Arbitrary many walkers meet infinitely often in a subballistic random environment

We consider $d$ independent walkers in the same random environment in $\mathbb{Z}$. Our assumption on the law of the environment is such that a single walker is transient to the right but subballistic. We show that - no matter what $d$ is - the $d$ walkers meet infinitely often, i.e. there are almost surely infinitely many times for which all the random walkers are at the same location.


Introduction and statement of the main results
1.1. Motivation and main result. Meetings, or collisions, of random walks have been studied since the early works of Pólya [P21]. Indeed, Pólya [P21] proves that almost surely, two independent simple random walks meet infinitely often in Z or Z 2 , but only a finite number of times in Z 3 . Proving this result was his main motivation for proving his celebrated result on transience and recurrence of simple random walks in Z d , d ≥ 1 (see Pólya [P84] "Two incidents"). Moreover, Dvoretzky and Erdös [DE51] stated that almost surely, in Z, 3 independent simple random walks meet (simultaneously) infinitely often whereas 4 independent simple random walks meet only a finite number of times.
More recently, the question whether two or three independent random walks meet infinitely often or only a finite number of times has been studied on some other graphs, for example wedge combs, percolation clusters of Z 2 and some trees. See e.g. Krishnapur and Peres [KP04], Chen, Wei and Zhang [CWZ08], Chen and Chen [CC10], [CC11], Barlow, Peres and Sousi [BPS12] and Hutchcroft and Peres [HP15]. For some applications of collisions of random walks in physics and in biology, we refer to Campari and Cassi [CC12].
Meetings/collisions have also been considered recently for random walks in random environments (RWRE) in Z. It was proved by Gantert, Kochler and Pène [GKP14] that almost surely, d independent random walks in the same (recurrent) random environment meet infinitely often in the origin for any d ≥ 2. In other words, the d walkers together are a recurrent Markov chain on Z d . Related questions have been studied by Gallesco [Ga13] and Zeitouni [Z01, p. 307]. A necessary and sufficient condition for several independent (recurrent) Sinai walks and simple random walks to meet infinitely often (even if the walkers together are a transient Markov chain) has been proved by Devulder, Gantert and Pène [DGP18]. Systems of several random walks in the same random environment have also been investigated in some other papers, see e.g. Andreoletti [A08], and Peterson et al. [PS10], [PZ09] and [JP17], but these papers do not consider collisions, which are our main object of study.
The aim of the present paper is to investigate if there are infinitely many meetings in the (zero speed) transient case, that is, when each random walk is transient with zero speed in the same random environment. In this case, the d walkers together are clearly a Let ω := (ω x ) x∈Z be a collection of i.i.d random variables, taking values in (0, 1) and with joint law P. A realization of ω is called an environment. Let d ≥ 2, and (x 1 , . . . , x d ) ∈ Z d . Conditionally on ω, we consider d independent Markov chains S (j) n n∈N , 1 ≤ j ≤ d, defined for all 1 ≤ j ≤ d by S (j) 0 = x j and for every n ∈ N, y ∈ Z and z ∈ Z, (1) We say that each S (j) n n∈N is a random walk in random environment (RWRE). So, S (j) := S (j) n n∈N , 1 ≤ j ≤ d are d independent random walks in the same (random) environment ω. Here and throughout the paper, N denotes the set of nonnegative integers, including 0.
Notice in particular that S (j) n n∈N , 1 ≤ j ≤ d, is not Markovian under P (x 1 ,...,x d ) . We sometimes consider S n := S (1) n , n ∈ N. We then denote by P x ω the quenched law for a single RWRE starting at x ∈ Z, and write P ω instead of P x ω when x = 0, and P(·) = P ω (·)P(dω) the corresponding annealed law. We denote by E, E (x 1 ,...,x d ) ω , E, E ω and E x ω the expectations under P, P (x 1 ,...,x d ) ω , P, P ω and P x ω respectively. We also introduce We assume that there exists ε 0 ∈ (0, 1/2) such that This ellipticity condition is standard for RWRE. Solomon [S75] proved that (S n ) n is almost surely recurrent when E(log ρ 0 ) = 0, and transient to the right when E(log ρ 0 ) < 0. In this paper we only consider the transient, subballistic case. More precisely, we assume that In this case, E(log ρ 0 ) < 0, so S n n∈N is a.s. transient to the right, and such a κ is unique. Furthermore, it has been proved by Kesten, Koslov and Spitzer [KKS75] that, under the additional hypothesis that log ρ 0 has a non-arithmetic distribution, S n is asymptotically of order n κ as n → +∞ (we do not make this assumption in the present paper).
Our main result is the following.
Then P (x 1 ,...,x d ) almost surely, there exist infinitely many n ∈ N such that Notice in particular that each random walk S (j) is transient, however d independent random walks S (j) , 1 ≤ j ≤ d meet simultaneously infinitely often, for every d > 1. Theorem 1.1 remains true for −1 < κ < 0 by symmetry (i.e., by replacing ω j by 1 − ω −j for every j ∈ Z). To the best of our knowledge, this is the first example of a transient random walk, for which arbitrary many independent realizations of this random walk meet simultaneously infinitely often.
Also, notice that we do not require a non-arithmetic distribution in our Theorem 1.1.
In this paper, we use quenched techniques. For more information and results about quenched techniques for RWRE in the case 0 < κ < 1, we refer to Enriquez, Sabot and Zindy [ESZ09a], [ESZ09b], Peterson and Zeitouni [PZ09] and Dolgopyat and Goldsheid [DG12]. For a quenched study of the continuous time analogue of RWRE, that is, diffusions in a drifted Brownian potential expressed in terms of h-extrema, we refer to Andreoletti et al. [AD15] and [ADV16].
Our results may be rephrased in terms of the capacity of the diagonal {y ∈ Z d : 1.2. Sketch of the proof and organization of the paper. We introduce, in Section 2, the potential V for the RWRE, and we provide some useful estimates concerning this potential. In particular, we prove a key lemma, Lemma 2.5, which will enable us to control the invariant probability measure of a RWRE reflected inside a typical valley.
In Section 3, we show that P-almost surely, there exist successive valleys [a i , c i ], i ≥ 1, for the potential V , with bottom b i and depth at least We also show (see Propositions 3.4 and 3.5) that almost surely for infinitely many i, (a) the depth of the i-th valley [a i , c i ] is larger than f i + z i for some deterministic z i , and there exists no valley of depth at least f i before it, and (b) the invariant probability measure of a RWRE reflected inside this valley is uniformly large at the bottom b i of the i-th valley (due to Lemma 2.5). We denote the sequence of these indices i by (i(n)) n∈N . The valleys number i(n), n ∈ N, are called the very deep valleys. See Figure 1 for the pattern of the potential for a very deep valley.
The advantage of such a very deep valley is that thanks to (a), with high probability, the particles S (1) , . . . , S (d) will arrive quickly to the bottom b i(n) of this valley (see Lemma 4.1), and stay together a long time inside this valley (see Lemma 4.2). This will ensure, thanks to a coupling argument (detailed in Subsection 4.2) and to (b), that S (1) , ..., S (d) will meet simultaneously in the valley of bottom b i(n) with a quenched probability greater than some strictly positive constant. We conclude in Subsection 4.3 that with a quenched probability greater than some strictly positive constant, S (1) , ..., S (d) will meet simultaneously for an infinite number of n. This, combined with a result of Doob, will ensure that P (x 1 ,...,x d ) almost surely, S (1) , ..., S (d) will meet simultaneously for an infinite number of n.
Finally, we prove in Section 5 two identities in law related to random walks conditioned to stay nonnegative or conditioned to stay strictly positive.
2. Potential and some useful estimates 2.1. Potential and hitting times. We recall that the potential V is a function of the environment ω, which is defined on Z as follows: Here and throughout the paper, log denotes the natural logarithm. For x ∈ Z, we define the hitting times of x by (S n ) n by where inf ∅ = +∞ by convention. We also define for x ≥ 0 and y ∈ Z, We recall the following facts, which explain why the potential plays a crucial role for RWRE.
Fact 2.1 (Golosov [G84], Lemma 7 and Shi and Zindy [SZ07] eq. (2.5) with a slight modification for the second inequality). We have, The following statements about hitting times can be found e. Fact 2.2. Let a < b < c be three integers.Then where we write a ∧ b = min(a, b), and where (11) is obtained from (10) by symmetry.

2.2.
Excursions of the potential. We now define by induction, as Enriquez, Sabot and Zindy in [ESZ09a] and [ESZ09b], the weak descending ladder epochs for V as e 0 := 0, In particular, the excursions Recall that, by classical large deviation results (see e.g. [ESZ09a] Lemma 4.2, stated when log ρ 0 has a non-lattice distribution but also valid in the lattice case), there exists a function I such that I(0) > 0 and One consequence of such large deviations is that E(e 2 1 ) < ∞, since for every k ≥ 1, We also introduce the height H i of the excursion [e i , e i+1 ] of the potential, that is, Recall that due to a result of Cramér (see e.g. [K97], Theorem 3 in the lattice and in the non-lattice case, see also [P15, Prop 2.1]), there exists C M > 0 such that where Γ = R if V is non-lattice and where Γ = aZ if the potential V is lattice and if a ∈ R * + is the largest positive real number such that P(V (n) ∈ aZ, ∀n ≥ 1) = 1. The same result is true with > instead of ≥, with C M replaced by a positive constant C M ( = C M in the lattice case). As a consequence, there exists a constant C M ≥ 1 such that We will use in the rest of the paper the following result. It is already known when the distribution of log ρ 0 is non-lattice and we extend it for the lattice case.
where Γ = R if V is non-lattice and where Γ = aZ if the potential V is lattice and if a ∈ R * + is the largest positive real number such that P(V (n) ∈ aZ, ∀n ≥ 1) = 1. Rroof. When V is non-lattice, the result is proved in [I72, Thm 1]. When V is lattice, the proof is the same as the proof of [I72, Thm 1] with the use of (15) instead of [I72, Lemma 1]. However, since the proof is short, we write it below with our notations.
First, notice that for z > 0, by the strong Markov property applied at stopping time e 1 . Hence, where α(z) := e κ(z−y) P sup .
Observe that, in the lattice case, (17) is not true for h → ∞ in R (it is even not true in 1 2 Γ) but that, in any case, under Hypotheses (2) and (3), there exist C (0) 2.
3. An estimate for the hitting time of the first valley. Let us define the maximal increase of the potential between 0 and x, and then between x and y, by We also introduce The following inequality (the proof of which uses our assumptions and Proposition 2.3) is useful to bound the expectation of the hitting time of the first valley of height h for (V (x), x ≥ 0). It is proved in [ESZ09a] when the distribution of log ρ 0 is non-lattice and we prove that it remains true in the lattice case.  (2) and (3), there exists where E |0 denotes the expectation under the annealed law P |0 of the RWRE (S n ) n on N starting from 0 and reflected at 0 (that is, with ω 0 replaced by 1).
Proof. Assume (2)  So it only remains to prove that k=0 e V (k) . We simplify the rest of the proof as follows. For x ∈ R, we denote by x the integer part of x.
by the strong Markov property. Hence for large h, But the expectation in the right hand side of the previous line is (16), so there exists C > 0 such that for large h, as h → +∞ since 0 < κ < 1. This ends the proof of (22) also when log ρ 0 has a lattice distribution, for large h and then for all h > 0 up to a change of constant. So Fact 2.4 is proved in both cases.
2.4. An estimate for the invariant measure. The following lemma will be useful to control the invariant probability measure of a RWRE reflected inside our valleys, and to show that the invariant measure at the bottom of some of these valleys (introduced below in Proposition 3.5) is uniformly large.
Lemma 2.5. Under hypotheses (2) and (3), there exist constants C 2 > 0 and h 0 > 0 such that for every h > h 0 , and We first notice that for all x > 0 and h > 0, Let h 0 := 8κ/I(0), where I(0) > 0 is introduced in (13). Hence, using e −V (x) ≤ 1 and in the first inequality, then, due to the Markov property at x, we have for all h > 0 and x ≥ 0, where we used (16) and (13) in the last two inequalities. Finally, using (25) and summing (26) and (27) over x, we get for every h > h 0 , since I(0) > 0, which proves (23) for all h > h 0 . We now turn to (24). Due to Proposition 5.1, the left hand side of (24) is equal to due to the strong Markov property in the last line, and since V (−·) is a random walk transient to +∞, and so the expectation in the previous line is finite and This proves (24) and then the lemma, up to a change of constants.

Construction of the very deep valleys
We fix some ε ∈]0, 1−κ 2κ [. We first build a succession of very deep valleys for the potential, with probability 1. To this end, we set for i ≥ 1, We now define (recall that H k and e k were respectively defined in (14) and (12)), We also introduce for i ≥ 1 (see Figure 1), We also define for i ≥ 1, Remark 3.1. Since the r.v. e σ(i)+1 , i ≥ 1, are stopping times with respect to the σ-field σ(V (1), ..., V (n)), n ≥ 0 , the random variables are mutually independent by the strong Markov property.
For every i ≥ 1, we consider the following sets In particular, the set Ω 4 will be useful in Lemma 4.4 to ensure that the coupling in Section 4.2 happens quickly, which is itself necessary in Lemma 4.6. We now estimate the probability of these events. • Control on Ω (i) 1 . Observe that for i ≥ 1, by the strong Markov property applied at stopping time e σ(i−1)+1 . Thus, setting n i = 2κ 0 On the first hand, for every m > 0 and every positive integer n, we have since z i ≥ 0. On the other hand, for every i ≥ 1, Also, by definition of e j , H j , σ(·) and b i , Thus, by definition of σ(·), f i and N i , and by (19), for some constant C 5 > 0, Moreover, setting K i := ie κf i , we have for i large enough, where we used the fact that e k+1 − e k , k ≥ 1 are i.i.d. so that E(e ) = E(e 1 ) and var(e ) = var(e 1 ) for ∈ N, and are also i.i.d., the Bienaymé-Chebychev inequality since E(e 2 1 ) < ∞ (as explained after (13)), 1 − x ≤ e −x for x ∈ R, and (19). The quantities appearing in (36) and in (35) are summable (since ε > 0) and so, due to (34), i≥1 P (Ω Hence, Ω summing over all the possible values of a i + 1 when it is less than or equal to by the Markov property. Hence, using P( Thanks to the control on Ω (i) 2 and since κ+κ 0 > 0, this gives i≥0 P (Ω This event corresponds to the case where, after location x, the potential increases at least h before y and before becoming again smaller than or equal to its value at x. Note that Ω Since a i , α i , b i and γ i are not stopping times, we sum over their possible values, For large i, by the Markov property at times x and y, followed by (19), From this, we conclude that i≥0 P (Ω and Ω (i) 3 .
We now turn to Ω Proof. We have for i ≥ 1, due to the strong Markov property at stopping time β + i , where we used the Markov property at stopping time e σ(i−1)+1 and (19) in the last line. Finally, due to the Markov inequality and to Lemma 2.5, we obtain P Ω I e −κz i , for all i large enough, where we used the fact that for (V (x+m 1 (f i ))−V (m 1 (f i )), x ≥ 0) is, by Proposition 5.2 and the strong Markov property at stopping time ↑ T (f i ), independent of (V (x + m 1 (f i )) − V (m 1 (f i )), x ≤ 0) in the last inequality while applying Lemma 2.5. Moreover, due to Proposition 3.3 and once more the strong Markov property, The result follows then from the fact that e −κz i = 1/i. Proposition 3.5. The setΩ of environments such that there exists a strictly increasing sequence (i(n)) n∈N of integers satisfying the following properties for every n ∈ N has probability P Ω = 1: The valley number i(n), that is, [a i(n) , c i(n) ], n ∈ N, is called the n-th very deep valley.
Proof. Due to Proposition 3.3 and to the Borel-Cantelli lemma, P-a.s., there exists i 0 ≥ 1 (we denote by i 0 the smallest one) such that for every i ≥ i 0 the following holds true for every i large enough, and so for all i ≥ i 0 (up to a change of the value of i 0 ). Moreover, due to Remark 3.1, the events Ω (i) 5 ∩ Ω (i) 6 , i ≥ 1 are independent. So, Proposition 3.4 and the Borel-Cantelli lemma ensure that, P-almost surely, the set I (depending on the environment) of positive integers i ≥ i 0 such that: We construct now (i(n)) n∈N by induction as follows: i(0) := inf I and for n ≥ 1, i(n) := inf{j ≥ max(i(n − 1) + 1, n, n 1+ε 1/κ−1−3ε/2 ) : j ∈ I} .
(45) By construction (i(n)) n∈N is strictly increasing, satisfies (38)-(44) for every n ∈ N, and i(n) ∈ N for every n ∈ N on a eventΩ having probability 1.
In the rest of the paper, for every n ∈ N, i(n) denotes the random variable (uniquely) defined by (45), and depending only on the environment.

Quenched estimates for the RWRE
This section is devoted to the proof of the main result, Theorem 1.1. We first need to prove some preparatory lemmas, for which we use quenched techniques. See Figure 1  Also for x ∈ Z, let P x (·) := P x ω (·)P(dω) be the annealed law of a RWRE in the environment ω, starting from x. The first lemma of this subsection says that for large n ∈ N, a RWRE in the environment ω hits relatively quickly the bottom b i(n) of the n-th very deep valley.
Lemma 4.1. Let x ∈ Z. P x -almost surely, Proof. We start with the case x = 0. We consider the RWRE S reflected at 0, obtained by deleting the excursions of S in Z \ N. More precisely, let r 0 := 0 and We define formally S := (S k ) k∈N by S k := S r k for every k ∈ N. Its hitting times are denoted by τ (y) := inf{k ≥ 0 : S k = y}, y ∈ N. Notice that, under P 0 ω , S is a RWRE in the environment ω, reflected at 0 on the right side (that is, a RWRE with transition probabilities given by (1) in which we replace ω 0 by 1 and S (1) by S ).
Consequently, for every positive integer i, due to the Markov inequality and to Fact 2.4, we have (N i and f i being defined respectively in (29) and (30)), This gives i≥1 P[τ (T ↑ (f i ) − 1) > N i /40] < ∞. Hence, due to the Borel-Cantelli lemma, since, due to [S75], S k → +∞ P-almost surely as k goes to infinity because E(log ρ 0 ) < 0 in our setting. Therefore P-almost surely, for all i large enough, τ (T ↑ (f i ) − 1) ≤ N i /20 and, in particular, for n large enough, τ (b i(n) ) ≤ τ (T ↑ (f i(n) ) − 1) ≤ N i(n) /20 due to (41). This proves the lemma in the case x = 0.
We now assume that x < 0. We use the simple decomposition τ (b i(n) ) = τ (0)+[τ (b i(n) )− τ (0)]. Since the RWRE S is transient to +∞, τ (0) < ∞ P x -almost surely, and so τ (0) ≤ N i(n) /20 for large n, P x -almost surely. Also, for P -almost every ω, (S k+τ (0) − S τ (0) ) k∈N has under P x ω the same law as S under P 0 ω by the strong Markov property, so we can apply to it the results of the case x = 0. Hence, the hitting time of b i(n) by (S k+τ (0) − S τ (0) ) k∈N , that is, τ (b i(n) ) − τ (0), is less than N i(n) /20 for large n, P x ω -almost surely, for P-almost every ω. Summing these two inequalities proves the lemma in the case x < 0.
Finally, assume that x > 0. We use a simple coupling argument. Possibly in an enlarged probability space, we can consider our RWRE S starting from x in the environment ω, and a RWRE Z in the same environment ω but starting from 0, independent of S conditionally on ω. That is, for every ω, (S, Z) has the law of S (1) , S (2) under P (x,0) ω as defined in (1), so with a slight abuse of notation, we denote by P (x,0) ω the law of (S, Z) conditionally on ω. Denote by τ Z (x) the hitting time of x by Z, which is finite almost surely since Z 0 = 0 < x and Z is transient to +∞. We can now define Z k := Z k if k ≤ τ Z (x) and Z k = S k−τ Z (x) otherwise. Conditionally on ω, Z is, under P (x,0) ω , a RWRE in the environment ω, starting from 0. Hence we can apply to it and to its hitting times τ Z the previous result for x = 0, and so τ S (b i(n) ) ≤ τ Z (b i(n) ) ≤ N i(n) /20 for large n, P (x,0) ω -almost surely for P almost every ω, which proves the lemma when x > 0.
The next lemma says that with large probability, after first hitting b i(n) , each RWRE S (j) stays a long time between a i(n) and c i(n) .

4.2.
Coupling. Similarly as in [DGP18,Subsection 5.3] (see also [B86] and [AD15] for diffusions in random potentials), we use a coupling between S starting from b i(n) , and a reflected RWRE S, defined as follows. For fixed n and ω ∈Ω, we define ω a i(n) = 1, ω c i(n) = 0 and ω y = ω y every for y ∈ Z \ {a i(n) , c i(n) }. For x ∈ {a i(n) , . . . , c i(n) }, we consider a RWRE S in the environment ω := ( ω y ) y∈Z , starting from x, and denote its quenched law by P x ω . So, S satisfies (1) with ω instead of ω and S instead of S (1) . That is, S is a RWRE in the environment ω, reflected at a i(n) and c i(n) . Now, define the measure µ n on Z by µ n (a i(n) ) := e −V (a i(n) ) , µ n (c i(n) ) := e −V (c i(n) −1) , Recall that µ n / µ n (Z) is the invariant probability measure of S. Thus, an invariant probability measure for S 2k ) k∈N is ν n , defined by That is, P νn ω S 2k = x = ν n (x) for all x ∈ Z and k ∈ N, with P νn ω (·) := x∈Z ν n (x)P x ω (·). For fixed n and ω ∈Ω, we consider a coupling Q (n) ω of S and S such that: such that under Q (n) ω , S and S move independently until then S and S move independently again after τ exit . We stress that ω, τ S=S and τ exit depend on n, but we do not write n as a subscript.
In order to show that the coupling occurs quickly under Q (n) ω , i.e. that τ S=S is small, we prove the two following lemmas. As before, for x ∈ Z, τ (x) denotes the hitting time of x by S (not by S).
Lemma 4.4. For every ω ∈Ω, we have Proof. Let ω ∈Ω. We fix n ∈ N. First, notice that due to (40) and (42), Also, observe that Because of the definitions (47) and (48) of ν n and Q (n) ωalmost surely. Moreover, the possible increments of ( S n −S n ) n are contained in {−2, 0, 2}. Therefore the walks S and S cannot cross without meeting, hence S 0 < S 0 , τ S=S > m implies for m ≥ 0 that S m < S m . So, where we used (53) in the last inequality and where the previous inequality comes from the independence of S with S and its hitting times τ (·) up to τ S=S under Q (n) ω , and from the fact that Q (n) ω S 2k = x = P νn ω S 2k = x = ν n (x) for all x ∈ Z and k ∈ N. Analogously, we obtain that Finally, combining (54), (55) and (56) proves the lemma.
Finally, the following lemma says that with large Q (n) ω probability, S and S are equal between times N i(n) /2 and 2N i(n) : Proposition 4.5. For every ω ∈Ω, we have Proof. The first limit is a direct consequence of Lemmas 4.3 and 4.4. The second one follows from Lemma 4.6. There exists C 8 > 0 such that, for every ω ∈ Ω, Proof. We fix ω ∈ Ω. First, notice that for every n ∈ N, µ n (2Z , so using (47) and (43), Using the definition of the coupling (first (48), and then S j = S j for every τ S=S ≤ j < τ exit ), we get for every n ∈ N and k ∈ for every even k (by (48) and due to the remark after (47)), and using Proposition 4.5, we get due to (58). Since this is true for every ω ∈ Ω, this proves the lemma.
We now set N i(n) = N i(n) − 1 {(N i(n) −b i(n) )∈(1+2Z)} , for all n ∈ N and ω ∈ Ω, so that N i(n) and b i(n) have the same parity.

Appendix : about random walks conditioned to stay positive
The following proposition gives the law of the potential at the left of m 1 (h), and more precisely between ↑ T h (h) and m 1 (h) (defined in (20) and (21)) when ↑ T h (h) ≥ 0. It is maybe already known, however we did not find it in the literature (in which m 1 (h) is generally replaced by the local minimum of V before a deterministic time instead of our stopping time T ↑ (h), see e.g. Bertoin [B93]).
Proposition 5.1. Let V be a random walk given as in (4) by the sequence of partial sums of i.i.d. r.v. log ρ i , i ∈ Z, such that P[log ρ 0 > 0] > 0 and P[log ρ 0 < 0] > 0 (this result does not require Hypotheses (2) or (3)). Let Proof. We denote by the disjoint union. Let ψ : t∈N * R t → [0, +∞[ be a nonnegative measurable function with respect to the σ-algebra { t∈N * A t : ∀t ∈ N * , A t ∈ B(R t )}. In this proof, to simplify the notation, we set m 1 := m 1 (h) and ↑ T := ↑ T h (h). We have, where V +u denotes the process (V ( +u+x)−V ( +u), x ≥ 0), since the first 3 conditions in the expectation in (62) mean m 1 = + u whereas the last one means ↑ T = . Hence, where we used inf [0, ] V > V ( ) − h ≥ V ( + u) and V (i) − inf [0,i] V ≤ max ] , +u[ V − V ( + u) < h for < i < + u to prove that (63) ≤ (62), followed by the Markov property applied at times and + u in the last equality, where Thus, since (V (u − k) − V (u), k ≥ 0) has the same law as (V (−k), k ≥ 0), Applying this and (64) to ψ = 1, we get which is non zero due to our hypotheses. This, together with (62), (64) and (65) gives This proves the lemma.
Moreover, the following proposition is useful in the proof of Lemma 2.5 and Proposition 3.4. Notice in particular that we get an identity in law with a random walk conditioned to hit [h, +∞[ before ] − ∞, 0[, instead of ] − ∞, 0] in the previous proposition.
Proposition 5.2. Let h > 0. Under the same hypotheses as in Proposition 5.1, if moreover lim inf x→+∞ V (x) = −∞, Proof. Let ψ 1 and ψ 2 be two nonnegative functions, t∈N * R t → [0, +∞[, measurable with respect to the σ-algebra { t∈N * A t : ∀t ∈ N * , A t ∈ B(R t )}. To simplify the notation, we set m 1 := m 1 (h) and T ↑ := T ↑ (h). First, we introduce the strict descending ladder epochs for V as Notice in particular that m 1 = e L , where L := min{ ≥ 0, H ≥ h} < ∞ a.s. Hence, summing over the values of L, we get Since this is true for any ψ 1 and ψ 2 , this proves the proposition.