Random walk in random environment in a two-dimensional stratified medium with orientations
RRANDOM WALK IN RANDOM ENVIRONMENT IN ATWO-DIMENSIONAL STRATIFIED MEDIUM WITH ORIENTATIONS
ALEXIS DEVULDER AND FRANÇOISE PÈNE
Abstract.
We consider a model of random walk in Z with (fixed or random) orientation of thehorizontal lines (layers) and with non constant iid probability to stay on these lines. We provethe transience of the walk for any fixed orientations under general hypotheses. This contrastswith the model of Campanino and Petritis [3], in which probabilities to stay on these lines areall equal. We also establish a result of convergence in distribution for this walk with suitablenormalizations under more precise assumptions. In particular, our model proves to be, in manycases, even more superdiffusive than the random walks introduced by Campanino and Petritis. Introduction
In this paper we consider a random walk ( M n ) n starting from on an oriented version of Z .Let ε = ( ε k ) k ∈ Z be a sequence of random variables with values in {− , } and joint distribution µ . We assume that the k th horizontal line is entirely oriented to the right if ε k = 1 , and to theleft if ε k = − . We suppose that the probabilities p k to stay on the k th horizontal line are givenby a sequence of independent identically distributed random variables ω = ( p k ) k ∈ Z (with valuesin (0 , and joint distribution η ) and that the probability to go up or down are equal.More precisely, given ε and ω , the process ( M n = ( M (1) n , M (2) n )) n is a Markov chain satisfying M = (0 , with transition probabilities given by : P ε,ω ( M n +1 − M n = ( ε M (2) n , | M , ..., M n ) = p M (2) n and ∀ y ∈ {− , } , P ε,ω ( M n +1 − M n = (0 , y ) | M , ..., M n ) = 1 − p M (2) n . kIf =1 k p If =-1 k k-1k+1 k-1k+1 E E k k p k (1-p )/2 k (1-p )/2 k (1-p )/2 k (1-p )/2 k (1-p )/2 k (1-p )/2 k (1-p )/2 k (1-p )/2 k (1-p )/2 k (1-p )/2 k (1-p )/2 k (1-p )/2 k p k p k p k p Layer k Layer k
We also define the annealed probability P as follows: P ( . ) := (cid:90) P ε,ω ( . ) dη ( ω ) dµ ( ε ) . Date : November 1, 2018.2010
Mathematics Subject Classification.
Key words and phrases. random walk on randomly oriented lattices, random walk in random environment,random walk in random scenery, functional limit theorem, transience.This research was supported by the french ANR project MEMEMO2 2010 BLAN 0125. a r X i v : . [ m a t h . P R ] N ov WRE IN A STRATIFIED ORIENTED MEDIUM 2
We denote by E and E ε,ω the expectations with regard to P and P ε,ω respectively.Our model corresponds to a random walk in a two dimensional stratified medium with orientedhorizontal layers and with random probability to stay on each layer.The model with p k = 1 / and with the ε (cid:48) k s iid and centered can be seen as a discrete versionof a model introduced by G. Matheron and G. de Marsily in [17] to modelize transport in astratified porus medium. This discrete model appears in [2] to simulate the Matheron and deMarsily model. It has also been introduced, separately, by mathematicians with motivationslinked to quantum field theory or propagation on large networks (see respectively [3] and [4] andreferences therein).In [3], M. Campanino and D. Petritis proved that, when the p k ’s are all equal, the behaviorof the walk ( M n ) n depends on the choice of the orientations ( ε k ) k . First, they prove that thewalk is recurrent when ε k = ( − k (i.e. when the horizontal even lines are oriented to the rightand the uneven to the left). Second, they prove that the walk is almost surely transient whenthe ε (cid:48) k s are iid and centered. These results have been recently improved in [4]. Let us mentionthat extensions of this second model can be found in [8, 19], and that its Martin boundary iscomputed in [15].In order to take into account the different nature of the successive layers of a stratified porusmedium, it is natural to study the case where the p k ’s are random instead of being all equal. Inthis paper, we prove that taking the p k ’s random and i.i.d. can induce very different behaviorsfor the random walk.First, we prove that under general hypotheses, the random walk is transient for every deter-ministic or random orientations, contrarily to the results obtained by Campanino and Petritisin [3] and [4] for their model. Hence, even very small random perturbations of their (constant) p k ’s transform their recurrent walks into transient ones.Second, it was proved in [9] that when the p k ’s are all equal, the random walk is superdiffusive,and that the horizontal position at time n is, asymptotically, of order n / . This was conjecturedin [17] and was one main motivation for the introduction of this model. We prove that, dependingon the law of p , our model can be even more superdiffusive, with horizontal position at time n of order n δ , where δ can take all the values in [3 / , .More precisely, our results are the following. We start by stating our theorem about transience. Theorem 1.
Let ( p k ) k be a sequence of independent identically distributed random variables.Suppose here that p is non-constant and that E [(1 − p ) − α ] < ∞ (for some α > ). Then, forevery deterministic or random sequence ( ε k ) k , the random walk ( M n ) n is transient for almostevery ω . We now give a functional theorem under more precise hypotheses. In particular, we will assumethat p − p is integrable and that the distribution of p − p − E (cid:104) p − p (cid:105) belongs to the normal domainof attraction of a strictly stable distribution G β of index β ∈ (1 , , which means that P (cid:32) n − /β n (cid:88) k =1 (cid:18) p k − p k − E (cid:20) p − p (cid:21)(cid:19) ≤ x (cid:33) → n → + ∞ G β ( x ) , x ∈ R , (1)the characteristic function ζ β of G β being of the form ζ β ( θ ) := exp[ −| θ | β ( A + iA sgn ( θ ))] , θ ∈ R , (2)with A > and | A − A | ≤ | tan( πβ/ | . Notice that this is possible iff A = A tan( πβ/ (since p − p ≥ a.s., see e.g. [12, thm 2.6.7]). WRE IN A STRATIFIED ORIENTED MEDIUM 3 If β ∈ (1 , , we consider two independent right continuous stable processes ( Z x , x ≥ and ( Z − x , x ≥ , with characteristic functions E ( e iθZ t ) = exp[ − A | t || θ | β ] , t ∈ R , θ ∈ R . If β = 2 , we denote by Z a two-sided standard Brownian motion. We also introduce a standardBrownian motion ( B t , t ≥ , and denote by ( L t ( x ) , x ∈ R , t ≥ the jointly continuous versionof its local time. We assume that Z and B are defined in the same probability space and areindependent processes. We now define, as in [14], the continuous process ∆ t := (cid:90) R L t ( x ) dZ x , t ≥ . We prove the following result.
Theorem 2.
Let ( p k ) k be a sequence of independent identically distributed random variables withvalues in (0 , . Suppose here that E (cid:104) p − p (cid:105) < ∞ and that the distribution of p − p − E (cid:104) p − p (cid:105) belongs to the normal domain of attraction of a strictly stable distribution of index β ∈ (1 , (i.e.that we have (1) and (2) ).We also assume that ( ε k ) k satisfies one of the following hypotheses : (a) for every k , ε k = ( − k , (b) ( ε k ) k is a sequence of independent identically distributed centered random variables withvalues in {± } ; ( ε k ) k is independent of ( p k ) k .Then, setting δ := + β , the sequence of processes (cid:18)(cid:16) n − δ M (1) (cid:98) nt (cid:99) , n − / M (2) (cid:98) nt (cid:99) (cid:17) t ≥ (cid:19) n converges in distribution under the probability P (in the space of Skorokhod D ([0; + ∞ ) , R ) ) to ( γ − δ σ ∆ t , γ − / B t ) t ≥ with γ := 1 + E (cid:104) p − p (cid:105) and with : * σ = (cid:16) V ar (cid:16) p − p (cid:17)(cid:17) / in case (a) with β = 2 , * σ = (cid:18) E (cid:20)(cid:16) p − p (cid:17) (cid:21)(cid:19) / in case (b) with β = 2 , * σ = 1 in cases (a) or (b) with β ∈ (1 , . We remind that p − p has a finite variance if β = 2 (see e.g. [12, Thm 2.6.6.]), hence σ is finitein all cases.The proof of this second result is based on the proof of the functional limit theorem establishedby N. Guillotin and A. Le Ny [9] for the walk of M. Campanino and D. Petritis (with ( p k ) k constant and ( ε k ) k centered, independent and identically distributed).It may be possible that the transience remains true for every non degenerate distribution of the p k ’s on (0 , . Indeed, roughly speaking, taking the p k ’s closer to one should make the randomwalk even more transient; however this is just an intuition and not a mathematical evidence. Weprove our Theorem 1 under a very general moment condition, which covers all the cases of ourTheorem 2. In particular, the most superdiffusive cases, with δ > / , are obtained when thesupport of / (1 − p ) is not compact.The proof of our first result is built from the proof of [3, Thm 1.8] with many adaptations.The idea is to prove that, when ( ε k ) k ∈ ZZ is a fixed sequence of orientations, that is when µ is a WRE IN A STRATIFIED ORIENTED MEDIUM 4
Dirac measure, (cid:88) k ≥ P ( M k = (0 , < + ∞ . (3)In the model we consider here, contrarily to the models envisaged in [3], the second coordinateof ( M n ) n is not a random walk but it is a random walk in a random environment, since theprobability to stay on a horizontal line depends on the line, which complicates the model. Evenif a central limit theorem and functional limit theorem have been established in [11] and in [10]for M (2) n , the local limit theorem for M (2) n has not already been proved, to the extent of ourknowledge. Moreover, in Theorem 1 we do not assume that the distribution of p − p belongs tothe domain of attraction of a stable distribution. For these reasons, it does not seam simple tomake a precise estimation of P ( M n = (0 , as it has been done in [5]. We also mention thatthe random walk ( M n ) n is not reversible.It will be useful to observe that under P ε,ω and P , ( M (2) T n ) n is a simple random walk ( S n ) n on Z , where the T n ’s are the times of vertical displacement : T := 0; ∀ n ≥ , T n := inf { k > T n − : M (2) k (cid:54) = M (2) k − } . We will use several times the fact that there exists
M > such that, for every n ≥ , we have P ( S n = 0) ≤ M n − . Now, let us write X n the first coordinate of M T n . We observe that X n +1 − X n = ε S n ξ n , where ξ n := T n +1 − T n − corresponds to the duration of the stay on the horizontal line S n afterthe n -th change of line. Moreover, given ω = ( p k ) k ∈ Z , ε = ( ε k ) k ∈ ZZ and S = ( S k ) k , the ξ k ’s areindependent and with distribution given by P ε,ω ( ξ k = m | S ) = (1 − p S k ) p mS k for every k ≥ and m ≥ . With these notations, we have X n = n − (cid:88) k =0 ε S k ξ k . This representation of ( M T n ) n will be very useful in the proof of both the results.2. Estimate of the variance
To point out the difference between our model and the model with ( p k ) k constant consideredby M. Campanino and D. Petritis in [3], we start by estimating the variance of X n under theprobability P for these two models in the particular case when ε k = ( − k for every k ∈ Z andwhen (1 − p ) − is square integrable. Proposition 3.
Let ε k = ( − k for every k ∈ Z . (1) If the p k ’s do not depend on k , then V ar ( X n ) = E (cid:104) p (1 − p ) (cid:105) n . (2) If the (1 − p k ) − ’s are iid, square integrable with positive variance, then there exists C > such that V ar ( X n ) ∼ n → + ∞ Cn / .Proof of Proposition 3. We observe that E ε,ω [ ξ k | S ] = p S k − p S k and V ar ε,ω ( ξ k | S ) = p S k (1 − p S k ) . Moreover, S is independent of ω under P , hence E ( X n ) = 0 . We have WRE IN A STRATIFIED ORIENTED MEDIUM 5
V ar ( X n ) = n − (cid:88) k,(cid:96) =0 E [( ξ k − ξ k +1 ) ( ξ (cid:96) − ξ (cid:96) +1 )]= n − (cid:88) k =0 E (cid:104) ( ξ k − ξ k +1 ) (cid:105) + 2 (cid:88) ≤ k<(cid:96) ≤ n − E [( ξ k − ξ k +1 ) ( ξ (cid:96) − ξ (cid:96) +1 )]= n − (cid:88) k =0 E (cid:34) p (1 − p ) + p (1 − p ) + (cid:18) p − p − p − p (cid:19) (cid:35) + 2 n − (cid:88) k =1 ( n − k ) E [( ξ − ξ ) ( ξ k − ξ k +1 )]= Cn + 2 n − (cid:88) k =1 ( n − k ) E (cid:20)(cid:18) p S − p S − p S − p S (cid:19) (cid:18) p S k − p S k − p S k +1 − p S k +1 (cid:19)(cid:21) . This gives the result in case (1). Now, to prove the result in case (2), we notice that, since p y and p y (cid:48) are independent as soon as y (cid:54) = y (cid:48) , we have E (cid:20)(cid:18) p − p − p S − p S (cid:19) (cid:18) p S k − p S k − p S k +1 − p S k +1 (cid:19)(cid:21) = E (cid:20) p − p p S k − p S k + p S − p S p S k +1 − p S k +1 (cid:21) − E (cid:20) p − p (cid:21) = 2 (cid:32) E (cid:20) p − p p S k − p S k (cid:21) − E (cid:20) p − p (cid:21) (cid:33) = 2 (cid:32) E (cid:20) p (1 − p ) (cid:21) − E (cid:20) p − p (cid:21) (cid:33) P ( S k = 0)= 2 V ar (cid:18) p − p (cid:19) P ( S k = 0) . We conclude as H. Kesten and F. Spitzer did in [14, p. 6], using the fact that P ( S k = 0) ∼ ck − / (as k goes to infinity) for some c > . (cid:3) Proof of Theorem 1 (transience)
We come back to the general case. It is enough to prove the result for any fixed ( ε k ) k . Let ( ε k ) k ∈ Z be some fixed sequence of orientations. Hence µ is a Dirac measure on {− , } ZZ . Withoutany loss of generality, we assume throughout the proof of Theorem 1 that ε = 1 and α ≤ . Wehave (cid:88) k ≥ P ( M k = (0 , (cid:88) n ≥ P ( S n = 0 and X n ≤ ≤ X n +1 ) . Hence, to prove the transience, it is enough to prove that (cid:88) n ≥ P ( S n = 0 and X n ≤ ≤ X n +1 ) < + ∞ . (4)This sum is divided into 8 terms which are separately estimated in Lemmas 4, 5, 6, 7, 8, 11, 12and 13 provided δ , δ , δ , δ are well chosen. One way to choose these δ i so that they satisfysimultaneously the hypotheses of all these lemmas is given at the end of this section. WRE IN A STRATIFIED ORIENTED MEDIUM 6
For every y ∈ Z and m ∈ N , we define N m ( y ) := { k = 0 , ..., m − S k = y } . We will usethe fact that X n = S n + D n with D n := (cid:88) y ∈ Z ε y p y − p y N n ( y ) and S n := n − (cid:88) k =0 ε S k (cid:18) ξ k − p S k − p S k (cid:19) . Roughly speaking, the idea of the proof is that X n ≤ ≤ X n +1 implies that X n cannot bevery far away from 0, which means that D n and S n should be of the same order, but this isfalse with a large probability. More precisely, we will prove that, with a large probability, wehave | D n | > n − δ and | S n | < n + α + υ for small δ > and υ > (see the definition of B n and the end of the proof of Lemma 6). Now let us carry out carefully this idea.Let n ≥ . Following [3], we consider δ > and δ > and we define : A n := (cid:26) max ≤ k ≤ n | S k | ≤ n + δ and max y ∈ Z N n ( y ) < n + δ (cid:27) . Our first lemma is standard, we give a proof for the sake of completeness.
Lemma 4. (cid:88) n ≥ P ( A cn ) < + ∞ . (5) Proof.
Let p > . Thanks to Doob’s maximal inequality and since E ( | S n | p ) = O ( n p/ ) , we have E [max ≤ k ≤ n | S k | p ] = O ( n p ) and so, by the Chebychev inequality, P (cid:18) max ≤ k ≤ n | S k | > n + δ (cid:19) ≤ E [max ≤ k ≤ n | S k | p ] n p ( + δ ) = O ( n − pδ ) . According to [14, Lem. 1], we also have max y E [ N n ( y ) p ] = O ( n p ) and hence P (cid:18) max y N n ( y ) > n + δ (cid:19) ≤ n (cid:88) y = − n P ( N n ( y ) > n + δ ) = O ( n − pδ ) . The result follows by taking p large enough. (cid:3) Let δ > and set E ( n ) := { p ≤ − /n α + δ } . We have
Lemma 5. (cid:88) n ≥ P ( S n = 0 , E ( n ) c ) < + ∞ . (6) Proof.
Indeed, since S is independent of ( p k ) k ∈ ZZ , we have P ( S n = 0 , E ( n ) c ) ≤ M √ n P (cid:18) − p > n α + δ (cid:19) ≤ Mn δ α E (cid:20)(cid:18) − p (cid:19) α (cid:21) whose sum is finite. (cid:3) We also consider the conditional expectation of X n with respect to ( ω, ( S p ) p ) which is equalto D n = (cid:80) y ∈ Z ε y p y − p y N n ( y ) . We introduce δ > and B n := (cid:110) | D n | > n − δ (cid:111) . WRE IN A STRATIFIED ORIENTED MEDIUM 7
Let c n := n α ( + δ )+ δ and E ( n ) := (cid:26) ∀ y ∈ {− n / δ , n / δ } , − p y ≤ c n (cid:27) . Since p ∈ (0 , a.s., there exist < a < b < such that P ( a < p < b ) =: γ > . Let Λ n := { k ∈ { , . . . , n − } , a < p S k < b } ,P := { y ∈ ZZ, a < p y < b } , and ζ y := { a
Uniformly on E ( n ) ∩ E ( n ) , we have P ε,ω ( S n = 0 and X n ≤ ≤ X n +1 , A n , B n , E ( n ) , E ( n )) ≤ (cid:88) k ≥ P ε,ω ( S n = 0 and X n = − k, A n , B n , E ( n ) , E ( n ))(1 − n − / (2 α ) − δ ) k ≤ n / (2 α )+2 δ (cid:88) k =0 P ε,ω ( S n = 0 and X n = − k, A n , B n , E ( n ) , E ( n )) + O ( n − ) ≤ P ε,ω ( S n = 0 and − n / (2 α )+2 δ ≤ X n ≤ , A n , B n , E ( n ) , E ( n )) + O ( n − ) . (8)In order to apply an inequality due to S.V. Nagaev [18], we define X k := ε S k (cid:16) ξ k − p Sk − p Sk (cid:17) , recallthat S n = (cid:80) n − k =0 X k , and introduce B n := (cid:16) E ε,ω (cid:104) S n | S (cid:105)(cid:17) / = (cid:18)(cid:80) n − j =0 p Sj (1 − p Sj ) (cid:19) / . Wehave B n ≥ a (cid:88) y : p y ≥ a N n ( y )(1 − p y ) . Let C (2 n ) := (cid:80) n − k =0 E ε,ω (cid:104)(cid:12)(cid:12) X k (cid:12)(cid:12) | S (cid:105) . On A n ∩ E ( n ) ∩ E ( n ) , we have (cid:88) y : p y
Let L n := C (2 n ) /B n . On A n ∩ E ( n ) ∩ E ( n ) , we have L n ≤ γ a / (cid:88) y : p y ≥ a N n ( y )(1 − p y ) − / c n ≤ γ a / − a √ γ n c n ≤ − a )( γ a ) / n − + α + δ α + δ ≤ n − δ , (10)if n is large enough, since δ α + 3 δ < − α .Let us recall that V n = (cid:16)(cid:80) n − k =0 X k (cid:17) / . We can now apply Nagaev ([18], Thm 1), whichgives uniformly on A n ∩ E ( n ) ∩ E ( n ) , P ε,ω ( | S n | ≥ n δ V n | S ) ≤ (cid:18) n δ (cid:19) exp (cid:18) − n δ − c (cid:48) L n n δ ) (cid:19) + 2 exp (cid:32) − c (cid:48)(cid:48) L n (cid:33) = O (exp( − n δ )) (11)where c (cid:48) > and c (cid:48)(cid:48) > are universal constants. We recall that X n = (cid:80) n − k =0 ε S k ξ k = S n + D n .We have, for large n , on A n ∩ B n ∩ E ( n ) ∩ E ( n ) , P ε,ω ( − n α +2 δ ≤ X n ≤ , E ( n ) | S ) ≤ P ε,ω (cid:16) | X n − D n | ≥ n − δ − n α +2 δ , E ( n ) | S (cid:17) ≤ P ε,ω ( | S n | ≥ n δ n ( d + δ ) / , E ( n ) | S ) ≤ P ε,ω ( | S n | ≥ n δ V n | S ) , since α + 2 δ < − δ and since δ + δ α + δ + 3 δ < − α . Integrating this proves the lemma,by (8) and (11). (cid:3) Lemma 7. (cid:88) n ≥ P ( E ( n ) c ) < + ∞ . (12) Proof. According to [7, Thm 1.3] applied twice with u = γ / : first with the scenery ( γ − { a such that, for every n ≥ , we have P ( E ( n ) c ) ≤ exp (cid:16) − c n (cid:17) . (cid:3) Lemma 8. We have (cid:88) n ∈ N P ( E ( n ) \ E ( n )) < ∞ . (13) Proof. We recall that, taken ω and S , (cid:16) ξ k − p Sk − p Sk (cid:17) y,k is a sequence of independent, cen-tered random variables. For every integer ν ≥ , there exists a constant ˜ C ν > such that (cid:12)(cid:12)(cid:12) E ε,ω (cid:104) ( ξ k − p Sk − p Sk ) ν | S (cid:105)(cid:12)(cid:12)(cid:12) ≤ ˜ C ν (cid:16) − p Sk (cid:17) ν P -almost surely. Consequently, for every N ≥ , thereexists a constant C N > such that for all ≤ ν ≤ N , (cid:12)(cid:12)(cid:12) E ε,ω (cid:104) ( ξ k − p Sk − p Sk ) ν | S (cid:105)(cid:12)(cid:12)(cid:12) ≤ (cid:16) C N − p Sk (cid:17) ν . WRE IN A STRATIFIED ORIENTED MEDIUM 9 Hence, for every n ≥ and N ≥ , we have on E ( n ) : E ε,ω [( V n ) N | S ] = n − (cid:88) k =0 2 n − (cid:88) k =0 · · · n − (cid:88) k N =0 E ε,ω (cid:34) N (cid:89) i =1 X k i | S (cid:35) = n − (cid:88) k =0 2 n − (cid:88) k =0 · · · n − (cid:88) k N =0 E ε,ω n − (cid:89) j =0 X θ j ( k ,...k N ) j | S ≤ n − (cid:88) k =0 2 n − (cid:88) k =0 · · · n − (cid:88) k N =0 2 n − (cid:89) j =0 (cid:18) C N − p S j (cid:19) θ j ( k ,...,k N ) = ( C N ) N (cid:32) n − (cid:88) k =0 − p S k ) (cid:33) N ≤ ( C N ) N n dN where θ j ( k , k , . . . , k N ) := { ≤ i ≤ N, k i = j } . Consequently, on E ( n ) , P ε,ω ( V n > n d + δ | S ) ≤ n − ( d + δ ) N E ε,ω (cid:104) ( V n ) N | S (cid:105) ≤ ( C N ) N n − δ N = O ( n − ) by taking N large enough. Integrating this on E ( n ) yields the result. (cid:3) Lemma 9. We have on E ( n ) , uniformly on ω , S and on k ∈ Z : P ε,ω ( X n = − k | S ) = O (cid:16)(cid:112) ln( n ) n − (cid:17) . (14) Proof. On E ( n ) , we have : P ε,ω ( X n = − k | S ) = 12 π (cid:90) π − π E ε,ω (cid:2) e itX n | S (cid:3) e ikt dt ≤ π (cid:90) π − π (cid:12)(cid:12) E ε,ω (cid:2) e itX n | S (cid:3)(cid:12)(cid:12) dt ≤ π (cid:90) π (cid:89) y ∈ P ( χ p y ( ε y t )) N n ( y ) with χ p ( t ) := (cid:12)(cid:12)(cid:12) E ε,ω [ e itξ | p = p ] (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) − p − pe it (cid:12)(cid:12)(cid:12)(cid:12) = 1 − p (1 + p − p cos( t )) . Since χ p ( t ) is decreasing in p and since < a < p y < b < for y ∈ P , there exist < β < π/ and c > such thatfor a.e. ω, ∀ y ∈ P, ∀ t ∈ [0 , β ] , χ p y ( t ) ≤ − a (1 + a − a cos( t )) ≤ exp( − ct ) . Let us define a n := (cid:112) n ) / ( cγ n ) . Since n = (cid:80) y ∈ P N n ( y ) ≥ γ n on E ( n ) , we have (cid:90) βa n (cid:89) y ∈ P ( χ p y ( t )) N n ( y ) dt ≤ (cid:90) βa n exp( − ct n ) dt ≤ (cid:90) βa n exp( − ct γ n ) dt ≤ n − on E ( n ) . Moreover, (cid:90) a n (cid:89) y ∈ P ( χ p y ( t )) N n ( y ) dt ≤ a n WRE IN A STRATIFIED ORIENTED MEDIUM 10 and (cid:90) πβ (cid:89) y ∈ P ( χ p y ( t )) N n ( y ) dt ≤ (cid:90) πβ (cid:89) y ∈ P (cid:18) − p y − p y cos( β ) (cid:19) N n ( y ) dt ≤ π (cid:18) − a − a cos( β ) (cid:19) γ n/ , since p > a > for p ∈ P . (cid:3) Lemma 10. Suppose that δ (cid:48) := δ − δ − δ > and δ + δ < . Then, uniformly on p and ( S k ) k , P ( A n \ B n | S, p ) = O (cid:16) n − δ (cid:48) (cid:17) . Proof. Up to an enlargement of the probability space, we consider a centered gaussian randomvariable G with variance n − δ independent of ( ω, S ) . We have P ( | D n | ≤ n − δ | S, p ) P ( | G | ≤ n − δ ) ≤ P (cid:16) | D n + G | ≤ n − δ | S, p (cid:17) and so P ( | D n | ≤ n − δ | S, p ) ≤ P (cid:16) | D n + G | ≤ n − δ | S, p (cid:17) / , . Let ˜ χ be the characteristic function of p − p . Since p is non-constant, there exist ˜ β > and ˜ c > such that ∀ u ∈ [ − ˜ β ; ˜ β ] , | ˜ χ ( u ) | ≤ e − ˜ cu . Consequently, P (cid:16) | D n + G | ≤ n − δ | S, p (cid:17) = 2 n − δ π (cid:90) R sin(2 tn − δ )2 tn − δ E [ e itD n | S, p ] E [ e itG ] dt = 2 n − δ π (cid:90) R sin(2 tn − δ )2 tn − δ e it p − p N n (0) (cid:89) y (cid:54) =0 ˜ χ ( ε y N n ( y ) t ) e − t n − δ dt ≤ n − δ π (cid:90) R (cid:89) y (cid:54) =0 | ˜ χ ( ε y N n ( y ) t ) | e − t n − δ dt. Let δ > be such that δ := − δ − δ − δ > and let b n := n δ + δ − . On the one hand, wehave I := (cid:90) {| t | > ˜ βb n } (cid:89) y (cid:54) =0 | ˜ χ ( ε y N n ( y ) t ) | e − t n − δ dt ≤ (cid:90) {| t | > ˜ βb n } e − t n − δ dt ≤ n δ − (cid:90) {| s | > ˜ βn δ } e − s / ds ≤ n δ − e − ˜ β n δ / . On the other hand, we will estimate the following quantity on A n : I := (cid:90) {| t |≤ ˜ βb n } (cid:89) y (cid:54) =0 | ˜ χ ( ε y N n ( y ) t ) | e − t n − δ dt. Let us define F n := { y (cid:54) = 0 : N n ( y ) ≥ n / − δ / } and ρ n := F n . On A n , we have n − n / δ ≤ (cid:80) y (cid:54) =0 N n ( y ) ≤ ρ n n / δ + (2 n / δ − ρ n ) n / − δ and hence ρ n ≥ n / − δ / (if Applying [16, Lemma 3.7.5, p. 58] to the random variable Y := p − p − p − p which is not identically equalto 0 and whose characteristic function is | ˜ χ | , we get that for every r > and every t ∈ (cid:2) − r ; r (cid:3) , | − | ˜ χ ( t ) | | ≥ t E [ Y {| Y |≤ r } ] . We take ˜ β such that ˜ c := E [ Y {| Y |≤ ˜ β − } ] > . For every u ∈ [ − ˜ β ; ˜ β ] , we have | − | ˜ χ ( u ) || ≥ | − | ˜ χ ( u ) | | ≥ ˜ cu and so | ˜ χ ( u ) | ≤ − ˜ cu . WRE IN A STRATIFIED ORIENTED MEDIUM 11 n is large enough). Therefore, on A n , we have α n := (cid:80) y ∈ F n N n ( y ) ≥ n − δ − δ / . Now, usingthe Hölder inequality, we have I ≤ (cid:89) y ∈ F n (cid:32)(cid:90) {| t |≤ ˜ βb n } | ˜ χ ( ε y N n ( y ) t ) | αnN n ( y ) dt (cid:33) N n ( y ) αn ≤ sup y ∈ F n (cid:32)(cid:90) {| t |≤ ˜ βb n } | ˜ χ ( ε y N n ( y ) t ) | αnN n ( y ) dt (cid:33) ≤ b n sup y ∈ F n (cid:32)(cid:90) | v |≤ ˜ β | ˜ χ ( ε y N n ( y ) vb n ) | αnN n ( y ) dv (cid:33) . Let us notice that, if | v | ≤ ˜ β , we have on A n , | ε y N n ( y ) vb n | ≤ ˜ βn / δ n δ + δ − = ˜ βn − δ ≤ ˜ β, since δ > . Hence, on A n , we have I ≤ b n sup y ∈ F n (cid:32)(cid:90) {| v |≤ ˜ β } e − ˜ c ( N n ( y )) v n δ δ − αnN n ( y ) dv (cid:33) ≤ b n sup y ∈ F n (cid:32)(cid:90) {| v |≤ ˜ β } e − ˜ cN n ( y ) v n δ δ − − δ − δ / dv (cid:33) ≤ sup y ∈ F n b n n − δ − δ + δ δ + (cid:112) N n ( y ) (cid:18)(cid:90) R e − ˜ cs / ds (cid:19) ≤ √ n − + δ + δ (cid:90) R e − ˜ cs / ds. Hence, uniformly on A n and on p , we have P ( A n \ B n | ( S k ) k , p ) = O ( n δ + δ − δ ) . (cid:3) Lemma 11. Under the same hypotheses, we have (cid:88) n P ( S n = 0 , X n ≤ ≤ X n +1 ; A n ∩ E ( n ) \ B n ) < ∞ . Proof. According to Lemma 9, Lemma 10 and since P ( S n = 0) = O ( n − / ) and E [1 / (1 − p )] < ∞ , we have E (cid:34) ∞ (cid:88) k =0 P ε,ω ( S n = 0 , X n = − k, A n ∩ E ( n ) \ B n ) p k (cid:35) = E (cid:34) ∞ (cid:88) k =0 p k { S n =0 } ( A n \ B n ) ∩ E ( n ) P ε,ω ( X n = − k | S ) (cid:35) ≤ C (cid:112) (ln n ) n − E (cid:20) − p { S n =0 } P ( A n \ B n | S, p ) (cid:21) = O ( n − − δ (cid:48) √ ln n ) . (15) (cid:3) WRE IN A STRATIFIED ORIENTED MEDIUM 12 Lemma 12. If δ α < δ , we have (cid:88) n P ( S n = 0 , X n ≤ ≤ X n +1 , E ( n ) c , A n , E ( n ) , E ( n )) < + ∞ . (16) Proof. We notice that on E ( n ) ∩ A n , P ( E ( n ) c | S, p ) ≤ n − dα/ E n / δ (cid:88) y = − n / δ − p y ) N n ( y ) α/ (cid:12)(cid:12)(cid:12) S, p ≤ n − dα/ E (cid:88) | y |≤ n / δ ,y (cid:54) =0 − p y ) α N α/ n ( y ) + 1(1 − p ) α N α/ n (0) (cid:12)(cid:12)(cid:12) S, p (17) ≤ n − dα/ (cid:18) n / δ E (cid:20) − p ) α (cid:21) + n + δ α (cid:19) n (1 / δ ) α/ = O ( n − δ α/ ) , (18)since α ≤ , δ α < δ and d = + α + δ α + 3 δ + δ . Similarly as in (15), this yields E (cid:34) ∞ (cid:88) k =0 P ε,ω ( S n = 0 , X n = − k, E ( n ) c ∩ A n ∩ E ( n ) ∩ E ( n )) p k (cid:35) = O ( n − − δ α/ √ ln n ) . Hence we have P ( S n = 0 , X n ≤ ≤ X n +1 , E ( n ) c , A n , E ( n ) , E ( n )) = O ( n − − δ α/ √ ln n ) . (cid:3) Lemma 13. If δ < (1 − α ) , we have (cid:88) n P ( S n = 0 , X n ≤ ≤ X n +1 , E ( n ) \ E ( n )) < ∞ . Proof. Notice that on { − p ≤ c n } , we have P ( E ( n ) c | p ) ≤ n / δ P (cid:18) − p > c n (cid:19) ≤ n / δ c αn E (cid:20)(cid:18) − p (cid:19) α (cid:21) = O ( n − δ α ) . Similarly as in (15), since E [1 / (1 − p )] < ∞ , for δ small enough, we have E (cid:34) ∞ (cid:88) k =0 P ε,ω ( S n = 0 , X n = − k, E ( n ) \ E ( n )) p k (cid:35) = E (cid:34) ∞ (cid:88) k =0 p k { S n =0 } E ( n ) \ E ( n ) P ε,ω ( X n = − k | S ) (cid:35) ≤ C (cid:112) (ln n ) n − n − / (cid:32) E (cid:34) ∞ (cid:88) k =0 P ( E ( n ) c | p ) { (1 − p ) − ≤ c n } p k (cid:35) + E (cid:20) − p { (1 − p ) − >c n } (cid:21)(cid:33) = O ( n − − cδ √ ln n ) , where we can use Hölder’s inequality, to deal with the second term of the third line, since α > and δ < (1 − α ) . (cid:3) WRE IN A STRATIFIED ORIENTED MEDIUM 13 We take δ ∈ (cid:0) , − α (cid:1) (since α > ) and then δ > and δ > such that δ < , δ < , δ < − δ , δ α + δ < − α − δ , δ + δ < δ and finally δ such that δ < , δ α < δ and δ α + δ δ < − α − δ . Combining all the previous lemmas with these choices for δ , δ , δ , δ , we get (4), which provesTheorem 1. 4. Proof of Theorem 2 (functional limit theorem) We assume that ( p k ) k satisfies the conditions of Theorem 2. Lemma 14. Let ( ε k ) k be a (fixed or random) sequence with values in {− 1; 1 } . Let ( p k ) k be asin Theorem 2. Then, under P , the sequence of random variables n − δ (cid:98) nt (cid:99)− (cid:88) k =0 ε S k (cid:18) ξ k − p S k − p S k (cid:19) , t ≥ n converges in distribution (in the space of Skorokhod D ([0; + ∞ ) , R ) ) to (0 , t ≥ .Proof. We first notice that it is enough to prove that N − δ sup ≤ n ≤ N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n − (cid:88) k =0 ε S k (cid:18) ξ k − p S k − p S k (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) → N → + ∞ in probability.Let us define ˜ E ( N, v ) := (cid:40) N − (cid:88) k =0 − p S k ) ≤ N + β + v (cid:41) . We proceed as in formula (17) (with a conditioning with respect to S only, and α < β butclose enough to β ) to prove that P [( ˜ E ( N, v )) c | S ] ≤ N − cv on A N for c > and N large enough.Moreover, P ( A cN ) → N → + ∞ by Lemma 4, which gives lim N → + ∞ P (cid:16) ˜ E ( N, v ) (cid:17) = 1 . Now, taken ( ε, S, ω ) , (cid:16)(cid:80) n − k =0 ε S k (cid:16) ξ k − p Sk − p Sk (cid:17)(cid:17) n is a martingale. Hence, according to the maxi-mal inequality for martingales we have, for every θ > , P ε,ω sup n ≤ N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:32) n − (cid:88) k =0 ε S k (cid:18) ξ k − p S k − p S k (cid:19)(cid:33) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ θ N δ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) S ≤ n ≤ N E ε,ω (cid:20)(cid:80) n − k =0 (cid:16) ξ k − p Sk − p Sk (cid:17) | S (cid:21) θ N δ ≤ (cid:80) N − k =0 1(1 − p Sk ) θ N δ ≤ N + β + v θ N δ = 2 N − + v θ − , WRE IN A STRATIFIED ORIENTED MEDIUM 14 on ˜ E ( n, v ) , since δ = + β . Hence, we get P (cid:32) sup n ≤ N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n − (cid:88) k =0 ε S k (cid:18) ξ k − p S k − p S k (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ θN δ (cid:33) ≤ − P ( ˜ E ( N, v )) + 2 N − + v θ − . From this we conclude that lim n → + ∞ P (cid:16) sup n ≤ N (cid:12)(cid:12)(cid:12)(cid:80) n − k =0 ε S k (cid:16) ξ k − p Sk − p Sk (cid:17)(cid:12)(cid:12)(cid:12) ≥ θN δ (cid:17) = 0 . (cid:3) The next lemma follows from the proof of [9, Thm 4] when β = 2 . The proof of the generalcase β ∈ (1 , is postponed to Section 5. Lemma 15. Let β ∈ (1 , . Let S = ( S n ) n ≥ be a random walk on Z starting from S = 0 ,with iid centered square integrable and non-constant increments and such that gcd { k : P ( S = k ) > } = 1 . Let (˜ ε y ) y ∈ Z be a sequence of iid random variables independent of S with symmetricdistribution and such that ( n − β (cid:80) nk =1 ˜ ε k ) n converges in distribution to a random variable Y with stable distribution of index β . Then, the following convergence holds in distribution in D ([0 , + ∞ ) , R ) n − δ (cid:98) nt (cid:99)− (cid:88) k =0 ˜ ε S k , n − S (cid:98) nt (cid:99) t ≥ −→ n → + ∞ ( ˜∆ t , ˜ B t ) t ≥ , with δ = + β , where ( ˜ B t ) t is a Brownian motion such that V ar ( ˜ B ) = V ar ( S ) and with ( ˜ L t ( x )) t,x the jointly continuous version of its local time and where ˜∆ t := (cid:90) R ˜ L t ( x ) d ˜ Z x , with ˜ Z independent of ˜ B given by two independent right continuous stable processes ( ˜ Z x ) x ≥ and ( ˜ Z − x ) x ≥ with stationary independent increments such that ˜ Z , ˜ Z − have the same distributionas Y . Now, we prove a functional limit theorem for ( X (cid:98) nt (cid:99) , S (cid:98) nt (cid:99) ) from which we will deduce ourtheorem 2. Proposition 16. Under the assumptions and with the notations of Theorem 2, the sequence ofprocesses (cid:18)(cid:16) n − δ X (cid:98) nt (cid:99) , n − / S (cid:98) nt (cid:99) (cid:17) t ≥ (cid:19) n converges in distribution under P (in the space of Skorokhod D ([0; + ∞ ) , R ) ) to ( σ ∆ t , B t ) t ≥ .Proof of Proposition 16. We observe that X n can be rewritten X n = n − (cid:88) k =0 ε S k (cid:18) ξ k − p S k − p S k (cid:19) + n − (cid:88) k =0 ε S k p S k − p S k . According to Lemma 14, it is enough to prove, under P , the convergence n − δ (cid:98) nt (cid:99)− (cid:88) k =0 ε S k p S k − p S k , n − / S (cid:98) nt (cid:99) t ≥ n → n → + ∞ ( σ ∆ t , B t ) t ≥ (19)in distribution in D ([0; + ∞ ) , R ) . WRE IN A STRATIFIED ORIENTED MEDIUM 15 In case (b) , (cid:16) ˜ ε y := ε y p y − p y (cid:17) y is a sequence of independent identically distributed random vari-ables with symmetric distribution such that ( n − /β (cid:80) ny =1 ˜ ε y ) n converges in distribution to arandom variable with characteristic function θ (cid:55)→ exp( − A | θ | β ) , where A := E ( p / (1 − p ) ) / if β = 2 . Hence the result follows from Lemma 15.In case (a) with β = 2 , we observe that (cid:80) n − k =0 ε S k is equal to 0 if n is even and is equal to 1if n is odd. Hence, (( n − / (cid:80) (cid:98) nt (cid:99)− k =0 ε S k ) t ≥ ) n converges to 0 in D ([0; + ∞ ) , R ) and it remains toprove the convergence of n − / (cid:98) nt (cid:99)− (cid:88) k =0 ε S k (cid:18) p S k − p S k − E (cid:20) p − p (cid:21)(cid:19) , n − / S (cid:98) nt (cid:99) t ≥ n . Let us write λ for the characteristic function of p − p − E (cid:104) p − p (cid:105) . Since p − p has a finite varianceand λ ( ε y · ) behaves as λ at 0, we can follow the proof of the convergence of the finite distributionsof [9, prop 1], which gives the convergence in distribution in D ([0; + ∞ ) , R ) thanks to thetightness that can be proved for the first coordinate as in [14].Now, let us explain how case (a) with β ∈ (1 , will also be deduced from Lemma 15. Thiscomes from the following lemma. Lemma 17. Let β ∈ (1 , . Let S = ( S n ) n be a simple symmetric random walk on Z starting from S = 0 . Let (˜ a y ) y ∈ Z be a sequence of iid random variables such that E ( | ˜ a | ) < ∞ , independentof S . We have n − δ (cid:98) nt (cid:99)− (cid:88) k =0 ( − k ˜ a S k − (cid:88) y (˜ a y − ˜ a y − ) N (cid:98) nt (cid:99) (2 y ) , t ≥ −→ (0 , in distribution as n goes to infinity (in D ([0; + ∞ ) , R ) ), with δ := + δ .Proof of Lemma 17. Let us write e n := n − (cid:88) k =0 ( − k ˜ a S k − (cid:88) y (˜ a y − ˜ a y − ) N n (2 y ) . We notice that it is enough to prove that n − δ sup ≤ k ≤ n | e k | P −→ n → + ∞ . Let η > be such that η < β − (such a η exists since β < ). For every n ≥ , we considerthe set Ω (cid:48) n defined by Ω (cid:48) n := sup k ≤ n | S k | ≤ n + η , sup ≤ k ≤ n sup | y |≤ n 12 + η +1 | N k ( y ) − N k ( y − | ≤ n + η . Let us show that lim n → + ∞ P (Ω (cid:48) n ) = 1 . As in Lemma 4, we have, lim n → + ∞ P (cid:32) sup k ≤ n | S k | ≤ n + η (cid:33) = 1 . Now we recall that for any even integer m , sup y E [ | N n ( y ) − N n ( y − | m ] = O ( n m ) , WRE IN A STRATIFIED ORIENTED MEDIUM 16 as n goes to infinity (see [14, lem 3] and [13, p. 77]). Hence, using again the Markov inequalityand taking m large enough, we get P (Ω (cid:48) n ) ≥ − o (1) − n + η sup n,y E [ | N n ( y ) − N n ( y − | m ] n m + ηm = 1 − o (1) . On Ω (cid:48) n , using the fact that k − (cid:88) (cid:96) =0 ( − (cid:96) ˜ a S (cid:96) = (cid:88) y (˜ a y N k (2 y ) − ˜ a y − N k (2 y − , for every k = 0 , ..., n , we have | e k | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) y ˜ a y − ( N k (2 y ) − N k (2 y − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:88) | y |≤ n 12 + η +1 | ˜ a y | n + η . Hence, thanks to the Markov inequality, we get for θ > . P (cid:32) n − δ sup ≤ k ≤ n | e k | > θ (cid:33) ≤ (1 − P (Ω (cid:48) n )) + P (cid:88) | y |≤ n 12 + η +1 | ˜ a y | > θn + β − − η ≤ (1 − P (Ω (cid:48) n )) + 3 E ( | ˜ a | ) θ n − β + +2 η . Hence, for every θ > , we have lim n → + ∞ P ( n − δ sup ≤ k ≤ n | e k | > θ ) = 0 . (cid:3) Now we observe that the characteristic function of ˜ ε y := p y − p y − p y − − p y − is t (cid:55)→ | ˜ χ ( t ) | (where ˜ χ stands for the characteristic function of p − p ). The distribution of ˜ ε is symmetricand ( n − β (cid:80) nk =1 ˜ ε k ) n converges in distribution to a random variable with characteristic function θ (cid:55)→ exp( − A | θ | β ) . According to Lemma 15 applied with the random walk (cid:16) ˜ S k := S k (cid:17) k , wehave n − δ (cid:98) nt (cid:99)− (cid:88) k =0 ˜ ε ˜ S k , n − S (cid:98) nt (cid:99) t ≥ −→ n → + ∞ ( ˜∆ t , ˜ B t ) t ≥ , in distribution in D ([0; + ∞ ) , R ) , where ( ˜ B t ) t is a Brownian motion such that V ar ( ˜ B ) = andwith ( ˜ L t ( x )) t,x the jointly continuous version of its local time and where ˜∆ t := (cid:90) R ˜ L t ( x ) d ˜ Z x , with ˜ Z independent of ˜ B given by two independent right continuous stable processes ( ˜ Z x ) x ≥ and ( ˜ Z − x ) x ≥ , the characteristic functions of ˜ Z and of ˜ Z − being θ (cid:55)→ exp( − A | θ | β ) . Hence,we have n − δ (cid:98) nt/ (cid:99)− (cid:88) k =0 ˜ ε ˜ S k , n − S (cid:98) nt (cid:99) t ≥ −→ n → + ∞ ( ˜∆ t/ , ˜ B t/ ) t ≥ , and so (cid:32) n − δ (cid:88) y ˜ ε y N (cid:98) nt (cid:99) (2 y ) , n − S (cid:98) nt (cid:99) (cid:33) t ≥ −→ n → + ∞ ( ˜∆ t/ , B t ) t ≥ , WRE IN A STRATIFIED ORIENTED MEDIUM 17 with B t := 2 ˜ B t/ . Now we observe that ˜∆ t/ = (cid:90) R ˜ L t/ ( x ) d ˜ Z x = (cid:90) R L t (2 x ) d ˜ Z x = (cid:90) R L t ( x ) dZ x , where L denotes the local time of B and with Z x := ˜ Z x/ . Now Lemma 17 applied to (cid:16) p y − p y (cid:17) y ∈ Z gives (19), which proves Proposition 16 in the case (a) with β ∈ (1 , . (cid:3) Proof of Theorem 2. We recall that for every n , we have X n = M (1) T n and S n = M (2) T n . Moreover we observe that we have T n = n − (cid:88) k =0 ( ξ k + 1) , that can be rewritten T n = n − (cid:88) k =0 (cid:18) ξ k − p S k − p S k (cid:19) + n − (cid:88) k =0 (cid:18) p S k − p S k − E (cid:20) p − p (cid:21)(cid:19) + n (cid:18) E (cid:20) p − p (cid:21)(cid:19) . We recall that γ = 1 + E (cid:104) p − p (cid:105) and we define ( U n ) n such that U n := max { k ≥ T k ≤ n } . We notice that the sequences of processes n − (cid:98) nt (cid:99)− (cid:88) k =0 (cid:18) ξ k − p S k − p S k (cid:19) , t ≥ n and n − (cid:98) nt (cid:99)− (cid:88) k =0 (cid:18) p S k − p S k − E (cid:20) p − p (cid:21)(cid:19) , t ≥ n converge in distribution in D ([0 , + ∞ ) , R ) to 0. The first convergence follows from Lemma 14where we take ε k = 1 for every k ∈ ZZ . The second convergence is a consequence of [14,Thm 1.1] since n δ /n → as n → + ∞ . Hence (cid:0) n − T (cid:98) nt (cid:99) , t ≥ (cid:1) n converges in distribution to ( γt ) t , We conclude that (cid:16)(cid:0) n − U (cid:98) nt (cid:99) (cid:1) t ≥ (cid:17) n converges in distribution (in D ([0; + ∞ ) , R ) ) to ( t/γ ) t .Therefore, according to Proposition 16 and to [1, Lem p. 151, Thm 3.9], the sequence of processes (cid:18)(cid:16) n − δ X U (cid:98) nt (cid:99) , n − / S U (cid:98) nt (cid:99) (cid:17) t ≥ (cid:19) n converges in distribution (in D ([0; + ∞ ) , R ) ) to ( σ ∆ tγ , B tγ ) t ≥ . This means that (cid:32)(cid:18) n − δ M (1) T U (cid:98) nt (cid:99) , n − / M (2) T U (cid:98) nt (cid:99) (cid:19) t ≥ (cid:33) n converges in distribution (in D ([0; + ∞ ) , R ) ) to ( σ ∆ tγ , B tγ ) t ≥ .Moreover, we have B tγ = γ − / B (cid:48) t and Z x √ γ = γ − / (2 β ) Z (cid:48) x , where ( B (cid:48) t ) t ≥ is a standard Brow-nian motion, and ( Z (cid:48) x ) x ∈ R has the same distribution as ( Z x ) x ∈ R and is independent of ( B (cid:48) t ) t ≥ .Furthermore we have L tγ ( x ) = γ − / L (cid:48) t ( γ / x ) , t ≥ , x ∈ R , where ( L (cid:48) t ) t ≥ is the local time of ( B (cid:48) t ) t and so ∆ tγ = γ − (cid:90) R L (cid:48) t ( γ x ) dZ x = γ − δ (cid:90) R L (cid:48) t ( y ) dZ (cid:48) y . WRE IN A STRATIFIED ORIENTED MEDIUM 18 Hence ( σ ∆ tγ , B tγ ) t ≥ has the same distribution as ( σγ − δ ∆ t , γ − B t ) t ≥ .Now we observe that we have M (2) (cid:98) nt (cid:99) = M (2) T U (cid:98) nt (cid:99) and (cid:12)(cid:12)(cid:12)(cid:12) M (1) (cid:98) nt (cid:99) − M (1) T U (cid:98) nt (cid:99) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ξ U (cid:98) nt (cid:99) and that for every θ > and T > , P (cid:32) sup t ∈ [0; T ] n − δ ξ U (cid:98) nt (cid:99) ≥ θ (cid:33) ≤ nT (cid:88) k =0 P (cid:16) ξ k ≥ θn δ (cid:17) ≤ nT (cid:88) k =0 E [( ξ ) β − η ]( θn δ ) β − η = o (1) (20)for η > small enough, since δβ > and since (if η < β − ) E [( ξ ) β − η | p ] ≤ C − p ) β − η a.s.and E (cid:104) − p ) β − η (cid:105) < ∞ . This completes the proof of Theorem 2. (cid:3) Proof of Lemma 15 The proof is very similar to those in [14] and [9], with some adaptations.We define ˜ D n := (cid:80) y ∈ Z ˜ ε y N n ( y ) , n ∈ N . Lemma 18. If β ∈ (1 , , the finite dimensional distributions of ( ˜ D (cid:98) nt (cid:99) /n δ , S (cid:98) nt (cid:99) / √ n ) t ≥ con-verge to those of ( ˜∆ t , ˜ B t ) t ≥ . Before proving Lemma 18, we first introduce some preliminary results.We observe that n − /β (cid:80) ny =1 ˜ ε y converges in distribution to a stable random variable of pa-rameter β , with characteristic function (cid:101) ζ β ( θ ) := exp( − A | θ | β ) (for some A > ). We can nowcompute the characteristic function of the finite dimensional distributions of ( ˜∆ t , ˜ B t ) t ≥ . Lemma 19. Let k ∈ N ∗ , ( t , t , . . . , t k ) ∈ R k + and ( θ ( i )1 , θ ( i )2 , . . . , θ ( i ) k ) i =1 , ∈ R k . We have, E exp i k (cid:88) j =1 (cid:0) θ (1) j ˜∆ t j + θ (2) j ˜ B t j (cid:1) = E exp − A (cid:90) + ∞−∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k (cid:88) j =1 θ (1) j ˜ L t j ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) β dx exp i k (cid:88) j =1 θ (2) j ˜ B t j . (21) Proof. We condition by ˜ B and we proceed as in [14, Lem 5]. We get E exp i k (cid:88) j =1 θ (1) j ˜∆ t j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ B = exp − A (cid:90) + ∞−∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k (cid:88) j =1 θ (1) j ˜ L t j ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) β dx , which gives the result. (cid:3) This comes from the Hölder inequality since E (cid:2) ( ξ ) | p (cid:3) ≤ − p ) . WRE IN A STRATIFIED ORIENTED MEDIUM 19 For fixed k ∈ N ∗ and ( t , t , . . . , t k ) ∈ R k + , we define for every ( θ , θ , . . . , θ k ) ∈ ( R ) k , ψ n ( θ , θ , . . . , θ k ) := E exp − A (cid:88) y ∈ ZZ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k (cid:88) j =1 θ (1) j N (cid:98) nt j (cid:99) ( y ) n − δ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) β exp i k (cid:88) j =1 θ (2) j S (cid:98) nt j (cid:99) √ n and φ n ( θ , θ , . . . , θ k ) := E exp i k (cid:88) j =1 (cid:0) θ (1) j n − δ ˜ D (cid:98) nt j (cid:99) + θ (2) j S (cid:98) nt j (cid:99) √ n (cid:1) = E (cid:89) y ∈ ZZ ˜ λ k (cid:88) j =1 θ (1) j N (cid:98) nt j (cid:99) ( y ) n − δ exp i k (cid:88) j =1 θ (2) j S (cid:98) nt j (cid:99) √ n where ˜ λ ( θ ) := E [exp( iθ ˜ ε )] for every θ ∈ R and θ j = ( θ (1) j , θ (2) j ) for every j ∈ { , . . . , n } . Lemma 20. For every k ∈ N ∗ , ( t , t , . . . , t k ) ∈ R k + and ( θ , θ , . . . , θ k ) ∈ ( R ) k , lim n → + ∞ | ψ n ( θ , θ , . . . , θ k ) − φ n ( θ , θ , . . . , θ k ) | = 0 . Proof. As in [14, p. 7], we have − ˜ λ ( θ ) ∼ θ → A | θ | β since the distribution of ˜ ε belongs tothe normal domain of attraction of the stable distribution with characteristic function (cid:101) ζ β . Theremainder of the proof is the same as in [9, Lem 5] with δ instead of / and β instead of , since P ( n − δ sup y ∈ ZZ | (cid:80) kj =1 θ (1) j N (cid:98) nt j (cid:99) ( y ) | > ε ) → n → + ∞ for ε > by [14, Lem 4] and since we have E ( (cid:80) y ∈ ZZ | n − δ (cid:80) kj =1 θ (1) j N (cid:98) nt j (cid:99) ( y ) | β ) ≤ C < ∞ by [6, Lem 3.3]. (cid:3) We now prove Lemma 21. For every k ∈ N ∗ , ( t , t , . . . , t k ) ∈ R k + and ( θ , θ , . . . , θ k ) ∈ ( R ) k , n − δβ (cid:88) y ∈ ZZ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k (cid:88) j =1 θ (1) j N (cid:98) nt j (cid:99) ( y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) β , k (cid:88) j =1 θ (2) j S (cid:98) nt j (cid:99) / √ n n converges in distribution as n → + ∞ to (cid:90) + ∞−∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k (cid:88) j =1 θ (1) j ˜ L t j ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) β dx, k (cid:88) j =1 θ (2) j ˜ B t j . Proof. The proof is very similar to the one of [9, Lem 6], and to the proof of [14, Lem 6] whichdeals with the first coordinate. Throughout the proof, C denotes a positive constant, whichcan vary from line to line, and can depend on ( θ ( i ) j , i = 1 , j = 1 , . . . , k ) . For n ∈ N and realnumbers a < b and t > , we introduce the notation T nt ( a, b ) := (cid:90) (cid:98) nt (cid:99) /n { a ≤ S (cid:98) ns (cid:99) / √ n andtwo real numbers µ and µ . We define for M > , (cid:96) ∈ Z and n ∈ N , U ( τ, M, n ) := µ n − δβ (cid:88) y < − Mτ √ n or y ≥ Mτ √ n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k (cid:88) j =1 θ (1) j N (cid:98) nt j (cid:99) ( y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) β ,T ( (cid:96), n ) := k (cid:88) j =1 θ (1) j T nt j ( (cid:96)τ, ( (cid:96) + 1) τ ) = 1 n k (cid:88) j =1 θ (1) j (cid:88) (cid:96)τ √ n ≤ y< ( (cid:96) +1) τ √ n N (cid:98) nt j (cid:99) ( y ) ,V ( τ, M, n ) := µ τ − β (cid:88) − M ≤ (cid:96) We define c ( (cid:96), n ) := { y ∈ ZZ, (cid:96)τ √ n ≤ y < ( (cid:96) + 1) τ √ n } . As in [14], we have for µ (cid:54) = 0 , µ − A ( τ, M, n )= (cid:88) − M ≤ (cid:96) As in [9, Lem 2], we have ( T ( n ) t j ( (cid:96)τ, ( (cid:96) + 1) τ ) , S (cid:98) nt j (cid:99) / √ n ) j =1 ,...,k,(cid:96) = − M,...,M → (Λ t j ( (cid:96)τ, ( (cid:96) + 1) τ ) , ˜ B t j ) j =1 ,...,k,(cid:96) = − M,...,M WRE IN A STRATIFIED ORIENTED MEDIUM 22 in distribution, as n → + ∞ , where Λ t ( a, b ) := (cid:82) ba ˜ L t ( x ) dx for t > and a < b . Consequently, ( V ( τ, M, n )) n converges in distribution as n → + ∞ to V ( τ, M ) := µ τ − β (cid:88) − M ≤ (cid:96) Hence by choosing adequate M and τ we get for n large enough (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E exp i µ n δβ (cid:88) y ∈ ZZ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k (cid:88) j =1 θ (1) j N (cid:98) nt j (cid:99) ( y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) β + iµ k (cid:88) j =1 θ (2) j S (cid:98) nt j (cid:99) √ n − E exp( i (cid:98) V ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ η. Since this is true for every µ ∈ R , µ ∈ R and η > , this proves Lemma 21. (cid:3) Proof of Lemma 18 . Applying Lemma 21, we get the convergence of ψ n ( θ , . . . , θ k ) to the righthand side of (21) as n → + ∞ . This combined with Lemma 19 and Lemma 20 proves Lemma18. (cid:3) Proof of Lemma 15 . We now turn to the tightness. We know that ( ˜ D (cid:98) nt (cid:99) /n δ , t ≥ n and ( S (cid:98) nt (cid:99) / √ n, t ≥ n both converge in distribution in D ([0 , + ∞ ) , R ) to continuous processes(respectively by [14] and by the theorem of Donsker), and the finite dimensional distributions of ( ˜ D (cid:98) nt (cid:99) /n δ , S (cid:98) nt (cid:99) / √ n ) t ≥ converge to those of ( ˜∆ t , ˜ B t ) t ≥ by Lemma 18, hence the distributionsof ( ˜ D (cid:98) nt (cid:99) /n δ , S (cid:98) nt (cid:99) / √ n ) t ≥ are tight in D ([0 , + ∞ ) , R ) (this is a consequence of [1] Theorems13.2 and 13.4, Corollary p.142 and inequalities (12.7) and (12.9)). This proves Lemma 15. Acknowledgements : The authors thank Yves Derriennic for interesting discussions and references. References [1] Billingsley, P. Convergence of probability measures (1999), 2nd edn (New York: Wiley).[2] Bouchaud, J.-P.; Georges, A.; Koplik, J. ; Provata, A.; Redner, S. Superdiffusion in Random Velocity Fields .Phys. 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E-mail address : [email protected] Université Européenne de Bretagne, Université de Brest, Département de Mathématiques,29238 Brest cedex, France E-mail address ::