Gabriel Faraud
email: faraud@math.univ-paris13.fr

Estimates on the speedup and slowdown for a diffusion in a drifted brownian potential

 . Our results are in agreement with their results in the discrete case.

1 Introduction.

Process in random media have been introduced in order to study physical or biological mechanisms such as the replication of DNA. The first model, in discrete time, goes back to A. Chernov [START_REF] Chernov | Replication of a multicomponent chain, by the lightning mechanism[END_REF] and D. Temkin [START_REF] Temkin | One-dimensional random walks in a two-component chain[END_REF]. It is now well understood : see, for example [START_REF] Solomon | Random walks in a random environment[END_REF], [START_REF] Sinai | The limiting behavior of a one-dimensional random walk in a random medium[END_REF], or [START_REF] Kesten | A limit law for random walk in a random environment[END_REF]. A continuous time version of this process has been introduced by S. Schumacher [START_REF] Schumacher | Diffusions with random coefficients[END_REF], and studied by T. Brox [START_REF] Brox | A one-dimensionnal diffusion process in a Wiener medium[END_REF]. It can be described as follows.

Let (W (x)) x∈R be a one-dimensional brownian motion defined on R starting from 0, and, for κ ∈ R,

W κ (x) := W (x) - κ 2 x.
Let (β(t)) t≥0 be another one-dimensional brownian motion, independent of W . We call diffusion process with potential W κ a solution to the (formal) equation

dX t = dβ t - 1 2 W ′ κ (X t )dt. (1.1)
W ′ κ has clearly no rigorous meaning, but a mathematical definition of (1.1) can be given in terms of the infinitesimal generator. For a given realization of W κ , X t is a real-valued diffusion started at 0 with generator 1 2

e Wκ(x) d dx e -Wκ(x) d dx .

This diffusion can also be defined by a time-change representation :

X t = A -1 κ B(T -1 κ (t)) ,
where A κ (x) =

x 0 e W κ(y) dy, T κ (t) = t 0 e -2Wκ(A -1 κ (B(s))) ds, and B is a standard Brownian motion. A κ is the scale function of this process, and its speed measure is 2e -Wκ(x) dx.

Intuitively, for a given environment W κ , the diffusion X t will tend to go to places where W κ is low, and to spend a lot of time in the "valleys" of W κ . If the environment is drifted (κ > 0), the process will be transient to the right, but it will be slowed by those valleys (see figure 1). This will be explained more precisely in section 3.

For general background on diffusion processes and time-change representation we refer to [START_REF] Rogers | Diffusions, Markov processes and Martingales[END_REF][START_REF] Revuz | Continuous Martingales ans Brownian Motion[END_REF][START_REF] Itō | Diffusion processes and their sample paths[END_REF].

We will call P the probability associated to W , P W the quenched probability associated to the diffusion, and P := P ⊗ P W the annealed probability. T. Brox gave a result concerning the long time behavior of the diffusion in the case κ = 0. Namely, under the probability P,

X t (log t) 2 → U,
where U follows an explicit distribution.

The case κ > 0 was studied both by K. Kawazu and H. Tanaka ( [START_REF] Kawazu | A diffusion process in a brownian random environment with drift[END_REF]) and Y. Hu, Z. Shi, M. Yor ( [START_REF] Hu | Rates of convergence of diffusions with drifted brownian potentials[END_REF]) and exhibits a "Kesten-Kozlov-Spitzer" behavior: when κ > 1, the diffusion has a positive speed; when κ = 1, under P,

X t log t t → 4
in probability, while, when 0 < κ < 1,

X t t κ → V
in distribution, where V follows the inverse of a completely asymmetric stable law. We are interested in the deviations between X t and its asymptotic behavior, in the case 0 < κ < 1.

This questions have already been studied in the other cases, we refer to [START_REF] Hu | Moderate deviations for diffusions with brownian potentials[END_REF] for estimates in the case κ = 0, and to [START_REF] Talet | Annealed tail estimates for a brownian motion in a drifted brownian potential[END_REF] for large deviation estimates in the case κ > 1.

Our study will split into four different problems, indeed the quenched and annealed settings present different behavior, and for each of them we have to consider deviations above the asymptotic behavior (or speedup) and deviations under the asymptotic behavior (or slowdown).

We start with the annealed results. For u and v two functions of t, we note u ≫ v if u/v → t→∞ ∞.

Theorem 1.1 (Annealed speedup/slowdown) Suppose 0 < κ < 1, and u → ∞ is a function of t such that for some ε > 0, u ≪ t1-κ-ε , then there exist two positive constants C 1 and C 2 such that lim t→∞ log P (X t > t κ u) u

1 1-κ = C 1 , (1.2)
and if log u ≪ t κ , lim t→∞ uP X t < t κ u = C 2 .

(1.3)

Furthermore, the results remain true if we replace X t by sup s<t X s .

This is in fact a easy consequence of the study of the hitting time of a certain level by the diffusion. We set H(v) = inf{t > 0 : X t = v}. We have the following estimates.

Theorem 1.2 Suppose 0 < κ < 1 and ε > 0. For u → ∞ v → ∞ two functions of t such that for some ε > 0, u ≪ v 1-κ-ε , there exist two positive constants C 1 and C 2 such that

lim t→∞ -log P H(v) < v u 1/κ u 1 1-κ = C 1 , (1.4
)

and if log u ≪ v, lim t→∞ uP H(v) > (vu) 1/κ = C 2 . (1.5)
The proof of this result involves a representation of H(v) introduced in [START_REF] Hu | Moderate deviations for diffusions with brownian potentials[END_REF].

We now turn to the quenched setting. We have the following estimates for the speedup Theorem 1.3 (Quenched speedup) Suppose 0 < κ < 1, and u → ∞ is a function of t such that for some ε > 0, u ≪ t 1-κ-ε , then there exists a positive constants C 3 such that lim t→∞ log P W (X t > t κ u) u 1 1-κ = C 3 , Pa.s.. Furthermore the result remains true if we replace X t by sup s≤t X s .

As before, the proof of this will reduce to estimates on the hitting times.

Theorem 1.4 For u → ∞ v → ∞ two functions of t such that for some ε > 0, u ≪ v 1-κ-ε , then For the slowdown, our result is less precise.

Theorem 1.5 (Quenched slowdown) Suppose κ > 0. Let ν ∈ (0, 1 ∧ κ), then

lim t→∞ log(-log P W [H(t ν ) > t]) log t = 1 - ν κ ∧ κ κ + 1
, Pa.s.,

lim t→∞ log(-log P W [X t < t ν ]) log t = 1 - ν κ ∧ κ κ + 1 , P -a.s.. (1.7)
Corresponding results for Random Walk in Random Environment have been developed in a recent article from A. Fribergh, N. Gantert and S. Popov [START_REF] Fribergh | On the slowdown and speedup of transient random walks in random environment[END_REF]. Our proof of the last result is quite inspired from theirs.

The article will be organized as follows :

• In Section 2 we show Theorem 1.1 and 1.2,

• In Section 3 we show Theorem 1.5,

• In Section 4 we show Theorem 1.3 and 1.4.

2 The annealed estimate.

For any nondecreasing function u(t), we will denote by u -1 (t) := inf{v : u(v) > t} the inverse function of u. We start with some preliminary statements.

Preliminary statements.

We first recall the Ray-Knight Theorems, they can be found in chapter XI of [START_REF] Revuz | Continuous Martingales ans Brownian Motion[END_REF]. Let L x t be the local time at x before t of a brownian motion γ t , and τ t := L 0 .

-1 (t) the inverse function of L 0 t . Let σ(x) be the first hitting time of x by γ t .

Statement 2.1 (First Ray-Knight Theorem) The process {L a-t σ(a) } t≥0 is a squared Bessel process, started at 0, of dimension 2 for 0 ≤ t ≤ a and of dimension 0 for t ≥ a.

Statement 2.2 (Second Ray

-Knight Theorem) Let u ∈ R + , The pro- cess {L t τ (u) } t≥0 is a squared Bessel process of dimension 0, starting from u.
We have a useful representation of H(v), due to Y. [START_REF] Hu | Moderate deviations for diffusions with brownian potentials[END_REF]. Let

θ 1 (v) = H(v) 0 1 {Xs≥0} ds, and θ 2 (v) = H(v) 0 1 {Xs<0} ds, such that H(v) = θ 1 (v) + θ 2 (v).
Statement 2.3 Let κ ≥ 0 and v > 0. Under P, we have

(θ 1 (v), θ 2 (v)) law = 4 v 0 e Ξκ(s) -1 ds, 16Υ 2-2κ e Ξκ(v)/2
1 .

Where Υ 2-2κ (x y) denotes the first hitting time of y by a Bessel process of dimension (2-2κ) starting from x, independent of the diffusion Ξ κ , which is the unique nonnegative solution of

Ξ κ (t) = t 0 1 -e -Ξκ(s) dβ ′ s + t 0 - κ 2 + 1 + κ 2 e -Ξκ(s) ds, t ≥ 0. (2.1)
β ′ being a standard brownian motion.

We shall use the following lemma from [START_REF] Talet | Annealed tail estimates for a brownian motion in a drifted brownian potential[END_REF](Lemma 3.1).

Statement 2.4 Let {R t } t≥0 denote a squared Bessel process of dimension 0 started at 1. For all v, δ > 0, we have

P sup 0≤t≤v |R t -1| > δ ≤ 4 (1 + δ)v δ exp - δ 2 8(1 + δ)v .
We now turn to the proof of Theorem 1.2.

Proof of Theorem 1.2.

Our proof will be separated in two parts : in the first part we will deal with the positive part of H(v), θ 1 , then we will focus on θ 2 .

The positive part.

In view of statement 2.3, we set

Z t := e Ξκ(t) -1,
then Z t is the unique nonnegative solution of

dZ t = Z t (1 + Z t )dβ t + 1 -κ 2 Z t + 1 2 dt,
and

θ 1 (v) = 4 v 0 Z t dt.
We call

f (z) = z 1 (1 + s) κ s ds (2.2)
the scale function of Z t . We have

f (Z t ) = t 0 (1 + Z s ) κ+ 1 2 √ Z s dβ s .
By the Dubbins-Schwartz representation (see chapter V, Theorem (1.6) of [START_REF] Revuz | Continuous Martingales ans Brownian Motion[END_REF]), there exists a standard Brownian motion γ(t) such that

f (Z t ) = γ t 0 (1 + Z s ) 2κ+1 Z s ds := γ(ρ(t)). (2.3)
We introduce

α t = ρ(t) -1 = t 0 Z αs (1 + Z αs ) 1+2κ ds = t 0 f -1 (γ s ) [1 + f -1 (γ s )] 1+2κ ds := t 0 h(γ s )ds. (2.4)
We obtain easily the following equivalents

f (z) ∼ z→∞ z κ /κ, f (z) ∼ z→0 log z, f -1 (z) ∼ z→∞ (κz) 1/κ , f -1 (z) ∼ z→-∞ e z , h(z) ∼ z→∞ (κz) -2 , h(z) ∼ z→-∞ e z .
We continue with a lemma, whose proof is postponed. Let τ t be the inverse local time of γ.

Lemma 2.1 Let ǫ > 0, c h := ∞ 0 h(x)dx. Let w(t) → ∞, such that w(t)/t → 0. Then for t large enough, P ρ(t) > τ t (1-3ǫ)c h ≤ exp (-w) , and
P ρ(t) < τ t (1+3ǫ)c h ≤ exp (-w) . Let ṽ ≪ v, in view of (2.3), θ 1 (v) = 4 v 0 f -1 (γ ρ(s) )ds = 4 ρ(v) 0 f -1 (γ s ) 2 [1 + f -1 (γ s )] 1+2κ ds := 4 ρ(v) 0 g(γ s )ds.
(2.5) Using lemma 2.1, with probability at least 1e -ṽ ,

τ " v (1+3ǫ)c h « 0 g(γ s )ds ≤ θ 1 (v) 4 ≤ τ " v (1-3ǫ)c h « 0 g(γ s )ds.
(2.6)

One can easily check that g(x) ∼ ∞ (κx) 1 κ -2 , and g(x) ∼ -∞ e 2x . In view of this it is clear that the most important part of the preceding integral will come from the high values of γ u . To be precise, for w ∈

v (1+3ǫ)c h , v (1-3ǫ)c h
and some large constant A, we have

τw 0 g(γ s )1 γs<A ds = A -∞ g(s)L s τw ds law = w 2 A/w -∞ g(sw)L s τ 1 ds = w 2 -A log(w) 5 /w -∞ g(sw)L s τ 1 ds + w 2 A/w -A log(w) 5 /w g(sw)L s τ 1 ds := J 1 + J 2 .
(2.7)

Using statement 2.4, for some constant C > 0, P(J 2 > w log(w) 5 ) < Ce -w . Recalling that, under the assumption of theorem 1.2, v ≪ v u 1/κ , we get that, for any δ > 0, as t → ∞,

P J 2 > δ v u 1/κ ≤ Ce - v (log v) 10 .
We postpone the proof of the following Lemma 2.2 for every δ > 0, as t → ∞,

P J 1 > δ v u 1/κ ≤ Ce - v (log v) 10 .
As a consequence, for every δ > 0, as t → ∞,

P τw 0 g(γ s )1 γs<A ds > 2δ v u 1/κ ≤ Ce - v (log v) 10 . (2.8)
It remains to deal with τw 0 g(γ s )1 γs>A ds. Due to the equivalent of g, for every ǫ > 0, for A large enough

(1 -ǫ) τw 0 (γ s ) 1/κ-2 1 γs>0 ds -I ′ ≤ τw 0 g(γ s )1 γs>A ds ≤ (1 + ǫ) τw 0 (γ s ) 1/κ-2 1 γs>0 ds, (2.9)
where

I ′ := τw 0 γ 1/κ-2 u 1 γu<A du law = w 1/κ A/w 0 y 1/κ-2 L y τ (1)
dy by the same computations as above. Using statement 2.4, for some constant C ′ > 0, with probability at least (1e -C ′ v ), L y τ (1) is lesser than, say, 100 on [0, A/w]. Therefore

I ′ ≤ 100w 1/κ A/w 0 y 1/κ-2 dy < 1000A 1/κ-1 w.
(2.10)

By the same proof as on page 218 of [START_REF] Jeulin | Sur les distributions de certaines fonctionnelles du mouvement brownien[END_REF], the process

U s = τs 0 (γ u ) 1/κ-2 1 γs>0 du
is an asymmetric κ-stable subordinator, more precisely

E exp - λ 2 U s = exp (-sc κ λ κ ),
where c κ = π 2κ sin (πκ)

κ κ Γ(κ) 2
. From a result of de Bruijn (see p 221 of [START_REF] Bertoin | Lévy processes[END_REF]), there exists a constant C 0 such that

log P U s s 1/κ < 1 u 1/κ = log P U 1 < 1 u 1/κ ∼ ∞ -C 0 u 1 1-κ . (2.11)
Similarly, by standard estimates on stable laws, for u → ∞, there exists a constant C ′ 0 such that

P U s s 1/κ > u 1/κ ∼ ∞ C ′ 0 u .
(2.12)

This, together with (2.6), (2.8), (2.9) and (2.10), implies that, for u → ∞, u ≪ v 1-κ there exists positive constants C 1 and C 2 such that, lim t→∞

-log P θ 1 (v) < v u 1/κ u 1 1-κ = C 1 and for u ≪ e v , lim t→∞ uP θ 1 (v) > (vu) 1/κ = C 2 ,
where

C 1 = 4 κ 1-κ C 0 c 1 1-κ h and C 2 = 4 κ C ′ 0 c h .

The negative part.

To finish the proof of Theorem 1.2, we need to deal with θ 2 . Note that for ε > 0,

P H v < v u 1/κ ≤ P θ 1 (v) < v u 1/κ , (2.13)
hence the lower bound in (1.4) is direct. We now turn to the upper bound. We recall that u and v are two functions of t such that u ≪ v 1-κ-ǫ . This implies in particular that u ≪ v. Note that

P θ 1 (v) < (1 -ε) v u 1/κ , θ 2 (v) < ε v u 1/κ ≤ P H(v) < v u 1/κ .
(2.14) Using statement 2.3, we obtain

P θ 1 (v) < (1 -ε) v u 1/κ , θ 2 (v) < ε v u 1/κ = P Υ 2-2κ e Ξκ(v)/2 1 < ε v u 1/κ , θ 1 (v) < (1 -ε) v u 1/κ . (2.15)
By a scaling argument, we get, for a ≥ 1

P Υ 2-2κ √ a 1 < a = P Υ 2-2κ 1 1 √ a < 1 ≥ C > 0. (2.16)
We recall from section 2.2.1 the representation e Ξκ(v) -1 = f -1 (γ(ρ(t))).

Let 0 < ǫ < ε/1000, and δ < ε/3 we call A the event that the condition of lemma 2.1 is fullfilled, that is

A = τ v/(1+3ǫ)c h < ρ(v) < τ v/(1-3ǫ)c h ,
Set ǫ ′ ≤ (εκ) 1/κ /2, we introduce the event

B := sup τ v/(1+3ǫ)c h <s<τ v/(1-3ǫ)c h γ(s) < ǫ ′ v u .
Formula 4.1.2 page 185 of [START_REF] Borodin | Handbook of Brownian Motion. Probability and its application[END_REF] (and the Markov property) implies

P [B] ≥ e -ε ′ u
for some positive ε ′ . We recall from section 2.2.1 the representation

e Ξκ(v) -1 = f -1 (γ(ρ(t))),
where f -1 is an increasing function such that f -1 (z) ∼ ∞ z κ /κ. Therefore for t large enough, on B ∩ A, e Ξκ(v) < ε v u

P H(v) < v u 1/κ ≥ P(B)P Υ 2-2κ e Ξκ(v)/2 1 < ε v u 1/κ , θ 1 (v) < (1 -ε) v u 1/κ |B ≥ P(B)P Υ 2-2κ ε v u 1/κ 1 < ε v u 1/κ , θ 1 (v) < (1 -ε) v u 1/κ |B -P(B)P(A c |B).
Recalling lemma 2.1 we get

P(B)P(A c |B) < e -v log v .
On the other hand, Υ 2-2κ

ε v u 1/κ
1 is independent of B and θ 1 , and

P Υ 2-2κ ε v u 1/κ 1 < ε v u 1/κ > C
by (2.16); therefore the upper bound in (1.4) will follow as soon as we show that lim t→∞

-log P θ 1 (v) < (1 -ε) v u 1/κ |B u 1 1-κ ≤ C 1 + µ(ε),
where µ(ε) → 0 as ε → 0. We recall from equation (2.5) that

g(x) = f -1 (γ s ) 2 [1 + f -1 (γ s )] 1+2κ ,
where f has been defined in (2.2). We now recall from equation (2.6) that, on

A θ 1 (v) 4 ≤ τ " v (1-3ǫ)c h « 0 g(γ s )1 γs>0 ds, therefore P θ 1 (v) < (1 -ε) v u 1/κ |B ≥ P τ " v (1-3ǫ)c h « 0 g(γ s )ds < (1 -ε) v u 1/κ |B -P[A c |B].
Once again, P[A c |B] is easily bounded. On the other hand, by Ito's brownian excursion theory (see for example chapter XII of [START_REF] Revuz | Continuous Martingales ans Brownian Motion[END_REF]), for every l ∈ R, γ(τ l + t) is a brownian motion started at 0, independent of (γ(t)) t≤τ l . Therefore

P τ " v (1-3ǫ)c h « 0 g(γ s )ds < (1 -ε) v u 1/κ |B ≥ P τ " v (1+3ǫ)c h « 0 g(γ s )ds < (1 -2ε) v u 1/κ |B P    τ " v (1-3ǫ)c h « τ " v (1+3ǫ)c h « g(γ s )ds < ε v u 1/κ |B    .
The event in the first probability on the right hand side is independent from B, therefore the conditionnal expectation is equal to the expectation and we can apply the results of section 2.2.1 to get

-log P τ " v (1+3ǫ)c h « 0 g(γ s )ds < (1 -2ε) v u 1/κ |B ≤ (C 1 +µ 1 (ε)+o(1))u 1 1-κ .
On the other hand, using the Markov property,

P    τ " v (1-3ǫ)c h « τ " v (1+3ǫ)c h « g(γ s )ds < ε v u 1/κ |B    = P τ δv 0 g(γ s )ds < ε v u 1/κ | sup 0<t<τ δv γ s < ǫ ′ v u , where δ = 1 ((1-3ǫ)c h ) - 1 ((1-3ǫ)c h )
. Note that, as the positive and negative excursions are independent, τ δv 0 g -(γ s )ds and B are independent, therefore we only need to bound

P τ δv 0 g(γ s )ds < ε 2 v u 1/κ | sup 0<t<τ δv γ s < ǫ ′ v u = P ∞ 0 g(x)L x τ δv dx < ε 2 v u 1/κ |L α τ δv = 0 .
where α = ǫ ′ v u . Intuitively, it seems clear that ∞ 0 g(x)L x τ δv dx will have better chances to be small if L α τ δv = 0, we are going to give a rigorous proof of that. Note that, using the second Ray-Knight theorem (Statement 2.2), L x τ δv is a squared Bessel process of dimension 0 starting from δv. On the other hand, under P[•|L α τ δv = 0], L x τ δv is a squared Bessel bridge of dimension 0 between δv and 0 over time α (we refer to section XI of [START_REF] Revuz | Continuous Martingales ans Brownian Motion[END_REF] for the definition and properties of the Bessel bridge).

We are going to use Girsanov's theorem in order to compute the equation solved by the squared Bessel bridge of dimension 0. Let P x and P α x,0 be respectively the distributions of the Bessel process of dimension 0 started at x and the distribution of the Bessel bridge of dimension 0 between x and 0 over time α. Let E x and E α x,0 be the associated expectations. Let X t be the canonical process and F t its canonical filtration.

Using the Markov property, we get, for every F t -measurable function F ,

E α x,0 [F (X s , s ≤ t)] = E x [F (X s , s ≤ t), X α = 0] P x [X α = 0] = E x F (X s , s ≤ t) P Xt [X (α-t) = 0] P x [X α = 0] := E x [F (X s , s ≤ t)h(X t , t)] ;
where h(s, t) can be explicitely computed (see for example Corollary XI.1.4 of [START_REF] Revuz | Continuous Martingales ans Brownian Motion[END_REF]). We get

h(X t , t) = exp x 2α - X t 2(α -t)
.

Using Ito's Formula, we can transform this expression to get

h(X t , t) = exp - t 0 1 2(α -s) dX s + t 0 X s 2(α -s) 2 ds .
Recalling that, under P x , X t is a solution to

dX t = 2 X t dβ t ,
where β is a Brownian motion, we get

h(X t , t) = exp - t 0 √ X s (α -s) dβ s + t 0 X s 2(α -s) 2 ds .
Therefore, thanks to Girsanov's theorem (see for example Theorem VIII.1.7 of [START_REF] Revuz | Continuous Martingales ans Brownian Motion[END_REF]), under P α x,0 ,

X t = x + t 0 X s dβ s -2 t 0 X s (α -s) ds.
Coming back to our original problem, we obtain that, under P(•|L α τ δv = 0), L x τ δv is a solution to

X t = δv + t 0 X s dβ s -2 t 0 X s (α -s) ds.
while, under P, L x τ δv is a solution to

X t = δv + t 0 X s dβ s .
Therefore, as there is pathwise uniqueness for these equation (see for example Theorem IX.3.5 of [START_REF] Revuz | Continuous Martingales ans Brownian Motion[END_REF]), the comparison theorem (see [START_REF] Watanabe | A comparison theorem for solutions of stochastic differential equations and its applications[END_REF]) allows us to construct a couple (X (1) , X (2) ) such that X (1) follows the same distribution as L x τ δv under P, X (2) follows the same distribution as L x τ δv under P(•|L α τ δv = 0) and X (1) ≥ X (2) almost surely. Then one gets easily that the distribution of

∞ 0 g(x)L x τ δv dx under P(•|L α τ δv = 0
) is dominated by its distribution under P. Then the upper bound in (1.4) follows easily by the results of section 2.2.1.

We now turn to the proof of (1.5). We have the trivial inequality

P[θ 1 (v) > (vu) 1/κ ] ≤ P[θ 1 (v) + θ 2 (v) > (vu) 1/κ ] ≤ P[θ 1 (v) > (1 -ε)(vu) 1/κ ] + P[θ 2 (v) > ε(vu) 1/κ ],
therefore the lower bound is direct. To get the upper bound, note that θ 2 (v) is increasing, so we have to show that for every ε > 0, and some s > 0,

P Υ 2-2κ e Ξκ(v+s)/2 1 > ε (vu) 1/κ = o 1 u .
Recalling the diffusion Z t from the last part, we need to bound

P Υ 2-2κ Z v+s + 1 1 > ε (vu) 1/κ = ∞ 0 P Υ 2-2κ √ z + 1 1 > ε (vu) 1/κ dµ v+s (z), (2.17)
where µ v (y) is the distribution of Z v . By scaling,

P Υ 2-2κ √ z + 1 1 > ε (vu) 1/κ = P Υ 2-2κ 1 1 √ z + 1 > ε (vu) 1/κ z + 1 ≤ P Υ 2-2κ (1 0) > ε (vu) 1/κ z + 1 .
It is known (see for example [START_REF] Werner | Some remarks on perturbed reflecting brownian motion[END_REF] page 40) that Υ 2-2κ (1 0) has the same distribution as 1 2Γ where Γ follows a distribution Γ(κ, 1), therefore, easy computations leads to

P Υ 2-2κ (1 0) > ε (vu) 1/κ z + 1 ≤ 1 κΓ(κ)(2ǫ) κ (1 + z) κ uv ∧ 1.
Recalling (2.17), we have, for all A > 0

P θ 2 > ε(uv) 1/κ ≤ A 0 1 κΓ(κ)(2ǫ) κ (1 + z) κ uv dµ v+s (z) + ∞ A dµ v+s (z)
Using for example exercise VII.3.20 of [START_REF] Revuz | Continuous Martingales ans Brownian Motion[END_REF], the diffusion Z t has speed measure dm(z) = 2 (1+z) 1+κ dz, so by Theorem 54.4 of [START_REF] Rogers | Diffusions, Markov processes and Martingales[END_REF] and a change of variable in order to lift the natural scale assumption, for any φ bounded and measurable,

∞ 0 φ(z)dµ v+s (z) → s→∞ ∞ 0 φ(z)π(dz), with π(dz) = m(dz)
2κ . Therefore as s goes to infinity, and for some finite constant c(ε),

P θ 2 > ε(uv) 1/κ ≤ c(ε) uv log(1 + A) + (1 + A) -κ .
Now, taking A such that (1 + A) ≫ u 1/κ and log(1 + A) ≪ v (this is possible due to the assumptions on u and v), we get the upper bound in (1.5).

Proof of Theorem 1.1.

In this section we use the results for the hitting times to get the results for the diffusion itself. We begin with the proof of (1.2). We have the trivial inequality

P (X t > t κ u) ≤ P [H(t κ u) < t] ; by taking v = t κ u in Theorem 1.2, we get the upper bound in (1.2). The condition u ≪ v 1-κ becomes u ≪ t 1-κ .
To get the lower bound, note that, for every ε > 0,

log P (X t > t κ u) ≥ log [P [H((1 + ε)t κ u) < t] P (X t > t κ u|H((1 + ε)t κ u) < t)] ≥ -C 1 ((1 + ε)u) 1 1-κ + log P (X t > t κ u|H((1 + ε)t κ u) < t) . (2.18)
The bound in the first term coming from (1.4). To treat the second term, note that

P (X t < t κ u|H((1 + ε)t κ u) < t) ≤ E P (1+ε)t κ u W inf s>0 X s < t κ u = P inf s>0 X s < -εt κ u ,
by invariance of the environment. By [12],

P inf t>0 X t < -u ≤ Cx -3/2 exp -(κ/2) 2 x/2 ,
(note that c in K. Kawazu and H. Tanaka's article corresponds to -κ/2 in our setting). Therefore we get easily that, for t large enough,

P (X t < t κ u|H((1 + ε)t κ u) < t) < 1/2. (2.19)
The lower bound in (1.2) then follows from equations (2.18) and (2.19).

To prove (1.3), we use the fact that, for every ε > 0, We begin with the proof of Lemma 2.1. It will turn out that once the tools for this Lemma will we introduced, Lemma 2.2 will be quite obvious. We recall from equation (2.4) that

P H t κ u > t ≤ P X t < t κ u ≤ P H (1 + ε)t κ u > t + P X t < t κ u ; H (1 + ε)t κ u < t . Taking v = t κ /u,
α = ρ(t) -1 = t 0 h(γ s )ds,
where h is some positive, integrable function. We have

α τt = ∞ -∞ h(x)L x τt dx = tc h + ∞ -∞ h(x) L x τt -t dx.
Our result then follows as soon as we show that, for t large enough

P ∞ -∞ h(x) L x τt -t dx > 3tǫ < exp (-w) . Let s such that s → ∞ and t/s 4 ≫ w, then ∞ -∞ h(x) L x τt -t dx = s -s h(x) L x τt -t dx + ∞ s h(x) L x τt -t dx + s -∞ h(x) L x τt -t dx := I 1 + I 2 + I 3 .
By a scaling argument, and using the fact that h is bounded, we have

|I 1 | ≤ C s -s L x τt -t dx law = Ct s -s L x/t τ 1 -1 dx = Ct 2 s/t -s/t L y τ 1 -1 dy.
Then, for t large enough,

P(|I 1 | > tǫ) ≤ P sup y∈[-s/t,s/t] |L y τ 1 -1| > ǫ 2Cs ≤ 2P sup y∈[0,s/t] |L y τ 1 -1| > ǫ 2Cs
, the last bound coming from the symmetry of L y τ 1 in y. On the other hand, using statement 2.2, L y τ 1 is a squared Bessel process of dimension 0 started from 1, therefore using statement 2.4 with δ = ǫ 2Cs , v = s/t, we get

P(|I 1 | > tǫ) ≤ C ′ s exp - ǫ 2 t s 3 ≤ exp(-w).
It is clear that, for large t, P(|I 3 | ≥ tǫ) ≤ P(|I 2 | ≥ tǫ). To bound I 3 , we note that, for t large enough,

|I 3 | ≤ 2 s -∞ L x τt x 2 dx + t s -∞ 1 x 2 dx law = 2 t s + ∞ s/t L x τ 1
x 2 dx , by the same scaling argument. The first part is negligible, and, using statement 2.2,

∞ s/t L x τ 1 x 2 dx = ∞ s/t Z x x 2 dx,
where Z t is a squared Bessel process of dimension 0 started at 1. The following result from [START_REF] Pitman | Sur une decomposition des ponts de Bessel[END_REF] allows us to compute the Laplace transform of this random variable.

Statement 2.5 (J. Pitman and M.Yor) Let Z t be a squared Bessel process of dimension d, starting from x, and µ a positive (Radon) measure on (0, ∞) such that, for all n, µ(0, n) < ∞. Then one has

E exp -Z t dµ(t) = φ µ (∞) d/2 exp x 2 φ ′ µ (0) ,
where φ µ is the unique decreasing and convex solution of

1 2 φ ′′ = µ.φ on (0, ∞), φ(0) = 1.
We note η = s/t, and

A t = ∞ η L x τ 1
x 2 dx. The preceding statement implies that

E [exp -λA t ] = exp 1 2 φ ′ µ (0) ,
where φ µ is the solution of:

φ ′′ (x) = 2λ φ(x) x 2 1 x≥η .
A decreasing solution on (η, ∞) of this equation is

φ(x) = C x η 1- √ 1+8λ 2
.

The condition φ(0) = 1 and the fact that

φ ′ is constant on [0, η] implies that C 1 -1- √ 1+8λ 2 = 1, thus E [exp -λA t ] = exp 1 - √ 1 + 8λ 2(1 + √ 1 + 8λ)η ,
As this function is analytic, for some λ > 0 (not depending on t),

E [exp λA t ] = exp 1 - √ 1 -8λ 2(1 + √ 1 -8λ)η , then P(I 2 > ǫt) ≤ exp 1 - √ 1 -8λ 2(1 + √ 1 -8λ)η -λǫt ,
from which the result follows, as 1/η ≪ t.

Let us now prove Lemma 2.2. We recall from (2.7) that,

J 1 = w 2 -A log(w) 5 /w -∞ g(sw)L s τ 1 ds ≤ 2 -A log(w) 5 /w -∞ 1 s 2 L s τ 1 ds.
Then the proof follows easily as a corollary of the proof of Lemma 2.1.

3 Quenched slowdown.

We now turn to the proof of Theorem 1.5. As before we first recall some useful facts.

Preliminary statements.

We recall the time change representation of X t (see, for example [START_REF] Hu | Rates of convergence of diffusions with drifted brownian potentials[END_REF])

X t = A -1 κ B(T -1 κ (t)) ,
where

A κ (x) = x 0 e W κ(y) dy, T κ (t) = t 0 e -2Wκ(A -1 κ (B(s))) ds,
and B is a standard Brownian motion.

We also need a result about Sturm-Liouville equations. Let V (t) be a positive function of t ≥ 0, and V (t) = t 0 V (u)du. We are interested in the solution of the differential equation

z ′′ (t) = -λV (t)z(t), t ≥ 0, z(0) = 1, z ′ (0) = 0. (3.1)
We have the following statement from [START_REF] Bobkov | Muckenhoupt's condition via Riccati and Sturm-Liouville equations[END_REF] (corollary 3.2)

Statement 3.1 Let λ(V ) be the supremum of all λ > 0 for which a solution to the problem (3.1) is positive in [0, 1), then

sup 0<t<1 (1 -t) V (t) ≤ 1 λ(V ) ≤ 4 sup 0<t<1 (1 -t) V (t).
We recall the following inequality from lemma 1.1.1 of [START_REF] Csörgö | Strong approximations in Probability and Statistics[END_REF] Statement 3.2 Let γ(t) be a one-dimensional brownian motion, then

P sup 0≤s 1 <s 2 <t,s 2 -s 1 <u |γ(s 2 ) -γ(s 1 )| > x 2 ≤ c t u exp - x 2 9u .
We finish with a useful lemma Lemma 3.1 let a > 0, and µ a Radon measure on [0, a], and suppose there exists φ a positive solution of the Sturm-Liouville equation

φ ′′ = -φµ, t ≥ 0, φ(a) = 1, φ ′ (a) = 0. (3.2)
Let X t be a squared Bessel process of dimension δ, starting at x, then

E exp a 0 X t dµ(t) ≤ φ(0) -δ/2 exp 1 2 φ ′ (0) φ(0) x .
Remark: This lemma is a extension of Statement 2.5, but we do not get equality in this case.

Proof: Let F µ (t) = φ ′ (t)/φ(t), by the concavity of φ this is a right continuous and decreasing function, thus we can apply the integration by parts formula to get

F µ (t)X t = F µ (0)x + t 0 F µ (s)dX s + t 0 X s dF µ (s).
Using (3.2), we can compute the last part

t 0 X s dF µ (s) = t 0 X s dφ ′ (s) φ(s) - t 0 φ ′ (s)dφ(s) φ(s) 2 = - t 0 X s dµ(s) - t 0 X s F µ (s) 2 ds.
Recalling that M t = X tδt is a local martingale, we set

Z µ (t) = exp 1 2 t 0 F µ (s)dM s - 1 2 t 0 X s F µ (s) 2 ds ,
which is a positive local martingale, hence a supermartingale. Using the previous computation, we get

Z µ (t) = exp 1 2 F µ (t)X t -F µ (0)x -δ t 0 F µ (s)ds + t 0 X s dµ(s) .
As Z µ is a supermartingale, E[Z µ (a)] ≤ E[Z µ (0)] = 1. Therefore the result follows easily.

3.2 Quenched slowdown for the hitting time.

In this section we show (1.8). The idea of the proof is to decompose the environment in valleys of a certain size, then to study the process of the valleys visited and the time spent in the valleys. We first give a formal definition of what a valley is. For t > 0, v > 0 and i ∈ N , we set K 0 = -⌊t⌋, and

K i+1 = inf x > K i , W κ (K i ) -inf y∈[K i ,x] W κ (y) > 3 κ log⌊t⌋, W κ (x) ≥ sup y>x W κ (y) -1 .
K i is finite almost surely, due to the transience of the drifted brownian motion. The intervals [K i , K i+1 ] will be called "valleys". An example of such valleys is given in figure 2.

Ki 0 Wκ(x)

x 3 κ log⌊t⌋ Ki 0 +1 Ki 0 +2 Ki 0 -1 Ki 0 -2 . . .

Figure 2: Decomposition in Valleys

We introduce the sequence defined, for k ≥ 0 by

s 0 = 0 s k+1 = inf{t > s k , X t ∈ {K j , j ≥ 0}}. We call Y k = X s k , l t = max{i : s i < H(v)} and
ξ(i) = ♯ {j ∈ [0, l t ], Y j = K i+1 , Y j+1 = K i } .
We set i 0 = max{j, K j < 0} and i 1 = max{j, K j < v}. By convention we note

K i 1 +1 = v. Let B = i 1 -1 i=1 ξ(i) (3.3)
denote the number of times the "walk" Y k backtracks. Let θ(t) be the timeshift associated to the diffusion, we set for 0

≤ i < i 1 next(i) = inf{t ≥ 0 : X t = K i , H(K i+1 ) • θ(t) < H(K i-1 ) • θ(t)} and H next (i) = H(K i+1 ) • θ(next(i)) -next(i).
We have the following decomposition of H v :

H(v) = H init + H dir + H back + H lef t + H right ,
where

H init = H(K i 0 +1 ) if H(K i 0 +1 ) < H(K i 0 ) H(K i 0 ) + H next (i 0 ) • θ(H(K i 0 )) else , (3.4)
is the time the diffusion takes to get to K i 0 +1 ,

H lef t = t 0 1 Xt<K 1 dt, (3.5)
is the time the diffusion spends at the left of K 1 ,

H right = H(v) • θ(next(i 1 )) -next(i 1 ), (3.6)
is the time spent to get from

K i 1 to v H dir = i 1 -1 i=i 0 +1 H next (i) (3.7)
is the time used for the direct crossings of the valleys and

H back = i 1 -1 i=i 0 +1 lt j=0 1 Y j =K i+1 ,Y j+1 =K j × H(K i ) • θ(s j ) -s j + H next (i) • θ(H(K i ) • θ(s j )) (3.8)
is the time "lost" as a consequence of the different backtracks of Y k . We introduce

D i = sup K i <s<t<K i+1 W κ (t)-W κ (s)
, to which we will refer as the "depth" of the valley [K i , K i+1 ], and

N (s, t) = {i ≥ 1, [K i , K i + 1) ∩ [s, t) = ∅}.
Note that, as seen on figure 2 there are some valleys of depth 0.

We have the following lemmas, whose proof will be postponed Lemma 3.2 (environment estimates) Let v = t ν and ǫ > 0. P-almost surely, for m > m 0 , for t large enough, W ∈ Ω where

Ω = Ω(t, m) = A(t) ∩ G(t) ∩ G(v) ∩ B(t, m) ∩ K(t) ∩ L(t) and
A(t) = max i≤i 1 (K i+1 -K i ) ≤ (log(t)) 2 , G(u) = sup -u≤r<s≤u W κ (s) -W κ (r) ≤ 1 κ (log u + 3 log log u) , B(t, m) = m-1 j=1 ♯{i ∈ N (-v, v) : D i ≥ 1 κ log v k/m + 4 log log(v)} ≤ v 1-k m , K(t) = sup -t<t 1 <t 2 <t |t 2 -t 1 |<1 |W κ (t 2 ) -W κ (t 1 )| ≤ (log t) 1/2 log log t , L(t) = sup 0<r<s<v W κ (s) -W κ (r) > 1-ǫ κ log v .
Furthermore, whenever u → ∞, the event G(u) is fullfilled for u large enough.

We now turn to some quenched estimates: let [a, c] be an interval of R. We call

D + = sup x∈[a,c] max y∈[x,c] W κ (y) -min y∈[a,x) W κ (y) , (3.9)
D -= sup x∈[a,c] max y∈[a,x] W κ (y) -min y∈(x,c]) W κ (y) , (3.10)
and

D = D -∧ D + .
We also introduce

M := sup x∈[a,c] W κ (x) -min x∈[a,c] W κ (x)
P x W H(a) ∧ H(c) > Cu(M ∨ 1)(1 ∨ (c -a) 4 ))e D < e -u . (3.11)
We also have a bound on the number of backtracks. For f → ∞, f = O(t)

P W [B ≥ f ] ≤ C 3 e -f . (3.12)
Finally, if W ∈ Ω, for some constant γ, for every 1 ≤ i ≤ i 1 , and for t large enough,

P K i W H(K i+1 ) > uγ(log t) 20 e D i-1 ∨D i |H(K i+1 ) < H(K i-1 ) ≤ e -u , (3.13)
P K i W H(K i-1 ) > uγ(log t) 20 e D i-1 ∨D i |H(K i-1 ) < H(K i+1 ) ≤ e -u , (3.14) P 0 W H(K i 0 ) ∧ H(K i 0 +1 ) > uγ(log t) 20 e D i 0 -1 ∨D i 0 ≤ e -u . (3.15)
Thanks to these lemmas, we are able to finish the proof of Theorem 1.5.

Upper bound.

We recall v = t ν . Suppose Ω(t, m) is fulfilled, by the previous decomposition,

P W (H(v) > t) ≤ P W H init > t 5 + P W H dir > t 5 + P W H back > t 5 + P W H lef t > t 5 + P W H right > t 5 .
We begin with H init . We recall from (3.4) that H init is the time the diffusion takes to get to K i 0 +1 . Using the precedent estimates, on G(v), we have, for t large enough

D i 0 ∨ D i 0 +1 < 1 κ (log v + 3 log log v).
Thus, for every ǫ > 0,

P W H init > t 5 ≤ P 0 W H(K i 0 +1 ) > t 1-ν/κ 5 e D i 0 ∨D i 0 +1 ∩ H(K i 0 +1 ) < H(K i 0 ) + P 0 W H(K i 0 ) > t 1-ν/κ 10 e D i 0 ∨D i 0 +1 ∩ H(K i 0 ) < H(K i 0 +1 ) +P K i 0 W H(K i 0 +1 ) > t 1-ν/κ 10 e D i 0 ∨D i 0 +1 |H(K i 0 +1 ) < H(K i 0 -1 ) ≤ 3e -t 1-ν/κ-ǫ .
Similarly, we have

P W H right > t 5 = P K i 1 W H(v) > t 5 |H(v) < H(K i 1 -1 ) ≤ e -t 1-ν/κ-ǫ .
It is also a direct consequence of lemma 3.3 that, on A(t), i 0 > t 2(log t) 2 , whence, recalling the definition of B in (3.3),

P W H lef t > t 5 ≤ P W B ≥ t 4 log 2 t ≤ exp - t 4 log 2 t .
To deal with H dir , note that

H dir = i 1 -1 i=i 0 +1 τ (0) + (i),
where τ

+ (i) is the first crossing of the interval [K i , K i+1 ]. The τ (0)
+ (i) are independent random variables, and τ (0) + (i) follows the same law as H(K i+1 ) under

P K i W [•|H(K i+1 ) < H(K i-1 )
]. On the other hand, if H dir > t/5, then the process spends an amount of time greater than t/20m in the valleys of depth in

k κm log v + 4 log log v, (k + 1) κm log v + 4 log log v .
On Ω(t, m), the number of such valleys is at most v 1-k m , we call σ(k) the time spent in those valleys. By lemma 3.3, and the precedent remarks, for some constant C,

σ(k) C(log t) 11 v (k+1)/κm ⊳ 2v (1-k/m) + Γ 2⌈v (1-k/m) ⌉, 1 ,
where we note A ⊳ B for " A is stochastically dominated by B", and Γ(k, β) is the Gamma distribution of parameter (k, β).

For m large enough, one can check easily that ν(1-k/m) < 1-ν(k+1)/m for all k ≤ m, whence, for t large enough,

P W σ(k) ≥ t 20m ≤ P Γ 2v (1-k/m) , 1 > t 1-ν(k+1)/κm (log t) 12 ≤ 4 t ν(1-k/m) exp - t 1-ν(k+1)/κm (log t) 12 ≤ exp -2t 1-ν(k+2)/(κm) + log(4)t ν(1-k/m) .
Therefore, as t → ∞,

P W [H dir > t/5] ≤ m exp -t 1-ν(k+2)/(κm) ≤ m exp -t 1-(1+ 2 m ) ν κ .
We now deal with H back .

P W H back > t 5 ≤ m-1 k=0 P W H back > t 5 , B ∈ [t k/m , t (k+1)/m ] + P W [B > t].
By lemma 3.3, P W [B > t] < e -t , and

P W H back > t 5 , B ∈ [t k/m , t (k+1)/m ] ≤ C exp -t k/m . (3.16)
On the other hand,

H back = i 1 -2 i=1 ξ(i) j=1 τ (j) + (i) + τ (j) -(i),
where

• τ (j) -(i) is the j -th crossing of the interval [K i+1 , K i ].
• τ

(j) + (i) is the first crossing of the interval [K i , K i+1 ] after the j -th crossing of the interval [K i+1 , K i ].
The τ (j) +,-(i) are independent variables, and τ (j) + (i) follows the same law as H(K i+1 ) under

P K i W [•|H(K i + 1) < H(K i-1
)], and τ (j)

-(i) follows the same law as H(K i ) under P K i+1 W [•|H(K i ) < H(K i+2 )],
(with the convention that K i 1 +1 = v). Therefore, thanks to lemma 3.3, τ (j) +,-(i) Ce H (log t) 10 ⊳ 1 + e for some constant C and

H = max i∈N " -t (k+1) log 2 t m ,v « D i .
Then, for W κ ∈ Ω(n, m)∩G t (k+1) log 2 t m , on the event {B ∈ [t k/m , t (k+1)/m ]},

H back C(t (k+1)/mκ ∨ v 1/κ )(log t) 10 ⊳ 2t (k+1)/m + Γ(2t (k+1)/m , 1). Therefore, when 1 -1 κ ν ∨ k+1 m ≥ k+1 m , P W H back > t 5 , B ∈ [t k/m , t (k+1)/m ] ≤ C exp -C ′ t 1-1 κ (ν∨ k+1 m ) .
Putting this together with (3.16), we obtain

P W H back > t 5 , B ∈ [t k/m , t (k+1)/m ] ≤ C exp -C ′ t (1-1 κ (ν∨ k+1 m ))∨ k m -1 m .
Putting together all the estimates, we get

lim inf t→∞ log(-log P W [H(t ν ) > t]) log t ≥ min k∈[-1,m+1] k m ∨ 1 - 1 κ ν ∨ k + 1 m - 1 m ∧ 1 -(1 + 2 m ) ν κ ≥ 1 - ν κ ∧ κ κ + 1 - 3 (1 ∧ κ)m
, Pa.s..

By taking the limit as m goes to infinity, we get the upper bound for

P W [H(t ν ) > t], namely lim inf t→∞ log(-log P W [H(t ν ) > t]) log t ≥ 1 - ν κ ∧ κ κ + 1
.

We now turn to the proof of the lower bound.

Lower bound.

We suppose that L(t) is fullfilled, therefore there is one valley of depth greater than 1-ǫ κ log v before v. Let b be the bottom of this valley, and c such that b < c and

W κ (c) -W κ (b) = 1 -ǫ κ log v.
It is easy to see that H(v) ≥ H(c) -H(b), whence

P W [H(t ν ) > t] ≥ P b W [H(c) > t].
We can suppose, without loss of generality, that b = 0. By the time change representation from the preliminary statements, under

P W , H(c) = T κ (σ(A κ (c)))
, where σ(x) is the first hitting time of x by a brownian motion B. Therefore

H(c) = σ(Aκ(c)) 0 e -2Wκ(A -1 κ (Bs)) ds. = Aκ(c) -∞ exp (-2W κ (A -1 κ (x)))L x σ(Aκ(c)) dx = c -∞ exp (-W κ (u))L Aκ(u) σ(Aκ(c)) du.
The last equality coming from a change of variable in the integral. By a scaling argument, we get

H(c) law = c -∞ exp (-W κ (u))A κ (c)L Aκ(u)/Aκ(c) σ(1)
du.

We suppose W κ ∈ K(t), so

A κ (c) ≥ e Wκ(c)-(log t) 2/3 > t (1-2ǫ) ν κ ,
and A κ (-1) > -e -(log t) 2/3 . Hence

H(c) ⊲ t (1-3ǫ) ν κ inf x∈[Aκ(-1)/Aκ(c),0] L x σ(1) .
For t large enough, A κ (-1)/A κ (c) > -1/2. Therefore by the first Ray-Knight theorem (Statement 2.1)

P b W [H(c) > t] ≥ P inf x∈[-1/2,0] L x σ(1) > t 1-ν κ +ε ≥ P Z ′ 1 > 2t 1-ν κ +ε P sup u∈[0,1/2] |Z u | < t 1-ν κ +ε ,
where Z t is a squared Bessel process of dimension 0 started at 0 and Z ′ t is a squared Bessel process of dimension 2 started at 0. The last probability is greater than 1/2 for t large enough, and the first one is explicitly known (see for example [START_REF] Borodin | Handbook of Brownian Motion. Probability and its application[END_REF]). We obtain that, for all ε > 0,

P W [H(v) > t] ≥ exp - 1 2 t 1-ν κ +ε .
To obtain the other lower bound, note that, similarly to lemma 3.2, almost surely, there is a valley of depth at least 1-ǫ κ+1 log t in [-t κ/(κ+1) , 0], let b ′ be the bottom of such valley, and c ′ > b ′ such that

W κ (c ′ ) -W κ (b ′ ) ≥ 1 -ǫ κ + 1 log t.
We have

P W [H(v) > t] ≥ P W [H(b) < H(t ν )]P b W [H(c) > t].
Recalling the time change representation,

P W [H(b) < H(t ν )] = A κ (t ν ) A κ (t ν ) -A κ (b)
.

when W κ ∈ K(t), we can easily show that for every ǫ > 0, as n goes to infinity,

P W [H(b) < H(t ν )] ≥ exp -t κ κ+1 +ǫ
. By the same computations as for the first bound, we get

P b W [H(c) > t] ≥ exp -t κ κ+1 +ǫ .
Putting together both inequalities, we get

lim inf t→∞ log(-log P W [H(t ν ) > t]) log t ≤ 1 - ν κ ∧ κ κ + 1 ,
which finishes the proof of Theorem 1.5.

Quenched slowdown for the diffusion.

In this section we finish the proof of Theorem 1.5. The lower bound is trivial, since

P W [X t < t ν ] ≥ P W [H(t ν ) > t].
To get the upper bound, let m ∈ N, note that

P W [X t < t ν ] ≤ P W [H(t ν ) > t] + m-1 k=0 P W H t ν+ k m < t < H t ν+ k+1 m P t ν+ k m W [H(t ν ) < t] + P W [H(t ν+1 ) < t]P t ν+1 W [H(t ν ) < t]. (3.17)
Using the explicit distribution of the supremum before t of a drifted brownian motion (see page 197 of [START_REF] Borodin | Handbook of Brownian Motion. Probability and its application[END_REF]) and the Borel-Cantelli lemma, we can easily see that for every k ∈ {1, m}, the event

U k m (n) :=    sup (n+1) ν <s<t<n ν+ k m W κ (s) -W κ (t) ≥ κ 4 n ν+ k m   
is fullfilled for all n large enough, therefore so does

U n = m k=1 U k m (n).
Hence on U ⌈t⌉ , there exist t ν < a < b < t ν+ k m such that

W κ (a) -W κ (b) ≥ κ 4 t ν+ k m .
By the same computations as in part 3.2.2, we get that, on U (⌈t⌉),

P b W [H(a) < t] ≤ P W   e κ 8 t ν+ k m inf x∈[0,e -κ 8 t ν+ k m ] L x σ(1) < t   ≤ P W   inf u∈[1,1-e -κ 8 t ν+ k m ] Z u < te -κ 8 t ν+ k m   ,
where Z u is a squared Bessel process of dimension 2 started at zero. We have

P W inf u∈[1,1-e -κ 8 t α ] Z u < te -κ 8 t α ≤ P W Z 1 < 2te -κ 8 t α + P W   sup u∈[1,1-e -κ 8 t α ] |Z u -Z 1 | ≥ te -κ 8 t α   .
Using statement 3.2 with u = te -κ 8 t α and the fact that

√ Z 1-t -Z 1 is
the Euclidean norm of a two dimensional Brownian motion, we get

P W   sup t∈[1,1-e -κ 8 t α ] |Z t -Z 1 | ≥ te -κ 8 t α   ≤ 2 exp - t 10 .
On the other hand, by the exact distribution of Z 1 ,

P (Z 1 < x) = 1 -e -x/2 < x.
Therefore we get that for some constant

C P t ν+ k m W [H(t ν ) < t] ≤ P b W [H(a) < t] < e -Ct ν+ k m .
On the other hand, the bound for the hitting time implies that

P W H t ν+ k m < t < H t ν+ k+1 m ≤ exp -t (1-(ν+ k+1 m )/κ)∧( κ κ+1 )-1 m ,
indeed the bound is trivial when ν + k/m > κ.

The same arguments apply to the other terms of (3.17), whence

lim inf t→∞ log(-log P W [X t < t ν ]) log t ≥ min k∈[0,m] ν + k m ∨ 1 - ν + (k + 1)/m κ ∧ κ κ + 1 - 1 m .
Minimizing over k and taking the limit as m go to infinity, we get the desired upper bound.

Proof of the lemmas.

We begin with the estimates on the environment.

Proof of lemma 3.2.

Note that, as an easy consequence of statement 3.2, almost surely for t large enough i 1 < 2t. Therefore

A(t) ⊃ Ã(⌊t⌋) := max i≤2⌊t⌋+1 |K i+1 -K i | ≤ log 2 (⌊t⌋) . (3.18)
Let us show that

P[ Ã(n) c ] = O(1/n 2 ). (3.19)
We have

P[ Ã(n) c ] ≤ 2n+1 i=0 P[K i+1 -K i ≥ (log (n)) 2 ]. (3.20)
By invariance of the environment,

P[K 1 -K 0 ≥ (log (n)) 2 ] = P[ K1 ≥ (log (n)) 2 ],
where

K1 = min t ≥ 0 : -min s∈[0,n] W κ (s) ≥ 3 κ log n, W κ (t) > sup s≥t W κ (s) -1 .
On the other hand, conditionally to K i , the process W κ (K i + s) -W κ (K i ) is a drifted Brownian motion conditionned to have its supremum lesser than 1. Therefore

P[K i+1 -K i ≥ (log n) 2 ] = P[ K1 ≥ (log n) 2 | sup t≥0 W κ ≤ 1] ≤ P[ K1 ≥ (log n) 2 ] P[sup t≥0 W κ ≤ 1] . For κ > 0, P[sup t≥0 W κ ≤ 1] is a positive constant. It remains to bound P[ K1 ≥ (log n) 2 ], note that if W κ (log n) 2 < - 6 κ log n,
and sup

t≥(log n) 2 W κ (t) -W κ (log n) 2 < 3 κ log n,
then there exists one point 3), therefore K1 < (log n) 2 . Taking the complementary events, we get

x * before (log n) 2 such that inf t∈[0,x * ] W κ (t) < -3 κ log n and W κ (x * ) ≥ sup s≥x * W κ (s) -1 (see figure
P[ K1 ≥ (log n) 2 ] ≤ P W κ (log n) 2 > - 6 κ log n or sup t≥(log n) 2 W κ (t) -W κ (log n) 2 > 3 κ log n .
x Wκ(x)

(log n) 2 -3 κ log n -6 κ log n x *
P W κ (log n) 2 > - 6 κ log n = O(n -3 )
and

P sup t≥(log n) 2 W κ (t) -W κ (log n) 2 > 3 κ log n = P sup t≥0 W κ (t) > 3 κ log n .
By formula 1.1.4(1) from page 197 of [START_REF] Borodin | Handbook of Brownian Motion. Probability and its application[END_REF], the last probability is equal to n -3 . Therefore recalling equation (3.20), this finishes the proof of (3.19). Therefore, using the Borel-Cantelli lemma and (3.18), A(t) is fullfilled for every t large enough. We now turn to G. We consider the process

U t := sup -∞≤s≤t W κ (t) -W κ (s). (3.21)
Note that for n = ⌊t⌋, sup

-(n+1)≤t≤n+1 U t ≤ 1 κ (log n + 3 log log n) ⊂ G(t).
The process U t is called a Reflected Brownian Motion with drift. This kind of process appears naturally in some queueing system models. It is a positive and stationnary diffusion, with stationnary law the exponential law of parameter κ. It is also reversible in time, therefore we can reduce to proving that, as n goes to infinity, the event

sup 0≤t≤n+1 U t ≤ 1 κ (log n + 3 log log n) (3.22)
is fullfilled.

In [START_REF] Salminen | On busy periods of the unbounded brownian storage[END_REF] it is shown that the length of the excursions away from zero (or busy periods) of U t follows a gamma distribution Γ 1 2 , κ 2

8

, and that the supremum m 0 over one excursion of U t has an explicit law, given by P(m 0 > y) = 2e -κy (1e -κy ) 2 (κy -(1e -κy )).

(

Let C be some large constant. We call F (n) the event that U t makes more than Cn excursions between time 0 and time n + 1. We have

P(F (n)) ≤ P Γ Cn 2 , κ 2 8 < n + 1 = γ(Cn/2, (n+1)κ 2 8 ) Γ(Cn/2)
,

where γ(•, •) is the incomplete gamma function. By Stirling's formula,

P(F (n)) = O(((n + 1)κ 2 /8) Cn/2 (Cn/2e) -Cn/2-1/2 ) = o(n -4 )
for C large enough. Therefore by the Borel-Cantelli lemma, almost surely there exists n 0 such that F (n) is fullfilled for all n ≥ n 0 . On the other hand, we call G(k) the event that the maximum during the kth excursion is lower than 1/κ(log k + 3 log log k). Recalling (3.23), for k ≥ 10,

P G(k) c = P m 0 > 1 κ (log k + 3 log log k) ≤ 8 k(log k) 2 .
By the Borel-Cantelli lemma, we get that there exists k 0 such that G(k) is fullfilled for all k ≥ k 0 . Take n > n 0 ∨ k 0 , and such that

1 κ (log n + 3 log log n)
is greater than the supremum over the k 0 first excursions of U t . Then on F (n) ∩ n k=k 0 G(k) the event in (3.22) is fullfilled. This implies the result for G(t).

Let us turn to B(t, m). Let n = ⌊v⌋. We call, for 0 < a < 1

B(n, a) = ♯ i ∈ N [-(n + 1), n + 1] : D i ≥ a κ log n + 4 κ log log n < n 1-a .
Recalling the definitions of the K i and U t , we note that the event that two different K i belong to the same excursion of U t implies that the maximum during this excursion is at least 3/κ log n, therefore, by the same argument as before, when n is large enough, this does not happen. We can also suppose that U t makes less than Cn excursions between time -(n + 1) and n + 1. Thus, on these events,

♯ i ∈ N [-(n + 1), (n + 1)] : H i ≥ a κ log n + 4 log log n
is stochastically dominated by a Binomial(2n + 1, p), where

p = P m t ≥ a κ log n + 4 log log n < 2 n -a log n 2 .
Whence, using Chebyshev's exponential inequality,

P [ B(n, a) c ] ≤ exp -n 1-a exp ((2n + 1) log(1 + p(e -1))) ≤ exp 4np -n 1-a .
The estimate on p, together with the Borel-Cantelli lemma, implies that, almost surely for n large enough,

m-1 1 B(n, k/m) ⊂ B(t, m) is fullfilled.
We finally prove that L(t) is fullfilled for t large enough. Recalling the notations concerning U t from (3.21), we call f (n) the event that U t makes more that n (log n) 2 excursions before time n. Using the explicit distribution of the length of the excursions of U t , we have

P (f (n) c ) ≤ P Γ n 2(log n) 2 , κ 2 8 > n .
Recalling that a Γ(k, θ) distribution has expectation kθ and variance kθ 2 , by Bienaymé-Chebyshev's inequality, for n large,

P (f (n) c ) ≤ 10 n(log n) 2 .
Now the Borel-Cantelli lemma implies that f (n) is fullfilled for all n large enough. Now suppose that f (⌊v⌋) is fullfilled, Note that U t and sup 0<s<t W κ (t) -W κ (s) are equal after the first 0 of U t . Call L(t) the event that there exists one excursion of height at least 1-ǫ κ log(n + 1) between the second and the ⌊ n (log n) 2 ⌋-th excursion of U t . It is easy to see that

f (⌊v⌋) ∩ L(t) ⊂ L(t).
On the other hand, by (3.23),

P L(t) c ≤ P m t < 1 -ǫ κ log(n + 1) n (log n) 2 ≤ 1 -e -(1-ǫ) log n+1 n (log n) 2 .
This is summable, therefore we can apply the Borel-Cantelli lemma to get the result on L(t), then on L(t).

The result on K(t) is a direct consequence of statement 3.2.

We now turn to the quenched estimates.

Proof of Lemma 3.3.

We begin with the proof of (3.11). Without loss of generality we can suppose x = 0 and D = D + . We suppose |c -a| ≥ 1, the proof being similar when |c -a| ≤ 1.

Recalling from the preliminary statements the time change representation of X t , we get that, under P W , H(v) = T κ (σ(A κ (v))), where κ (A κ (c)x))), and X s is a Bessel process of dimension 2, started at 0. We call α := A κ (c) -A κ (a), and λ(V ) the supremum of all λ such that a solution to y ′′ (t) = -λV (t)y(t), t ≥ 0 y ′ (α) = 0, y(α) = 1 is positive in [0, α]. λ(V ) is usually known as the spectral gap, or Poincaré's constant associated to V.

A κ (x) = x 0 e W κ(y)
By a standard change of variable in the previous differential equation, and an application of statement 3.1, we get From Lemma 3.1 we get that E[exp λ(V )U ] is finite, but we need an explicit bound. Toward this goal we are going to extend the interval : let c ′ be such that (c ′a) = 2(ca) and let us extend W κ on [c, c ′ ] by a constant function (equal to W κ (c)). We call Ṽ

1 λ(V ) ≤ 32(A κ (c) -A κ (a)) 2 sup 0<t<1 (1 -t)
(x) = exp (-2W κ (A -1 κ (A κ (c) -x))), for x ∈ [A κ (c) -A κ (c ′ ), A κ (c) -A κ (a)
] and λ( Ṽ ) the supremum of all λ such that a solution to

y ′′ (t) = -λ Ṽ (t)y(t), t ≥ y ′ (α) = 0, y(α) = 1 (3.24) is positive in [A κ (c) -A κ (c ′ ), α].
By the same calculations as before we get

1 λ( Ṽ ) ≤ 32 sup a≤x≤c ′ x a e -Wκ(v) dv c ′ x e Wκ(v) dv ≤ 32(c ′ -a)e D + = 64(c -a)e D + .
For λ < λ( Ṽ ), let φ be a solution to (3.24) on [A κ (c) -A κ (c ′ ), α], then φ is a solution to (3.24) on [0, α], and by concavity,

φ(0) ≥ A κ (c ′ ) -A κ (c) A κ (c ′ ) -A κ (a) ≥ e -M 2 .
Together with lemma 3.1, we get

E W [exp(λH(a) ∧ H(c))] < 2e M .
This, together with Markov's inequality, finishes the proof of the first part of lemma 3.3.

In order to prove (3.12), note that, due to the time change representation, and for W ∈ Ω,

P K i W [H(K i-1 ) < H(K i+1 )] = K i+1 K i e Wκ(x) dx K i+1 K i-1 e Wκ(x) dx -1 ≤ max i≤i 1 (K i -K i-1 )
e 1+Wκ(K i ) e Wκ(K i-1 )-(log t) 1/2 log log t ≤ t -3/2 , (3.25)

using the fact, that, by definition of the K i , on

K(t) ∩ G(t), W κ (K i-1 ) ≥ inf K i-1 ≤x≤K i W κ (x) + 3 κ log t ≥ W κ (K i ) + 2 κ log t - 3 κ log log t.
Then we have to distinguish two cases : either the walk Y j gets to the level v in more than 3n steps or in less than 3n steps. In the first case there are at least n steps back before H(v), and in the second case the number of steps back is dominated by a Binomial(3n, n -3/2 ). Thus

P W [B ≥ f (t)] ≤ 3n n 1 n 3/2 n + P Binomial(3n, n -3/2 ) ≥ f (t) .
The result follows easily from Stirling's formula and Chebyshev's exponential inequality.

We now turn to the proof of (3.13), (3.14) and (3.15). We start with (3.13). First note that

P K i W H(K i+1 ) > uγ(log t) 20 e D i-1 ∨D i |H(K i+1 ) < H(K i-1 ) ≤ P K i W H(K i-1 ) ∧ H(K i+1 ) > uγ(log t) 20 e D i-1 ∨D i ) P K i W [H(K i+1 ) < H(K i-1 )] ,
As a direct consequence of (3.25), we have, Pa.s., for n large enough,

P K i W [H(K i+1 ) < H(K i-1 )] ≥ 1 2 .
We are going to use (3.11) in order to bound the numerator. Note that, due to the definition of the K i ,

sup K i-1 <s<t<K i+1 W κ (s) -W κ (t) ≥ D i-1 ∨ D i .
On the other hand, on

A(t) ∩ K(t), K i+1 -K i-1 ≤ 2(log t) 2 , and then sup x∈[K i-1 ,K i+1 ] W κ (x) - min x∈[K i-1 ,K i+1 ] W κ (x) < (log t) 3 .
Therefore, the result follows easily by application of (3.11).

We now turn to the proof of (3.14). As before,

P K i W H(K i-1 ) > uγ(log t) 20 e D i-1 ∨D i |H(K i-1 ) < H(K i+1 ) ≤ P K i W H(K i-1 ) ∧ H(K i+1 ) > uγ(log t) 20 e D i-1 ∨D i ) P K i W [H(K i-1 ) < H(K i+1 )]
.

The numerator is the same as in the proof of (3.13), so we only have to deal with the denominator. We recall from (3.25) that

P K i W [H(K i-1 ) < H(K i+1 )] = K i+1 K i e Wκ(x) dx K i+1 K i-1 e Wκ(x) dx -1
.

On K(t) ∩ G(t), we obtain easily 4 Quenched speedup.

P K i W [H(K i-1 ) < H(K i+1 )] ≥ e Wκ(K i )-Wκ(K i-1 )-log t (log t) 3 . Note that on A(t) ∩ K(t), W κ (K i-1 ) -W κ (K i ) ≤ (log t)
In this part we show Theorem 1.3. We first recall some facts.

Preliminary statements.

Our proof is mainly based on "Kotani's formula", expressed in [START_REF] Kawazu | A diffusion process in a brownian random environment with drift[END_REF],

Statement 4.1 (Kotani's lemma) Let λ > 0. Then for t ≥ 0

E W e -λH(t) = exp -2λ t 0 U λ (s)ds , P -a.s.,
where U λ (t) is the unique stationnary and positive solution of the equation

dU λ (t) = U λ (t)dW (t) + 1 + 1 -κ 2 U λ (t) -2λU λ (t) 2 dt.
(Here W (t) is the Brownian motion defined in the introduction).

We shall also use the following result from [START_REF] Hu | Moderate deviations for diffusions with brownian potentials[END_REF] (Lemma 2.4)

Statement 4.2 lim 1 r sup |x|<u L x τ (r) -r = 0, a.s.,
whenever u → ∞ and r ≫ u log log u.

Proof of Theorem 1.4.

We use the same time change method as in the annealed case, in order to get almost sure estimates for U λ . Let g(x) =

x 1 e 2/s+4λs s 1-κ ds.

One can easily check that g is a scale function of U λ . By the same arguments as in section 2.2.1, we get

t 0 U λ (s)ds = µ(t) 0 g -1 (γ(u)) 1-2κ exp - 4 g -1 (γ(u)) -8λg -1 (γ(u)) du,
where γ(u) is a standard brownian motion,

µ(t) = t 0 U λ (s) 2κ exp 4 U λ (s) + 8λU λ (s) ds, and
µ -1 (t) = t 0 g -1 (γ(s)) -2κ exp - 4 g -1 (γ(s)) -8λg -1 (γ(s)) ds.
We have the following lemma, whose proof is postponed

In order to get the lower bound, we introduce a small δ > 0. For the sake of clarity we call ε := v u 1/κ . Note that for λ > 0

E W e -λ * H(v) = E W e -λ * H(v) 1 H(v)<(1-δ)ε + E W e -λ * H(v) 1 (1-δ)ε≤H(v)≤(1+δ)ε + E W e -λ * H(v) 1 H(v)>(1+δ)ε := J 1 + J 2 + J 3 .
We are going to show that J

1 + J 3 ≪ E W e -λ * H(v) . We call F (x) = P W [H(v) < x]
. By the Cramer-Chernoff inequality, for x < ε, one gets

F (x) ≤ exp (λ(x)x -C(1 -o(1))vλ(x) κ ) = exp (-C(1 -o(1))v(1 -κ)λ(x) κ ) = exp -(1 -o(1))C 1 1-κ (1 -κ)κ κ 1-κ v 1 1-κ x κ κ-1 . (4.4) Recall that E e -λ * H(v) = e -C(1+o(1))(Cκ) κ 1-κ u 1 1-κ .
We deduce that for α

= 2(1 -κ) κ-1 κ , F (αε) ≪ E W e -λ * H(v) .
For this α, we have

J 1 ≤ F (αε) + (1-δ)ε αε e -λ * x dF (x) = e -(1-δ)ελ * F ((1 -δ)ε) + (1 -e -αε )F (αε) + λ * (1-δ)ε αε e -λ * x F (x)dx. (4.5)
Our goal is to use (4.4) in order to bound F in the last equation. The problem is that the o(1) in (4.4) depends on x. We are going to use the monotonicity of F (x) in order to get an uniform bound. Let η < δ/1000, n > κ α(1-κ)η . For 1 ≤ k ≤ n, we set x k = kε/n. Using (4.4), there exists v 0 such that, for all v > v 0 , and 1

≤ k ≤ n, F (x k ) ≤ exp -(1 -η)C 1 1-κ (1 -κ)κ κ 1-κ v x k v κ κ-1 . Note that for x k-1 < x < x k , x > αε, and v > v 0 , F (x) ≤ F (x κ ) ≤ exp -(1 -η)C 1 1-κ (1 -κ)κ κ 1-κ v 1 1-κ x + ε n κ κ-1
.

By the concavity of the function x → x κ κ-1 , and the condition ε > x > αε, we get easily

x + ε n κ κ-1 ≥ x κ κ-1 + 1 n κ α(κ -1) α κ κ-1 ε κ κ-1 ≥ (1 -η)x κ κ-1 .
We deduce that for every ε > x > αε,

F (x) ≤ exp -(1 -η) 2 C 1 1-κ (1 -κ)κ κ 1-κ v 1 1-κ x κ κ-1 := e G(x) (4.6)
Therefore, replacing F by e G in (4.5), and doing the integration by parts in the other direction, we get

J 1 ≤ e -(1-δ)ελ * e G((1-δ)ε) + (1 -e -αε )e G(αε) + λ * (1-δ)ε αε e -λ * x e G(x) dx = e G(αε) + (1-δ)ε αε e -λ * x de G(x) . (4.7)
Recalling the definition of α, e G(αε) ≪ E e -λ * H(v) , and the integral can be bounded by

C ′ v κ 1-κ (1-δ)ε αε x 1 κ-1 e -λ * x e G(x) dx.
Therefore, recalling (4.1), and (4.3) for estimates on E W e -λ * H(v) , and the expressions of λ(x) and G respectively in (4.2) and (4.6), one gets

J 1 E W e -λ * H(v) -1 ≤ o(1) + P sup x∈[αǫ,(1-δ)ǫ] exp -C 1 1-κ κ κ 1-κ v 1 1-κ (1 -η) 2 (1 -κ)x κ 1-κ + κε 1 κ-1 -ε κ κ-1 (1 + o(1)) .
where P is some polynom in (u, v) and the terms between the brackets come respectively from e G , e -λx and E W e -λ * H(v) -1 . By a change of variable in the sup, we get We now deal with J 3 . As before we get As before, we can take η small and get

J 3 ≪ E W e -λ * H(v) .
Therefore we get that, as v → ∞,

J 2 > 1 2 E W e -λ * H(v) .
Recall that J 2 = E W e -λ * H(v) 1 (1-δ)ε≤H(v)≤(1+δ )ε ≤ e -λ * (1-δ)ε P W [H(v) < (1 + δ)ε] .

Note that the preceding computations remain true for u ′ := (1+δ) κ u, whence lim inf

v→∞ log P W H(v) < v u 1/κ u 1 1-κ > (1 + δ) κ 1-κ ((1 -δ)κ -1)C 1 1-κ κ κ 1-κ .
Taking the limit as δ → 0, we get the result.

It remains to prove lemma 4.1, which is the purpose of the next section. g -1 (s) ν exp -4 g -1 (s) -8λg -1 (s) L s τr ds

:= I 1 + I 2 + I 3 ,
where a is such that a > 1/λ and e 4λa 4λa = log 1 λ log log log 1 λ .

We shall use the following consequence of the law of large numbers : let f : R → R such that R |f (x)|dx < ∞, then On the other hand

I 1 = 0 -∞ g -1 (s) ν exp - 4 g -1 (s) -8λg -1 (s) L s τr ds ≤ 0 -∞ g -1 (s) ν exp - 4 g -1 (s)
L s τr ds.

Using (4.11), it is not difficult to check that g -1 (s) ν exp -4 g -1 (s) is integrable on (-∞, 0), therefore an application of (4.10) lays

I 1 = O(r).
Let us now treat I 3 . Note that for y ≥ a, yλ → ∞ and for some constant As yλ → ∞, we get g(y) ≤ 2ce 4λy .

Therefore, for x ≥ g(a), 2ce 4λg -1 (x) ≥ x, so g -1 (x) ≥ 1 4λ log x 2c . Therefore, using (4.10) we get, for some constant c ′ > 0,

I 3 ≤ ∞ g(a)
(g -1 (x)) ν∨0 e -8λg -1 (x) L x τr dx ≤ c ′ ∞ g(a)

log(x/2c) 4λ To deal with I 2 , note that, by the definition of a and (4.12), r ≫ g(a) log log g(a).

Therefore we can apply statement 4.2 to get

I 2 = r(1 + o(1)) g(a) 0 g -1 (s) ν exp - 4 g -1 (s) -8λg -1 (s) ds.
By a change of variables g -1 (s) = y, as λ → 0, the last integral is equal to g -1 (s) ν exp -4 g -1 (s) -8λg -1 (s) L s τr ds

:= I ′ 1 + I ′ 2 + I ′ 3 .
I ′ 3 is similar to the precedent case, with ν < 0, so we get I ′ 3 = o(r). We have easily

I ′ 1 ≤ e -1 b 0 -∞ g -1 (s) ν exp - 3 g -1 (s)
-8λg -1 (s) L s τr ds.

The integral is a O(r) by the same proof as for I 1 , therefore

I ′ 1 = o(r).
By the same proof as for I 2 , we get e -8λ/u-u u 1+κ du.

I ′ 2 =
One can easily check that the integral is bounded, therefore this part goes to zero. This finishes the proof of lemma 4.1.

Figure 1 :

 1 Figure 1: A "valley".

=

  C 3 , Pa.s..(1.6)

We have Lemma 3 . 3 (

 33 quenched estimates) Let a, c, and D be as above, and W ∈ Ω, then for some constant C, and u > 1 max x∈[a,c]

Figure 3 :

 3 Figure 3: K1

e

  -2Wκ(A -1 κ (B(s))) ds, and σ(x) is the first hitting time of x by a Brownian motion B. ThereforeH(a) ∧ H(c) = σ(Aκ(a))∧σ(Aκ (cκ (A -1 κ (x)))L x σ(Aκ(a))∧σ(Aκ (c)) dx.We are going to use the second Ray-Knight Theorem (Statement 2.2) : note that L x σ(Aκ(a))∧σ(Aκ (c)) ≤ L x σ(Aκ(c)) , and that L x σ(Aκ(c)) is stochastically dominated by the local time at x before σ(A κ (c)) of a Brownian motion started at a. Therefore H(a) ∧ H(b) ⊳ Aκ(c)-Aκ(a) 0 V (s)X s ds, where V (x) = exp (-2W κ (A -1

t 0 e( 1 ( 1 e

 011 -2Wκ(A -1 κ (Aκ(a)+s(Aκ(c)-Aκ(a)))) ds = 32(A κ (c) -A κ (a)) sup 0<t<1 t) Aκ(a)+t(Aκ (c)-Aκ(a)) Aκ(a) e -2Wκ(A -1 κ (u)) du = 32(A κ (c) -A κ (a)) sup 0<t<1 t) d(t) a e -Wκ(v) dv,whered(t) = A -1 κ (A κ (a) + t(A κ (c) -A κ (a))). Easy computations show that (1t)(A κ (c) -A κ (a)) = Wκ(v) dv ≤ 32(ca)e D + .

J 1 E

 1 W e -λ * H(v) < o(1)+ P exp -(Cv)

( 1 -

 1 η) 2 s κ κ-1 (1κ) + κs -1 + o(1) .

( 4 . 8 )

 48 For η and o(1) very small, infs∈[α,(1-δ)] (1η) 2 s κ κ-1 (1κ) + κs -1 + o(1)is positive by concavity of the function s → s κ κ-1 (1κ) , therefore as an easy consequence J 1 ≪ E W e -λ * H(v) .

J 3

 3 < e -(1+δ)ελ * F ((1 + δ)ε) + λ * ∞ (1+δ)ε e -λ * x F (x)dx for β > 0, as F (x) ≤ 1 λ * ∞ βε e -λ * x F (x)dx ≤ e -βλ * ε = exp -β(Cκ) therefore for some β depending on κ, R(ε) := λ * ∞ βε e -λ * x F (x)dx ≪ E W e -λ * H(v) .by the same argument as for J 1 , we get that, for any ε < x < βε, for v large enough, F (x) ≤ e G(x) ; thereforeJ 3 -R(ε) ≤ e -(1+δ)ελ * e G((1+δ)ε) + λ * βε (1+δ)ε e -λ * x e G (x)dxBy the same computation as we did to get to (4.8), we haveJ 3 E W e -λ * H(v) < o(1)+ P exp -(Cv) (1κ) + κs -1 + o(1) .

4. 3

 3 Proof of Lemma 4.1.Let ν = 1 -2κ, andD ν = τr 0 g -1 (γ(s)) ν exp -4 g -1 (γ(s)) -8λg -1 (γ(s)) ds. = ∞ -∞ g -1 (s) ν exp -4 g -1 (s)-8λg -1 (s) L s τr ds.

  some constant c > 0, for all x ≤ 0 and λ < 1

  κ ds ≤ g(y) ≤ c

ν∨0 x - 2 2 ∞ 1 log

 221 L x τr dx ≤ c ′ (g(a)) -1/

u 1 -

 1 (ν+κ) du.Recalling the definition of ν we have ν + κ = 1κ > 0, thenI 2 = r(1 + o(1)) Γ(1κ) (4λ) 1-κ .This finishes the proof of the first part of lemma 4.1, as 1κ > ν ∨ 0. To treat the case ν = -2κ, let b < 1 be such that b → 0 and -g(b) = o r log log r . As before, we separate the integral as follows

y

  1+κ dy.The first part converges, by dominated convergence, to

INTRODUCTION.

Acknowledgement : We are thankful to Yueyun Hu for pointing this subject to us and for many helpful discussions, and to an anonymous referee for detailed and constructive remarks on the manuscript.

research partially supported by the ANR project MEMEMO.

-8λg -1 (γ((s)) ds.

Then, whenever λ → 0 and r ≫ log(1/λ) log log(1/λ),

and for some positive constant D,

Let us use this lemma to finish the proof of Theorem 1.4. We get easily that µ -1 (τ r ) = D -2κ (r). Whence, for some constant D ′ ,

almost surely, as λ → 0 and t ≫ log(1/λ) log log(1/λ). Therefore, under the same assumptions, for some constant D ′′ ,

Thus, going back to Kotani's lemma, for t > 0, and for some constant C, we get, as

)λ κ v), P -a.s.. (4.1) By application of Chebyshev's inequality, for λ as before,

We call λ(x) the value of lambda that minimizes λx -Cvλ κ . It is clear that λ(x) is a decreasing function of x, such that

Let λ * = λ v u 1/κ , we get easily the expression

One can easily check that λ * → 0, v ≫ log(1/λ * ) log log log(1/λ * ). Therefore we can apply the precedent estimate to get lim sup

Quenched Speedup for the diffusion.

In this section we prove Theorem 1.3. The upper bound is a trivial consequence of Theorem 1.4 , since

To get the lower bound, let ε > 0. Note that

Note that almost surely, for t large enough, we can find

It is clear that

. By the same computations as in 3.2.2, one gets easily that

for t large enough. Taking the limit as ε → 0, this finishes the proof of Theorem 1.3.


Jonathan J Healey

Instabilities and transition in three-dimensional flows with rotation GLOBAL INSTABILITY OF FLOWS ACROSS A JUNCTION

Global instability in infinite domains

Hydrodynamic stability theory originally developed through the study of flows where the streamwise development of the flow could be neglected. Under this parallel-flow approximation linearized disturbance equations can be reduced to ordinary differential equations, like the Rayleigh or Orr-Sommerfeld equations, by representing an arbitrary disturbance by a superposition of waves. The results of these 'local' stability theories are assumed to be applicable close to the chosen point in the flow where the velocity profile has been extracted from the streamwise-developing flow, and made parallel by ignoring its streamwise-dependence. The issue of how these local stability results can be interpreted to predict the stability of the whole flow is referred to as the 'global' stability problem.

When the basic flow varies strongly in the streamwise direction nonparallel effects cannot be ignored, local results are meaningless, and the global stability problem must usually be approached by solving partial differential disturbance equations numerically. But when the basic flow only varies slowly in the streamwise direction, the local theory is the correct leading order approximation in a WKB theory that allows weak nonparallel effects to be included. In this case global stability properties can be inferred from local stability properties.

Global instability is related to local absolute instability, which describes the growth or decay of disturbances in the rest frame. Consideration of the upstream and downstream responses to periodic forcing at a point in a weakly inhomogeneous flow leads to the conclusion that the global mode is given at leading order by the absolute instability at a saddle point where ∂ω 0 /∂x = 0, where ω 0 is the local absolute frequency and x the streamwise coordinate (this saddle is a turning point in the WKB theory), see [START_REF] Soward | The linear stability of the flow in the narrow gap between two concentric rotating spheres[END_REF], [START_REF] Chomaz | A frequency selection criterion in spatially developing flows[END_REF] and the review by [START_REF] Huerre | Open shear flow instabilities[END_REF]. If Re(ω 0 ) does not depend on x, i.e. if there is no detuning, then the saddle ∂ω 0 /∂x = 0 lies on the real x-axis, and the intuitively reasonable expectation that the x station with greatest absolute growth rate should act as the wave-maker for the flow holds. Less obvious is the fact that when there is detuning the saddle will lie at a complex value of x and then the global instability given by ω 0 at the saddle will have smaller growth rate than that of the most unstable absolute instability on the real x-axis.

2 Global instability in semi-infinite domains [START_REF] Chomaz | Models of hydrodynamic resonances in separated shear flows[END_REF] considered global instability in a semi-infinite domain in which the absolute growth rate decreases with downstream distance, as a model for wake flow, and they found that the global mode was given by ω 0 at the upstream boundary. [START_REF] Healey | Model for unstable global modes in the rotating-disk boundary layer[END_REF] considered global instability in a domain in which the absolute growth rate increases with downstream distance, but the detuning effect is strong enough to maintain global stability. [START_REF] Davies | Global stability of the rotating-disc boundary layer[END_REF] argued that this global stabilization by detuning explained the global stability observed in the numerical solutions of the linearized Navier-Stokes equations for disturbances to the rotating-disc boundary-layer obtained in [START_REF] Davies | Global behaviour corresponding to the absolute instability of the rotating-disk boundary layer[END_REF], and observed in experiments on the same flow by [START_REF] Othman | Experimental investigation of absolute instability of a rotating-disk boundary layer[END_REF]. However, [START_REF] Healey | Model for unstable global modes in the rotating-disk boundary layer[END_REF] showed that imposing a downstream boundary condition in this type of flow in a region of local absolute instability can create a global instability driven by the absolute instability at the downstream boundary. In rotating-disc flow the radial coordinate is the relevant streamwise coordinate, suggesting that the edge of the disc could be crucial in creating global instability.

Global instability in semi-infinite domains joined by a junction

The present work is motivated by these findings. They encourage us to explore the local stability properies near the edge of the disc in more detail. In particular, we seek to form a global mode in which the disturbance solution in the flow over the disc is matched to the disturbance solution in the flow beyond the edge of the disc.

Existing experiments on the stability of the rotating disc have not included measurements of velocity profiles for the flow beyond the edge of the disc. [START_REF] Smith | A note on a wall jet negotiating a trailing edge[END_REF] used interactive boundary-layer theory to calculate the flow separating from the edge of a thin disc when there is von Kármán flow [START_REF] Th | Über laminare und turbulente Reibung[END_REF] over both the upper and lower surfaces of the disc. However, most stability experiments use a thick disc to reduce the effects of vibration, and the substantial mountings and motor arrangements below the disc are likely to prevent von Kármán flow from being established on the lower surface. Although a detailed study of the flow beyond the edge of a rotating disc used in a stability experiment would be desirable, in the present work we take a similar approach to that taken EUROMECH Colloquium 525, 21-23 June 2011, Écully, France Instabilities and transition in three-dimensional flows with rotation in [START_REF] Healey | Model for unstable global modes in the rotating-disk boundary layer[END_REF], and explore instead the qualitative phenomena that arise when one considers the global instability of flow across a junction using the Ginzburg-Landau equation. Nonetheless, the stability and propagation properties of the flows on either side of the junction are taken from stability calculations for a simplified model of von Kármán flow over the disc, and for two simplified, but plausible, flows that could exist beyond the edge of the disc.

Results

Our most important finding is that in some circumstances the global mode can be more unstable than the local absolute instability on either side of the junction. For example, a stable flow upstream of a junction connecting to a downstream flow that is only convectively unstable can nonetheless be globally unstable, despite the absence of any local absolute instability. This arrangement can arise in a wide variety of practical applications, not just at the edge of a rotating disc, e.g. when stable flow in a nozzle becomes a convectively unstable jet flow after leaving the nozzle. The nozzle then acts as a junction between a channel (or pipe) flow and the jet. Another example is when a boundary layer separates from a surface: the separation point then acts as a junction between (possibly) stable attached flow, and convectively unstable separated flow. This applies both to the case of separation from a sharp edge and separation from a smooth surface. The origins of this unexpected behaviour will be explained.


Alexander Ni

Instabilities AND TRANSITIONS IN THREE-DIMENSIONAL FLOWS WITH ROTATION Structural Transitions and Stability Issues in Swirling Compressible Flows

A theory similar to that of Laval Nozzle is developed for swirling compressible flows in annular axisymmetric ducts. On the basis of this theory structural transitions in these flows and their stability are examined. The goal of the study is to provide a simple unified view of the topic, which gives insight into the key physical features, and the overall parametric dependence of the transitions behavior.

Vortex breakdown and shock waves are well-known flow structural transitions in many flows of technological interest. Being a useful flow element in gas turbine combustors as a flame holder, a vortex breakdown may stimulate a flow separation and corresponding undesirable losses in diffusers. Control of a vortex breakdown and shock wave is a serious technical challenge. Theory of shock waves is well advanced. In classical gas dynamics shock wave represents a surface where flow parameters change by jump, being connected by Hugoniot-Rankine conditions. In some cases this approximation does not work, and internal shock wave structure must be considered. The general theory of a vortex breakdown describing quantitatively all its aspects still does not exist despite the importance of the problem, and the efforts expended on it over a period of more than fifty years. This is due to extreme complexity of the flow.

Internal compressible swirling flows are of a growing interest for needs of modern technologies: supersonic compressors, supersonic separators in oil-processing industry etc. One of the main challenges of these technologies is a reduction of pressure losses. It makes an understanding of flow transitions from subsonic to supersonic regimes and vice versa to be the key issue for proper aerodynamic channel shape design. Currently a main theoretical instrument for complex flow prediction is computational fluid dynamics. However, analytical methods like the Laval Nozzle theory still preserve their importance for flow evaluations. Their advantage is in ability to capture main physical factors governing complex flows. In the paper a theory similar to that of Laval Nozzle is developed for swirling compressible flows. The presented analysis predicts existence of both structural transitions corresponding to vortex breakdown at low Mach numbers, and those analogous to shocks in supersonic velocities range. As distinct from non-swirled flows the "evolutionarity" (stability) of transitions is controlled by inertia waves. This fact was postulated for vortex breakdown in incompressible flows by Benjamin in [START_REF] Benjamin | Theory of the vortex breakdown phenomenon[END_REF] and is generalized in the article to the case of compressible flows. It is established that for each type of transition a wave mode responsible for its "evolutionarity" exists. Benjamin's concept of flow transitions is linked closely with duct geometry. Benjamin treated a flow transition as a jump between flow states having same flow radial boundaries and flow integrals. These states can be connected virtually in a continuous way by introduction a duct of an appropriate shape. It is shown that the virtual duct must contain a critical cross-section (its geometrical interpretation is given in the text) where the speed of a corresponding to the transition wave mode turns to zero. The number of possible critical duct cross-sections coincides with the number of possible transitions and depends on inlet flow parameters distribution. In all previous publications only first eigenmodes were considered to propagate in swirling flow. We show that higher orders eigenmodes are also possible. The proposed in the article approach allows easily investigate flows with arbitrary inlet density, entropy, velocity distributions. It is particularly important for analysis of flows in combustors where variations of these parameters are large. The approach provides guidelines for design of transitions-free channels.

Stability analysis of swirling flows relative 3D helical perturbations shows strong influence of losses to flow stability.


Tobias Luginsland

Leonhard Kleiser

Instabilities and transition in three-dimensional flows with rotation LARGE EDDY SIMULATION OF SWIRLING JET FLOW UNDERGOING VORTEX BREAKDOWN INCLUDING NOZZLE MODELING

Scientific background and objectives

Swirling jets undergoing vortex breakdown occur in many technical applications, e.g. vortex burners, turbines and jet engines. At the stage of vortex breakdown the flow is dominated by a conical shear layer and a large recirculation zone around the jet axis. This highly nonlinear flow state additionally features hysteresis behaviour, bistability, i.e. different flow states occuring at the same values of control parameters, and bifurcation. Despite decades of intense research there is still no consensus found about the underlying mechanism leading to this abrupt change of flow [1]. In their experimental investigation Liang & Maxworthy [2] suggest the existence of a pocket of absolute instability in the wake of the jet triggering a global mode overwhelming the whole flow. In a recent investigation [3] a co-rotating, counter-winding helical mode is found with its 'wave-maker' located in the inner region of the jet. A preceding investigation of our group [4] showed that our DNS/LES code CONCYL is capable of capturing the fundamental physics of swirling jets undergoing vortex breakdown, see also [5]. The results showed good qualitative agreement with experimental data for a self-excited and a forced jet. The fact that the breakdown occurred directly after the inflow plane was considered to possibly prevent the development of self-excited instabilities due to the presence of an inflow sponge layer. To account for more realistic inflow boundary conditions, in the present work a rotating nozzle is included in the computational domain [6]. A Re=5000, Ma=0.6 swirling jet with the azimuthal velocity component as high as the streamwise component (swirl number S=1.0) is investigated by LES along the lines of [4]. The simulation is done on a domain with size L r × L z = 10R × 20R in the radial and streamwise directions with a nozzle of length L n = 6R included, where R is the nozzle radius. The grid resolution is N θ × N z × N r = 36 × 180 × 216 grid points. Our work is in progress and preliminary results are presented and will be compared to recently published results in the literature.

Simulation methodology and Results

Our simulation code, recently parallelized for massively parallel architectures, solves the compressible Navier-Stokes equations in cylindrical coordinates [7]. Spatial derivatives are calculated using high-order finite difference schemes (up to 10th order) for the streamwise and radial directions while a Fourier spectral method is employed in the circumferential direction. Time integration is done applying a 4th-order 6th-stage LDDRK scheme developed for investigating aeroacoustics of jets [8]. As a subgrid scale model we use our in-house Approximate Deconvolution Model (ADM). Boundary conditions are chosen in analogy to [4] and the nozzle is modelled as an isothermal wall. Figure 1 shows the development of θ, t -averaged profiles for all three velocity components. At the jet axis a deficit in the streamwise velocity develops, with the profile changing from jet-to wake-like within the downstream part of the nozzle indicating beginning vortex breakdown. At the same downstream position the azimuthal velocity profile start to deviate from the initial solid-body rotation. Behind the nozzle both velocity components are largest within the azimuthal shear layer accompanied with a rising radial velocity component related to the massive spreading of the jet. The recirculation is strongest around z = 7.5R representing the core of the recirculation bubble, which has a length of about 2R. Within the nozzle pressure and density (not shown) increase in the downstream direction, adapting to far-field values right behind it. Instantaneous contours of the streamwise, w, and azimuthal, v, velocity components are plotted in figure 2. The local recirculation near the centerline is much stronger compared to the averaged data. The rapid breakup of the conical shear layers is visible. In the recirculation zone the fluid counter-rotates with respect to the mean flow, an effect also reported by Facciolo et al. [9]. Compared to results stated in [4] the spreading angle of the jet is larger. An analysis of the instantaneous stagnation point location shows that it travels upstream, shifting up-and downstream around z = 6.5R precessing around the jet axis in a counter-clockwise manner. Iso-surface plots of the pressure show a helical, counter-winding mode, co-rotating with the mean flow, similar to structures observed by Billant et al. [10] for the post-breakdown stage.

Figure 1 :

 1 Figure 1: Axial development of mean streamwise, azimuthal and radial velocity (top to bottom). The grey bar indicates the position of the nozzle wall.

Figure 2 :

 2 Figure 2: Contour plots of the instantaneous streamwise (left) and azimuthal (right) velocity at 20 flow-through times.


Instabilities and transition in three-dimensional flows with rotation

INSTABILITIES IN TAYLOR-COUETTE-POISEUILLE FLOW WITH POROUS WALLS

Denis MARTINAND 1 , Nils TILTON 1 & Eric SERRE 1 & Richard M. LUEPTOW 2 1 LM2P2 UMR 6181 CNRS Aix-Marseille Université, Marseille, France. 2 Department of Mechanical Engineering, Northwestern University, Evanston IL, USA.

We consider Taylor-Couette-Poiseuille flow developing between an outer, fixed, impermeable cylinder and a concentric, inner, rotating, permeable cylinder with radial suction, see figure 1. This system is useful for dynamic filtration because the shear due to the rotating cylinder and, in the case of supercritical flow, the Taylor vortices, wash contaminants away from the permeable cylinder and prevent fouling. The fluid mechanics of this system are not fully understood due to the coupling between the axial pressure drop, which drives the axial Poiseuille flow, and the transmembrane pressure difference which drives the suction. In addition to the filtration flux, this coupling induces axial variations of the velocity field. These variations eventually modify the nature of the subcritical flow which can evolve from suction to injection (cross flow reversal) or consume the whole axial flow (axial flow exhaustion). Moreover, owing to the axial and radial flows and their variations along the axial direction, the stability of this flow strongly departs from that of Taylor-Couette flow.

Analytical approach

Because filtrating devices utilize membranes with small permeability and suction, we propose an asymptotic solution to the subcritical flow assuming a slow axial variation of the velocity and pressure fields [START_REF] Tilton | Pressure-driven radial flow in a Taylor-Couette cell[END_REF]. The transmembrane suction is coupled with the pressure through Darcy's law. The obtained analytical approximation correctly captures the axial variations of the velocity field, e.g. the two aforementioned possible behaviors. This laminar flow is then used as a base state to study the appearance of centrifugal instabilities in the form of Taylor vortices developing in the most common configuration, i. e. when the mean axial flow decreases downstream due to filtration but do not reverse. According to the theory of nonlinear global modes in slowly varying open flows [START_REF] Pier | Bifurcation to fully nonlinear synchronized structures in slowly varying media[END_REF], the unstable state is expected to form a front at the axial location where the flow undergoes a transition from local convective unstability to local absolute instability. This front acts as a wavemaker and selects the frequency of the vortices.

Numerical approach

These analytical results for the subcritical and supercritical flows are then compared with dedicated spectral direct numerical simulations implementing Darcy's law on the inner cylinder. These numerical simulations require some special care to correctly handle the axial and transmembrane flows, together with solving the pressure field, in order to avoid numerical noises prone to trigger uncontroled extrinsic instabilities.

Results

As seen in figure 2, global synchronized modes governed by local absolute instabilities are retrieved in numerical simulations. These numerical results, concerning for instance the features of these instabilities such as the location of the front or the frequency, are in good agreement with the analytical prediction based on previous

Figure 1 :

 1 Figure 1: Sketch, not to scale, of the annular geometry and laminar base flow.

EUROMECHFigure 2 :

 2 Figure2: Temporal evolution of the axial velocity as a function of the axial coordinate, the rotating Reynolds number is set to 120, the axial Reynolds number varies from 4.32 (inlet) to 3.24 (outlet). (--): Boundary between the locally convectively and absolutely unstable regions


Innocent Mutabazi

R Guillerm

A Prigent

V Lepiller

FRE31O2 S Malik Lomc

Instabilities and transition in three-dimensional flows with rotation FLOW INSTABILITIES IN A VERTICAL DIFFERENTIALLY ROTATING CYLINDRICAL ANNULUS WITH A RADIAL TEMPERATURE GRADIENT

Introduction

Flow in a differentially rotating cylindrical annulus with a radial temperature gradient is encountered in many industrial applications [START_REF] Kreith | Convection Heat Transfer in rotating systems[END_REF]. It has been investigated experimentally and numerically by few authors [START_REF] Snyder | Experiments on the stability of Couette motion with a radial thermal gradient[END_REF][START_REF] Weidman | On the stability of circular Couette flow with a radial heating[END_REF][START_REF] Kuo | Taylor-Couette flow with buoyancy: onset of spiral flow[END_REF][START_REF] Lepiller | Weak temperature gradient effect on the stability of the circular Couette flow[END_REF][START_REF] Guillerm | Etude expérimentale des instabilits thermo-hydrodynamiques dans un systme de Couette-Taylor[END_REF]. For small values of the control parameter,away from the boundaries, the base flow has two velocity profiles each of which is unstable. The rotation induces circular Couette flow which is potentially unstable to centrifugally driven perturbations leading to longitudinal vortices. The radial temperature gradient induces a baroclinic vertical flow (ascending near the hot surface and descending near the cold one). This flow has a velocity profile with an inflexion point and it is potentially unstable to transverse oscillatory perturbations. The present study is concerned with experimental and linear stability analysis of this flow when the driving forces (control parameters) are increased in magnitude.

Experimental setup and stability analysis

The experimental setup consists of two coaxial cylinders of height L = 55.9 cm and a gap d = 0.5 cm. The aspect raio of the system is Γ = 111.8 and the radius ratioη = 0.8. The inner cylinder is rotating at angular frequency Ω while the outer cylinder is fixed. Inside the inner cylindrical tube was circulating a water low maintained at controlled temperature T 1 and the outer cylinder was immersed into a large thermal bath maintained at controlled temperature T 2 . The working fluid is a deionized water with a kinematic viscosity ν = 10 -2 cm 2 /s at T = 295K. The flow is described by three control parameters : the Prandtl number P r = ν/κ, the Grashof number Gr = αδT gd 3 /ν 2 which measures the magnitude of the temperature gradient on the flow and the Taylor number T a = (Ωad/ν)(d/a) 1/2 related to rotation and therefore counting for centrifugal effects. The flow was visualized by adding a suspension of Kalliroscope AQ1000 in 2% by volume. To visualize the temperature fields, we seeded the flow with SR25C5W thermochromic liquid crystals from Hallcrest of about by 0.05%. They occur in form of encapsulated spheres of mean diameter 75m and their response time to a temperature change is about 3ms. These particles have also been used to determine the velocity components, using a Basler IEEE-1394 camera, by tacking pictures of the cross section with a time delay of 82 ms.

Experimental results

A radial temperature gradient imposed on the cylindrical surfaces of the flow annulus induces a large convective cell with particules ascending near the hot wall and descending near the cold one. For a fixed value of Grashof number, we have increased the rotation rate until we obtained a bifurcation to a new state formed of pattern of helicoidal vortices (Figure 1-a). All critical states are represented in the diagram (Gr, T a) in Figure 1-b. For chosen state, we have measured the velocity field and the temperature in the cross section (Figure 2).

Linear stability analysis

The flow is governed by the Navier-Stokes equations, energy equation and mass conservation equations written in cylindrical coordinates together with no-slip boundary conditions and isothermal cylindrical surfaces. Neglecting the end effects near the top and bottom plates, the base flow velocity profile is v = V (r) e θ + W (r) e z where V (r) and W (r) are the circular Couette profile and the baroclinic axial velocity component respectively; their expressions can be found in [START_REF] Lepiller | Weak temperature gradient effect on the stability of the circular Couette flow[END_REF]. We have performed linear stability of this flow assuming an infinite length of the system. We found that critical modes are nonaxisymmetric oscillating vortices. The variation of critical values of T a with Gr are in a very good agreement with experimental ones. The corresponding state diagrams are well superimposed.

Conclusion

We have performed thourough investigation of stability of the flow between differentially rotating annulus with a radial temperature gradient. We have found a good agreement between experimental results, those from linear stability analysis and from numerical simulations [START_REF] Yoon | Effect of radial temperature gradient on Taylor vortex flow[END_REF].

Figure 1 :

 1 Figure 1: a) Pattern obsevred for T a = 24 and Gr=706. b) Diagram of bifurcations in the plane Gr, T a).

Figure 2 :

 2 Figure 2: a) Space-time diagram of the pattern obsevred for Ta= 24 and Gr=706. b) Temperature distribution in the gap, c) Vorticity distribution in the gap.


Valery Okulov
email: vaok@mek.dtu.dk

DEVELOPMENT OF HELICAL VORTEX THEORY

.

At present the theory of 2-D helical vortex dynamics are based on various analytical components, which holds true for all values of the helix pitch, such as (i) the 2-D Biot-Savart law for helical filaments represented by Kapteyn series [START_REF] Hardin | The velocity field induced by a helical vortex filament[END_REF] or rewriting in a form with singularity separation [START_REF] Okulov | Determination of the velocity field induced by vortex filaments of cylindric and weak conic shapes[END_REF] (fig. 2); (ii) solutions of helical vortex tubes with finite core, governed by series expansion of helical multipoles [START_REF] Fukumoto | The velocity field induced by a helical vortex tube[END_REF]; (iii) relations between the induction of vortex filaments and the self-induced velocity of helical vortex tubes [START_REF] Boersma | On the self-induced motion of a helical vortex[END_REF] resulting in a closed analytical solution of the helix motion [START_REF] Okulov | On the stability of multiple helical vortices[END_REF]; (iv) analytical representation of Goldstein's solution for the circulation of a helical vortex sheet in equilibrium [START_REF] Goldstein | On the vortex theory of screw propellers[END_REF]; (v) Kelvin's N-gon stability problem of point vortices generalized to multiple helical vortices [START_REF] Okulov | On the stability of multiple helical vortices[END_REF][START_REF] Okulov | Stability of helical tip vortices in a rotor far wake[END_REF].

Figure 1 .

 1 Figure 1. Examples of 2-D flows with rectilinear (a) and helical vortices (b); 3-D paths of fluid particles induced by single helical vortex filament in a cylinder (c).

Figure 2 .

 2 Figure 2. Isolines of the azimuthal velocity field induced in cross-section by helical vortex filament: (a) inaccurate simulation by finite part of the Kapteyn series with unreal singularities on cylinder supporting the helix; (b) accurate simulation after separation of the singularity along the helix filament in the Kapteyn series [3].

Figure 3 .

 3 Figure 3. Two possibilities for equilibrium of helical vortex pairs and its prototypes from point vortex pairs: (a) centre or axis of rotation lies on the same hand from both point vortices; (b) centre of or axis rotation lies between ones

Figure 4 .

 4 Figure 4. The circulation ratio β of the two helical vortices in equilibrium rotation as function of their relative proximity for different values of helical pitch h: 1.5 (dash-dotted lines); 2(dashed lines); 5 (dotted lines); infinity (solid lines). Circle and point indicate equilibrium of the helical pairs with different levers.