 1 1 ∈ dx and F " is P x 1 (X 1 ∈ dx). The law of X 1 has a density p 1 (x 1 ,y) with respect to the Lebesgue's measure and p 1 (x 1 ,y) depends continuously on x 1 and y. It only vanishes when y = 0 or 1. Then inf

(x 1 ,y)∈[r,1-r]×[r,1-r] p 1 (x 1 ,y) > 0
Denoting this minimum by ǫ ′ , we have, for all

x 1 ∈ [r,1 -r], ϑ 1 (dx) ≥ ǫ ′ 1 [r,1-r] (x)dx.
Conditionally to F , the particles are independent from each other, so that

P ipp (x 1 ,...,x N ) ((X 1 1 ,...,X N 1 ) ∈ dy 1 ...dy N |F ) = N i=1 P ipp (x 1 ,...,x N ) (X i 1 ∈ dy i |F ),
where P ipp (x 1 ,...,x N ) (X i 1 ∈ dy i |F ) is greater than P ipp (x 1 ,...,x N ) (X i 1 ∈ dy i and F ) and then greater than ϑ 1 (dy i ). Finally, we have

P ipp (x 1 ,...,x N ) ((X 1 1 ,...,X N 1 ) ∈ dy 1 ...dy N |F ) ≥ ǫ ′ N 1 [r,1-r] N (y 1 ,...,y N )dy 1 ...dy N ,
so that [r,1 -r] N is a small set. For all x ∈]0,1[ N , and all n ≥ 1, the probability of being in C at time n + 1 starting from x at time n is bounded below by the probability p C > 0 for (Y 1 ,...,Y N ) to enter C at time n + 1, starting from 0 at time n. Hence, at each time n ≥ 1, (X 1 ,...,X N ) returns to C at time n + 1 with a probability greater than p C > 0. That implies that the return time to C for the 1-skeleton of the interacting particle process with N particles is bounded above by a time of geometrical law, independent of the starting point x ∈]0,1[ N , and then satisfies condition (35). 2

The chain (X 1 n ,...,X N n ) n∈N is aperiodic. Moreover, if the Lebesgue measure of a subset A ⊂]0,1[ N is strictly positive, then P ipp x (τ A < ∞) > 0 for all x ∈]0,1[ N , where τ A is the first hitting time on A for the chain (X 1 n ,...,X N n ) n∈N . Thanks to [8, Theorem 2.1 p.1673], if such a Markov chain has a small set which satisfies (35), then it is geometrically ergodic. As a consequence, Lemma 17 allows us to conclude the proof of Proposition 16.

Convergence to the QSD

We are interested in proving the following result, which is the second part of Theorem 10 Proposition 18 The sequence of random measures X N N ≥2 converges in law to the deterministic measure ν, QSD of the process X.

Proof of Proposition 18 : For each r ∈]0,1/4[, we define γ r as a non-negative continuous bounded function from ]0,1[ to R, equal to 1 on D 2r and equal to 0 on D c r . We have

X N (D c r ) ≤ X N (1 -γ r ), ∀r ∈]0,1/4[.
Thanks to Proposition 16, the sequence of random measures (µ N (t,dx)) N ≥2 converges in law to X N when t tends to +∞. That implies

E µ N (t,1 -γ r ) -→ t→+∞ E X N (1 -γ r ) , ∀r ∈]0,1/4[.
We denote by µ ′N (t,dx) the empirical measure of (Y 1 t ,...,Y N t ). Let us choose γ r monotone on ]r,2r[ and ]1 -2r,1 -r[. From the coupling inequality,

µ N (t,1 -γ r ) ≤ µ ′N (t,1 -γ r ) for all t ∈ R + and r > 0. Then E µ N (t,1 -γ r ) ≤ E µ ′N (t,1 -γ r ) ,
which tends to 0 when t tends to +∞ and r to 0, uniformly in N. As a consequence,

E(X N )(1 -γ r )-→ r→0 0, where E(X N ) is the deterministic measure defined by E(X N )(A) = E(X N (A)), for all measurable set A. That yields E(X N )(D c r )-→ r→0 0,
uniformly in N. The family of intensity measures (E(X N )) N is then tight. This is a sufficient condition for the family of random variables (X N ) to be tight, as shown in [16, Corollary 2.2]. We conclude that it exists a sub-sequence (X φ(N ) ) which converges in law to a random probability measure X . Choose µ N (0,dx) = X N (dx). The non-degeneracy property is fulfilled. Thanks to Proposition 14,

(µ φ(N ) (t,dx)) t∈[0,T ] law -→ N →∞ (P X (X t ∈ dx|X t = ∂)) t∈[0,T ] , ∀T > 0 in the Skorokhod topology D([0,T ],M 1 (]0,1[))
. The limiting process (P X (X t ∈ dx|X t = ∂)) t∈[0,T ] being almost surely continuous, µ φ(N ) (t,dx) law -→ N →∞ P X (X t ∈ dx|X t = ∂), ∀t > 0, with respect to the weak topology of M 1 (]0,1[). By stationarity, the random probability measures µ φ(N ) (t,dx) and X φ(N ) have the same law. Making N tend to ∞, we deduce that P X (X t ∈ dx|X t = ∂) and X have the same law too. This looks like a QSD, but X is a priori a random measure and we need the following result to conclude.

Lemma 19 For all m ∈ M 1 (]0,1[), lim t→∞ P m (X t ∈ A|τ > t) = ν(A), where τ is the killing time of the process X and ν its unique QSD.

Proof of Lemma 19 : Let m be a probability measure on ]0,1[. ∃λ 0 > 0, φ 0 and φ 0 two continuous maps vanishing on 0 and 1 such that, for all x ∈]0,1[ (see R.G. Pinsky's explanations [22, Thanks to Lemma 19, P X (X t ∈ dx|X t = ∂) converges almost surely to the Yaglom limit when t → +∞, and so do X . Finally, X is the unique QSD of the process and the proof is complete. 2 4 Numerical applications

The logistic case

We apply our result to the logistic Feller diffusion with values in ]0, + ∞[, driven by the stochastic differential equation

dZ t = Z t dB t + (rZ t -cZ 2 t )dt, Z 0 = z > 0, (38)
and killed when it hits 0. Here B is a 1-dimensional Brownian motion and r,c are two positive constants. We define P 0 as the law of 2 √ Z., which is killed at 0 and satisfies the SDE

dX t = dB t - 1 2X t - rX t 2 + cX 3 t 4 dt, X 0 = x ∈]0, + ∞[.
For each ǫ > 0, we define the law P ǫ and denote its QSD by ν ǫ . As proved in [START_REF] Cattiaux | Quasi-stationarity distributions and diffusion models in population dynamics[END_REF], (H1) and (H3) are fulfilled in this case. Thanks to Theorem 1 and denoting by ν the Yaglom limit associated with P 0 , we have

ν ǫ law -→ ǫ→0 ν.
In the numerical simulations below, we set ǫ equal to 0.001.

By Theorem 10, we have

X N -→ N →+∞ ν ǫ ,
where X N is the empirical measure of the system studied in Section 3. In the numerical simulations, we set N = 1000 and, because of the randomness of X N , we approximate E(X N ) using the Ergodic theorem: we compute 1 10000 10000 t=1 µ N (t,dx). The graphic below (see Figure 3) shows this approximation for different values of r and c.

As it could be wanted for, greater is c, closer is the support of the QSD to 0. We thus numerically describe the impact of the linear and quadratic terms on the QSD.

The Wright-Fisher case

We illustrate the result of Section 2.3 by an application to the Wright-Fisher diffusion with values in ]0,1[ conditioned to be killed at 0. This diffusion is driven by the SDE dZ t = Z t (1 -Z t )dB t -Z t dt, Z 0 = z ∈]0,1[, and killed when it hits 0 (1 is never reached). In [START_REF] Huillet | On Wright-Fisher diffusion and its relatives[END_REF], the author proves that the QSD of this process exists and has the density 2 -2x with respect to the Lebesgue measure.

Define P 0 as the law of X . = arccos(1 -2Z . ), where Z is defined as above. P 0 is the law of the diffusion with values in ]0,π[, driven by the SDE dX t = dB t -1 -2 cos X t 2 sin X t dt, X 0 = x ∈]0,π[, killed when it hits 0 (π is never reached). For all ǫ ∈]0,π/2[, define P ǫ and ν ǫ as in Section 2.3. The drift of the diffusion is q(x) = 1-2 cos Xt 2 sin Xt , ∀x ∈]0,π[. Let us show that it satisfies the hypotheses of Theorem 9.

  1)dy → +∞ 0 m(y) min (e -Q(y) ,1)dy, then +∞ 0 m(y) min (e -Q(y) ,1)dy = 0.

3. 1 . 1

 11 Coupling's construction Define Q = sup x∈]0,1[ |q(x)| (we have Q < +∞ by hypothesis) and fix i ∈ {1,...,N}. The process Y i is defined with values in [0,1/3] by the SDE

Figure 2 :

 2 Figure 2: The process Y 1

Figure 3 :

 3 Figure 3: E(X N ) for the diffusion (38), with different values of r and c

Figure 4 :

 4 Figure 4: Wright Fisher case, ǫ = 0.001 and N = 1000

  Define the time ζ = inf {0 ≤ t ≤ τ ∞ , Y i t > X i t }and let us work conditionally to ζ < τ ∞ , then we have, by right continuity of the two processes, Y i ζ ≥ X i 2n ,t 2n+1 [, and, for h small enough, one can assume that ζ + h < t 2n+1 . Then, for all

	t ∈ [ζ,ζ + h], d(X i t -Y i t ) = (Q -q(X i t ))dt + dL 1/3 t , where Q -q(X i t ) ≥ 0 and L 1/3 t increasing process due to the reflecting property of the boundary 1/3 for Y i (note that is an
	Y i

Proposition 11

The coupling inequality

[START_REF] Lambert | The branching process with logistic growth[END_REF]

is fulfilled for all t ∈ [0,τ ∞ [.

Proof of Proposition 11 :

ζ a.s.

We first show that ζ is a jump time for the particle i. Assume the converse.

If ζ = 0, then X i ζ = Y i ζ and, if ζ > 0, then X i and Y i are continuous in a neighbourhood [ζ -h,ζ +h] of ζ, where h > 0, and we have Y i t ≤ X i t , for all t ∈ [ζ -h,ζ[, so that, by left continuity Y i ζ ≤ X i ζ , and then Y i ζ = X i ζ . Therefore, we have X i ζ ∈]0,1/3] and ∃n ≥ 0 such that ζ ∈ [t ζ > 0, so that, for h > 0 small enough, Y i t > 0 for all t ∈ [ζ,ζ + h[). Then X i -Y i

stays non-negative between times ζ and ζ + h, what contradicts the definition ζ. The time ζ is then a jump time for the particle i. If X i ζ-= 0, then, by definition of ζ, Y i ζ-= 0 and, by left continuity of the process Y

  Hypotheses 2 and 3]):lim t→∞ e λ 0 t P x (τ > t) = C 1 φ 0 (x),Here e λ 0 t P x (τ > t) is uniformly bounded above in the variables t and x (see [3, Proof of the equality 7.2, p27]), then, by dominated convergence, one can integrate with respect to m under the limit in (36), lim t→∞ e λ 0 t P m (τ > t) = C 1

					φ 0 (x)m(dx).
					D
	The same holds for (37):
	lim t→∞	e λ 0 t P m (X t ∈ A, τ > t) = C 2	D	φ 0 (x)	A	φ 0 (y)dy m(dx).
	Then, by Fubini's Theorem,
		lim t→∞	P m (X t ∈ A, τ > t) P m (τ > t)	=	C 2 C 1 A	φ 0 (y)dy,
	that is, from (36) and (37) with A =]0,1[,
		lim
							(36)
		lim

t→∞ e λ 0 t P x (X t ∈ A, τ > t) = C 2 φ 0 (x) A φ 0 (y)dy. (

37

) t→∞ P m (X t ∈ A, τ > t) P m (τ > t) = A φ 0 (y)dy D φ 0 (y)dy

which is nothing else but ν(A) (see

[START_REF] Pinsky | On the convergence of diffusion processes conditioned to remain in a bounded region for large time to limiting positive recurrent diffusion processes[END_REF] Proposition 1.10]

). 2

Acknowledgments I am extremely grateful to my Ph.D. supervisor Sylvie Méléard for his carefully and essential help on the form and the content of my first paper. I would like to thank Pierre Collet for his explanations on some spectral theory tools and Krzysztof Burdzy for his indication on the proof of the interacting particle process existence.

We have, ∀x ∈]0,π[, q(x) 2 -q ′ (x) = 8 cos 2 x + 2 cos x + 2 cos x + 1 4 sin 2 x + 1

which is positive and tends to +∞ both in 0+ and π-. It implies that hypothesis (HH1) is fulfilled. For all x ∈]0,π[,

.

That implies

Finally, hypotheses (HH2) and (HH3) are satisfied and Theorem 9 can be applied.

In the following numerical simulation (see Figure 4), we set ǫ = 0.001 and N = 1000. We compute E(X N ), which is an approximation of ν ǫ , and then of ν, with the method used in the logistic case (see Part 4.1).

The simulation is very close to the QSD (2 -2x)dx, which shows the efficiency of the method.

Remark 7 In the simulations, we have chosen to simulate a system with N = 1000 particles, because of empirical consideration. The question of convergence speed will be studied in further works.


Gaëlle Daniele
email: gaelle.daniele@isa-lyon.fr

Florent Lafay

Carine Arnaudguilhem

Sandra Mounicou

Alain Geffard

Isabelle Bonnard

Odile Dedourge-Geffard

Christelle Bonnefoy

Emmanuelle Vulliet

Affiliations

Evaluation of the accumulation of the iodinated contrast agents diatrizoic acid and iohexol in Dreissena polymorpha mollusks Authors

Keywords: Iodinated contrast media, bioaccumulation, Dreissena polymorpha, ICP MS, LC-MS/MS

Mollusks are very sensitive to aquatic environmental alterations and then, are important bioindicators for monitoring the contamination of water bodies. Iodinated X-ray contrast media (ICMs) are ubiquitously present in the aquatic environment, primarily due to their high consumption for diagnosis purposes, high injection levels, low biodegradability, and low removal rates by wastewater treatment plants. Although these compounds are assumed to be of low toxicity, aquatic organisms are continuously exposed to these agents, which may result in adverse effects as ICMs can act as iodine source and disrupt the endocrine system. Thus, the evaluation of their environmental risk, especially on aquatic fauna is of great interest.

To this end, we first compared the accumulation behavior, based on iodine analysis, of two ICM exhibiting different osmolality, diatrizoic acid and iohexol in Dreissena polymorpha bivalves exposed under laboratory conditions at concentrations of 0, 100, and 1000 µg/L during 4 and 7 days. This study was the first to provide information on iodine concentration in whole soft tissues and several organs in control zebra mussels. Moreover, it showed, after exposure, an increase of iodine content mainly in the digestive glands, followed by gills and gonads, highlighting that ICMs actually enter the organisms. Thus, bioaccumulation of ICMs studies were then performed, by liquid chromatography coupled to tandem mass spectrometry, on entire mollusks and digestive glands of organisms exposed at 0, 10, 100, and 1000 µg/L of both ICMs during 21 days, followed by 4 days of depuration. These first data on ICMs concentrations in zebra mussels, showed a clear accumulation of ICMs in mussels as a function of relative exposure level, as well as a rapid depuration. Osmolality did not seem to have a significant impact on the accumulation level, but a slight difference was observed on the accumulation pattern between both ICMs.

Introduction

The contamination of aquatic environments by pharmaceutical products is of high ecological concern due to their potential health impacts on aquatic organisms. Among these compounds, iodinated Xray contrast media (ICMs) have been used since the 50s in medical imaging to artificially increase contrast and improve investigations of organs, tissues, or tumors (Pérez and Barceló, 2017). The annual production of ICMs is estimated at 5 000 tons, and 40 to 80 million doses (highly concentrated aqueous solutions -up to 200 g/application) are injected annually, by the venous or arterial route [START_REF] Fabbri | Iodinated X-ray contrast agents: Photoinduced transformation and monitoring in surface water[END_REF]. In order to reduce their harmful effects on health and to be excreted in less than 24 hours, ICMs have been designed to be as inert and stable as possible (90% of them are rejected by the human body in non-metabolized form) and highly hydrophilic. Most ICMs are triiodobenzene derivatives incorporating highly polar functional groups to increase their water solubility and lower their toxicity [START_REF] Christiansen | X-ray contrast media-an overview[END_REF]. They are generally classified on the basis of their chemical structure, as ionic/non-ionic and monomer/dimer; diatrizoic acid being the most common ionic ICM, whereas iohexol, iopamidol, iopromide, and iomeprol being non-ionic ones. ICMs with low ionic strength and osmolality (as dimeric and non-ionic ones) exhibit lower intrinsic toxicity [START_REF] Christiansen | X-ray contrast media-an overview[END_REF][START_REF] Kim | Which iodinated contrast media is the least cytotoxic to human disc cells?[END_REF].

After being rejected by the human body in urine, ICMs are eliminated in wastewaters. However, their removal rates in wastewater treatment plants (WWTPs) are low, due to both poor adsorption to activated sludge and biological degradation. Their removal by ozonation vary from 35 to 55% or even 20% for ionic ICMs; only iopromide is degraded up to 60 -70% [START_REF] Ternes | Ozonation : a tool for removal of pharmaceuticals, contrats media and musk fragrances from wastewater?[END_REF][START_REF] Pal | Emerging contaminants of public health significance as water quality indicator compounds in the urban water cycle[END_REF][START_REF] Fabbri | Iodinated X-ray contrast agents: Photoinduced transformation and monitoring in surface water[END_REF][START_REF] Xu | Distribution and relevance of iodinated X-ray contrast media and iodinated trihalomethanes in an aquatic environment[END_REF]. Thus, several µg/L of ICMs are found in natural waters, groundwater, and drinking water [START_REF] Sengar | Comprehensive review on iodinated X-ray contrast media: Complete fate, occurrence, and formation of disinfection byproducts[END_REF][START_REF] Watanabe | Ubiquitous detection of artificial sweeteners and iodinated X-ray contrast media in aquatic environmental and wastewater treatment plant samples from Vietnam, the Philippines, and Myanmar[END_REF][START_REF] Xu | Distribution and relevance of iodinated X-ray contrast media and iodinated trihalomethanes in an aquatic environment[END_REF][START_REF] Zemann | Sources and processes affecting the spatio-temporal distribution of pharmaceuticals and X-ray contrast media in the water resources of the Lower Jordan Valley, Jordan[END_REF] with concentrations up to 12.9 µg/L, 9.6 µg/L, and 2.7 µg/L, respectively [START_REF] Sengar | Comprehensive review on iodinated X-ray contrast media: Complete fate, occurrence, and formation of disinfection byproducts[END_REF].

As a consequence, aquatic organisms are continuously exposed to these ICMs. However, if the occurrence of these compounds in the environment is relatively well documented, very few studies concern their potential impacts and fate in aquatic organisms. Indeed, no toxic effects were observed after an exposure to 10 g/L of ICM, in fish, crustacean, or algae [START_REF] Steger-Hartmann | Environmental risk assessment for the widely used iodinated X-ray contrast agent iopromide (Ultravist)[END_REF], which suggested that ICM were not of ecological concern [START_REF] Williams | The CompTox Chemistry Dashboard: a community data resource for environmental chemistry[END_REF]. Furthermore, it is commonly believed that these compounds will not bioaccumulate in fish and wildlife due to their negative log Kow values. Nevertheless, ICMs were recently detected in brain and gonads of fish, which are essential organs for fish hormone regulation [START_REF] Arnnok | Selective uptake and bioaccumulation of antidepressants in fish from effluent-impacted Niagara river[END_REF]. Moreover, ICMs have structural similarities with the natural thyroid hormones thyroxine and triiodothyronine especially involved in reproduction, brain, and body development (Venturi et al., 2012). Indeed, Singh et al.

(2019) studied the potential binding of ICMs to nuclear receptors in the endocrine system based on in silico calculations and showed stronger binding of iohexol compared to the natural hormones.

In addition, ICMs represent an iodine source that could potentially interact with the natural regulation of iodine in aquatic organisms. Iodine is an essential trace element, and a wide range of concentrations can be observed in aquatic organisms. Concentrations below limit of quantification (LOQ : 0.02 µg/kg, ICP-MS) to 1210 µg/100 g (muscle) were measured in a large panel of fish species [START_REF] Bogard | Nutrient composition of important fish species in Bangladesh and potential contribution to recommended nutrient intakes[END_REF]Nerthus et al., 2018). Similarly, concentrations of 26-120 µg/100 g (raw edible part : muscle) of iodine were measured in shrimp/prawn [START_REF] Bogard | Nutrient composition of important fish species in Bangladesh and potential contribution to recommended nutrient intakes[END_REF]. [START_REF] Moniruzzaman | Nutritional evaluation of some economically important marine and freshwater mollusc species of Bangladesh[END_REF] observed high concentrations of iodine comprised between 58-66 mg/100 g in 4 mollusk species of marine water (Saccostrea cuccullata, Crassostrea virginica, Meretrix meretrix, Andara granosa) and 50-53 mg/kg in 3 species of freshwater (Pila globose, Lamellidens marginalis, Lamellidens corrianus). [START_REF] Moniruzzaman | Nutritional evaluation of some economically important marine and freshwater mollusc species of Bangladesh[END_REF] underlined that significantly higher amount of iodine was found in oyster species than the freshwater bivalves and snail. The iodine content in the different species tends to be largely dependent on environmental conditions (FAO/WHO, 2004). However, no information is currently available in the literature regarding the influence of ICMs exposure on iodine regulation by aquatic organisms, particularly by species recognized as bioaccumulators of essential or non-essential elements.

In this context, this study proposes to evaluate the impact of two ICMs of different osmolality regularly detected in water, diatrizoic acid and iohexol, on the bivalve mollusk Dreissena polymorpha. This sedentary mussel is commonly found in a wide habitat diversity in Europe and North America (de Lafontaine et al., 2000). In addition, this mussel has a high filtration rate and high bioaccumulation potential that make it a widely used indicator for field biomonitoring studies and laboratory experiments [START_REF] Bourgeault | Lessons from a transplantation of zebra mussels into a small urban river: An integrated ecotoxicological assessment[END_REF][START_REF] Daniele | Development of a multi-residue analysis of diclofenac and some transformation products in bivalves using QuEChERS extraction and liquid chromatography-tandem mass spectrometry. Application to samples from mesocosm studies[END_REF][START_REF] Hani | 1H-NMR metabolomics profiling of zebra mussel (Dreissena polymorpha): A field-scale monitoring tool in ecotoxicological studies[END_REF]. Based on laboratory exposures, the first objective of this work concerned the study of the possible effects of ICMs exposure on iodine concentration and its distribution into the different organism's tissues by ICP-MS analysis. Based on these first results, the second aim was to study the accumulation kinetic of both ICMs in whole soft tissues and in the digestive glands, the most accumulative organs, by LC-MS/MS. This experiment was conducted during a 21-day exposure to different concentrations of ICMs followed by a 4-day depuration step.

Materials and Methods

Chemicals and reagents

UPLC-MS grade acetonitrile (ACN) and methanol (MeOH) were purchased from Biosolve Chimie (Dieuze, France). Tetramethylammonium hydroxide (TMAH 25%), formic acid (purity ≥ 99 %) and heptane (purity ≥ 90%) were acquired from Sigma-Aldrich (Saint-Quentin Fallavier, France). Water (Optima LC/MS grade) was bought from Fisher Chemical (Illkirch, France) for LC-MS/MS analysis.

Ultrapure water (18.2 m) produced from distilled water with a Milli-Q Elix water purification system (Millipore, Molsheim, France) was used for the other experiments.

Analytical standards on diatrizoic acid (DTZ, purity ≥ 98.0 %) and iohexol (IO, purity ≥ 95.0 %) were obtained from Merck. Phenacetin-ethoxy-1-13 C (98 atom % 13 C) used as LC-MS/MS injection standard was purchased from Isotec (Ohio, USA). Iodate (calibrant) and tungsten (internal standard) standards at 1000 mg/L (PlasmaCAL) used for ICP MS quantification were purchased from SCP Science (Courtaboeuf, France). Certified Reference material (ERM BB422) of lyophilized fish muscle with 1.4 ± 0.4 mg/kg of iodine was obtained from LGC Standards (Molsheim, France).

Stock solutions and calibration standards

Standard stock solutions of DTZ and IO were prepared individually at 1000 µg/L in MeOH and stored at -20°C during 3 months. Appropriate dilutions of these solutions were used for quantification purposes by LC-MS/MS. An 8-points matrix-matched calibration was performed by spiking "blank matrices" free of target compounds to determine the ICM levels in the whole mollusk or digestive glands.

For total iodine determination, stock elemental standards of iodine (I) and tungsten (W) at 500 and 25 µg/L respectively were prepared in water. A calibration curve ranging from 0 -5000 ng/L as I with 500 ng/L of internal standard (W) in 0.1% TMAH was used for quantification.

Zebra mussel maintenance and experimental design

For all experiments, adult zebra mussels (of 18-22 mm) were collected from the Lac du Der ( 51290Giffaumont-Champaubert, France, 48°33'035" N; 4°45'011" E), considered as reference site, and brought back to the laboratory where they were kept in 40 L glass tanks of aerated Cristalline Aurele drinking water, at 14 °C. The acclimation period lasted three weeks in darkness. The mussels were fed with two microalgae species (Scenedesmus obliquus and Chlorella pyrenoidosa, using 1,000,000 microalgae of each species per mussel) twice a week and the water was also changed twice a week.

Two consecutive exposure experiments of mussels to ICMs were conducted. The first one was devoted to ICP-MS analysis of iodine and corresponded to a short exposition to ICMs during 7 days.

The second experiment, a chronic exposure during 21 days followed by 4-days of depuration, was mainly devoted to the quantification of ICMs by LC-MS/MS. For both experiments, the same mussel density (40 per liter) was used. A stock solution of 1 mg/ml of DTZ or IO in ultrapure water was used for contamination and it was renewed each two days (experiment 1) or each week (experiment 2).

Water was sampled (5 mL) just before and after changes to monitor exposure concentration. At each sampling date the volume of exposure medium was adjusted to the number of mussels remaining in tanks. The mussels were fed twice a week as in the maintenance conditions (see before), a day before water change. In both experiments, a larger number of zebra mussels were exposed than those used for the study presented here. The remaining organisms were dedicated to other analyzes not considered in this article. For all exposures, no mortality or biological effect on the organisms were observed.

For the first experiment, zebra mussels were exposed in January 2019 to 0 and 100 µg/L of DTZ or IO.

Eighty mussels were placed in each tank (3: one control and two exposed). To complete the experimental design, other mussels were exposed with the 1000 µg/L DTZ or IO. For both short exposures, on day 4 and 7, 10 mussels per condition were sampled individually to recover gills, digestive glands, gonads, and the remaining soft tissues. Samples were weighted (wet weight, ww), shock frozen in liquid nitrogen, and stored at -80°C until analysis by ICP-MS (see 2.5).

For the chronic experiment, zebra mussels were exposed in January 2020 to 0, 10, 100, and 1000 µg/L of DTZ or IO. Four hundred mussels were placed in each tank (7: one control and six exposed). Mussels were exposed during 21 days then placed in a new tank with clean water for a 4-day depuration step (day 21 to day 25). At day 0 (control), day 3, 7, 10, 14, 21 (exposure step), and day 22 and 25 (depuration step), 20 mussels were sampled from each condition for LC-MS/MS analysis (see 2.6). Among these, whole soft tissues from five mussels were recovered individually, and digestive glands from the remaining 15 mussels were dissected and pooled by 3 (corresponding to 5 pools of 3 glands). All samples were weighted (ww) shock frozen in liquid nitrogen and stored at -80°C until analysis. The design of the experiment is presented in Figure 1.

Sample preparation and extraction protocols

Alkaline iodine extraction from tissues for total ICP MS analysis

The protocol was adapted from [START_REF] Leufroy | Determination of total iodine in French Polynesian foods: Method validation and occurrence data[END_REF]. Each organ (n= 10) were directly extracted individually in a 1.5 mL-eppendorf tube. Organs were blended at 3000 rpm with an G50 electric homogenizer (G-Biosciences, VWR, France) in 0.25 mL of 5% TMAH for gonads, 0.5 mL of 5% TMAH for gills and digestive glands, and 0.5 mL or 0.75 mL for tissues depending of their mass (0.5 mL for tissue with mass < 200 mg and 0.75 mL for masses > 200 mg). Then, eppendorf tubes were placed individually into a water bath in 50-mL tubes for 3 hours at 90°C in a heating block (DigiPREP MS, SCP Science, Courtaboeuf, France). After cooling down, extracts were centrifuged at 10 000 rpm for 15 min and the supernatant was diluted 50 fold in H 2 O to adjust the concentration of TMAH to 0.1%, after addition of W for a final concentration of 500 ng/L. Extracts were filtrated through a 0.45 µm cellulose acetate membrane before ICP MS analysis.

To validate the extraction procedure and analysis, an aliquot of 50 mg of the Certified Reference Material (ERM BB422) was extracted following the same protocol with 0.5 ml of 5% TMAH and diluted 50-fold in H 2 O before analysis.

ICMs analysis in water

DTZ and IO concentrations in water were monitored at each water change (every two days) to measure actual exposure concentrations in the tanks. Water samples were thawed at room temperature, homogenized and diluted 10 and 100 times respectively for the 100 and 1000 µg/L exposures; water sample resulting from the exposure at 10 µg/L was injected without dilution. Then, sample were analyzed by LC-MS/MS after addition of 50 µL of phenacetine-ethoxy-1-13 C at 100 µg/L as injection standard.

Mollusk and digestive glands extractions for ICMs accumulation studies

Extraction was performed on 5 individual mollusks or 5 pools of 3 digestive glands. An aliquot of 150 mg and 50 mg of grounded and homogenized fresh mollusk and digestive glands respectively, was extracted by solid liquid microextraction in 2 mL-screw tubes with 1.2 mL ACN/MeOH (50:50, v/v) and 500 µL of heptane. The tube was shaken 1 min with a Beadbug homogenizer (4000 rpm). Next, the tube was centrifuged for 5 minutes at 10 000 rpm. Then, 800 µL of the supernatant were transferred in a 15 mL-tube. A second extraction was performed with 1 mL of ACN/MeOH mixture, and then 1 mL was removed and transferred in the 15 mL-tube. The gathered extract was vortexed before being evaporated to dryness in a nitrogen flow at 40 °C. The dry residue was dissolved in 150 µL of H 2 O containing the injection standard, and 5 µL of this mixture was injected in LC-MS/MS for analysis.

ICP MS analysis

Analysis were performed on an 7700x ICP MS from Agilent Technology (Tokyo Japan) equipped with a Scott chamber cooled at 2°C, a micromist nebulizer delivering 400 µL/min, nickel sampling and skimmer cones, and a collision cell with H 2 gas (3.8 mL/min). The RF power was set to 1550 W. Lens voltages, torch position and sample gas flow (c.a. 1.09 mL/min) were optimized daily. Quantification was performed on 127 I, 182 W and 183 W (internal standard) isotopes with an integration time of 0.09 s/isotope.

Liquid chromatography tandem mass spectrometry (LC-MS/MS)

The analysis of DTZ and IO were performed using a liquid chromatographic system H-Class UPLC from Waters (Saint-Quentin en Yvelines, France) coupled to a Xevo TQ-S triple-quadrupole mass spectrometer (Waters) equipped with a StepWave ion guide. The chromatographic column was a Kinetex PS-C18 (100 × 2.1 mm; 2.6 µm) from Phenomenex (Le Pecq, France). The mobile phase was composed of (A) 0.1% formic acid in ultrapure water and (B) ACN with 0.1% formic acid. The following gradient was used for iohexol: 0-0.6 min, 1% B; 0.6-3.5 min, 1-52% B; 3.5-4.0 min, 52-100% B, 4.0.-6.2 min, 100 %B; 6.2-9.0 min, 1% B. For diatrizoic acid the gradient was as follows: 0-0.6 min, 0% B; 0.6-3.5 min, 0-52% B; 3.5-4.0 min, 52-100% B, 4.0.-6.2 min, 100 %B; 6.2-9.0 min, 0% B. The flow rate was maintained at 0.3 mL/min during 3.5 min, then increased up to 0.5 mL/min until 6.2 min, before returning to its initial rate. The oven temperature was fixed at 45 °C and the injection volume was set at 5 µL and 10 µL for tissues and water samples, respectively. The multiple reaction monitoring (MRM) mode was employed for the MS/MS detection to be more sensitive. The electrospray source was set in the positive ionization mode (ESI+). The optimum ESI-MS/MS settings were obtained by direct infusion of each individual ICM into the electrospray source.

The declustering potential (DP) and collision energy (CE) were optimized to get the highest signal intensities for each ICM. Two transitions were selected in the MRM mode for each ICM (Table S1).

The transition that presented the most intense signal (MRM1) was chosen for quantification and the second one (MRM2) for confirming the ICM detection. Ionization source parameters were optimized for maximum intensity of the precursor ions as follows: capillary voltage 3200 V, desolvation temperature 500°C (DTZ) and 550°C (IO), source offset 50 V, source temperature 150°C, nitrogen desolvation and nebulizer gas flows of 900 L/h and 150 L/h, respectively.

Results and discussion

Due to their high occurrence in natural waters and to their structural similarities with the natural thyroid hormones, ICMs represent a source of iodine that could potentially interact with the natural regulation of iodine in aquatic organisms. Thus, the study of their accumulation in bioaccumulative aquatic organisms such as bivalves is of great interest. First, we investigated the accumulation of ICMs in a global approach, by studying the accumulation of iodine in the different organs. Then, we studied the accumulation kinetics of the two ICMs both in whole soft tissues and in the most accumulative organ.

Effects of ICMs exposure on iodine concentrations in mussels

To study the effect of ICMs exposure on organism's iodine concentration, Dreissena polymorpha were exposed to acute and high concentrations of DTZ and IO (100 µg/L and 1000 µg/L). In the exposure tanks, the average actual iodine concentrations measured were of 42.2 µg/L and 64.82 µg/L, 414.9 µg/L and 537.9 µg/L for the 100 µg/L and 1000 µg/L exposures to IO and DTZ, respectively. For both experiments, a mean concentration of 3.8 µg/L of iodine was observed in the control condition due to natural iodine.

Figures 2 and3 show the concentrations of iodine after 4 and 7 days of exposure to DTZ and IO at 0, 100 and 1000 µg/L, respectively in the four organs, i.e gills, digestive glands, gonads, and remaining soft tissues. In this study basal iodine concentrations were determined in control organisms at 206 ± 104, 158 ± 79, 92 ± 105, and 359 ± 334 ng/g for gill, digestive gland, gonad, and remaining tissues, respectively, which corresponded to 260 ± 199 ng/g in the entire mussel. Actually, no information is available in the literature regarding basal iodine concentrations in freshwater mussels, Dreissena polymorpha. The iodine concentrations measured in the different organs/tissues are in accordance with those registered in other shellfish or shrimp (ranged from 308 to 1300 ng/g, FAO/WHO 2004;Oehlenschläger 2012;Broad et al., 2015). However higher iodine concentrations were registered in marine bivalves (Bay of Bengal : [START_REF] Moniruzzaman | Nutritional evaluation of some economically important marine and freshwater mollusc species of Bangladesh[END_REF]sea of Marmara : Özden et al., 2009) and freshwater ones from Bengladesh [START_REF] Moniruzzaman | Nutritional evaluation of some economically important marine and freshwater mollusc species of Bangladesh[END_REF] with concentrations comprised between 6.9 and 135.03 µg/g (wet weight).

Moreover, we can notice that among organs of control zebra mussels, similar iodine basal concentrations are observed in gills and digestive glands and lower concentrations are observed in the gonads (Figure 2 and 3). Digestive glands and gills are commonly considered in studies on metals exposure, in link to their rule in metals accumulation (according to the route of exposure: food vs water), metals metabolism and detoxification. However, in this study, the highest concentrations of iodine are observed in the remaining soft tissues. These are mainly composed of the gland of byssus, muscle and mantle. Among these organs, the mantle could represent an iodine-rich organ according to its rule in shell production.

In the exposed organisms, no accumulation of iodine was observed in gills during DTZ-and IO exposures at 100 µg/L (Figure 2 and see Table 1). However, at 1000 µg/L, an iodine accumulation can be observed, especially for DTZ exposure, with significant differences in iodine concentrations compared to controls after 4 days of exposure. This could be explained by the high osmolality of DTZ compared to IO which may migrate through membranes more easily. To note, the iodine concentration is quite stable between 4 and 7 days of exposure to DTZ, suggesting an equilibrium of the organisms with their exposure medium from day 4. In the digestive glands, the iodine concentration increases with time for both molecules. Iodine accumulation can be significantly detected in the digestive glands after 7 days of exposure at 100 µg/L of DTZ and from 4 days at 1000 µg/L for both DTZ and IO exposures. In the gonads, iodine accumulation can be detected significantly (Table 1) during all DTZ exposure conditions. Iodine concentration in gonads increases over time (between 4 and 7 days of exposure) whatever the concentration of DTZ in their medium.

Interestingly, in the remaining soft tissues (showing the highest iodine concentration in controls) no iodine accumulation was observed during ICMs exposure whatever the condition (duration and/or concentrations used). At 100 µg/L of ICM, corresponding to a 10-(IO) or 15-(DTZ) times increase in iodine concentration in the medium compared to the control, the results suggest a lack of ICM accumulation in the mussel or a regulation of iodine by the mussel to compensate for iodine supply via the ICM. At 1000 µg/L of ICM, the results support the hypothesis of an ICM accumulation and an excess of mussel homeostatic capacities leading to an increase in iodine concentrations in their tissues. In the same way, despite the high basal iodine concentration in the remaining tissues (including mantle), the results at the high exposure condition underline the role of the digestive gland in contaminant accumulation/detoxification with a twofold increase in iodine concentration. So, in the second part of this study, a particular interest was given to the digestive gland in addition to whole soft tissues.

Accumulation of diatrizoic acid and iohexol in Dreissena polymorpha

The second part of this work concerned the study of the accumulation of DTZ and IO in mussels (whole soft tissues or digestive glands) during a 21 days-chronic exposure followed by a 4 daysdepuration phase to 0 (control), 10, 100, and 1000 µg/L (nominal concentration) of both ICMs.

Concentrations of DTZ and IO were stable throughout the exposure, so that bivalves were actually exposed over 21 days to an average concentration of 9.5 ± 1.1, 89.5 ± 4.9, and 760 ± 60 µg/L for DTZ, and of 9.3 ± 1.0, 123 ± 11, and 1032 ± 84 µg/L for IO at the 95% confidence level. A total of 5 individual bivalves and 5 pools of 3 bivalve digestive glands was extracted and analyzed at each sampling time. Accumulation of both ICMs in whole soft tissues and digestive glands during exposures to 0 (control), 10, 100, and 1000 µg/L are presented on Figures 4 and5, respectively.

Whatever the sampling date, DTZ and IO were never detected in the organisms (whole soft tissues and digestive glands) in the control condition (LQ <0.2 ng/g and <0.3 ng/g in whole tissues and digestive glands, respectively).

Whatever the biological sample (whole soft tissues: Figure 4 or digestive glands: Figure 5), DTZ and IO concentrations increased significantly (p < 0.05) according to the exposure concentration. In addition, we can note that the ICM concentration in the digestive glands are 3-fold higher than in whole soft tissues, especially for DTZ. This can be explained by the low accumulation of the ICMs in gills, gonads, and remaining soft tissues which constitute the whole soft tissue compared to digestive glands alone (dilution effect, see above). For DTZ, maximum mean concentrations (for one exposure date) in whole soft tissues and digestive glands are 1.3 ± 1.2 and 3.3 ± 0.9 ng/g, 13 ± 4 and 29 ± 8 ng/g, 130 ± 58 and 249 ± 54 ng/g, respectively, for organisms exposed to 10, 100, and 1000 µg/L. For IO, maximum mean concentrations in whole soft tissues and digestive glands are 1.1 ± 0.7 and 3.2 ± 0.9 ng/g, 17 ± 6 and 25 ± 8 ng/g, 254 ± 55 and 275 ± 51 ng/g, respectively, for organisms exposed to 10, 100, and 1000 µg/L. The higher concentration of IO compared to DTZ can be explained by a higher measured concentration of this ICM in the exposed water for the nominal concentrations 100 and 1000 µg/L. Moreover, in whole soft tissues (Figure 4), the accumulation patterns appear different according to the ICM considered. Indeed, for all DTZ exposure concentrations, compared to control (Day 0) an accumulation of DTZ was observed from day 3, then the concentrations were stable until the end of the exposure (day 21). In the case of IO, the concentration increased significantly from day 3 to day 21 (comparison of mean concentrations between day 3 and day 21 for 10 µg/L: p = 0.04, for 100 µg/L: p < 0.01; and for 1000 µg/L: p < 0.01)(Table 2). In the digestive glands (Figure 5), a similar pattern to that of whole soft tissues was observed for IO with a significant increase (except in 10 µg/L condition) in concentration between day 3 and day 21 (100 µg/L: p < 0.01; 1000 µg/L: p = 0.01)(Table 3). For DTZ, after an accumulation observed on day 3 in the digestive glands, the concentration still increased significantly (except for condition 1000 µg/L) after 14 days of exposure (highest concentrations observed during the exposure step; 10 µg/L: p < 0.01; 100 µg/L: p < 0.01).

During the depuration step, we can observe an important and significant decrease (see Table 2 of p values comparison between day 21 and day 22) in ICM concentration for both whole soft tissues (Figure 4) and digestive glands (Figure 5) after only one day (day 22) of depuration. Nevertheless, except for 10 µg/L in whole soft tissues, the concentration of both ICM is relatively stable between day 22 and day 25 (p > 0.05, see Table 2). Furthermore, we can note that for the 100 and 1000 µg/L exposures, after the depuration phase, the concentrations of DZT in whole soft tissues (Figure 4) and digestive glands (Figure 5) are closer to the concentrations of the controls compared to IO. These results are in accordance with the half-lives of both ICMs that are evaluated at 121min for IO (Olsson et al., 1993) and at about 30-60 min for DTZ (McEvoy, 2005) in humans. The bioaccumulation factor (BAF) was calculated for each ICM, in whole soft tissues and digestive glands (Table 4). It reflects the body's ability to bioaccumulate a given compound from the environment, through its respiratory and skin surfaces, excluding exposure through food. It was calculated as follows:

where Ctissue (µg/kg) is the mean ICM concentration determined for wet whole soft tissues or digestive glands during the exposition (day 3 to day 21), and Cwater (µg/L) is the mean water concentration measured during the corresponding exposition. For both ICMs and whatever the experimental condition (exposure concentration and organ/tissue considered) the BAF are low and lower than 1, underlying the absence of bioaccumulation of these molecules in the studied bivalves. Nevertheless, we can notice a slightly higher BAF for DTZ in digestive glands, which is in accordance with the greater iodine accumulation observed in the acute exposition (see section 3.1). These results confirm the generally accepted view that these compounds do not bioaccumulate in fish and wildlife because of their negative log Kow values (-1.28 to -3.05) reinforced by the fact that to date, no environmental risks have been associated with these compounds in humans and wildlife [START_REF] Steger-Hartmann | Environmental risk assessment for the widely used iodinated X-ray contrast agent iopromide (Ultravist)[END_REF][START_REF] Williams | The CompTox Chemistry Dashboard: a community data resource for environmental chemistry[END_REF]. However, a recent study unexpectedly detected high levels of the highly polar chemical iopamidol in brain and gonad tissues of various fish species caught in the Niagara River with concentrations as high as 1059 ng/g (dry basis), resulting in a bioaccumulation factor of approximately 46,000 [START_REF] Arnnok | Selective uptake and bioaccumulation of antidepressants in fish from effluent-impacted Niagara river[END_REF].

This suggests that the bioaccumulation of ICMs is receptor-mediated and, based on a in silico approach, that they may act as endocrine disrupting chemicals in the environment [START_REF] Singh | Binding of iodinated contrast media (ICM) and their transformation products with hormone receptors: Are ICM the new EDCs?[END_REF]. Nevertheless, in our study, if bioaccumulation is not observed, the accumulation curves show clearly that there is an intake of these ICMs by the organisms, with higher registered concentrations in digestive glands. Although the accumulation pattern seems to differ in whole soft tissues depending to the ICM and DTZ shows a slightly higher bioaccumulation factor, no clear difference appears depending on the osmolality of the two ICMs studied.

For the experiments at 10 and 100µg/L of ICMs, analysis by LC-MS/MS showed the intake of ICMs by mussels, particularly in the digestive glands, without an effect on the iodine concentration in this organ (see section 3.1). With the total ICP MS analysis, not only the natural iodine background in the organs did not allow to detect any significant accumulation at low exposure concentrations, but as iodine is an essential element, the intake of iodine at low ICMs exposure concentrations, could be et al., 2009;[START_REF] Bonnefille | Metabolomics assessment of the effects of diclofenac exposure on Mytilus galloprovincialis: Potential effects on osmoregulation and reproduction[END_REF][START_REF] Hani | 1H-NMR metabolomics profiling of zebra mussel (Dreissena polymorpha): A field-scale monitoring tool in ecotoxicological studies[END_REF]. The aim of this approach is the characterization of a set of metabolites, or metabolome, which is the collection of low-molecular weight molecules (typically with mass lower than 1500 Da) present in a cell, tissue or organism at a given time. Being the result of gene expression and the end-product of the metabolism, the metabolome is considered representative of the physiological state of an organism [START_REF] Weckwerth | Metabolomics in systems biology[END_REF]. In parallel, potential biological effects of ICMs could be assessed by the study of their interaction and functional connections with metabolites, proteins and other biomolecules within organisms and cells. This type of study defined as metallomics [START_REF] Haraguchi | Metallomics as integrated bimetal science[END_REF][START_REF] Mounicou | Metallomics: The concept and methodology[END_REF][START_REF] Williams | Chemical selection of elements by cells[END_REF], will help us in understanding the homeostasis of iodine as well as other endogenous elements.

4.Conclusion

This study was the first to provide iodine concentration in the whole soft tissues or organs of control freshwater zebra mussels. A basal iodine concentration of 260±199 ng/g was determined in the

Figure 1 :

 1 Figure 1: Design of the experiment

Figure 2 :

 2 Figure 2: Iodine concentration (µg/kg wet weight) in (a) gills, (b) digestive glands, (c) gonads and (d)

Figure 4 :

 4 Figure 4: Concentrations of diatrizoic acid (DTZ, blue) and iohexol (IO, green) in Dreissena polymorpha

  compensated by a loss of native iodine by homeostasis. At the highest exposure condition (1000 µg/L), the ICMs quantification by LC-MS/MS, showed a change in the iodine concentration (an increase), particularly in the digestive glands. This increase in the iodine concentration could reflect an effect of ICMs exposure on iodine regulation/metabolism of the mussel. For both iodine and ICMs analyses, the original results obtained in this study highlight an accumulation of ICMs in aquatic organisms. The lack of information actually in the literature concerning the ICMs ecotoxicity using early biological markers (molecular responses) makes it difficult to propose an hypothesis on direct toxic effects of accumulated ICMs on mussels or indirect effects on the base of iodine metabolism disturbance. In this way it would be interesting to study the potential effect of ICMs on mussels by omics approaches, i.e metabolomics and metallomics. Initially developed in the clinical and biomedical fields, metabolomics has gained interest in ecotoxicology during the last decade(Bundy

  whole mussel Dreissena polymorpha, shared among the gills, digestive glands, gonads, and remaining tissues (composed of the gland of byssus, muscle, and mantle) at 206±104, 158±79, 92±105, and 359±334 ng/g, respectively. Iodine concentration increased in gills, digestive glands, and gonads when bivalves were exposed to high concentrations of ICMs, showing that both high and low osmolality ICMs can effectively penetrate the tissues of filter-feeding organisms such as the bivalves Dreissena polymorpha. Despite the high concentration of iodine in the remaining tissues, no accumulation of iodine was observed in these tissues in the exposed organism. Further studies are necessary to precise the potential rule of mantle in the high iodine concentrations in control organism without accumulation during ICMs exposure. Based on the total iodine analysis on the different organs, the digestive gland was chosen for accumulation studies as it exhibits the highest iodine accumulation via ICMs exposure. This study offers the first determination of ICM concentrations in both mollusk's whole soft tissues and digestive glands by selective and sensitive LC-MS/MS analysis. The level of accumulation was quite similar for diatrizoic acid and iohexol in whole soft tissues and in digestive glands in a manner dependent on the relative level of exposure, with however a slightly different accumulation pattern in whole soft tissues, and dependent of the molecule. Although bioaccumulation factors are low (inferior to 1), the present findings show that ICMs can enter in bivalves that may use ICM as iodine source. Further studies are needed to finely study biological effects at molecular level through metallomic and metabolomic approaches.

  ).

		Exposure		Diatrizoic acid		Iohexol
	Days Concentration Gills	Digestive glands	Gonads Tissues Gills	Digestive glands	Gonads Tissues
	4	100 µg/L 1000 µg/L	0.95 0.05	0.34 <0.01	0.02 0.03	0.29 0.83 0.53 0.20	0.90 <0.01	0.95 0.14	0.32 0.82
	7	100 µg/L 1000 µg/L	0.80 0.05	0.05 <0.01	0.01 <0.01	0.52 0.79 0.79 0.05	0.03 <0.01	0.15 0.08	0.37 0.32
	Table 1: Student p-values relative to Dreissena polymorpha exposed 4 and 7-days to 100 and 1000
	µg/L of diatrizoic acid and iohexol. p-values < 0.05 (grey cells) indicate significative differences
	between exposed and control organs.

Table 2 :

 2 Day-by-day comparison of Student p-values relative to whole soft tissues of Dreissena polymorpha exposed to 10, 100, and 1000 µg/L of diatrizoic acid and iohexol. p-values < 0.05 (grey cells) indicate significative differences between two conditions.

		.97	0.67	1.00	0.85	1.00	0.16
	10-3	0.95	0.95	0.84	0.90	0.73	0.17
	14-3	1.00	0.86	0.78	0.08	0.08	0.06
	21-3	0.95	0.96	1.00	0.04	< 0.01	< 0.01
	22-3	0.42	< 0.01	0.06	0.27	0.92	1.00
	25-3	0.73	< 0.01	< 0.01	0.94	0.96	1.00
	10-7	1.00	1.00	0.93	1.00	0.96	1.00
	14-7	1.00	1.00	0.62	0.63	0.23	1.00
	21-7	1.00	0.17	1.00	0.44	0.01	0.01
	22-7	0.90	0.02	0.11	0.94	0.63	0.38
	25-7	0.99	< 0.01	0.01	0.27	0.72	0.07
	14-10	1.00	1.00	0.15	0.54	0.77	1.00
	21-10	1.00	0.50	0.83	0.37	0.10	0.01
	22-10	0.94	0.01	0.69	0.90	0.15	0.39
	25-10	1.00	< 0.01	0.12	0.33	0.20	0.07
	21-14	1.00	0.34	0.78	1.00	0.80	0.02
	22-14	0.68	0.02	< 0.01	0.99	0.01	0.18
	25-14	0.92	< 0.01	< 0.01	0.01	0.01	0.02
	22-21	0.93	< 0.01	0.06	0.96	< 0.01	< 0.01
	25-21	1.00	< 0.01	< 0.01	< 0.01	< 0.01	< 0.01
	25-22	1.00	0.85	0.83	0.03	1.00	0.97

Table 4 :

 4 Bioaccumulation factors (BAF) in Dreissena polymorpha's whole soft tissues or digestive glands after 21-days exposure to 10, 100, and 1000 µg/L of diatrizoic acid and iohexol

				DTZ		IO
				Whole soft	Digestive	Whole soft	Digestive
			BAF (L/kg)
				tissues	glands	tissues	glands
	Exposure	concentration	10 µg/L 100 µg/L 1000 µg/L	0.10 0.12 0.12	0.23 0.24 0.24	0.08 0.09 0.15	0.30 0.15 0.16

Acknowledgement

The results of this study were obtained within the framework of the "ACTIONS" research project. This