Olivier Raimond
email: olivier.raimond@u-paris10.fr

Bruno Schapira
email: bruno.schapira@math.u-psud.fr

EXCITED BROWNIAN MOTIONS AS LIMITS OF EXCITED RANDOM WALKS

We obtain the convergence in law of a sequence of excited (also called cookies) random walks toward an excited Brownian motion. This last process is a continuous semi-martingale whose drift is a function, say ϕ, of its local time. It was introduced by Norris, Rogers and Williams as a simplified version of Brownian polymers, and then recently further studied by the authors. To get our results we need to renormalize together the sequence of cookies, the time and the space in a convenient way. The proof follows a general approach already taken by Tóth and his coauthors in multiple occasions, which goes through Ray-Knight type results. Namely we first prove, when ϕ is bounded and Lipschitz, that the convergence holds at the level of the local time processes. This is done via a careful study of the transition kernel of an auxiliary Markov chain which describes the local time at a given level. Then we prove a tightness result and deduce the convergence at the level of the full processes.

1. Introduction 1.1. General overview. Self-interacting random processes play a prominent role in the probability theory and in statistical physic. One fascinating aspect is that behind an apparent simplicity, they can be extremely hard to analyze rigorously. Just to mention one striking example, it is still not known whether once reinforced random walks on a ladder are recurrent in general (see however [Sel] and [Ver] for a partial answer and the surveys [MR] and [Pem] for other problems on reinforced processes). A major difficulty in these models is that we loose the Markovian property and in particular the usual dichotomy between recurrence and transience can be broken. A famous example where this happens is for vertex reinforced random walks on Z: it is now a well known result in the field, first conjectured and partially proved by Pemantle and Volkov [PemV], that almost surely these processes eventually get stuck on five sites [Tar]. For analogous results concerning self-attracting diffusions, see [CLJ], [HRo] and [R].

Beside this very basic, yet fundamental, problem of recurrence, a question of particular interest is to understand the connections between the various discrete and continuous models. In particular an important challenging conjecture is that selfavoiding random walks on Z 2 converge, after renormalization, toward the SLE 8/3 (see [LSW] for a discussion on this and [DCS] for some recent progress). There are in fact not many examples where invariance principles or central limit theorems were fully established. But for instance it was proved that random walks perturbed at extrema converge after the usual renormalization toward a perturbed Brownian motion (see e.g. [Dav] and [W]).

In this paper we are interested in the class of so-called excited random processes, which are among the most elementary examples of self-interacting processes. By this we mean that the interaction with the past trajectory is as localized as possible: the evolution of these processes at any time only depend on their local time at their present position. A discrete version was introduced relatively recently by Benjamini and Wilson [BW] and a generalization, called multi-excited or cookie random walks, was then further studied by Zerner [Z] and many other authors (see in particular [MPRV] and references therein). Closely related models were also considered in [ABK], [BKS], [K] and [KRS]. Dolgopyat [D] observed that in dimension 1, in the recurrent regime, and after the usual renormalization, multi-excited random walks also converge toward a perturbed Brownian motion (we will give a more precise statement later). However, as we will see below, the latter are not, in some sense, the most natural continuous versions of excited processes. Somewhat more natural ones were introduced two decades ago by Norris, Rogers and Williams [START_REF] Norris | Self-avoiding random walk: a Brownian motion model with local time drift[END_REF], in connection with the excluded volume problem [START_REF] Norris | An excluded volume problem for Brownian motion[END_REF], and as a simplified model for Brownian polymers. They were later called excited Brownian motions by the authors [RS].

The aim of this paper is to show that excited Brownian motions can be approached in law by multi-excited random walks in the Skorokhod space, i.e. in the sense of the full process. For this we need to use a nonstandard renormalization, namely we need to scale together and appropriately the sequence of cookies, which govern the drift of the walk, the space and the time. Now let us give more details, starting with some definitions:

A multi-excited or cookie random walk (X ε (n), n ≥ 0) is associated to a sequence ε := (ε i , i ≥ 1) ∈ (-1, 1) N , of cookies in the following way: set

p ε,i := 1 2 (1 + ε i ),
for all i ≥ 1, and let (F ε,n , n ≥ 0) be the filtration generated by X ε . Then X ε (0) := 0 and for all n ≥ 0,

P[X ε (n + 1) -X ε (n) = 1 | F ε,n ] = 1 -P[X ε (n + 1) -X ε (n) = -1 | F ε,n ] = p ε,i , if #{j ≤ n : X ε (j) = X ε (n)} = i.
We notice that the case of random cookies has also been studied in the past, for instance by Zerner [Z], but here we consider only deterministic ε.

On the other hand excited Brownian motions are solutions of a stochastic differential equation of the type:

dY t = dB t + ϕ(L Yt t ) dt, where B is a Brownian motion, L •
• is the local time process of Y and ϕ : R → R is some measurable (bounded) function.

So at a heuristic level the discrete and the continuous models are very similar: the drift is a function of the local time at the present position. But the analogy can be pushed beyond this simple observation. In particular criteria for recurrence and nonzero speed in both models (see respectively [KZ] and [RS]) are entirely similar (see below). Our results here give now a concrete link. We first prove that when ϕ is bounded and Lipschitz, the local time process of X ε , conveniently renormalized, converges to the one of Y , exactly in the same spirit as in Tóth's papers on selfinteracting random walks (see [T2]). Then we obtain a tightness result and deduce a convergence in the Skorokhod space at the level of the processes (see Theorem 1.4 below). For proving the convergence of the local time processes we use a standard criterion of Ethier and Kurtz [EK] on approximation of diffusions. To show that we can apply it here we introduce some auxiliary Markov chains describing the local time on each level and we make a careful analysis of the transition kernels of these Markov chains (see Section 2).

Acknowledgments: We thank an anonymous referee for encouraging us to prove the main result without the recurrence hypothesis and with weaker assumptions on the regularity of ϕ.

1.2. Description of the results. For a ∈ Z and v ∈ N, set τ ε,a (v) := inf {j : #{i ≤ j : X ε (i) = a and X ε (i + 1) = a -1} = v + 1} . Consider the process (S ε,a,v (k), k ∈ Z) defined by S ε,a,v (k) = #{j ≤ τ ε,a (v) -1 : X ε (j) = k and X ε (j + 1) = k -1}.
In particular, when τ ε,a (v) is finite, S ε,a,v (a) = v. We will say that X ε is recurrent when all the τ ε,a (v)'s are finite. A criterion for recurrence when ε i ≥ 0 for all i or when ε i = 0 for i large enough is given in [Z] and [KZ] (namely in these cases, X ε is a.s. recurrent if, and only if,

i ε i ∈ [-1, 1]).
Assume now that ϕ is bounded and let ε n = (ε i (n), i ≥ 1) be defined by

ε i (n) := 1 2n ϕ i 2n for all n ≥ 1 and all i ≥ 1.
Since ϕ is bounded, if n is large enough then ε n ∈ (-1, 1) N and X εn is well defined.

Then for a ∈ R and v ≥ 0, set

Λ (n) a,v (x) := 1 n S εn,[2na],[nv] ([2nx]) for all x ∈ R. (1)
We give now the analogous definitions in the continuous setting. First for a ∈ R, let τ a (v) := inf{t > 0 : L a t > v} for all v ≥ 0, be the right continuous inverse of the local time of Y at level a. Again we say that Y is a.s. recurrent if all these stopping times are a.s. finite. This is equivalent (see [START_REF] Raimond | Excited Brownian motion[END_REF]Theorem 1.1]) to the condition C + 1 = C - 1 = +∞, where

C ± 1 := ∞ 0 exp ∓ x 0 h(z) dz z dx,
and where

h(z) := z 0 ϕ(ℓ) dℓ for all z ≥ 0.
In particular when ϕ is nonnegative or compactly supported this is equivalent to

∞ 0 ϕ(ℓ) dℓ ∈ [-1, 1]. Then set Λ a,v (x) := L x τa(v)
. The Ray-Knight theorem describes the law of (Λ a,v (x), x ∈ R) (a proof is given in [START_REF] Norris | Self-avoiding random walk: a Brownian motion model with local time drift[END_REF] when v = 0, but it applies as well for v > 0), when τ a (v) is a.s. finite, and we recall this result now. To fix ideas we assume that a ≤ 0. An analogous result holds for a ≥ 0. So first we have Λ a,v (a) = v. Next, Λ a,v is solution of the stochastic differential equation:

dΛ a,v (x) = 2 Λ a,v (x) dB x + 2(1 + h (Λ a,v (x))) dx for x ∈ [a, 0], (2) dΛ a,v (x) = 2 Λ a,v (x) dB x + 2h (Λ a,v (x)) dx for x ∈ [0, ∞), ( 3
)
where B is a Brownian motion, and (3) holds up to the first time, say w + a,v , when it hits 0, and then is absorbed in 0 (i.e. Λ a,v (x) = 0 for x ≥ w + a,v ). Similarly (Λ a,v (a -x), x ≥ 0) is solution of (3) (with a drift -2h instead of 2h and an independent Brownian motion) up to the first time, say w - a,v , when it hits 0, and then is absorbed in 0.

For d ≥ 1, we denote by D(R, R d ) the space of càdlàg functions f : R → R d endowed with the usual Skorokhod topology (see for instance Section 12 in [Bil]). The space D(R, R) will also simply be denoted by D (R). It will be implicit that all convergences in law of our processes hold in these spaces.

Our first result is the following theorem:

Theorem 1.1. Assume that ϕ is bounded and Lipschitz. Assume further that for n large enough, X εn is recurrent and that Y is recurrent. Then for any finite set I, any a i ∈ R and v i ≥ 0, i ∈ I,

(Λ (n) ai,vi (x), x ∈ R) i∈I L =⇒ n→∞ (Λ ai,vi (x), x ∈ R) i∈I .
As announced above, this theorem gives the convergence of a sequence of excited random walks toward the excited Brownian motion (associated to ϕ) at the level of the local times. We will also extend this result in a non homogeneous setting, i.e. when ϕ is allowed to depend also on the space variable. We refer the reader to Section 3 for more details. Actually we will need this extension to prove Theorem 1.1 in the case |I| ≥ 2. This will be explained in Section 4.

Note that if ϕ is compactly supported or nonnegative (and bounded Lipschitz), and if ∞ 0 ϕ(ℓ) dℓ ∈ (-1, 1), then Y is recurrent and X εn as well for n large enough. A consequence of Theorem 1.1 is the following Corollary 1.2. Under the hypotheses of Theorem 1.1, for any finite set I, any a i ∈ R and v i ≥ 0, i ∈ I,

1 4n 2 τ εn,[2nai] ([nv i ]) i∈I L =⇒ n→∞ τ ai (v i ) i∈I . For u i ∈ R, i ∈ I, denote by θ (i)
ui some independent geometric random variables with parameter 1 -e -ui , independent of X εn . Denote also by γ (i) ui , i ∈ I, some independent exponential random variables with parameter u i , independent of Y . Then as in [T1], we can deduce from the previous results the Corollary 1.3. Under the hypotheses of Theorem 1.1, for any λ i ≥ 0, i ∈ I,

1 2n X εn (θ (i) λi/(4n 2 ) ) i∈I L =⇒ n→∞ Y (γ (i) λi ) i∈I .
Finally we get the following:

Theorem 1.4. Assume that ϕ is bounded and Lipschitz. For t ≥ 0, set X (n) (t) := X εn ([4n 2 t])/(2n), which is well defined at least for n large enough. Then

(X (n) (t), t ≥ 0) L =⇒ n→∞ (Y (t), t ≥ 0).
Note that, as opposed to the previous results, we do not assume in this last theorem that X εn and Y are recurrent.

To obtain this result we need to prove the tightness of the sequence X εn ([4n 2 •])/(2n), n ≥ 1. This is done by using a coupling between different branching processes, similar to those which were used for proving Corollary 1.2. The convergence of finitedimensional distributions follows from Corollary 1.3 and an inversion of Laplace transform.

As for Theorem 1.1 an extension of this result to the non homogeneous setting can be proved (see Theorem 7.1 at the end of the paper).

Let us mention now a related result of Dolgopyat [D]. He proved a functional central limit theorem for excited random walks when ε is fixed, and in the recurrent regime; more precisely when ε i ≥ 0 for all i and α := i ε i < 1. In this case the limiting process is a perturbed Brownian motion, i.e. the process defined by

X t = B t + α sup s≤t X s -inf s≤t X s for all t ≥ 0,
with B a Brownian motion.

We will first prove Theorem 1.1 in the case |I| = 1 in Section 2. In section 3, we will extend the result to the non homogeneous setting and in Section 4 we will deduce the result in the general case |I| ≥ 1. Corollaries 1.2 and 1.3 will be proved respectively in Section 5 and 6, and Theorem 1.4 in Section 7.

2. Proof of Theorem 1.1 in the case |I| = 1

We assume in this section that |I| = 1. Let a ∈ R and v ≥ 0 be given. To fix ideas we assume that a ≤ 0. The case a ≥ 0 is similar. Moreover, we only prove the convergence of Λ (n) a,v on the time interval [a, ∞), since the proofs of the convergence on (-∞, a] and on [0, +∞) are the same. Ethier and Kurtz. It is now a standard fact and not difficult to check (see however [BaS] or [KZ] for more details) that for all a ∈ N -and v ∈ N,

A criterion of

(1) the sequence (S ε,a,v (a), . . . , S ε,a,v (0)) has the same law as (V ε,v (0), . . . , V ε,v (-a)), where (V ε,v (k), k ≥ 0) is some Markov chain starting from v, which is independent of a, (2) conditionally to w = S ε,a,v (0), the sequence (S ε,a,v (k), k ≥ 0) has the same law as some Markov chain ( V ε,w (k), k ≥ 0), starting from w, which is independent of a, (3) the sequence (S ε,a,v (a -k), k ≥ 1) has the same law as (

V -ε,v+1 (k), k ≥ 1),
where by definition (-ε) i = -ε i for all i ≥ 1.

Moreover, the sequences (S ε,a,v (a -k), k ≥ 1) and (S ε,a,v (k), k ≥ 0) are independent. The laws of the Markov chains V ε,v and V ε,v will be described in Subsection 2.2 in terms of another Markov chain W ε , see in particular (8) and ( 9). Note that this idea to use the Markovian property of the process S ε,a,v goes back at least to Kesten, Kozlov and Spitzer [KKS].

In the following, in order to lighten the presentation we will forget about the dependence on the starting point (which does not play any serious role here) in the notation for V ε and V ε . Thus V ε and V εn should be understood respectively as V ε,v and V εn, [nv] , where the v will be clear from the context, and similarly for V ε and V εn .

Now we first prove the convergence of Λ

(n) a,v on [a, 0].
The proofs of the convergence on [0, +∞) and on the full interval [a, +∞) are similar and will be explained in Subsection 2.6.

So on [a, 0], Λ

a,v can be decomposed as a sum of a martingale part

M (n) a,v and a drift part B (n) a,v : (4) Λ (n) a,v (x) = [nv] n + M (n) a,v (x) + B (n) a,v (x
) for all x ∈ [a, 0], with the following equalities in law:

M (n) a,v (x) = 1 n [2nx]-[2na] k=1 V εn (k) -E[V εn (k) | V εn (k -1)] , (5)
and

B (n) a,v (x) = 1 n [2nx]-[2na] k=1 E[V εn (k) | V εn (k -1)] -V εn (k -1) . (6) Let also A (n) a,v be the previsible compensator of (M (n) a,v ) 2 .
We have the equality in law:

(7) A (n) a,v (x) = 1 n 2 [2nx]-[2na] k=1 E[V εn (k) 2 | V εn (k -1)] -E[V εn (k) | V εn (k -1)] 2 , for all x ∈ [a, 0].
We will deduce the convergence of Λ

(n) a,v from a criterion of Ethier and Kurtz [EK], namely Theorem 4.1 p.354. According to this result the convergence on [a, 0] in Theorem 1.1 follows from Propositions 2.1 and 2.2 below. In addition we need to verify that the martingale problem associated with the operator 2λd 2 /(dλ) 2 + 2(1 + h(λ))d/dλ is well posed. This follows from Theorem 2.3 p.372 in [EK] (with the notation of [EK] take r 0 = 0 and r 1 = +∞).

Proposition 2.1. Let R > 0 be given. Set τ R n := inf{x ≥ a : Λ (n) a,v (x) ≥ R}. Then for a ≤ x ≤ 0 ∧ τ R n , B (n) a,v (x) = 2 x a (1 + h(Λ (n) a,v (y))) dy + O 1 √ n ,
where the O(n -1/2 ) is deterministic and only depends on a and R.

Proposition 2.2. Let R > 0 be given. Then for a ≤ x ≤ 0 ∧ τ R n , A (n) a,v (x) = 4 x a Λ (n) a,v (y) dy + O 1 √ n ,
where the O(n -1/2 ) is deterministic and only depends on a and R.

These propositions will be proved in the Subsections 2.2-2.5.

2.

2. An auxiliary Markov chain. Let ε and v ≥ 0 be given. We express here (see in particular ( 8) and ( 9) below) the laws of

V ε = V ε,v and V ε = V ε,v in terms
of the law of another Markov chain W ε . A similar representation already appeared in Tóth's paper [T1] on "true" self-avoiding walks. So let us first define (s ε,i , i ≥ 0) by s ε,0 = 0 and for i ≥ 1,

s ε,i := i j=1 1 {Uj ≥pε,j } ,
where (U j , j ≥ 1) is a sequence of i.i.d random variables with uniform distribution in [0, 1]. This s ε,i is equal in law to the number of times the excited random walk jumps from level k to k -1, for some arbitrary k ∈ Z, after i visits at this level k.

For m ≥ 0, set W ε (m) := inf{i ≥ 0 : s ε,i = m}. Then W ε (m) is equal in law to the number of visits to level k before the m-th jump from k to k -1. Moreover, (W ε (m), m ≥ 0
) is a Markov chain on N starting from 0 and with transition operator Q ε defined for any nonnegative or bounded function

f by Q ε f (r) = ℓ≥1 f (r + ℓ)2 -ℓ (1 + ε r+1 ) . . . (1 + ε r+ℓ-1 )(1 -ε r+ℓ ), for all r ∈ N. Furthermore it is immediate that the law of V ε (k + 1) conditionally on {V ε (k) = m} is equal to the law of W ε (m) -m + 1: L(V ε (k + 1) | V ε (k) = m) = L(W ε (m) -m + 1). (8) Similarly the law of V ε (k + 1) conditionally on { V ε (k) = m} is equal to the law of W ε (m) -m: L( V ε (k + 1) | V ε (k) = m) = L(W ε (m) -m). (9)
By convention we denote by Q 0 the transition operator associated to the sequence (ε i , i ≥ 1), where ε i = 0 for all i. In other words

Q 0 f (r) = E[f (r + ξ)] for all r ∈ N,
where ξ is a geometric random variable with parameter 1/2, i.e. P(ξ = ℓ) = 2 -ℓ , for all ℓ ≥ 1. Note that E(ξ) = 2 and V(ξ) = 2. In particular, if u is defined by u(r) = r for all r ∈ N, then for all m ≥ 1,

Q m 0 u(0) = E[ξ 1 + • • • + ξ m ] = 2m, where ξ 1 , . . . , ξ m are i.i.d. geometric random variables with parameter 1/2. Note also that for all m ≥ 1, E[W ε (m)] = Q m ε u(0). Thus (8) shows that (10) E[V ε (k) | V ε (k -1)] -V ε (k -1) = Q Vε(k-1) ǫ u(0) -Q Vε(k-1) 0 u(0) + 1,
for all k ≥ 1. So in view of ( 6) and ( 10), our strategy for proving Proposition 2.1 will be to estimate terms of the form

Q m ε u(0) -Q m 0 u(0). Note that since x < τ R n
by hypothesis, we can restrict us to the case when m ≤ Rn + 1. Likewise

(11) E[V 2 ε (k) | V ε (k-1)]-E[V ε (k) | V ε (k-1)] 2 = Q Vε(k-1) ε u 2 (0)-(Q Vε(k-1) ε u(0)) 2 ,
for all k ≥ 1. So in view of ( 7) and ( 11) we will have also to estimate terms of the form

Q m ε u 2 (0) -(Q m ε u(0)) 2
, for proving Proposition 2.2.

Some elementary properties of the operators

Q ε and Q 0 . For f : N → R, we set |f | ∞ = sup r∈N |f (r)|, Lip(f ) = sup r =r ′ |f (r) -f (r ′ )| |r -r ′ | , and Lip 2 (f ) = sup ℓ∈N Lip(∆ ℓ f ), where ∆ ℓ f (r) = f (r+l)-f (r). Naturally we say that f is Lipschitz if Lip(f ) < +∞. Note that for any f , Lip 2 (f ) ≤ 2Lip(f ) ≤ 4|f | ∞ . Set D := {h : N → R : there exists f, g Lipschitz, such that h(r) = f (r) + rg(r)} . For ε : N → (-1, 1), let R ε := Q ε -Q 0 . Note that when |ε| ∞ ≤ 1/2, R ε h and Q ε h are well defined for any h ∈ D. Observe also that Q ε 1 = Q 0 1 = 1, where 1 is the constant function on N. In particular R ε 1 = 0. Moreover, for any Lipschitz f , |Q ε f | ∞ ≤ |f | ∞ , and
|Q ε f -f | ∞ ≤ CLip(f ), ( 12
)
where C = ℓ≥1 ℓ(4/3) -ℓ . As a corollary we get the Lemma 2.3. There exists a constant C > 0 such that for all |ε| ∞ ≤ 1/2, all j ≥ 0 and all Lipschitz functions f ,

|Q j ε f -f | ∞ ≤ CjLip(f ). Proof. Write Q j ε f -f = j i=1 Q i-1 ε (Q ε f -f ),
and then use (12) for each term of the sum.

Set for all r ≥ 0 and ℓ ≥ 1,

ε r,ℓ := -ε r+ℓ + ℓ-1 i=1 ε r+i ,
and define R ε by

R ε f (r) = ℓ≥1 f (r + ℓ)2 -ℓ ε r,ℓ . This R ε is a linearized version of R ε = Q ε -Q 0 ,
and also the first order term in the expansion of R ε as |ε| ∞ → 0. The next result is immediate.

Lemma 2.4. Assume that |ε| ∞ ≤ 1/2. Then for any h ∈ D and any r,

R ε h(r) = - ℓ≥1 ε r+ℓ h(r + ℓ)2 -ℓ - ∞ i=ℓ+1 h(r + i)2 -i . In particular R ε 1 = 0 since ∞ i=ℓ+1 2 -i = 2 -ℓ .
We also get the following Lemma 2.5. There exists a constant C > 0 such that for all |ε| ∞ ≤ 1/2 and all h ∈ D, with h(r) = f (r) + rg(r),

(13) R ε h(r) = f ε (r) + rg ǫ (r),
where f ε and g ε satisfy

(i) |f ε | ∞ ≤ C(Lip(f ) + |g| ∞ ) × |ε| ∞ (ii) |g ε | ∞ ≤ CLip(g) × |ε| ∞ (iii) Lip(f ε ) ≤ C(Lip 2 (f ) + Lip(g)) × |ε| ∞ + C(Lip(f ) + |g| ∞ ) × Lip(ε) (iv) Lip(g ε ) ≤ CLip 2 (g) × |ε| ∞ + CLip(g) × Lip(ε).
Proof. By using that h(r) R ε 1(r) = 0 for all r, we get

R ε h(r) = - ℓ≥1 ε r+ℓ (h(r + ℓ) -h(r))2 -ℓ - ∞ i=ℓ+1 (h(r + i) -h(r))2 -i .
Thus (13) holds with

f ε (r) = - ℓ≥1 ε r+ℓ (f (r + l) -f (r))2 -ℓ - ∞ i=ℓ+1 (f (r + i) -f (r))2 -i - ℓ≥1 ε r+ℓ g(r + ℓ)ℓ2 -ℓ - ∞ i=ℓ+1 g(r + i)i2 -i ,
and

g ε (r) = - ℓ≥1 ε r+ℓ (g(r + ℓ) -g(r))2 -ℓ - ∞ i=ℓ+1 (g(r + i) -g(r))2 -i .
All assertions follow immediately. For instance we can write

|g ε | ∞ ≤ Lip(g) × |ε| ∞ × ℓ≥1 ℓ2 -ℓ + ∞ i=ℓ+1 i2 -i ,
which implies (ii) and one can prove similarly (i), (iii) and (iv).

Next we have

Lemma 2.6. There exists a constant C > 0 such that for all |ε| ∞ ≤ 1/2 and all h ∈ D, with h

(r) = f (r) + rg(r), R ε h(r) -R ε h(r) = f ε (r) + rg ε (r),
where f ε and g ε satisfy

|f ε | ∞ ≤ C(Lip(f ) + |g| ∞ ) × |ε| 2 ∞ , |g ε | ∞ ≤ CLip(g) × |ε| 2 ∞ . Proof. Recall that (R ε -R ε )h(r) = ℓ≥1 h(r + ℓ)2 -ℓ (ε r,ℓ -ε r,ℓ ),
where for all r and ℓ,

ε r,ℓ := (1 + ε r+1 ) . . . (1 + ε r+ℓ-1 )(1 -ε r+ℓ ) -1. Since h(r)R ε 1(r) = h(r) R ε 1(r) = 0 for all r, we get (R ε -R ε )h(r) = ℓ≥1 (h(r + ℓ) -h(r))2 -ℓ (ε r,ℓ -ε r,ℓ ) = f ε (r) + rg ε (r). with f ε (r) = ℓ≥1 (f (r + l) -f (r))2 -ℓ (ε r,ℓ -ε r,ℓ ) + ℓ≥1 g(r + ℓ)ℓ2 -ℓ (ε r,ℓ -ε r,ℓ ),
and

g ε (r) = ℓ≥1 (g(r + ℓ) -g(r))2 -ℓ (ε r,ℓ -ε r,ℓ ).
But for any r and any ℓ ≥ 1,

|ε r,ℓ -ε r,ℓ | ≤ (1 + |ε| ∞ ) ℓ -1 -ℓ|ε| ∞ ≤ ℓ 2 (1 + |ε| ∞ ) ℓ-2 |ε| 2 ∞ (14) ≤ ℓ 2 (3/2) ℓ-2 |ε| 2
∞ . The lemma follows.

Lemmas 2.5 and 2.6 imply Lemma 2.7. There exists a constant C > 0 such that for all |ε| ∞ ≤ 1/2 and all h ∈ D, with h

(r) = f (r) + rg(r), R ε h(r) = f ε (r) + rg ε (r),
where f ε and g ε satisfy

|f ε | ∞ ≤ C(Lip(f ) + |g| ∞ ) × |ε| ∞ , |g ε | ∞ ≤ CLip(g) × |ε| ∞ .
We will need also the following Lemma 2.8. For all h ∈ D, with h(r) = f (r) + rg(r), and all i ≥ 0,

Q i 0 h(r) = f i (r) + rg i (r), where f i and g i satisfy (i) |f i | ∞ ≤ |f | ∞ + 2i|g| ∞ (ii) Lip(f i ) ≤ Lip(f ) + 2iLip(g) (iii) |g i | ∞ ≤ |g| ∞ (iv)
Lip(g i ) ≤ Lip(g). Moreover, for all r,

|Q i 0 h(r) -h(r + 2i)| ≤ √ 2i(Lip(f ) + |g| ∞ + (r + 2i)Lip(g)).
Proof. Just recall that for all i and r,

Q i 0 h(r) = E[h(r + ξ 1 + • • • + ξ i )]
, where ξ 1 , . . . , ξ i are i.i.d. geometric random variables with parameter 1/2. Thus,

Q i 0 h(r) = f i (r) + rg i (r), where f i (r) = E[f (r + ξ 1 + • • • + ξ i )] + E[(ξ 1 + • • • + ξ i )g(r + ξ 1 + • • • + ξ i )],
and

g i (r) = E[g(r + ξ 1 + • • • + ξ i )].
Claims (i), (ii), (iii) and (iv) follow immediately frome these expressions (by using also that

E[|ξ 1 + • • • + ξ i |] = 2i). Next write |Q i 0 f (r) -f (r + 2i)| ≤ E[|f (r + ξ 1 + • • • + ξ i ) -f (r + 2i)|] ≤ Lip(f )E[|ξ 1 + • • • + ξ i -2i|] ≤ √ 2iLip(f ),
by using Cauchy-Schwarz inequality and the fact that E(ξ i ) = 2 and V(ξ i ) = 2, for all i. We also have

|Q i 0 (h -f )(r) -(r + 2i)g(r + 2i)| ≤ E[|(ξ 1 + • • • + ξ i -2i)g(r + ξ 1 + • • • + ξ i )| +E[|(r + 2i)(g(r + ξ 1 + • • • + ξ i ) -g(r + 2i))|] ≤ √ 2i (|g| ∞ + (r + 2i)Lip(g)) .
Lemma 2.9. There exists a constant C > 0 such that for all |ε| ∞ ≤ 1/2, all i ≥ 0 and all Lipschitz f ,

|(Q i ε -Q i 0 )f | ∞ ≤ CiLip(f ) × |ε| ∞ . Proof. First write Q i ε -Q i 0 = i-1 j=0 Q i-j+1 ε R ε Q j 0 . (15)
Then by using that

|Q i-j+1 ε f | ∞ ≤ |f | ∞ ,
for all j ≤ i -1, we get (using Lemma 2.7 with g = 0),

|(Q i ε -Q i 0 )f | ∞ ≤ i-1 j=0 |R ε Q j 0 f | ∞ ≤ C i-1 j=0 Lip(Q j 0 f )|ε| ∞ .
We conclude the proof of the lemma by using that Lip(Q j 0 f ) ≤ Lip(f ) for all j.

Lemma 2.10. There exists a constant C > 0 such that for all |ε| ∞ ≤ 1/2, all i ≥ 0 and all h ∈ D, with h(r) = f (r) + rg(r),

|Q i ε h(r)| ≤ |f | ∞ + (r + Ci)|g| ∞ . Proof. We have |Q i ε h(r)| ≤ |f | ∞ + |g| ∞ |Q i ε u(r)|
, where u is defined by u(r) = r for all r ∈ N. Now, Lemma 2.9 implies that

|Q i ε u(r) -Q i 0 u(r)| ≤ Ci|ε| ∞ . We conclude by using that Q i 0 u(r) = r + 2i.
Lemma 2.11. There exists a constant C > 0 such that for all |ε| ∞ ≤ 1/2, all i ≥ 0 and all h ∈ D, with h(r) = f (r) + rg(r),

|(Q i ε -Q i 0 )h(r)| ≤ Ci (Lip(f ) + |g| ∞ + (r + i)Lip(g)) × |ε| ∞ .
Proof. We have Q j 0 h(r) = f j (r) + rg j (r) and R ε Q j 0 h(r) = f j,ε (r) + rg j,ε (r). Lemma 2.10 implies that

|Q i-j+1 ε R ε Q j 0 h(r)| ≤ |f j,ε | ∞ + (r + C(i -j + 1))|g j,ε | ∞ . Lemma 2.7 implies that |f j,ε | ∞ ≤ C(Lip(f j ) + |g j | ∞ ) × |ε| ∞ and |g j,ε | ∞ ≤ CLip(g j ) × |ε| ∞ . Lemma 2.8 implies that |f j,ε | ∞ ≤ C(Lip(f ) + |g| ∞ + 2jLip(g)) × |ε| ∞ , and |g j,ε | ∞ ≤ CLip(g) × |ε| ∞ . Now, i-1 j=0 |f j,ε | ∞ ≤ Ci(Lip(f ) + |g| ∞ + iLip(g)) × |ε| ∞ , and i-1 j=0 (r + C(i -j + 1))|g j,ε | ∞ ≤ C(r + i)i × Lip(g) × |ε| ∞ .
Using then (15), this proves the lemma.

Our last result in this subsection is the following (recall that u(r) = r for all r ∈ N):

Lemma 2.12. There exists a constant C > 0 such that for all m ≥ 0 and all

|ε| ∞ ≤ 1/2, |Q m ε u -Q m 0 u - m i=1 Q m-i 0 R ε u| ∞ ≤ C(m|ε| 2 ∞ + m 2 |ε| ∞ Lip(ε)).
Proof. First observe that for all j ≥ 0,

Q j 0 u = u + 2j. Since R ε is linear and R ε 1 = 0, we get R ε Q i-1 0 u = R ε u for all i ≥ 1. Thus m i=1 Q m-i 0 R ε u = m i=1 Q m-i 0 R ε Q i-1 0 u.
Next we have

Q m ε u -Q m 0 u - m i=1 Q m-i 0 R ε Q i-1 0 u = m i=1 Q m-i 0 (R ε -R ε )Q i-1 0 u + m i=1 (Q m-i ε -Q m-i 0 ) R ε Q i-1 0 u.
By using Lemma 2.6 and the fact that Lip(Q i 0 u) ≤ Lip(u) = 1 for all i, we get

m i=1 |Q m-i 0 (R ε -R ε )Q i-1 0 u| ∞ ≤ C m i=1 Lip(Q i-1 0 u)|ε| 2 ∞ ≤ Cm|ε| 2 ∞ .
Then by using Lemma 2.9 we obtain

m i=1 |(Q m-i ε -Q m-i 0 ) R ε Q i-1 0 u| ∞ ≤ C|ε| ∞ m i=1 (m -i)Lip( R ε u).
Using Lemma 2.5 with f (r) = 1 and g(r) = 0, we have g ε (r) = 0 and ( 16)

Lip( R ε u) = Lip(f ε ) ≤ CLip(ε).
This proves the lemma.

2.4. Proof of Proposition 2.1.

Recall that ε n = (ε i (n), i ≥ 1), with ε i (n) = ϕ(i/2n)/(2n).
Since ϕ is bounded, we can always assume by taking large enough

n if necessary, that |ε n | ∞ ≤ 1/2. Note also that Lip(ε n ) = O(1/n 2 ). Assume now that m = O(n).
Then Lemma 2.12 shows that

Q m εn u -Q m 0 u = m-1 i=0 Q i 0 R εn u + O 1 n .
Next write

m-1 i=0 Q i 0 R εn u(0) = m-1 i=0 R εn u(2i) + m-1 i=0 (Q i 0 R εn u(0) -R εn u(2i)).
By using Lemma 2.8 (applied to f = R εn u and g = 0) we get

m-1 i=0 |Q i 0 R εn u(0) -R εn u(2i)| ≤ √ 2 m-1 i=0 Lip( R εn u) √ i ≤ Cm 3/2 Lip(ε n ) ≤ C √ n .
On the other hand, set

a ℓ := -ℓ2 -ℓ + ∞ j=ℓ+1 j2 -j = 2 -ℓ+1 . ( 17
)
Then by using Lemma 2.4 we get

m-1 i=0 R εn u(2i) = m-1 i=0 ∞ ℓ=1 a ℓ (ε n ) 2i+ℓ = ∞ ℓ=1 2 -ℓ+1 × 1 2n m-1 i=0 ϕ 2i + ℓ 2n . But ∞ ℓ=1 2 -ℓ+1 = 2
, and since ϕ is Lipschitz and bounded

1 n m-1 i=0 ϕ 2i + ℓ 2n = m/n 0 ϕ(s) ds + O ℓ n .
Thus putting the pieces together we get

Q m εn u(0) -Q m 0 u(0) = h m n + O 1 √ n . ( 18
)
Finally we get the equalities in law, for a ≤ x ≤ 0 ∧ τ R n ,

B (n) a,v (x) = 1 n [2nx]-[2na] k=1 (E[V εn (k) | V εn (k -1)] -V εn (k -1)) = 1 n [2nx]-[2na] k=1 1 + Q Vε n (k-1) εn u(0) -Q Vε n (k-1) 0 u(0) = 1 n [2nx]-[2na] k=1 1 + h Λ (n) a,v a + k -1 2n + O 1 √ n = 2 x a 1 + h(Λ (n) a,v (y)) dy + O 1 √ n ,
where the second equality follows from ( 10) and the third one from ( 18) and the relation between Λ

(n)

a,v and V εn given in (1) and at the begining of Subsection 2.1. This finishes the proof of Proposition 2.1. 2.5. Proof of Proposition 2.2. We assume throughout this subsection that m = O(n). Then on the one hand by using Lemma 2.12, we get

Q m εn u(0) = 2m + m i=1 Q m-i 0 R εn u(0) + O 1 n . Moreover Lemma 2.5 shows that |Q i 0 R εn u(0)| ≤ | R εn u| ≤ C|ε n | ∞ = O(1/n) uni- formly in i. Thus (Q m εn u(0)) 2 = 4m 2 + 4m m i=1 Q m-i 0 R εn u(0) + O(1).
On the other hand we have for all ε,

Q m ε u 2 = Q m 0 u 2 + m i=1 Q m-i ε R ε Q i-1 0 u 2 .
A variance calculus shows that

Q i-1 0 u 2 = u 2 + 4(i -1)u + 4(i -1) 2 + 2(i -1), which implies that for all ε, R ε Q i-1 0 u 2 = R ε u 2 + 4(i -1)R ε u, since R ε 1 = 0. Thus Q m ε u 2 (0) = 4m 2 + 2m + E ε,m + F ε,m
, where

E ε,m = m i=1 Q m-i ε R ε u 2 (0), and
F ε,m = 4 m i=1 (i -1)Q m-i ε R ε u(0).
We now prove the following Lemma 2.13. We have

E εn,m = m i=1 Q m-i 0 R εn u 2 (0) + O(1), (19)
and

F εn,m = 4 m i=1 (i -1)Q m-i 0 R εn u(0) + O(1). (20) Proof. We have (R ε -R ε )u 2 (r) = ℓ≥1 (2rℓ + ℓ 2 )2 -ℓ (ε r,ℓ -ε r,ℓ ) for all r.
Thus, by using ( 14), we see that there exists a constant C > 0 such that

|(R εn -R εn )u 2 (r)| ≤ C|ε n | 2 ∞ (r + 1) ≤ C (r + 1) n 2 for all r. (21) This implies that |E εn,m - m i=1 Q m-i εn R εn u 2 (0)| ≤ m i=1 Q m-i εn |R εn u 2 -R εn u 2 |(0) ≤ C n 2 m i=1 Q m-i εn f (0)
with f (r) = r + 1. By using Lemma 2.3, applied to f (r) = r + 1, we see that there exists C > 0 such that

|E εn,m - m i=1 Q m-i εn R εn u 2 (0)| ≤ C n 2 m i=1 (1 + m -i) ≤ C m 2 n 2 = O(1).
Recall the formula for a ℓ given in (17) and let

b ℓ := -ℓ 2 2 -ℓ + ∞ i=ℓ+1 i 2 2 -i . Then Lemma 2.4 shows that (22) R εn u 2 (r) = f n (r) + rg n (r),
where

f n (r) = ℓ≥1 b ℓ (ε n ) r+ℓ g n (r) = 2 ℓ≥1 a ℓ (ε n ) r+ℓ .
Next by using Lemma 2.11, we get for all j = m -i and 1

≤ i ≤ m, |(Q j εn -Q j 0 ) R εn u 2 (0)| ≤ Cj (Lip(f n ) + |g n | ∞ + jLip(g n )) × |ε n | ∞ ≤ Cj (|ε n | ∞ + (j + 1)Lip(ε n )) × |ε n | ∞ ≤ C m n 2 + m 2 n 3 = O 1 n .
This proves (19). Now Lemmas 2.6 and 2.9, together with ( 16), show that (for j = m -i)

|Q j εn R εn u -Q j 0 R εn u| ∞ ≤ |Q j εn (R εn -R εn )u| ∞ + |(Q j εn -Q j 0 ) R εn u| ∞ = O |ε n | 2 ∞ + j|ε n | ∞ Lip(ε n ) = O 1 n 2 .
This proves (20) and finishes the proof of the lemma.

We can now write

Q m εn u 2 (0) -(Q m εn u(0)) 2 = 2m + m-1 j=0 Q j 0 R εn u 2 (0) +4 m-1 j=0 (m -j -1)Q j 0 R εn u(0) -4m m-1 j=0 Q j 0 R εn u(0) + O(1).
By using Lemmas 2.8 and the form of R εn u 2 given by ( 22) (and using that Lip(

f n ) = O(n -2 ), Lip(g n ) = O(n -2 ) and |g n | ∞ = O(n -1 )) we get for j ≤ m -1, Q j 0 R εn u 2 (0) = R εn u 2 (2j) + O(n -1/2 ).
By using Lemmas 2.8, the fact that R εn is Lipschitz and bounded, and ( 16), we get

Q j 0 R εn u(0) = R εn u(2j) + O(n -3/2 ).
Therefore

Q m εn u 2 (0) -(Q m εn u(0)) 2 = 2m + m-1 j=0 R εn u 2 (2j) -4 m-1 j=0 (j + 1) R εn u(2j) + O(n 1/2 ). Lemma 2.4 shows that R εn u 2 (2j) = 4j ℓ≥1 a ℓ (ε n ) 2j+ℓ + O 1 n , and
R εn u(2j) = ℓ≥1 a ℓ (ε n ) 2j+ℓ . Thus Q m εn u 2 (0) -(Q m εn u(0)) 2 = 2m + O(n 1/2 ).
Then (11) shows that for a ≤ x ≤ 0 ∧ τ R n ,

A (n) a,v (x) = 1 n 2 [2nx]-[2na] k=1 Q Vε n (k-1) εn u 2 (0) -(Q Vε n (k-1) εn u(0)) 2 = 2 n [2nx]-[2na] k=1 Λ (n) a,v a + k -1 2n + O 1 √ n = 4 x a Λ (n) a,v (y) dy + O 1 √ n .
This finishes the proof of Proposition 2.2.

2.6. Proof of the convergence on [0, +∞). The proof of the convergence of Λ

(n)

a,v on [0, +∞) is essentially the same as the proof on [a, 0]. Namely we can define

M (n) a,v , B (n)
a,v and A

(n) a,v , respectively as in ( 5), ( 6) and ( 7) with V everywhere instead of V . Let also

w (n,-) a,v := 1 2n sup{k ≤ 0 : S εn,[2na],[nv] (k) = 0} w (n,+) a,v := 1 2n inf{k ≥ a : S εn,[2na],[nv] (k) = 0}. (23) Then Λ (n) a,v (x) = Λ (n) a,v (0) + M (n) a,v (x) + B (n) a,v (x) for all x ∈ [0, w (n,+) a,v ). Moreover (9) shows that E[ V ε (k) | V ε (k -1)] -V ε (k -1) = Q Vε(k-1) ǫ u(0) -Q Vε(k-1) 0 u(0), and
E[ V ε (k) 2 | V ε (k -1)] -E[ V ε (k) | V ε (k -1)] 2 = Q Vε(k-1) ε u 2 (0) -(Q Vε(k-1) ε u(0)) 2 ,
for all k ≥ 1. Then by following the proofs given in the previous subsections we get the analogues of Proposition 2.1 and 2.2: Proposition 2.14. Let R > 0 and T > 0 be given. Then for 0

≤ x ≤ T ∧ τ R n ∧ w (n,+) a,v , B (n) a,v (x) = 2 x 0 h(Λ (n) a,v (y)) dy + O 1 √ n ,
where the O(n -1/2 ) is deterministic and only depends on a, T and R.

Proposition 2.15. Let R > 0 and T > 0 be given. Then for 0

≤ x ≤ T ∧ τ R n ∧ w (n,+) a,v , A (n) a,v (x) = 4 x 0 Λ (n) a,v (y) dy + O 1 √ n ,
where the O(n -1/2 ) is deterministic and only depends on a, T and R.

So according again to the criterion of Ethier and Kurtz (Theorem 4.1 p.354 in [EK]), we deduce the convergence in law of Λ

(n) a,v on [0, +∞). Actually one can deduce the convergence on [a, +∞) as well. For this we just need to observe that the criterion of Ethier and Kurtz applies in the same way for non homogeneous operators. For reader's convenience let us recall the main steps of its proof. First Propositions 2.1, 2.2, 2.14 and 2.15 imply the tightness of the sequence (Λ (n) a,v , n ≥ 1) on [a, +∞). Next Itô Formula shows that any limit of a subsequence is a solution of the non-homogeneous martingale problem (see the definition in [EK] p.221) associated with the operator

G x f (λ) := 2λf ′′ (λ) + 2(1 + h(λ))f ′ (λ) if x ∈ [a, 0] 2λf ′′ (λ) + 2h(λ)f ′ (λ) if x ∈ [0, +∞).
Then Theorem 2.3 p.372 in [EK] (with their notation replace t by x, x by λ and take r 0 = 0 and r 1 = +∞) shows that this martingale problem is well posed (in particular it has a unique solution). This proves the desired convergence on [a, +∞). Since the proof of the convergence on (-∞, a] is the same as on [0, +∞), this concludes the proof of Theorem 1.1.

Extension to the non homogeneous setting

We give here an extension of Theorem 1.1 when |I| = 1 and ϕ is allowed to be space dependent. Apart from its own interest, we will use this extension to prove Theorem 1.1 when |I| ≥ 2, see the next section.

We first define non homogeneous cookies random walks. If

ε = (ε i,x , i ≥ 1, x ∈ Z), is given, we set p ε,i,x := 1 2 (1 + ε i,x ),
for all i and x. Then X ε is defined by

P[X ε (n + 1) -X ε (n) = 1 | F ε,n ] = 1 -P[X ε (n + 1) -X ε (n) = -1 | F ε,n ] = p ε,i,x , if #{j ≤ n : X ε (j) = X ε (n)} = i and X ε (n) = x.
Similarly non homogeneous excited Brownian motions are defined by

dY t = dB t + ϕ(Y t , L Yt t )
dt, for some bounded and measurable ϕ. Such generalized version of excited BM was already studied in [START_REF] Norris | Self-avoiding random walk: a Brownian motion model with local time drift[END_REF] and [RS]. In particular Ray-Knight results were obtained in this context and a sufficient condition for recurrence is given in [RS] (see below). Now let ϕ be some fixed bounded càdlàg function. Assume that for each n ≥ 1, a function

ϕ n : Z × [0, ∞) → R, càdlàg in the second variable, is given. Consider ∆ n (x) := sup ℓ |ϕ n ([2nx], ℓ) -ϕ(x, ℓ)|,
and assume that

∆ n → 0 in D(R) when n → ∞. (24) Assume further that sup k,ℓ |ϕ n (k, ℓ)| < 2n for n large enough and define ε n = (ε i,x (n), i ≥ 1, x ∈ Z) by ε i,x (n) = 1 2n ϕ n x, i 2n , ( 25
)
for all i ≥ 1 and x ∈ Z. Say that ϕ is uniformly Lipschitz in the second variable if

sup x∈R sup ℓ =ℓ ′ |ϕ(x, ℓ) -ϕ(x, ℓ ′ )| |ℓ -ℓ ′ | < +∞. (26) Finally define Λ (n)
a,v and Λ a,v as in the homogeneous setting (see the introduction). We can state now the following natural extension of Theorem 1.1: Theorem 3.1. Let ϕ be some bounded càdlàg function satisfying (26). Assume that for n large enough, X εn is recurrent and that Y is recurrent. Assume further that (24) holds. Then for any a ∈ R and v ≥ 0,

(Λ (n) a,v (x), x ∈ R) L =⇒ n→∞ (Λ a,v (x), x ∈ R) .
The proof of this result is exactly the same as the proof of Theorem 1.1. Note that as at the end of the previous subsection, we need to use here a non homogeneous version of Ethier-Kurtz's result (Theorem 4.1 p.354 in [EK]). This time we just have to verify that the martingale problem associated with the operator

G x f (λ) := 2λf ′′ (λ) + 2(1 + h(x, λ))f ′ (λ) if x ∈ [a, 0] 2λf ′′ (λ) + 2h(x, λ)f ′ (λ) if x ∈ [0, +∞),
is well posed, where h(x, λ) = λ 0 ϕ(x, µ) dµ, for any x and λ. But again this follows from Theorem 2.3 p.372 in [EK].

In particular the above theorem applies to the following situation, which we will use in the proof of Theorem 4.1. Assume that ϕ : R × [0, ∞) → R satisfies the hypotheses of Theorem 3.1 and that a sequence (ϕ n , n ≥ 1) converges to ϕ as in (24). Assume in addition that for each n ≥ 1, a function (λ(n, x), x ∈ Z) is given. Set λ n := λ(n, [2n•]) and assume further that there exists λ ∈ D(R) such that

λ n → λ in D(R) when n → ∞. (27) Set ϕ λ (x, ℓ) := ϕ(x, λ(x) + ℓ) for all x ∈ R and ℓ ≥ 0, and ϕ ′ n (x, ℓ) := ϕ n (x, λ(n, x) + ℓ) for all x ∈ Z and ℓ ≥ 0. Note that if (27) holds, then ϕ ′ n ([2n•],
•) converges to ϕ λ as in (24). Let now ε n,λn = (ε i,x (n, λ n ), i ≥ 1, x ∈ Z) be defined by

ε i,x (n, λ n ) := 1 2n ϕ ′ n x, i 2n .
Let Λ (n,λn) and Λ (λ) be the processes associated to ε n,λn and ϕ λ as in the introduction. The following is an immediate application of Theorem 3.1:

Corollary 3.2. Assume that the hypotheses of Theorem 3.1 and (27) hold true. Then with the above notation, for any a and v,

Λ (n,λn) a,v (x), x ∈ R L =⇒ n→∞ Λ (λ) a,v (x), x ∈ R .
Remark 3.3. Actaully the result of this corollary holds as well in the slightly more general setting where v is not fixed. More precisely, if

v n converges to v when n → ∞, then Λ (n,λn) a,vn also converges in law toward Λ (λ)
a,v . The proof is exactly the same, since this setting is covered by Ethier-Kurtz's result.

To finish this section, we recall some sufficient condition for recurrence of X ε and Y proved respectively in [START_REF] Zerner | Multi-excited random walks on integers[END_REF]Corollary 7] and [START_REF] Raimond | Excited Brownian motion[END_REF]Corollary 5.6] in the non homogeneous case. We notice that it applies only when for all i and x, ε i,x , respectively ϕ, is nonnegative. We only state the result in the continuous setting, the result for X ε being analogous. So if for x ∈ R,

δ x (ϕ) := ∞ 0 ϕ(x, ℓ) dℓ, then Y is recurrent as soon as lim inf z→+∞ 1 z z 0 δ x (ϕ) dx < 1.

Proof of Theorem 1.1 in the general case

We actually prove the result in the non homogeneous setting.

Theorem 4.1. Under the hypotheses of Theorem 3.1, for any finite set I, any

a i ∈ R and v i ≥ 0, i ∈ I, Λ (n) ai,vi (x), x ∈ R i∈I L =⇒ n→∞ (Λ ai,vi (x), x ∈ R) i∈I .
Proof. When |I| = 1, the result is given by Theorem 3.1. The general case can then be proved by induction on the cardinality of I. To simplify the notation we only make the proof of the induction step when the cardinality of I equals 2, but it would work similarly in general. So let a, a ′ , v and v ′ be given. All we have to prove is that for any continuous and bounded functions H and H,

E H Λ (n) a ′ ,v ′ H Λ (n) a,v → E H (Λ a ′ ,v ′ ) H (Λ a,v ) , (28)
when n → ∞. Consider the events

A n := Λ (n) a,v (a ′ ) < v ′ ,
for n ≥ 1, and

A := {Λ a,v (a ′ ) < v ′ } .
Observe that conditionally to Λ

a,v and on the set A n we have the equality in law:

Λ (n) a ′ ,v ′ -Λ (n) a,v = Λ (n,Λ (n) a,v (a+ •)) a ′ -a,v ′ -Λ (n) a,v (a ′ ) , (29)
with the notation of Corollary 3.2. This identity is straightforward. Maybe less immediate is the analogous equality in the continuous setting, so we state it as a lemma:

Lemma 4.2. Let a, a ′ , v and v ′ be given. Conditionally to Λ a,v and on A, we have the equality in law:

Λ a ′ ,v ′ -Λ a,v = Λ (Λa,v (a+ •)) a ′ -a,v ′ -Λa,v (a ′ ) . (30)
Proof. One just has to observe (see also (2) in [RS]) that conditionally to Λ a,v and on A, the law of (Y t+τa(v) , t ≥ 0) is equal to the law of an excited BM starting from a and associated to the non homogeneous function ϕ defined by

ϕ(x, ℓ) = ϕ(x, Λ a,v (x) + ℓ).
The lemma follows.

It follows from (29) that for any continuous and bounded H,

E H Λ (n) a,v + (Λ (n) a ′ ,v ′ -Λ (n) a,v ) Λ (n) a,v 1 An = H n Λ (n) a,v 1 An , where H n (λ) := E H λ + Λ (n,λ(a+•)) a ′ -a,v ′ -λ(a ′ )
, for any λ in the Skorokhod space D(R) such that λ(a ′ ) ≤ v ′ . Define similarly H by

H(λ) := E H λ + Λ (λ(a+•)) a ′ -a,v ′ -λ(a ′ )
, for any λ ∈ D(R) such that λ(a ′ ) ≤ v ′ . Now Corollary 3.2 (see also the remark following it) shows that for any sequence of functions λ n , satisfying λ n (a ′ ) ≤ v ′ , and converging to some λ in D(R), H n (λ n ) converges toward H(λ). Moreover, by using the Skorokhod's representation theorem (see Theorem 6.7 in [Bil]), we can assume that Λ

(n) a,v converges almost surely toward Λ a,v . We claim that 1 An also converges a.s. to 1 A . To see this it suffices to prove that

P[Λ a,v (a ′ ) = v ′ ] = 0. But the set {Λ a,v (a ′ ) = v ′ } is included in the set {e a ′ (v ′ ) = 0}
, where e a ′ (v ′ ) denotes the excursion of Y out of level a ′ starting from τ a ′ (v ′ -), and this last event has probability 0 (this is well known to be the case for the Brownian motion, and can be deduced for Y by an absolute continuity argument, see also [RS]).

So if H and H are two continuous and bounded functions, we deduce from the dominated convergence theorem that

E H Λ (n) a ′ ,v ′ H Λ (n) a,v 1 An → E H (Λ a ′ ,v ′ ) H (Λ a,v ) 1 A , (31)
when n → ∞. By using the same argument we see that the convergence in (31) also holds if we replace A n and A respectively by A c n and A c . Then (28) follows and this concludes the proof of Theorem 4.1. On the other hand, the occupation times formula ( [RY] p.224) gives

τ a (v) = R Λ a,v (
y) dy.

Now Theorem 1.1 shows that for any a i , v i , i ∈ I, and any fixed A > 0, the following convergence in law holds:

A -A Λ (n) ai,vi (y) dy i∈I L =⇒ n→∞ A -A Λ ai,vi (y) dy i∈I .
So Corollary 1.2 follows from the following lemma (recall that w

(n,±) a,v
is defined in (23)):

Lemma 5.1. Let ǫ > 0, a ∈ R and v ≥ 0 be given. Then there exists A > 0, such that P |w (n,±) a,v | ≥ A ≤ ǫ, for all n large enough.

Proof. We prove the result for w (n,+) a,v . The proof for w (n,-) a,v is the same. First observe that w + a,v is nonnegative and a.s. finite: it is equal to sup{Y t : t ≤ τ a (v)} and τ a (v) is a.s. finite since Y is recurrent. So for any ǫ > 0, there exists A > a such that P[w + a,v ≥ A] ≤ ǫ. Moreover by using Theorem 1.1 and Skorokhod's representation Theorem, it is possible to define Λ (n) a,v and Λ a,v on the same probability space, such that for any η > 0,

P sup 0≤x≤A |Λ (n) a,v (x) -Λ a,v (x)| ≥ η ≤ ǫ,
for n large enough. Thus

P[T n (η) ≥ A] ≤ 2ǫ, ( 32
)
where

T n (η) = inf{x > 0 : Λ (n) a,v (x) ≤ η}. Recall now that on [0, +∞), Λ (n) a,v (•) is equal in law to V εn ([2n•])/n (see the begin- ning of Section 2.1). But since |ε n | ∞ = O(1/n), ( V εn (k), k ≥ 0)
is stochastically dominated by a Galton-Watson process (W n (k), k ≥ 0) with offspring distribution a geometrical law with parameter 1 -p n = 1/2 -c/n, for some constant c > 0 (with the convention that if G is a random variable with such geometrical law, then

P(G = k) = p n (1 -p n ) k for all k ≥ 0, in particular E(G) = (1 -p n )/p n < 1). Moreover, when W n (0) = 1, the probability for W n to extinct before time [nA] can be computed explicitly. If f (n) k (•) is the generating function of W n (k), then this probability is equal to f (n)
[nA] (0). An expression for f (n) k (0) is given for instance in [AN] p.6-7:

f (n) k (0) = 1 -m k n 1 -s n m k n -s n for all k ≥ 1,
where

m n = p n 1 -p n = 1 + 4c n + O 1 n 2
, and

s n = {1 -m n (1 -p n )}/p n = 1 - 4c n + O 1 n 2 .
It follows that f

(n) [nA] (0) = 1 -c ′ /n + O(1/n 2 ), with c ′ = 4c/(1 -e -4cA
) > 0. Now the law of W n starting from [ηn] is equal to the law of the sum of [ηn] independent copies of W n starting from 1. Thus if W n (0) = [ηn], the probability for W n to extinct before time [nA] is f (n)

[nA] (0) [ηn] . If η is small enough and n large enough, this probability is larger than (1 -ǫ). By using now that V εn is stochastically dominated by W n , (32) and the strong Markov property, we get

P w (n,+) a,v ≥ 2A ≤ P w (n,+) a,v ≥ 2A and T n (η) ≤ A + P [T n (η) ≥ A] ≤ P [W n ([nA]) > 0 | W n (0) = [ηn]] + 2ǫ ≤ 3ǫ.
This concludes the proof of the lemma.

Proof of Corollary 1.3

To simplify notation we only prove the result when |I| = 1 but the general case works the same. First note that for any λ, the law of Y γ λ has for density the function a → λE[L a γ λ ]. Indeed, for any bounded and measurable function φ,

E[φ(Y (γ λ ))] = E ∞ 0 φ(Y (s))λe -λs ds = E 0<s<t λ 2 φ(Y (s))e -λt ds dt = E R ∞ 0 λ 2 φ(a)L a t e -λt dt da = R E[φ(a)λL a γ λ ] da,
where in the third equality we have used the occupation times formula (see Corollary (1.6) p.224 in [RY]).

We now follow the argument given by Tóth in [T2]. First observe that if

τ ε,a (v) := inf {j : #{i ≤ j : X ε (i) = a and X ε (i + 1) = a + 1} = v + 1} ,
then exactly as we proved Corollary 1.2, we can show that τ εn,[2na] ([nv])/(4n 2 ) converges in law toward τ a (v) for any a ∈ R and v ≥ 0. Next observe that for any a ∈ Z and k ∈ N,

P[X εn (k) = a] = v∈N {P[τ εn,a (v) = k] + P[ τ εn,a (v) = k]} .
Thus for any a ∈ R, 2nP X εn (θ λ/(4n 2 ) ) = [2na] = 2n(1 -e -λ/(4n 2 ) ) k≥0 e -kλ/(4n 2 )

× v∈N P[τ εn,[2na] (v) = k] + P[ τ εn,[2na] (v) = k] ∼ λ 2n v∈N E e -λ τ εn ,[2na] (v) 4n 2 + E e -λ τ εn ,[2na] (v) 4n 2
, since 2n(1 -e -λ/(4n 2 ) ) ∼ λ/(2n). Note now that

1 n v∈N E e -λ τ εn ,[2na] (v) 4n 2 = ∞ 0 E e -λ τ εn ,[2na] ([nv]) 4n 2 dv,
and that for any

v ∈ R + , Corollary 1.2 implies E e -λ τ εn ,[2na] ([nv]) 4n 2 → E e -λτa(v) ,
when n → ∞. The same remark applies with τ instead of τ . Thus by application of Fatou's lemma, for every a ∈ R,

lim inf n→∞ (2n)P X εn (θ λ/(4n 2 ) ) = [2na] ≥ λ ∞ 0 E e -λτa(v) dv. (33)
But notice that for every a ∈ R and v ≥ 0,

E e -λτa(v) = λ ∞ 0 e -λs P[τ a (v) ≤ s] ds = λ ∞ 0 e -λs P[L a s ≥ v] ds = P[L a γ λ ≥ v]. Therefore λ R ∞ 0 E e -λτa(v) dv da = λ R E[L a γ λ ] da = λE[γ λ ] = 1. (34)
On the other hand for any n,

R (2n)P X εn (θ λ/(4n 2 ) ) = [2na] da = 1. ( 35
)
It follows now from (33) (34) and ( 35) that for almost every a ∈ R,

lim n→∞ (2n)P X εn (θ λ/(4n 2 ) ) = [2na] = λE[L a γ λ ].
The corollary is then a consequence of Sheffé's lemma.

Proof of Theorem 1.4

We actually prove the following extension of Theorem 1.4 in the non homogeneous setting: Theorem 7.1. Let ϕ be some bounded càdlàg function satisfying (26). Let also (ϕ n ) n≥1 be a sequence of bounded càdlàg functions converging to ϕ as in (24). Let ε n be defined by (25) and for t ≥ 0, set X

(n) (t) := X εn ([4n 2 t])/(2n). Then (X (n) (t), t ≥ 0) L =⇒ n→∞ (Y (t), t ≥ 0).
Proof. We first assume that Y is recurrent and that X εn is recurrent as well at least for n large enough. We will see below how one can then remove this hypothesis by using a truncation argument. In this proof we will use Corollaries 1.2 and 1.3, and their extension to the non homogeneous setting (these being straightforwards).

Tightness: We first need to show that the sequence (X (n) (•), n ≥ 1) is tight. All we have to prove (see e.g. Lemma (1.7) p.516 in [RY]) is that for each T > 0, α > 0 and η > 0, there are n 0 and κ > 0, such that for n ≥ n 0 ,

P sup t≤s≤t+κ |X (n) (s) -X (n) (t)| ≥ η ≤ ακ for all t ≤ T. (36)
We first prove the above inequality for t = 0. For this it suffices to find κ > 0 such that

P τ εn,[2nη] (0) ∧ τ εn,[2nη] (0) ≤ 4n 2 κ ≤ ακ,
for n large enough (η > 0 and α being arbitrary and fixed), since the analogous result for η < 0 is similar (use the same proof with the process -X (n) instead of X (n) ). In fact it suffices to prove that P τ εn,[2nη] (0) ≤ 4n 2 κ ≤ ακ/2, (37) since the result with τ εn,[2nη] (0) in place of τ εn,[2nη] (0) is similar. A basic coupling shows that if sup ϕ ≤ C, for some constant C > 0, then the probability on the left hand side of (37) is bounded by the analogous probability we would get by taking ϕ constant equal to C. But it is well known (this follows also from Corollary 1.2) that as n tends to ∞, the left hand side in (37) converges toward P[τ η (0) ≤ κ] and that this last term is a o(κ) for Brownian motion with constant drift (see for instance Proposition (3.7) p.105 in [RY]). This proves (37). To obtain (36) it suffices to observe that after time t, X (n) is equal in law to a renormalized nonhomogeneous cookie random walk starting from X (n) (t) and evolving in a shifted cookie environment (see also ( 29)). So we can apply the same proof and we obtain the same result with the same constants everywhere. This finishes to prove the tightness of (X (n) (•), n ≥ 0). It remains to prove the convergence of the finitedimensional distributions:

Convergence of finite-dimensional distributions: To simplify notation we prove the result for one-dimensional distributions, but it works the same in general. So let (W t , t ≥ 0) be some limit in law of (X (n k ) (t), t ≥ 0), for a subsequence (n k , k ≥ 0). Then for any bounded and measurable function φ,

E φ X εn k (θ λ/(4n k 2 ) ) 2n k ∼ k→∞ λ ∞ 0 e -λt E φ(X (n k ) (t)) dt → k→∞ λ ∞ 0 e -λt E[φ(W t )] dt = E[φ(W γ λ )].
On the other hand Corollary 1.3 shows that the term on the left hand side converges toward E[φ(Y (γ λ ))]. Since this holds for any λ and any φ, we deduce that W t and Y (t) have the same law for every t ≥ 0 (see [START_REF] Feller | An introduction to probability theory and its applications[END_REF]Theorem 1a p.432]). This proves the convergence of one-dimensional distributions.

This finishes the proof under the additional hypothesis of recurrence and it just remains to explain how one can remove this hypothesis. For this we use a truncation argument. For any R > 0 and n ≥ 1, let ϕ R (x, ℓ), resp. ϕ n,R (x, ℓ), be the functions equal to ϕ(x, ℓ), resp. ϕ n (x, ℓ), when x ∈ [-R, R), resp. when |x| ≤ 2nR, and equal to zero otherwise. It is immediate that ϕ n,R still converges to ϕ R as in (24). Denote now by X

R and Y R the processes associated respectively to ϕ n,R and ϕ R . Since ϕ and the ϕ n 's are bounded, and since they are equal to zero outside of [-R, R), these processes are recurrent. So we just have proved that X (n) R converges in law toward Y R . Note now that up to the time ( τ εn,[2nR] (0)∧τ εn,[-2nR] (0))/(4n 2 ), (with the notation from section 6), X (n) R and X (n) are equal in law. Similarly up to the time τ R (0) ∧ τ -R (0), Y R and Y are equal in law. But Corollary 1.2 shows that τ εn,[2nR] (0)/(4n 2 ) and τ εn,[-2nR] (0)/(4n 2 ), converge in law respectively toward τ R (0) and τ -R (0). By using also the monotonicity in R of these random times, we deduce that for any T > 0, P τ εn,[2nR] (0) ∧ τ εn,[-2nR] (0) ≤ 4n 2 T → 0, when R → ∞, uniformly in n. It follows immediately that for any T > 0, X (n) converges in law to Y on the time interval [0, T ]. Since this is true for any T > 0, this concludes the proof of Theorem 7.1.

Remark 7.2. The notation Λ a,v (x) is taken from Tóth and Werner [TW]. We notice by the way that here also the set Λ = {(Λ a,v (x), x ≥ a)} (a,v)∈R×[0,∞) , forms a family of reflected/absorbed coalescing processes. In [TW] the Λ a,v 's were moreover independent Brownian motions (reflected or absorbed in 0 depending on the time interval) and therefore Λ was called (in their Section 2.1) a FICRAB (for family of independent coalescing reflected and absorbed Brownian motions). Such family of coalescing Brownian motions seems to have been first studied by Arratia [Arr] and is now better known under the name of Brownian web (see for instance [FINR]). Here the situation is slightly different: first each Λ a,v is some diffusion which is not a Brownian motion and before they coalesce two Λ a,v 's are not independent. For instance if v < v ′ , then (Λ a,v , Λ a,v ′ ) satisfies the following system of stochastic differential equations:

   dΛ a,v (x) = 2 Λ a,v (x) dB x + 2(1 {a≤x≤0} + h(Λ a,v (x))) dx dΛ a,v ′ (x) = 2 Λ a,v (x) dB x + 2 Λ a,v ′ (x) -Λ a,v (x) d B x +2(1 {a≤x≤0} + h(Λ a,v ′ (x))) dx, (38)
for all x ∈ [a, +∞), where B and B are two independent Brownian motions. This result follows from (30) and the Ray-Knight theorem (see for instance [START_REF] Raimond | Excited Brownian motion[END_REF]Theorem 6.1]). Note that we could describe similarly the law of (Λ ai,vi , i ∈ I), for any finite set I, and any (a i , v i ), i ∈ I. In [TW], the family Λ was called a sequence of forward lines and the dual sequence, the sequence of backward lines Λ * = Λ * a,v (•) : (a, v) ∈ R × [0, ∞) , was defined by Λ * a,v (x) = sup {w : Λ -x,w (-a) < v}, (39) for all x ≥ a and v ≥ 0. As in [TW] we can define Λ * here and we have also (Λ * a,v (x), x ≥ a) = (Λ -a,v (-x), x ≥ a), in law (see Theorem 2.3 in [TW]). It is important to observe that (40) (Λ a,v (x), x ∈ R) is a function of ((Λ a,v (x), x ≥ a), (Λ * -a,v (x)), x ≥ -a)). We notice now some other notable differences with the situation in [TW]. First if we denote by Q h the law of Λ, then the law of Λ * is Q -h . In particular Λ and Λ * do not have the same law (in other words Λ is not self-dual), except if h = 0. Moreover, for any a (say a < 0) and v ≥ 0, the process Λ a,v will almost surely not hit 0 in the time interval [a, 0]. The reason is that in the time interval [0, τ a (v)] the excited BM will cross each level x ∈ [a, 0] and strictly increase its local time on these levels (by using the absolute continuity between the laws of a standard BM and the excited BM). Similarly given any a < a ′ , v and v ′ , we have Λ a,v (x) = Λ * -a ′ ,v ′ (-x) = Λ a ′ ,v ′ (x) for all x ∈ [a, a ′ ] almost surely. Let us also notice that couples of processes such as (Λ a,v (x), a ≤ x ≤ 0) and (Λ * 0,v ′ (x), 0 ≤ x ≤ -a), if a < 0, are conjugate diffusions (see [T3] for a definition). Similarly (Λ 0,v (x), x ≥ 0) and (Λ * a,v ′ (x), x ≥ -a), if a < 0, are also conjugate. Now we can sketch another proof of Theorem 4.1 which bypass the use of Corollary 3.2 and uses instead these notions of forward and backward lines. The idea is to first prove that (Λ (n) ai,vi (x), x ≥ a i ), i ∈ I This can be done by using Ethier-Kurtz's result (Theorem 4.1 p.354 in [EK]), ( 29) and (38). One can next define analogues Λ (n) and Λ (n), * respectively of Λ and Λ * , in the discrete setting and it then suffices to use (39) (and its discrete counterpart), ( 40) and ( 41) to deduce the desired convergence. Since we already gave another proof, we omit the details here.

5.

  Proof of Corollary 1.2 Note that for any a ∈ R and v ≥ 0, τ εn,[2na] ([nv]) = [2na] + 2 k∈Z S εn,[2na],[nv] (y) dy + o(1).

  ai,vi (x), x ≥ a i ), i ∈ I} . (41)


Marion Rabiet
email: marion.rabiet@unilim.fr

Christelle Margoum

Véronique Gouy

Nadia Carluer

Marina Coquery

Pesticide transfer dynamics and fluxes in the stream of a small vineyard watershed -Assessing the effect of sampling strategy on fluxes estimation

Keywords: Agricultural watershed, Diuron, Flood, Sampling strategy, Pesticides fluxes, Surface water, Vineyard

    Pesticide transfer dynamics and fluxes in the stream of a small vineyard watershed. Assessing the effect of sampling strategy on fluxes estimation M. Rabiet, C. Margoum, V. Gouy, N. Caluer, Marina

Introduction

The intensive use of pesticides for crop protection against diseases led to the widespread presence of these compounds in all the compartments of the environment (soil, water, air). These environmental problems are particularly highlighted in vineyards, where pesticides play a major role in grape production. Large-scale studies, that monitored pesticides losses to surface waters, show that the transfert of pesticides is largely influenced by the intrinsic properties of the compounds but also by the weather conditions, the soil types and land use [START_REF] Kreuger | Pesticides in stream water within an agricultural catchment in southern Sweden, 1990-1996[END_REF]. Several authors emphasize the significance of rainfall-induced surface runoff as a major source of pesticides in streams [START_REF] Richards | Pesticide concentration patterns in agricultural drainage networks in the Lake Erie basin[END_REF]; [START_REF] Kuivila | Concentration, transport and biological effects of dormant spray pesticides in the San Francisco Estuary, California[END_REF][START_REF] Ng | Atrazine and metolachlor losses in runoff events from an agricultural watershed: the importance of runoff components[END_REF][START_REF] Kreuger | Pesticides in stream water within an agricultural catchment in southern Sweden, 1990-1996[END_REF][START_REF] Capel | The behaviour of 39 pesticides in surface waters as a function of scale[END_REF][START_REF] Bach | Input pathways and river load of pesticides in Germany -a national scale modeling assessment[END_REF]Schulz, 2001a;Schulz, 2001b[START_REF] Holvoet | Monitoring and modeling pesticide fate in surface waters at the catchment scale[END_REF].

The aim of the present study is on the one hand, to better estimate the spatial and temporal variability of pesticide concentrations in a small stream draining an agricultural watershed and on the other hand, to assess the role of floods on the mobilisation of pesticides and total fluxes in surface water. Moreover, the effect of the sampling strategy on the evaluation of pesticides loads was estimated using various methods.

Materials and method

Study area

The investigation site is the Morcille catchment's, located about 70 km north of the city of Lyon in France (Figure 1). Its small size (8 km²) allows to define a study area where transfers are more easily comprehensible. The head of the basin is essentially covered by forest, whereas downstream, it is mainly dedicated to vineyard. On the whole, the vine covers about 70% of the catchment total area. No industrial activities are present on the site. The hydrology of the Morcille river varies considerably over a year, with water flow ranging from 0.005 to more than 0.5 m 3 /s. It is characterized by significant hydrological events (strong and fast floods, especially in spring and summer). In this area, climatic, geological, topographical and agricultural conditions concur to create very dynamic hydrological conditions and a high erosion rate.

Water sampling and analysis

A punctual monitoring survey was performed from March 2007 to March 2008, in order to achieve a better understanding of surface water exposure levels to pesticide during a whole hydrological year. The spatial variability was approached owing to the choice of three sampling sites along the Morcille river from upstream (1), intermediate [START_REF] Capel | The behaviour of 39 pesticides in surface waters as a function of scale[END_REF] to downstream (3) (Figure 1). In addition, water samples were also collected during storm events at the intermediate site (site 2) during spring and summer 2007. Sampling was performed using two refrigerated automated samplers. Additionally, a weekly composite sampling was performed in July and August 2007 with a refrigerated automated sampler.

Eight pesticides and metabolites were chosen for monitoring from different families of herbicides and fungicides currently used for vineyard treatments: azoxystrobin, carbendazim, dimetomorph, procymidone, tebuconazol, diuron and two of its metabolites:

Results and discussion

Punctual monitoring of pesticide concentrations along the Morcille River

The presence of pesticides was highlighted throughout the investigation period with a large seasonal variation, in relation with the pesticides application calendar (Figure 2). Indeed, total dissolved pesticide concentrations ranged from below detection quantification to 16.4 µg/L. The maximum total pesticide concentrations are observed in June and July, which corresponds to the increase of storms occurrence, as well as to fungicides treatments. Pesticide concentrations declined in August but remained above quantification level throughout the year. Concentrations were 0.3 µg/L, in March 2007 and 2008, corresponding to a residual backgroung linked to previous vineyard treatments. Water contamination by pesticides largely overstepped the application period, as shown by the high concentrations measured in October (up to 7.8 µg/L at site 2). The weekly loads of the studied pesticides, calculated with punctual weekly concentrations, ranged from 0 to 337 g over the study period, which led to an annual load of pesticides of 1525 g. Pesticides fluxes transiting during June and July represent more than 70 % of the annual load. On a spatial point of view, a large increase of pesticide concentration was observed from upstream to downstream, which is coherent with the increasing vineyard surface proportion downstream.

Storm events monitoring during spring and summer 2007

A large mobilisation of pesticides in the river was observed during the floods, with concentrations rising rapidly and usually peaking almost at the same time as water discharge. With the exception of 2 extreme values, maximum concentrations during floods were between 7.3 and 21.1 µg/L, with an average value of 12.7 ± 4.3 µg/L. Depending on the flood, pesticide maximum concentrations varied within only a factor of 3, whereas maximum water discharge varied by a factor of up to 30. In fact, pesticide maximum concentration was not a simple function of water discharge; it is likely that other processes interfere, such as pesticide availability in the soil, dilution and dominant hydrological processes (such as surface runoff or subsurface flow), which induce various residence times. During floods, the total dissolved pesticides fluxes ranged from 1.0 to 60 g.

Effect of sampling strategy on the estimation of pesticides fluxes in the river

In order to assess the effect of the sampling strategy on fluxes estimations, different methods were applied on the data collected in August 2007. Monthly fluxes were estimated using weekly, bimonthly and monthly simulated punctual monitoring by choosing either the first, second, third or fourth weekly sample as a reference for the calculation (Figure 3). In addition, pesticides monthly fluxes were calculated with the weekly average composite samples results. Lastly, pesticide monthly loads were estimated in base flow and storm flow using punctual monitoring achieved during base flow conditions and automatic sampling achieved during floods, respectively. According to the method used, monthly loads ranged from 18.6 to 130.5 g. The highest estimation corresponds to the weekly composite sampling, which we considered as the most accurate value. Monthly loads, estimated from punctual surveys (weekly, bimonthly, monthly) vary between 16.9 and 40.7 g, largely below the "most realistic" flux value (i.e., 130.5 g). In fact, in August 2007, around 83% of the total pesticides fluxes were transported during floods, which, in total lasted for less than 5 days.

Conclusion

The occurrence of pesticides is closely related to the application calendar and the hydrological conditions. Pesticides concentrations were higher during the application season, with maximum concentrations occuring during June and July. Some pesticides were quantified in surface water for several months after application, and throughout the year for diuron, indicating their potential to persist in soils and shallow groundwater. Along the river, we observed a large increase of pesticide concentrations from upstream to downstream, which could be related to the increasing vineyards surface downstream. Our study of pesticides behaviour during floods show the major role of hydrological conditions in the transport of pesticides. In August 2007, floods contributed to more than 80% of the total load of pesticides transiting through the stream. These results provide further evidence that floods are the dominant pathways for the transport of pesticides in the Morcille stream.

A major conclusion of this study is that punctual monitoring of surface water is not adapted for an accurate evaluation of the contaminant exposure level. This is especially the case in dynamic hydrosystems such as the Morcille stream, where such sampling methods largely underestimate the variability of pesticide concentrations and 5th European Conference on Pesticides and Related Organic Micropollutants in the Environment, 22/10/2008, Marseille, FRA pesticides fluxes. These observations have critical consequences for the monitoring requirements of pesticides, in particular for the implementation of the Water Framework Directive.

Figure 1 .

 1 Figure 1. Location of the Morcille watershed and the sampling sites along the stream.

Figure 2 .Figure 3 .

 23 Figure 2. Total dissolved pesticide concentrations (µg/L) along the Morcille stream from March 2007 to March 2008.

5th European Conference on Pesticides and Related Organic Micropollutants in the Environment, 22/10/2008, Marseille, FRA

Acknowledgments

Thanks for support from the Rhône-Alpes Region ( LTER "Zone atelier bassin du Rhône" and Cluster Environment), as well as for financial help from the French national research structure: ANR-EcoGER (programme Papier) and EC2CO (programme Padyma). We would like to thank N. Ball and B. Motte for technical field assistance and C. Guillemain for analytical optimization and analyses of pesticides. Thanks are due to A. Albinet for helpful correction.