Emilie Soret
email: emilie.soret@math.univ-lille1.fr

Stephan De Bièvre
email: stephan.de-bievre@math.univ-lille1.fr

Stochastic acceleration in a random time-dependent potential

Introduction

Our goal in this paper is to make progress on the rigorous analysis of the stochastic acceleration of a classical particle moving through a random time-dependent potential. The full problem can be described as follows. A particle moves in R d , and its position q(t) obeys the following law of motion: q(t) =i λ i ∇V (q(t)r i , ωt + φ i ), q(0) = q 0 , q(0) = v 0 .

(1.1) T m is a real valued potential which is bounded and of compact support in its first variable in the ball of radius 1 2 centered at the origin. The frequency vector ω ∈ R m is fixed, so that the particle moves under the influence of a potential V (q(t)r i , ωt + φ i ) that is quasi-periodic in time, when it is close to the scattering center r i . The scattering centers r i ∈ R d are a countable and locally finite family of (random or deterministic) points that satisfies a "finite horizon" condition, that we shall not explicitly describe. The phases φ i and the coupling constants λ i are i.i.d random variables in T m respectively R.

Here V ∈ C ∞ R d ,
1 Such a particle undergoes successive scattering events (also refered to as collisions) when crossing one of the balls of radius 1 2 centered on the r i , and executes a uniform straight line motion otherwise. When the potential V is time-independent, the particle's kinetic energy is preserved in the scattering events and is therefore uniformly bounded in time. We are interested in the case when V does depend on time, in which case the kinetic energy is expected to grow in time. This is the phenomenon known as "stochastic acceleration". It has been extensively studied by various authors in a variety of models (see for example [START_REF] Goudon | Stochastic acceleration in an inhomogeneous time random force field[END_REF], [START_REF] Peter | Stochastic acceleration[END_REF] and [START_REF] Vanden | Some remarks on the quasilinear treatment of the stochastic acceleration problem[END_REF]) and has been the subject of some controversy concerning the precise rate of growth. We refer to [START_REF] Aguer | Classical motion in force fields with short range correlations[END_REF] for further background.

In [START_REF] Aguer | Classical motion in force fields with short range correlations[END_REF] and [START_REF] Aguer | Comportements asymptotiques dans des gaz de lorentz inélastiques[END_REF], the above model was analysed numerically and partial arguments were given to argue that, asymptotically in time (d ≥ 2), E( q(t) ) ∼ t 1/5 , E( q(t) ) ∼ t, where the expected value is with respect to the (λ i , φ i ) and to an initial distribution of particle velocities.

In this paper, we shall consider a simplified model for the particle's motion, in which its possible recollisions with the same scatterer are ignored. Within that framework, we give a complete and rigorous analysis of the asymptotic behaviour of q(t) corroborating the t 1/5 law above for d > 5 (Theorem 2.1).

The model is described in detail in Section 2. It treats the successive scattering events as independent, leading to a Markov chain description for the particle's momentum and position at each scattering event. We therefore establish that the t 1/5 law is indeed obtained from successive random scattering events with a smooth potential. The numerics in [START_REF] Aguer | Classical motion in force fields with short range correlations[END_REF] suggests this behaviour is not altered by possible recollisions but we do not prove this here.

Our work relies first of all on the analysis of the single scattering events for a high energy particle that was given in [START_REF] Aguer | Classical motion in force fields with short range correlations[END_REF] and [START_REF] Aguer | Comportements asymptotiques dans des gaz de lorentz inélastiques[END_REF]. This yields a sufficiently sharp description of the transition probabilities of the Markov chain at high momenta to allow us to control the asymptotic behaviour of the energy of the particle in this Markov chain dynamics. For that purpose we then adapt techniques developed in [START_REF] Dolgopyat | Motion in a random force field[END_REF] in the context of a related problem on which we shall comment below.

The paper is organised as follow. In Section 2 we introduce the model that we consider and we describe the behaviour of the kinetic energy by a Markov chain where each step corresponds to a passage trough a scattering region. In Section 3, we state a technical result (Theorem 3.1) for a class of Markov chains which includes the one described in Section 2 and we show how it implies our main result, Theorem 2.1. In Section 4, we show that correctly rescaled and under some technical conditions, each Markov chain of this class converges weakly to a transient Bessel process (see Theorem 4.1). This Averaging Theorem is a key element of the proof of Theorem 3.1. Sections 5, 6 and 7 contain the three steps of the proof of Theorem 3.1. An appendix concludes this paper with in particular the proof of Theorem 4.1.

Acknowledgments. The authors thank B. Aguer, M. Rousset, T. Simon and D. Dereudre for helpful discussions. This work is supported in part by the Labex CEMPI (ANR-11-LABX-0007-01).

The Markov chain model

The solution (q(t), q(t)) of (1.1) can be viewed as a stochastic process on the probability space generated by the (λ i , r i , φ i ). To each trajectory (q(t), q(t)) one can associate a sequence (t n , v n , b n , r in , λ in , φ in ) i∈N . Here t n is the instant the particle arrives at the n-th scattering region with incoming velocity v n = q(t n ); r in is the n-th scattering center visited by the particle, λ in and φ in are, respectively the associated coupling constant and phase; b n is the impact parameter (Figure 1). More precisely, we have

e n = v n ||v n || , q(t n ) = r in - 1 2 e n + b n , b n • v n = 0, ||b n || ≤ 1 2 .
Figure 1: A particle at time t n arriving with velocity v n and impact parameter b n on the n-th scatterer, centered at the point r in .

The change in velocity experienced by a sufficiently fast particle at the n-th scattering event can be written

v n+1 = v n + R(v n , b n , φ in , λ in ) (2.1)
where, for all

v ∈ R d , b ∈ R d with v • b = 0, and (φ, λ) ∈ T m × R, R(v, b, φ, λ) = -λ +∞ 0 dt ∇V (q(t), ωt + φ); (2.2)
here q(t) is the unique solution of

q(t) = -λ∇V (q(t), ωt + φ), q(0) = b - 1 2 v ||v|| , q(0) = v.
(2.3)

We will always suppose the potential V satisfies the following hypothesis:

Hypothesis 1. V ∈ C ∞ (R d , T m
) is bounded and of compact support in its spatial variable in the ball of radius 1/2 at the origin. The potential V and all its derivates are bounded, and we write

0 < V max := ||V || ∞ < +∞. Moreover, (ω • ∇ φ )V = 0. Equation (2.1) determines v n+1 in term of v n , b n , φ in , λ in . To determine t n+1 , b n+1 , λ in+1
, φ in+1 , one would need to solve a geometric problem which consists in finding the location r in+1 of the next scatterer visited by the particle. We shall present and study a simplified model of the dynamics in which this problem is eliminated. For that purpose note first that, once the particle leaves the n-th scatterer, it travels with a constant velocity v n+1 over a distance ℓ n before meeting the n + 1-th scatterer.

Hence t n+1 = t n + ℓn vn+1 , where we ignored the duration of the scattering event itself. Furthermore q n+1 = q n + v n+1 (t n+1t n ) where q n = q(t n ).

Starting from this description of the dynamics and ignoring possible recollisions, we now model the solution ( q(t), q(t)) of (1.1) by a coupled discrete-time Markov chain in momentum and position space as follows. Each step of the chain is associated to one scattering event. Thus, starting with a given initial velocity v 0 ≫ 1, we define iteratively the velocity v n and the time t n just before the n-th scattering event through the relations:

v n+1 = v n + R (v n , κ n ) t n+1 = t n + ℓ v n+1 q n+1 = q n + v n+1 (t n+1 -t n ),        (2.4) where κ n = (b n , φ n , λ n ).
(2.5)

Here the random variables b n are chosen independently at each step of the Markov chain and follow a uniform law in B(0, 1 2 ) conditionally to b n • v n = 0. The variables λ n and φ n are also sequences of independent random variables and identically distribued in [-1, 1] and T m respectively.

Finally, note that we have added a very last simplification to this Markov chain by replacing the random variables ℓ n by the mean distance ℓ between two scatterers successively visited by the particle. In this way the geometric problem associated to the distribution of the scatterers in the space is completely eliminated.

This Markov chain provides a simplified but still highly non-trivial model for the original dynamical problem given in (1.1). Note that the momentum change undergone by the particle during collisions is entirely encoded in the momentum transfer function R(v, b, φ, λ) (see (2.2)) in both the original problem and the above Markov chain. The main simplifications in (2.4) come from the fact that we ignore geometric considerations (the spatial distribution of the r i ) as well as possible recollisions.

To state the main result of this paper, we introduce trajectories (q(t)) t∈R+ where for all n ∈ N, q(t n ) = q n is a solution of (2.4) and for all t ∈ [t n , t n+1 ]

q(t) = q(t n ) + (t -t n )v n+1 .
(2.6)

Theorem 2.1. Suppose d > 5. Then for all ν > 0 and v 0 ∈ R d , there exist c(ν) > 0 depending on ν and C(v 0 , ν) > 0 depending on both v 0 and ν such that

lim ||v0||→+∞ P ∀t > 1 v 0 , c(ν)t 1 5 -ν ≤ || q(t)|| ≤ C(v 0 , ν)t 1 5 +ν = 1. (2.7)
The proof of Theorem 2.1 is given in Section 3 where the role of the condition d > 5 will be explained. In order to establish this theorem, we have to analyze the behaviour of the first equation of (2.4),

v n+1 = v n + R(v n , κ n ),
(2.8) for v n large. For that purpose, we need to understand the behaviour of the momentum transfer R(v n , κ n ) in (2.2). Using first order perturbation theory, we can write (see [START_REF] Aguer | Classical motion in force fields with short range correlations[END_REF]),

R(v, κ) = - λ ||v|| +∞ -∞ dy ∇V b + (y - 1 2 )e, ωt ||v|| + φ + O ||v|| -3 , with b•v = 0. As V is sufficiently smooth, we have the following expansion for K ∈ N, (v, κ) = (v, b, φ, λ) ∈ R 2d × T m × R with b • v = 0: R(v, κ) = K k=1 α (k) (e, κ) ||v|| k + O(||v|| -K-1 ), e = v ||v|| , with α (1) (e, κ) = -λ +∞ -∞ dy ∇V b + (y - 1 2 )e, φ .
Note that e • α (1) (e, κ) = 0. Then, if we look at the energy transfer

∆E(v, κ) = 1 2 (v + R(v, κ)) 2 -v 2 , (2.9) we have ∆E(e, κ) = L ℓ=0 β (ℓ) (e, κ) ||v|| ℓ + O(||v|| -L-1 ), (2.10)
where 2) . Consequently, the first term in (2.10) is equal to 0 and ∆E(e, κ) ∼ ||v|| -1 . The following theorem (see [START_REF] Aguer | Classical motion in force fields with short range correlations[END_REF]) describes the average energy transfer during a unique collision.

β (0) = e • α (1) = 0 and β (1) = e • α (
Theorem 2.2. For all unit vectors e ∈ R d , α (1) (e) = 0 = α (2) (e). Moreover, for all v ∈ R d

∆E(v) = B v 4 + O( v -5 ), (∆E(v)) 2 = D 2 v 2 + O( v -3 ), where B = d -3 2 D 2
and

D 2 = λ 2 C d T m dφ R 2d dq 0 dq ′ 0 ||q 0 -q ′ 0 || 1-d ∂ t V (q 0 , φ)∂ t V (q ′ 0 , φ) > 0,
where C d is the volume of the sphere of radius 1 2 in R d-1 . In particular, for all unit vectors e ∈ R d and for all ℓ = 1, 2, 3, β (ℓ) (e) = 0, B = β (4) (e), and D 2 = (β (1) (e)) 2 > 0.

Theorem 2.2 and (2.9) yield

∆||v n || 3 = 3β (1) n + 3 β (4) n + 1 2 β (1) n 2 ||v n || 3 + O 0 ||v n || -1 + O ||v n || -4 . ( 2
ξ n = ||v n || 3 3D , ω n = β (1) n D , and γ = 1 3 B D 2 + 1 2 = 1 6 (d -2) ≥ - 1 6 , ( 2
.12) and using (2.11), we obtain the discrete Markov chain with values in R

ξ n+1 = ξ n + ω n + γ ξ n + O 0 ξ -1 3 n + O ξ -4 3 n , (2.13)
with ω n = 0, ω 2 n = 1. To understand the behaviour of the system's kinetic energy, it remains therefore to study the Markov chain (ξ n ) n , a task we turn to in the following sections. In particular, Theorem 3.1 is a technical result valid for a class of Markov chains including (ξ n ) n defined by (2.13).

Strategy of the proof

We start with a remark that explains the origin of the condition d > 5 in Theorem 2.1. Note that under Hypothesis 1, a global solution of (2.3) always exists. Nevertheless, the integral in (2.2) may not converge. Indeed, it is conceivable that for given v = q(0), the solution satisfies q(t) ≤ 1 2 for all t > 0 large. In other words, the particle may not leave the scattering region after having entered it: it is trapped. In this case the integral in (2.2) may not converge. As shown in [START_REF] Aguer | Classical motion in force fields with short range correlations[END_REF], and as is intuitively obvious, this will not happens if v is large enough (meaning v > 12|λ| ∇V ∞ , see [START_REF] Aguer | Classical motion in force fields with short range correlations[END_REF]). The particle will then exit the scattering region after a finite time of order v -1 . We will show below below that for d > 5 (this means γ > 1 2 in (2.12)), the Markov chain (2.12) is transient. This implies an initially fast particle never slows down so that there is no trapping and the chain is well defined.

We will consider a slightly more general class of Markov chains, which may be of interest on its own, and which is defined as follows. Let (ω k ) k∈N be a family of bounded, i.i.d. real random variables, with zero mean and whose variance equals 1:

E(ω k ) = 0, E(ω 2 k ) = 1, ∃M ≥ 1, |ω k | ≤ M. (3.1)
We will denote their common probability measure by µ. Let F : R + * × [-M, M ] → R + * be a measurable function satisfying the following properties:

Hypothesis 2. ∃γ ∈ R, 0 < ξ -< ξ + , α > 0, β > 1, such that F is continuous on [ξ -, +∞[×[-M, M ] and, for all ξ > ξ + , F (ξ, ω) = ξ + ω + γ ξ + G 0 (ξ, ω) + G 1 (ξ, ω), (3.2)
and where the functions G 0 and G 1 are such that, for large ξ,

sup ω |G 0 (ξ, ω)| = O ξ -α and sup ω |G 1 (ξ, ω)| = O ξ -β , (3.3)
with α > 0 and β > 1. Moreover, E (G 0 (ξ, •)) = 0.

We will study the asymptotic behaviour of the Markov chains

ξ k+1 = F (ξ k , ω k ), ξ 0 > 0. (3.4)
Note that the Markov chain described by (2.13) satisfies Hypothesis 2. The following result is the main technical ingredient for the proof of Theorem 2.1.

Theorem 3.1. Suppose γ > 1 2 . Then (i) For all 0 < p ≤ 1, for all ν > 0, there exists ξ * > ξ + such that for all ξ 0 ≥ ξ * , we have

P ∀k ∈ N, ξ 0 + k 1 2 1-ν ≤ ξ k ≤ ξ 0 + k 1 2 1+ν ≥ 1 -p, (ii
) For all ν > 0, we have

lim ξ0→+∞ P ∀k ∈ N, ξ 0 + k 1 2 1-ν ≤ ξ k ≤ ξ 0 + k 1 2 1+ν = 1.
This asymptotic behaviour can be anticipated from the following observation. Let us consider the special case where F is of the form (3.2) for all ξ > 0 (and not only for large ξ) and drop the two last x 10 errors terms, i.e F (ξ) = ξ + ω + γ ξ . This is possible if 2 √ γ > M , as is easily checked. In that case, one readily finds that

5 γ = 3 γ = 1 2 k → ξ 2 0 + (2γ + 1 + 1 4 γ (1-M 2 √ γ ) 2 )k k → ξ 2 0 + (2γ
ξ 2 k+1 = ξ 2 k + 2γ + ω 2 k + 2ω k ξ k + γ ξ k + γ 2 ξ 2 k , so that E(ξ 2 k+1 ) = E(ξ 2 k ) + (2γ + 1) + γ 2 E(ξ -2 k ). It follows that, for all k ≥ 2, ξ 2 0 + (2γ + 1)k ≤ E(ξ 2 k ) ≤ ξ 2 0 + (2γ + 1 + 1 4 γ (1 -M 2 √ γ ) 2 )k.
(3.5)

It shows that, indeed, E(ξ 2 k ) behaves as k in this simple case. Of course, this information on the second moment of ξ k does not imply the statement of the Theorem 3.1, even in this case. Conversely, the statement of the Theorem 3.1 does not allow to draw conclusions on the moments of ξ k , since we have no control on the trajectories on a set of small probability.

Another way to anticipate the asymptotic behaviour of ξ k is to notice that the Markov chain

ξ k+1 = ξ k + ω k + γ ξ k
can be thought of as a time discretized version of the stochastic differential equation satisfied by a Bessel process R t of dimension 2γ + 1:

dR t = dB t + γ R t dt,
where B t is a standard Brownian motion and γ > -1 2 . It is of course well known (see [START_REF] Revuz | Continuous martingales and Brownian motion, volume 293 of Grundlehren der Mathematischen Wissenschaften[END_REF]) that R t ∼ (2γ + 1)t when γ > -1 2 . In Section 4, a rigorous version of this observation constitutes the first step of the proof of Theorem 3.1. Indeed, we introduce a family of rescaled processes R ǫ t and then show that the R ǫ t converge, as ǫ → 0, to a Bessel process with R(0) = 1 (Theorem 4.1). We note that the transience and recurence of various time and space discretized versions of the Bessel process are discussed in [START_REF] Alexander | Excursions and local limit theorems for Bessel-like random walks[END_REF] and [START_REF] Csáki | Transient nearest neighbor random walk and Bessel process[END_REF] but no results on their asymptotic behaviour are obtained there.

Observe that in Hypothesis 2 no assumption is made on the behaviour of the chain when ξ < ξ -. Such information is unavailable in the application we have in mind as we already indicated, and it is therefore important to see what can be said without it. Clearly, one cannot hope to obtain general results valid for all γ, without such additional information. Indeed, if γ is too small, the trajectories will reach the region ]0, ξ + ] with probability one, and the asymptotic behaviour of the chain will then depend crucially on the behaviour of F in that region. This can be seen for example when γ = 0, and F (ξ, ω) = ξ + ω, for all ξ ≥ ξ -. In that case, we are dealing with an ordinary random walk for ξ > ξ -, which is recurrent. If then ξ -> M and F (ξ, ω) = 0 for all ξ < ξ -, it is clear that, with probability 1, lim k→+∞ ξ k = 0 (and

E(ξ k ) → 0). On the other hand, if F (ξ, ω) = |ξ + ω|, ∀ξ, ω, then E(ξ k ) ∼ k 1/2
and lim sup k ξ k = +∞, with probability 1. In short, when γ is small, the chain is recurrent and one needs a "non-trapping" condition of the trajectories in the region [0, ξ + ] to ensure the asymptotic behaviour of ξ k is still of the form k 1/2 . Once we have Theorem 3.1, we can show Theorem 2.1.

Proof of Theorem 2.1. Theorem 3.1 ii) and (2.12) yield that for all ν > 0 lim

v0 →+∞ P ∀n ≥ 0; (3D) -1 3 v 0 3 3D + n 1 2 -1+ν 3 ≤ q(t n ) -1 ≤ (3D) -1 3 v 0 3 3D + n 1 2 -1-ν 3 = 1.
(3.6) Furthermore, by (2.4) we have, for all n ≥ 1,

t n = ℓ n j=1 v j -1 , t 0 = 0. (3.7)
Combining (3.6) and (3.7), straightforward estimates show that for all ν > 0, the following bounds on t n hold, lim

v0 →+∞ P ∀n ≥ 2; c 1 (ν)t 6 5+ν n ≤ n ≤ C 1 (v 0 , ν)t 6 5-ν n-1 = 1. (3.8)
Here c 1 (ν) > 0 and C 1 (v 0 , ν) > 0 are two positive constants depending only on ν and (v 0 , ν) respectively. This implies, by part ii) of Theorem 3.1, that for all ν > 0 lim

v0 →+∞ P ∀n ≥ 2; c 2 (ν)t 1-ν 5+ν n ≤ q(t n ) ≤ C 2 (v 0 , ν)t 1+ν 5-ν n-1 = 1. (3.9)
Then, as for all t ∈ (t n-1 , t n ], q(t) = q(t n ) (see (2.6)), it follows from (3.9) that lim

v0 →+∞ P ∀t > ℓ v 1 ; c(ν)t 1-ν 5+ν ≤ q(t) ≤ C(v 0 , ν)t 1+ν 5-ν = 1. (3.10) Using (3.6), this result is easily extended to all t > 1/ v 0 .
The rest of this paper is devoted to the proof of Theorem 3.1. The strategy is the following. We will consider, in Section 4, a family of Markov processes R ε n = εξ n , indexed by ε = ξ -1 0 . We show that after an appropriate rescaling of the time variable, the limit of this new family as ε → 0 is a Bessel process of dimension 2γ + 1 = d + 1 3 when γ > 1 2 (d > 5 in the initial problem). This yields Theorem 4.1. The proof of this averaging theorem is given in Appendix A. In Section 5, implementing a strategy developed in [START_REF] Dolgopyat | Motion in a random force field[END_REF] for a similar problem, we define an auxiliary process η ℓ ∈ Z and corresponding stopping times τ ℓ such that, roughly, ξ τ ℓ ∼ 2 η ℓ (see Figure 3). In other words, the increments of the process (η ℓ ) ℓ are ±1, and ∆τ ℓ = τ ℓ+1τ ℓ is the time the process (ξ n ) n needs to double or half its value. In Section 6, we use Theorem 4.1, properties of the Bessel process and the Porte-Manteau Lemma to show that, provided γ > 1 2 and η 0 is large enough, (η ℓ ) ℓ is a submartingale. We then control ∆τ ℓ . Basically, we show (Proposition 5.1) that there exists µ > 0 such that η ℓ ∼ µℓ + η 0 , and ∆τ ℓ ∼ 2 2η ℓ .

In Section 7, we use the results of Sections 5 and 6 to conclude the proof of Theorem 3.1.

We end this section with a further comment on [START_REF] Dolgopyat | Motion in a random force field[END_REF]. The authors of that paper study a similar model, in which however the force does not derive from a potential. In other words, it is not irrotational. In that case they show that, provided q(0) is large enough, and for d ≥ 4, q(t) ∼ t 1/3 , q(t) ∼ t 4/3 , with high probability. Note that the energy growth is faster here than when the force derives from a potential as in our case: it grows as t 2/3 as compared to t 2/5 in the latter situation. This faster growth allows the authors of [START_REF] Dolgopyat | Motion in a random force field[END_REF] to show the spatial trajectories of the particles do not self-intersect, so that recollisions do in fact occur only with very low probability. This in turn allows them to control the growth of q(t) . The situation under study in this paper is very different. As argued and shown numerically in [START_REF] Aguer | Classical motion in force fields with short range correlations[END_REF], the slower growth of the energy when the force does derive from a potential leads the particle to turn on a short time scale, so that self-intersections of the trajectory do occur and the growth of q(t) , as t, is slower than the power t 6/5 one could naively expect. In fact, the numerics of [START_REF] Aguer | Classical motion in force fields with short range correlations[END_REF] indicates q(t) ∼ t. We will come back to this aspect of the problem in a further publication.

A scaling limit

Let ε * > 0, to be fixed later. We introduce ε = ξ -1 0 , and define, for ε < ε * ,

R ε n := εξ n . Note that R ε 0 = 1, independently of ε. It then follow from (3.2) that R ε n satisfies R ǫ n+1 = G(ǫ, R ǫ n , ω n ) where, for x > R ε + = εξ + G(ǫ, x, ω) = x + ǫω + ǫ 2 γx -1 + ε α+1 G ε 0 (x, ω) + ε β+1 G ε 1 (x, ω) ,
where

G ε 0 and G ε 1 are such that sup ω∈Ω |G ε 0 (x, ω)| = O x -α and sup ω∈Ω |G ε 1 (x, ω)| = O x -β ,
with α > 0 and β > 1. Moreover, E (G ε 0 (x, •)) = 0. We then construct a continuous time process by linear interpolation, as follows. For n ∈ N,

t n = nε 2 and for t ∈ [t n , t n+1 ], R ǫ (t) = t n+1 -t ǫ 2 R ǫ n + t -t n ǫ 2 R ǫ n+1 . Theorem 4.1. Fix T ∈ R + * . If γ > 1/2, the processes (R ǫ t
) t∈[0,T ] converge weakly, as ǫ → 0, to the Bessel process of dimension 2γ + 1, and with initial condition 1.

The condition on γ guarantees that the limiting Bessel process is transient, does not explode in finite time and does not reach zero. This is an important element of the proof which is given in Appendix A.

In addition, we will need the following result.

Lemma 4.2. Let γ > -1 2 , and let R be a Bessel process of dimension 2γ + 1 with R(0

) = 1. Let, for a -< 1 < a + , T a-,a+ = inf{t ≥ 0 | R(t) ∈]a -, a + [}, T a-= inf{t ≥ 0 | R(t) < a -}, T a+ = inf{t ≥ 0 | R(t) > a + }. (i) Then, for all T ≥ 0, 0 < P(T a-,a+ > T ) < 1. (ii) If in addition γ > 1 2 , P(T a-> T a+ ) = a 1-2γ - -1 a 1-2γ - -a 1-2γ +
In order not to break the flow of the main argument, the proofs of Theorem 4.1 and Lemma 4.2 are given in Appendix A.

An auxiliary process

Let L > 0, η ∈ Z and define the intervals

J η = [2 η -L, 2 η + L]. We consider the subset N L := Z J η of R +
* and we will study how the Markov chain (ξ k ) k visits successively N L by introducing an auxiliary process (η ℓ ∈ Z) ℓ and corresponding stopping times τ ℓ , so that ξ τ ℓ ∈ J η ℓ , (see Figure 3). We start with a technical remark. Note that in Hypothesis 2, ξ + can always be replaced by a larger value. It turns out to be convenient to work under the following further condition on ξ + : ξ + ≥ |γ| M . Under this hypothesis, one easily checks that

∀ξ k > ξ + , ξ k+1 ∈ (ξ k -C M,γ , ξ k + C M,γ ) , (5.1)
where

C M,γ = 2M + C M α |γ| α + C M β |γ| β , C > 0.
This expresses the rather obvious fact that, for large enough ξ k , the step size of the random walk is small compared to ξ k .

Let us now define the process (η ℓ ) ℓ∈N precisely. First, set

η + = min{η ∈ N | 2 η > 2 max{ξ + , C M,γ }} > 1, (5.2)
where the last inequality follows from the observation that M ≥ 1 (See (3.1)). In view of (5.2), one can choose L satisfying C M,γ < L < 2 η+-1 , from which it follows that, for all η, η ′ ≥ η + , η = η ′ , we have J η ∩ J η ′ = ∅. Note that, in view of (5.1), the process ξ k cannot jump across one of these intervals without visiting it. In this way, for all ℓ, η ℓ+1 = η ℓ ± 1,as we will see.

We are now in a position to define the process η ℓ , and the associated stopping times τ ℓ recursively, as follows. We restrict ourselves to initial conditions ξ 0 for which there exists an integer η 0 so that ξ 0 ∈ J η0 , with η 0 > η + . Note that if ξ 0 is not in such an interval, by Lemma 6.1 we can control the time that the procces spend before entering in J η0 . Then, define τ 0 = 0 and

τ 1 := inf{k > τ 0 | ξ k ∈ J η0-1 ∪ J η0+1 }.
We define

η 1 = η 0 + 1, if ξ τ1 ∈ J η0+1 , and η 1 = η 0 -1, if ξ τ1 ∈ J η0-1 . Figure 3: (ξ k ) k visiting N L . Here, η ℓ+1 = η ℓ -1.
We then proceed recursively. Suppose that, for some ℓ ∈ N, τ 0 , η 0 , τ 1 , η 1 , . . . τ ℓ , η ℓ have been defined, with

ξ τ k ∈ J η k , for all 0 ≤ k ≤ ℓ. If η ℓ = η + , we define τ ℓ+1 = τ ℓ and η ℓ+1 = η ℓ . Otherwise we define τ ℓ+1 = inf{k > τ ℓ | ξ k ∈ J η ℓ -1 ∪ J η ℓ +1 }, η ℓ+1 = η ℓ + 1, if ξ τ ℓ+1 ∈ J η ℓ +1 , and η ℓ+1 = η ℓ -1, if ξ τ ℓ+1 ∈ J η ℓ -1 .
We will show in this section that the process (η ℓ ) ℓ is asymptotically a submartingale, with high probability, and that η ℓ ∼ µℓ, for some µ > 0 (see Proposition 5.1 (ii)). In Section 6, we will combine this result with estimates on the dwell times τ ℓ+1τ ℓ between successive visits of the original process ξ k to N L , which we show to be of order (2 η ℓ ) 2 ∼ 2 2µℓ , to conclude that τ ℓ ∼ 2 2µℓ , and hence ξ τ ℓ ∼ √ τ ℓ .

(See Proposition 6.2 (ii)&(iv) for a precise statement.) It will then remain, in Section 7, to interpolate between the stopping times τ ℓ to obtain Theorem 3.1.

Note that the sequence (τ ℓ ) ℓ is increasing, and we have the following dichotomy: either the sequence (τ ℓ ) ℓ is strictly increasing, lim ℓ→+∞ τ ℓ = +∞, and ∀ℓ ∈ N, η ℓ > η + , or ∃L * ∈ N and T * > 0 so that τ ℓ = T * and η ℓ = η + , forall ℓ ≥ L * .

There is no reason to think the process η ℓ is still a Markov process, specifically (η ℓ ) ℓ describe the behaviour of (ξ k ) k on interval:

P (η ℓ+1 = η ± 1|η ℓ = η, η ℓ-1 = η ± 1) = P ξ τ ℓ+1 ∈ J η±1 |ξ τ ℓ ∈ J η , ξ τ ℓ-1 ∈ J η±1 = P ξ τ ℓ+1 ∈ J η±1 |ξ τ ℓ ∈ J η .
Actually, it depends on if ξ τ ℓ is rather on the left than on the right of J η ℓ .

To control its asymptotic behaviour, we will show it is, with high probability, a submartingale, if η 0 is sufficiently large, and control its jump probabilities P(η ℓ+1 = η ℓ ± 1 | η ℓ , . . . , η 0 ). (See Proposition 5.1 (i).) We note that the transience of the chain (ξ k ) k is essential in the arguments of this section; it is, as we shall see, ensured by the condition that γ > 1 2 . The main properties of the process η ℓ are summarized in the following proposition.

Proposition 5.1. (i) Suppose γ > 1 2 . For all δ > 0 there exists η > η + such that for all ℓ ∈ N * and for almost all η 0 , η 1 , . . . , η ℓ-1 ≥ η + , η ℓ > η, we have

|P (η ℓ+1 = η ℓ ± 1|η ℓ , . . . , η 0 ) -p ± | < δ,
(5.3)

where p + = 2 2γ-1 -1 2 2γ-1 -2 1-2γ > 1 2 and p -= 1 -p + .
(ii) For all 0 < p ≤ 1 and for all δ > 0, there exists η * > η + so that for all η 0 ≥ η *

P (|η ℓ -µℓ -η 0 | ≤ δ (ℓ + η 0 ) , ∀ℓ ∈ N) ≥ 1 -p,
where µ = 2p + -1 > 0.

(iii) For all 0 < p ≤ 1, there exists η * > η + so that for all η 0 ≥ η * ,

P ∀ℓ ∈ N, η ℓ ≥ η 0 2 ≥ 1 -p.
We start with two preliminary observations. First, in what follows our notation will not distinguish between on the one hand the random variable P (A|η ℓ , . . . , η 0 ), viewed as a function on the underlying probability space or on N ℓ+1 , and on the other hend the values it takes at points in N ℓ+1 , also denoted by (η ℓ , . . . , η 0 ) ∈ N ℓ+1 . Second, we will often make use of the following useful property of the process ξ k , which is a consequence of its Markovian nature:

inf ξ k-1 ∈I P(A ∈ F + k |ξ k-1 ) ≤ P(A ∈ F + k |ξ k-1 ∈ I, C ∈ F k-2 ) ≤ sup ξ k-1 ∈I P(A ∈ F + k |ξ k-1 ), (5.4)
where I is an interval, F k-2 is the sigma-algebra generated by the

ξ k ′ , 0 ≤ k ′ ≤ k -2 and F + k the sigma-algebra generated by the ξ k ′ , k ′ ≥ k.
Proof. (i) Let η 0 , η 1 , . . . , η ℓ > η + . We then have

P (η ℓ+1 = η ℓ + 1|η ℓ , . . . , η 0 ) = 0<i1≤•••≤i ℓ P (η ℓ+1 = η ℓ + 1|η ℓ , . . . , η 0 ; τ ℓ = i ℓ , . . . , τ 1 = i 1 ) ×P (τ ℓ = i ℓ , . . . , τ 1 = i 1 | η ℓ , . . . , η 0 ) .
(5.5)

Here and in what follows, the values of η 0 , . . . , η ℓ and of the multi-indices i j are restricted to values for which the set on which we condition has non-zero probability. Introducing, for all i ∈ N and for all η > η + ,

τ +,i (η) = inf{k ≥ 0|ξ i+k > 2 η+1 -L} and τ -,i (η) = inf{k ≥ 0|ξ i+k < 2 η-1 + L},
we can write, for all η 0 , η 1 , . . . , η ℓ > η + , and for all 0

< i 1 < • • • < i ℓ , P (η ℓ+1 = η ℓ + 1|η ℓ , . . . , η 0 ; τ ℓ = i ℓ , . . . , τ 1 = i 1 ) = P (τ +,i ℓ (η ℓ ) < τ -,i ℓ (η ℓ )|η ℓ , . . . , η 0 ; τ ℓ = i ℓ , . . . , τ 1 = i 1 ) = P (τ +,i ℓ (η ℓ ) < τ -,i ℓ (η ℓ )|ξ i ℓ ∈ J η ℓ , . . . , ξ i1 ∈ J η1 , ξ 0 ∈ J η0 ; τ ℓ = i ℓ , . . . , τ 1 = i 1 ) . (5.6)
It then follows from (5.4) and the homogeneity of the process ξ k that inf

ξ0∈Jη ℓ P(τ +,0 (η ℓ ) < τ -,0 (η ℓ )|ξ 0 ) = inf ξi ℓ ∈Jη ℓ P (τ +,i ℓ (η ℓ ) < τ -,i ℓ (η ℓ )|ξ i ℓ ) ≤ P (τ +,i ℓ (η ℓ ) < τ -,i ℓ (η ℓ )|ξ i ℓ ∈ J η ℓ , . . . , ξ i1 ∈ J η1 , ξ 0 ∈ J η0 ; τ ℓ = i ℓ , . . . , τ 1 = i 1 ) ≤ sup ξi ℓ ∈Jη ℓ P (τ +,i ℓ (η ℓ ) < τ -,i ℓ (η ℓ )|ξ i ℓ ) = sup ξ0∈Jη ℓ P (τ +,0 (η ℓ ) < τ -,0 (η ℓ )|ξ 0 ) .
Inserting this into (5.6) and using the result in (5.5) finally yields inf

ξ0∈Jη ℓ P (τ +,0 (η ℓ ) < τ -,0 (η ℓ )|ξ 0 ) ≤ P (η ℓ+1 = η ℓ + 1|η ℓ , . . . , η 0 ) ≤ sup ξ0∈Jη ℓ P (τ +,0 (η ℓ ) < τ -,0 (η ℓ )|ξ 0 ) . (5.7)
We will now use the Porte-Manteau Theorem again to conclude the argument. For ease of notation, we shall write τ ± (η) = τ ±,0 (η) in what follows. Let η > η + and ξ ∈ J η . We consider the set E ξ (η) defined as follows:

E ξ (η) = {τ + (η) < τ -(η); ξ 0 = ξ}, so that inf ξ∈Jη P (τ + (η) < τ -(η)|ξ 0 = ξ) = inf ξ∈Jη P (E ξ (η)) , sup ξ∈Jη P (τ + (η) < τ -(η)|ξ 0 = ξ) = sup ξ∈Jη P (E ξ (η)) .
(5.8) Noting that

E ξ (η) = {∀k < τ + (η), ξ k > 2 η-1 + L; ξ 0 = ξ},
one sees, with the notation of Section 4 (R ε k = ξ k ξ0 , and

ε = ξ -1 0 ), that, provided ξ ∈ J η , E ξ (η) ⊃ {∀k < τ + (η), R ε k > 1 2 σ -(η); R ε 0 = 1}, where σ -(η) = 2 η + 2L 2 η -L .
Note that σ -is a decreasing function of its argument which tends to 1 as η → +∞. Let η * > η + , to be chosen later, as a function of δ in (5.3). Let η > η * ; it then follows that

E ξ (η) ⊃ {∀k < τ + (η), R ε k > 1 2 σ -(η); R ε 0 = 1} ⊃ {∀k < τ + (η), R ε k > 1 2 σ -(η * ); R ε 0 = 1}.
In order to apply the Porte-Manteau Theorem, we need to replace the stopping time τ + (η) of ξ k by an appropriately chosen stopping time of the continuous time process R ǫ (t) introduced in Section 4. We will proceed in two steps. First we replace τ + (η) by a stopping time τ ε + for the discrete time process R ε k , which is defined as follows:

τ ε + = inf{k ≥ 0|R ε k > 2σ + (η * )}, where σ + (η * ) = 2 η * -L 2 2 η * -L
is also decreasing and tends to 1 as η * → +∞. One checks that τ ε + ≥ τ + (η), for all η > η * , so that for all η > η * , and for all ξ ∈ J η ,

E ξ (η) ⊃ {∀k < τ + (η), R ε k > 1 2 σ -(η * ); R ε 0 = 1} ⊃ {∀k < τ ε + , R ε k > 1 2 σ -(η * ); R ε 0 = 1}.
We next consider two stopping times T ε + and T ε -for the continuous time processes (R ε (t)) t , defined as follows

T ε + = inf{t ≥ 0|R ε (t) = 2σ + (η * )} and T ε -= inf{t ≥ 0|R ε (t) = 1 2 σ -(η * )}.
It then follows from the definition of (R ε (t)) t by linear interpolation of the (R ε k ) k between the times t k = kε 2 , and the fact that

τ ε + is an integer that (τ ε + -1)ε 2 < T ε + < τ ε + ε 2 , so that 1 2 σ -(η * ) < R ε (τ ε + -1)ε 2 < 2σ + (η * ) and R ε τ ε + ε 2 > 2σ + (η * ). Hence {∀t < τ ε + ε 2 , R ε (t) > 1 2 σ -(η * ); R ǫ (0) = 1} ⊃ {∀t < T ε + , R ε (t) > 1 2 σ -(η * ); R ǫ (0) = 1}.
Finally, we may conclude that, for all η > η * and for all ξ ∈ J η , with ǫ = ξ -1

P (E ξ (η)) ≥ P ∀t < T ε + , R ε (t) > 1 2 σ -(η * ); R ǫ (0) = 1 .
(5.9)

We will now apply the Porte-Manteau Theorem to get a lower bound on the right hand side of this inequality. For that purpose, we first remark that, for all ε and η * > η + ,

P ∀t < T ε + , R ε (t) > 1 2 σ -(η * ); R ε (0) = 1 = P ∀t < T ε + , R ε (t) > 1 2 σ -(η * ); R ε (0) ∈] 5 6 , 7 6 [ , Figure 4: The linear interpolation of the (R ε k ) k between t k and t k+1 yields that T ε ± happens before ε 2 τ ε ± . because P R ε (0) ∈] 5 6 , 1[∪]1, 7 6 = 0. The set {ϕ ∈ C([0, T ], R + )|∀t < T a+ , ϕ(t) > 1 2 σ -(η * ); ϕ(0) ∈ ] 5 6 , 7 6 [} where T a+ = inf{t ≥ 0|ϕ(t) = a + },
P ∀t < T ε + , R ε (t) > 1 2 σ -(η * ) ≥ P ∀t < T a+ , R(t) > a -= P(T a+ < T a-) = a 1-2γ - -1 a 1-2γ - -a 1-2γ + ,
(5.10) where we use the notation of Lemma 4.2 with a -= 1 2 σ -(η * ), a + = 2σ + (η * ) and where (R(t)) t is a Bessel process of dimension 2γ + 1 and initial condition R(0) = 1. Since σ ± (η * ) → 1 when η * → +∞, there exist η * large enough, depending only on δ and L, so that,

P(T a+ < T a-) ≥ p + - δ 2 , where p + = 2 2γ-1 -1 2 2γ-1 -2 1-2γ > 1 2
, since γ > 1 2 . It then follows from (5.10) that there exists ε so that

inf ε<ε P ∀t < T ε + , R ε (t) > 1 2 σ -(η * ) ≥ p + -δ,
Combining this with (5.7), (5.8) and (5.9), we obtain inf

η ℓ >η P (η ℓ+1 = η ℓ + 1|η ℓ , . . . η 0 ) ≥ p + -δ,
where η = max{η * , log 2 (ε -1 + L)}. This is the desired lower bound on the jump probability of the process η ℓ .

To control the upper bound in (5.7), we proceed in the same manner. First, for all ξ ∈ J η , ε = ξ -1 ,

E ξ (η) = {∀k < τ + (η), ξ k > 2 η-1 + L; ξ 0 = ξ} ⊂ {∀k < τ + (η), R ε k > 2 η-1 + L 2 η + L ; R ε 0 = 1} ⊂ {∀k < τ + (η), R ε k > 1 2 ; R ε 0 = 1}.
Now, let η * > η + , to be chosen later, and let η > η * . Consider the stopping time τ ε

+ = inf{k ≥ 0| R ε k > 2σ + (η * )} where σ+ (η * ) = 2 η * -L 2 2 η * + L
. Note that σ+ is increasing and converges to 1 when η * → +∞. One readily checks that τ ε + ≤ τ + (η) and hence

{∀k < τ + (η), R ε k > 1 2 ; R ε 0 = 1} ⊂ {∀k < τ ε + , R ε k > 1 2 ; R ε 0 = 1} ⊂ {∀t < T ε + , R ε (t) > 1 2 ; R ε (0) = 1}, where T ε + = inf{t ≥ 0| R ε (t) = 2σ + (η * )} and (τ ε + -1)ε 2 ≤ T ε + < τ ε + ε 2 . Finally, we have E ξ (η) ⊂ {∀t < T ε + , R ε (t) > 1 2 ; R ε (0) = 1}.
(5.11)

Set ã+ = 2σ + (η * ). Now, we can again use the Porte-Manteau Theorem and Theorem 4.1, because the

set {ϕ ∈ C ([0, T ], R + ) | ∀t ≤ T ã+ , ϕ(t) > 1 2 ; ϕ(0) = 1} where T ã+ = inf{t ≥ 0| ϕ(0) = 1, ϕ(t) = ã+ } is closed. This leads to lim sup ε→0 P ∀t ≤ T ε + , R ε (t) > 1 2 ≤ P ∀t ≥ T ã+ , R(t) > 1 2 (5.12)
where (R(t)) t is as before a Bessel process of dimension 2γ + 1 and initial condition R(0) = 1. Defining ã-= 1 2 , and using Lemma 4.2, we have

P ∀t ≤ T ã+ , R(t) > 1 2 = P T ã-> T ã+ = 2 2γ-1 -1 2 2γ-1 -ã1-2γ + .
(5.13) It follows from (5.12) and (5.13) that there exists ε depending on δ so that

sup ε<ε P ∀t ≤ T ε + , R ε (t) > 1 2 ≤ p + + δ.
Combining this with (5.7) and (5.11), we see there exists η > η + so that sup

η ℓ >η P (η ℓ+1 = η ℓ + 1| η ℓ , . . . , η 0 ) ≤ p + + δ,
which is the desired upper bound.

(ii) Let 0 < δ < µ and 0 < p ≤ 1. We first write down the Doob decomposition (see [START_REF] Stewart | Markov processes[END_REF]) of η ℓ explicitly:

η ℓ = η 0 + M ℓ + A ℓ ,
where

M ℓ = ℓ j=1 (η j -E (η j |η j-1 , . . . , η 0 )) , and A ℓ = ℓ j=1 (E (η j |η j-1 , . . . , η 0 ) -η j-1 ) .
As is well known, and easily checked, M ℓ is a martingale with respect to the natural filtration induced by the process η ℓ , a fact we will use below. Now,

| η ℓ -(η 0 + µℓ) |≤| M ℓ | + ℓ j=1 | [(E (η j |η j-1 , . . . , η 0 ) -η j-1 ) -µ] | .
It then follows from part (i) of the Lemma that, for all δ > 0, there exists η > η + so that,

∀ℓ ∈ N, η ℓ-1 , . . . , η 0 > η ⇒| η ℓ -(η 0 + µℓ) |≤| M ℓ | + δ 2 ℓ .
(5.14)

Figure 5: A typical trajectory of η ℓ , on F L , as defined in (5). Now, for any L > 0, define

F L = {|M ℓ | ≤ δ 2 ℓ, ∀ℓ ≥ L}.
Then, on F L , and provided η 0 > η + L > η + + L, so that η L-1 , . . . , η 0 > η, one has

| η L -(η 0 + µL) |≤ δL,
so that in particular η L > η 0 > η + L. This in turn implies that η j > η, for all 0 ≤ j ≤ 2L. We can therefore apply (5.14) for all L ≤ ℓ ≤ 2L to conclude that on F L , and provided

η 0 > η + L > η + + L, one has | η ℓ -(η 0 + µℓ) |≤| M ℓ | + δ 2 ℓ ≤ δℓ.
(5.15)

Proceeding recursively, one then concludes that (5.15) holds on F L , for all L ≤ ℓ. For 0 ≤ ℓ ≤ L, one has from (5.14) that

| η ℓ -(η 0 + µℓ) |≤| M ℓ | + δ 2 ℓ ≤ 2L + δℓ.
Hence, if we choose η 0 > 2L δ , we can conclude that,

∀ 0 ≤ ℓ ≤ L; | η ℓ -(η 0 + µℓ) |≤ δ(η 0 + ℓ).
From this, and (5.15), we conclude that, for all δ > 0 and all L > 0, if

η 0 > η * = max{η + L, 2L δ }, then P (∀ℓ ∈ N, | η ℓ -(η 0 + µℓ) |≤ δ(η 0 + ℓ)) ≥ P(F L ).
(5.16)

It remains to show that, given δ and p, there exists L so that

P(F L ) ≥ 1 -p
to conclude the proof. For that purpose, let us introduce the quadratic variation of M ℓ ,

D 2 ℓ = ℓ i=1 (M i -M i-1 ) 2 .
The Burkholder inequality (see [START_REF] Karatzas | Brownian motion and stochastic calculus[END_REF]) then says that, for all ℓ ∈ N, exists a constant C > 0

E ( max 0≤i≤ℓ |M i |) 4 ≤ CE(D 4 ℓ ).
The definition of the η ℓ and of the martingale M ℓ immediately imply that, with probability one, |M ℓ -M ℓ-1 | ≤ 2, for all ℓ. This implies immediately that E(D 4 ℓ ) ≤ 4ℓ 2 . Hence, by the Tchebychev inequality,

P |M ℓ | > δ 2 ℓ ≤ P max 0≤i≤ℓ |M i | > δ 2 ℓ ≤ 2 4 δ 4 ℓ 4 C4ℓ 2 = C δ 4 ℓ 2 ,
where C is a numerical constant. It then follows that

P(F L ) ≥ 1 - ℓ≥L P(|M ℓ | > δ 2 ℓ) ≥ 1 - C δ 4 L .
Choosing L = C δ 4 p , the result now follows from (5.16). (iii) This is an immediate consequence of (ii).

Estimates on the dwell times τ ℓτ ℓ-1

As explained in the introduction of Section 5, having obtained the asymptotic behaviour of η ℓ , we now need to control the stopping times τ ℓ and show that with high probability they behave, roughly, as τ ℓ ∼ ξ 2 τ ℓ ∼ 2 2η ℓ . We turn to this task in this section, the main result of which is stated in Proposition 6.2 (ii)&(iv). For that purpose, we will first estimate the dwell times τ ℓτ ℓ-1 (Proposition 6.2 (i)&(iii)). Roughly speaking, this is the time the process needs to move from ξ τ ℓ-1 to either 2ξ τ ℓ-1 or to 1 2 ξ τ ℓ-1 . As we will see in Lemma 6.1, the latter can be estimated from above and from below using Theorem 4.1, together with the Porte-Manteau Theorem and Lemma 4.2 (ii), a task we now turn to.

Let us define, for all n 0 ∈ N, for all b -< 1 < b + , and for all r > 0, the stopping time

K n0 = inf{k ∈ N | ξ n0+k ∈]b -r, b + r[}. (6.1) If ξ n0 ∈]b -r, b + r[, K n0 = 0. Otherwise, K n0 > 0: n 0 + K n0 -1 is then the last instant that the process is still inside the interval ]b -r, b + r[.
The following lemma gives the bounds on K n0 that we shall be needing. Lemma 6.1. Suppose Hypothesis 2 holds and that γ ≥ 1/2. (i) There exists ξ * > ξ + and 0 < q -< 1, so that, for all m ∈ N * , for all n 0 ∈ N,

sup r≥ξ * sup ξn 0 ∈]b-r,b+r[ P mr 2 < K n0 | ξ n0 ≤ q m -< 1. (ii) Let b ′ -, b ′ + be such that b -< b ′ -< 1 < b ′ + < b + .
Then there exists ξ * > ξ + and 0 < q + < 1 so that, for all n 0 ∈ N,

sup r≥ξ * sup ξn 0 ∈]b ′ -r,b , + r[ P K n0 ≤ r 2 | ξ n0 ≤ q + < 1. (6.2)
Proof. (i) We first treat the case with m = 1. Let r > 0, ξ n0 ∈]b -r, b + r[. The homogeneity of the Markov chain implies it is enough to consider n 0 = 0. Consider the set

{K 0 > r 2 } = {∀ 0 ≤ k ≤ r 2 , ξ k ∈]b -r, b + r[} = {∀t k ≤ ε 2 r 2 , R ε (t k ) ∈]εb -r, εb + r[}.
where we used the notation of Section 4. Since (b + r) -1 < ε < (b -r) -1 , it follows that

{K 0 > r 2 } ⊂ {∀t k ≤ ε 2 r 2 , R ε (t k ) ∈] b - b + , b + b - [} ⊂ {∀t k ≤ 2T, R ε (t k ) ∈] b - b + , b + b - [},
where

T = 1 2b 2 + . Now choose r > (b - √ T ) -1 so that ε 2 < T . Then, if K satisfies t K ≤ 2T < t K+1 , we have T < t K . Since R ε (t)
is constructed by linear interpolation between the R ε (t k ), we can then conclude that

{∀t k ≤ 2T, R ε (t k ) ∈] b - b + , b + b - [} ⊂ {∀t ≤ T, R ε (t) ∈] b - b + , b + b - [} ⊂ {∀t ≤ T, R ε (t) ∈ [ b - b + , b + b - ]}, so that ∀r > (b - √ T ) -1 , ∀ξ 0 ∈]b -r, b + r[, P({K 0 > r 2 }) ≤ P(∀t ≤ T, R ε (t) ∈ [ b - b + , b + b - ]). (6.3) The set {ϕ ∈ C([0, T ], R * ) | ∀t ∈ [0, T ], ϕ(t) ∈ [ b- b+ , b+ b-]
} is closed, so we can apply the Porte-Manteau Theorem, together with Theorem 4.1 to conclude that

∃ε * > 0, ∀ε < ε * , P({∀t ≤ T, R ε (t) ∈ [ b - b + , b + b - ]}) ≤ q -:= 1 2 (1 + q-), (6.4)
where q-:= P(∀t

∈ [0, T ], R t ∈ [ b- b+ , b+ b-]})
. By Lemma 4.2, q-< 1 so that q -< 1. It then follows from (6.3) and (6.4) that

∀r > ξ * = max{(b - √ T ) -1 , (b -ε * ) -1 }, ∀ξ 0 ∈]b -r, b + r[, P ({K 0 > r 2 }) ≤ q -< 1.
This proves (6.1) for m = 1.

It remains to show the case m > 1. This will follow from the Markov property of the chain, as follows. We write ∆ =]b -r, b + r[. Let us introduce K * = ⌊r 2 ⌋, where ⌊•⌋ denotes the integer part. First note that

P(K n0 > mK * | ξ n0 ) = P(ξ n0+1 ∈ ∆, . . . , ξ n0+mK * ∈ ∆ | ξ n0 ) = Π m-1 s=0 Π K * i=1 ξn 0 +sK * +i∈∆ P(dξ n0+sK * +i | ξ n0+sK * +i-1 ) = Π m-1 s=0 ξ n 0 +(s+1)K * ∈∆ L K * (dξ n0+(s+1)K * , ξ n0+sK * ), (6.5)
where

L K * (A, y) := ξ n 0 +(s+1)K * ∈A ξ n 0 +(s+1)K * -1 ∈∆ . . . ξn 0 +sK * +1∈∆ Π K * i=2 P(dξ n0+sK * +i | ξ n0+sK * +i-1 )P(dξ n0+sK * +1 | ξ n0+sK * = y) = P(ξ n0+(s+1)K * ∈ A, ξ n0+sK * +i ∈ ∆, 1 ≤ i < K * | ξ n0+sK * = y),
which does in fact not depend on s, nor on n 0 , because the Markov chain is homogeneous. Now remark that, when A = ∆, and y ∈ ∆, one finds

L K * (∆, y) = P(K n0 > K * | ξ n0 = y).
It then follows from (6.5) and from (6.1) for m = 1 that, for all m ∈ N * ,

P(K n0 > mK * | ξ n0 ) ≤ q m
-. This completes the proof of (i).

(ii) The argument is analogous to the first part of (i). Again, because of the homogeneity of the chain, it is enough to prove the result for n 0 = 0. Let

ξ 0 ∈]b ′ -r, b ′ + r[⊂]b -r, b + r[. We then have {K 0 ≤ r 2 } = {∃k ≤ r 2 , ξ k ∈ ∆} ⊂ {∃t k ≤ 1 b ′ - 2 , R ε (t k ) ∈] b - b ′ - , b + b ′ + [} ⊂ {∃t ∈ [0, T ], R ε (t) ∈] b - b ′ - , b + b ′ + [} where we set T = 1 b ′ - 2 . The set {ϕ ∈ C([0, T ], R * ) | ∃t ∈ [0, T ], ϕ(t) ∈] b- b ′ - , b+ b ′ +
[} is closed, so we can apply the Porte-Manteau Theorem, together with Theorem 4.1 and Lemma 4.2 to obtain (6.2) with

q + = 1 2 (1 + q+ ), q+ = P(∃t ∈ [0, T ], R t ∈] b - b ′ - , b + b ′ + [}) < 1.
To state the main result of this section, we introduce "good" sets where the dwell times are suitably controlled and that we will show to be of high probability. Let η 0 > η + , δ > 0 be given, as well as two increasing sequences (k ± ℓ ) of positive integers, with 0 < k - ℓ ≤ ℓ. Define furthermore the sequence

a ℓ = 2 2[(1-δ)η0+(µ-δ)(l-1-k - ℓ )] 2 -δk - ℓ . (6.6)
Then we introduce

G 1 = {∀ℓ ∈ N, |η ℓ -µℓ -η 0 | ≤ δ(ℓ + η 0 )} = ∩ ℓ G 1 (ℓ), G 2 = {∀ℓ ∈ N * , τ ℓ -τ ℓ-1 ≤ k + ℓ-1 2 2η ℓ-1 } = ∩ ℓ G 2 (ℓ), G 3 = {∀ℓ ∈ N * , ∃ℓ -k - ℓ ≤ k ≤ ℓ, τ k -τ k-1 ≥ a ℓ } = ∩ ℓ G 3 (ℓ) and G = G 1 ∩ G 2 ∩ G 3 .
If we set δ = 0, and k - ℓ = 0, k + ℓ-1 = 1, and use (6.6), then one can easily check that on G, τ ℓ ∼ 2 2µℓ and ξ τ ℓ ∼ 2 µℓ , this mean that ξ τ ℓ ∼ √ τ ℓ which is the power law we are trying to establish. But in that case, we cannot hope to prove a suitable lower bound on P(G). To do so, we need to make the set G a little bigger, by taking δ > 0 and choosing suitable growing sequences k ± ℓ . This will allow us to show P(G) is close to 1 in the following proposition, using Proposition 5.1 and Lemma 6.1, and at the same time to get suitable bounds on τ ℓ in function of 2 2η ℓ ∼ ξ 2 τ ℓ . Proposition 6.2. (i) ∀0 ≤ p < 1, and for all δ > 0, ∃η > η + so that ∀η 0 > η and for all sequences (k + ℓ ) ℓ∈N , we have

P(G 1 ∩ G 2 ) ≥ 1 - +∞ ℓ=1 p k + ℓ-1 - - p 2 , (6.7)
where p -is defined in Lemma 6.1 (i).

(ii) Let 0 < ν < 1, 0 < p ≤ 1. Then there exists δ > 0 so that, for all 0 < δ ≤ δ and k + ℓ = 2 δ(ℓ+η0) , there exists η such that, ∀η 0 ≥ η,

{G 1 , ∀ℓ ∈ N, τ 1-ν ℓ ≤ 1 2 2 2η ℓ } ⊃ G 1 ∩ G 2 P G 1 , ∀ℓ ∈ N, τ 1-ν ℓ ≤ 1 2 2 2η ℓ ≥ P(G 1 ∩ G 2 ) ≥ 1 -p.
(iii) ∀0 ≤ p < 1, ∀δ > 0, ∃η > η + so that for all η 0 ≥ η and for all sequences 0 < k - ℓ ≤ ℓ, one has

P(G 1 ∩ G 3 ) ≥ 1 - +∞ ℓ=1 q k - ℓ + - p 2 , (6.8)
where q + is defined in Lemma 6.1 (ii).

(iv) ∀0 ≤ p < 1, ∀δ > 0, ∃η > η + so that for all η 0 ≥ η

{G 1 , ∀ℓ ∈ N, τ ℓ ≥ a ℓ } ⊃ G 1 ∩ G 3 P(G 1 , ∀ℓ ∈ N, τ ℓ ≥ a ℓ ) ≥ P(G 1 ∩ G 3 ) ≥ 1 -p, provided k - ℓ = min{δ(ℓ + η 0 )
, ℓ} and a ℓ is given by (6.6).

We point out that, in order to get a sharp upper bound on the τ ℓτ ℓ-1 in part (i) of the lemma, one would like to take the k + ℓ small, or at least bounded, in the left hand side of (6.7). But this estimate is useful only if the k + ℓ are large for all ℓ and tend to +∞ as ℓ → +∞. This is indeed needed for the sum in the right hand side to converge to a small number.

Proof. (i) First note that it follows from Proposition 5.1 (ii) that, for all 0 ≤ p < 1 and all δ > 0, there exists η * > η + so that, for all η 0 ≥ η * , P(G 1 ) ≥ 1 -p 2 . Hence

P(G 1 ∩ G 2 ) ≥ 1 - p 2 -P(G 1 ∩ G c 2 ). (6.9) Now,
P(G 1 ∩ G c 2 ) ≤ +∞ ℓ=1 P(G 2 (ℓ) c ∩ G 1 ) ≤ +∞ ℓ=1 P(G 2 (ℓ) c ∩ G 1 (ℓ -1)) ≤ +∞ ℓ=1 P(G 2 (ℓ) c |G 1 (ℓ -1)) (6.10)
and, for all ℓ ∈ N * ,

P(G 2 (ℓ) c |G 1 (ℓ -1)) ≤ sup η ℓ-1 ∈I ℓ-1 P(G 2 (ℓ) c |η ℓ-1 } (6.11)
where we used the observation that G 1 (ℓ -1) = {η ℓ-1 ∈ I ℓ-1 }, where

I ℓ-1 = [η 0 (1 -δ) + (µ -δ)(ℓ -1), η 0 (1 + δ) + (µ + δ)(ℓ -1)].
Now, proceeding as in the beginning of the proof of Proposition 5.1,

P(G 2 (ℓ) c |η ℓ-1 ) = P(τ ℓ -τ ℓ-1 > k + ℓ-1 2 2η ℓ-1 | η ℓ-1 ) = i P τ ℓ -τ ℓ-1 > k + ℓ-1 2 2η ℓ-1 | η ℓ-1 , τ ℓ-1 = i P(τ ℓ-1 = i | η ℓ-1
). (6.12)

We have, for all i, ℓ and η ℓ-1 ,

P(τ ℓ -τ ℓ-1 >k + ℓ-1 2 2η ℓ-1 | η ℓ-1 , τ ℓ-1 = i) = P(inf{k|ξ i+k ∈]2 η ℓ-1 -1 + L, 2 η ℓ-1 +1 -L[} > k + ℓ-1 2 2η ℓ-1 | ξ i ∈ J η ℓ-1 , τ ℓ-1 = i) ≤ sup ξi∈Jη ℓ-1 P(inf{k|ξ i+k ∈]2 η ℓ-1 -1 + L, 2 η ℓ-1 +1 -L[} > k + ℓ-1 2 2η ℓ-1 | ξ i ) ≤ sup ξi∈Jη ℓ-1 P(inf{k|ξ i+k ∈]2 η ℓ-1 -1 , 2 η ℓ-1 +1 [} > k + ℓ-1 2 2η ℓ-1 | ξ i ),
where we used (5.4). We now remark that inf{k|ξ i+k ∈]2 η ℓ-1 -1 , 2 η ℓ-1 +1 [} = K i , where K i is defined in (6.1), with b -= 1 2 , b + = 2 and r = 2 η ℓ-1 . It therefore follows from Lemma 6.1 (i) and from what precedes that, provided 2 η ≥ ξ * , we have for all η ℓ-1 ≥ η,

P(τ ℓ -τ ℓ-1 > k + ℓ-1 2 2η ℓ-1 | η ℓ-1 , τ ℓ-1 = i) ≤ p k + ℓ-1 - .
Using this in (6.10), we find that, for all η ≥ log 2 ξ * ,

P(G 2 (ℓ) c |η ℓ-1 ) = P(τ ℓ -τ ℓ-1 > k + ℓ-1 2 2η ℓ-1 |η ℓ-1 ) ≤ p k + ℓ-1 - ,
which, when inserted into (6.9)-(6.11), yields the result provided η ≥ max{η * , log 2 ξ * }.

(ii) Let ν > 0, 0 ≤ p < 1. Let δ > 0. On G 1 ∩ G 2 , a simple calculation using k + ℓ = 2 δ(ℓ+η0) yields τ ℓ = ℓ k=1 τ k -τ k-1 ≤ (2 2µ -1) -1 2 2[(1+ 3 2 δ)η0+(µ+ 3 2 δ)ℓ] . Introducing δ = (5 -3ν) -1 min 2µν, 2ν -η -1 + (1 -ν) log 2 |2 2µ -1| -1 ,
it now easily follows that, if δ ≤ δ, and

η 0 ≥ η + , then, on G 1 ∩ G 2 , τ 1-ν ℓ ≤ 1 2 2 2η ℓ ,
which is the desired estimate. To see it occurs with high probability, we use (6.7) to check that there exists η > η + , depending on δ and p, so that, for all η 0 ≥ η, one has

P(G 1 ∩ G 2 ) ≥ 1 -p.
(iii) As in (i), we argue that, for all 0 ≤ p < 1 and all δ > 0, there exists η * > η + so that, for all η 0 ≥ η * ,

P(G 1 ∩ G 3 ) ≥ 1 - p 2 - +∞ ℓ=1 P(G 3 (ℓ) c ∩ G 1 (ℓ -1) ∩ • • • ∩ G 1 (ℓ -k - ℓ -1)). (6.13)
For ease of notation, we introduce, for ℓ -

k - ℓ ≤ k ≤ ℓ, G 4 (k, ℓ) = {τ k -τ k-1 < a ℓ }, and G 5 (k, ℓ) = G 4 (k, ℓ) ∩ G 1 (k -1).
Remarking that

G 3 (ℓ) c = {∀ℓ -k - ℓ ≤ k ≤ ℓ, τ k -τ k-1 < a ℓ } = ∩ ℓ k=l-k - ℓ G 4 (k, ℓ),
and introducing

G - 5 (k, ℓ) = ∩ k-1 k ′ =ℓ-k - ℓ G 5 (k ′ , ℓ),
we can then write

P(G 3 (ℓ) c ∩ G 1 (ℓ -1), . . . , G 1 (ℓ -k - ℓ -1)) = P(∩ ℓ k=ℓ-k - ℓ G 5 (k, ℓ)) =   k - ℓ -1 k=0 P G 5 (ℓ -k, ℓ)|G - 5 (ℓ -k, ℓ)   P(G 5 (ℓ -k - ℓ , ℓ)).
(6.14)

Now, for ℓk - ℓ < k ′ ≤ ℓ, we have

P(G 5 (k ′ , ℓ)|G - 5 (k ′ , ℓ)) = P(G 4 (k ′ , ℓ)|G 1 (k ′ -1), G - 5 (k ′ , ℓ))P(G 1 (k ′ -1)|G - 5 (k ′ , ℓ)) ≤ P(G 4 (k ′ , ℓ)|G 1 (k ′ -1), G - 5 (k ′ , ℓ)) ≤ P(∆τ k ′ < a ℓ |η k ′ -1 ∈ I k ′ -1 , G - 5 (k ′ , ℓ)) ≤ sup η j ∈I j ℓ-k - ℓ -1≤j<k ′ P(∆τ k ′ < a ℓ |η k ′ -1 , η k ′ -2 , . . . η ℓ-k - ℓ -1 , ∆τ k ′ -1 < a ℓ , . . . , ∆τ ℓ-k - ℓ < a ℓ ), (6.15)
where we introduced ∆τ j = τ jτ j-1 . As before, we write

P(∆τ k ′ < a ℓ |η k ′ -1 , η k ′ -2 , . . . η ℓ-k - ℓ -1 , ∆τ k ′ -1 < a ℓ , . . . , ∆τ ℓ-k - ℓ < a ℓ ) = 0<ij -ij+1<a ℓ P(∆τ k ′ < a ℓ |η k ′ -1 , η k ′ -2 , . . . η ℓ-k - ℓ -1 , τ k ′ -1 = i k ′ -1 , . . . , τ ℓ-k - ℓ -1 = i ℓ-k - ℓ -1 ) × P(τ k ′ -1 = i k ′ -1 , . . . , τ ℓ-k - ℓ -1 = i ℓ-k - ℓ -1 |η k ′ -1 , η k ′ -2 , . . . η ℓ-k - ℓ -1 , ∆τ k ′ -1 < a ℓ , . . . , ∆τ ℓ-k - ℓ < a ℓ ). (6.16)
It remains to estimate

P(∆τ k ′ < a ℓ |η k ′ -1 , η k ′ -2 , . . . η ℓ-k - ℓ -1 , τ k ′ -1 = i k ′ -1 , . . . , τ ℓ-k - ℓ -1 = i ℓ-k - ℓ -1
). For that purpose, we make the observation that

{η k ′ -1 , η k ′ -2 , . . . η ℓ-k - ℓ -1 , τ k ′ -1 = i k ′ -1 , . . . , τ ℓ-k - ℓ -1 = i ℓ-k - ℓ -1 } = {ξ i k ′ -1 ∈ J η k ′ -1 , C} where C = {η k ′ -2 , . . . η ℓ-k - ℓ -1 , τ k ′ -1 > i k ′ -1 -1, . . . , τ ℓ-k - ℓ -1 = i ℓ-k - ℓ -1 }. Indeed, on the set where ξ i k ′ -1 ∈ J η k ′ -1 and τ k ′ -1 > i k ′ -1 -1, we do have τ k ′ -1 = i k ′ -1 . Hence P(∆τ k ′ < a ℓ |η k ′ -1 , η k ′ -2 , . . . η ℓ-k - ℓ -1 , τ k ′ -1 = i k ′ -1 , . . . , τ ℓ-k - ℓ -1 = i ℓ-k - ℓ -1 ) = P(inf{t|ξ i k ′ -1 +t ∈]2 ηi k ′ -1 -1 + L, 2 ηi k ′ -1 +1 -L[} < a ℓ |ξ i k ′ -1 ∈ J η k ′ -1 , C) ≤ sup ξi k ′ -1 ∈Jη k ′ -1 P(inf{t|ξ i k ′ -1 +t ∈]2 ηi k ′ -1 -1 + L, 2 ηi k ′ -1 +1 -L[} < a ℓ |ξ i k ′ -1 ) (6.17)
where we use the observation that C ∈ F i k ′ -1 -1 , and (5.4). We now wish to use Lemma 6.1 to conclude. For that purpose, first note that, there exist 0 < b -< 1 < b + so that for all r ≥ 2 η+ ,

1 2 r + L < b -r < r < b + r < 2r -L.
Clearly, one can think of b -as being close to 1 2 and of b + as being close to 2. With the notation of (6.1), and r = 2 ηi k ′ -1 , one has

K i k ′ -1 < inf{t|ξ i k ′ -1 +t ∈]2 ηi k ′ -1 -1 + L, 2 ηi k ′ -1 +1 -L[}, so that P(inf{t|ξ i k ′ -1 +t ∈]2 ηi k ′ -1 -1 + L, 2 ηi k ′ -1 +1 -L[} < a ℓ |ξ i k ′ -1 ) ≤ P(K i k ′ -1 < a ℓ |ξ i k ′ -1 ).
Recalling that ℓk - ℓ < k ′ ≤ ℓ, one checks readily that a ℓ ≤ r 2 , so that (6.2) implies, there exists η * so that, for all η 0 ≥ η * , for all η k ′ -1 ∈ I k ′ -1 and for all

ξ i k ′ -1 ∈ J η k ′ -1 , P(inf{t|ξ i k ′ -1 +t ∈]2 ηi k ′ -1 -1 + L, 2 ηi k ′ -1 +1 -L[} < a ℓ |ξ i k ′ -1 ) ≤ P(K i k ′ -1 < a ℓ |ξ i k ′ -1 ) ≤ P(K i k ′ -1 < r 2 |ξ i k ′ -1 ) ≤ q + < 1, (6.18) provided b ′ -, b ′ + are chosen so that b -r < b ′ -r ≤ r -L ≤ r + L ≤ b ′ + r < b + r,
for all r ≥ 2 η+ , which is always possible. (One should think of b ′ ± as being close to 1.) Inserting (6.15)-(6.18) into (6.14) yields

P(G 3 (ℓ) c ∩ G 1 (ℓ -1), . . . , G 1 (ℓ -k - ℓ -1)) ≤ q k - ℓ + ,
provided η 0 ≥ η = max{η * , η * }. Inserting this in (6.13) yields (6.8).

(iv) This is now an immediate consequence of (iii).

7 Proof of Theorem 3.1 (i) Let p > 1 and 0 < ν < 1. Let 0 < ν < ν. It then follows from Proposition 6.2 that, there exists δ and η so that, for all 0 < δ < δ and for all η 0 ≥ η, one has

P(G) ≥ 1 -p,
where G = G 1 ∩ G 2 ∩ G 3 and where k + ℓ = 2 2δ(η0+ℓ) , k - ℓ = min{δ(η 0 + ℓ), ℓ}. Note that G depends on η 0 and δ. In addition, on G, the following inequalities hold for all ℓ ∈ N * :

|η ℓ -µℓ -η 0 | ≤ δ(ℓ + η 0 ), τ 1-ν ℓ ≤ 1 2 2 2η ℓ , τ ℓ ≥ a ℓ = 2 2[(1-δ)η0+(µ-δ)(ℓ-1-k - ℓ )] 2 -δk - ℓ .
Since (2 η0 +τ

1 2 ℓ ) 1-ν ≤ 2 (1-ν)η0 +τ 1-ν 2 ℓ
, one easily infers from the first two inequalities that, on G, provided δ ≤ min{ δ, ν}, one has for all η 0 ≥ max{η, (2(νδ)) -1 }, and for all ℓ,

2 η0 + τ 1 2 ℓ 1-ν ≤ 2 η ℓ . (7.1)
Similarly, using the first and third inequality above, one shows that on G, provided δ ≤ min{ δ, ν}, one has for all η 0 ≥ max{η, (2(νδ)) -1 }, and for all ℓ,

2 η ℓ ≤ 2 η0 + τ 1 2 ℓ 1+ν . (7.2)
We are now ready to conclude the proof. By the definition of the stopping time τ ℓ , and using that η ℓ ≥ η ℓ+1 -1, as well as (7.1)-(7.2),we have for all k ∈ [τ ℓ ; τ ℓ+1 [

1 4 2 η0 + k 1 2 1-ν ≤ 1 4 2 η ℓ+1 ≤ 2 η ℓ -1 + L ≤ ξ k ≤ 2 η ℓ +1 -L ≤ 22 η ℓ ≤ 2 2 η0 + k 1 2 1+ν
Finally, remarking that 2 = (2 η0 )

1 η 0 ≤ (2 η0 + k 1 2 ) 1 η 0
one obtains the result if one chooses η large enough so that ν + 2 η ≤ ν.

(ii) This is an immediate consequence of (i).

A Appendix

In this Appendix we prove Theorem 4.1 and Lemma 4.2. Recall that we consider a family of continuous and piecewise linear stochastic processes (t → R ε (t), t ∈ [0, 2T ]) 0<ε<ε * defined as follows. For each 0 < ε < ε * , where

ε * ≪ 1 R ε (t n ) = εξ n , t n = nε 2 , R ε (t) = t n+1 -t ε 2 R ε (t n ) + t -t n ε 2 R ε (t n+1 ), t ∈ [t n , t n+1 ].
Here (ξ n ) n is defined in (3.4). Note that the initial value R ε (0) = 1 is independent of ε and non-random. Each realization of the process (R ε (t)) t∈[0,2T ] belongs to

C := (C ([0, 2T ] : R + ) , || • || ∞ ) .
Let B(C) designate the Borel sets of C.

The method used in the proof of Theorem 4.1 is standard. It is in particular described in [START_REF] Goudon | Stochastic acceleration in an inhomogeneous time random force field[END_REF]. It can be decomposed in 3 steps.

Step 1 For each η > 1, we introduce the process X ε which is R ε stopped at η -1 or at η (see (A.1)). We show that the process X ε admits convergent subsequences as ε → 0 by showing it is precompact.

Step 2 We show the limits of the converging subsequences are solutions of the martingale problem associated to a Bessel process in dimension 2γ + 1, stopped at η -1 or η. As the latter is well-posed, we conclude that the limits have the distribution of the preceding stopped Bessel process and that it is not only the subsequences which are converging but the entire family.

Step 3 We show that the convergence result still holds when we delete all the stopping times, which means we tak η → +∞. The transience of the Bessel process in dimension strictly larger than 2 is an essential ingredient in this part of the proof.

Proof of Theorem 4.1.

Step 1: Precompactness of the stopped proccess Let η ≫ 1 and ε * = (ηξ + ) -1 , then for all 0 < ε < ε * , η -1 > εξ + . We introduce for all 0 < ε < ε * the stopping time

τ ε := inf{t ∈ [0, T ]; R ε (t) ∈ (η -1 , η)} (A.1)
with the convention inf{∅} = 2T . We then introduce the stopped process

∀t ∈ [0, 2T ], X ε (t) = R ε (t ∧ τ ε ).
In other words, once X ε reaches η or η -1 it stays constant. The assumption ε * = (ηξ + ) -1 guarantees that for all n, ξ n

= 1 ε R ε (t n ) > ξ + . Introduce A ε f (x) = a ε (x)f ′′ (x) + b ε (x)f ′ (x) + c ε (f, x),
where

a ε (x) = ε 2 + ε 4 γ 2 x 2 + ε 2α E G ε 0 (x, ω) 2 + ε 2β E G 1 (x, ω) 2 + ε β+3 E (G 1 (x, ω)) , (A.2) b ε (x) = ε 2 γ x + ε β+1 E (G 1 (x, ω)) , (A.3) c ε (f, x) = K n=3 f (n) (x)E εω + ε 2 γ x + ε α+1 G ε 0 (x, ω) + ε β+1 G ε 1 (x, ω) n + O εf K+1 , (A.4) and D ε * = {f ∈ C [η -1 , η] ∩ C ∞ (η -1 , η) , lim x→η ±1 A ε f (x) = 0}.
Then we have the following lemma.

Lemma A.1. The operator (A ε , D ε * ) is a core for the infinitesimal generator of the stopped proccess (X ε (t n )) n .

See [START_REF] Stroock | of Grundlehren der Mathematischen Wissenschaften[END_REF] and [START_REF] Mandl | Analytical treatment of one-dimensional Markov processes[END_REF] for the proof.

Hence, as for all

f ∈ D ε * , the process f (X ε (t n )) - n-1 j=0 A ε f (X ε (t j ))
n∈N is a martingale, it is easy to check by (A.2)-(A.4) and (3.3) that for all f ∈ D ε * there exists a constant 0 < C f < +∞ depending only on f such that the process (f (X ε (t n )) -C f t n ) n is a sub-martingale. As well, for all δ > 0 there exists ε such that for all ε < ε we have

P (|X ε (t j ) -X ε (t j-1 )| > δ) = 0,
) → (0, ε * ) such that t → X ϕ(ε) (t), t ∈ [0, 2T ] 0<ε<ε * ⇀ (t → X ϕ (t), t ∈ [0, 2T ]) ,
where the symbol ⇀ refers to convergence in distribution.

Step 2: Convergence and limit Introduce

τ ϕ := t ∈ [0, 2T ]; X ϕ (t) ∈ (η -1 , η)
with the convention inf{∅} = 2T . As it evolves in the compact [0, 2T ], the sequence (τ ε ) 0<ε<ε * is also tight, however the limit of a subsequence (τ ψ(ε) ) 0<ε<ε * converging is not a stopping time.

Theorem A.2. The processes (X ϕ (t ∧ τ ϕ )) t∈[0,2T are solution of the martingale problem associated to the infinitesimal generator (L, D * ) where

L := 1 2 d 2 dx 2 + γ x d dx and
D * := f ∈ C [η -1 , η] ∩ C ∞ (η -1 , η) ; lim x→η ±1 Lf = 0 .
Note that (L, C ∞ (R)) is the infinitesimal generator of a Bessel process of dimension 2γ + 1. The condition on D * : lim x→η ±1 Lf (x) = 0 yields that the Bessel process stays constant once it reaches the points η ±1 . We call the points η ±1 as being adhesif (see [START_REF] Mandl | Analytical treatment of one-dimensional Markov processes[END_REF]).

We introduce (R(t)) t∈[0,2T ] a Bessel process of dimension 2γ + 1 such that R(0) = 1 and

τ := inf t ∈ [0, 2T ]; R(t) ∈ (η -1 , η)
with the convention inf{∅} = 2T . Then (R(t ∧ τ )) t∈[0,2T ] is generated by (L, D * ) (see [START_REF] Mandl | Analytical treatment of one-dimensional Markov processes[END_REF]).

As martingale problems associated to Bessel processes are well-posed, this theorem implies that all the (X ϕ (t ∧ τ ϕ )) t∈[0,2T ] have the same distribution which is the one of (R(t ∧ τ )) t∈[0,2T ] . In particular τ and the τ ϕ have the same distribution.

Proof of Theorem A.2. The process

ϕ(ε) f (t n ) = f X ϕ(ε) (⌊ t n ∧ τ ϕ ϕ(ε) 2 ⌋ϕ(ε) 2 ) - ⌊ tn ∧τ ϕ ϕ(ε) 2 ⌋ j=1 A ϕ(ε) f X ϕ(ε) (t j ) is a F X ϕ(ε) n -martingale for all f ∈ D ϕ(ε) *
. Nevertheless, for all f ∈ D * it is only a submartingale. By Doob decomposition (see [START_REF] Stewart | Markov processes[END_REF]) we can write

M ϕ(ε) f (t n ) = M art ϕ(ε) f (t n ) + O ϕ(ε) f (t n ) (A.6)
where M art

ϕ(ε) f (t n ) n is a F X ϕ(ε) n -martingale and O ϕ(ε) f (t n )
n is deterministic and tend to 0 as ε → 0.

Then, applying the Representation Theorem of Skorohod (see [START_REF] Billingsley | Probability and measure[END_REF]), there exist a probability space Ω, F, P and processes t → Xϕ(ε) (t), t ∈ [0, 2T ] 0<ε<ε * , and Xϕ (t) t∈[0,2T ] respectively of same distribution than t → X ϕ(ε) (t), t ∈ [0, 2T ] 0<ε<ε * and (X ϕ (t)) t∈[0,2T ] , there exists also τ ϕ stopping time for Xϕ with the same distribution than τ ϕ and such that sup t∈[0,2T ] Xϕ (t ∧ τ ϕ ) -Xϕ(ε) (t ∧ τ ϕ ) ---→ ε→0 0, P-a.s.

Lemma A.3. i) Let 0 ≤ t 1 < t 2 ≤ 2T , then for all f ∈ D * we have

lim ε→0 f Xϕ(ε) ⌊ t 2 ∧ τ ϕ ϕ(ε) 2 ⌋ϕ(ε) 2 -f Xϕ(ε) ⌊ t 1 ∧ τ ϕ ϕ(ε) 2 ⌋ϕ(ε) 2 - ⌊ t 2 ∧τ ϕ ϕ(ε) 2 ⌋-1 j=⌊ t 1 ∧τ ϕ ϕ(ε) 2 ⌋ A ϕ(ε) f Xϕ(ε) (t j ) = f Xϕ (t 2 ∧ τ ϕ ) -f Xϕ (t 1 ∧ τ ϕ ) - t2∧τ ϕ t1∧τ ϕ
Lf Xϕ (s) ds, P-a.s.

ii) The limit process M ϕ f (t) A ϕ(ε) f Xϕ(ε) (t j ) for all f ∈ D * .

t2∧τ ϕ t1∧τ ϕ Lf Xϕ (s) ds - ⌊ t 2 ∧τ ϕ ϕ(ε) 2 ⌋-1 j=⌊ t 1 ∧τ ϕ ϕ(ε) 2 ⌋ A ϕ(ε) f Xϕ(ε) (t j ) ≤ t2∧τ ϕ t1∧τ ϕ Lf Xϕ (s) ds -ϕ(ε) 2 ⌊ t 2 ∧τ ϕ ϕ(ε) 2 ⌋-1 j=⌊ t 1 ∧τ ϕ ϕ(ε) 2 ⌋ Lf Xϕ (t j ) +ϕ(ε) 2 ⌊ t 2 ∧τ ϕ ϕ(ε) 2 ⌋-1 j=⌊ t 1 ∧τ ϕ ϕ(ε) 2 ⌋ Lf Xϕ (t j ) - ⌊ t 2 ∧τ ϕ ϕ(ε) 2 ⌋-1 j=⌊ t 1 ∧τ ϕ ϕ(ε) 2 ⌋ Lf Xϕ(ε) (t j ) +ϕ(ε) 2 ⌊ t 2 ∧τ ϕ ϕ(ε) 2 ⌋-1 j=⌊ t 1 ∧τ ϕ ϕ(ε) 2 ⌋ Lf Xϕ(ε) (t j ) - 1 ϕ(ε) 2 ⌊ t 2 ∧τ ϕ ϕ(ε) 2 ⌋-1 j=⌊ t 1 ∧τ ϕ ϕ(ε) 2 ⌋ A ϕ(ε) f Xϕ(ε) (t j ) .
The first term tends to 0 when ε → 0 as an approximation of the integral. The second term tends to 0 as ε → 0 too because of the convergence almost sure of Xϕ(ε) to Xϕ . For the third term, we use [START_REF] Stroock | of Grundlehren der Mathematischen Wissenschaften[END_REF] and show that for each f ∈ D * , 1 ϕ(ε) 2 A ϕ(ε) f ---→ ε→0 Lf uniformly on the compact subsets of [η -1 , η].

ii) We have to show that for all 0 ≤ s < t ≤ 2T and all 0 ≤ s 1 < • • • < s d ≤ s, we have, for all

application φ ∈ C ∞ c R d , E M ϕ f (t ∧ τ ϕ ) -M ϕ f (s ∧ τ ϕ ) φ Xϕ (s 1 ∧ τ ϕ ), • • • , Xϕ (s d ∧ τ ϕ ) = 0.
For that we use point ii) of Lemma A.3, (A.6) and conclude with the Dominated convergence Theorem.

Then Theorem A.2 is obtain by returning to (Ω, F , P).

By Theorem A.2 the two stopping time τ := inf{t ∈ [0, T ]; R(t) ∈ (η -1 , η)} and τ ϕ have the same distribution, we'll write τ for both.

We end this step with the following corollary.

Corollary A.4. The family of processes (t → X ε (t ∧ τ ), t ∈ [0, 2T ]) 0<ε<ε * converge weakly to (R(t ∧ τ )) t∈[0,2T ] .

Proof. We have to check that for all φ : C ([0, 2T ] : R + ) → R continuous and bounded, lim ε→0 E φ X ϕ(ε) (t ∧ τ ) = E (R(t ∧ τ )) .

For that we use the tightness of t → X ϕ(ε) (t ∧ τ ), t ∈ [0, 2T ] 0<ε<ε * in a reductio ad absurdum.

Step 3: Suppression of the stopping times For the moment, we have the following weak convergence

(t → R ε (t ∧ τ ε ∧ τ ), t ∈ [0, 2T ]) 0<ε<ε * ⇀ (R(t ∧ τ )) t∈[0,2T ] .
In this last step, we want to delete all the stopping times in order to expand the convergence to the whole family (R ε ) ε . We will use the transience of the Bessel process for d > 2 (here d refers to the dimension of the Bessel process) and then deleting the stopping times remains to make η → +∞.

Proposition A.5. Let γ > 1 2 , then (t → R ε (t), t ∈ [0, T ]) 0<ε<ε * ⇀ (R(t)) t∈[0,T ] .

Proof. For d > 2 (which means γ > 1 2 ), the transience of the Bessel process (R(t)) t∈[0,2T ] yields that P (∀t ∈ [0, 2T ]; 0 < R(t) < +∞) = 1, and then, lim η→+∞ P (τ = 2T ) = 1. Let a decreasing subsequence ϕ : (0, ε * ) → (0, ε * ) such that τ ϕ(ε) 0<ε<ε * converge weakly to τ ϕ(0) (which is not a stopping time), it's easy to show that P τ ≤ τ ϕ(0) = 1 and then lim η→+∞ P τ ϕ(0) ≤ 2T = 1.

As τ ϕ(0) = lim ε→0 τ ϕ(ε) , we can deduce that for all p > 0, there exist η c ≫ 1 and ε(η c ) > 0 such that for all η > η c and ε < ε(η c ), we have P τ ϕ(ε) ≥ T > 1p. From that, we can deduce that for all φ : C ([0, T ] : R

+ ) → R E φ R ϕ(ε) (•) -φ (R(•)) ≤ 2||φ|| ∞ p + E φ R ϕ(ε) (• ∧ τ ϕ(ε) ∧ τ ) -φ (R(• ∧ τ )) .
But by Theorem A.2

E φ R ϕ(ε) (• ∧ τ ϕ(ε) ∧ τ ) -φ (R(• ∧ τ )) → ε→0 0.
Then, there exists η c ≫ 1 such that for all η > η c , for all φ : C ([0, T ] : R + ) → R,

lim p→0 lim ε→0 E φ R ϕ(ε) (•) -φ (R(•)) = 0. (A.7)
Now, assume that R ε doesn't converge weakly to R, then there exist φ : C ([0, T ] : R + ) → R and a decreasing subsequence ψ : (0, ε * ) → (0, ε * ) and there exists δ > 0 such that

E φ R ψ(ε) (•) -φ (R(•)) > δ.
As τ ψ(ε) is tight, it admits converging subsequences τ ϕ(ψ(ε)) and (A.7) implies that E φ R φ(ψ(ε)) (•)φ (R(•)) → ε→0 0, which is a contradiction and then ends the proof of both Proposition A.5 and Theorem 4.1.

Figure 2 :

 2 Figure 2: Illustration of (3.5) with different values of γ. Mean of 500 simulations of a Markov chain (ξ k ) k .

P

  (|X ε (t j ) -X ε (t j-1 )| > δ) = 0, (A.5) which assure by Theorem 1.4.11 of[START_REF] Stroock | of Grundlehren der Mathematischen Wissenschaften[END_REF], the precompactness of the family (t → X ε (t), t ∈ [0, T ]) 0<ε<ε * . This yields the existence of decreasing functions ϕ : (0, ε *

  .11) Here ∆||v n || 3 = ||v n+1 || 3 -||v n || 3 where (v n ) n is the stochastic process defined by (2.8) and O 0 ||v n || -1 designates a term of O ||v n || -1 with zero average. Introducing

  Xϕ is time-continuous as limit of Xϕ(ε) which are time-continuous. Then, for all f ∈ D * f Xϕ (t ∧ τ ϕ )f Xϕ(ε) ⌊

	Proof.	i) t ∧ τ ϕ ϕ(ε) 2 ⌋	≤||f ′ || ∞	Xϕ (t ∧ τ ) -Xϕ ⌊	t ∧ τ ϕ ϕ(ε) 2 ⌋
						+ Xϕ ⌊	t ∧ τ ϕ ϕ(ε) 2 ⌋ -Xϕ(ε) ⌊	t ∧ τ ϕ ϕ(ε) 2 ⌋
						---→ ε→0	0	P -a.s.
	We have, now to control	t2∧τ ϕ t1∧τ ϕ Lf Xϕ (s) ds -	⌊ j=⌊ t 2 ∧τ ϕ ϕ(ε) 2 ⌋-1 t 1 ∧τ ϕ ϕ(ε) 2 ⌋
						t	,
						M ϕ f (t) = f Xϕ (t ∧ τ ϕ ) -	t2∧τ ϕ t1∧τ ϕ	Lf Xϕ (s) ds
	is a F	t	Xϕ	-martingale.

We remark that the condition γ > 1 2 just appears in the last step, in order that the Bessel process (R(t)) t∈[0,T ] does not reaches 0 or explodes in finite time. If γ ≤ 1 2 then the Bessel process is reccurent and we are not able to delete the stopping times with this method.

In the proof of Theorem 3.1, we need some estimation of exit time for a transient Bessel process. In particular, we need to know the probability for a Bessel procces starting at 1 to reach 2 before 1 2 . Lemma 4.2 Let γ > -1 2 , and let R be a Bessel process of dimension 2γ + 1 with R(0

or also [START_REF] Stewart | Markov processes[END_REF].

(ii) This is readily shown using the Optional Stopping Theorem, as follows. Consider the process M (t) = R(t) 1-2γ . Note that 1 -2γ ≤ 0 for γ ≥ 1 2 but since for these values of γ the Bessel process is almost surely positive (see [START_REF] Revuz | Continuous martingales and Brownian motion, volume 293 of Grundlehren der Mathematischen Wissenschaften[END_REF]), M (t) is well defined. Considering

it is clear that P T a+ < T a-= P T M + < T M -. By the Ito lemma, it is easily checked this is a local martingale. It then follows from the Optional Stopping Theorem that

On the other hand,

From (A.9) and (A.10), we then obtain

which yields the desired result.


Marc Hässig

Yann Rolland

Marc Sosson

Riad Hassani

Gultekin Topuz

Ö Çelik

Muriel Gerbault

Ghazar Galoyan

Carla Muller

L Sahakyan

Rolland Yann

Ömer Faruk Çelik

Carla Müller

Ara Avagyan

The Armenian and NW Anatolian ophiolites: new insights for the closure of the Tethys domain and obduction onto the South Armenian Block and Anatolian-Tauride Platform before collision through dynamic modeling

published or not. The documents may come

In the Lesser Caucasus three main domains are distinguished from SW to NE: (1) the South Armenian Block (SAB), a Gondwanian-derived continental terrane; (2) scattered outcrops of ophiolites coming up against the Sevan-Akera suture zone; and (3) the Eurasian plate. The Armenian ophiolites represent remnants of an oceanic domain which disappeared during Eurasia-Arabia convergence. Previous works using geochemical whole-rock analyses, 40Ar/39Ar and paleontological dating have shown that the ophiolite outcrops throughout this area were emplaced during the Late Cretaceous as one non-metamorphic preserved ophiolitic nappe of back-arc origin that formed during Middle to Late Jurassic. From these works, tectonic reconstructions include two clearly identified subductions, one related to the Neotethys subduction beneath the Eurasian margin and another to intra-oceanic subduction responsible for the opening of the back-arc basin corresponding to the ophiolites of the Lesser Caucasus. The analysis of the two stages of metamorphism of the garnet amphibolites of the ophiolite obduction sole at Amasia (M1: HT-LP peak of P = 6-7 kbar and T > 630 • C; M2; MP-MT peak at P = 8-10 kbar and T = 600 • C) has allowed us to deduce the onset of subduction of the SAB at 90 Ma for this locality, which age coincides with other paleontological ages at the obduction front. A preliminary paleomagnetic survey has also brought quantification to the amount of oceanic domain which disappeared by subduction between the SAB and Eurasia before collision. We propose a dynamic finite element model using ADELI to test the incidence of parameters such as the density of the different domains (or the interval between the densities), closing speed (or speeds if sporadic), the importance and interactions of mantle discontinuities with the subducting lithosphere and set a lithospheric model. Our field observations and analyses are used to validate combinations of factors. The aim is to better qualify the predominant factors and quantify the conditions leading to the onset of obduction, the paradox of dense oceanic lithosphere emplaced on top of a continental domain, after subduction and prior to collision.

The results of this modeling are also compared to new observations of the assumed eastward extension of this ophiolitic nappe in NW Anatolia. Analyses of the Refahiye ophiolites show similar geochemical signatures as the Armenian ophiolites, due to a similar setting of formation (back-arc). The impact of the obduction of such a vast oceanic domain is not to be taken for granted when considering the following collision stage.


Marc Hässig

Yann Rolland
email: yrolland@unice.fr

Marc Sosson

Ghazar Galoyan

Lilit Sahakyan

Gultelin Topuz

Ö Çelik

Ara Avagyan

Carla Muller

Marc Hassig

Rolland Yann

Omer Farouk

Large scale obduction of preserved oceanic crust: linking the Lesser Caucasus and NE Anatolian ophiolites and implications for the formation of the Lesser Caucasus-Pontides Arc

come   L'archive ouverte pluridisciplinaire

During the Mesozoic, the Southern margin of the Eurasian continent was involved in the closure of the Paleotethys and opening Neotethys Ocean. Later, from the Jurassic to the Eocene, subductions, obductions, micro-plate accretions, and finally continent-continent collision occurred between Eurasia and Arabia, and resulted in the closure of Neotethys. In the Lesser Caucasus and NE Anatolia three main domains are distinguished from South to North: (1) the South Armenian Block (SAB) and the Tauride-Anatolide Platform (TAP), Gondwanian-derived continental terranes; (2) scattered outcrops of ophiolite bodies, coming up against the Sevan-Akera and Ankara-Erzincan suture zones; and

(3) the Eurasian plate, represented by the Eastern Pontides margin and the Somkheto-Karabagh Arc. The slivers of ophiolites are preserved non-metamorphic relics of the now disappeared Northern Neotethys oceanic domain overthrusting onto the continental South Armenian Block (SAB) as well as on the Tauride-Anatolide plateform from the north to the south. It is important to point out that the major part of this oceanic lithosphere disappeared by subduction under the Eurasian Margin to the north. In the Lesser Caucasus, works using geochemical whole-rock analyses, 40Ar/39Ar dating of basalts and gabbro amphiboles and paleontological dating have shown that the obducted oceanic domain originates from a back-arc setting formed throughout Middle Jurassic times. The comprehension of the geodynamic evolution of the Lesser Caucasus supports the presence of two north dipping subduction zones: (1) a subduction under the Eurasian margin and to the south by (2) an intra-oceanic subduction allowing the continental domain to subduct under the oceanic lithosphere, thus leading to ophiolite emplacement. To the West, the NE Anatolian ophiolites have been intensely studied with the aim to characterize the type of oceanic crust which they originated from. Geochemical analyses have shown similar rock types as in Armenia, Mid Ocean Ridge Basalt (MORB) to volcanic arc rocks and Intra-Plate Basalts (IPB). Lithostratigraphic comparisons have shown that the relations between the three units, well identified in the Lesser Caucasus, are similar to those found in NE Anatolia, including the emplacement of stratigraphically conform and discordant deposits. New field data has also shed light on an outcrop of low-grade metamorphic rocks of volcanic origin overthrusted by the ophiolites towards the south on the northern side of the Erzincan basin, along the North Anatolian Fault (NAF). We extend our model for the Lesser Caucasus to NE Anatolia and infer that the missing of the volcanic arc formed above the intra-plate subduction may be explained by its dragging under the obducting ophiolite with scaling by faulting and tectonic erosion. In this large scale model the blueschists of Stepanavan, the garnet amphibolites of Amasia and the metamorphic arc complex of Erzincan correspond to this missing volcanic arc. We propose that the ophiolites of these two zones originate from the same oceanic domain and were emplaced during the same obduction event. This reconstructed ophiolitic nappe represents a preserved non-metamorphic oceanic domain over-thrusting up to 200km of continental domain along more than 500km. Distal outcrops of this exceptional object were preserved from latter collision which was concentrated along the suture zones.


Nadjette Bengourna

Florence Despetis

Laurent Bonnet

Rémi Courson

Pierre Solignac

Hamid Satha

Nathalie Olivi-Tran
email: nathalie.olivi-tran@univ-montp2.fr

Textural, Structural and Electrical Characterizations of EMI MAc Silica Ionogels and Their Corresponding Aerogels

Keywords: Ionogels, structure and texture, conductivity 1

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Ionic liquids (IL) are molten salts which are liquid at a temperature lower than 100 o C.

Since the first IL synthesis, chemists and material scientists have been highly interested in this new type of conducting liquids because of their numerous properties and potential applications like green solvent, electrolytes, sensors, catalysts or batteries [Galinski et al. (2006)][ [START_REF] Rakita | Ionic Liquids as green solvents: Progress and Prospects[END_REF]][ Keskin et al. (2007)]. It is also possible to synthesize nano-composites materials by mixing IL with silica precursors, via a sol gel route. The first ionogel (for gel synthesized with an IL) was obtained in 2000 by S.Dai [Dai et al. (2000)]. Several investigations led to a variety of materials [Vioux et Asymmetric cation, N,Ndialkylimidazolium or 1-butyl-3-methylimidazolium associated with anions such as tetrafluoroborate, were the most commonly IL used for a few years [Gupta et al. (2012)] [Shi et al. (2005)]. Here we chose to work with another imidazolium cation, 1-ethyl-2-methylimidazolium bonded to the anion acetate (EM IM Ac). This IL exhibits a high electric conductivity (2500µS/cm) at ambient temperature [www.ilcochemie.de][ Francois (2006)].

The present study is focused on the synthesis and the characterization of EM IM Ac silica ionogels and their corresponding aerogels.

Textural and structural characterizations as nitrogen adsorption/desorption and SAXS were carried out on the aerogels. Hence, porous volume and texture were obtained together with the structure of the silica solid network.

Then, by a 4-wire-electrodes set up, we measured the temporal response of our ionogels to a voltage pulse and we modeled these responses by a RLC circuit. Up to now, only three studies on ionogels conductivity were performed [START_REF] Neouze | [END_REF]

EXPERIMENTAL SET-UP

Synthesis of the ionogels and aerogels

Chemicals

The ionic liquid: 1-Ethyl-3methylimidazolium acetate (EM IM Ac) of purity > 90%, and the silica precursors, tetramethoxysilane T M OS (98%) and methyltrimethoxysilane M T M S (98%), were from Aldrich. Technical grade ethanol was used to extract water and IL from the gel before the CO 2 supercritical drying. Because of the hydrophilicity of our IL, this last was heated under vacuum (70 o C, 10 mbar) for 48h before each synthesis. Gels were prepared via an hydrolytic one step process by mixing two solutions: solution (1) with T M OS and M T M S in different molar ratios and solution (2) containing IL and acidic water (HCl concentration equal to 0.65N ). The molar ratio nIL/nSi was set between 0.3 and 0.6 and for each sample, the hydrolysis molar ratio nwater/nSi was equal to 16. After stirring, the mixture was poured into the conductivity cell and in molds for further characterizations of the ionogels. Gelling occurred after a time which depended on the IL ratio and the M T M S ratio [START_REF] Brinker | Sol-gel Science Academic Press[END_REF]]. Gels were transparent disks with a diameter of 16mm and a height of 4 mm without any cracks. All these steps were done at 22±2 o C. The knowledge of the texture and the structure of ionogels imply their drying under conditions which allow to obtain dry materials exhibiting properties very close to these of the ionogels. For these reasons, drying must be carried out under supercritical conditions to avoid capillary stresses which would lead to dense and cracked materials. For the same reasons and to avoid the dissolution/redeposition phenomenon which occurs at high temperature and which can induce textural modifications at a very low scale [Quinson et al. (1992)], we chose to dry the gels using supercritical CO 2 so-called COLD process. The wet gels were aged during 12h. For CO 2 supercritical drying, the liquid that impregnates the wet gels must exhibit a total miscibility with CO 2 . So, the wet gels, which contained water and the IL, were first soaked into ethanol for 15 days to obtain gels with pores mainly filled by ethanol. Then the ethanol was exchanged with liquid CO 2 at 8 o C and 6 M P a during 5 hours. Finally, temperature and pressure were raised to 38 o C and 10 M P a and after 10min, the pressure was released slowly during 12h. Samples obtained were translucent (figure 1) without any cracks and exhibited a shrinkage which depended on the composition of the gel.

Characterization methods

Textural and structural characterizations of the aerogels

The bulk densities of the samples were evaluated from their weight and linear dimensions.

For the textural characterization, nitrogen adsorption/desorption isotherms at 77K were carried out on a Micromeritic ASAP2010. Samples were out-gassed under vacuum during 16h at 50C. The specific surface area was obtained using BET theory. Its accuracy was about 4%. For rigid porous materials, when the nitrogen relative pressure P/P 0 gets close to 0.99, the volume of adsorbed nitrogen must correspond to the porous volume [Reichenauer et al. (2001)]]. The dwell time, which is the interval of time allowing a pressure change of 0.01%, was set to 10 seconds. The pore size distributions were derived from BJH method from the desorption branches of the isotherm. SAXS (small angle X ray scattering) and WAXS (wide angle X ray scattering) experiments were performed with an in-house setup of the Laboratoire Charles Coulomb, using a X-ray tube GeniX3D from Xenocs and a Schneider 2D image-plate detector prototype, in order to estimate the fractal dimension of the materials.

Electrical conductivity measurements

These measurements were performed on ionogels using a set up described on figure 2.

The cylindrical cell allowing these measurements consisted of a Teflon mold containing the ionogel and of 4 Pt-electrodes partially immersed in the ionogel and passing through the sides of the mold. The 4 electrodes were P t wires with a section of 0.5 mm. These wires were immersed in the bulk of gels along 7mm and fixed to the mold to make sure that all the measurements were done in the same conditions. The mold was cylindrical with a depth of 5 mm and a diameter of 15 mm. The 4 wires were connected to a pulse generator (Aligent 33220A, 20MHz, Arbitrary waveform generator) and to an oscilloscope (TDS2022C, 200MHz, Tektronik).

A rectangular voltage pulse was applied between 2 (channel 1) of the 4 electrodes. Then the potential difference was measured between these 2 electrodes (channel 1) and between the 2 other ones (channel 2). First, we measured the electrochemical domain of stability of the EMIMAc. We found that for an applied potential equal to 4V and an offset equal to -1.375V , there was oxido-reduction of the IL at room temperature. We checked that for an applied potential equal to 2.5V and an offset equal to -1.25V , the IL was stable.

Straightforwardly, we chose to study the electrical temporal response of our ionogel samples at this voltage. The voltage pulses had an absolute value of 3.75V (applied potential 2.5V plus offset 1.25V ), a frequency of 50Hz and pulse durations of either 20ns or 100ns. This allowed us to measure the electrical response of the solvent (IL, HCl, water) contained in the porous network of the ionogels. Indeed, as the diameter of the electrodes was smaller than the depth of the set up (i.e. smaller than the height of the ionogel cylinder), these electrodes were always in contact with the IL contained in the pores. As we applied rectangular voltage pulses to the ionogel, we were able to analyze the temporal response of the sample.

In order to analyze the role of the silica network, a blank was first done on EM IM Ac + HCl solution. Then electrical voltage measurements were performed on the ionogel samples within 1 hour after gelling occurred.

RESULTS

Textural and structural Characterizations

The relative shrinkage which occurred during the different steps, from the gel to the aerogel, and the characteristics of the aerogels are presented in table II. Most of the samples exhibited low densities and high specific surface areas. The specific surface area showed a tendency to increase up to a maximum value with a higher ratio of nM T M S/nT M OS but it did not change with the IL concentration. A divergence between the measured porous volume V mp and the calculated one V cp was observed. As reported in the literature [Reichenauer et al. (2001)], when the density of the aerogel is low, it is more difficult to explore the whole porous volume by adsorption/desorption. We must also note that some IL remained inside the aerogels (∝ 5% in weight) as already described in the literature [Karout et al. (2009)].

Nitrogen adsorption isotherms are given in Figure 3 for three compositions. The pore size distributions, reported in Figures 4, are calculated from the isotherms. Aerogel made without M T M S displayed a larger pore size distribution. This distribution is increasingly narrow and shifted to lower diameters when the M T M S ratio is increasing (Figure 4

left).

However, distributions do not vary when the IL ratio increases (Figure 4

right).

All the ionogels synthesized are fractal ones even if the fractal domain is relatively small (one decade)(figures 5).The fractal dimension is enclosed between 2.2 and 2.5, values which are characteristic of acid catalyzed aerogels. Samples C,D and E exhibit the same behavior with a larger fractal domain than sample A or B, and a fractal value around 2.5.

Conductivity: electrical voltage characterization

The maximum of the measured amplitudes on both channels 1 and 2 are presented in table III for all the samples. Examples of electrical voltage temporal responses of the ionogels lead to figures 6 and 7.

• For an input pulse duration of 20ns, the maximum of the response on channel 1 for all the ionogel samples was close to the value of the input voltage, within experimental fluctuations • For an input pulse duration of 100ns, the maximum of the response on channel 1 for all the ionogel samples was larger than the input voltage

• For both durations, on channel 2, all the responses were damped.

• For the system EMIMAc + HCl solution, the maximum of the response on channel 1 was lower than the input pulse but with oscillations whatever the pulse duration. An electrical modeling of these responses is presented in section Discussion.

DISCUSSION

The presence of M T M S (hydrophobic groups) in the sol postpones the hydrolysis and condensation reactions and thus increases the gelation time [Schwertfeger et al. (1992)]. For samples C to E, the IL ratio increases and the associated pH increase enhances polycondensation reactions [Pope & Mackenzie (1986)] and thus the gelling time decreases [Schwertfeger et al. (1992)].

It is worth noting that the shrinkage after drying is rather low (< 10%)for all the investigated samples except sample A. For sample A, which is synthesized without M T M S, the soaking governs the shrinkage amplitude [START_REF] Brinker | Sol-gel Science Academic Press[END_REF]]. Capillary tensions induced by IL exchange are probably overcoming the strength of the silica network. For the other samples, the M T M S induces a repulsive wall/liquid effect (stable Si -CH 3 groups on the surface of SiO 2 particles) and thus reduces the capillary tensions [Schwertfeger et al. (1992)]. Consequently shrinkage decreases slightly as the increase of the MTMS content.

For samples A, B and C, the methyl modification obviously acts on the specific surface area which increases with the MTMS ratio. For the sample without M T M S, the fractal dimension is slightly smaller and the fractal domain is narrower. M T M S reduces crosslinkage and make the clusters grow slowly [Venkateswara et al. (2001)]. Regarding the aerogel texture, it is clear that while nM T M S/nT M OS value increases, the pore size distribution decreases towards the smaller pore sizes and becomes narrower [START_REF] Neouze | Ionogels obtenus par confinement dun liquide ionique dans une matrice base de silice[END_REF]].

Hence, EM IM Ac and M T M S induce noteworthy changings in the textures or structures of the aerogels, the effects on conductivities are shown below. Though the bulk density of samples C, D and E varies from 0.16 to 0.2 g.cm -3 depending on the IL ratio, their specific surface area and pore size distribution do not depend on the density. Their structures are very similar (Figure 5

right and table II).

The electrical voltage response of our ionogel samples is typical of a RLC series circuit.

We show here in figures 8 a few examples of RLC series circuits signals to which a rectangular voltage pulse is applied.

The shape of the response which is taken at the capacity component (C) in the simulation depends on the duration of the voltage pulse: if the duration of the input pulse is too short, the sample is in the electrical voltage transient state, this is the case for pulse durations equal to 20ns in the experiments and 0.1s in the simulations. For longer pulse durations (here 100ns in the experiments and 0.4s in the simulations), we are in the permanent regime.

In the case of responses measured on channel 2 (see section Results), the damping is due to the direction of the applied potential (perpendicular to channel 2).

What is worth noting, is the difference between the responses of the ionogels samples and that of the HCl+IL solution: the highest amplitudes of the ionogels electrical voltage responses on channel 1 are always larger than the responses of the HCl+IL solution. For the ionogels synthesized with M T M S, channel 2 shows maxima which are larger than the HCL+IL solution; but for sample A (without M T M S), channel 2 is lower than the blank.

Finally, we characterized the texture and the structure of EM IM Ac silica ionogels as well as their temporal conductivity. We showed that depending on the amount of M T M S and EM IM Ac, the structures, textures and electrical voltage responses were different.

CONCLUSION

We synthesized ionogels from T M OS, M T M S and EM IM Ac. Aerogels were obtained from supercritical drying of our ionogels. The texture and the structure were performed on these aerogels by Nitrogen adsorption/desorption and SAXS. An electrical characterization was performed on the ionogels with a 4-wire-electrodes set-up. This characterization allowed us to model the ionogels by a RLC series circuit. The texture and structure of the aerogels and the electrical voltage characteristics of the ionogels all depended on the different amounts of T M OS, M T M S and EM IM Ac used in the synthesis.

  al (2010)][Neouze et al. (2005)][Karout et al. (2007)][Klingshirn et al. (2005)] implying numerous characterizations: porous volume, fractal dimension, conductivity of the ionogel bulk measured with two plate electrodes [Gupta et al. (2012)[Neouze et al. (2006)][Karout et al. (2009)][Zhang et al. (2010)][Zhou et al. (2004)][Karout et al. (2009)] .

  ][Neouze et al. (2005)][Goeury et al. (2001)] and none of them gave the electrical temporal response of ionogels.

•

  Samples A, B, C, D, E have the same response on channel 1 within experimental variations • Samples B,C, D, E have the same response on channel 2 within experimental variations • Sample A has a response on channel 2 which is damped with respect to samples B, C, D, E and EMIMAc + HCl solution.

FIG. 1 .FIG. 2 . 12 FIG. 3 .FIG. 4 .

 121234 FIG. 1. Photographs of synthesized aerogels, left: sample (A) ; right: sample (C)

13 FIG. 5 . 14 FIG. 7 .

 135147 FIG. 5. SAXS/WAXS curves. left: Effect of the MTMS ratio; right: Effect of IL ratio

TABLE II .

 II Sample n IL /n Si molar ratio n M T M S /n T M OS Shrinkage observed after each step of the drying and characteristics of the aerogels. * V cp is calculated from the bulk density and the skeletal density of silica which has been estimated in the literature to about 2 g.cm -3 in silica aerogels[Ayral et al. (1992)]

			Sample			20ns		Sol pH Gelation time 100ns
					Channel 1 Channel 2 Channel 1 Channel 2
			A	A	0.3 0 3.56	0.13	4.5	4.08	25 min 0.29
			B	B	0.3 0.2 3.58	0.68	4.5	4.08	35 min 1.54
			C	C	0.3 0.4 3.56	0.67	4.5	4	2h40 1.52
			D	D	0.45 0.4 3.52	0.58	5	3.88	1h40 1.44
			E	E	0.6 0.4 3.52	0.536	5.5	3.84	1h	1.38
		TABLE I. Compositions and gelation times for the studied samples EMIMAc+HCl 3.46 0.36 3.56 0.92
	Sample Shrinkage	Shrinkage	Bulk	den-	Calculated *	Desorption	V mp /V cp (%) Specific sur-
		after soaking	after drying	sity ±0.002	pore	vol-	BJH	pore	face	area
		±0.1 (%)	±0.1 (%)	(g.cm -3 )	ume V cp ±	volume	(m 2 .g -1 )
								0.1(cm 3 .g)	V mp	±
										0.1(cm 3 .g)
	A	13.6	14.3		0.288		3.0		2.82	95	640 ± 25
	B	3	8.1			0.192		4.7		3.14	67	885 ±35
	C	6	9.3			0.198		4.5		2.94	65	1090 ±40
	D	6.6	7.8			0.173		5.3		3.29	62	1040 ±40
	E	6.4	7			0.167		5.5		3.22	59	1050 ±40
							10

TABLE III .

 III Voltage maxima at different conditions (in V )